JPL PUBLICATION 85-29, VOLUME 111 

Proceedings of the Workshop on 

Identification and Control of 

Flexible Space Structures : 

Volum.2 Ill 

G. Rodriguez 

Editor 

April 1, 1985 

National Aeronautics and 

Space Administration 

Jet Propulsion Laboratory 

California Institute of Technology 

Ps”,dena, California 
JPL PUBLICATION 85-29, VOLUME Ill 

Proceedings of the Workshop on 

identification and Control of 

Flexible Space Structures 

Volume Ill 

C. Rodriguez 

Editor 

Apid 1, 1985 

Naticnal Aeronautics and 

Space Administration 

Jer 'nipulsion Laboratory 

Calik -ma Institute of Technc!cgy 

%sadma, California 
This publication was prepared by the Jet Propulsion Laboratory, Californ:a Institute 

of Technology, under a contract with the Nat:onal Aeronautics and Space 

Administration. 
ABSTRACT 

These proceedings repon the rerPltr of a w9rkshop on identification and control of 

flexible space structures held in San Mego, CA, July 4-6, 1984. The workshop was 

co-sponsortd by the jet Propulsion Labor4tory and the NASA Langley Research Center, 

and preceded the 1984 American Control Conference held at the same location. The 

main objectives of the wortrb;op were to provide a forum to exchange ideas in erplorinJ 

the most advanced mdeb, estimation, identification and control methodologies to 

flexible space structures. The workshop nrponded to the rapidly &rowing interest 

within NASA in large space systems (space station, platforms, antennas, flight 

experiments) currently mer design. The workshop contitted of meys, tutorials, 

contributed papers, and discussion sessions in the fo'llouiq general 8TC.l: missions of 

current interest - spclce platforms, antennas, and wt erptrimmts; controYrtrpctan 

interactions - modeling, integrated design aod optimization, control .pd stabilization, 

ad shape control; uncertainty nunaltement - parameter identification, model error 

estimation/comptaution, ad adaptive control; end experimental evrhmtion - grolaboratory demonstrations .pd wt experiment des-. P4pert .ad lectures on them 

topics were presented at a total of fotuteen SCssionsr hchdirrg three panel dircumions. 
TABLE !IF CONTENTS 

VOLUME I 

SESSION I: INTRODUCTION TO WORKSHOP 

Chsirrnen: L. W. Taylor, Jr., NASA Langley Research Center 

G. Rodrigunz, Jet ,repulsion Laboratory ........ 1 

NASA SPACE CONTROLS RESEARCH & TECHNOLOGY PROGRAM 

D. E. McIver and R. W. Key .................... 1 

AFhrAL CONTROL TECHNOLOGY PROGRAMS 

V. 0. Hoehne ........................... 13 

SESSION 11: CONTROL OF SPACE STATIONS 

Chairman: J. B. Dahlgren, Jet Propulsion Laboratory ....... 37 

MULTIVARIABLE CONTROL OF A SOFT COUPLED SPACE STATION 

J W. Sunkel and A. F. Hott .................... 37 

A DUAL SPIN SPACE STATION DESIGN 

M. A. Paluszek .......................... 5i 

AUTOMATIC ASSEMBLY OF SPACE STATIONS 

P. K. C. Wartg ........................... 67 

SPACE STATION DYNAMIC MODELING, DISTURBANCE ACCWODATION, AND 

ADAPTIVE CONTROL 

S. J. Wang, C. H. Ih, Y. H. Lin, and E. Mettler .......... 103 

SESSION 111: CONTROL OF LARGE ANTENNAS 

Chairman: V. 0. Hoehne, Air Force Wright Aeronautical 

Laboratories ..................... 141 

DYNAMIC PERFORYANCE OF SEVERAL LARGE ANTENNA CONCEPTS 

G. C. Anaersen, L. B. Garrett, and R. 2. Calleson ......... 141 

ANTENNA POINTING OF LARGE FLEXIBLE TELECOMMUNICATIONS SPACECRAFT 

B. Govin and A. Bousquet ..................... 165 

DESIGN AND EVALSATION OF CONTROL SYSTEMS FOR LUGE COMMUNICATIONS 

SATELLITES 

M. E. Steiber.. ......................... 183 

CONTROL OF LARGE ANTENNAS BASED ON ELECTROMAGNETIC PERFORMANCE 

CRITER XA 

Y. H. Lin, M. Hamidi, and M. Wnshadi ............... 199 

VIBRATION CONTROL EXPERIMENT DESIGN FOR THE 15-m HOOP/COLUMN ANTENNA 

F. M. Ham and D. C. Hyland .................... 229 

A HARDWARE DEMONSTRATION OF DISTRIBUTED CONTROL FOR A FLEXIBLE 

OFFSET-FEED ANTENNA 

D. B. Schaechter and N. C. Nguyen ................. 253 

V 
SESSION IV: DYNAMICS AND CONTROL EXFERIMENTS 

Chairman: R. A. Russell, NASA Headquarters 269 ........... 

CONTROL OF FLEXIBLE STRUCTURES 

R. A. Russell.. ......................... 269 

SPACE STATION CONFIGURATION AND FLIGHT DYNAMICS ID 

E. Mettler and M. d. Milman .................... 299 

LARGE SPACE STkUCTURE FLIGHT EXPERIMENT 

D. C. Schwab, S. J. Wang, and C. C. Ih .............. 345 

TIME-OPTIMAL BANG-BANG SLEW OF RIGIDIZED SCOLE CONFIGURATION 

J. G. Lin and L. W. Taylor, Jr. .................. 38 3 

SESSION V: CONTROL/STRUCTURE INTERACTION EXPERIMENTS 

Chairman: J. A. Breakwell, Lockheed, Palo Alto Research Lab ... 401 

EXPERIMENTS IN STRUCTURAL DYNAMICS P_ND CONTROL USING A GRID 

R. C. Montgomery ......................... 4CTHE EXPERIMENTAL 2OMPUTER CONTROL OF A TWO-DIMENSIONAL 

HYPERBOLIC SYSTEM 

Y. Yam, J. H. Lang, D. H. Staelin, and T. L. Johnson ....... 413 

RATIONALE FOR AN EXPERIMENTAL TEST FOR FLEXIBLE SPACE 

STRUCTURE ATTITUDE CONTROL 

Th. Lange, G. Heimbold, B. Schaefer, and K. Kolzach ........ 433 

NUMERICAL AND EXPERIMENTAL EVALUATION FOR SINGLE-AXIS CONTROL OF AN LSS 

LABORATORY MODEL 

Y. Ohkami, 0. Okamoto, S. Yoshimura, T. Kida, and I. Yamguchi . . 447 

ON THE MEASUREMENT OF MATERIAL DAMPING IN A SIMULATED SPACE 

ENV I RONMENT 

D. L. Edberg ........................... 463 

VOLUME I1 

-______- SESSIOK VI: FLEXIBLE MULTIBODY DYNAMICS IWD CONTROL 

Chairman: S. M. Joshi, NASA Langley Research Center ....... 1 

A REDCCING TRANSFORMATION FOR DYNAMICS MODELING OF A CLUSTER OF 

CONTIGUOUS FLEXIBLE STRUCTURES WITH CONSTUINTS 

R. P. Singh and P. W. Likins ................... 1 

VIBRATION/LIBRATION INTERACTION DYNAMICS DURING THE ORBITER BASED 

DEPLOYMENT OF FLEXIBLE MEMBERS 

V. J. Modi and A. M. Ibrahim ................... 7 

DESIGN OF MULTiVARIABLE CONTROLLERS USING TUE INTEGRATED ANALYSIS 

CAPABILITY (IAC) 

J. A. Bossi, G. A. Price, and S. A. Winkleblack .......... 23 

vi 
A STRUCTURAL DYNAMICS APPROACH TO THE SIMULATION OF SPACECRAFT 

CONTROL/STRUCTURE INTERACTION 

J. W. Young. ........................... 39 

ATTITUDE CONTROL TRADEOFF STUDY BETWEEN THE USE OF A FLEXIBLE BEAM 

AND A TETHER CONFIGURATION FOR THE CONNECTION OF TWO BODIES 

IN ORBIT 

S. H. Graff ............................ 61 

SESSION VII: CONTROL OF DISTRIBUTED PARAMETER SYSTEMS 

Chairman: P. K. C. Wang, University of California, Los Angeles . . 83 

SOME ASWPTOTIC PROBLEMS IN THE OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS 

J. L. Lions.. .......................... 83 

FINITE CONTROL IN UNDERDAMPED DISTRIBUTED PARAMETER SYSTEMS 

D.J.Inman............. ............... 101 

NEW DIRECTIONS IN ASYMPTOTICALLY STABLE FINITE-DIMENSIL ;&L ADAPTIVE 

CONTROL CF LINEAR DISTRIBUTED PARAMETER SYSTEMS 

M.J.Balas.. .......................... 111 

A FACTORIZATION APPROACH TO THE LINEAR REGULATOR OUADRATIC COST 

PROBLEM 

M. H. Milman ........................... 133 

APPROXIMATION TECHNIQUES FOR PARAMETER ESTIMATION AND FEEDBACK CONTROL 

FOR DISTRIBUTED MODELS OF LARGE FLEXIBLE STRUCTURES 

H. T. Banks and I. G. Rosen .................... 145 

SPACE STATION PARAMETRIC MODELS 

M. Hamidi and S. J. Wang ..................... 157 

SESSION VIII: INTEGRATED MODELING, DESIGN AND OPTIMIZATIOE 

Zhairman: G. T. Tseng, Aerospace Corporation .......... 20 1 

APPROXIMATION OF OPTIMAL INFINITE DIMENSIONAL COMPENSATORS FOR 

FLEXIBLE STRUCTURES 

J. S. Gibson, 1). L. Mingori, A. Adamian, and F. Jabbari ...... 201 

CONTROL OF A FLEXIBLE SPACE ANTENNA: A FINITE DIMENSIONAL 

PERSPECTIVE BASED ON DISTRIBUTED PARAMETER THEORY 

D. L. Mingori, J. S. Gibson, P. Blelloch, and A. Adamian ..... 219 

AN INTEGRATED CONTROL AND MI.\IIMUM MASS STRUCTURAL OPTIMIZATION 

ALGORITHM FOR LARGE SPACE STRUCTURES 

A. Messac, J. Turner, and K. Soosaar ............... 231 

CHARACTERISTIC ELASTIC SYSTEMS OF TIME-LIMITED OPTIMAL MANEUVERS 

A. J. Hale and R. J. Lisouski ................... 26 7 

ISAAC (INTEGRATED STRUCTURAL ANALYSIS AND CONTROL) VIA CONTINUilM 

MODELING AND DISTRIBUTED FREQUENCY DOMAIN DESIGN TECHNIQUES 

C. L. Gustafson, M. Aswani, A. L. Doran, and G. T. Tseng ..... 28 7 

vi i 
OPTIMIZATION OF CONTROLLED STRUCTURES 

M. Salarna, M. Hamidi, and L. Demsetz ............... 311 

SESSION IX: PARAMETRIC AND NONLINEAR CONTROJ 

Chairman: M. Hamidi, Jet Propulsion Laboratory ......... 

PARAMETRIC STIFFNESS CONTROL OF FLEXIBLE STRUCTURES 

F. C. Moon and R. €I. Rand ..................... 

VIBRATIONAL STABILIZATION Or FLEXIBLE STRUCTURES 

M.Zak .............................. 

STIFFNESS CONTROL OF LARGE SPACE STRUCTURES 

J. L. Fanson, J. C. Chen, and T. K. Caughey ............ 

SUB-OPTIMAL CONTROL OF NONLINEAR FLEXIBLE SPACE STRUCTURES 

T. J. Dehghanyar, S. F. Masri, R. K. Miller, G. A. Bekey, and 

T. K. Caughey ........................... 

SESSION X: MODELING AM) MODEL REDUCTION 

Chairman: J. Sesak, General Dynamics Co. ............ 

AN OVERVIEW OF LATEST MODEL REDUCTION AND CONTROL METHODS OF LARGE 

FLEXIBLE SPACE STRUCTURES 

J. M. Santiago, W. J..Lange, Jr., and M. Jamshidi ......... 

FREQUENCY DOMAIN CONTROL DESIGN OF XRGE SPACE STRUCTURES; A PRACTICAL 

APPROACH 

R. Harding and A. Das ....................... 

A CONTROL CONCEPT FOR LARGE FLEXIBLE SPACECRAFT USING ORDER 

REPUCTION TECHNIQUES 

G. Thieme and H. Roth ....................... 

WIDEBAND DISTURBANCE ACCOMMODATION IN PRECISION FLEXIBLE SPACE 

STRUCTURES 

D. R. Hegg and G. J. Kissel .................... 

VOLUME 111 

-- SESSION XI: MODEL REDUCTION AND ROBUSTNES 

Chairman: A. F. Tolivar, Jet Propulsion Laboratory ....... 

A STUDY ON THE CONTROL OF THIRD GENERATION SPACECRAFT 

E. J. Davison and W. Gesing ............ ....... 

SENSOR/ACTUATOR SELECTION FOR THE CONSTRAINED VARIANCE CONTROL 

PROBLEM 

M. L. r Irenzo and R. E. Skelton ................. 

EIGENVALUE PLACEMENT AND STABILIZATION BY CONSTRAINED OPTIMIZATION 

S. M. DeCaro and D. J. Iwan ................... 

329 

329 

343 

351 

365 

381 

38 1 

397 

415 

433 

21 

37 

viii 
MATRIX TRANSFER FUNCTION DESIGN FOR FLEXIBLE STRUCTURES-- 

AN APPLICATION 

T. J. Brennan, A. V. Cornpito, A. L. Doran, C. L. Gustafson, and 

C.L.Wong ............................ 47 

ROBUST CONTROL DESIGN FOR LARGE SPACE STRUCTURES 

W. L. Eastman and J. A. Bossi ................... 63 

ON THE STABILITY OF COLLOCATED CONTROLLERS IN THE PRESENCE OF 

UNCERTAIN NONLINEARITIES AND OTHER PERILS 

S. M. Joshi ............................ 83 

- SESSION XII: ADAPTIVE CONTROL 

Chairman: R. W. Montgomery, NASA Langley Research Center .... 99 

ADAPTIVE CONTROL--ACTUAL STATUS AND TRENDS 

I. D. Landau ........................... 99 

A NONLINEAR DUAL-ADAPTIVE CONTROL STRATEGY FOR IDENTIFICATION AND 

CONTROL OF FLEXIBLE STRUCTURES 

F.E.Thau ............................ 117 

STABLE DIRECT ADAPTIVE CONTROL OF LINEAR INFINITE-DIMENSIONAL SYSTEMS 

USING A COMMAND GENERATOR TRACKER APPROACH 

M. J. Balas, H. Kaufman, and J. Wen ................ 127 

SELF-TUNING ADAPTIVE CONTROLLER USING ONLINE FREQUENCY IDENTIFICATION 

W. W. Chiang and R. H. Cannon, Jr, ................ 145 

ADAPTIVE FILTERING FOR LARGE SPACE STRUCTURES--A CLOSED-FORM SOLUTION 

H. E. Rauch and D. B. Schaechter ................. 161 

ROBUST ADAPTIVE CONTROL 

K. S. Narendra and A. M. Annaswamy ................ 175 

SESSION XIII: ESTTMATION/IDENTIFICATION 

Chairman: L. W. Taylor, Jr., NASA Langley Research Center .... 197 

MAXIMUM LIKELIHOOD ESTIMATION WITH EMPHASIS ON AIRCRAFT FLIGHT DATA 

K. W. Iliff and R. E. Maine .................... 197 

OPTIMAL SENSOR LOCATIONS FOR STRUCTURAL IDENTIFICATION 

F. E. Udwadia and J. A. Garba ................... 24 7 

COMBINED STAT5 AMI PARAMETER ESTIMATION FOR A STATIC MODEL OF THE 

MAYPOLE (H30P/COLUMN) ANTENNA SURFACE 

H. T. Banks, P. K. Lamm, and E. S. Armstrong ........... 26 3 

EXPERIMENTAL VERIFICATION OF IDENTIFICATION ALGORITHMS FOR CONTROL OF 

FLEXIBLE STRUCTURES 

B. Sridhar, J.-N Aubrun, and K. R. Lorell ............. 28 3 

AN EIGENSYSTEM REALIZATION ALGORITHM (ERA) FOR MODAL PARAMETER 

IDENTIFICATION AND MODEL REDUCTION 

J. N. Juang and R. ‘3- Pnppa .................... 299 

ix 
A RESIDUALS APPROACH TO FILTERING, SMOOTHINC AND IDENTIFICATION FOR 

STATIC DISTRIBUTED SYSTEMS 

G. Rodriguez ........................... 

NUMERICAL EXPERIMENT. 'ION WITH MAXIMUM LIKELIHOOD IDENTIFICATION IN 

STATIC DISTRIBUTED SYSTEMS 

R. E. Scheid, Jr., and G. Rodriguez ................ 

- SESSION XIV: FUTURE RESEARCH DIRECTIONS 

Moderator: H. A. Rediess, 8. R. Textron ............. 

DISCUSSION ............................... 

smopsrs ............................. 

UNCERTAINTY MANAGEMENT METHODOLOGY FOR LARGE SPACE STRUCTURES 

W. E. Vander Velde ..................... 

SOME ESTIMATION AND IDENTIFICATION PROBLW IN RANDOM FIELDS 

A. V. Balakrishnan ..................... 

MODEL ERROR STRUCTURE AND THE INSEPARABILITY OF THE CONTROL 

AND IDENTIFICATION PROBLEMS 

R. E. Skelton.. ...................... 

FUTURE RESEARCH TECHNOLOGY DIRECTIONS FOR SPACE STATION ... F. Carlisle ....................... 

THE ROLE OF EXPERIMENTS IN THE DEVELOPMENT OF CONTROL 

TECHNOLOGY 

H. A. Rediess ........................ 

AUDIENCE DISCUSSION ........................ 

TECHNICAL EVALUATION REPORT OF THE WORKSWP ON IDENTIFICATION AND 

CONTROL OF FLEXIBLE SPACE STRUCTURL 

H. A. Rediess and A. Nayak .................... 

APPENDIX A 

PROGRAMSCHEDULE . -. . 

-_---- APPENDIX B 

ATTENDEEWPARTXCIPANTS ...................... 

319 

377 

403 

40 3 

40 3 

40 7 

413 

41% 

422 

425 

431 

439 

443 

453 

X 
A STUDY ON THE CONTROL OF 

THIRD GENERATION SPACECRAFP 

E. J. 

University 

Ravbon and 

of Toronto 

W. a 

Toronto, Ontnrio, Canada MSS iA4 

ABSTRACT 

An overview of some studies which have recently been carried out in El]-[33 

on the control of third-generation spacecraft, as modelled.by the MJT space 

vehicle configuration, is made. 

has appendages which cannot in general be assumed to be rigid. In particular, it 

is desired to design a controller for MSAT which stabilizes the system and satisfies certain attitude control, shepe control, -and possibly station-keeping requirements; in addition, it is desired that the resultant controller should be 

robust and avoid any undesirable "spill-over effects". 

ler obtained should have minimum complexity. 

late the problem as a robust servomechanism problem [SI-[7], and thence to obtain 

existence conditions and a controller characterization to solve the problem. 

tion and appears to be quite satisfactory. 

This spacecraft is highly non-symmetrical and 

In addition, the controlThe method of solution adopted to solve this class of problems is to formuThe final controller obtained for MSAT has a distributed control configuraINTRODUCTION 

This paper summarizes studies carried out in [l]-[3] on control system 

(1) Large mass 

(2) High power 

(5) Large non-symmetric flexible appendages 

(4) 

structures known as third-generation spacecraft. Such spacecraft have: 

Precise communication RF bean control requircments. 

In particular, the class of spacecraft represented by the Mobile Conmrunicatioiis 

Satellite (MSAT) is used as a reference for these studies. "his spacecraft has 

non-symetric appendages which cannot be assmed to be rigid (see Figure 1). 

shape-control ani possibly station-keeping control for such third eeneration 

spacecraft (referred to as LFSS',, which may be listed as follows: 

Tlicre are a number of control problems Rssociated with the attitude-control, 

A. TLe LFSS Control Problem 

Problem 1: 

long the imaginary axis, 

complish in this case is to stabilize. the rigid body modes of the LFSS, and at 

the same tine to stabilize the elastic modes of the LFSS. 

LFSS stabilization problem. 

*This work was supported by the.Department of Communicatioiv, Ottawa, Canada under 

-- - -1_- 

Lightly Lamped, Oscillatory Plan9 

A LFSS has eigenvalues either at the origin or approximately disrributed nOne of the basic objectives that a controller must acThis is called the 

---- -.__I_ _I_ - .------. 

contracts DOC-CR-SP-82-007, DOC-CR-SP-83-002, DOC-CR-SD-84-002. 

1 
Problem 2 : Modelling 

framework for the model!ing of the low frequency elastic modes of the LFSS in a 

reasonably accurate way, but that the high frequency elastic modes cannot be expected to be determined accurately, i.e. there will always be errors present in 

nodelling the high frequency elastic modes of the LFSS. In addition, the calculation of dampening effects on the LFSS can only be done with great uncertainty. 

In modelling a LFSS,expcrience has shown that dynamic analysis may provide a 

Problem 3: 

known "infinite dimensional" system reprl jntation of a LFSS. 

adopts this infinite dimensionality representation seriously from an engineering 

standpoint, there is no question that the number of system elastic modes present 

in a LFSS is always larger than the number which any design model of a LFSS can 

accommodate. In trying to control the modelled rigid and elastic modes, it is 

essential that the controller should not cause-these mmdelled high frequency 

clastic modes to become unstable. 

The Infinite Dimensional Plant - The "Spill-Over Problem" 

'l'iie classical modelling of elastic structures as continua results in the well 

Whether or not one 

This is called the "Spill-Over Problem". 

l)robl,m 4: The Sensor/Actuator Placement Problem 

ware is not in gzneral specified. 

part of the LFSS control problem is in determining the number and location of 

sensor/actuators on the LFSS. 

The LFSS is intrinsically distributed, and the configuration of control hardThus, unlike many conventional control pr..blems, 

Problem 5: 

second generation spacecraft, p-ecludes single-input, single-output contr31 design. 

Some type of multivariable control design method is mandatory to deal with the 

severe intcraction occurring in the system. 

Requirement for Multivariable Control Theory 

The concept of "third generation" spacecraft, unlike the first and some 

Problem 6: 

Th, 2 is a practical limiation on the quantity of hardware that can be distributed over the LFSS vehicle, 

full state feedback is available, and that the number of actuators/sensors used 

must be limited, i.e. one must minimize any unnecessary sensor/actuators required 

for LFSS control. 

Minimization of Number of Sensors/Actuators 

This implies in particular that one cannot assume 

The following problem definition is now given: 

B. The LFSS Robust Servoinechanism Problem 

sional linear time invariant model: 

Assxme that a LFSS can be exactly described by the following finite dimen- 

. 

x = AX + BU + Ew 

y = Cx + Fw 

Ym m = C x t Fmum 

2 
r n where xcR is the state, ucRm is the control (actuator inputs), ymcR are the 

are assume3 to be constant unmeasurable disturbances applied to the structure, 

error in the system where yref is a constant set-point. 

(1) may include an arbitrarily large number of elastic modes (but not infinite). 

is given by: 

measured (sensor) outputs, and ycR r are the outputs to be regulated. Here wR n 

A are assumed to be constant unknown measurement errors and e = y-yref is the 

Thus, it is assumed that 

Assume now thct an approximate model of (l), called the design model for (l), 

-- x = Ax + Ell + Elk1 

y = tx + Tu 

where xrRn is the state of the design model, and where %<n. 

to find a controller based on the design model (2j, such that when it is applied 

to (l), the system is asymptotically stable, i.e. no spill-over occurs, and such 

that: 

I? is desired now 

h 

hi! lim e(t) = 0 , Vx(O)eRn, VwR s2 , Vu ER 

m tThis is called the LFSS Robust Servomechanism Problem, which includes the following subproblems: - (lj Stabilization 

(2) Station-keeping 

(3) Attitude control 

(4) Shape control. 

THE MSAT CONTROL PROBLEM 

The MSAT spacecraft is illustrated in Figure 1. It consists of four components, one of wich is rigid (the bus) and three of which are flexible (the solar 

array, the tower, and the reflector). The tower-reflector-hub hinge point is 

assumed to have a gimbal (see Figure 2). 

The coordinates assumed for each of these substructures are as follows: 

(1) Bus - three rigid rotations (e ,e ,eZ)’ XY 

(2) Tower - relative displacement of tower tip to tower root 

(f - 16 1, f -16 * , f - 16 3) 

- relative angular displaceeent of reflectcr with respect to 

frame fiyed at tower root (with zero gimbal angles) 

(a1 ,ac2 ,a3) 

(3) Reflector - zwo gimbal angles at tower-reflector-hub hinge point (61,f3,)’ & 

The actuators which are assumed to be available are as follaws: 

Eight thrusters fi, i=1,2,. . . ,8, four from thrusters on the bus and four 

from thrusters at the reflector hinge point, aligned as shown in Figure 2. 

(1) 

3 
(2) 'i'wo torquers at the reflector hub, one about each gimbal axis (g 

Figure 2). 

In this case, a design model and an evaluation model was developed in [4], 

in which the design model has 18 states consisting of S rigid body modes (corresponding to the three rigid rotations of the bus and two gimbal angles of the 

reflector) together with 4 elastic modes, and the evaluation model has 32 states 

consisting of 5 rigid body modes and 11 elastic modes. Table 1 gives the eigenvalues of the open loop system for the two models. 

included the effect of dampening terms D, DE (see Table 1). 

The models used in this study 

With Damping 

TABLE 1: @pen Loop Eigenvalues of MSAT Vehicle 

With Damping 

Okj0.124 -0.0009232j0.124 

Okj0.239 -0.00170 2j0.240 

02 jO.556 -0.00856 2 j0.556 

02 jO.780 I -0.0211 f jO.779 Elastic 

Body I 

Modes 

-_ -- 

10 times) 10 times) Modes 

Evaluation Model 1 

Jith Damping 

Term DE 

' Excluded 

1 (repeated 

10 times) 

02 jO.124 

0% j 0.151 

02 j0.239 

02j0.556 

02 jO.690 

02 jO.780 

02 j 1.55 

02j3.14 

02j 3.96 

02 j9.95 

02 j 14.0 

Included 

10 times) 

-0.000923+ j 0.124 

-0.0008532 j 0.151 

-0.00171 2j0.239 

-0.00856 2 j 0.556 

-0.00553 2j0.690 

-0.0211 2j0.780 

-0.0751 2jl.SS 

-0.0280 2j3.14 

-0.0528 2j3.96 

-0.524 2j10.1 

It may be noted that the elastic modes of the evaluation de1 interweave with 

the elastic modes of the design model. 

A. 

for the MSAT vehicle. 

controller to be designed for the MSAf vehicle: 

Ikqu i remen t I 

problems : 

Description of Problem to be Salved 

11, this case it is desired to solve the LFSS Robust Servomechanism Problem 

In particular, there are two separate requirements for the 

bind a controller, based on the MSAT design model, which solves the following 

Stability: stabilize the 5 rigid body modes and the 4 elastic modes of thc 

Attitude control: regulate ex, 8 BZ to desired constant set points ex , system. ref 

ref respectiveiy, in the presence of unknown constant disturb- eref 

ances 

Y' 

Y ' OZ 

4 
the presence of unknown constant disturbances. 

above requirerents, and not cause any instability to occur with respect 

to any of the vehicle's elastic des which are not included in the 

design model. 

Controller complexity: 

actuators which are required to solve the problem. 

Discrete controller implementation: it is desired that the controller, when 

implemented digitally, should nct require an excessively large sampling 

rate to maiitain stability. 

Spill-over problem: it is desired that the controller should satisfy the 

it is desired to minimize the nuniber of sensors and 

Requirement I1 

evaluation model, and verify that all objectives above are satisfied. 

Apply the controller obtained, based on the MSAT design model, to the HUT 

The outputs to be regulated in this case are given by: 

U. Assumptions Made in Problem Formulation 

In this problem, it is assumed that there is no requirement for controlling 

the w w w rigid body des. (Note: this assumption is not essential, e.g. 

(21, 131 also deals with the case of station-keeping.) It is also assumed that 

there is no need to include any gyroscgpic teras in the design and evaluation 

nodels. 

x' y' z 

UETHOD OF SOLUI'IW AD@PTED To OBTAlN 

A CONTROLLER To SOLVE PROBLEM 

The method of approach adopted to solve this problem was based on using the 

results of the "robust servomechanism problem" [5]-[7], in conjunctim with a 

parameter optimization method [8] to determine the controller's Tarameters. e.?. 

see [9] which solves a special case of the above problem when the sens3r.c slnd 

actuators are collocated, using a decentralized control configuratim. In this 

case, existence conditions for a solution to the problem were obrained, and a 

necessary controller structure developed. In particular, it was found that any 

coxitroller which solves the PIAT problem specifications nust consist of a "servocorpensator" 151 (unique) , together with a stabilizing conqensator (non-unique) . In this study, the simplest possible stabilizing compensator, i.e. a stabilizing 

compensator consisting of only proportional and rate feedback terms, was used. 

In this case, in order to satisfy the existence conditions obtained for a 

solution to exist tu the problem, it wzs necessary to choose the following inputs 

(actuators) and measurable outputs [sensors) for the controller : 

Outputs (sensors) : 

5 
Inputs (actuators) : 

* *t*t 

where g 

fl,f2, ..., f ,f 

, fl’ f2, fs, f correspond to various combinations of the thrusters 

c, 6 

3 (see Figure 2), as described in Appendix I. 78 

In this case, tIIe following distributed controller was obtained as a soluticn to the NSAT robust servomechanis8 problem, based on the MSAT design model: 

- K2s 

Y 

0 

z 

e 

B2j 

’8 -6 ’ 

e -G 

- Kg - 

5 ez-er 

xx 

YY 

where s denotes the Laplace Trabsfoxm operator, where 

and where K1, K2, Kg, K4, R5, y are given as follows: 

- 

K1 - 

- 

-0.000439 1 1.43 0.500 24.7 1.34 -0.0460 

0.0255 4.64 1.12 15.6 . 

0.00326 38.0 -0.231 14.5 

-6.81 -0.000957 0.00981 -0.000483 18.6 

59.0 -0.00916 0.0216 0.0127 

(7) 

6 
K, = 3 

- 

-3.00877 1 28.5 10.0 494 26.7 -0.920 

0.510 92.8 22.3 312 

0.0653 760 -4.63 290 -0.191 

-136 -0.0191 0.1% -0.00965 372.2 

- 

1180 -0.183 0.432 0.254 -48.1 

?.14x10-~ 2.50~10-~ 1 - 24X10-2 6.68~ -2.30~ 1 O-' 

1. ~SX~O-~ 2. 32x1ci3 5.58~ 7.80~10-~ -2.19~10-~ 

.3.41~10-~ -4.79~10-~ 4.90~10-~ -2.41~10-~ 9.31~10-~ 

1.63~10-~ 1.90~10-~ -1.16~10-~ 7.26~10-~ -4.78~10-~ 

2.95~10-~ -4.58~10-~ 1.08~10-~ 6.34X10+ -1.20Xio-~ 

- 

-0.464 

-0.0144 

-0.00438 0.201 

0.136 

-0.000753 1.13 

0.0268 0.225 

- C.0273 -0.226 

-3 

y = 2.0x10 

This controller is just a multivariable generalization of the classical 

three term controller used in classical control. The controller has minimal complexity , the sense that it has minimum order feedback dynamics and has the 

ninimum number of actuators/sensors required in order to solve the problem. 

is to be noted that no a priori assumption On the distributed structure of (7) 

was made - the distributed structure of the controller (7) arose from the 

analysis automatically. 

It 

PROPERTIES OF PROPOSED CONlROLLER 

The main features of the proposed controller when applied to the MSAT design 

illrodel .a evaluation model will now be described. The main features of interest 

are : 

(1) 

(2) 

The fgllowing results are obtained: 

The stabilization properties c,f the proposed controller. 

The steady state regulation pioperties of the proposed controller. 

A. 

cofit;-aller (7) to the HAT design model and evaluation models. 

Eigenvdues of Closed Loop System Using Proposed Controller 

Table 2 gives a listing of all eigenvalues obtained by applying the proposed 

7 
TABLE 2: Listing of Closed Loop Eigenvalues Using Proposed Controller (7) When 

App?ied to MSAT Design and Evaluation Models 

-- 

Standard Design Model 

-0.00047: j 0.0085 t 

-0.00242 j0.016 1 

-0. 00512 jO.022 rigid body 

des -0.0097+j0.030 t 

-0. (:lo+ j0 -031 1 

-0.OO014+ j 0.124 + 

-0.0172 j 0.557 I 

elastic body 

modes -0.00612 jO.24G 

-0.029+J0.780 I 

-5 .oxlo-k 

-5.0x10-4 T 

-5.ox10-' I 

-5. o~io-~ 

-5.0~10-" I 

servo-compensator 

modes 

-5.0x10-k 

-5 .Oxlo-' 1 

-2 .oxloI 

f eedforward 

controller 

modes 

-2. OX 10- 

-2.0~10- 

-2 o~io-~ 

-2.0~10-) I 

-2.0~10-~ 1 

Evaluation Model 

-0.00047-+ j 0.0085 t -0.0024ij0.016 I 

-3.0051+j0.023 rigid body 

des 

-0.00972 j 0.030 I 

-0.010+ jO.031 1 

-0.00014+j0.124 

-0.00020+j0.151 

-0.00612 j 0.240 

-0.0172 j 0.557 

-0.00791j0.690 elastic I body 

des -O.O029+jO. 780 

-0.129ki1.35 I 

-0.0672j3.16 

-0.0692 j3.95 

-2 -5.j 8.88 

-0.512 j 11.3 

-1.7X10-' 

-5 .Oxlo" t 

-5.0~10-~ I 

-5.OXl Oe4 1 

servo-compensator 

modes -5.0~10-~ 

-5. 0X10-4 

-5.0~10-~ 1 

-2.0~10-~ t -9 0x10'3 I 

-2.0~10- 

-2.0~10-~ 

-2.0~10'~ I 

f eedforward 

controller 

modes 

-. 

-2. OX 10- J 

It is obseriied that the resultant closed loop system is asymptotically stable 

for both the design and evaluation models, i.e. no undesirable spill-over effects 

occur. 

mainly associated with the servo-compensator modes. 

expect for the case of tracking, that the dominant time response of the system 

would be associated with the feedforward controller modes, i.e. TCdornt500 sec 

f 8 pin., and for the case of disturbance rejection, that the dominant time of 

the system would be associated with the servo-compensator modes, i.e. 

2000 seci0.6 hrs. This result is verified in the simulation studies to follow. 

It is also observed that the dominant time constant of the system is 

This implies that one would 

- 

TCdom ' 

8 
B. Steady-State Values of Outputs Using Proposed Controller: Tracking Case 

tion tracking, when the proposed coniroller (7) is applied to the MSAT design and 

evaluation model. 

tically regulated to their correct values as desired. 

Table 3 gives a summary of results obtained for the case of unit step funcIt is observed that all 9 outputs of the system are asymptoTABLE 3: Steady-State Values of Outputs Using Proposed Controller (7) When 

Applied to Design and Evaluation Model - Tracking Case 

1 0 0 

0 1 0 

0 0. 1 

0 0 0 

0 0 0 

0 0 0 

0 0 0 

0 0 0 

0 0 0 

Note: Any Inumber/<10-16 is assumed to be zero. 

C. Steady-State Values of Outputs Using Proposed Controller: Disturbance 

Reject ion Case 

Tables 4 and 5 give a summary of all results obtained for the case of disturbance rejection, when the proposed controller is applied to the MSAT design 

and evaluation models respectively. 

function change occurs for different disturbances corresponding to ic ,& , . . . , 

Zo,r9 defined inTable 6. 

are asymptotically regulated to zero, and that the remaining two outputs are 

approximately equal to zero in all cases, as is Jesired. 

D. Sampling Rate Requirements for Digital Implementation of Proposed Controller 

If it is assumed that the proposed controller (7) is to be implemented 

digitally, then it is necessary that the sensor outputs and actuator signals be 

updated at a fast enough rate so as to guarantee closed loop stability, when the 

the controller is applied to the evaluation model. In this case, on assuming 

that the sensor and actuator signals are updated at the same rate, it. was found 

that a sampling rate of at least 0.1 Hz must be used to implement the proposed 

controller. 

In this case, it is assumed that a unit step 

It is observed that the first 7 outputs of the system 

12 

. 

This requirement is not demanding. 

9 
TABLE 4: Steady-State Values of Outputs Usinn Proposed Controller (7) When 

Applied to MSAT Design Model - Disturbance Rejection Case 

s, .=I 

1 

0 !%I O I O lo 

i,;l €p E2-l $4 t,.l fo=l f0=l 

0 0 .o 0 0 0. 0 

0 

0 

0 0 0 0 0 0 0 

14.lOl 0 0 0 0 0 0 010 

ololoiololololo 

0 

0 

0 

Note: Any )number1<10-16 is assumed to be zero. 

0 0 0 0 0 

'0 0 01 0 0 0 .: 0 

-c 'I'AB1.E 5: Steady-State Values of Outputs Using Proposed Controller (7) Uhen 

Applied to MSAT Evaluation Model - Disturbance Rejection Case 

- - - - - €,-1 P2=1 €,=l f,=l fo=l ffl 

-1 8q1 8ql b gcl'l gc2-1 

1 ex 

0 Io I 

1 

5,. I 0 0 

0 0 

0 0 

0 0 

, 9L 

f-'6, 0 0 

%+"1 

52*52 

0 I -1 

' f-Ib3 1 4xlO-j -3XlC-8 

f 5, 0 L 

0, -3x10-' SX~O-' 3 

Note: Any Inumber1<10-16 is assumed to be zero. 

gc3 

0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 

0 0 o 4x10-' -4x10-' 0' 0 -3x10-6 -8x10-' 

0 0 0 -7x10'8 7x10-8 0 0 1~10-~ 3~10-~ 

10 
SIMlLATIONS OBTAINED USING PROPOSED CONTROLLER 

TO SOLVE USAT PROBLEM 

This section gives some typical simulations of the closed loop system obtained by using the proposed controller (7) applied to the MSAT design and evaluation models. Additional simulation studies are given in [3]. 

A. 

that there are no disturbances present, and that a unit step function change of 

+1 occurs in the set point for 8 at t=O, i.e. 8 

troller is applied to both the design and evaluation model in this case. 

observed that the system's response is almost decoupled, i.e. the output Ox is 

approximately equal to its desired value of +l.at t+50 min, and that all other S 

outputs are barely excited. 

example. 

Example No. 1 (Attitude Control: 8rf=1) 

In this example, it is assumed that the system has zero initial conditions, 

ref- ref- -1, By -0, 8 

X X Z 

Figure 3 gives a plot of all 9 output variables y given by (4) when the conIt is 

Figure 4 gives a plot of the 7 control variables u given by (5) for this 

D. Example No. 2 (Disturbance Rejection: f,=1) 

In this example, it is assumed that the system has zero initial conditions, 

that all set points are identically equal to zero, and that a unit step function 

change of +1 occurs at t=O corresponding to a disturbance thrust ?,=1, where f5 

is defined in Table 6. This example would correspond to a misaligned thruster 

associated with the proposed controller. 

applied to both the design and evaluation model in this case. 

that the elastic modes of the vehicle are now excited, and that the output variables are asymptotically regulated to zero in approximately 2.7 hours, which is 

consistent with the closed loop eigenvalues of the system given in Table 2. 

Figure 6 gives a plot of the 7 control variables u for this example. 

Figure 5 gives a plot of all 9 output variables y when the controller is 

It is observed 

c. Exanple No. 3 (Disturbance Rejection: $=l) 

This example is similar to Example No. 2 except that it is assumed that 5 

unit step function of +1 occurs at t=O corresponding to a disturbance thrust f9=l, 

where f9 is defined in Table 6. 

trary constant disturbance which may affect the system. 

applied to both the design and evaluation models in this case. It is observed 

that the elastic modes of the vehicle are now also excited as they were in 

Example No. 2, and that the output variables are satisfactorily asymptotically 

regulated with the same time constant as in Example No. 2. 

Figure 8 gives a plot of the 7 control variables u for this example. 

This disturbance is representative of an arbiFigure 7 gives a plot of all 9 output variables y when the controller is 

11 
TABLE 6: -- Definition of Disturbances Assumed 

Fl"~sz~'F6''O''9 

gsl %* 

gcl 'gc2,gc3 

- - 

.- 

-- - 

- 

Disturbance forces corresponding to the thrusters 

f ,f ,f ,f ,f ,f respectively of Figure 2 

respectively about the gimbal axis 8,,f3; 

axis respectively 

125609 

Disturbance torques corresponding to gfi ,gg 

Disturbance torques in the bus about the x,y,z 

ROBUST PROPERTIES OF CONTROLLER DESIGN METHOD 

A study of the robustness properties of the proposed controller design method 

was carried out 131. 

using the proposed method to different design models of MSAT. 

that the proposed design method appears to be quite insensitive to the type of 

design model used, e.g. all controllers obtained, when based on MSAT design models 

which had at least two dominant elastic body modes included, produced stable 

closed loop systems and give satisfactory tracking/regulation, when applied to 

the MSAT evaluation model. Other studies showed that the controller is robust 

with respect to evaluation models of arbitrary complexity. 

This was done by comparing the controller designs obtained 

It was concluded 

CONCLUSIONS 

'This paper gives a brief summary of the work performed in [1]-[3]. In these 

studies, the control system design of a third-generation spacecraft, as modelled 

by the hSAT space configuration is studied. This spacecraft is highly nonsymmetrical and has appendages which cannot, in gensral, be assumed to be rigid; 

the elasticity of these appendages makes the control system design particularly 

demanding. 

stabilizes the system and satisfies certain attitude control, shape control ard 

possibly station-keeping requirements. 

tant controller should be robust and avoid any "spill-over effects", i.e. it 

should satisfy the problems' specifications based on only an approximate design 

model for biSAT being available. In addition, the controller obtained should have 

minimum - complexity, i.e. a minimum number of sensors/actuators should be used. 

The method of solution adopted to solve this class of problems was tc formulate the problem as a robust servomechanism problem and thence to obtain existence 

conditions and a controller characterization to solve the problem. In this case, 

the contraller obtained must contain a servo-compensator together with a stabilizing compensator. 

The final controller obtained for MSAT has a distributed control configuration, and appears to be quite satisfactory, i.e. extensive testing of the controller shows that the controller is indeed robust with respect to the choice of 

the design model, and that it satisfies all specifications of the problem statement. 

In particular, it is desired to design a controller for MSAT which 

In addition, it is desired that the resul12 
ACKNOULEDGEiMENTS 

lhe authors are grateful to P. Hughes, A.H. Reynaud and 1.1. Stieber for their 

valuable assistance in this work. 

Comunications, Ottawa, Canada under the Scientific Authorities : 

i..F. Stieber. 

This work was sponsored by the Department of 

H. Reynaud an3 

REFERENCES 

111 Lavison,E.J., Iiuglles,P.C., "A Study to Determine Control Techniques Applicable 

to Third-Generation Spacecraft", DOC Report No.: DOC-CR-SP-82-007, March 19C2. 

\2] bavison,E.J., "A Study to Develop Control Techniques Applicable to ThirdGciieration Spacecraft1*, DOC Report No. : DOC-CR-SP-63-002, February 1963. 

i;] Lavison,E.J., "A Study to Develop Control Techniques Applicable to ThirdGeneration Spacecraft - Part III", DOC Report No.: DOC-CR-SP44-002, 

1.ebruax-y 19C4. 

[I] I;uglies,P.C., Sincarsin,G.B., W3AT Structural Dynamics Model for Control 

Systeni Design", DOC Report No. : DOC-CR-SP-82-054, August 1982. 

Invariant Multivariable Systems':, IEEE Trans. on Automatic Control, Vol. AC-21, 

5 ] bavison, E. J . , "Robust Control of a Servomechanism Problem for Linear TimeIJo. 1, 1976, pp. 25-34. 

[O] Davison,E.J., Goldenberg,A., "Robust Control of a General Servomechanism 

Problem: The Servo-Compensatort1, Etutomatica, Vol. 11, 1975, pp. 461-471. 

~'l] Davison,E.J., The Robust Decentralized Control of a General Servomechanism 

Problem", IEEE Trans. _- on - Automatic --. Control, Vol. AC-21, No. 1, 1976, pp. 14-X. 

IC] Davison,E.J., Ferguson,I .J., "Design of Controllers for the Multivariable 

Robust Servomechanism Problem Using Parameter Optimization Methods1', IEEE 

Trans. -- on Automatic Control --' Vol. AC-26, No. 1, 1981, pp. 93-110. 

[SI iiest-Vukovich,G., Davison,E.J., tlughes,P., The Decentralized Control of 

Large Flexible Space Structures1', IEEE Trans. on Automatic Control, 1984 to 

appear. 

13 
APPENDIX I 

Definitions of gf ,f;,f;,f;,€i 

7 

* 

is defined jn terms of thrusters fl, f,, fg, f4 as follows: gc3 L 

8.66 * 

ifg rO = [ 8. 6i "3 c3 

** 

f5, f are deiined in terms of thrusters f5, f6, f,, f8 as follows: 6 

** 

fqr fj, f4 as follows: 1' fl, f are definea in terms of thrusters f 2 

* * 

if flrO and f220 

* * 

if fl 2 0 and f2 < 0 

* * 

if fl 5 0 and f2Z 0 

* * 

ii fl < 0 and f2 < 0 

14 
IORIWAL PAQE'IS 

OE ?OH QUALITY 

. AY 

I 

::ip,urc 1: The MSAT configuration - a tyF?cal third generation spacecraft. 

15 
-Thrusters (4) 

Figure 2: Assumed control inputs for MAT spacecraft (taken from [SI). 

16 
ORJGdVAL PAQE'fi 

W POOR QUALITY 

Figure 3: Plot of 9 regulated outputs jr for exanple no. 1. 

, sec) 

17 
+O 

Y 

-I 

Figurn 5: Plot of 9 regulated outputs y for exaaple no. 2. 

! 

: t (sec) 

Figure 6: Plot of 7 control inputs u for example no. 2. 

18 
... .. 

I. 

+ t(sec) 

0 lo4 

Figure ?: Plot of 9 regulated outputs y for exanple no. 3. 

I: 

I. 

"1 .I; 

I 

I 

.. 

.. 

(sec) 

Figure 8: Plot of 7 control inputs u for example no. 3. 

19 
SENSOR/ACTUATOR SELECTION FOR TEE 

CONSTRAINED VARIANCE CONTROL PROBLEM 

M. L. DtlRccI13D 

Air Force Acldwly, coloedospiags,CO~1 

RESdQI 

sctrodofAcronwtKs - dAstnnrautics 

purdut university 

west Lafkyere. IN 47907 

This paper considers the problem of designing a 1:rm- controllc- *x system subject to inequality variance constraints. P. quadratic penalty function 

z,,pmach is used to yield a liwar controller. Both the weights in the quadratic penalty function and the locations of sensors .and actuators are selected by 

successive approximations to obtain an ootimal design which satisfies the input/ 

mtput variance constraints. The method is applied to l!&A's 64 mter HoopColumn Space Antenna for satellite comunications. In addition the solution for 

tbe control law, the min feature of these results is the systemtic deternination of actuator design requirements which allw the given input/output perforFance constraints to be satisfied. 

I. INRODUCTIOY 

Consider the task of controlling the linear, stochastic systc: 

W = diag I... Wii ...], V = diag I... Vii .... 1, 

such that these four control desian goals are net: 

e. 2 L A 

(I) E$i2(t) = lir E yi (t) - < ai , i = 1, ..., k 

t- 

- 2 2 A- 2 E u. (t) = lir E ut (t) 5 pi , i = 1, ..., n 

t- 01 

L1 PRECEDING PAGE BLANK NOT FILMED 
(11) ~nty ‘i < sensors are used 

fm the acbnissibte set of sensors described fkm (IC) 

. (111) ~nty rr! m actuators ar~ used 

- 

m 

(3) 

(4) 

.. 

b(i + w) = 1 bi(ui + w.) 1 

i=l 

frm the admissibte 8et of n actuators described fra (la) 

R 

B(u + W) = 1 bi(ui + w.) 1 

i=l 

( IV) *e contmt ii( t) is a tinear ptuzction of the psent and 

past measurements F(T), T - < t. 

Hany engineering control design problems can be stated with performance 

constraints of the form (I). For example, large space telescopes are feasible 

only if the IUS pointing errors (~j~~)~’~ are within certain bounds 

(EJ~~)~’~ 5 ai) so as to achieve diffraction-limited performance (ui) of the 

optics. The designer may also have the freedom to choose from a number of 

different types of sensors and actuators at a number of different locations. The 

locations and the types of actuators (sensors) determine the vectors bi (mi) in 

(4) and (IC). 

P. straight-forward approach to acconrodate the bounded input/output problem 

(I) yields nonl inear controllers 1-21, violating goal (IV). A straight-forward 

approach to accarmodate gcals (IV and (I) is to use a penalty function method 

[3-53, minimizing 

22 
rhile adjusting Q and R until (I) is satisfied. These successive approximation 

schemes [3-51 presume a fixed measurement/control structure, and hence do not 

satisfy goals (XI) and (III). It is important to unify the treatment of all 

four goals (I-IV) since it has been shown [6-73 they are inherently interdependent problem. In particular, for the isotated problems; [6] has shown the 

optimal sensor and actuator selection for LQG problems (5) with fhed (Q,R), and 

[3-5] have adjusted Q and 9 to satisfy the constrained-variance problem (I) with 

fixed sensors and actuators (i.e. fixed B, A). 

Unfortunately, the optimal answer for the -sinul taneous solution of both 

groblems turns out not to be the juxtaposition of results [6] and [3-51, due to 

the interdependence of the two problems. 

problem (I-IV) , rhich we call the Constrained Variance Sensor/Actuator Selection 

(CVSAS) problem. Section11 describes the approach. Section I11 gives the formulas for sensor and actuator effectiveness to deal with goals (11) and (111). 

Section IV presents the numerical algorithm for iteratively dealing with goal (I). 

Section V gives the algorithm for solving the entire problem (1-IV). Section VI 

illustrates ths application to the Hoop-Column Antenna. 

The purpose of this paper is to present a unified treatment of the entire 

11. APPROACH 

The solution of the ?roblem with inequality constraints (I) is generally 

not unique. To be a bit mre specific than statement (I) we define two variations 

of the problem. The first is called the "Constrained-Input Variance" option of 

the CVSAS. In this option the input constraints in (I) are binding and the 

output constraints in (I) are : SZGed. 

CIVSAS: The Constrained-Input Vartance, Sensor/Actuator Selection Problem 

Satisfy (XI), (XII), and with aZZ input-constraints binding, 

T minimize (recat2 yi = ci XI, 

If however, there is no i for which ai *EJi2 > 1 then minimize 

23 
with aZZ input constmints binding (6). 

Definition: 

wilt mean the minh constraint violation in the sme of the minimum value of 

V in (7) with input constmints binding (6). 

The CIVSAS problem is useful when one wishes to determine the best perforrance achievable for a given power limitation on the input devices (actuators). 

That is, for a given set of pi the CIVSAS finds the minimum achievable output 

performance. 

!l'he phwe %tinhat achievabte output perfomwncetr for the CIVSAS 

Y 

The second variation of the CVSAS problem is called the Constrained Output 

variance Sensor/Act!ator Selection (COVSAS). . 

COVSAS: The Constrained-Output Variance, Sensor/Actuator Selection Problem 

Satisfy (I I), (I1 I ), and with all output constnainte binding 

i = 1, ..., k , -2 2 

u Eli = 1 , (8) i 

minimize 

If however, there is no i for which pi-2Ewui2 > 1 then minimiae 

- 

m 

i=l 

Vu = 1 pi-*E mi u 

with all output cmtraints binding, (8). 

Definition 2: The phrase %tinimwn achievable input performance" for the COVSAS 

wiZZ mean the minimum constraint hotation in the sense of (9), with all output 

constraints binding (8). 

The COVSAS is useful when one wishes to determine the necessary capabilities 

(design requirements) of the actuators in order to achieve the specified output 

performance. That is, for a given set of ui the COVSAS finds the minimum 

achievable input performance. 

24 
I I I. SENSOR/ACTUATOR EFFECTIVENESS 

In this Section we tem orarily assume that Q and R in (5) are specified 

diagonal matrices Q = diag p... qi ...I, R = diag [... ri ....I, and we wish to 

determine a ranking of the effectiveness of the admissible set of sensors and 

actuators for the LQG problem described by (1) and (5). To help with this task 

a price or "cost" is assigned to each input and output by decomposing the total 

system cost function (5) into contributions from each in t and output. This 

task is called ''input or output cost analysis" and from I? 61 we have the results 

m k 11 

i=l i=l ' i-1 i=l 

wv th where viu, viY, vi , vi is the contribution in v of, respectively, the i 

control ui, output yi, noise wi, or noise vi, and 

where P, K, i and L satisfy 

(12a) T -1 .A T -1 o = PAT t AP - PR v rv t BC~B~, [fl, .;., f,~ = F = PF v 

T 1T T 

(12b) o = KA t A K - KBR- B K t c QC, [g,, ..., g,] = G~ = -KBR-~ 

The effectiveness of the ith sensor is measured by 

25 
and the effectiveness of the ith actuator is measured by 

act = *U vi - vi W 

vi 

These terms Vi 

output costs Viu, Viw, Viv which are involved in the performance of each sensor 

and actuator. 

calculated by Viw, but the effect of an actuator involves both Viu and Viw since 

the actuator is noisy, and this dependence is accounted for in (14)). To see 

that' V. and ViaCt gives the appropriate measure of the effect of deleting 

the iti sensor or the ith sensor or the ith sensor or the ith actuator, refer 

to the numerical work in [7]. 

and ViaCt represent the particular combinations of the input/ 

(The distinction here is that the effect of the input wi can be 

Two results from [6! add insight into the use of (13), (14). 

Theorem 1, [6,7]: 

For a specified !Q,R), the optimal value of the LQG; perfomce metric (5) 

cannot be reduced by the deletion of any of the admissibte smors Zi, ...) E. 

Theorem 2, [6,7]: 

For a specified (0,R) the optimal value of the LQG psrfomce metric (5) 

- can possib2y be reduced by the deletion of sane of the admissibZe actuators 

ui, i = 1, ..., n?. 

These theorems partially explain why the sensor effectiveness Vi is a 

th much simpler calculation than ViaCt. Since the magnitude of the gain on the i 

sensor signal 1 Ifi 1 l2 = 1 Imil lpp vii -2 -t o as vii + -, an extremely noisy sensor 

simply will not affect the optimal LQG controller. Hence, the effectiveness of 

the ith sensor can be calculated by the input cost VIv. Section V will show how 

to use (13) and (14) in the solution of the COVSAS problem. 

2 

26 
IV. THE COVLQG ALGORITHM 

Now we cite an algorithm (COVLQG) to solve the COVSAS problem under the 

temporary assumption that 

actuators are used (8 = B and 19 = b!). The COVLQG algorithm will first be stated 

and then its theoretical properties will be discussed. 

= a and fi = m. That is, all admissible sensors and 

- TheCOVLQG algorithm (i.e. the COVSAS with a = a, 6 = ml: 

Step A: Compute P from (12a). If ai-21 1ci I 1; > 1 STOP. NO sotution to 

the COVLQG problem dsts. Otherwise initialize 

-2 -2 qi(o) = ai , ri(o) = vi . 

Discussion of Step A: The lower bound on Eji'2 in an LQG problem is 

Eji2.? I lei! 1, (from the well known lower bound tr CPC on V in (5)), and this 

result is indebendent of the choice of C - > 0, R > 0. 

2 T 

Step 0: Compute 

-l YiY Y i: qi > 0 2 E y. = Qi -1 

-1 u 

~-u~~ = ri 'i 

-2 2 

Using (11)s (12). If Egi = 1 U i: qi > O and if 

-2 E u .* > 1 V i = 1 , . . m, STOP. 

In the COVLQG option all necessary control effort is 

!The COVLQG solution has been found. "i -1 - 

Discussion of Step B: 

applied to force the constraints Edi 

the stopping criterion of Step B indicates a solution of the COVLQG problem is 

given by Theorem 5 of [7]. 

to be binding. A foml proof that 2 

5 ai 

Step C: Q and R update equations: Let the iteration index be j unri set 

-2 

< qi(j+l) < cui , qi(j+l) = cui -2 ~j~~~q~(j~ i = I, ..., k. If (fai 2)-1 

(E < 3 smaZZ specified comt;ant) then sot qi(j+l) = 0. 

-2 Eji2 = 1 U ii qi > 0, then set ri(j+l) = [ii2E mi ~~]"~r~(j), U i:

ui E-ui < 1. For a22 other 1, sat ri(J+l) = ri(j). Retwn to Step B. 

If 

ai -2 2 

Discussion of Step C: The ri(j+l) of Step C are clearly adjusted toward the 

stopping condition of Step B (pi 

€-ut2 to increase. The justification for setting qi = 0 when either qi(j+l) -+ 0 

-2 €=ui2 - > l), since a reduction in ri causes 

27 
or when qi(j+l) + m is as follows: The tendency of qi toward zero indicates a 

lack of output controllability due to a degenerate rank of C (rank C k). In 

this case, the algorithn ceases to attempt the impossible (i.e. to force two 

dependent outputs to arbitrary values) by removing this particular yi (the least 

critical one as indicated by the smallest qi + 0) frm the cost function by 

setting its coefficient qi = 0. Now let rank C k. The tendency of qi toward 

m can result only when a stabflitable, detectable system is not output controllable, (even though C = k) and an uncontrollable output converges ta a value which 

violates its constraint i oi2). The constraint is violated the smallest 

alnount possible since tn this case the corresponding qi -t m on successive 

iterations of the update equations. When this condition is determined, such 

yi's are removed from the cost function on futu-re iterations (by setting qi = 0) 

since it now has been established that they cmmt be brought within specifica2 

tion.Eji2 - < 0 i' 

A sipilar algorithm exists for the Constrained Input Variance LQG problem 

(CIVLQG) and details are given in [7]. 

V. THE COVSAS ALGORITHK 

The sensor/actuator effectiveness formulas (13), (14) derived in Section 111 

and the COVLQG alsorithm~ of Section IV are now integrated to solve the COVSAS 

problem posed in Section 11. 

COVSAS A1 qori thm: 

Step 1. Specify CA,B,C,!.I,V,~,~,U ,u I. Run COVLQG algorithm using k 22 

actuators, m sensors. 

Step 2. Compute Vi , Vi act from (13), (14) rmd rank sensors and 

actuators according to their effectiveness: 

. se~s ... le sens , VIsens L "2 - 

act , act ... 2 Vm "1 L vp - 

act 

Delete the Remor and actuator with the lmest effectivensss values 

Visens, Viact, provided such detetim does not came loss of 

28 
controZlabiZity or observabi1ity.j Unless R c 

lntess m < 6 + 1, reset m to m-1. 

+ 1, reset 1-0 E-1. 

€gi = 1 V i = 1, ..., k -2 2 If ai II 

< 

2 1" -2 2 

vi E-ui I( jt1)iteration and Y i: vio2 E u. > 1, if [T 1 

i=l -1 

return to Step 1. Otherwise STOP. A 1" -2 2 [e 1=1 .I pi E-ui j(jt1)iteration 

soZution to the COVSAS has been found. 

Piscussion of Step 2: HunericaI experience wi;h this algorithm suggests that 

more than one sensor and mre than one actuator may be deleted on each iteration. 

In fact, tdr many cases the same resblt can be obtained by reducing E to 'i and 

m to 1 on the first iteration. However, this quicker convergence can sornetimes 

converge only to suboptimal answers, and the algorithm above is written in its 

most conservative form (deleting only one sensor and/or actuator per iteration) 

where convergence to optimal values is more reliable [7). 

VI. CONTROL OF A SPACE ANTENNA 

Fig. 1 depicts the Hoop-Column Antenna arrangwient for a proposed NASA 

comnunications satellite. Stationed in a geosynchronous orbit, the objective of 

the antenna control system is to regulate the orientation and focus of the 

satellite antenna relative to its muitiple feed horns (at node 10). Table 1 

1 ists the 24 1 inear and angular displ{:,ements which make up the outputs yi , 

i = 1, ..., k, where k = 24. Table 2 lists the 39 admissible sensors and Table 

3 lists the 12 admissible actuators. Note that ARX2 stands for angular rate 

about the x axis at node 2. AX2 stands for atigular displacement about axis x at 

node 2. 110-22 stands for a rectilinear displacement between nodes 10 and 2 in 

the z direction, The specifications for the outputs are ui = 22.8 are seconds 

for i = 1, ..., 6, and ai = .158 mn for i = 7, ..., 24. The specifications for 

the inputs ui are pi = 10 dn-em, i = 1, ..., 12. The actuator noise is described 

by rJ = diag [ . . Wi . . .], Wi = .l (dy-cm) , Y i = 1, . . . 12. The sensor noise 

is V = diag [... Vi ...I, Vi., = 7.615~10~~ rad2, i = 1,2,3,13,14,15, Vi = 

2.5m7 m2, i = 4, ..., I?, 16, ..., 27, Vli = 4.76xN5 (rad/sec) , i = 28, 

, , . , 39. 

of sensors to 12 = 1. The dynarics of the antenna structure were described by 

10 elastic modes and 3 rigid body modes. The square of the frequencies 

2 

2 

It is des', ,sed to 1 hit the number of actuators to 6 -- 6 and the number 

+Observabil ity, controllability checks are particularly simple for flexible 

space structures using the tests in [8]. That is, rank tests of matrices 

[B; AB, ... An%], [C T , A TT C , ... ATn-'CT] can be amided. 

29 
2 

w , i = 1, ..., 10 of the elastic nodes are i 

9 22 “) = (.40579, 7.2090, 7.2362, 13.27/, 90 (wl , W2 9 *e*, 

44.834, 132.14, 147.66, 445.01, 448.69, 775.86;) (raJ/sec) 2 . 

Pore complete information for the antenna model pay be found in [7]. 

The results of the COVSRS algorithm applied to the Hoop-Column Antenna are 

mmrized in Table 4. The 6 actuators deleted from the admissible set of 

Table 3 are (listed in order of deletion): y12, u9, u6, ul0, u7, u4. The 27 

sensors deleted (in order of ieletion) are: z15, z3, 26, tI2, z, zI3, z2, zl, 

224s 227, 249 25, 218, 221s 2309 2399 2339 27, 28, 231, 223, ‘209 2359 225, 222, 

‘1.6’ . 

control effort is less using only 6 actuators, (6x5.C21 = 30.12) than using 12 

actuators (12~3.275 = 39.30 

fewer actuatcrs, since for several actuators the noise effec; Vib is greater 

than the signal effect Viu in (14) (note th; negat’ve valhes of Viact in Table 

Notice that even though the output constraints are still binding the total 

30.12). Thus, betber perfo-mance is possible with 

4) 

Perhaps the lpost important information from the Cn‘:”4S is the detemina+ fof the minimum achievable actuatw specification ?ram Table 5 bat all 

of the 24 outputs are held within their design consctdiriLd (cI = 22.8 are secs. 

for angles and ui = .158 mn for rectilinear displacements) hy acti9tors which 

must be designL1 for the capabilities of TABLE 5. That is, the gi.den nutput 

specifications, ai are Vossible to meet if ui is changed I=> actuators are 

redesigned) (from Table 5) to u1 = 73, p2 = 26, p3 = 105, u4 = 26, p5 = 32, 

p6 = 39. 

VII. CONCLUSI~r!S 

Presented is an algorithm COVSAS which integrates the following tasks: 

Selects sensors and actuators from an admissible set. 

Designs a linear feedback controller which satisfies output variance 

cons trai nts . Determines a3tuator design requirements which a1 low the output variance 

constraints to be satisfied. 

30 
Numerical yoperties of the convergence of this algorithm are given for NASA's 

Hoop-Column Antenna. Additional theoretical properties of convergence of this 

algorithm are given in [7]. 

REFERENCES 

[l] H. Tgivonen, "Pinimum Variance Control of First Order Systems with a 

Constraint on the Input Arplitude" -- IEEE Trans. Auto. Ctntrol, Yol. AC-26, 

No. 2, April 1981, pp. 556-558. 

[2] G.C. Goodwin, "Amp1 itude Constrained Rinirmm Variance Controller, ti 

Letters, Vol. 0, pp. 181-182, 1972. 

[3] P.K. tlakila, T. G!csteriund, and H.T. Toivonen, "Constrained Linear Quadratic 

Gaussian Control," 21st IEEE CDC, Orlando,- FL., Dec. 1982. 

tmn. 

[A] H. Toivonen, "Vat-iance-Constrained Self-Turiing Control ," Automatica, to 

appear . 

[5! R.E. Skei ton and F.L. DeLorenzo, "Weight Selection for Covariance Constrained 

LQS Regulators: Larga Space Structure Application," Automatica, to appear. 

[6] R.E. Skelton and F.L. DeLorenzo, "Selection of Noisy Actuators and Sensors 

in Linear Stochastic System," J. Larqe Scale Systems, Theory and Appli- 

-- cations, Forth-Holland pub1 ishing, to appear. 

[7] KL. DeLorenzo, "Selection of Noisy Sensors and Actuators for Regulation of 

Linear Systems," Ph.D. Thesis, t'ay 1983, School of Aeronautics and Astronautics, Purdue tiniversity, Vest Lafayette, IN. 47907. 

[Sj P.C. Hughes and R.E. Skelton, "Controltability and Observability of Linear I atrix-Second-Order Systems," 3. Applied Mchanics, Vol. 47, No. 2, June 

1980, pp. 415-420. 

31 
Figure 1: Hoop Column Antenna 

32 
Table 1: Hoop Column Output Description 

Output # 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

Sensor 

Number 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

TY ne 

Inertial Angle 

II 

II 

Relative Angl e Between 

Inertial Angle 

Relative Linear Disp. BeGeen 

11 

I! 

II 

I1 

I1 

0 

II 

II 

I1 

II 

I1 

0 

I1 

I1 

I1 

I* 

II 

Nodal Location 

2 

2 

2 

10 and 2 

10 

6 and ll 2 

9 and 2 

10 and 2 

I01 and 10 

107 and 10 

113 and 11 1C 

119 and 10 

n 

n 

I( 

H 

II 

I1 

I1 

n 

II 

II 

Table 2: Hoop-Cclumn Sensor Labels 

Sensor Sensor 

Label Number Label Number 

Ax2 

AY 2 

AZ2 

X6-X2 

Y 6-Y 2 

26-22 

X9-X2 

Y9-Y2 

29-22 

XlO-x2 

YlO-YZ 

210-22 

AX10 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

AY 10 

AZlO 

XlOl -x10 

Y 101 -Y 10 

ZlOl-210 

X107-X10 

Y107-Y10 

2107-210 

Xll3-X10 

Y 113-Y 10 

2113-210 

x119-x10 

Y 119 -Y10 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

35 

39 

Direction 

X 

Y 

Z 

X 

Y 

Z 

X 

Y 

X 

Y 

X 

Y 

X 

Y 

Z 

X 

Y 

Z 

X 

Y 

Z 

x 

Y 

2 

Lahi 

2119-210 

ARX2 

ARY 2 

ARZ2 

ARX6 

ARY 6 

AR26 

ARX9 

ARY9 

ARZ9 

ARXlO 

ARYlO 

ARZlO 

33 
Table 3: Hoop Column Actuator Description 

Actuator torque akt 

axis at 

Me location 

= TX2 

= TY2 

= TZ2 

= TX6 

u1 

u2 

u3 

"4 

US = TY6 

= TZ6 

= TX9 

= TY9 '8 

u = TZ9 9 

'6 

u7 

UIO = T 1! 10 

ull = T Y 10 

u12 = T Z 10 

34 
Table 4: Hoop Column Output Constrained COVSAS Results 

Identified Ave Input 

Iteration Identified Actuators Value Number of 

knber (7.6) Sensors /Actua tors 

sensors @e") (Vy) 

1 AZlOi .OW41 16) TZ10( -1.362) 3.2?5 39/12 

AZ2( .000297) lZ9 ( -1.369) 

Z6-Z2( 0) 

Z9-Z2( @} 

z10-22 (0) 

2 kY 1 ( .063362) lZ6(-2.1605) 3.592 34/ 10 

AxIO( .003358) 

AY 2 ( .oO226) 

AX2(. 0[)226) 

Z113-Z10( .OO1942) 

Z119-Z10( .001884) 

3 X6-X2( .01457) TX10( -1.2055) 3.699 28/9 

Y6-Y2( -01455) 

ZlC7-Z10( .0108) 

Z101-Z10( .0110) 

4 ARZZ ( .02844 TX3(-1.2917) 3.997 24/8 

ARZlO ( .02232) 

ARZ6(. 02238) 

5 X9-X2 ( .0986) TX6(-1.4793) 4.377 21/7 

6 ARX6(. 07648) ---- <, .829 19/6 

Y9-Y2( .0839) 

ARX2( .07648) 

7 Y 107-Y lo(. 13395) ---- 4.857 17/6 

8 X119-X10(. 1557) ---- 4.905 1516 

XRY9 ( .1098) 

X113-X10(. 1555) 

X:Ol-XIO(. 1551) 

9 -e-- ---- 5.021 1216 
Output Y 

1 (AX2) 

Z(AY2) 

3 (AZ2) 

4( AXlO-AX2) 

5( AY10-AY2) 

7( X6-XZ) 

8( Y6-YZ) 

9 (X9-X2) 

10( Y9-Y2) 

S(AZ10) 

11( x10-X2) 

12( Y10-Y2) 

14( ~lol-Yloj 

13 (X101-X10) 

15fZlOl-ZlO) 

16( X107-X10) 

17( Y107-Y10) 

18( 2107-210) 

19 (X113-X10) 

20( Y 113-Y 10) 

21( 2113-210) 

22 (X119-X 10) 

23 (Y 119-Y 10) 

24 ( Z 1 19 -Z 10) 

Tab1 e 5 : Ou tpir t -cons trai ned Speci fi cations 

2 ‘2i 

.015 sec 

,015 sec 

11.588 sec 

.001 sec 

.001 sec 

12.000 sec 

.Of0 m 

.010 m 

.om m 

.om mn 

.158 mn 

.158 mn 

.lo4 m 

.158 mn 

.007 m 

.158 m 

.156 m 

.m m 

.122 mn 

.158 m 

.001 m 

.158 m 

.391 m 

.001 m 

2 

i 

Actuator # (minimm achievable) 

1 TX2 72.91 dn-cm 

2 TY2 26.145 dn-cm 

3 TZ2 105.47 dn-m 

4 TY6 26.138 dn-cm 

5 TY9 31.750 dn-cm 

6 TYlO 38.812 dn-an 

36 
' N85-31198 

EIGENVALUE PLACEMENT AND STABILIZATION 

BY CONSTRAINED OPTIMIZATION 

S. M. DcCuo* and D. J. Inman** 

University at Bliffalo 

State University of New Yo& 

Buffalo, NY 14260 

ABSTRACT 

A pole placement algorithm is proposed which uses constrained non-linear 

optimization techniques on a finite dimension4 model of a linear n degree of 

freedom system. 

assigned; r being the rank of the sensor coefficient ma+,rix. 

combining feedback control theory methods with optimization techniques, one can 

ensure the stability characteristics of a system, and can alter its transient 

response. 

Low order feedback control is assumed where r poles may be 

It is shown that by 

One common method of approaching the problems of controlling the vibration 

cf a structure is '3 employ eigenvallie (pole) placement methods. 

have attracted the attention of numerous authors over the past twenty-five years, 

including W. M. Wonham [6], E. J. Davison [31, S. Srimthkumar [SI, A. Y. Andry 

et a1 [ 11, [ 21 and many others. 

Such solutions 

In exploring pole placement in dynamical systems, an inadequacy of stabilitx 

considerations in contemporary algorithms was noted and thus motivated this work. 

It appears that the problem has not been solved or even addressed in many 

approaches. 

If a system is controllable, one has the ability to place a predetermined 

number of poles. 

limit on the number of poles that may be assigned. As is well known, the rank of 

the sensor coefficient matrix determines how maw poles may be placed exactly. 

These poles may be noted as the contrcllable eigenvalues of the system, while the 

remaining may be labelied uncontrollable. 

Thus, when pole placement techniques are employed, there is a 

*Present Address: Member Technical Staff 

AT&T Bell Laboratories 

Whippany Road Rcom 3C-249 

Whippany, New Jersey 07981 

**Research supported by AFOSR Grant Number AFOSR 82-0242 

37 
Thus, due to restrictions inherent to every system, every pole may not be 

Therefore, one does not have control over tine full order of 

Wnen moving the ailowable eigenvalues, those which are not placed 

desirably placed. 

the system. 

will also be affected, with the possibility of generating an unstable state. 

Since an urstable system is undesirable, the ability to place a predetermined number of poles, while forcing the system to remain stable would be 

quite desirable to the designer. 

tory assignment of the desired modes, but unfortunately can drive the remaining 

eigenvalues unstable. Thus, requiring iteration of the algorithms, campromising 

the desired choice of eigenvalues or eigenvectors, until a stable response 

results. With the large number of modes reqaired in modelling flexible structures, these methods become costly and time consuming. 

Many pole placement methods yield sa.cisfacHence, a pole placement method is proposed which constrains the ,unspecified 

modes to be stable by taking advantage of constrained optimization techniqucs. 

It appears that no previous work has gusranteed stable unplaced poles or has 

assured the magnitude of relative stability. 

Several numerical examples will be presented, and results will be compared 

with those of Srimthkumar 151. 

PROPOSED SOLUTION 

The systems studied in this paper are of the mechanical type, which are 

second order by nature, incorporating mass, stiffness and damping parameters, 

where only the class of discrete systems shall be investigated. 

Assuming small motions aboclt the equillibrium point implies linearization 

of the equations of motion, which become 

The forcing function vector, F(t), may then be described as 

where [V] and [PI are the velocity and position feedback matrices, respectively. 

- q(t) is the coordinate vector, while i(t) and - i(t) are the first an& second time 

derivatives of this vector. 

[MI is known as the mass or inertia matrix, [D] 1.s called the damping natrix, 

and [SI is the stiffness matrix. 

scopic cr Coriolos matrix, and [HI is the circulatory matrix. 

The matrix [GI may be referred to as the gyroThe [MI, [D], [SI, [GI and [HI matrices are assume6 to be time-invariant, 

and therefore are represented by constant values, all be;ng of nth order, where 

n represents the number of degrees of freedom of the system. 

Using normal sttrte space methods by letting 

38 
the n-dimensional system becomes the following &-dimensional model : 

-*j 

- 

E)' 

r(t) = [Clj CZ1&(t) (2) 

where [MI is assumed to have an inverse and [ 21 g(t) is a representatioc of 

the system's forcing f'unctior!, F(t). 

- 

.A 

More simply, equation (2) may be expressed as follows: 

where 

~(t) is ',he output vector, [Cj is a constmt sensor coefficient matrix, and [K] 

is the feedback gain xatrix. 

cient matrix of actuator dynamics, and u(t) is the control vector. 

conditions hold : 

[B' my now be described as the constan2 coeffiThe following 

39 
And a more revealing representation is shown in Figure 2. 

FIGURE 2 

Equation (2) may be zewritten as follows: 

If we define 

40 
and describe equation (4) as follows: 

where 

and 

Then, the set of equations must satisfy the eigenvalue problem, i.e., 

2n 

Z the 2n eigenvalues 

2n 

E the corresponding eigenvectors. {q) i=l 

v. may then be defined to correspond to the above partitioning ss follows: 

1 

yielding 

] k]+ 

which implies 

- -% 

substituting, 

[ *lKCl -;-- 

I 1 B1KC2 ----- 

Io I 

'7 1 kl 

2 -1 -1 ti% = -M (D+G)ciq - M (S+H)q + B KC t w '+ B KC w 11i-i 12Yi 

If we define {A3,) i = 1,2,...,r as the r eigenvalues to be placed, 

equation (6) may be expreosed as 

WA~ = -M'~(D*G)wA - M-'(s+H)w + B~KC~WA + B~KC~W 

41 
w = w w I ....e I w 1 

[-1 I -2 1 I-- 

where 

and A = diag(A X ..., X ) 1’ 2’ r 

By taking adl*an’ nge of the generalized left in-rerse theorem, 

[K] = [BTB1] -1 [B1][WA T 2 +M -1 (D+G)WA + M-1(S+H)W][C1WA+C2W]-1, 

which is the equation describing the gain matrix needed to obtain thoso eigenvalues desired. 

A single objective function was then determined from the set of equations 

described by equLtion (71, where the values of [K] were determined by minimi-dng 

that abjective function. 

real part of the eigenvalues of the closed loop system were all negative, These 

constraints were also modified, as was desired, to inc:ease the stability argin. 

The constraints impoqed on the system were that the 

Exarcple 1: 

FIGURE 3 

Specifications: 5 = m2 = 1 

s1 = 4 

s* = 1 

dl = 2 

d2 = 1 

42 
Eigenvalues of unforced system: 

-1.666207 t 1.41334i 

- .333783 i: -.83265i 

[c] = [o -' * o] lJ00 

Desired eigenvalues: 

Resulting eigenvalues using the propos,sd method with no additional factor for 

relative stability: 

-4.00000G + Oi 

-1.335420 + Oi 

- .249608 + Oi 

-',000000 + oi 

Resulting eigenvalues using the proposed method with &dJed fdctor of relative 

stability : 

Resulting eigenvalues using Sririathkumar method: 

9.1256 + oi 

- .8141 + Oi 

-4.0000 4. oi 

-3.0000 + C'i 

Note t5at the method proposed here yields the desired .eigenvalue.; xnd that 

the unspecified eigenvalues remain stable, wkereaa .LD the* Srinathkmr method 

an unspecified eigenvalue ie moved into the right half plane. 

43 
FIGURE 4 

Specificatims: y = 4 

m =y=rah=l 2 

sl=s =s =s4=1 23 

d =a =.5 12 

Eigenvalues of unforced system: 

-.004055 i: 1.647953i 

-.17C649 t 1.131418i 

-.062364 t .355674i 

-.075432 t .730441i 

1 10030300 

00010000 [cl = [ 

Desired eigenvalues: x = -.4 t .5 

192 

44 
Resultirg eigenvalues using the proposed method, where a factor for relative 

stability was added: 

-. 289342 2 1.378583 

-.14545; +, l.ln345i 

-.400907 2 .500003i 

-.197840 2 .425944i 

Example 3: 

Specifications: g = m2 = 1 

SI = 3 

Eigenvalues of unforced system: 

22.074313i 

-+ .8350001 

[cl = [l 1 0 03 

Desired eigenvalue: .5 + Oi 

Resulting ' igenvdues using the propoeed method, where factor for relative 

stability was added: 

- ,170373 2 1.809Og?i 

-1.617157 + Oi 

- .5OOOOC + Oi 

CONCLUSION 

A pole placement ilgorithm has been proposed which used constrained nonlinear programming techniques for a flnite dimensional model of a linear n degree 

of freedom systexc. It has been shcwn that by constrsini.lg the eigenvalues of the 

45 
full order system while simultaneously placing those allowable, one can ensure 

the stability characteristics of a systen?, and can alter its transient response. 

Results of the Srinathkumar met'nod were presented for &le 1, and shoved 

how this metiiod yielded the desired eigenvalues q.fte accurately, yet unfortunately forced the originally stable system unstable, therefore resulting in an 

undesirable response. 

Xo previous work has guaranteed stable unplaced poles or has assured the 

magnitxde or' relative stability. 

11 [I] Andry, A.N.. Shapiro, E. Y.,andChung, J. C., Eigenstructure Assignment 

for Lineer Systems," IEEE Wansactions on Aeyospace and Electronic Systems, 

Val. 19, No. 5, Septenber 1983, pp. 711-729. 

[2] Andry, A. N., Shapiro, E. Y., and Chung, J. C., "Modal Control of Vibrating 

Systems," F'roceedings of the First International Xodal Analysis Conference 

and hkibit, Orlaneo. 

[3] Lavison, E. J., "On Pole Assigment in Linear Systems with Incomplete State 

Feedback," IEXE Transactions on Automatic Control, June 1970, p~. 348-351. 

[4] Meirovitch, L., Computational hiethods ,& Structurd Dynamics, Sijthoff and 

NoorcUhof f , Rockville , Maryland, 1980. 

[5] Srinathkumar, S., "Eigenvalue/Eigenvector Assignment Using htput Feedback," 

IEEE Transactions on Automatic Control, Vol. AC-23, Bo. 1, February 1978, 

pp. 79-81. 

[6] Wortau, W. M., *'On Pole Assignment in Multi-InpQt Controllable Linear 

Syctems," IiEE Transactions on Automatic Control, Vol. AC-12, Ne. 6 

December 1967, pp. 660-665. 

46 
> N85-31199 

MATRIX TRANSFER FC?NC'IION DESIGN FOR 

FLEXIBLE STRUCTURIB-AN APFLICATION 

A. V. CorpiBo, A. L. Dorm, C. L. 

ArlspaceCocporatioa 

Lth hngtk, CA 9ooo9 

T. J. and C. L. Wq 

ABSTRACT 

The application of utrix transfer fuaction desiga techniques to the 

prchlem of di8turbaaca rejection on a flexibl" space structure in dmartrated. 

The design approach is based on paraeteritiag a class of stabiliriag 

compensators for the plant and forrulating the design specification. as a 

con8traiaed miniriation problem ia tema of theac patautera. 

yield8 a matrix transfer fuaction representatioa of the colpensator. 

space realization of the capeasator is coastructed to iavestigate performance 

and stability on the nominal md perturbed rodel.. The application is ude to 

the ACOSS (Active Control of Space Struttures) optical mtructure. 

The rolution 

A state 

I. INTRODUCTION 

The problem of flexible space structure coatrol has motivated a great 

deal of re8earch for theoreticians aad practitioaers of rultivariable coatrol 

design. 

significant gap betueea the ultivariable theory and the coatrol derign 

implementation. 

of problem specification. 

constraint8 into the specific mathematical cost functionals required by moot 

design methods may be impossible in many carer. 

design methodology may not be traceable to the parameters which describe the 

In spite of the effort. directed ia thio area there still reuiar a 

This gap stew from tuo aources. The first difficultv io one 

Translatioa of corrplcx spater'requircwnts aad 

Free parameters in the chosen 

47 
system in terns of desired performance, plant uncertainty, hardware 

limitations, etc. 

design techniques is the lack of reliable clgnrithms and software to perform 

the sophisticated mathematical manipulations required hy these techniques. 

Recent years have shown very considerable advances in this field (see [l]) but 

mch remains to be done. 

A second roadblock to the implementation of modern control 

Most of the XIXO (aulti-input/iauu?ti-output) compensators which have 

actually iefL the textbook and been calculated in computers are based on state 

space methods, and, in particular, LQG (LinearQuadratic-Guassian) design 

theory. 

design techniques ae well as the availability of reliabie algorithms to solve 

matrix Riccati equations and the ease of performing mst state space 

manipulations. 

systems have been avoided. The extensions of classical frequency domain 

concepts to flIflO systems have not been totally satisfying and calculation3 

involving matrices of transfer functions present an entirely new set of 

problems. Nonetheless, Erequency domain design is still appealing and certain 

feedback notions cannot be adequately expressed without reference to transfer 

functions. 

This is due in part to the long hiatory of development of these 

Frequency domain techniques for calculating HIM0 feedback 

We have carried through a compensator design for a flexible structure 

based on transfer function parameterization techniques. General theories of 

feedback control system parameterization have been developed by several 

authors (121, [31, and 141). 

selection of a set of numerical quantities, along with an acceptable range of 

values, which span a class of possibly acceptable compensators and, wi 7 

which, one is able to adequately express the system requirements in tenus of 

costs and constraints. A particularly simple parameterization for stable 

plants was introduced by Zames, [4], and exploited for the unity feedback 

configuration of Figure 1 bv Deroer and Chen [Sl. 

parameterization we will implement here. The details are in section IV. 

Previous examples of this design approach can be found in (61 and [7]. 

The goal of a parametric approach is tne 

This is the 

48 
11. ACOSS STRUCTURE 

The ACOSS optical structure was developed by the Charles Stark Draper 

Laboratcries as a control design test specimen to evaluate the design 

approaches devzloped for the DARPA ACOSS program, (81. 

exhibit the closely gpaced, lov frequency mode distribution expected on some 

future space systems. 

having 84 dynamic degrees of freedom (see Figure 2). 

nominal structure, two perturbed structures were defined to represent plant 

uncertainty. 

It was designed to 

Thz structure is provided as a finite element model 

In addition to the 

The perturbed Pbdels represent mass and atiffness variation8 of 

approximately 10%. The nominal model is denoted PO, the perturbed models are 

P2 and P4. 

The performance goal is expressed in terms of a line of sie5t error on a 

focal plane on the lower section of the truss as shown in Figure 2. The error 

has two angular components and a defocus component resulting from deviations 

in the optical path due to structural vibrations. 

determine the optical path. 

structure. Two disturbances are defined au 'he structure as shown in the 

figure. 

propagating from the equipment panel and we assume it ha, a flat PSD out to 

5 Ht. 

system. 

into the line of aight is still unacceptably high. The control problem is to 

further reduce this residual with active scructural control. 

Three rigid airrots 

Theae are aarumed to be rigidly mounted to the 

For our design problem ue are only considering the disturbance 

The equipment panel is isolated from the structure by a spring-damper 

The residual disturbance propagation through this isolation system 

IIX. HODEL SELECTION AND ACTUATOR PLACEMENT 

For the current design problem we chose a 5 modo model of the structure, 

selecting those modes having mort eignificant influence between disturbance 

and line of sight. 

internally balanced coordinates. 

The modal influence was determined based on ideas from 

For a description of internally balanced 

49 
coordinates see 191 and for an application to modal coordinates see [lOl. 

Given a second order model description, 

Mode 7 

I 

Frequency (Hz) .15 

.. + 2 s.u.i. + wi2qi = g; 3 i 1,. ..,n qi 111 

8 12 13 21 

.26 .58 .58 2.3 

- 

ith natural damping 5 frequency w , inputs E, and outputs yB an index 

i i YO t 

ranking the modes can be calculated as the approximate “second 

order modes,” ( [ 101 by 

Using the modal disturbance influence matrix for the &.‘a and the line 

1 

of sight measurement matrix for the h.’s the 5 highest rank modes are 

tabulated in Table 1. Agreeing with our intuition, these turn out to be two 

isolator rotations, two isolator translations, and the first bending mode of 

the upper truss. 

-1 

A description of the modes of the structure can be found in 

[SI. 

Table 1. Design Modes 

The line of sight measurement matrix ie a function of 21 nodal degrees 

of freedon. Prom among these 21 degrees of freedom we c%ose to locate three 

force actuators (assumed to be of the mornenture excharige or proof mass type) to 

control the three line of sight measurements. 

appeal is again made to the approximate second order modes of equ9tior. (3). 

To make this selection an 

50 
If the forcing function on the right of (1) is g .u where 8.. is the 

influence of the jth actuator, .i = 1,...,21 on the ith amde, i 

then we denote the corresponding second order mode by U.. and define 

il 13 

I1...,5, 

1J 

2 52 

a = 1 aij i=l j (4) 

Here, a. is a measure of the influence of the jth actuator on the 

1 

line of sight for the selected 5 mode model. 

force directions span the three spatial directions and have large a. with 

respect to the total 21 possible actuators.- Two of the actuators selected are 

located on the cornera of the primary mirror and the third is on the lower 

truss. 

We chose three actuators whose 

J 

To complete the description of the design plant we assumed the 

availability of direct measuremmta of the line of sight. 

were used for the control design. 

the design plant in modal coordinates, 

No other sensors 

We now have a state space description of 

2 = Fx + Gu + Dd (5) 

y = Hx (6) 

whece 0 it :he actuator comrrmnd and d is the disturbance input. 

Fcr calculation of the compensator we need a transfer function 

representa.ian of the design plant. 

constructiiq state opace realizations of the compensator is a polynomial 

matrix coprime factorization [11,12', that is, P 

coprimc polynomial matrices. 

factorization from a state apace description can be found in [131. 

The convenient representation for 

ND-' where N and D are 

An algorithm to construct a coprime 

51 
IV. DESIGN PROBLEM 

The feedhack configuration used for the design is shown in Figure 1. 

The closed loop systein is referred to as d. 

plant, a 3x3 transfer function given from the state space equations by 

H(sI-F)-lG. The inputs are u and u2 with the reference input, 

identically zero. 

represented by y 

through the transfer function P = H(s1-F) 

an additive disturbance, Pd, at the plant out.put. 

P is the open loop design 

1 

The outputs are y1 and y2 with the line of sight 

The disturbance propagates into the line of sight 

c1 -1 2' 

D and may thus be represented as cy 

The closed loop system transfer function is defined to be 

Stability of H can be taken to be closed loop stability. H may be 

expressed in a simple parameterized form as 

YU YU 

where Q is referred to as the Zamee parameterization, (41, with 

Q C(I + PC)-' . (9) 

We state here the fundamental result from [5] which is the basis of this design 

approach. 

- Fact: For P exponentially stable and strictly proper, Q is exponentially 

stable and proper if and only if 

52 
(i) C is proper and 

(ii) H is exponentially stable and proper. 

YU 

When this is the case the cornpeneator ir given by 

In other vords designing stabilizing compensators for in equivalent to 

specifying exponentially stable, proper Q. 

From (8) we see that the 1/0 map, that. is, the transfer function from 

uI to y2 is 

Given an invertible plant transfer function, P, one can see from the 

relation (1L) that a parametrization of the closed loop system by Q is 

equivalent to a parameterization by H. Moreover, for P exponentially ntable, 

Q exponentially stable implies the same for H. But since 

it becomes cleat that exponential stability of H only implies exponential 

stability of Q when P has no watable zeros. 

additional condition on H, namely that H ha8 the same right half plane zero 

structure as P, then parameterization by such H is equivalent to parameterization by exponentially stable Q. 

additional constraint of properness of Q is required and.will result in an 

excess pole over zero constraint on H which depends on P. 

However, by imposing an 

If a proper compensator is desired the 

Parameterization by the I/O map, 11, may rimplify the deuign problem and 

allow the designer to more directly specify his design obiective. 

example, for 3 dirturbance attenuation problem, the cloned loop disturbance to 

output map, or sensitivity map, ir simply given as (I - H). In addition, in 

some applications, a decoupled 1/0 map is desirable and one ie directly able 

For 

53 
to parameterize a uiagonal H. This is the approach we take for this design. 

Calculating the transmission zeros of cur design plant using the QZ algoritba 

114) we find that there are no zeros in the right half plane so we may freely 

specify H as diag(hi, i=1,2,3) with each h. of the fom 

1 

where g is a gain and 

2 pnW = tJ2 + 25; w 8 + wn 

nn (14 1 

This parameterization has 21 parameters consisting of the gains, and 

second order damping and frequency terms. 

We set for ourselves a deeign goal of minimizing clo8ed loop response to 

the disturbance over a low frequency band of 5 Hz. 

a constrained optimizLtion problem as follows: 

To achieve this we define 

Minimize 

subject to 

:Stability 

0.04 < wn, tod < w :9andwidth b 

j 

hi(0) 1 :Low frequency r oiae rejection 

54 
The matrix under the norm of J is diagonal so we aimp'v take the 

Euclidean vector norm of the diagonal. The minimization a& .ne cost at 5 Hz 

and the DC unity gain constraint will result in disturbance rejection across 

the 5 Hz band. The P term in the cost weights the diagonal terms in (I - H) 

according to the way the disturbance propagates through the structure. 

k 

In general (I - H) is the ratio of the relative uncertzhty in the 1/0 

map to the relative uncertainty in the open loop plant. More qreciselv 

where 

for a "perturbed" plant $ rhich reeultr in a perturbed 1/0 map 6. 

minimization of J reduces the impact of plant uncertainties on closed loop 

s y e t em performance. 

In effect, 

Having specified the opiimization problem one can use numerical or 

analytical means to solve it. 

minimum to this problem analytically. 

dependent on the bandwidth, Ob, 

satisf Lea 

Omitting the details, we calculated a local 

The achievable performance is clcarlv 

For A given U,, the local minimum 

5s 
4 

W d =i 

2 R 

W 

n 

Given this solution we can adjust the bandwid,h of each of the three 

loops to achieve a desired performance level. 

each channel we have the following parameter values: 

To achieve 0.04 reduction in 

(23 1 

V, COMPENSArdR REALIZATICN 

Having arrived at parameter valuee we have specified rhe derired I!O 

map, H. The compenaator which will produce this I!n is 

Sines the I/O map is given by diag(hi 1 with each hi of the form 

i 

'i di 

n I- (26) 

56 
the compensator becomes 

C = P-ldiag (5) d.- . (27) 

We have already expressed P as a polynomial matrix coprime factorization, 

-1 P = ND . Thus (27) becomes 

Since the degree of d. - n. is 4, w can factor this polynomial into two 

quadratics as 

1 1 

di - n. * d. d. , i = 1,2,3. 

l1 l2 1 

Hence (29) can be rewritten a8 

(29) 

- 

By inspection, diag(d. IN ir column-rtductd [121, ad has column Lagtees 

ll - 

eqc,ling those of D. Consequently I) {diag(di )N}-' ir proper and has a rtate 

1 - 

space realization [12, 

space realization, the 

for c* 

- 

See. 6.41. NOW, since diag(n./d. alro har a state 

two realization. can be carcaded to yield a realization 

1 12. 

57 
VI. RESULTS 

Having computed a state space description of th; compensator we are now 

able to determine closed loop stability for various versions of the plant 

simply by extracting the eigenvalues from the closed loop state equations 

derived from Figure 1. 

P2, and P4) the five mode description remains stable under feedback by our 

compensator. 

We find tnat for el1 three versions of the plant (PO, 

To investigate the robustness of the design with respect to unmodeled 

dynamics we appended additional modes to the plant rnodel and found that the 

closed loop system became unstable in almost all cases. 

this problem we discovered that though the 5 mode design plant was minimum 

phase, the addition of almost any other mode or set of modes resulted in a 

nonminimum phese plant. Information about these unscible zeros was not 

available in the design plant so the resulting compensato- tended to place 

closed loop poles at these zeros. 

one of modeling or model redxtion. 

must hrve information about the righL h.mlf plane zeros of the plant. 

Upon investigation of 

Thus the stability problem experienced is 

In general, any control de8ign approach 

The performance of the closed loop system remained very consistent with 

the predic&ions made during the design stage. 

at 5 hz of the two angular components of the line of sight is given as a 

fraction of open loop response for the three models by: 

The steady state RSS response 

The broadbaad dirturbance attenuation is illustrated on the Bode plots 

of Figures 3 and 4 which compare open and closed loop response. 

sipnificant portion of th- 5 He band the performance improvement is 3 to L 

orders of magnitude. 

Across a 

58 
VII. CONCLUSIONE 

We have demonstrated the applicability of a transfer fmction parameterization design approach for problems of broadband disturbance attenuation on 

flexible space structuren. 

with a great deal of flexibility to meet system requirements by the choice of 

parameter set and selection of cost function and constraints. 

implementation of this technique requires difficult numerical calculations 

involving matrix transfer functions, algorithms and software for these types 

of problems are already elserging. The success of this approach is dependent 

This methodology provides the control designer 

Although the 

on an appropriate parameter selection in which to express the problem 

specifications. This suggests research, probably application specific, which 

addresses the issues of problem description and requireaents interpretation in 

the control design process. 

REFERENCES 

[ 11 A. J. Laub, "Numerical Aspects of Control Design Computations," 

Computer-Aided Design and' Analysis of Digital Guidance and Control 

Systems, NATO AGARD-LS-128, July 1983. 

21 L. Pernebo, "An Algebraic Theory for the Design of Controllers Linear 

Multivariable Feedback Systems," IEEE Trans. Aut. Contr. , Val. Ac-26, 

pp. 171-194, February 1981. 

[ 31 D. C. Youla, H. A. Jabr, and J. J. Bongiotno, Jr., "Modern Wiener-Hopf 

Design of Optimal Controllers - Part 11: The Multivariable Ca#e," IEEE 

Trans. Aut Contr., Vol. AC-21, pp.319-338, June 1976. 

[ <] G. Zsmes, "Feedback and Optimal Sensitivity: Modal Reference Transformations, Multiplicative Aeminorms, and Approximate Inverses," IEEE 

Trans. Aut. Contr., Vol. AC-26, pp. 301-320, April 1981. 

[ 51 C. A. Desoer and M. J. Chen, "Design of Multivariable Feedback Systems 

with Stable Plant," IEEE Trans. Aut. Contr., Vol. AC-26, pp. 408-415, 

April 1981. 

[ 61 F. P. Pernandez, "Multivariable Controller Design Using Parameter 

Optimization," M.I.T. master's thenis, May 1983. 

59 
REFERENCES (Continued) 

I 71 C. L. Gustafson and C. A. Desoer, "Controller Design Cor Linear 

Multivariable Feedback Systems with Stable Plants, Using Optimization 

with Inequality Constraints," Int. J. Contr., Vol. 37, No. 5, 

pp. 881-907, 1983. 

[ 81 Charles Stark Draper Laboratory, "ACOSS Six (Active Control of Space 

Structures)," Interim Report, RADC-TR-80-377, January 1981. 

91 B. C. Moore, "Principal Component Analysis in Linear Systems: 

Controllability, Observability, and Mode1 Reduction," IEEE Trans. Aut. 

Contr., Vol. AC-26, pp. 17-32, February 1981. 

[lo] C. 2. Gregory, Jr., "Reduction of Large Flexible Spacecraft Models Using 

Internal Balancing Theory," Proceedings of the 1983 AIM Guidance and 

Control Conference, Gatlinburg, Tennessee. 

[ll] b. #. Callier and C. A. DesOer, Multivariable Feedback Systems, 

SpringerVerlag: New York - Heidelberg - Berlin, 1982. 

[121 T. Kailath, Linear Systems, Prentice Hall: Englevoad Cliffs, N.J., 1980. 

1131 R. V. Patel, "Computation of Matrix Fraction Descriptions of Linear 

Time-Invariant Systems," IEEE Trans. Aut. Contr., Vol. AC-26, 

pp. 148-161, February 1981. 

[14] A. 3. Laub and B. C. Moore, "Calculation of Transaission Zeros Using QZ 

Techniques," Automatica, Vol 14, 1978, pp. 557-566. 

60 
E = &j 

- C P "1 

- y2 

FIGW. 1. BLOCK DIAGW OF 2 

61 
1 

IITUDE (db) 

FIGURE 3. PO RESPONSE 

1 ab0 

FPSQUENCY 

(rad / sec 1 

FIGURE 4. P2 RESPONSE 

62 

FREQUENCY 

(rad /set) 
ROBUST CONTROL DESIGN FOR LARGE 

SPACE STRUCTURES 

w. L. Em$ma 

McDoanct!~TechnicalservictsCo. 

Houston, TX 77W8 

J. A. Bami 

University of Washington 

seattlc, WP 98195 

INTRODUCTION 

-, !ne control design probleix for the class of future spacecraft referred to as 

lwge space structures (US) is Sy n3w well known [l-31. The issue is the reduced order control of a very high-order, lightly damped syPtem with uncertain 

syste- parameters, particularly in the high frequency modes. This paper presents a design methodology which incorporates robust?iess considerations as 

part of the design process. Canbining pertinent results fra multivariable 

system theory and optimal control and estimation, LQG eigenstructure assignment [4] and LQG frequency-shaping, C5-73 were used to improve singulai- value 

robustness measures in the presence of control and observation spillover. 

The design technique is summarized as follows. A low order LQG compensator is 

synthesized using the technique of recursive eigenstructure aaaigment to 

place closed-loop eigenvalues where desired. 

singular value performance margin and for singular value gain margin with 

respect to plant uncertainties (e.g., modeled dynamics). The compensator is 

then resynthesized using frequency-shaping concepts to improve the singular 

value robustness measures. The recursive eigenstructure assignment technique 

allows regulator close-loop eigenvalue placement at the desired locations for 

the plant and as required for frequency-shaping. Forthermore, the frequencyThis design is evaluated for 

53 
shaped compensator eigenvalues can also be assigned, thus assuring LQG compensator stability, as well as estimator stability. 

This procedure using robust freyuency-shaped compensation was applied to the 

design of the controller for a representative large space structure. Results 

are presented as singular value &de L?ots. Canparisons are made to a recent 

study8 utilizing the sam large space structure model. 

LQC CONTROL DESIGN FOB LSS 

Control design plant modelling for LSS utilizes a high-order structural model, 

typically obtairied by f inite-element programs such as NASTRAN. 

tions of cmputer implementation require that the f inite-element model be 

reduced to a design model. 

into prilcary and residual modes, where the primary modes are to be used for 

control design. The modal truncation can be baseJ on engineering judgement or 

on a selection criterion such as modal cost analysis [9]. 

The limitaOne approach is to truncate the high-order model 

The system model has the form 

64 
where xp are the primary modes and XR are the revidual modes. An observer-based control design for +he primary modes then has the form 

A 

'P = AP ';p + Bpu + G(y - Cp qp) 

Using LQC design, the gains (h, G) are selected to minimize quadratic performspillover and observation spillover respectively. These terms have the potential fur interacting through the observer (2) to produce instability. 

LQG theory guarantees that the reduced-order closed loop system is stable with 

Of ('P-BpK) and (Ap-CCp). However, no such guarantee holds for 

the compensator, 

u = Hy (3) 

which has the Of (Ap-BpK"CCp). This fact can be fatal for LSS 

reduced-order centrol, unless measures are taken to emure system robustness. 

ROBUSTNESS MEASURES FOR LSS 

For multivariable fesdback systems the emerging singular val~- robustnsss 

theory can be used to develop measures for stability and performance. Kosut, 

et al,a applied this theory to the large space structure control design problem, treating the residual dynamics as a perturbation. For a system with a 

65 
stable nominal feedback system (based on the reduced model) and stable perturSctions (due to the residual dynamics), sufficient conditions for stability 

are obtained when the singular valde stability measures exceed the maximum 

per,drbation due to model uncertainty. Fig. 1 defines the terminology for a 

large space structure control system. For an additive perturbation, rig. 2a, 

the Sufficient conditions for stability are 

\;here 7 (0) indicates the maximum singular value and - o (0) indicates the minimum singular value. (Singular values of the cmplex rnaL-ix i. are the positive 

square roots of the eigenvalues of A*A, where (e)* indicit?s conjugate 

tran-spose.) If Cc(s) is minimum phase and invertible, a multiplicative perturbation can be formed, Fig. 2b, and the sufficient conditions for stability 

are then 

66 
where the jw arguments have been suyressed. 

ating frequency region (i.e., the ftcontrol bandwidth") is provided when the 

per forraance measure 

Good performance within the operP'4 = - 0 CI + CCH] 

is large. 

plot analysis; the measures (5) are ge~!eralizations of Nyquist inverse polar 

The stability measures (4) are generalizations of Nyquist polar 

plot analysis. The need for large performance measure (6) is a generalization 

of the desirability of large loop gains. 

ROBUST COMPENSATION DESIGN 

The stability and performance measures presented above require staL lity of 

tqe nominal feedback system. In a previous work C41, the authors presented a 

recmsive design procedure which assigns the closed-loop eigenstructure in 

linear quadratic regulators. At each stage, the requi-ad solution for the 

steady state tticcati matrix which shifts 3 pole or pCIe pair t@ specified 

values is obtained. Far pole pair placement, a free parameter in the solution 

Fermi ts selection of closed-loop eigenvectors. 

ioarized in Arpendix 1. 

This design procedure is sumilsing duality, the procedure also applies to estiinator design. Ely extension, 

the procedure can be used to design stable compensators by considering the 

closed-loop rzgulator dynamics matrix (A-BK) as the open-loop system and picking the estimator gain to place the comperisator eigemtructure of (A-BK-CC). 

67 
Crnpensator robustness can be enhanced through the use of frequency-skaped 

control and estimatim [5,6]. In frequency-siiaped estimation, a frequency-domain performance index is comidered, 

where w is the disturbance and v is the sensor noise. Sensor r.3ise frequency-shaping is re?lized by treating v as an.auta cerrc:ate;i mise 53urce of 

the form 

uhere vl(jw) is a white noise process. In the approach used here, Q(jw) is 

determined by pcle placement, equivalent to iAjecting ficti tiou: process 

nois:. 

weighting over the entizd spectrum. 

R1/2(jw) must be proper (not strictly proper) to maintain sensor noise 

Then define a pseudo-measurement 

68 
This dynanic mdel is appended to the system dynamics to form the frequencyshaped estimator, 

where z’ is obtained frm (10). The gains fi.and Cv in be Jicked to pla.:. the 

eigenvalues of (11) at those of the frequency-shaping filter (IO) and the others as required for performance. A dual result can be used to develop frequency-shaped gains for the regulator. 

Because frequency-shaping ada., .states to the compensato:, an efficient choice 

of the loops to be shaoed is desirable. 

loop selection based on the singular vectors or the return ratic mairices CcH 

or HG,. 

singular vector corresponding to F(A) will get the largest amplificaLion by A. 

Similarly, a vector in the direction of On, the singular vector correvpcnding 

to - o(A) will get smallest amplification, Therefore, if the component of y in 

tk direction which is closest to q1, la reduced by a fi!ter before it enters 

A, ;(A) is effectively reduced. - o(A) 

of y closet tc qn befora it enters A. 

introduces transmission zeros into the cmpennater transfer function, 

Kim [7] has developed a procedure for 

!le 2onjeectured that ar. input vector y in the d:rection OF 91, the 

increased by increasing the cornpoi... 

It ,A be shown that frequencv-sha9ing 

69 
DESIGN KETHODGLOGY 

The riiscuasion which nc; been presented above suggests the f9llowing design 

methodoi 3g: : 

;. Cmpsnzator aesign for performance of the reuced order system. 

2. EVL .scion of the stability margins (4,5) against the perturbation due to 

the residua: dynamics. 

3. Selecti.cn of frequency-shsping filters to enhance stability robustness. 

4. Synthesis c: .cque~cy-shapsd ccmpensator to incorporate performance and 

stab i 1 i t y marg; ns . 

The recursive eigeastructure design algorithm can tz used for the Gesirpls. 

EXAMPLE. 

Tne design methodology was app1iP.i to a contrcl design for the ACOSS-1 model, 

hlk~ ur.ed in the comparis?n study C91. 

state-space data are listed ic Appendix 2. As in the comparison study +.he 

first eight structura: modes were retaiaed. A regulator was designed with 

ciosed-loop poles c?t, 20% danpifig; a compensator was designed with poles at 

The model is illustrated and the 

riticd darnpini;. Fig. 3 illustrates stability measure (5) for the loop 

broKen at the 7uLDut.. Perfcrmance s adequate at iow frequencies but stability robustrrc.: .'s inadequate above 1 Hz. 

79 
To improve stability robustness, frequency-shapea estimation was incorporated 

in all three output loops using second-order low-pass filters. Fig. 4 illustrates the recovery of stability robustness while still retaining good low 

frequency perfornance, Fig. 5. 

DISCUSSION 

In the comparative study by Kosut, et a1 C81, both LQG modal control and a 

frequency-shaped control were investigated (along with others). 

was found to have pcor perzormance as well as poor stability rJbustness. 

quency-shaped control was found to have adequate stability robustness, but 

poor low f reqbeilcy performance. 

LQG control 

FreThe methodology presented here addresses both of these issues. Performance is 

achieved by pole placemenL design of the compensator, achieving good loop 

gains at low frequency. Stability robustness is achieved by adding frequency-shasing without sacrificing low ftequency performance, since the gain 

of the frequency-shaping filters is one at low frequencies. 

CONCLUSIONS 

A design meth dology for control systms for large space structures has been 

proposed which incorporates both performance and stability robustness concerns 
as an integral part of the design process. 

ing the poles of the compensator. 

quency-s!,aping the compensator to satisfy a frequency domain stability 

robustness test. 

Performance was achieved by placStability robustness was achieved by freAn ex.mple #as F: sented which applied the rnethod~lgy to a system with the 

loop broker. at the output. A full design study wcu!d also require examination 

of the system wi:h the loop broker. at the input, using regdlator frequency-shaping to enhance robustness. 

REFERENCES 

1. Seltzer, S. !I., "Dynamics and Control of Large Space Structures: A.1 Overview," Jnl. of Astro. Sciences, Voi. XXVII, No. 2, pp 95-101. -- -- 

2. Likins, P., "The New Generation of Dynamic Interaction Problems," -- Jnl. of 

- Astro. Sciences, Vol. XXVII, No. 2, pp 103-113. 

3. Salas, M., "Trends in Large Space Structure C?ntrol Tneory: Fondest 

Hopes, Wildest Dream," IEEE Transactions on Automatic Control, - Vol. 

AC-27, NO. 3, pp 522-535. 

4. Eastman, W. L., and Bossi, 3. A., "Design of Linear Quadratic Regulators 

xitk Assigned Eigenstructure,tt Int. Jdl. of Coctrol, Vol 39, No. 4, 

731 -742. 

72 
5. Cupta, N. K., "Frequency-Shaped Cost Functionals: Extension of Linear-Quadratic-Csussian Desibm Methods," Jnl. of Guidance and Control, Vol. 

3, No. 6. 

6. Cupta, N. li., "Robust Control/Estimator Design by Frequency-Shaped Cost 

Functionals," Froc. of 20th IEEE Control and Decision Conf., San Diego, 

CA, 1981. 

7. . Kim, B. , "Frequency-Shaped Estimation and Robust Controller Design for 

Aircraft Flight Ccntrol in Wind Disrurbances," Ph. D. Thesis, University 

of Washington, Dept. of Aeronautics and Astronautics, Seattle, MA., 1983. 

8. Kosut, R. L., Salzwedel, H., and Emami-Naeini, A., "Robust Control of 

Flexible Spacecraft, Jnl. of Guidance and Control, Vol. 6, No. 2. 

9. Skelton, R. E., "Cost Decmpositicn of Linea- Systems with Application to 

Model Reduction," Int. Inl. of Control, Vol 32, No. 6,. pp. 1031-1055. 

10. Sdlheim, 0. A., "Design of Optimal Control Systems wit.h Prescribed 

Eigenvalues," Int. Jnl. of Contral, Vol. 15, No. 1. 

11. Strunce, R., Lin, J., Hegg, D., Henderson, T., "Actively Controlled Structure Theory," Final Report, Vol. 2 of 3, R-1338, Charles Stark Draper 

Lab., Cambridge, Mass, Dec. 1979. 

73 
APPENDIX 1 

Recurs i ve E i gens t ruc t UT e Des i gn 

The steady-state optimal Control law for the linear, time-invariant, 

cont roll ab1 e system : 

= AX + BU 

which minimizes the quadratic performance index, 

J = 112 je- LXT QX + UT Ru] dt 

is linear state feedback 

u = Kx = - R-1 BT sx 

(A. 1) 

(A.2) 

(A. 3) 

where S is the solution of the steady-state Riccati equation, 

In this aFpendix we smmarize an interaciive design technique bihich solvns 

(A.4) to provide specified eigenvalues of the closed-loop system dynamics 

matrix A+BK and which also permits sane freedom in selecting closed-loop 

74 
eigenvectors. The method is reported elsewhere [4] in detail. It extends the 

procedure of Soiheim ClGj in which, for fixed R, the elements of Q providing 

the required pole placement are calculated directly. 

The desigr. technique is recursive; at each stage, the system dynamics matrix A 

in (A.1) incorporates previous state feedback. 

i ng eigens tructure calculation: 

We then implement the followrfhere A = T-l AT is block diagonal, T is the real eigenvector matrix of A, and 

ii = T'lBH-1BTT-T is symmetric and positive semi-definite. is identical to A 

except for a block of shifted poles, 

eigenvectors t,o closed-loop eigenvectors; it is defined as the "stage" 

eigenvector matrix. ,C is the Riccati matrix in the open-loop diapralized 

coordinate system; 3 is chaeen to shift a single poie or a pair of poles. 

corresponding gain matrices, K, de .ermined for each stage are subsequently 

added to obtain a final gain which achieves the same closed-loop pole locations. 

X is the transformation frm open-loop 

.- 

The 

To provide the required pole shift, the only non-zero elements of S correspond 

to the entries of A which are to be shifted. With this ohoice of S, the characteristic equation factors into the product sf terms for the unshifted poles 

and a term for the desired shifted poles. Thu-, 
where I is the indox set for the unshifted poles, and D(s) contains explicit 

elements of S, H, and the block of A which is to be shifted. Matching the 

coefficients of powers of s in D(s) to tile equivalent terms in the clcsed-loop 

characteristic equation provides a set of equations in the required elements 

of 5. 

satisfies 

For t.he single pole shift Ajj = X to K, the only non-zero element of 

For double pole placement it can be shown that the three required elements of 

lie on the intersection of two quadric surface in a mathematical space having the three 2 elements as coordinates. 

solation for Q has b similar geometric interpretation.) 

submatrix of H is positive definite, the surfaces are a plane and a hyperboloid of one or two sheets; the i;lters%slm, if it exists, is always an ellipse. If the relevant submatrix of :; is singular, the surfaces are planes, 

and the intersection is a line. The different points comprising the solution 

all provide the desired eigenvaly-. placement, but with different eigenvectors. 

(It can also shown that a direct 

If the corresponding 

In ref. 4 a solution for 5 is presented which takes advantage of the quadriG 

surface geometry LO define a free paraiieter that allow design freedom In the 

choice of closed-loop eigenvectors. The solution for the stage eigenvector X 

partitions into two sets of equations. The first is a homogeneous Lyapunov 

equation for the submatrix correspmding to the shifted pole block in A. For 

5. r?le pair shift, the submatrix is 2x2. Hence, depending upon the nature of 

76 
the closed-loop pole3 (real or complex), one or two elemedts of the submatrix 

mhy be chosen arbitrarily; the remaining elements then depend :.TL the choice of 

elments of g. The other equation is a non-homogeneous Lyapunov equation in 

the remaining elements of the columns of X containing the 2x2 submatrix; its 

solution depends upon the 2x2 submatrix, the elements of 5 and n, and certain 

elements of H. 

The closed-loop system eigenvector matrix is then TCL - TX, 

depends upon 5, which varies with the choice of the free parameter. 

The sGlution of x 

Therefore, by recursively shifting pole pairs, design freedom exists to select 

closed-loop eigenvectors while providing required pole placements. 

The procedure outlined above lends itself to a recursive procedure for pracdcal multivariable regulator design. 

lows: 

The steps in the procedure are as fol1. System (A.1) is placed !.n modal form. 

2. The designer seiects the control weighting matrix R, then H is calculated. 

3. The designer selects a real pole or pair of poles to be shifted and their 

desired location; I 8 pair, he also aelects the free parameter which 

determines the closet?-loop eigenvectors. 

4. The stage gain is calculated and the closed-loop system is placed in modal 

form . 

77 
5. Steps 3 and 4 are repeated for other poles until the designer is satiafied. 

6. The total system gain is obtained by adding the stage gains. 

Clearly by duality, the same process can be applied to estimator dssiy?, permi t ting the development of multivariable compensators. 
APPENDIX 2. 

The ACOSS-1 flexible spacecraft model was developed by the Charles Stark 

Draper LabOratorY-lO It is representative of many radar and optical control 

problems, but is small enough to be tenable for research studies. The atructire is a tetrahedral truss supported by three right-angle bipc&. The truss 

members are flexible in the axial direction only. The model has 12 modes; for 

control design, only eight are assumed to be known. 

79 
Residual Dynamics 

4 

I +AG 

Fig. 

- 

G 

Compensator 

1 - LSS Control System 

J t 

I J 

Fig. 2a - Additive Perturbation 

I 

i 

Fig. 2b - Mu1 tip1 icative Perturbatir. 

80 
.I 1 10 

f 

Fig. 3 Reduced - Order Control Stability Margin 

.I 1 10 

f 

Fig. 4 Frequency - Shaped Control Stability Margin 

81 
100 

10 

J 

.1 

61 

.oo I 

.ooo 1 

.I 1 

f 

Fig. 5 Loop Gzins 

82 
ON THE STABILITY OF COLLOCATED 

CONTROLLERS IN THE PRESENCE OF 

UNCERTAIN NONLINEARITIES AND 

OTHERPERILS 

S. M. Joeai 

NASA Langley Research Center 

Hampton, VA 23665 

ABSTRACT 

Robustness properties are investigated For tuo types of controllers for 

large flexible space structures, which use-collocated sensors d actuators. 

The first type Is s~l attitude controller which uses negative definite feedback 

of .easured attitude and rate, uhlle the second type Is a &ping enhancement 

controller which uses only velocity (rate) feedback. It is proved that collocated attitude controllers preserve closed-loop global asymptotic stability 

when linear actuator/sensor dynamics satisfyiqg certain phase conditiom are 

present, or monotonic increasing nonlinearities are present. For velocity 

feedback controllers, the global asyrptotic stability is proved under much 

weaker conditions. In particular, they have 90. phase margin and can tolerate 

nonlinearities belongiqg to the lo,-) sector in tbe actuator/sensor characteristics. The results significantly enhance the viability of both types of 

collocated controllers, especially when the available infomatlcm about the 

large space structure parameters is inadequate or inaccurate. 

INTROWCTION 

Large flexible space structures are Infinite-dimensional systems wlth very 

8-11 inherent energy dissipatim (damping). Becam of practical limitatlone, 

only finite-dlm!nsional controllers and point actuators and sensor8 mst be 

used for controlling large spa= structures (LSS) . In addition, coneIder8ble 

uncertainty exists in the knowledge of the parameters. For these reaaons, the 

design of a stable controller for a large space structure (LSS) is a 

challenging problem. 

A class of controllers, terwd "collocated controllers" [l], represents an 

attractive controller because of ita guaranteed stabil'ity properties In the 

presence of plant uncertaiatier. Collocated attitude (a) controllers are 

designed to control the rigid-body attitcde ae well as the structural des, 

while collocated direct velocity feedback (CDVPB) controllers are designed only 

for enhancerent of structural damping. Both types of collocated controllers 

guarantee stability regardless of the number of -des in the LSS rodel and 

uncertainties in the knowledge of tk parameters [I], [Z]. A CA controller 

basieally consists of compatible oensor/actuator pairs placed at tlic same 

c -a 83 
locations, and utilizes negative definite feedback of position and velocity 

(e.g., LSS attitude and attitude! rate). A CWW controller 131 is a special 

case of the CA controller *re only rate feedback is used for damping 

enhancement without affectiw tbe rigid-body modes. It has been proved in 

references [I], 121, (31 that, the closed-loop system is always stable in the 

sense of Lyapunov, and is also asymptotically stable (AS) under certain 

additional conditions. 

Although collocated controllers have attractive stability proper:ies with 

perfect (i.e., linear, instantaneous) sensors and actuators, the sebsurs and 

actuators available in practice tend to have noniinearities and phase lags 

associated with them. In order to be useful in practical applications, the 

controller should be tolerant to nonlinearities (e.g., saturation, relays, 

deadtones, etc.), and to phase shifts (e.g., actuator dpaarics and/or corputational delays). Uncertainties usually exist in the Lnowledge of the nonlinearities ad the phase lags. For these reasons, this paper investigates tire 

closed-loop stability of collocated controllers In the presence of deled 

sensor/actuator dynamics and noaliooaritics. f& siturtioa ia matheaatically 

described by including an operator in the feedback path. The actual input 

c't) *.s given by: 

where % is the ideal (desired) input, A? is a mmmticipative, linear or 

nonlinear, tire-varying or invariant operator. For CA controllers, it is 

proved that the closed-loop system ie globally asymptotically stable if 

1) 2 is linear, time-invariant (LTI) and stable with a rational transfer 

2) If Pe consiats of timeinvariant, strictly monotonic increasing 

nonlinearities belonging to the (0, 0) sector. (A functian Mu) is said 

to belong to the [k.h) sector if MO) = 0 and ku2 Mu) < hu2 for all 

0 * 0). 

matrix H(s) which satisfies certain freqwncy-dorain conditions, or 

For CDVPB controllers, it is proved that global asymptotic stability is 

preserved when 

11 is a stable nonlinear dynaric operator and satisfies certaln passivity 

conditions, or 

2) $f( is a stable LX operator with phase within 290' 

3) $4 consists of mr hear gains belonging to the [O,-) sector. 

These analytical 1' .ults significantly enhance the stability and 

robustness propertice of collocated controllers, and therefore increase their 

practical aypllcabili cy. 

84 
PROBLEM PORMJIATION 

Th linearized equations of motiaa of a large flexible space structure 

(using torque actuators) are given by: 

.. 0 T 1 

AX+BX+~= 1 riUi 

I= 1 

where 

(2) 

(3) 

where Os, $s denote th three rigid-body Buler angles, % is tbe 

nuber of structural des, q1 denotes tbe modal aplitude of itb structural 

mode (i = 1, 2, 1, Is denotes the 3 x 3 ament of Inertia mtrix, 

de-s ope mtrix at the lth (3-uis) actuator location. 

It is uaumed tha m, 3-d~ torque actuators are used. It 11 denotes the I 

x 0 identity mmtrix, d diag( ) denotes a block-di.gona1 matrix. D la a 

symetric positive definite or 8ddefInIte matrix wbich represents the 

inherent structural damping. Since - damping, 1y) mtter haw mall, is 

always present, e IUIUDE D > 0 throughout this paper. A Is m % x % 

diagonal ratrix of squared structural frequencies 

"'7 7 Qi is the 3 x 

22 2 

*nq' A = diag (*,, w2, . . (9) 

Assrning that I, Saxis attitude md rate 8ensots (e.g., star trackers ad rate 

gyros) are placed at tht locatiomm of tht actuators, tk meuured 3-axis 

attitude Jai and rate yri at actuator location 1 (ignoring noise) are 

given by: 

0 

=rx yri i (11) 

denoting 

85 
0 TT TT 

Ya [Yaln Yay - , Y,l 

Yr = [Yrl, TT Yr2' B Y,l TT 

(14) 

(15) 

where u, ya, yr are 3m x 1 vectors, and I' is a 3m x (nq + 3) matrix. The 

control law for the collocated attitude controller is given by: 

u= 

c ucp + Ucr 

u CP = SPYa 

"cr = Xryr 

(16) 

(17) 

(18) 

where uc represents the c-nd input, ucp and uer represent comand 

attitude and rate inputs, and Gp, Gr are 3a x 3m feedback gain matrices. 

For CDVFB controllers, the rigid-body rates are removed from the feedback 

signal by subtracting attitude rates at two locations. Consequently, the model 

used for damping enhancement has the form: 

(19) q+$+Aq=#u 'T 

where 7 consists of appropriate differences between the mode-slopes. The 

control law is given by: 

u C = 'GrYr 

where 

(20) 

The control laws given above for CA and CDVFB controllers have very 

attractive robustness properties. It was shown in [l], [2) that, if D > 0, 

Gp = GpT > 0, and Gr = GrT > 0, then the closed-system is 

asymptotically stable (AS). The stability result holds regardless of the 

number of modes in the model, and regardless of inaccuracy in the knowledge of 

the parameters. In real life, however, nonlineariries and phase lags exist in 

the sensors and actuators, which invalidate these robust stability properties. 

The real problem then is to investigate the closed-loo stability for the case 

where the actual input is given by Eq. (11, where is a nonacticipative, 

linear or conlinear, time-varying or invariant operator. The situation is 

shown in Figure 1. Our approach is to make use of input-output stability 

concepts and Lyapunov methods. We assume throughout the paper that the problem 

is well-posed, and that a unique solution exists. We start by defining the 

terminology and the concepts, which are adopted from [4]. 

86 
MTHWTICAL PBELIHINARIES 

Consider the linear vector space h2 of real square-integrable 

n-vector functions of time t, defined as: 

where Rn is the linear space of ordered n-tuples of real numbers, and It+ 

denotes the interval 0 - < t < -. The 6calar product is defined ae 

For gEh2, its norm is defined as 

Define the truncation operator PT such that 

g(t) 0 L t - < T A 

BTW = P#t) = 

Io t>T 

Def lne the extended space &e2: 

(25) 

Thus he2 Is a linear vector spce of functions of t vhose truncations are 

square-integrable m (0.1) for all T < me For gl, g2 € b2, define 

the truncated Inner product 

Consider an operator $f :be2+be2. 2 fr oafd to be strictly 

passive if there exist finite constants B and 6 > 0 such that 

(28) 2 

< $?e, ioT 3 + awz, v T 0, tfg Lne 

fl is passive if 6 - 0 in (28). 

87 
ROBUSTNESS OF COLLOCATED ATTITUDE CONTROLLERS 

Stability With Dynamlc Operator in the Loop 

We consider the case where the operator is linear and time-invariant 

Ue denote $! g 

g(z,; g) where zo is the initial state vector of x, and assume m = 1 

(LTI), and has a finite-dimensional state-space representation. 

by 

for simplicity (i .e. , one 3-axis actuator). 

Theorem 1. Suppose Is a non-anticipative, strictly slable, completely 

obsepable, LTI operator whose transfer matrix is H(s) = cI+H(a), where E > 0 

and H(s) is a proper, minimum-phase, rational mtrix. Under these conditions, 

the closed-loop system given by Eqs. (l), (2), (lo), (ll), (16)-(18) is 

asymptotically stable (AS) if 

where * denotes the conjugate transpose. 

- Proof - Define the function 

Since C > 0, A > 0, V(t) 

t, and Ging (11, (lo), (11)b (1614181, 

0 for all t 2. 0. Differentiating V with respect to 

.T T -1 

v=- 2x Bx - 2ucr Gr x[zo; ucl 

where $? also depends on its initial state zoo Since k! is linear, 

where h,(t) is the unforced response of &? due 

Since &Is strictly stable, Ihl, is finite for any 

Substituting (32) in (31) and integrating frm 

to nonzero initial 

finite zo. 

0 to T, since V(T) 2 

where 

(31) 

(32) 

state. 

0, 
C8 . In (34). 8 denotes the derivative operator. (?is" is technical;? noncausal; 

however, this difficulty can be overcome & defining the derivative of 8 truncation at T to be equal to that of the -truncated function.) Using Parseval's 

theorem, 

The matrix in the brackets is positive (from Eq. 29), and we have 

-1 2 

<U cr' Gr Rp~cp>T 2 IUcrIT 

which yields (from (33) 

-1 2 0 V(0) -2 <;, &>T -2 E lu cr I T -2 <uCr, Gr ho>T 

(35) 

(36) 

DTD 

wherein we have used the fact that x Bx - q Dq. Therefore, 

where I Ig denotes the spectral nom of a matrix, and X, denotes the 

smallest eigenvalue. Eq. (37) can be written as 

where c1 = d< and c2 = Ihol Therefore, lim (t) = 0, and t& ucr 

it) - 0. Wnoting the rigid-body att1tudet* a = (os, Be, $ )T, this 

implies that lim a (t) = 0. 

t-, 

Taking the limit of the closed-loop equation as 

t- 

- 

(39) 

89 
where the overhead bar denotes the limit as t+=. From (391, flucp = 0 and 

Since fz is observable and its output tends to 

zero, its state vector tends to zero as t-, and the Bystem is asymptotically 

stable. 

= 0, which yields = 0. 

The followirig corollary essentially states that , for diqonal Gp, Gr, 

and H, it is sufficient that the phase lag of &jw) is less than the phase lead 

introduced by the controller. 

Corollary - 1. I. Suppose Gp, Gr and H are diagonal and satisfy the 

assumptions of Theorem 1 . Then the closed-loop system is globally 

asymptotically stable if 

-1 WGri -tan - < Arg G- Pi 

where Arg( ) denotes 

For the case 

-1 Oeri 

. Pi 

{ai(j")} 180' -tan - for all real w G (40) 

condition (40) becomes 

Gr i - G- > l/ai 

Pi 

Thus, for the case of first-order sensor/actuator 

(41) 

dynamics, the system is 

asymptotically stable if the ratio of rate-to-proportional gain is at least 

equal to the magnitude of the actuator pole. 

In Theorem 1 and Corolla y 1.1, the transfer function of ,@ was assumed to 

be of the form: H(s) = tI + ks), where E > 0. That Is, a direct transmission 

term, no matter how small, was present. Prom Theorem 1, the closed-loop system 

is AS for any E. > 0. Therefore, the closed-loop eigenvalues are all in the 

open left half-plane (OLHP). Beca?lsc CIL~ continuity, ii is obvious that, when 

E = 0, the eigenva1':nz rill not cross the imaginary axio. That is, the elgenvalue# Z2ii be in the closed left half-plane (CLIIP). Theorem 2 given below 

considers the case when E - 0. It essent lly ohowe that, if the closed-loop 

system with no elastic modes is AS with 

Theorem 2. Suppose 2 is a non-anticipative, strictly stable, completely 

observable, LTI operator with rational transfer matrix H(a) which is proper and 

minimum-phase. If the closed-loop system for the rizid body model alone (i.e., 

Eqs. 'I), (2), (10). (Il), (16)-(18) with nq = 0) is AS, then the entire 

clos, 

# in the loop, then so is 

with elastic modes, provided that (29) is satiafied with H replacing 

Loop system (i.e., with nq f 0) is AS provlded that 

H(jw) ("Gr - jG ) + ("GI + 

Proof . Considering the rigid-body 

P 

- 

* 

jGp) H (jw) 2 0 for all real w (42) 

equations, 

90 

(43) 
where uq = -G a - GrG and uq = -Gp#q-Gr4i. 

function from 4 to a ia given by P. Thur the. tranrfer 

# 

Since the clo6ed-loop rigid-body 8yrtem ir rerictly stable by umaption, H(8) 

is rtrictlp stable ad finite-gain, which irplier 

. . 

where y ir the gain of H md ir 

proof of Theorem 1, = can arrive 

replaced by b. Since %r - Sr (a + 

. lu I < cllqlr + c21hmIT 

cr T - (45) 

where c1 and c2 are poritive constantr. Completing squarer aa in (38) and 

noting that is. finite, it can be proved that 441~ is bounded for all 

remainder of the proof is similar to that of Theorem 1. 

T > 0, ad that lw(t)=O. Rm (45), +r ale0 tend8 to zero a8 two Tb 

Corollary - 2.2 With the same asrumptiom as in Theorem 2, if Gp, +, and E 

are diago 1, then the closed-loop spotem tr AS if (40) ir uatirfied with E' 

replacing T. 

Prom Corollary 2.2, for the caie here Hii(r) $/(r + 3) with 

ki, ai > 0, the clored-loop aaymptotic etability io aarured if Gpi 5 

aiGri for i = 1, 2, . . , 80 

The significance of the rerultr of thie rectim io that the stability can 

be assured by -king the ratio of the rate-to-proportional gain+ mfficlently 

large. One has to know only the renror/rctuatot ch.r.cterietice. .ad the 

knowledge of the plant parameters ir not required. Thim mmlt i. c#pletely 

consistent with the rerult obtained in 151 for ein&e-input, single-output 

system, for mall Gp and +, wing a root-locur argument. 

the, loop. 

, 

Thl next rcctioa considerr the cam where ncral1neariti.r are prerent in 

Stability in the Prerence of Nonlinearltiem 

Suppose Eq. (1) ir replaced by 

u - Muc) (46) 

where I, io an tvector, one-to-oni, tir-invariant function, $: en", as 

f OllOW8: 

91 
For this case, the stability of the closed-loop system can be investigated 

using Lyapunov methods. A function d(v): R1+R1 is said to belong to the 

(0, -1 sector if d(0) = 0 and vb(v) > 0 for v # 0. d is ,said to belong to the 

[Os -) sector if d(0) - 0 and VMU) 0 for v f 0. [Pig. 21 Many nonlinearities encountered in practice, such as saturation, relay, dead-tones, belong to 

the [Os -1 sector. As in the previous section, we assum that the problem is 

well-posed, and that a unique solution exists, and we consider the case with 

one 3-axis actuator for simplicity. 

Theorem 3. Consider the closed-loop system given by Eqs. (2), (l.O), (ll), 

(16)-(18), and (46), where Gp and Gr are positive definite and diagonal, 

and each $i is in the (0, -1 sector and is strictly monotonic Increasing for 

i = 1, 3, . , m. Then the closed-loop system is globally asymptotically 

stable. 

- Proof. Define 

U 

-1 $i( v)dv 

i= 1 

denote the iith and ith elements of Gp and 

ucp respectively. This form is the well-lmo~ "Lure'-type" Lyapunov 

functfon (6). From Eqs. (4) and (61, xTCx + :TAX - 0 anly when a - 0, 

q - q - 0. That is, this quantity can be zero when a 0 0. However, when q = 

0, Ucp* - G a, which is nonzero when a f 0. Thus the third tela 011 the 

right hand sfb, of (48) is positive (since $1 is in the (0, -) sector) for a 

f 0. Therefore, V is positive definite. From (481, using (21, (461, 

( 1614 18) , 

where Gpi and %pi 

Since u 'cpi = GpiC;:Ucri* we have from (49): 

Since $1 Is strictly monotonic increasing, 

*To i -2q Dq (51) 

0 

V-0 only when GO and ucri = 0, which implies - 0. Considering the 

closed-loop equation, 

92 
which yields $i (ucpi) 0 and. q = 0. Since $i(J) = 0 only at v - 0, 

this implies that a - 0. Thus V 5 0 only at the origin, and the system is 

globally asymptotically stable. 

Thus the collocated controller is guaranteed to be globally asymptotically 

stable in the presence of monotonic increasing nonlinearities. This p .._ .. of the nonifnearities is also called "incremental passivity." As se' n t.1 

previous section, if the nonlinearities are replaced by dynamic operat 8, mere 

incremental passivity is not sufficient for stability. 

ROBUSTNESS OF VELO$ITY FEEDBACK CONTROLLERS 

Stability with Dynamic Operator in the Loop 

Consider the case where a nonlinear dynamic operator p(zo; v) is 

present in the loop. & is represented by the following state-space 

model : 

Suppose 

w(t) = p(z, t) (54) 

where v and w are 3m x 1 vectors which are the input and the output of . Define the operator 

We define 

finite zO. 

Theorem 4. Consider the system given by Eqs. (l), (191, (20) (211, where the 

o erator 2 has the state-space representatlon given by ;L3), (54). Suppose 

G&R is passive and ,g is uniformly observable, finite-ga! n, internally 

stable, continous operator. Then the closed-loop system is globally 

asymptotically stable. 

to be internally stable If I fl (to; 0) I is finite for any 

- Proof. Defining 

V(t) - qTAq + iTi (56) 

V(t) > 0 for all t 0. Differentiating V(t) with reapect to t and uslng Eqs. 

(19),720), (21) and (11% 

= -2q .T Dq . - 2uT G-l R(z0; UCr) 

cr r (57) 

Integrating from 0 to T , elnce V 0, 

93 
which yields (after manipulation) 

where B is a constant (see Eq. 28). 

By using 2 procedure similar to that in the proof of Theorem 1, it can be 

proved that Wql is bounded, and that the system is globally asymptotlcally 

stable. 

The following corollary is an imediate consequence of Theorem 3. 

Corollary 4.1 If & is a strictly stable, completely observable, LTI operator 

with rational, mimimum-phasr: transfer matrix H( e), the closed-loop system of 

Eqs. (l), (191, (201, (21) is asymptotically stable provided that 

* 

ti(jw)Gr + GrH (jw) - > 0 for all real w (60) 

Note that the above condition is equivalent to passivity of Gr-',# . Corollary 4.2. Under the assumptions as in Corollary l.ly if Gr and 

are diagonal, the closed-loop system of Eqs. (11, (191, (20), (21) is 

asymptotically stable if 

Re[Hi(jw)] > 0 for all real w - 

As a result of Corollary 4.2, CDVFB controllers can tolerate etable 

f irst-order dynamics in the loo^. If Ri(8) - e'JCi, we have 

He[Hi(jw)] > 0 for -90" bi - < 90'; therefore, CDVFL controllers have 90' 

phase margi;: 

Stability in the Presence of Nonlinearities 

Suppose the operator fl in (1) is replaced by an m-vector nonlinear 

function J, as in Eq. (471, except that J, Is slowed to be time-varying. The 

following theorem gives eufiicient conditione for global asymptotic stability. 

Theorsm 5. Consider the closed-loop system given by Ego. (i), (191, (201, 

(21), where Gr is diagonal and positive definite, and each $i belongs to 

the [0, -) sector. Then the closed-loop system is globally neymptotically 

stable. 

94 
- Proof. Starting with V aa in Eq. (561, 

Thus f < 0, and f f 0 only if 6 E 0, which can happen (from the equations of 

motion) only when q E 0. Therefore, the system Is globally asymptotically 

stable. 

The next theorem considers a dpecisl case when nonllnearitiea and 

first-order dynamics are simultaneously present In the loop, as sho-a in Fig. 

3. 

Theorem 6. Consider the closed-loop system iven by Eqs. (11, (191, (201, 

(21), where Gr > 0 is diagonal. Suppose $! - diag ( #I, Rz, . . . . . where 

where each $1: R1+R1 is a time-invariant, dlfferentiable function 

belonging to the (0, =) sector, and there exists a canstant K < such that lJl[l < K over the interval (--, OD). Suppose qi is an LTI operator whose 

transfer function is: Gi(s! = ai(1 + pis)-l, ai 3 0, pi > o for i = 

1, 2, . ,m. Then the system is globally asymptotically stable. 

- Proof. Starting with V 88 in Eq. (56) and proceeding a8 In the proof of 

Theorem 4. we have 

when 

Using sean value theorel, Eq. (64) can be written a8: 

is the unforced response of 91 due to nonzero initial state. 

vhere u^ lies in the interval. bounded by ( 0 ; Ucr i 1 

+ g,i. Noting that the operator $1 

simplifying, we have 

9iiO; ucri33- 16 paroive IO], and 

95 
where 

3a 

1- 1 

'go' = 1 'ecil < - 

The remainder of the proof is similar to that of rheorem 4. 

CONCLUDIH; REMARKS 

Robustness properties were investigated for tsv types of controllers for 

large space structures, whlch use collocated sensors and actuators. The first 

type is the collocated attitude (CAI controller, whlch controls the rigid-body 

attitude and the elastic motion wiqg negative. definite feedback of measuredattitude and rate. The second type of controller is tkz collocated direct 

velocity feedback (CWW) controller for damping enhancement. Such controllers 

are known to provide closed-loop asymptotic stability regardless of the wrrber 

of modes and parameter values, provided that the actuators and renoore are 

perfect. This robust stability property was extended further in this paper by 

proving that the global asymptotic stability le prcaerved eveo vhcn sensor#/ 

actuators are not perfect. Tbe CA controller preserves global asymptotic 

stabljty when the sensorejactuatore are represented by (1) linear, tiwinvariant dynamics which satisfy attain sirple phase conditions, or (1%) 

time-invariant, monotonic increasirg nonlinearities belonging to the (0, a) 

Rector. The QWPB controller preserves global asymptotic stability mder much 

weaker conditions. In particiiar, CWW controllers have 90" phaw margin and 

are tolerant to time-varying nonlinearities in the [O, 0) sector. There global 

asymptotic stability results are valid regardless of the number of modes in t;ot 

model and regardless of parater valuer. Therefore. it can be concluded that 

these controllers offer viable methods for robust attitude control or damping 

enhancement, eopecially when the parameters are not accurately known. An 

important application of the collocated attitude controller would be during 

deployeent or assembly of a large apace structure, then the dpnaric characteristics are changing, and during initial operating phase, when tb dynamic 

characteristfcs are not known accurately. A robust collocated contr9ller can 

provide stable interim control which caa perhapr be replaced later by 'a 

high-periormance controller designed wing parameterr estimated OIL orbit. 

REFERENCES 

1. Joehi, S. M.: A Controller Design Approach for Large Flexible Space 

Structures. NASA CR-165717, May 1981. 

2. Elliott, L. E., Wngori, D. L., and Iwens, R. P.: Performance of Robust 

Output Feedback Controller for Flexible spacecraft. Proc. 2nd Symposiwn on 

Dynamics and Control of Large Flexible Spacecraft, Blacksburg, Va., June 

1579, ppo 409-420. 

96 
3. Balas, U. J.: Direct Velocity Feedback Control of Large Space Structures. 

Journal of Guidance ad Control, Vol. 2, Uay-June 1979, pp. 252-253. 

4. Desoer, C. A., and Vidyasagar, M.: Feedback Systems: Input-Output 

Propert le8 . Academic Press, N.Y . , 1975. 

5. Joshi, S. U.: Design of Stable Peedback Cantrollers for Large Space 

Structure#. Proc. Third VPI&SU/AIlu Syrpoeiu on Dynadcs and Control of 

Large Flexible Spacecraft , Blacksburg, Va., June 15-17, 1981 . 6. Alzerrsn, n. A., and Gantmacher, P. Be: Absolute Stability of Regulator 

Systems. BoldeD-Day, Inc., Sau Francisco, 1964. 

Figure 1. - Collocated Contoller 

97 
Figure 2.- Nonlinearity belonging to the IO,-) sector 

Figure 3.- Linear dynemi-a and nonlinearltiee slmltaneoualy In the loops 

98 
ADAPIWE CONTROL-ACTUAL 

STATUS AND TRENDS 

I. D. I.--ub-- 

Reservch Dhctar at C.N.R.S. 

GRECO "Systcms AdapmW' 

pnd 

ENSlEG 

B. P. 46,38402 - S;rintMtrtirrd')iaes, Errnce 

Leboratoire d'Automraquc * deGnnoMe 

Important progress in research and application of Adaptive Control 

Systems has been achieved in the last ten.years. 

are currently used in applications will be reviewed. Theoretical aspects 

currently under iavestigation and which are related to the application of 

adaptive control techniques in various fields will be briefly discussed. 

Applications in various areas will be briefly reviewed. 

techniques for vibrations monitoring and active vibration control will 

be em:hasized. 

The techniques which 

The use of adaptive 

I. ZNTBODUCTION 

The use of adaptive control techniques is motivated by the need of 

automatically adjusting the parameters of the controller when plant parameters and disturbances are unlrnavn or change with the, in order to achieve 

(or to maintain) a certain index of performance for the controlled system. 

While cdis problem can be reformulated as a nonlinear stochastic control 

problem (the unknown parameters are considered as auxiliary states) the 

resulting solutions are extremely coqlicated. Therefore, in order to obtain 

something useful, it .La necessary to Pake'approxiratioae. Adaptive control 

techniques can be viewed as approxlmatlons for nonlinear Stochastic .control 

problems. 

Regulators (STR) can be considered as two approximations among other possible 

approxhtlans. 

been fctr~nsively studied and they are well understood. 

have bt.n proven to be usable in practice and aa Important number of bucce8sful Lpplications have been reported. Hovcver, some important theoretical 

r tblems still need further investigation and wre experience utiliting 

these techniques in practice should be gained. 

Model Reference Adaptive Controllers (MMC) and Self-Tuning 

These two approaches to adaptive cotrroi probleur have 

Theee approaches 

As mentioned earlier the naAC and STR approachee can be considered 

as possible approximations for the solutions of some nonlinear stochastic 

control problems. However, when maklng approximations, some hypothesi8 

should ',e coaeidered which can justify these approxlmations. The basic 

hypc%hesis for MRAC and S?R Is of an algebraic nature: for any possible 

vcr'kues of the plant (and disturbance) parameters, there exists a linear 

has the pre-specified characteristics. 

.ontroller with a fired complexity such that the plant plus the Controller 

The adaptive control loop 

99 
will only search for the values of the tuned parameters of a controller 

whose structure has been fixed using a standard control design technique. 

The HRAC and S'J'R techniques have been initially developed independently. 

Subsequently, connections between these two techniques have been invesgigated 

and emphasized. 

(1983). For certain classes of probleme these two approaches are equivalent. 

It is Important to note that the development of these two adaptive control 

tecaiques is largely based on the deep understanding of certain types of 

linear algebraic control design techniques and of an appropriate interpretation of the controller design strategy. 

See Egardt (19801, Landau (1981), Landau (1982), Astrom 

A brief review of the underlying concepts and configurations used for 

HRAC and STR is given in Section 11. The linear tracking and regulation 

problem is reviewed la Section XI1 and this allows the definition of the 

structure of the controller. The structures of various adaptive control 

schemes are presented in Section IV. 

are discussed in Section V. 

research trends Rre indicated in Section VII. 

The parameter adaptation algorithms 

Applications..are listed in Section VI. Current 

11. MIDEL REFERENCE ADAPTIVE CONTBOUERS AND SELF-TUNING REGULATORS 

- BASIC PRINCIPLES. 

Figure 2.1 Illustrates the basic philosophy for designing a linear 

controller. 

istics of a dynamic system which is a "realization" of the desired inputoutput behavior of the closed loop control system. 

designed such that the closed loop control system is characterized by the 

same parameters as those of the "desired" dynamle system. 

The desired performance is specified in terms of the characterThe controller is 

Since desired performance correspon6s in fact to the output of the 

"desired" dynamic system which is pre-specif ied , the design problem can 

be recast as in Fig. 2.2. 

such that the error between the output of the plant and the output of the 

reference model (the dynamic system which has the desired characteristics) 

ie identically null for identical initial conditione and such that an 

eventual initial error will vanish with a certain dynamics. 

The objective is now to design a controller 

These two Interpretatloas of the linear control design in the case 

Figure 2.3 is aa extension of the 

The difference 

of a plant with unknown or varying parameters lead to two adaptive control 

schemes, shown in Pigs. 2.3 and 2.4. 

scheme given in Pig. 2.2 and is culled (explicit) WC. 

between the output of the plant and the output of the reference model is 

a measure of the difference between the real performance and the desired 

one. 

adaptation algorithm) to directly adjust the parameters of the controller. 

This is a "direct" adaptive control scheme. 

This information is used through an "adaptation:rnechaniem" (parameter 

Figure 2.3 is an extension of the scheme considered in Fig. 2.1 in 

the sense that a suitable controller can be designed if a plant model 

is estimated on-line based on the current input-output data available. 

This scheme is called STR and it is inspired by the separation theorem 

100 
€3 Plant. 

Figure 2.3 

I' 0 c 

' 

I 

I 

I 

L-: 

Imp1 ici t Refercence Model 

I 

----d Erpl ici t Rrference 

tloda 1 

Figure 2.4 , 

PRECEDING PAGE BLANK NOT FILMED 

103 161- 102- 
A. Minimum Phase Plants 

Consider the S.I.S.O. discrete linear time invariant plant described 

by : 

a) deterministic environment: 

b) stochastic environment: 

A(q-l)y(k+d) = B(q-l)u(k) + C(q")u(k+d) 

where: 

-n A(s') = 1 + a 1 q-l + ... + anq .. 

B(q-l) = bo+b1q-' + . . . + bmq-m bo # 0 

-1 -1 -n C(q 1- = 1 + clq + ... + c*q 

CR(q-') y(k+l) = 0 

where : 

(3.3) 

(3.4) 

is an asymptotically stable polynomial. 

In order to design the controllerr we will consider two strategies, 

one using an explicit reference model as part of the control system and 

the other using a 1-seep ahead predictor of the plant output which together 

with the controller will form an implicit reference model. 

Strategy 1: Explicit Reference Model 

One considers an explicit reference model given by: 

M where y (k) is the output of the explicit reference model. 

objective is: 

The design 

104 
is the plant model error. 

regulation objective specified by Eq. (3.4) (for uM(k) f 0, E(k) - y(k)). 

Equation (3.1) with d = 1 can be rewritten as: 

It is obvious that Eq. (3.7) Includes the 

where 

* -1 -1 

B (9 1 B(q 1 - bo 

and Eq. (3.7) becomes: 

(3.11) 

* '-1 

CR(q-')e(k+l) = R(q")y(k) + bou(k) + B (q )u(k) - %(q-')y,(k+l) = 0 (3.12) 

which yields the desired control 

Introducing the notation: 

T O,(k) = [u(k-1) . . . u(k-m), y(k) . . . y(k-n+l)l 

eo = [bl ... bm , rl ... T 

Equation (3.15) can be written: 

or in an equivalent form: 

-1 M 

CR(q )y (k+U = eT4W 

where : 

eT - [b eT] 00 

(3.13) 

(3.14) 

(3.15) 

(3.16) 

(3.17) 

(3.18) 

(3.19) 

The resulting control scheme is given in Fig. 3.1. 

105 
I- I 

M.R. 

I 0 regglation 

Figure 3.1 

106 
Strategy 2: Implicit Reference Model. 

This strategy is directly inspired by the separation theorem: one 

first designs an appropriate predictor for the plant output, and then a 

control will be computed such that the output of the predictor behaves as 

the desired output in tracking. 

First step: 

the 1-step ahead prediction error E(k+l) is defined by: 

(predictor design). Fe predictor will be designed such that 

;(k+l) y(k+l) - $(k+l) (3.20) 

where $(k+l) is the predictor output and will vanish according to: 

C,(q-') E(kt1) - 0 ; k - > 0 (3.21) 

Using Eq. (3.91, one obtains from Eq. (3.21) that the 1-step ahead predictor 

is characterized by: 

(3.22) -1 A * -1 T cR(q )y(k+l) - bou(k) + R(q-l)y(k) + b (q )u(k) - 6 OW 

* -1 where R(q"). B (q 

and (3.19) respectively. 

), 8, O(k) are given by Eqs. (3.101, (3.11), (3.181, 

Seconi step: 

that y(k+l) - p(k+l); where F(k+l) is the desired output given by Ey. (3.6). 

One f inally obtains: 

(computation of the control). The control is computed such 

and the control is given by Eq. (3.17) as expected. 

Because of the output of the predictor is equal to the output of the 

explicit reference model, the predictor plus the control will form an 

"implicit reference model." 

B. Tracking and Regulation in Stochastic Environment 

We will examine first the behavior of the controller designed in the 

previous section when the plant is subject to a stochastic disturbance of 

the type contddered in Eq. (3.2). For d - 1 Eq. (3,2).becomes: 

A(q") y(k+l) - B(q-l) u(k) + C(q'l) w(k+l) (3.24) 

Using the control given in Eq. (3.13) one obtains: 

CR(q-l) y(k+l) - CR(q-') yM(k+l) + C(q-') w(k+l) (3.25) 

107 
Neglecting the effect of the deterministic disturbance (which vanisher 

with the dynamics defined by CR(q-1)) one can re-write Eq. (3,251 as: 

y(k+l) I yM(k+l) + cO w(k+l) 

c,(q-l) 

(3.26) 

Equation (3.26) shows the presence of two reference models: a 

deterministic one for tracking by D(q-l) whose input is the reference 

signal u (k) and a stochastic one for regulation defined by 

input is the white noise sequence o(k+l). 

CT(q-l) 

M 

In general the objective of the design in a stochastic environment is 

to obtain a minimrim variance tracking and regulation, 1.e.: 

From Eq. (3.26) it results straightforwardly that the Objective of 

Eq. (3.27) le achieved If one chooses: 

which leads to: 

For the case d > 1, the control can no longer be computed directly 

using the strategies given above since thio will lead to a non-caural 

controller (future values of the output and input are involved for the 

computation of the control at the instant k). 

by using a polynomial identify which allows us always to express the output 

y(k+d) in terms only of y(k); y(k-1) ... and u(k), u(k-1) ... This problem can be avoided 

.. 

Consider the following polynomial identity: 

C,(q-') - A(q-') S(q-') + q'dR(q'l) (3.30) 

-1 which has a unique solu' 'on for the polynomials S(q-') and R(q 

deg S(q'l) - d-1 where 

) for 

108 
-1 -d+l S(q-3 = 1 + S1q ... sd,lq 

R(q-') = r1+r2q -1 ... rnq -u+l 

Using the identity of Eq. (3.30) in Eq. (3.9) for d > 1 one obtains: 

Equation (3.7) for d > 1 becomes: 

which yields the desired control 

(3.31) 

(3.32) 

(3.33) 

(3.34) 

(3.35) 

(3.36; 

The control has he same str cture as for the case d = 1 except that the 

polynomials R(q- E ) and B (q -Y ) are different, as well as 8, and Oo(k): 

Note that the strategy presented above achieves a poleszeros placement. 

C. Non-minimum Phase Plants 

-1 In this case one can no longer aa8uriG that B(t ) is asymptotically 

The basic control strategy (algebraic approach) is 

stable and therefore the zero8 of the plant transfer function can no 

longer be cancelled. 

the poles placement technique without zeros cancelllag. 

for the design of the controller is the Bezout identity: 

The balric relation 

109 
and the controller has the structure: 

1 if B(1) = 0 

B(1) elsewhere 

For a survey of the control strateg,?s for non-minimum phase plants, see 

Landau, M'Saad, Ortega (lrq3). 

IV. STRUCTURES OF ADAPTIVE CONTROL CYSTEMS 

In adaptive control scheues the fixed controller designed €or the case 

of knobrr parameters is replaced by an adjustable controller having the same 

parameter vector which for the case of thedesign considered for minimm 

phPse plants is given by: 

-ructure, i.e., the fixed parameter vector will be replaced by an adjustabl-e 

and the correspondiag control law will be given (either in deteiministic or 

stochastic environment) by: 

or : 

'4.3) 

See Fig. 4.la. 

Note that in the case of schemes using an im?licit (prediction) 

reference model (STR) the plant predictor will be replaced by an adaptive 

predictor gcverned by: 

and the control will be computed according to the etrategy in the linear 

case with known parametere which will lead to Eq. (4.3). See Fig. 4.lb. 

V. PARAIGTER ADAPTATION ALGORITHMS 

Various approaches have been considered for the development of parameter 

adaptation algorithms (PAA). 

given by : 

A fairly general structure for the PA4 is 
Figure 4.la 

Y (k) 

Figure 4. I ’ 

111 
(5.2) 

Using the matrix inversion le-: 

A 

where eik) is the adjustable parameter vector, F(k) is the matrix adaptation 

gain, +(k) is the measurement or the observation vector and uo(ksl) ard 

v(k+l) are the "a priori" and the "a posteriori" adaptation errors respscttively. The_"a priori" adaptation error is a measurable quantity which 

depends on e(i) up to the instant k, and the "a posteriori" adaptation error 

which enter: in the adaptation algorithm is not directly measurable (it 

depends on e(k+l)) but can be expressed in term of the "a priori" adaptation 

error as indicated in eq. (5.2). 

Different choices for Al(k) and Xz(k) are possible leading to different 

types of variations of the adaptation gains. 

adaptive control systems in various situations depend upon the choices of 

these two parameters. 

The performances of the 

For details see Landau, Lotano (1981) and Landau (1983). 

VI. APPLICA'i' iONS 

There are already a significant number of applications of adaptive 

control systems as well as a few mrcial products. 

Astrb (1983). Landau (1981), Landau, Tomhuka, Auelander (1983), Narendra, 

Mcnopoly (1980). Unbehauen (1980). 

For refereaces, see 

The adaptive control schemes can be used in three modes of operation: 

1) Auto-tuning of a linear controller in the case of plants with unknown 

but constant parameters. 

2) Building a gain schedule for unknown plants with dynamics depending 

on operating points. 

3) Adapting in real-time the controller for plants with unknown and timevarying parame t ers . 

112 
An important remark to be made is that adaptive control algorithms 

cannot be used in p actice without a priori analysis of the control problem 

corresponding to each tentative application. This analysis should give 

answers to two categories of questions regarding (a) the need of adaptive 

control and (b) specific design requirements. 

The main areas of applications are: 

- Grinding 

- Drying furnaces 

- Cement mills 

- Chemical reactors 

- Distillatim columns 

- Diesel and explosion engines 

- Keating and ventilation 

- Paper machines 

-. Power systems 

- Electrical drives 

- Autopilots for ships 

- Robotics 

- Heat exchangers 

- pH-control 

- Actlve vibration control 

An adaptive active vibrations control is described in Mote, Rahimi 

(1983). It uses tirst a recursive parameter estlmation technique for 

estimating in real-time the parametric model of the composite vibration 

signal for circular plates (the vibrations frequencies). Then the parameters 

of the transfer from control heat to vibration frequency are estimated online and used for computing in real time the controller parameters. 

VII. THEORY 

The most complete theory is available! today for the adaptive control 

of minimum phase plants achieving a poles-zeros placerer\t. For this type 

of plant, tracking and regulation vith independent objectives can be 

achieved both in deterministic and stochastic environments. Both MRAC 

and STR approaches lead in this case to "direct" adaptive control scheaes. 

The besic assumptions for the design of adaptive control systems far 

minimum-phase plants in deterministic and stochastic environments are 

mrized next. 

- Exact knowledge oi the plant delay (d). 

- Knowledge of an upper bound for the degree of A(q-') which is the 

denominator of the plant transfer function. 

- The zeros of the plant transfer function must lie within the unit circle. 

- A lower botnd of the magnitude of the leading coefficient of the plant 

transfer function should be known. 

- The sign of the leading coefficient of the numerator plant transfer 

function is useful to be known (in order to avoid large adaptation 

transients). 

- The stochastic disturbances are modeled by ARMA processes. 

- Asymptotic type convergence is considered. 

113 
However, ir practice some of these assumptions caanot be reasonably 

satisfied, in particular, the need for knowing an upp:r bound for the 

denoainator degree (which in many cases simply does nc.t exist) and the 

requirement that the disturbance is of AWA type. 

The use of reduced order models in adaptive control design is one of 

the main research topics today, and interesting results have been obrainec 

leading to improved design techniques. See Ioannou (1983), Ortega, Idnusu 

(1983), Kosut (1983). 

The case of disturbances which cannot be modeled by ARl4A ;--ocesses has 

also been considered. 

(1982). 

See, for example, Samson (1983). Peterson. Narendra 

Another aspect is the extension of the adaptive control design for 

the multi inputs - aulti outputs systems. 

extension raises important parameterization problems for liIwD plants. 

survey of the various designs available can be found in Dion, Dugard (1983). 

More a priori knowledge on the plant structure than ia the SLSO case is 

required, and the research is directed towards the development of adaptive 

control schemes requiring less a priori structural information. 

form of HIM0 transfer matrix plays a key role in understanding the multivariable case. 

Except for trivial cases, the 

A 

The Eerrite 

The case of adaptive control of non-minlmum phase plants Is more 

complicated both from the point of view of the caplexfty of the adaptive 

control schemes and of the analysis. 

techniques for this type of plant is given In Landau, M'Saad. Ortega (1983). 

Xost of the schemes are of *'indirect" type, and the major question to be 

answered in order to show the convergence of the system is whether the 

estinaeed plant model converges twards the model with satisfactory properties (stabilizable). Global convergence results have been obtained, but 

vith the requirement of using an additional persistent excitation signal, 

see Goodwin. Teoh, Innis (1982). The robustness of the adoptive control 

designs for non-minimm phase plants with respect to model reduction and 

ill-modeled disturbances has also been studied, see, for example, 

Praly (1983). 

A survey of the adaptive control 

REFERENCES 

Astrlf91, K.J. (1983; neory and applications of adaptive control, Auttmatica 

Vol. 19, No. 5. 

Dion, J.M., Dugard, L., Pameterization far multivariable acaptive systems, 

Proc. IFAC Workshop "Adaptive Systems in Control \and Signal Processing," 

Pergarnon, Oxford. 

Egardt, B. (1980) Unification of some discrete timu, adaptive control schemes, 

IEEE Trans. A.C., AC-PS, 710. 

Godwin, C.C., Teoch, E.K., McInnis, B.C. (1982) Globally convergent 

adaptive controllers for linear systems having arbitrary zeros, Rep. 

University of Newcastle. 

114 
Ioannou, P. (1983) Robust redesign of adaptive control in the presence of 

disturbances in udeled dynamics, Proc. IFAC Workshop "Adaptive 

Systems in Control and Signal Prc)cessing," Pergamon, Oxford. 

Kosut, R.L. (1983) Robustness issues IE adaptive control. Ibid. 

Landau, I.D. (1981) Model reference adaptive controllers and stochastic 

self-tuning regulators - A unified approach, Trans. ASUE, Journ. 

D.S.M.C., VO~. 103, pp. 404-416. 

Landau, I.D. (1982) Combining model reference abaptive controllere id 

stochastic sd€-tuning regulators, Automatica, Vol. 18, No. 1. 

Landau, I.D., H'Saad, H., Ortega. B. (1983) A6am.i.va controllc- - 5r discrete 

tlme systems with arbitrary zeros, A survey. ?roc. IFAC Workshop 

"Adaptive Systeas in Control and Signal YKG c-3singr" Pergamon, Oxford. 

Landau, I.D., Lozano, E. (1981) Unification of discrete tire explicit model 

reference adaptive control designs, Aut-tica, Voi. 17, No. 4, 

pp. 593-611. 

Landau, I.D. (1983) Lecture notes on adaptive control. 

Landau, I.D., Tomizuka, H., Auslander, D. (1983) Adaptive Systems in Control 

and Signal Processing, Pergamn, Oxford. 

Mote, C.D., Rahimi, A. (1983) Real time vibration control of rotating 

circular blades by tqerature control and system Identification. 

Proc. IFAC Workshop "Adaptive Systens in Control and Signal Processing," 

Pergamon, Oxford. 

Narendra, K.S., Monopoly, R. (1980) Applications of adaptive control, 

Academic Press, N.Y. 

Ortega, R., Landau, I.D. (1983) Robustness of discrete adaptive controllers: 

Quantification and inprwement, Proc. I=-CDC, San Antonio. 

Peterson, B., Narendra, K.S. (1982) Bounded error adaptive control, IEEE 

Trans. A. C. , AC-27. 

Praly, L. (1983) Robustness of indirect adaptive control based on pole 

placement design, Proc. IFAC Workshop "Adaptive System in Control 

and Signal Processing," Pergamon, Oxford. 

Samson, C. (1983) Stability analysis of adaptively controlled systems 

subject to bounded disturbances. Automatics, Vol. 19, No. 1, 

pp. 81-86. 

Unbebauen, 8. (1980) Methods and applications of adaptive control, SpringerVerlag, Berlin. 

115 
i 885-31203 

A NONLINEAR DUAL-ADmiVE 

CONTROL STRATEGY FOR 

IDENTIFICATION AND CONTROL 

OF FLEXIBLE STRUCTURES" 

F. E. Thmo 

City Cokge of The City Univenrity of New Yo& 

New York, NY 10031 

ABSTRACT 

A technique is presented for obtaining a control law to 

regulate the modal dynamics and identify the modal parameters 

of a flexible structure. The method is based on using a 

min-max performance index to derive a control law which may 

be considered to be a best compromise between optimum 

one-step control and identification inputs. Features of the 

approach are demonstrated by a computer simulation of the 

controlled modal response of a flexible beam'. 

I. INTRODUCTION 

A class of indirect adaptive control systems proposed for the 

control of large space structures [l] is based on a modal 

decomposition of the system dynamics and may incorporate one 

or more on-line testing schemes [2] to determine when 

successful parameter identification has been achieve. The 

control strategy used in calculating the actuator inputs must 

achieve adequate regulation or tracking performance and, at 

the same time, prov;de inputs to allow adequate parameter 

identification. A on:ro1 system designer is thus faced with 

the problem of 'Pvising a control strategy to ensure 

acceptable system perfbrmance even when on-line parameter 

identifiability tests hasre failed because the system 

* This work was supported by NASA under Grant NAG-1-6. 

117 
configuration has changed or the environment in which the 

system operates has changed. 

In this paper we formulate and examine the performance of a 

nonlinear dual-adaptive control scheme in which a 

sampled-data controller is designed to select a best 

compromise between an input signal that is optimum for 

mean-square system regulation and an input signal that is 

optimum for parameter identification. Dual control theory, 

originally formulated by Feldbaum C3,41, has been studied in 

[5-71 and in the references cited therein. A key concept 

introduced by Feldbaum is the dual control strategy based on 

a performance index that takes into account the fact that 

future observations on the process will be made. A 

controller mav be able to "probe" the system for state and 

parameter estimation improvement, which then may improve 

future regulation and tracking performance. In many 

si'tuations where the dual nature of stochastic control is not 

taken into account the controller becomes "cautious" C5,61 

and tends to "turn-off". This undesirable phenomenon is 

avoided by the approach described below. 

11. FORMULATION OF AN ADAPTIVE PERFORMANCE INDEX 

The discrete-time dynamics for each mGde is assumed to be 

described by the ARMA model 

y( t)+aly( t - l ) + a e y ( t - 2 ) = blu( t-l)+b~u( t-2)+e( t) 

where y( t) denotes modal displacement, u( t) denotes modal 

force, and e(t) is a sequence of independent, 

equally-distributed, normal (0,d) random variables. It is 

assumed that e(t) is independent of y(t-l),y(:-2),..., 

u(t-l),u(t-2),... and that the parameters a~,a2,b~,b2 

are unknown constants. If we let Yt denote the information 

available to the controller at time t, 

I Yt = y(t) ,y( t-1) ,. . ., u(t-1) ,u( t-2),. .. { 

x(t) denote the modal parameter vector and e(t) denote,a 

modal measurement vector, 

xT(t) = (a1 sa2 sblsb2); 

eT(t) f (-y(t-l),-y(t-2) ,u(t-l),u(t-e) 

where ( . )f denotes vector or matrix transpose, then (1) 

may be rewritten as 

y( t) = el( t)x( t)*e( t) 

(3) 

(4) 

118 
where the constant parameter "dynamics' satisfies 

x(t+l) = x(t) 

It can than be shown, following the analysis of L81, that the 

conditional distribution of x(t+2) given Yt+ 

with mean x(t+2) and covariance matrix P(t+2 t where x(t) and 

P(t) satisfies the difference equations 

is normal 

(5) 

Furthermore, the control law that minimizes the regulation 

c ri teri on 

V,(u(t)) = E{y*(t+l)IYt) (9) 

is given by 

(10) xi ( t+ 1) x3 ( t+ 1 1 +P3i ( t+l 1 ei ( t+l 1 

xi( t+ 1 +P 33 ( t+ 1 1 

u(t) = - 

T where r) denotes the sum over i I 1 to 4 with the value 3 

excluded. 

To provide bounded modal in uts that improve parameter 

amplitude will not become excessively large, the controller is 

designed to optimize, at each sampling instant t, the following 

performance criterion: 

identification accuracy whi ! eguaranteeing that the modal 

subject to the constraints 

where 

119 
0 

Vc denotes an acceptable or desired level of regulation cost. 

VI(u(t)) denotes and identification cost function of u(t), 

VI(u(t)) = trace [~(t+2;j (14) 

denotes ar,d acceptable or desired level of identification 

0 

VI 

cost. The maximization indicated in (11) yields a function 

V(u(t)) which, although not convex, is interpreted as 

specifying, for each admissible u(t), the most costly linear 

combination of relative re ulation and relative identification 

cost. blinimization of V(u3 thus yields the modal input that 

minimizes this most costly combination of relative 

identification and regulation performance. 

111. SIMULATION RESULTS 

Since Vc(u(t)) and trace P(t+2) are relatively simple 

functions of u(t) the numerical solution of the one-step optimization problem (11)-(13) at each sampling time is quite 

feasible. Results of simulation studies described below 

illustrate an interesting feature of this a proach: since the 

parameters involved in the evaluation of Vc P u(t)) and 

VI(u(t)) depend on system measurements, the optimum 

distribution of relative cost, A (u) depends on on-line 

measurement data and hence, at each sampling instant, the 

weighting between identification and regulation will change depending on the on-line system performance. This is in 

contrast toC91 in which a fixed weighting between absolute 

control and identification cost Is used at each sample time. 

In the simulation study we compare the performance of three 

c on t rol sy stems : 

a) A constrained adaptive controller that minimizes (9) 

subject to the control magnitude constraint. 

b) An optimum identification controller that minirbiires 

(14) subject to the control magnitude constraint. 

c) The one-step dual-adaptive controller based on 

(11)-(13). 

In Figures 1-3 we present simulated modal response data for the 

first flexible mode of the Langley beam experiment described in 

[lo] where we assume here that a single actuator is used. The 

accumulated on-line regulation cost, VT, shown In Figure 1 is 

defined as 

120 
and the on-line identification cost, PT, is defined as 

PT(N) = trace [P(N)l 

where P(N) is calculated on-line using (8). Note that for the 

first 10 to 15 sampling times the regulation cost of the 

dual-adaptive controller is close to that of the constrained 

minimum-variance controller and the fdentiffcatfon cost of the 

dual-adaptive control system is close to that of the 

constrained one-step optimum identificatf on controller. 

Figure 2 indicates that the dual-adaptive controller's actuator 

signals switch between its linfts, t0.5, more frequently than 

do the actuator signals of the otheT controllers. 

due to the lack of any energy constrai.nt in the above problem 

f ormul ati on. 

This may be 

A future study will examine the performance of the 

energy-constrained dual-adaptive controller in comparison with 

energy- c ons t ra i ned mi ni mum- va ri ance and one- step optimum identification controllers. The relative regulation cost and 

relative identification cost defined in (13) are plotted in 

Figure 3 where 

0 

Vc (N) = Q 2 N 

is the accumulated control cost that would be achieved if the 

parameters of the system where known precise4 and if an 

unconstrained control law were used; ~2 I: 10-i was used in 

the simulation runs. A constant value Vf 10-4 was chosen 

as indicating the acceptable level of parameter 

identification. Figure 3 indicates that, dependin on on-line 

at one sawpling instant can have widely differing shapes from 

their respective distributions at other sampling times. This 

leads to the on-line variations in the dual-adaptive control 

strategy mentioned earlier. 

measurements, the one-step identification and regu ! ation cost 

The simulation results indication that the one-step, constrained dual - adapti ve control 1 er has the feature of 

providing, based on measured data, sy-ctem inputs .that result in 

parameter identffication whfle maintaining bounded modal 

amp1 i tude response. 

121 
REFERENCES -

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

Montgomery, R. C. and Thau, F.E., "Adaptive and Learning Control of Large Space Structures," AIAA Guidance and 

Control Conference, June, 1980. 

Sundararajan, N. and Montgomery, R. C. , "Adaptive Control 

of a Flexible Beam Using Least Square Lattice Filters," 

1983 Large Space Structures Workshop, Blacksburg, Va. 

Feldbaum, A. A., "Dual Control Theory I - IV, Automation 

and Remote Control 1960, vol. 21, pp. 874, pp. 10~3; 1~b2, 

vol. 22 , PP. 1, PP. 103. 

Feldbaum, A. A. , Optimal Control Systems, Academic Press, 

New York, 1965. 

Bar-Shalom, Y;, "Stochastic Dynamic Pro ramin 

and Probing, 

Caution 

IEEE Trans., Vol. AC - 28, No. 3; Oct. 

1981, pp, 1184 - 1195. 

Uittenmark, B., "Stochastic Ada tive Control Methods: A 

Survey,"Int. J. Control, Vol. 2!, No. 5, pp. 705 - 730, 

1975. 

Bar-Shalom, Y. and Tse, E., "Dual Effect, Certainty Equivalence, and Separation in Stochastic Control , "IEEE Trans., Vol. AC - 19, pp. 494 - 500, Oct. 1974. 

Astrom, K. 3. and Wittenmar!, Bo, "Pi-obleas of 

Identification and Control, 3. Hath. Anal and Appl, 

Wittenmark, e., "An Active Suboptinfl Dual Cotttroller for 

Systems with Stochastic Parameters, Auto. Contr. Theory 

Appl., Canada, Vol. e, pp. 13-19, 1975. 

VOI. 34, pp. 90 - 113, 19-11. 

10. Thau, F. E., Monto ornery, R. C., and Horner, G.C., 

"On-L i ne Struc tura! Parameter I: denti f f cati qn, 

Guidance and Control Conference, Au ust, 1981, 

P rac . AIAA 

Albuquerque, New Mexico, pp. 5aO-53 !I . 

122 
500 

450 

400 

350 

300 

250 

200 

150 

100 

VT 

50 

0 

3 -x to 

40 

PT 

1 I I I I I 

0 10 20 30 40p 

SAMPLE nM 

(a) ONE-STEP OPTIMUM IDENTIFICATION 

(b) MINIMUM-VARIANCE ADAPTIVE 

(e) DUALADAPTIVE 

I 

0 

1_11111 

10 20 30 40 50 

SAMPLE TIME 

Fig. 1. On-Line Regulation and Identification Cost 

for Three Feedback Control 1 ers 

123 
0.1 

YO 

-0.1 

0.2r 

U 

-0.2 L 

I 1 I I 1 I 

0 IO 20 30 40 50 

SAMPLE TIME 

(a) ONE-STEP OPTIMUM IDENTIFICATION 

(b) MIN IMUM-VMI ANCE ADAPTIVE 

(c) DUAL-ADAPTIVE 

0.5 

uo 

4.5 

0.5 

uo 

-0.5 

0 IG 20 30 40 50 

SAMPLE TIME 

Fig. 2. Modal Displacement and Modal Force 

for First Flexible Mode 

124 
42 

41 

N=13 

1 ! I I I 1 1 Ill !A 

O f +03 

-0.5 

U U. 

Fig 3. Relative Control Cost RC 5- V,(u)/Vi apd Relative 

Identification Cost RI = V,(u)/Vi for 

3 Sample Times 

125 
5 R85-31204 

STABLE DIRECT A -.IsIIvE CONTROL OF 

LINEAR INFINFlrlEDDIMJ!,iJSIONAL SYSTEMS USING 

A COMMAND GENERATOR TRACKER APPROACH 

M. 1. Bmb, Ii, Kdhm, aad J. Wa 

Rcrrsstkcr pdyoechnic Institute 

Troy, NY 12180 

Ue present a c-d generator tracker approach to model following control 

of linear distributed parameter systems (EPS) whose dynanics are described on 

infinitedimensional Bilbert spaces. 

controllers capable of exponentidly stable tracking of the refereme trajectorjes when certai3 ideal trajectories xre known to exist for the open-loop DPS; 

we present ccnditions for the existence of these ideal trajectories. An adaptive 

version of this type of controller is also presented aid shown to achieve (in 

sane cases, asyaptotically) atabze finitediwnsional control of the infinitedimensional DPS. 

This rethod generates finitedimensional 

I. INTRODUCTION 

By a distributed parareter system (DPS), ve mean a system whose dynamical 

behavior with respect to external disturbances is described by paitial differential equations. 

especially if high performance is demanded, e.g., a slmple electrical circuit 

at vary high frequencies. 

equation) approximations often suffice to describe the system behavior of mmy 

engineering systm. Indeed, such gpproximations are necessery for DPS controller designs to be irplemented with on-line digital computers. Nevertheless, 

the distributed parameter nature of control prnblems should not be discarded 

prematurely; otherwise, control. approaches can be generated vhich look good on 

paper but are not sufficiently robust to operate with the xtue system. This 

has been illustrated in computer shulation and in even a few laboratory demonatrations of flexible structures, yet, it continues to be ignored in some parrs 

of the control community. 

a DPS viewpoint is essential. 

Of course, everything is a DPS if it is carefully scrutinized, 

However, lumped parameter (ordinary differatfa1 

To understand the controller-structure interaction, 

- 

The most rerious difficulty of the DPS vi2wpoint is that it requires the 

mathematical ideas of infinite-dhensional function spaces aad unbocnded operators on these spaces; €or example, see [1]-[2]. Savtral results in the past 

have been posed within this mathematical framework with the required mathematical 

rigor [3]. Yet, the necessary practichl constraints were interpreted so that 

the results would be relevant to structural dynamicists and control system 

engineers and would make the maximum use of their experience and intuition. 

With these ideas in mind, the concept of model following appears to be a 

procedure that yields a useful finite dimensional controller that might be 

designed taking Into account the distributed natcre of the system dynamics, 

whereas early model folloving control systems required the satisfaction of 

certain “Perfect Hodel Following“ conditions which necessitatzd the. we of a 
reference model having the same order as that of the process [4], the more 

recent output model following controller or CoPrand Generator Tracker (CGT) as 

developed by Broussard [SI allows the use of a model of arbitrary order, provided 

that the number of controls is equal to the number of outputs being controlled. 

This concept in fact served as the basis for a finite dimensional aflaptive 

controller that was used for controlling large structural systems [6, 71. 

Thus since the CGT algorithm makes it possible to use a finite dimensional 

reference model which subsequently gives a finite dimensional controller regardless of the pr3cess order. 

troller which produces stable closed-loop operation with the class of liiiear 

distributed parameter systems considered here. The difficulties of stable 

adaptive distributed parameter control are detailed in, e.g., [SI-191 and the 

references containcd therein. In Sixtiors 2 and 3 the nonadaptive model 

following controller is developed aid analyzed; in Section 4, the adaptive 

verslon is presented zind shown to produce a stable closed-loop. 

axid future c'irecticns are presellted in Section 5. 

This provider the basis for a direct adaptive conConclusions 

2. PROBLM FORHULATIQN 

2.1 Process Description 

lhe distributed paraaeter systems (DPS) of interest will be modeled by the 

following state space form: 

0 = Av(t) + Bf(t) ; v(o) = v i at (2. la) 

where the state v(t) is in an infinite-dimensional real Hiibert space H with 

inner product ( , 0) and corresponding norm I I 1 I . The bounded input-output 

operators B and C have the same finite rank P, and f(t), y(t) represent the 

- iquts for P line2r actuators and the outputs from P linear sensors, respectively. Thus, 

P 

i= 1 

Bf(t) = 1 bi f,(t) (2.2) 

and 

T 

y(t) = [yl(t), ..., yp(t)l 

Yj(t) = (c 

with 

v(t)) ; 1 2 j - <P (2 3) j' 

where bi and c belong to H. In infinite-dimensional theory, the operator A 

is a closed, livear, unboundad (differential) operator with domain D(A) dense in 

H. Furthermore, (2.1)-(2.3) represeuts some well-posed physical system, which 

in mathematical terms is the weak formulation of (2.1): 

j 

128 
t 

vCt> = U(t) v0 + J U(t-r) Bf(t)dt 

0 

y(t) = W(t) f t 2 0 

(2.4) 

where v 

operators generated on H by A. This latter means: 

is any initial state in H and U(t) is the C -semigroup of bounded 

0 0 

V(0) = I (2.5b) 

lim [U(t) - I] v = 0 : v in H 

t4+ 

Av = [lin - U(t)-l~v ; v iu MA) t 

t+o+ 

(2.5~) 

(2.5d) 

NGte that the semigroup U(r) evolves the initial cGnditions v forward in the. 

When v is in D(A) and f(t) has continuous first derivative, v(t) also is differentiabfe, lies in D(A) for t 2 0, and satisfies (” 1). 

and any sqaare-integrable f(t) V-11 satisfy the weak formulation (2.4) and yield 

states v(t) in H for all t 10. Consequently, (2.4) is easier to work with in 

infinite-dimensions and is more likely to represent the actual physical system 

being modeled by (2.1;. This form, (2.1) or (2.4), models most practical 

interior control problems for linear DPS where the actuator and sensor influence 

functions are given by b and c respectively. 

For example, control of the damped wave equation on a region Sls R 

0 

However, any vo and K 

i j’ 

n 

by a 

single actuator and sensor is described by (for E > 0): 

.-, L 

- Aou(x,t) = b(x) f(t) a u(x,t) + E au(x,t) 

2 at 

’; 

y(t) = I c(x) u(x,t) dx 

R 

(2.6a) 

(2.6b) 

where u(x,t) is the displacement from equilibrium of Sl and the influence functions b and c can be taken as approximations of Dirac delta functions at the 

location of the actuator and sensor. The operator A. is the Laplacian given by 

a‘u(x,t) 

2 A u(x,t) = 1 

0 e=i ax, 

on D(A ) E !u(x,t)EHoiu(x,t) is smooth and u(x,t) = 0 on the boundary of n). 

The domain D(Ao) is dense in Ho f L (n) with the usual Inner product (e,*),. 

This can be put into the form (2.1) by choosing the state v(t) = [u(x,t), 

2 0 

129 
'"1 x H with the auergy inner product: au(x,t)p in = 0 

- at 

The operator A in (2.1) becomes 

A = -3 

and the rest follows. 

Another important exaaple is the mathematical setting for large structural 

systems (LSS) which may be described as a continurn by the following system of 

partial differential equations: 

m(x;u (x,t) + Dout(x,t) + Aou(x,t) = F(x,t) (2.10) 

tt 

where u(x,t) represents a vector of instantaneous displac-nts nf the structure 

R from its equilibrium position due to transient disturbances and the applied 

force distribution F(x,t). The displacements can be translational and ratational, and the forces can be generalized to include torques, as well. The mass 

density m(x> is positive and bounded on n. 

The internal restoring force term 'Lou is generated by a tine-invariant, 

symmetric, non-negative differential operator A appropriate to the LSS. The 

dmain D(Ao! of A. contains all smooth functions satisfying the LSS bornpry 

conditions and is dense in the infinite-dimensional Hilbert space Eo = L (0) 

with the usual Inner product (*, -IO and associated norm I I I Io. 

the operator A is assumed to have discrete spectrum, i.e., isolated resonances; 

0 

In post cases, 

C 

this can be expressed by the following eigen-problem: 

2 Ao4k = "k 0, (2.11) 

where w 

vibration mode shapes. 

rarely known for an actual LSS. 

are the vibration mode frequencies and +,(x) are the corresponding k 

Of course, exact expressions for this modal data are 

The damping term Dout is camposed of a skew symetric part, which represents gyroscopic damping due to any on-board rotors or constant spin rate of 

the whole LSS, and a small symmetric part which represents the internal structural damping and is thought to provide very low mode damping. 

The applied force distribution is 

1 30 
where F 

nonlinearities) and Fc represents the contrcl forces due to P actuators: 

represents the external disturbance forces on the LSS (and possible D 

P 

F C =Bf= 0 E bi(X) fi(t) 

i=l 

(2.13; 

where the actuator amplitudes are fi(t) and the actuator influence functions 

are bi(x) in Ho. 

approximate 6(x-x 1; however, they do not have to be point devices. 

These are usually localized or point devices so that they 

i 

ObservatinE are obtained by P sensors 

y=Cu+Eu (2.14) 0 Ot 

where y,(t) = (c.,uo) + (e.,u 

position sensors and e for velacity sensors in H . Again, these are usually 

lccalized or point devices but they do not have to be. 

1 < ,i < P, with influence functions c for - J J to' - - j 

j 0 

The LSS dynamics are defixied by (2.10) and (2.14) csn be put into the 

infinite-dimensional state space form: 

bv(t1 

at (2.15a) 

I i y(t! = Cv(t) ; v(0) = v9 (2.1%) 

-A-- - Av(t; + Bfie) + I'f,(t) 

with (A,B,C! as in (2.3) and the persistent disturbance term rfD(t) obtained 

from F in (2.12). Impulsive disturbances in the structure are model.ed by the 

initial condition v . D 

0 

The Iiille-Yosida Theorem (e.g. [l], Theo. 8, 9, p. 153), provides conditiom under which an operator A generates a C -semigroup U(t) satisfying: 0 

I lU(t) I I 2 Ke-at, t 0 (2.16) 

where K 

the resolvent operator R(X,A) 

1 and u real. The necessary fnd sufficlent conditions are given for 

(XI-A)- : 

(2.17) 

for all real X 

bounded operator on HI. The spectrum of A, a(A) = p(A) is much more complicated in infinite-dimensions, but, in finite-dimensions, it consists only of 

the (finite number of) eigenvalues of A. We say that A is exponentially stable 

when (J > 0 in (2.16), i.e., the semigroup U(t) generated by A deczyb exponentially at the rate a. 

dimensions, but no others provide the safety of a stability margi3 a; therefore, 

- a in the resolvent set of A, p(A) = iX complex /R(A,A) is a 

There are many other types of stability in kfinite131 
this seems to be the kind of stability of most practical interest for cngineering applications where there is alvays same uncertainty in the model of JPS. 

2.2 Model Following Control Frob]-em Formulation 

Given the DPS as defined in (2.1), it is desired to find a finite dimensional controller so that the output y(t) ''foll~ws" a desirable output trajectory ym(t). 

(asymptotically) stable reference model : 

This output trajectory is to be generated by the finite dimensional 

q = A q + B,um m (2.18a) 

(2.180) 

where 

q is the model state vector having dimension N, 

u is a step 3r reference level conmand with dimension P, 

y m 

P 

is the output trajectory also having the dimension P, 

and Am, Bn are matrices with appropriate dimensions. 

the dimension of bcth y, and urn is the same as the dimension of the process 

input f and the process output y as defined in (2.1). 

chosen. 

It should be noted that 

Usually qo = 0 will be 

The output model following control problem to be solved is the development 

of an algorithm that defines the process input f(t) so that the following two 

model following conditions (MFC) are satisfied: 

3. DEVELOPMENT OF THE NONADAPTIVE MODEL FOLLOWING CONTF.OLLER 

3.1 Solution Definition 

In a manner similar to Broussard's developmegt of the Copand Generator 

Tracker (%TI [5], the concept of an ideal state v , control f 

trajectory y will be introduced. 

aatisfy the process dynamics (2.1) and that the ideal output y 

and output 

be identical 

It is required that these tgajectories 

132 
tc the model output y,. Thus: 

* * * 

AV (t) + Bf (t) 

* * * 

y (t) = LhJ (t) ; v io) = v 

0 

(3.la) 

(3. lb) 

* 

where the ideal state v (t) is (as with v(t)) in the infinite dinensiond 

Hilbert space H. 

Fur t he mor e 

* 

In a manner similar to that in [SI, it will be assumed that v (t) and * 

f (t) are linearly related to the model state :vector q(t) and command vector 

u (t) as follows: 

m * 

-J (t) = All q(t) + S12 um 

f (t) = s21 q(t) + s22 urn 

* 

The bounded linear operators Sll, S,,, S21, S22 will not be determined to 

satisfy MFC 1. 

(3.3) 

(3.4j 

To this effect, differentiation of (3.3) with respect to t and substitution of (3.1) and (2.18) gives: 

where 

(3. Sa) 

* * 

Replacing v and f on the right side of (3.5) by (3.3) and (3.4) gives: 

Sll Amq : B u '11 m m 

A("p + s12 IJ 1 + B(S2p + S2? Urn) = (3 6) 

Now since (3.6) must be valid for all q and urn, it is necessary that: 

11 + BS21 SI1 Am = AS 

Sll Bm = ASl2 + BS22 

(3.7) 

(3.8) 

133 
Finally the incorporation of (3.2) yields 

Thus : 

order 

order 

which 

* 

Y (t) = CSllQ + CS12 um = ym = cmq 

(3.10) 

(3.11) 

In summary then eqs. (3.71, (3.8), (3.10) and (3.11) must be*solved in 

to find S,l and S22 which in turn define the ideal control f 

Recall however, that both MFC 1 and MFC 2 must both be satisfied. In 

to satisfy MFC 2, it is useful to consider the equation for the error 

e=v -v (3.17) 

of Eq. (3.4). 

* 

* 

is in D(A) when vo and vo are both in D(A). Differentiation of (3.12) 

with respect to time gives: 

* 

ae av 

a: at -- -- 

* * = AV + Bf - (Av + Bf ) 

* = Ae +, h(f - f ) (3.13) 

This equation suggests that the actual model following control f be defined as: 

* 

f =: f + G(y - ym) 

* * 

=f +GC(V-V) 

* 

=f +GCe (3.14) 

Substituticn of (3.14) into (3.13) gives: 

l = (A + B G c)e (3.15) 

where G:RP-+RP is a bounded linear operator. 

(A + B G C) generates an exponentially stable Co-semigroup, then the control f 

as defincd by (3.14) will satisfy the conditions for model following. 

Thus if G is chosen such that ' 

It is important to note that this controller is clearly finite dimensional. 

For implementation it is only necessary to "build" a finite dimensional reference model and form the proper linear combination of its state vector and comrr.and 

vector. 

such that the decay of any transient caused by initial plant model output error 

is sufficiently fast. We summarize the above discussion as 

The gain operator G fs also finite dimensional and should be chosen 

134 
Theorem 1: If (A,B,C) is exponentially output stabilizable and there exist 

bounded linear operators S 11, S12, S21, and S22 such that (3.7) - (3.8) and 

(3.10) - (3.11) are satisfied, then the model following control (3.4) and (3.14) 

satisfies the model followir-g cmditions MFC (1) and (2) and lim [v(t)-v (t)] = 

0 when both v and v belong to D(A). t0 0 

From [lo], lie see that (A,B,C) is exponentially output stabilizable if and 

only if i$ f N(C)I and HR F N(C) form a pair of stabilizing subspaces for (A,B). 

Note that dim = ? which is the number of sensors (or actuators) used. "he 

conditions for existence of the ideal trajectories (3.1) will be developed in 

the next subsection. 

3.2 Existence of Ideal Trajectories 

* 

The existence of ideal trajectories v (t) for the DPS (2.1) is determined 

by solutions S.. to the operator equations (3.7) - (3.8) and (3.10) - (3.11). 

These can be rewritten as 

1; 

(3.16) 

wher N 

.e S :R +D(A) and S12:Rp+D(A) are bounded operators with finite-rank and 

P +R and S, :R+R are matrices of appropriate dimension. Note that 

(3.16) describes a kind of aggregation (in the sense of hki) for the infinitedimensional system (2.1) in&o a finite-dimensional system (2.17). 

of the ideal trajectories v (t) in (3.1) guarantees that such an aggregation is 

possible, i.e. the DPS (2.1) generates the ideal trajectories which correspond 

to those of the finite-dimensional model (2.18). 

N '4 PP 

&2 

The existence 

* 

In most situations, the ideal initial condition will be vo = 0; hence, 

from (3.5b) we would choose q, = 0 and SI2 = 0, which correctly corresponds to 

(3.11). This reduces the other operator equations to the following: 

A = A Sll + B S21 '11 m 

'11 Bm = s22 

'11 - 

- 

we have the following : 

(3.17a) 

(3.17b) 

(3.174 

Theorem 2: 

closed curve r containing a(Am) in its inTsrior and a(A) in itb exterior, then, 

given any linear operator S 21 

operator S :RN+D(A) given by 

If the spectra c(X) and a(A ) are separated by a Sm00th simple 

:R'+RP, there exists a uniqrle bounded linear 

11 

135 
(3.18) 

N ior any q in k . 

- PROOF: From (3.17a), it follows that for any AE.(J(A)() u(A,): 

SII R(A,Am) - R(A,A) B S21 R(A,Am) R(A,A) Sll (3.19) 

But integration of (3.19) over the curve I' produces: 

because : encloses the finite number of singularities of A 

the spectrum of A. 

D(A), and this is the desired result. d 

and excludes all of 

Clearly, since R(X,A):H+D(A), Sll mustmhave its range in 

Once, we have specified the matrix S21, the unique operator Sll is determined. Satisfaction of (3.17~) could most easily be done by defjning Cm to be 

C Sll. The determination of the matrix S for (3.17b) could be done from 22 * -1 * 

SZ2 = (B B) B Sli Bm (3.20) 

as long as B 

full ram P an2 so the inverse of B B exists. 

is c sen so that a solution exists. Note that the operator B has * m 

Although the above existence result does not really require the number of 

actuators and sensors to be equal, this will be needed in the later sections. 

Also, the following alternative existence reeult requires it: 

Theorem 3: 3 Let zezo belong to p(A) and C A-l B be nonsingnlar on R , then 

Sll satisfies: 

(3.21) sll = 51 A* + *12 

Tke proof of Theo. 3 can be obtained by straightforward computation using (3.16). 

Furthermore, tote that 

136 
which is the same as (3.17a); however, Theo. 3 gives a wider range of solutions 

thar. Theo. 2 since S need not be zero. The solution of (3.21) can be handled 

when zero belongs to 

12 

(A ) because we then have the following: m 

which has a unique solution SI1 whenever the u(Arn-') and u(n ) are separated 

by a smooth simple closed curve (see proof of Theo. 2). 

11 

4. THE ADAPTIVE MODEL FOLLOWING CONTROLLER 

4.1 Development of the Adaptive Controller 

Tlie nogadaptive control law (3.14) requires exact knowledge of the gain 

operators G, S and S These may be known to exist via mathematical structure 

of the DYS (A,6:;) in ($?i) (e.g. Theos. 1, 2, 3) but they may not be available 

in an explicit form. Consequently, we Mould need an adaptive version of (3.14): 

f(t) = S21(t) q(t) + SZ2(t) urn + G(t) ey(t) (4 1) 

where 

* 

e :y-ym=y-y Y 

We assume throughout See. 4.0 thaf the hypotheses of Theo. 1 are satisfied 

for the DPS (2.1). Take e(t)*Ev(t) - v (t) and, frm (2.1), (3.1),(3,3) and 

(4.2), obtain (for vo and vo in D(A)): 

ae(t) = A,e(tS t. BAK(t) rit) 1 at 

where 

A i A + BGC generates en exponentially stable Co-eemigroup Uc(t) and 

C 

Wt2' and AK(t) S K(t) - K where 

0 r(t) E 

137 
where E - , z belongs 'io RW2', and r 

matrices on R . Note that (since K is coilstant): 

I' are both positive definite 

dKI 

dtP P' . .I 

0 

AiI(t) = $(t) = -rI -1 ey(t) (r(t),*) 

where 

AKI(t) E I: (t) - K . I 0 

(4.5) 

The closed-loop adaptively controlled DPS io given by (4.3) and (4.5): 

a&) - - - I at= e(t) + F (t, e(c)> 

(4.6) 

- 

e(o) = e i 

i 

I- 0 

V+2P The state I(t) of (4.6) resides in a new Hilbert space E where 

R ) with B (H,,H ) represeqting the Schmidt c?.ass oE compact linear operators 

from H into H with inner product (A,B) I tr A E where "tr" denotes thc trace 1 2 

I of the operator; cap. [ll] pp 262-264 for further details. The inner product uti 

H is formed by s-mining those of H and B we shall use the same symbols for all 

E H .A B2(R' 9 

P 

* 212 

2' 

138 
inner products ( * , *) and their corrzaponding noms 1 I I I . tion F(t, *) :H+H is continuous; hence, -- The nonlinear func- 

_. 

c(c) = i(t) Go; t I > 0 (4.7) 

where G(t) is 

- 

U<t)h = 

wl:i re 

- 

u (t) = C 

the nonlinear semigroup defined on % by (for any h in E) : 

- 

il t (t)h + I Ec(t-T) F(T,c(T)h)dt (4.8) 

v (t) 07 

I 0 Li 

C 1 is the linear C -semigroup generated on by io I 0 C 

(4.6). The above follows from [12] Lema 5.2 p. 186 where further details on 

nonlinear semigroups are also available; consequently, the clcsed-loop infinitedimensional system (4.5) AS well-posed on H. .' 

4.2 Closed-Loop Stability 

The stability analysis of the nonlinear infinite-dimensional syctem (4.6) 

requires the extension of Lyapunov theory to infiniredimensional spaces. This 

has been done in [12]-[13] and we summarize the necessary elements here: 

- Def: 

there exists 

t - > 0. 

Y implies lim 

t-- 

(4.5). Usually we can take & = 0. We say an equilibrium poitt is unstable 

whenever it is not stable. 

The equilibrium point 4 is stable for the system (4.6) if for every c > 0 1 /e(t) - $1 I < E for all 

If, in addition to stability, there is a y > 0 such that I le(o) - $1 I < 

> 0 such that 1 Ie(o) - 4;: < 6 implies 

I Iz(r)-$ll = 0, then 4 is said to be asymptotically stable for 

?ef: 

and ir(e) 5 o for all e in ii wnere 

A continuous functional V:%R is a Lyapunov function for (4.6) if V(o) = 0 

v (e( t))-v(e). irCe, lim sup c (4.9) 

t*+ 

L 

where e is in % and e(t) = y(t)s as given in (4.7). 

-- J.emma 1: If V:H+R is a Lyapunov fmctron for (4.6) with the property that 

(4.10) 

for all f= such that 11.11 5 h (where 0 

f :[O,h]+R 

equilibrium point is stable for (4.6). 

h c m) and fl is or' class M, (i.e. 

+ with fl(o) = 0 and fl strictly increasing on [O,h], then the zero 1 

Le= 2: If in addition to the hypotheses of Lema 1, the Lyapunov function V(*) 

has tt. property: 

139 
(4. lla) 

(4. llb) 

where f 

stable $or (4.6). 

is also of class h$, then the zero equflibrium point is asymptotically 

The proofs of knnae 1 and 2 can be found in 1131. 

Lyapuncv' s Direct Method on infinite-dimenaional spaces. 

nese results constitute 

We now have the following stability result for our adaptively controlled 

closed-loop svsteni (4.6) : 

TheGrem 4 : Assume the foiioving : -- 

(a) In (4.3), Ac 5 A + BGC satisfies 

(Ac V, Pv) + (h, ACv) -(QV,v) (4.12) 

for all v in D(A) where P ad Q are syretric positive operators on E such tht 

(for some (I, 8 pcsitive constants): 

11.1 l2 5 (V,b) L L? I Ivl l2 (4.13a) 

(4.13b) 2 !VI I 5 (QV,~) (:.e Q is coercive) i ! 

for i?ll v in H, 

(b) B P = C, 

(c) 

(4.14) 

* 

* 

the hypotheses for Theo. 1 are satisfied, and both vo and vo belong to D(A), 

Tel - 

chen V(L) E (e,Pe) + (OKI, rI AKI)svith 5(t) I KI(t) - KO and e I 

Lyapunov function for (4.6) snd the zero equilibrium point is stable. 

- PROGF : Re ca 11 that 

AK(t) = AKI(t) + Kp(t) (4.15) 

Now, clearly V is a continuous functional from 

V(o) = 0. 

differentiable. Hence, from (4.6) md (4.121, 

into R (due to (4.13aj with 

Furthermorc, since fr is a quadratic functional, it is Frechet 

c(e) = -(Qe,e) + 2l~ 

where p 5 [(Pe, BAKr) + (AKI,FI i5)1 

From (4.161, (4.4~1, and (4.151, we have 

(4.17) 

140 
* 

p = (B Pe, OKr) - (AKI, e (r,*)) 

Y * * = (B Pe, AKr) - (r, A5 ey) 

= !B Pe, AKIr) + (B Pe, K r) - (r, AK e ) 

= (AKIr, [B Pe - e 1) + (K r, B Pe) 

* * * 

P IY * * 

(4.18) Y P * * 

where we have used (A,B) E tr A B = tr(BA ). 

(4.18), yields 

Furthermore, using (4.14) in 

tron (4.4b). Consequently, using (4.19) in (4.17), we obtain 

(4.19) 

(4.20 

where a = (r ) and we have used (4.13b). p - ‘min p 

Also, using (4.13a), we have 

2 In other words, f,(C) E [l + Xmin (r I )I5 

aboge satisfies the hypotheses of Lemma 1 and the desired result is obtained. 3 

which is of class I$. Therefore, the 

* 

Note that the use of a proportional adaptive gair (4.4b) produced the second 

term in (4.20) ; however, this term is not essential and the above argument 

could be shplified by omitting (4.4b) from the edaptive gain laws. 

The hypotheses (a) and (b) correspond to the Kalman - Yakubovich conditions 

in infinite-dimeqsional spaces. 

(Av, v) 2 wl Ivl I 

(A,B ,Chould be equivalent to satisfaction of hypotheses (2) ; hovever, there 

would be no guarantee that ? and 0 could be found in (4.12). such that (4.14) 

could be obtaineG. In finite-dimensional spaces, the Kalplan - Yakubovich 

conditions are equivalent to the strict positive realness of the transfer 

function Tc(s) = C(sI-Ac) 

115-118. 

relationship in infinite-dimensional spaces. For example, [17] asserts that 

ReTc(jW) must be coercive,which would be quice a bit stronger than what is 

required in finite-dimensions. 

gat ion. 

F’rom [13] Theo. 4.7, if for some real w, 

for all v in D(A), then exponential output stabilization of 

-1 B, i.e. Re Tc(jo) > 0 fcr all real w; see [14] pp. 

A r.uulber of papers, e.g. [lS] - [17], have been kitten on this 

This is an area rhat requires further investiAs pointed out in [9], we cannot immediately conclude asymptotic stability 

from (4.20) since it does not satisfy the hypotheses of Lemma 2. In finltedimensional space, we could apply the LaSalle Invariance Pxinciple to obtain 

141 
asymptotic stability as is done in 161; however, in infinitedimensional spaces, 

it is not the case that "bounded sets are prec-act" and this is essential for 

the LaSalle result. 

The following result ([I31 Theo. 5.4 p. 188) may be helpful: 

Lema 3: Let in (4.6) generate the linear Co-semigroup E,(t) on and F 

is any-bounded, continuous function such that (4.6) generates a nonlinear semigroup U(t) on H (as given in i4.8)), then all bounded orbits ~7f (4.6) are precompact if either 

(a) U (t) is rompact operator for all t 0 

-- C - 

- 

C 

or 

- 

(b) U (t) is exponentially stable and the function 

bounded sets into recaapact ones) 

is campact (i.e. maps C 

PA 

Due to the form of lo' ] , it is not possible to satisfy (b); however, 

(a) may be satisfied, for example by operators A which generate holomorphic 

semigroups. This latter is determined by the form of damping operator in a 

flexible structure. Again, this is a topic for further investigstion. An 

alternative adaptive gain law: 

C 

(4.21) i,(t)v = -I' -1 (e (r,v) + KI(t)v) 

Y 

yields: 

which does not quite give asymptotic stabtlity but might be modified to do so. 

5. CONCLUSIONS 

In this paper, we have presented a direct adaptive controller for linear 

distributed parameter systems (DPS) described on infinite-dimeneional Hilbert 

spaces. The controller is based on a caasand generator tracker approach used 

in finite-dimecsional spaces, e.g. [6] where it is shown to be asymptotically 

stable. We have shown here that, under certain conditione on the open-loop 

loop DPS, ideal trajectories do exist and the adaptive codtroller is stable, 

i.e. the output and gain errors remain bounded. 

A in (2.1) generates a holomorphic Co-semigror;p is hposed, then we can also 

conclude asymptotic stability which guarantees asymptotic tracking or node1 

following. 

If the further condition that 

A number of issues have been opened for further investigation: 

(1) use of dynamic rather than output feedback stabilization; 

142 
generation of asymptotic ideal trajectories by the open-loop DPS; 

connections between the Kalman-Yakubovich conditions and the inputoutput description of the 93s; 

development of altermtive adaptive gain laws which produce asymptotic 

stability of the closed-loop system; 

exploration of reasmable conditions under which taSalle's Invariance 

Principle can be used to determine asymptotic sta5flity of the closedloop system. 

ACKNOWLEDGPWTS 

This research vas supported in part by the National Science Foundation 

ufider Grant No. ECS-80-16173 and the Air Force Office of Scientific Research 

'*?L;er Grant No. AFOSR-83-0124. Any opinions, findings, and conclusions or 

reLommendaticqs expressed in this publication are those of the author and do 

necessarily reflect the views of NSF or AFOSR. 

REFERENCES 

R. Curtain and A. Pritchard, Functional Analysis in Modern Applied Hathematics, Acedemic Press, NY, 1977. 

A. Balakrishnan, Applied Functional Analysis, Springer-Verlag, NY, 1976. 

M. Balas, "Toward A More Practical Control Theory for Distributed Parameter 

Systems", in Control and Dynamic Systems: Advances in Theory and Applica- 

--' tions Vol. 18, Z.T. Leondes, editor, Academic Press, NY, 1982. 

H. Erzberger, "On the Use of Algebraic Methods in the Analysis and Design 

of Model Following Control Systems", NASA TN D-4663, 3.963. 

J. Broussard and M. O'Brien, "Feedforvard Control to Track the Output of 

a Forced Model", 17th IEEE Conf. on Decision and Control, Jan. 1979, pp. 

1144-1155. 

I. Bar-Kana, H. Kaufman, F!. Balas, "Model Reference Adaptive Control of 

Large Structural Systems", &+ J. Guidance, Control, and Dynan, Ics, MarchApril 1983, pp. 112-118. 

I. Bar-Kana, H. Kaufman? "Some Applications df Direct idaptive Control to 

Large Structural Systems", AIM J. Guidance, Control and Dynamje, to 

appear. 

P. Balas and C.R. Johnscn, Jr., "Adaptive Control of Distributed Parameter 

Systems: The Ultimate Reduced-Order Problem", IEEE Conf. on Decision and 

Control, Fc. Lauderdale, F1. 1979. 

H. Balas, "New Directions in Asymptotically Stable Finite-Dimensional 

Adaptive Control of Linear Distributed Parameter Systems", This Proceedings. 

143 
[lo] H. Balas, "Linear Distributed Parameter Systems: Closed-Loop Exponential 

Stability With A Finite-Dimensional Co-itroller", Automatica, to appear. 

ill1 T. Kato, Perturbation Tlieory For Linear Operators, Springer-Verlag, NY, 

1966. 

(121 J. Walker, Dynamical Systems and Evolution Equations: Theory and Applica- 

- tions, Plenum Press, NY, 1980. 

J. Walker, "On the Application of Liapunov's Direct Method to Linear 

Dynamical Systems", J. ?lath. Analy. b Appl., Vol. 53, 1976, pp. 1.87-220. 

[131 

i141 S. Lefschetz, Stability of Nonlinear Control Systems, Academic Press, 

NY, 1962. 

[151 P. Hiris and H. Megan, "Remarks on Elyperstability of Infinite-Dimensioaal 

Linear Systems;", Int. J. Control, VOI. 35, 1982, pp. 739-747. 

1161 . D. Wexler, "On Frequency Doaain Stability for Evolution Equations in 

Hilbert Spaces Via the Algebraic Riccati Equation", SUM J. Math. Analy., 

Vol. 11, 1980, pp. 969-983. 

[171 A. Likhtarnikov and V. Yakubovich, "The Frequency Theorem for Equatloos 

of Evolutionary Type", Sibirskii Uathematicheskii Zhurnal, Vol. 17, 1976, 

pp. 1069-1085. 

144 
'5 N85-31205 

SELF-TUNING ADApTlvE CONTROLLER USING 

W. W. Cw and R. B. CIIMOa, Jr. 

Stanford University 

Palo Alto, CA 94304 

ONLINE FREQUENCY IDENTIFICATION 

ABSTSACT 

A real-time adaptive controDn bu beem designed ud tested smcceddly 00 8 fourth order laboratory 

dynamic system which feat- very low strrrcturl damping and a mon-cobdcd -r pair. The 

controller, impkmented in a digitd miaicomp.ta, couistr'd a stsk estimator, 8 set of state kedbuk 

e. and 8 Requency-- (FU) for d time pMmtter idecrtilutbm. Tk FLL cam dckct 

parameter and its counterput in the state estimator, and corrcct the eathator puuncter in d time. The 

adaptation dgwithm cam comct tk com?roikr error ud stabihe the system for mocc tLu 50$6 *uirtion 

in the plant natural frequency, compared ritk 8 10% Mty mugb in kqac8q vuhtkm hrahd-gain 

controller havig the SuDe palormuce at the nomid plant conditiin. After it ba kcLcd to the eomct 

plant freqaency, the adaptive controkr works aa well u the bdqpb eolltrolkr does de8 there is no 

parameter mismatch. The is dcmo.strrtd -, 

ad can ab0 be proven with simpk rootbcos methods. 

the closed-loop natural freqaency of the system Si coatrogsd, t.kmhte tb mbatcb betreerr.-t 

rapid c-e of tkb drpti*e 

I. INTRODUCTION 

A controller using Kalmu 5lter and hn state Wb8ck usually has good pertornuncC, provided a 

very accurate model of the plant is known. Bot such coltrdkn ue very Seuitie to parameter rui.ti, 

especially when the plant has very low inherd damping, and rhem the masor is not colocated with the 

actuator. 

A twodisk laboratory model, consisting of two inertia dias coanccted 8 torsion rod, rkb bas a 

structural damping of 0.004, and with separated seasor ud actortor locatio- WM constmcted to krt mct.l 

adaptive controller designs. The form of the quatiom of moth of the model is known due to the eaee of 

analysis of the lumped system; but the bek of ucmrak knowkdge about the natd structural fmquency 

during controller design corresponds to a phnt parameter uactrkio~ or rui.tmn; and this mcertaiuty is 

what the adaptive controller handles. 

It has been proposed by Kopf, Brown, h4arsh (Ref.1) and Mac& (Mt) to use a Phm: ;.0eLd-L00p 

to implement tuned damping and notch dltered command toque, so that the ftedback control force azx the 

structural frequency can be adjusted propedy according to the natural frequency of the plant, Bosca%ul 

and Cannon (Ref.3) have impkmented such a kind of controller for the tw&k experimental aystem. 

Under the same match project, a dillennt appd using a Frequency-Locked-Loop (FLL) to Mente 

the plant bequency waa developed. Thi- oaper dwriba in detail how the FLL identides the Inknown plant 

parameter and updates the controller in red time. 

145 
II. DESCRIPTION OF THE TWO-DISK PLANT AND FIXED-GAIN CONTROUEbt 

The plant to be coutrolled b a mechanical system which consists of two h0riCent.l steel dish connected 

by a vertical elastic steel rod. The two dish ue supported by bearinga which dbw rotational motion ody. 

A low-friction DC motor is attacked to the bwer disk, and RVDT sensor detects the angular position of 

the upper disk. 

If structural damping is negkted*, the state eqiiation of motion of thb system can be expd a~ 

where tl and ts are the position state3 of the rigid body mode aod the stmctural osciIl8tion mode respectively, ts and t4 are rates of those states respectively; w,, ia the natural frequency, J is the tokl moment of 

inertia of the two dish, and Y b the control toque from the DC motor. 

The Sensor output b 

# = 21 + 2s: 

A fintsrder high-paw !Utet with 100 Hz cutoff frequency is used to Merenthte the position semr output 

and provides the pseudo-rate of the top disk. 

. (2) 

If all the parameten of the plant are known accurately, a0 LQG design (Ref.4) rill result in a set of state 

feedback gains C for regulation and estimator gains I, br state estimrrtion. However, if the pht natd 

frcqacncy w,, is not known by the controller designer, and a value We is aaed m the estimator, the stability of 

the whok system has to be analyzed by augmenting the system state eqrutha with thosc of the estimator 

states, and finding the modd frequencies and dunpimp of the system (Ref.5) 

Using the same penalty weightings for control effort and state errors, an LQG design produces Merent 

feedback gains C and L for dserent natural frequencies w, of the plant. Analysb shows that the stabity 

of the whok sipstern is lesa sensitive to those feedback gak tbu to the parameter we used in the estimcrtor, 

since an error in the latter parameter corruponds to a modeling error, whik ~kbio~ in the dormer ones 

comspond to ditlcrent weighting in the LQG design procar. In the experiment descni here, feedbreit 

gains C and L are chosen for the nomind plant frequeoc;, and are kept constant m order to demonstrate 

the adaptation of the controller by correcting we in the estimator. 

.hm the analysis of the augmented system state quatiom, the kqueney We of the most unstable 

c.6sd-loo~ mode can be found as a function of w,, and we, if .II other puuneten ue kept constant. Thir 

'UdCth. 

we = J(we,wm), (3) 

ulll &t f the closed-loop performance of the adaptation proeas, and has to be taken into account in the 

design yrocess. The twodisk model has a nomind frequency of 13.3 rad/see, md the function described t 

equatio 1 (3) can be shown approximately na in Fig. 1, can be appmxhakd as 

We - Wn = (Wa - We) + 0.6 (4) 

for /wI - we/ < l.brad/~~. 

m. FREQUENCY IDENTIFICATION USING FRXQUENCY-LOCKED-LOOP 

A Phasc-Cocked-Loop (PLL) WM initidly propd to be d to detect the vibration frequency. PLLs 

have been used widely in locking onto high-frequency signals in electrical engineerbg applications, but it - 

It is actually 0.001 

146 
is only beginning to be uaed h locking onto lor-frequency &nab in mceh.nical systems. A PU bos tba 

ability to identify the phrw and frequency of a signal eoatuninattd by a nktively large amount of nok .t 

other trtguencies. Several signal components at Merent trepuencke can be ideatfled by using wed PLLa. 

The traditional PLLs are nonlinear elements for which the perforname is hucd to analyse and predict; 

and they have Limited locking ranp due to their nonlinearity? Besides, PLLs ace more aeeositive to the 

phase than to the frequency of their driving signal, which makes them olrsgitabk for frequency ideatibeatior 

because the identieeation will be disturbed by the phose in the eemr syld every time a new mition 

command or an external disturbance is applied to the system, even though a PLL haa identW the comet 

plant frequency already. 

A modification is made to a PLL to eliminate ita sensitivity to phase in the input signal and make the 

input/output relation bear in a larger tracking range, so that it works better for frquetcy identitleation, 

while retaining the other virtues of PLLs. The dnal product, caUed a lkquency-Loeked-hp (FLL), ia 

shown schematically in Fig. 2, and its input/output relation can be seen from the tnoetioaal block diagram 

in Fig. 3, where w, is the frequency of the input &pal and wo ia the output signal - the frequency detected by 

the FLL. Also shown in the same block diagram ue w,, the starting osciliation frequency; Au, the cometion 

on the output; and we,, the error of the output of the F'LL. 

The character of the block G(4) can be chosen arbitrarily by the designer cia bng M it c.0 update the 

output frequency of the FLL accorcliig to its error 0,. If a hpk integrator 0 is choscn M the element 

C(r), then the FLL will have a pole at -K where 

Parameters a and b should be determined with the folbwing restriction 

In the present case, 

and the linear sear& range is choscn to be 

wm = 13.snn/Sce, 

The pole loeation 8 = -K should be determined m the molt of a compmmhe ketrcen speed of nsponae 

and noise rejection, at the nomind locking frequency range. In thio case, the parametera of the FLL ut 

chosen rn 

to work in the range of 1 to 3 Hz. 

o=6.0, 6=4.0, C=#).O, w Kz1.67, (9) 

With parameters chose0 M above, the block diagram in Fig. S can be aimpUded to the tdr hroction 

Fig. 4(s) ahma the test molt of the FLL output when the frequency of the $pa& signal is changed stepwiscly. 

The mpow for rmd input change (the hat change in Fig. 4(a) ) io similu to the step mponee of a dntorder dlter with pole at -K, M ahown io Fi. 4(b). The rerponse for a larger input change (the w-ond 

147 
change in Fig. 4(a) ) experienced some nonlioeuity at the beginnkg beczrrse its internal strocton b mot 

linear; however, the FLL still tracked the input signal and provided the comet output in a mmondde time. 

IV. CORRECTION OF PARAMETER ERROR IN TIlE CONTROLLER 

Because eigenvalues are properties of the system, they are independent of the instantaneous due d 

state variables and are indueneed only by changes of parameters. The rehtion between we aud we, m shown 

in Eqn. 4, CM be expressed am in Fig. 5. Using the Uemnce between (u,, - 0.6) and we to mpdate - 

thmuKh the integrator $ - the parameter w, in the controller, the dared loap dyalmirr of the parameter 

variation, identfieation, and correction can be expremed ua in Fa. 6. The ~icriStic equation of the 

closed parameter adaptation loop is 

or, 

a(# + K) + (0 + 2K)B = 0, 

which can be written in Evan's form as 

The root locus of Eqn. 13 vs. the positive value of R with K = 1.67 is shown in Fig. 7, and tbe value cf 

H = 9.9 is chosen obviously to maximihe the adaptation de. The change of the dope io Fig. 1 cornponds 

to a variation in the gain in Eqa. 4, and Eqn. (11) C.D be m&ed M 

rH K + (8 + X)(r + K) = O, 

where 2 > r > 0 , and the mot bus shown in Fig. 8 quarantees the stability of the rystem over the range 

of the gain V. 

Any sensor measurement, controller state variable, or linear combidion thereof can be &asen as the 

input signal to drive the FLL, so long as the signal contains the modal traquency of merest (the kger the 

better!!. The error between the sensor rate and the estimate of it is &own to drive the FLL, since there is 

less error signal if all parameters in the controller are correct. 

The FLL must be turned off if its input sigd is too sdl, in order to reject the Muence hm random 

noise. 

A PDP-11/23 minicomputer was wed to implement the controller and the FLL at 2S HG sample rate. 

The test results of thio adaptive system an summerid in the dotbring section. ' 

V. EXPERlMENW RESULTS 

Fig. 9 rhows the natd 0rCill.tiOn d the uncontrdled disk system. The kquency d oeehtbn in 

2.11 Hs. with 0.004 damping. (The bng-period motion b cwcd becaase the disk b Long fkom the 

ceiling with a long steel wire to reduce the ud.l tht 01 beuir;p. "him mod. t apphated u a rigid 

body mode in the controller design adyds.) 

Fig. 10 shows the step nrponse of 8 nonulapthe control system designed with the LQC method. The 

response is very good (Fig. 10) when there b no modcling error in the controlbr daip. However, u Fig. 11 

shows, the system becomes unstabk when there is 10% modeling error in frequency in the designing of the 

nonadaptive controller. 

148 
When the FLL is used in the adaptive control, the system em detect and eomct a controkr'r pMmekt 

error of 50% or more ia frqaency. Fign. 13 (a) through (f) show the KpIor ontpot in Werent teata. 'fke 

instability due to the initid parameter erm L ahown when the control syatem waa juat tuned oa, urd tke 

system was then stabilized after the uhptatios algorithm had corrected tbe controlkr'r error. The initid 

turn on of the control system aad the time rbeo podtion eommlada ue &aged an mar&& on thow 

recordings. 

Fig. 13 shows the comparison of the impulsive disturbance nspoose, between the nonadaptive controller 

with no modeling error and the adaptive one after its parameter emr ha been corrected. The compuison 

shows almost no diaercnce betweep their perfomamea. 

VI. DISCUSSON 

(A) Frequency-Locked-Loop 

The F1.t is a nonlinear element, but its input/output relation b almoat hear. It behaves linearly for 

40% changes ID input ripd frequency, and rtin work for loosC chow in frequency in the nonlinear mn. 

The test recorded in Fig. 4 attests to the dteuaaion above.. The hear rage can 1. chosen by aekting 

parameters pnperly. 

,The FLL still worka when the amplitude of its input si& h, aa weak aa two quantiution intermla of 

the A/D ccnverter, if it L free of noise and biru, but in mal rpptiUtbns it mast be tmed od at d &vel 

of input signd to reduce the effect of noiac. 

The FLL can idetitifv the plant characteristic in a sd rindow ol the &cqaency rpectrum, 10 tba: the 

effects of other parts of the ayrtem dynsmier do not have to be wten into aecoan4 if they are rot critical 

to the overall performance. It em only detect modes that ere either only slightly damped or unatabk, since 

they can provide oscillatory sign& for detection; however, heavily damped modes ue uadiy robuat to 

parameter uncertainty and dw't need adaptive control 

(B) Parameter Error Correction Loop 

The parameter emr correction acheme w be determined by root-bcua analyam, or even by the LQG 

method, siuce the FLL haa a linear chluaeteristie. 

Fig. 12 shows some small-smplitnde vibration brrilding up due to the WI d ripd to lock the FLL, but 

the pvameter estimate emc wm soon corrected and vibration suppd. 

By examining the reaponre to command hge and to dbtnrb.nees, it b band that the Se&lhning 

Adaptive Controller behaved dmwt the aune a8 tbe comt hed opthd controller, except tor bhe few 

cycles of vibration at the beginning when the pvnmcter error WIU being corrected. 

It is better to use the cmr of an eatimated aenaor output to drive the FLL, mince it is undisturbed by 

the control force during a new command cbange if the model is correct. 

Both the identiaeatioa and error cometion are running in red time while' the controller ia doinc Its job. 

Any change in the plant can be tracked and adapted to rapidly. 

VU. CONCLUSION 

The uae of FLL in identifying ryrtem vibration hquency and adapting controllet parameters b promising, AH kinds of eootrolkn, roch u Kalmu~ alter UI~ state feedback, band-pur, or notch dlten CUI bave 

their parameter enom corrected in a rimilu way. It L expected that ayrtem with II).DY vibration modea can 

149 
be handled with several FLLI. 

ACKNOWLEDGMENTS 

The research wporttd upon here wm spomred by the National Aeronautics and Space AdmSistntbm 

through the Langley Research Center and the Jet Propubion Laboratory. The author^ appreciated the 

advice and kelp from Dm E. Rosenthd who was doing ibe other part of t!ie research under the Mme NASA 

cont :act. 

REFERENCE 

(1) Edward H. Kopf, Thomas K. Brown and Elbert L. Muah, 'Flexible Stator COIA~ on the GJileo 

Spacecraftp, paper 79161, MS/W Astrodyaamica SpecwiSt Coafereacc, rroviacetorn, MASS/ Jane 

25-27, 1979. 

(3) Dan E. Roscntbal and Robert H. Cannon, Jr., "Experiincntr with Adaptive Control of F!&k SLnetures", to be published in the Journal of The Astronautical SC~BCCI, 1984. 

(4) h. E. Brywn, Jr. 6 Y-C Ho, 'Applied Optimal Control", Hemisphere Publishing Co., IQ'lb. 

(5) Daniel B. DeBra, 'Estimatbn in Satellite CoatroP, Proeecdiop of the Nmth Intermti -.A Symposium 

on space Technology and Seknee, Tokyo, 1971. 

150 
-3 -2 -1 

Odt : (na/scc) 

151 
rP U 

" L 

152 
. 

153 
154 
-Re 

H =99 

(0 = -sa) 

-1 L 

U 

I\ 

.I 

n 

-E -K 

?=l 

I 

Fig. 8 Variation of Pok of the Clod Parunttet Loop .*mu t, the Ch.ogbg Sbpc io Fig. 1. 

155 
156 
157 
T.O. C.C. C.C. 

158 
T.O. : initial turned o.. 

C.C. : step command &agec. 

Fig. 12 Step Response of the Adaptive Cortmllcr with FLL Detcctbg W kde)hg Emorb Pk.r hcq.cncl: 

(8) 80 ern?. 

(b) -lO%enor. 

(c) +25% error. 

(e) -So%em. 

(2) +50% error. 

(d! -25%err0r. 

159 
Fig. 13 Comparison of the Impobirt Disturbance Rcrp0.r~ betveen (a) tbe Nonadaptive Controller with No 

McrdcIing Error snd (b) the Adaptive One .ikr Its Parameter Error H.s Been Corrected. 

160 
To.85-31206 

ADAPTIVE FILTERING FOR LARGE SPACE 

STRUCTURES-A CLOSED-FORM SOLUTION 

€I. E. R.uch azd D. B. Wmchter 

Lockheed Palo Alto Research Laboratory 

Palo Alto, CA 94304 

ABSTRACT 

In a previous paper Schaechter proposes using an extended Kalman filter to 

The present paper shows that the time-varying gains for 

estimate adaptively the (slowly varying) frequencies and damping ratios of a 

large space structure. 

estimating the frequencies and damping ratios can be determined in closed-form 

so it is not necessary to integrate the matrix Riccati equatiots. 

tain approximations, the time-varying adaptive gain can be written as the.product of a constant matrix times a matrix derived from the components of the 

estimated state vector. This is an important savings of computer resources and 

allows the adaptive filter to be implemented with approximately the same effort 

as the non-adaptive filter. The success of this new approsch for adaptive 

filtering has been demonstrated using synthetic data from a ."wo node system. 

After cerI. INTRODUCTION 

Adaptive esthation and control techniques are being studied for their 

future application to the real-time control of large space structures, where 

uncertain or changing parameters may destabilize standard control system 

designs. In a recent paper Schaechter €11 proposes using an extended Kalman 

filter to estimate adaptively the (slow y varying) frequencies and damping 

ratios of a large space structure. For a system with N states and M (slowlyvarying) parameters the extended Kalman filter requires integration of 3n N+M by N-W 

nonlinear matrix Riccati equation to determine the covariacce and gain for the 

fllter. 

the nonlinear matrix Riccati equation to be replaced by integration of a smaller 

set of linear matrix equations. The N states of the system are estimated using 

constmt gains determined off-line. The time-varying gains for estimating the 

(slowly varying) s? of M parameters are determined on-line by integrating an 

M by N set of linear matrix equations. 

Schaechter introduces approximations which allow the integration of 

The contribut2on of the work presented here is to show that the timevarying gains for estimating the (slowly varying) frequencies and damfing 

ratios can be determined in closed-form so it is not necessary to integrate 

the M by N set of linear matrix equations. This is an important savings of 

cmputer resources and allows the adaptive filter to be -'.mplemented with 

approximately the same effort as the non-adaptive filter. In particular, 

after certain approximations the time-varying adaptive gain can be written 

as the product of a constant matrix times a matrix derived from the components 

of the estimated state vector. The constant matrix ?.s determined off-line 

just as the constant gains for estimating the state are determined off-line. 

161 
The success of this new approach for adaptive filtering has been demonstrated on a computer simulation usicg synthetic data from a two mode system. 

Work in progress is applying the new approach to a much larger system using 

experimental data. 

resillts are presented in the paper. 

The theoretical development and preliminary experimental 

11. FORMULATION WITHOUT ADAPTIVE FILTERING 

The standard state variable formulation of the dynamic equations of motion 

are shown below where the dot indicates derivative, x is the state vector, u is 

the control vector, z is the measurement vector, and w and v are dynamic noise 

and measurement noise. [2] 

0 

x = FX + GU +rw (1 1 

When'the dynamic system is pregisely known, a state estimator of the following 

form may be constructed where x indicates the estimate of the state x and K is 

the gain matrix. 

* * A 

x 0 FX + GU +K(z - Hx) A (2) 

The differential equation for the estimation error ;= x - x ie obtained by 

subtracting Eq. (2) from Eq. (1). 

The differentia: matrix equation for the covariance of the estimation error P 

follows where R and Q are from the covariance of thz measurement noise v an 

the dynamic noise v. 

P o EET) 

T + = (F - KH)P + P(F - KH) 

+ rqrT + KRK~ 

The optimal gain matrix K is chosen to minimize t,he trace of the estimate 

error covariance to give the usual result 

T -1 K=PPHR (5) 

Notice that €or a precisely known dynamic system, the estimatlon gains may be 

precomputed, even in the event of a time varying system. The analysie used 

with the adaptive filter cloecly parallels the development without adaptive 

filtering. 

111. ADAPTIVE FORMULATION AND SOLUTION 

Adaptive control may be required when ;he model in Eq. (1) is unknown, 

uncertain, or dependent upon a changing eystem configuration. The modifications that need to be made in Eq. (1) in order to include the effect8 of an 

162 
uncertair. parameter are given below where the vector parameter a has a dynamics 

matrix C with dynamic noise w a' 

x = F(a)x + Gu -t rw 

a= Ca+w 

a 

z=Hx+v (6) 

As can be seen from Eq. (61, the system dynamics are now a function cf the vector parameter a. In this formulation, the vector parameter a represents small 

changes from a nominal value ro the average value of a is zero. These parameters are assumed to be slowly varying so that they may be adjoined to the 

state vector. An adaptive state estimator may be written so both the st'ate 

vector and the vector of Faraaeters are updated using the measurements. 

1 A m 

x = F(a)x + Gu + Kx(z - Hx) 

I A 

s. = Ca + Ka(z - Hx) 

Let the symmetric matrices P, and Pa represent the covariance of Ct.1 mror in 

the estirates for x and a, respecttvely, and let the rectangular matrAr. Pax 

represent the cross-covariance of the errors in the estimates of x and a. 

It is necessary to calculate these covariance matrices in order to determine 

the optimal gains K, and Ka. The optimal gains are selected to minimize the 

trace of the covariance of the estimation error and have the following values, 

T -1 Kx = PxH R 

r -1 K = PaxH R 

a 

N n 

Proceeding as before, and assuming the esiimation error a = a -. a is small, 

gives the vector differential equation for the error, 

k = - KaH + Cz + wa - Kav 

The n:s,rix differential equatio.18 for the covariance are: 

163 
dP /dt = X 

+ 

+ 

dPax/dt = 

+ 

+ 

dPa/dt - 

where. Fa = 

Pa(Fa;lT- KaHPx 

T 

K,iRKx m 

CPa + P$’ + Qa 

KaHpax * - Pax(KaH)T + Ka RK,T 

aF/ aa 

and R, Q, and Qa are covariances of v, rw, and w 

(without delta function). 

a 

The remaiader of this analysts will show approximations which can be used to 

reduce the coniputational effort needed to calculate the covariance matrices 

and the optimal gains when the covariance mal-rix P, is very sm.11 (of 

order cl and the covariance matrix Pax Is also very small (of urddr e). 

gain K, will be very small (of order c) because it is calculated frog Pax. 

The differential equaiion €or the covariance matrix P, will. involve somf mall 

terms, but most of the terns are larger and constant. If the last two tams 

in the differential equation for P, are neglmted (beczuse they are small te:ms 

of order E), it is possible to calculate the steady-state constant valde of the 

covariance P,. From the cons’,ant value sf th? covariance P, the constant gain 

K, can be determined. As one might suspect, the constant gain 5 hias ,he saae 

value as it would have if there were no errors in estimating the paramecers e. 

The 

Becmse the covariance matrices Pax and P, are of order e, many of the tei 

in the differential equation for Pa :+? of order e squared. 

terms in the differential equation for Pa are neElectci? (because they ate v ’I 

small cerms of order 5 squared), it is pc -sible to calculate the steady.-state 

value of the covariance P, (to order e). 

steady-state value obtained for Pa is the Silme valuc which’ would have heru 

obtained if K, were zero. 

If the last 

As one might suSpect, the constant 

All that remains is to calculate the time-varying covariance Ea si- st the 

needed gain Ka can be determined. Because the gain hss been ‘1 .: A tu 

equal P,HTR-I, the last two terme i:: the differential equation foi Pax cancel 

out. For the remaining analysis it will be aacumed there Are N btaie varjables so the first N/2 variables (designated he N/2 length 1 vtor x*) cor- 
respond to mode position, ani the last N/2 variables (designated by the N/2 

length vect r x**) correa+ocd to velocity of mode position. 

equatioirs 'or the dynamics of the mode variabies without any forcing or disturbing terms are presented below where A* correspond9 to the damping terms 

(-2.90) and A** corresponds to the frequency terms (-w-). Notice both A* and 

A** are diagonal N/2 by N/2 matrices. 

The differential 

dx/&t = Fx 

dx*/dt = x** 

dr:**/dt - A*x*+A**x** 

Let there be LJ parameters in the vector a and arrange the order of the parameters a so that the first N/2 garmeters axe the same as the elements of the 

dicgcnal matrix A* and the last N/2 parameters are the same as the elements 

of the diagonal matrix A**. Fu:;thermore, assume the N-by-N symmetric covariance htrix Pa aSbvCidted with these. parameters is diagonal and composed 

of diagor.al sub-matri-as Pa* and Pa**, With these assusptions, the partial 

derivative can be w I in a partLcularly simple way where x* and xx* 

represent diagonal G~-.~L~~s with the di3gonal elements equal to the vectors 

X* and x** 

16.5 
One further assumption is that che dynamzcs matric C (for the parameters a) is 

diagonal and equal to the scalar co times the identity matrix I. 

assumptions, the differential equation for the cross covariance Pax can be 

written as follows where 

With those 

x is a diagonal matrix made UF of the elements of x. 

dPax/dt = Pax(F-KxH+C) T 

- 

+ (Pa***[X3 IT 

where 

and 

c = coI 

The remai.rder of the analysis will deal with the cross-covariance matrix P, 

which is the transpose of the covariance matrix Pax. The differential equation for the cross-covariance Pxa cag be written as follows: 

dPxa/dt = F* Pxa + P a *** ['I 

where F* = F - Kx H + C (14) 

The linear matrix differential equation for Pxa has particularly desirablc characteristics. All the terms in the differential equation are known constants 

(because the gain Kx and the covariance Pa arf known and constant) except for 

driving terms due to estimates of the stat_e If the approximation is made 

that the derivative of the forcing terns x is equal to the dynamics matrix F 

times G, then, except for trmsieut terms, the solution to the linear matrix 

differential equation for Pxa can be written in closed form as a linear com- 

. ;pa;ion of the torcing terms 2. This is similar to the result in elementary 

iinear differential equations where the general solution is composed of the 

sum of the homogeneous solutfofi due to the unforced different,al !quation and 

the particular solv'.ion due to the forcing function. 

Because the forcing funct'ion [;I is a diagonal matrix, the first element $1 is 

the forciig te? for the firs; column of the solution for the matrix P 

second element x2 is the forcing term for ?he second column of the matrix Pax, 

and so on. 

P2X. The linear matrix-vector differential equation for the i-th column can 

be wrltten as follows where Pairis a scalar which is the i-th element of the 

dipqonal matrix Pa and ;1 's, a scalar which is the i-th element of 2 and Pf is 

is the i-th col-nn of the matrix Pa*** which is all zeroes except for entries 

eqiial to tne diagonal tlements of Pa 

x. 

the 

Let Pi be a vector which represents the i-th column of the matrix 

av' 

166 
A 

dPi/dt = F* Pi + Pi* xi (15) 

The solution for the vector Pi is assumed to be composed of the sum of-two 

vectors. 

(corresponding to the estimate of the position of the mode) and the second 

vector is the constant vector Gi times the scalar & 

estimate of the velocity of the appropriate mode). 

The first vector is the constant vector Ei times the scalar xj 

(corresponding to the 

where for i 5N/2 then j = i and k = i + N/2 

for i >N;2 then j = i-N/2 and K = i 

The derivative of the vecfor Pi can be ca!culated direct1 

and 

derivative of the vector x is equal to F x with A: 

represent the j-th element of the representative aiagona 

up F. 

if it assumed the *' being scalars which 

matrices which make 

dPi/dt = E i1 ds /dt + Gidjik/dt 

= Fi< + G i A*< j j + GiAj**< 

Substituting the expression for the assumed form of the vector Pi and the 

expression for the derivative of the vector Pi into the differential equation, 

gives the following equations where 6ij is a discrete delta function which is 

unity if i equals j and zero otherwise. 

* 1 

= F* E x + F*Gi% + 6 P *X + dikPk* xk ij ijj 1 

A 

Collecting all terms which multiply the scalar x 

equation. 

gives one vector equation 

and cglleccing all terms which multiply the scal a r Gk gives a second vector 

The- 2 are two vector equations and two unknown vectors Ei and Gi. 

G iJ A.* = F*Ei A SijPj* 

Ei + G A ** = F* Gi + 6ikPk* 

ij 

(19) 

167 
The expression for Ei obtained fiom the second equation is substituted into the 

first equations to give a single equation with the unknown vector Gi 

Since n * and A ** are both scalars, it is possible to solve directly for the 

unknown vector 'Gi where I is the identity. j 

In the same way, the expression for Gi obtained from the first equation is siibstituted into the second equation to give a single equation with the unknown 

vector Ei. 

It is again possible to solve directly 

Thus the cvo unknown vector quantities 

quantities so the covariance vector Pi 

matrix P can be determined. 

ax 

[-15 (F*-IA.**)P.*+6. A *Pk* ij J J ik j (23) 

Ei and Gi can be determined from known 

and the approximation for the covariance 

168 
IV. SIPJUTION RESULTS WITH TWO MODES 

The new, simplified adaptive formulation was first tested vith a single 

mode system. After encouraging results were obtained with one mode, a tgo-mode 

system was examined. 

shown in Figure 1. 

The two-mode system used in the simulation studies is 

FIG. 1 TWO-MODE SYSTEPThe system consists of two masses, M, three springs, K, and three viscous 

dampers, B. For this study, M=l, K=l, and B10.10. Control forces may be 

applied to both masses, random external forces disturb both masses, and noisy 

measurements of the position of both masses are available. 

are used for estimating the state vector, and for estimating the parameter 

vector. The differential equations representing this system are: 

The'measurements 

z1 = x1 + v1 

z2 * x2 + v2 

The natural frequencies and damping ratios of this system are: 

w1 = 1 c1 = 0.05 

w2 - 1.732 c2 = 0.0869 

where the low frequency mode is the common mode motion of the two masses. 

spectral densities of both the process and measurement disturbances (Q and R) 

are 0,0163. 

during a sixty second computer simulation. 

The 

Tw43 hundred position measurements of both the rnames were made 

This sixty second duration was 

169 
selected to assure about ten oscillations of the lowest frequency mode. 

sample rate was selected to give about ten samples per cycle of the highest 

frequency mode. 

that are to be estimated was 250 seconds (so Co is 1/250). 

larger than the time constants of the system. 

value is important in order to allow the adaptive filter to average values 

over several cycles of the system. The fol' )wicg table gives a summary of 

the test cases. In each case, both the standard, non-linear extended Kalman 

filter, and the siuiplified extended Kalman filter described in this paper 

were run in order to make comparisons. In all of the cases studied thus 

far, these two cases were indistinguishable, except for a small, initial 

transient. This transient effect is attributed to beginning the standard 

extended Kalman filter covariance integration with values slightly different 

from the steady state values. 

The 

The value of the correlation time constant for the parameters 

This value is much 

The selection of a "large" 

r 

CASE 1 w1 unknown 5 known no noise 1 

w2 unknown c2 known 

CASE 2 W1 unknown Cl unknown no noise 

w2 unknown c2 unknown 

CASE 3 unknown C1 unknown 

5 unknown 

-. ~ L 

r.oise 

2 

7- -. 

The results are shown i the following figures and are discussed below. 

In Case I the starting estimates for the natural frequencies were chosen 

to be 10% in error with w1 estimated to be 0.9 (rather than 1.0, and 9 

estimated to be 1.559 rather than 1.732). The damping parameters were 

exact, and no noise was present in the system. The results for the estimate 

of W1 (Fig. 2) show that the modal frequency is very readily identified from 

the measurements, inspite of the 10% initial error in the estimate. 

system response diminishes, less information is available for updating the 

parameters. Consequently, with no new information coming into the system, 

the parameter estimate begins to return to its nominal value (0.9) with the 

selected time constant of 250 sec. 

As the 

The estimate of 9 behaves similarly. 

In Case 11, the objective was the same as in Case I with the additional 

problem of simultaneously estimating the damping parameters. 

estimates of the damping parameters were zero. The results of :hc poor initial guess of the damping parameter are evidept in Fig. 3. 

of the modal frequency tends to be lightly damped, but in all other aspects, 

the estimate of W1 appears to have the 

Case I. As has been found in past 

The initial 

The estimate 

that were present ~II 

estimate of the dampii; 

170 
parameter itself is quite poor. 

measux emen t contains very 1 it t 1 e damping info nuat ion. 

This is due to the fact that the position 

Cast! I11 is identical to Case I1 with the addition of both process and 

measurement disturbances. Surprisingly, this case yielded the best results, 

as is evident in Figures 4 and 5. 

visible in the figures. However, in contrast with the previous two cases, 

the process noise continues to excite the system after the transient effect 

of the initial conditions have subsided. The result is that the measurements 

continue to provide informatj.on on the parameters for the duration of the 

simulation. 

less perturbed by the external disturbance, the improvement in the natural 

frequercy estimate of mode two is not as dramatic. 

The effects of the .ioise are clearly 

Since the higher frequency mode is more heavily damped, and is 

CONCLUSIONS 

This paper has developed approximations which allow dramatic reductions 

in the on-line computational requirements of the extended Kalman filter. 

Numerical simulations of this technique have validated the approach for two 

simple spring-mass systems. It was found that the full non-linear extended 

Kalman filter and the closed-form adaptive filter developed in this paper gave 

virtually identical results. Work is currently in progress to apply this 

ipproach to a much larger system using experimental data, rather than simulated 

data. 

REFERENCES 

(1) SZHAECHTER, D. B., "Adaptive Control for Large Space Structures", AIAA 

Guidance and Control Conference, Gatlint1,rg. Tennessee, paper no. 83-2246, 

August 15-17, 1983. 

(2) BRYSON, A. E. and HO, Y. C., AppXed Optimal Control, Blaisdell, Waltham 

Massachusetts, 1969. 

171 
TI 

FIGURE 2 Estimate of o1 (CASE I) 

FIGURE 3 Zstimiite of w1 (CASE 11) 

172 
TIME 

FIGURE 4 Estimate of w1 (CASE 111) 

TRUE w2 

TIME 

FIGURE 5 Estimate of w2 (CASE 111) 

173 
ROBKST ADAPTIVE CONTROL 

K. S. Nartndra and A. M. Ammwamy 

Yale University 

New Haven, CI' 06520 

ABSTRACT 

The paper discusses swreral concepts and results in robcst adaptive control 

and is organized in three parts. 

Different formulations of the problem and theoretical solutions that have been 

suggested are reviewed here. 

role of persistent excitation in robust adaptive systems and the use of hybrid 

control to inprove robustness. 

research are suggested which combine different approaches currently known. 

The first pal. "ir.;eys existing algorithms, 

The second part contains new results related to the 

In the third part promising new areas for future 

1. INTROEUCTICN 

The stable adaptive control of linear time invariant plants, in what is now 

termed "the ideal case", was resolved in 1980 [l-41. 

[1-4] regarding the plant to pro\-e global stability are quite stringent. 

cally, knowledge of the sign of the high frequency gain K , the relative degree 

n* and an upper bound n on the order af the plant transfe! function are assumed 

to be known. 

lie in the left half plane, the plant parameters are constant (though unknown) and 

the system is disturbance free. IIowevar, in practice, these assumptions are rarely met. No actual plant is truly linear, finite dimensional or noise free. Further, in practical situations, the rationale for using adaptive control is to compensare for large variations in plant parameters, In the presence of such deviations from ideal conditions, the algorithm suggested in [l-41 no longer assure 

the boundedaess of the signals in the adaptive loop. 

interest in the. past few years in what is termed robust adaptive control to 

achieve satisfactory performance in the prese .;e of both wdelinp, and operating 

uncertainties. 

been made in this direction, presents some new results for improving robustness 

and discusses promising directions for future research. 

The assumptions made in 

SpecifiFurther it is assumed that the zeros of the plant transfer function 

This accounts for the wide 

This >aper attempts to survey some of the modest gains that have 

Adaptive systems are special classes of nonlinear systems and many questions 

which arise in such systems can be stated as problems in the stability theory of 

differential equations. In particular, questions of robustneds can be addressed 

using amply discussed results cn practical stability and total stability. Since 

such results are bound to find increased application in adaptive systems,some of 

the more frequently used concepts, definitions and theorems are collected in 

section 2. 

Recent years have witnessed many contributions to the robustneso problem. 

Among these some assume additional prior information regarding the UP !rtainties 

to suitably mcdiry the adaptive algorithms [S-9 ] while others assi. that the 

reference inputs possess properties which make the ideal system E ,,onentially 
stable. 

the true situation deviates in specific ways from the ideal. 

lytical results which are currently known are presented in section 3. 

In all cases it is shorrn that boundedness of solutions is assured when 

Some of these anaSections 4 and 5 contain some new results on persistent excitation and hybrid auaptive control which are relevant to the problem of robustness. In sectiun 4 a nonlinear error equation of second order is discussed in detail. While 

the ideal system is uniformly asymptotically stable it io shown that unbounded 

solutions can result if the disturbance is sufficiently large. It is also shown 

that by increasin,: the dezree of persistent excitation of the reference input the 

overall system can be made practically stable. 

trgl algorithms recently introduced by the authors [lo]. The same algorithms can 

also be modified to adaptively conLro!. discreti plants by updating control parameters infrequently. Some plausible arguments are given towards the end of the 

section as to why such algorithms -lay be more robust than continuous algorithms 

when external bounded disturbances are present. 

Section 5 discusses hybrid conFinally, in section 6, possibie :lays of combining known methods are discussed 

in the.hope that: it will stimulate research in these new directions. While no 

hard results exist in these areas the suggestions are based on extensive simulat ion studies. 

2. MATHEMATIC-4L PRE',IHINARIES AND STAdILITY RESULTS 

Some well known concepts and results of stability theory which find frequent 

application in the analysis of adaptive systems are included in this section. 

Ziile they can be readily found in any good text [11-13] we present them here fox 

easy reference as well as to place some of the problems discussed.in the following sections in proper perspective. We start with the definiLions of uniform 

asymptotic and exponential stability of the solution x = 0 of an equation 

3 = f(x,t), f(0,t) = 0. We assume that f is continuous and satisfies condicions 

which guarantee the existence and uniqueness of solutions and continuity of their 

dependence on the initial conditions. The general solution of che differeiitial 

equation is denoted as p(t,x ,t ) with p(tO,xO,tO) = xo. 00 

differential equation J = f(x,t) is uniformly asymptotically stable if it is uniformly stnble and for some E 0 and all c2 > 0 there is a T(c1,c2) > 0 such that 

IIxo JJ < cl implies11 p(t,xo,tO) II < 

(ii) Definitjon (Exponential Stability) : The equilibrilim state of the equation 

(i) definition (Uniform Asymptotic Stability): The equilibrium x = 0 of the 

1 

for all t >c t,, 4 T. 

= f(x,t) is exponentially stable if two positive constants a and t3 which are 

A linear time-invariant system with f(x,t) = Ax where A is a constaat nlatrix 

is asymptotically stable if the eigenvalue. of A arc in the open left t .If of the 

complex F'Lane. Asymptotic, uniform asymptotic and exponential stabiiity are 

equivalent in this case. 

does not iaply uniform asymptotic stability whereas the l.atter is equivaleric to 

exponential stability. For linear systems, all stability properties hold in the 

large. In general, for nonlinear systems exponentfal stability implies uniform 

For linear time-varying .,yatems,asymptotic stability 

176 
asymptotic stability but not vice versa. 

t, all stability properties are uniform. 

If f(x,t) is autonomous or periodic in 

In robust adaptive control we are interested in deducing the propertie nf 

the solutions of a perturbed system (S ) from the tehavior of the solutions of an 

unperturbed system (S) . These are deskbed by the differential equations 

a 

A = f(x,t) (S) 3 x a fht) + g(x,t) (SP) (1) 

Let the equilibrium state of (S) be exponentially stable. 

for sufficiently small b and 6,andllxll< 6,then the equilibrium star- of (S ) is 

also exponentially stable 1111. In physical situations the condition g(0,tY = 0 

required above is not generally met and this gives rise to the concept of total 

stability . 

If Ilg(x,t)ll < L Y 11 

(iii) Definition (Total Stability) [’lj: The equilibrium state x = 0 of (S) is 

totally stable if for every E > 0 two positive purnbers 6 (E) and 62(~) exist such 

that every solution p(t,x 

1 

t ) of (S ) satisfiesIlp(t,xo,tO)I( < E, t >c tO provided 

IIxob < 61 and)lg(x,t) I< 0, 62’ o P 

In the Russian literature this is also referred to as stability under persistent disturbances. 

implies total stability [ll] and is frequently used to prove robustness of adaptive systems in the presence of sufficiently small perturbations. 

magnitude constraint on ~(x 11 in the definition of total stability has been relaxed 

by Ande:-son and Johnstone ?8] at the expense of stronger conditions on f(x,t). 

The uniform asymptotic stability of the li- Jetturbed system 

Recently the 

In practical s:ftrters we are inta-rested In the uniform boundednees of the aolutioas ir? the presence of perturbat!ons as well as in the magni:Jdes of this bmnd. 

This leads to the concept of practical s t a b i l i t y d e f a e d below. 

(iv) Definition (Practical Stability) [12]: Let Qo - {XI Ilxfl < 6,) be open 

set in 1p 

If the solutions of (S ) lie wittin R close1 bounded set Q =)Qo for xo E Q, then 

n 

0‘ and 6* > 0 a constant such thatIlg(x,t)ll < for all x and L 2 t 

P 

the system (S) is said- to be practiL-21~ stable. 

Total stability assurz8 the existence of Q, and 62 relative to which the eystern (SI is Dractically stable but provides no way of estimating the size of Q or 

the magnitude of 6 In adaptive control applications this is not adequate. ‘One 

is more interested in determining an estimate of Q from a knowledgd of 6 

2’ 

2’ 

An alternarive method for tredting the effect of pertur. tions is by considering them arl bounded Independent functiona of time. 

concept of bounded input - bounded output (BIBO) stability. 

This leads to the WI C1 known 

(v) DefiniLlon (BIB0 Stability): A system 4 - f(x,u,t) with i(”,O,t) = 0 is 

BIBO stable if for every a >c 0 and every a 3 0 there is a 3 = e(a,a) wch that 

JIpu(~,xO,tO)ll i 6 for all T >c t for every ?.nitial condition (x 0’ t c ) with 

I/.+,..’! f a and syp IJu(t> 116 a, where p (-, 

i@#>.!I- u( *), 

0 

t ) 18 tne solution of the system with 

u xoD 0 

177 
A liilciar system x = A(t)x + b(t)u is L'dG stable if the homogeneous part 1.8 

This is a property which is frequently used i:: 

In contrast 

A similar 

unifcrmly asymptoticallv stable. 

robust adaptive control using the concept of perristent excitattoc. 

to the above, uniform asymptotit stability of a nonlinear system daes not imply 

BIB0 stabi1it.r. An example of this bas given by Desoar et a1 [14]. 

situ;.tion arises in the discussion of robustness of a second order nonlinear eyetern in erction 4. 

Stability Problems in Adaptlve Syste 3: Thc study of the stability cr adaptive 

systems (2s shwn In the following stctions) can be conveniently carried out usiul 

a set of nmlinear time-varying error differential equations. Even in the ".'Ai.al'' 

or disturbance free case the time-variations arise due to the presence of the 

reference input r(*). 

stability questions which arise in adapt 

-- 

The foilowing are qome noteworthy features of many of the 

e systems. 

(i) In the !deal case, a Lyapunov function V > 0 with '5 . J can ne found. The 

r.egntive semi-definiteness of 3 cannot he avoided and is D resu: i. ci the adaptivi 

law used. 

(ii) h:' a result of (i) even the unforced (autonomousj P. cea is uniformly stable. Even when tne reference input is persistently excirins, 3 6 0 buc the system 

can be shown ta be uniformly asymptotically stable [15]. 

theorem cannot be directly applied Lo prove this since the system is nonautonomous. 

We note that LaSalle's 

(iii) Since the system is exponentially sta. !.e with a persistently exciting reference input, Malkin's theorem can be used to conclude that the solutions will be 

bounded for some initial set Q and perturbation of magnitude $ & However, velry 

:ittle can be said directly about either Qo or &2. 

boundedness of solutions udng Lyapunov's Direct method for given bounus on perturbations is no longer trivial. Sowe of these caws are <:onshered in section 3. 

In section 4 it is shown that even v' m the reference inpu. is persis*.entiy exciting, if the diaturbance is l-cge thc. ;~lutiona can be unbounded. klccrnlitely, for 

a given bound on the disturbance the prsiatent excitation cm Ee made sufficiently large to assure the boundednest of the solutions. 

0 2' 

(iv) Aaother CL .sequence of the eemidefiniteness of 9 is that assuring even the 

3. ZECENT XESULTS Ih ADAPTIVE CONTROL 

In this secti.n we attempt, to survey briefly some 01 the theoretical results 

currently known ir. the area of robust adaptive control. The aim of the section ia 

to provide an understanding of the qualitative idea? that l6.d to these resuih 8 4 

well as the analy .ical toola used In derl.*;'-ng them. Sincr the ideal system rclrma 

the starting point of all perturbatiols a.:Ayses, we shall.briefly outline the 

statement of the oroblen end tne proof of stability in chis case. Further, whiie 

several stable ad,iptive algoriL,im have bee:r suggested fn the literature, we shall 

discuss the proof of stability using only 

proof8 using all the other algorithms follow along similar lilies. 

2 algLrithm proposed in [16]. The 

a) ge;l System: The plant 

e 

x =A,x 4- 

P PP 

to be contrdied is describe3 by the rate eqwticns 

w 

bu 

P yP = cP P 

178 
and a reference mo&l is ciescribed by 

. .F 

x m =Ax mm +bmr ; ym=cLx mm (3) 

where v, , u and y 

x anu 'ym are the st4tt: anb outpt of the model. The transfer functions of 

the plant and mdel aceare rzspecti-rely the state input and output of the plant and 

m 

T -1 % K 2 (s) 

r, .\s; 

T 

Rm(s) Wp(s) = cp (s1-Ap)-lb = Lk ; hm(s) = cm (sI-Am) bm * 

r. P 

The following assumptions a:- r i-le regarding W (s) and W (8) P m 

(i) Z (s) , R (8) and d -) are mrtc polynomi '-s of degrees m, n and n*=n-m 

P P 1 

P m (ii) 7 (s) and R (9) ax, . tricLly SCdbk polynomials 

and (iii) r is a piecewise cantinuous unuormly bounded reference input. 

The objective is to co trJ the plant in such a fashion that the output error 

between plant and model e 

nals and parameters of the system remain uniformly bounded. 

that knowledge of :he exact relative degree n* of the plant, an upper bound n on 

its order, tho sign of the gain K 

given in ,ii) are needed to solve 'he problem. n* enables the model to be constructed while the value of n detLLmines the order of the controller to be used. 

The sign of K and the constraint on Z (s) are needed to prove the stability of 

s 

= v - y, twds to zero asymptotically, while the sigIt is now well known 

19 

and the condition that 2 (5) be Hurwitz as 

P P 

the overall spstem. P 

Structure of Controller: 

K - Km = 1. 

ing equations is used: 

In the following we shall assume that K is known and 

To meet the control objective a controller describe1 by the followI' 

where F Is an asymptotically stable run matrix, (P,g) is controllable, 

w = [w(1)Tsb(2) ] and 6(t) is a 2n dimensional parameter vector which is to be 

adjusted adaptively. 

exists such that the transfer function of the plant together with the controller 

matches that of the model exactly, when e(t) E e*. 

is to adjust 9(t) in such a mpwier that :he overall system is globally stable and 

lim e,(t) = 0. 

tsidered, we discclss below the general case when W !s) has a relative degree 

n* 3 2. 

of the plant can be expressed as 

T T 

It is well known [17] that a unique constant vector 8* 

The aim of the adaptive law 

While several dpecial cases of the adaptive control problem have been conP A If e(t) - e* - +(t), then + is the parameter errqr vector and the output 
The Adaptive Scheme: 

is added to e (t) to generate an augmented error E (t) If 

To generate the adaptive law an auxiliary error signal y (t) 

a 

(6) 

1 1 

then 

vhere W (s)Iw = i. 

mented mrrror E (t) and the signal C(t) and is given by 

The adaptive law for updating e(t) then depends on the aug- 

- El(t)S(t) 

1 + sT(t)r(t) 

1 

G(t) = i(t) = (8) 

rhis has been shown to result in global stability of the adaptive loop [16]. 

Proof of Global Stability: 

T 

If V(4) = 1/2 ()i +s the adaptive lav (8) yields 

2 

-El (0) 

1 + sT(t)s(t) i(6) = 

from which it follows that 

(i) # and are uniformly bounded 

(ii) E L2 

2 and (lii) El(t) = u(t)[l + ~~(t)s(t)]~’~, u 8 l. 

Since the complete proof is too long and involved to be included here in its 

entirety we merely outline the principal steps involved. 

(a) Since the parameter vector is bounded by (i) it is first shown that 

SUPIYp(T) 1 ‘L sup 1 P’, (TI(: % SUP ((wll Q SUP 1 I;(?) u (10) 

T,<t TSt Tdt Tdt 

Here is an equivalence relation and Implies that the corresponding signals in 

(10) grow at the same rate [is]. 

(b). Since 6 e L it can be shown that ya(t) grovs at a. slower rate than 2 

sup Bw(T)~ denoted by ya(t) - o[sup M~(T) fl 1. 

T,<t T<t 

(c) From (5), (9-iii) and (11) it follows that 

z 

(d) Since v E L using equation f4) we conclude that 

sup\\ w(2) (T) 11 - o sup 11 W(T) 11 which contradicts (10). 

Tdt Tdt 

Hence all the signals in the system are uniformly bounded and lim el(t) = 0. 

t-w 

(11) 

180 
The importance of demonstrating the boundedness of b(t) and 4 E L 2 in the 

proof of ftability is worth noting. 

that lim O(t) = 0. which serves the same purpose.] The former assures that the 

relevant si nals in (10) grow at the same rate while the latter is used to prove 

that Iy (t)r andllw2(t)ll should grow at different rates if the adaptive control 

P 

is used, leadiag to a contradiction. 

[In some cases it may be possible to show 

tAsymptotic Stability of the Ideal System: Once the boundedness of all the signals 

in the adaptive system has been established, interest shifts to the convergence 

of the parc3eter vector 8(t) to its desired value 8* or equivalently of +(t) to 

the null vector. Since the adaptive law (8) can be represented as 

i(t) = 

the conditions that have to be imposed on r(t)*.to accomplish this is of interest. 

Following the results of Morgan and Narendra [lg] if s(t) is persistent41 + sT(t)stt) 

ly exciting lim $(t) * 0 and the convergence is exponential. Since Wm(s)Iu = s, 

ta sufficient condition for ~(0) to be persistently exciting is that u(*) is persistently exciting [lS]. Hence conditions under which w(*) will be persistently 

exciting have beell investigated by several authors [15,20-22]. 

Persistent Excitation (PE) of w(t) and w*(t): 

the parameter vector to the null vector were stated in ’rems of the PE of u(t). 

However since w(t) is a dependent variable within the adaptive loop, very little 

can be said directly about its persistent excitatfon. Hence attempts were made to 

express this condition in terms of the PE of signals in the model which are at the 

discretion of the designer. Since the adaptive system and wdel transfer functions 

are identical when e(t) E 8*, the model can be parametrized in such a fashion that 

a signal w* in it would correspond to the signal w(t) in the adaptive loop. Further since the model is time inverian , conditions on r(t) which would assure the 

PC of w*(t) can be derived. If u(t) w(t) - u*(t), the adaptive law assures that 

lim w(t) = 0. Hence, in the ideal case the PE of u*(t) ensures the PE of w(t) 

tand hence the convergence of the parameter vector 8(t) to its true value. 

Early results on the convergence of 

% 

Ctaments: 

(i) The above arguments have focussed attention on several interesting questioas related KO persistent excitation and transforPrations’.under which the property is preserved [15]. 

(ii) The convergence of z<t) tci 0 is used above to show the PE of w(t) and hence 

the convergence of $(t) to 0. This ia no longer possible when an external disturbance is present since even the boundedness of the signals is not assured in 

such a case. 

(iii) From the results of several authors it is now known that an almost periodic 

181 
reference input with n-distinct frequencies results in the PE of w(t). 

b) Adaptation Under Perturbations: 

tion (3a) assumed ideal conditions. 

The adaptive control system described in secThe plant was linear and time-invariant and 

no external disturbances were present. In addition, considerable prior knowledge 

of plant transfer characteristics vas assumed to help in setting up a reference 

model and deriving stable adaptive laws. As mentioned earlier, plants are rarely 

strictly linear or finite dimensional and in many practical situations the need 

for adaptive control arises due to large parameter variations. Also, external input and output disturbances are invariably present in real systems. 

is a definite need to extend the theory developed for the ideal case to situations 

with modeling errors and external disturbances. 

been proposed in recent years to achieve robustness in the presence of such perturbations are briefly reviewed in this section and sow new results are reported 

in sections 4 and 5. 

Hence there 

Some of the schemes that have 

The basic adaptive system in the ideal case is only uniformly stable. 

implies that bounded perturbations can theoretically produce unbounded outputs. 

When the reference input is persistently exciting, the nonlinear system is uniformly asymptotically stable in the large and exponentially stable when the initial state x . 

BIB0 results to be derived using theorems cf the type described by blkin, provided the perturbations are sufficiently small. However, as pointed out in section 2, very little can be said using such an approach about the effect of bounded 

perturbations of a specified aaxipupp amplitude on the global behavior of the solutions of the adaptive system. 

global methods have also been used to derive results in robust adaptive system. 

The principal concepts involved in deriving some of these are discussed below. 

This 

lies in 8 finite ball around the origin. The latter fact allows 0 

In addition to such perturbation methods a few 

(i) Use of Dead-Zone - IS]: The problem statement is 

ideal system with the exception that y 

turbance. Using the same adaptive law (8) as in the 

tions can be expressed as 

= c Tx + v 

P PP 1 

9T(t)r(t) + v(t) = €(t) 

at) = 

1 + sT(t)rs(t) 

and 

-I-& ( t 15 (t) 

similar to that given for the 

where v is a bounded disideal case, the error equa1 

(14) 

(adaptive law) 

where v is an equivalent output disturbance due to vl. 

to the presence of v(t) in (14). 

in the right direction. 

its desired value. 

is of the order of the vound v of v(t). The modification in the algorithm suggested in [5] is to use a dead-zone so that the adaptive parameters are not adjusted when c(t) lies inside it. 

a linear time-invariant mode when le(t)l 6 v + 6 (for some constant 6 > 0) and 

an adaptive mode otherwise. 

a system with bounded signals. 

time. 

Invariant system in a finite time after which the output error will lie entirely 

in the dead-zone and hence adaptation ceases entirely. 

The difficulty arises due 

Otherwise the parameter vector my be adjusted away from 

This implies that problems of convergence may arise when c(t) 

T T When sgn[+ 5 + v] = sgn[4 r;] the adaptation is 

0 

Hence the overall system operates in two modes-- 

In (51 it is sh8wn that such an algorithm results in 

Further, adaptation takes place for only a finite 

This implies that in practice the system will converge to a linear time182 
(ii) 

was taken by Kreisselmeier and Narendra [6). While the statement of the problem 

as well as the structure of the controller are identical to that in ($1, it is 

assumed that no knowledge of a bound on the disturbance is available. Instead, 

it is assumed that the desired vector e* has a norm less than a specified value 

Ile*ll max. 

S:{el llell c.11 e* llmx}. The adaptive law used to update e(t) is identical to that 

in the ideal case when 8 lies in the interior of S and is modified when it reaches 

the boundary of S, or lies outside it. In [a] it is shown that such a scheme results in the boundedness of all signals in the system. 

Bound on 1 e* u : An alternate approach to the bounded disturbance problem 

Hence the search procedure can be confined essentially to the set 

Apart from the obvious differences between the schemes suggested in [SI and 

[6], there are mathematical diiferencee in the proof that are worth stressing. 

As in [1-4], the proofs of stability in [5] use limiting arguments as t + 03 to 

show that 4 E L2. Such a procedure cannot be used in [6], since #(t) does not 

tend to any limit ?s t -t 0. 

behavior of the Fjsten over a finite interval. 

es in IS] and [6] complement each other and can be > ombined to have wider application in adapti\e systems in the future. 

Hence all arguments are based on the analysis of the 

As shown in sectton 6 the approach- 

(iii) The o-modification Scheme: In approaches (i) and (ii) certain prior information is assumed to implement the adaptive laws. In contrast to this, a scheme 

suggested by Ioannou and Kokotovic [7] assures boundednass of all signals in the 

system, without any asscmptions regarding the bounds on either the disturbance or 

the control parameters. However, to the authors' knowledge, the naethod has been 

shown to result in global boundedness only for the special case when the reference model is strictly positive real. 

The method is based on the following simple ideas. If V(e,$) is a quadratic 

Lyapunov function candidate, the time derivative *(e,$), along a trajectory, is 

generally a quadratic futction of e and hence is negat ve se definite. Whe a 

disturbance is rtesent, V(e,+) has the general form -e Qe + e av, where Q = 2, 0, 

a is 6 constant vector and v is a bounded disturbance. Very little can be conclL-led regarding stability from this and accounts for the modifications suggested 

in [5] and [6). In [7], an additional term -ae is used in the adaptive law, as a 

result of which t(e,$) becomes negative definite outside a bounded region in the 

(e,4) space. From this it is concluded that all signals in the system are bounded. 

*9 

(iv) The methods outlined in sections 

3b(i-iii) deal with the global behavior of the adaptive systems when bounded perturbations are present. 

adaptive control problems where the assumptions made regarding the system deviate 

slightly from the ideal. While [ 81 addresses primarily the problem of timevarying plant parameters the authors claim that the same methods with remarkably 

littlezhange allow examination of the effect of measurement noise, plant nonlinearity and undermodelling of the plant order. 

Adaptive Systems and Time-Varying Plants: 

In contrast to this Anderson and Johnstone [ 81 examine 

As in our discussions in section (3a), the authors first consider the ideal 

system and demonstrate uniform or exponential stability in the presence of persistent excitation. For the various types of perturbations considered,their aim 

is then to show that the resulting equations can be cast in such a form that the 

183 
total stability of the overall system can be demonstrated using modifications of 

Malkin's theorems. 

ily for establishing !he existence of robustness in the presence of sufficiently 

small perturbations rather than for providing guidance in the choice of the control input to assur: bomdedness of solutions when the class of perturbations is 

specified. 

iwwever, as mentioned earlier, the theorems are Gseful primar4. PERSISTENT EXCITATION AND ROBUSTNESS 

In the last section, we discussed two approaches of studying the robustnass 

problem in adaptive systems. The approach in 34 assumed that the Perturbations 

were sufficiently small and derived BIB0 results local in nature, using Malkin's 

theorem, vhereas in 3a-3c the approach was global in nature and used additional 

information regarding plant dynamics and the external perturbations. Also, the 

first approach made use of the PE of the reference input which vas not needed in 

the second. 

In this section, we present some new results which demonstrate global boundedness of all signals in the adaptive system in the presence of bounded disturbances'when the reference input is sufficiently persistently exciting. 

that by analyzing a set of nonlinear error differential equations, we can establish the global robustness behavior of the adaptive systems. In particular, it is 

shown that if the persistent excitation of the model output is larger than the 

disturbance, the solutions will be globally bounded and that if the maximum qlitude of the disturbance is greater than that of the model output, the system can 

have unbounded solutions. 

tive control of a first order plant and studying the corresponding second order 

nonlinear differential equations in detail. 

applicable to the general adaptive control probhm. 

We show 

The basic idea is stated here by considering the adapThe same methodology is 

Nonlinear Error Equations: 

ing reference model and the resulting error equations are as follows: 

The plant to be adaptively controlled, the correspondPlant: . +u+v; 

yP = aPYP = + 

Model : . 

Y, = -Ym + r 

. Error Equations: el = -el + $yp + v 

Adaptive Law: 4 = -elYp 

is the output ' el where r is the referfnce input, v is a bounded input disturbance 

1IYp m 

error defined as e 

when v(t) 0, by considering 

- y and .$ is the parameter error. In the ideal case, 

it immediately follows that the system is uniformly stable and if y (t) is persistently exciting, the system is exponentially stable. When a disturbance 

184 
v(t) is Present, it is tempting to proceed as in the ideal case and require y (t) 

nentially stable resulting in a bounded error vector for bounded perturbations. 

Since stability of the overall system has not been established, y (t) cannot be 

assumed to be bounded and proving that it is PE becomes speciousOP Hence we have 

to express the right hand side of (15) in terms of the model output y (t) which 

is an independent variable rather than the dependent variable y (t). %is restilts 

in (15) to be persistently exciting so that the unperturbed system is expo- P 

in the following nonlinear error differential equations: P 

By analyzing the above nonlinear differential equation, we demonstrate the global 

behavior of the adaptive system in the presence of v(t). 

The Ideal :ystem: In the absence of external: perturbations, the nonlinear system 

can be shown to be uniformly asymptotically stable in the large as follows: 

W(el,$) = 7 [el + 4 1, the time-derivative W[rl,$] = -el2 5 0. Hence the system 

e,(t) and $(t) are uniformly bounded for all t > t 

2T Since e E L and g1 is bounded, lim e,(t) = 0. 1 

can be considered to be the input to the linear part which is exponentially stable 

if y,(t) is PE. 

as t -+ 

ial time t and the magnitude of the initial conditions, the system is u.a.s.l. 

It is also worth noting that when y (t) is PE, the linear part of (18) is expoIf 

122 

if U[el(tO),$(tO)] < -. 

The nonlinear vector [Oe,,-e, ] 

t- 

+ 0 as t + m, this input tends to zero and hence x(t) + 0 

- 0' 2 

As e 

h 'T where x = [e,,$] . Since all the arguments ere independent of the init0 

m 

nentially stable but the nonlinear system is exponentially stable only when the 

initial state x(t ) lies in a finite ball around the origin aud not &loball;? exponent ial ly s t abl? . Perturbed System: 

tem, we shall discuss three cases where the perturbed nonlinear system (17) is 

autonomous. 

To provide some insight into the behavior of the nonlinear sysCase (i) y,(t) : 0: 

lim $(t) = -m and lim e,(t) = 0. 

',* tCase (ii) y (t) E ymx: m 

LaSalle's theorem, is u.a.s.1. since the largest invariant set in E = {xlei2 - 0) 

is the origin. However, since the system is nonlinear, it no longer follows that 

a bounded input will result in a bounded output. If, for example, v(t) 5 -v 

When v(t) E 0, thP system is uniformxy stable. If v(t) 5 vmX, 

The unforced system in this case is autonomous and, by 

max * 

185 
we can show that lirn e (t) = - and lim g(t) = -m. 

t- YIMX t-+oo 

where v 

Case (iii) ym(t) z y > vwx: The system is Lagrange stable. When 

v(t) 5 -vmax, the system has an equilibrium state at (0, - ) which is 

u.a.s.1. Similarly when v(t) E v the system has an equilibrium state at 

(0, - - ). 

max > Ymax, 

max 'Yrnax V 

lMX 

ymax 

IMX 

V 

IMX 

'max 

The above special cases reveal that the behavior of the nonlinear system is 

very much dependent on y and v In particular, when y (t) : y and 

v(t) : -v 

tions are bounded when y > v The results also carry over to the general max max. 

case when both v(t) and y (t) are time-varying and are stated in the following 

main theorem of the paper? 

Theorem I:Let ly (t) I 

citing signal in the sense described in [23]. 

T ,E 

with [t2*t2 + 60]c [t ,t + To] and - Jt2+6hm(r)drl 

(a) 

UX max m WX 

the system has unbounded solutions when v and all solumax * max >ymax 

(Fig. 1) 

Iv(t)l vmax and y (t) be a smooth persistently ex- 

> 0, there exists a t2 E [tlstl + To], 

m I Yma,' m 

This implies that positive numbers 

c0. Then 

and 6 exist such that given any t 00 0 

To t2 

= sgn(el(t))vmax lepl < Ymax 

11 

If Ymx < Vmx* by choosing an input v(t) as 

v(t) = -sgra(ym(t))vmx when (el(t)l 2 ymaX 

A a(t) when a(t) f 0 and is equal to unity when a(t) = 0, there 

1 

= 1m- where sgn(a(t)) 

exist initial conditions for which llm $(t) = -00 and e (t) approaches asymptotically the region le 1 

(b) If cO > v 

t-- 

+ E, where E is an arbitrary positive constant. 1 2 ymax 

+ 6 where 6 is any arbitrary positive constant, then all the 

UlaX 

solutions of the differential equation (17) are bounded. 

Proof : 

a) Let D1 be the open domain enclosed by the line el - -v and the curve 

mAx 

e +v 

with 4 < 0. When y,(t) 2 y,, and v(t) ,= -vmX all solutions 1 mi;x 

4- - 

el + ymax 

that start on the boundary a(D ) enter D Since the system is autonomous and 

contains no singularities in D,,all solutions originating D are unbounded and 

1 1' 

I 1 

ymax. 

lirn $(t) - -00~ lim el(t) = - 

t* tFor a time-varying signal y (t) the proof of unboundedness is related to the 

above autonomous case. Consider*the solution of the differential equation with 

186 
UUX , with y,(t) E ymx and v(t) E -v 

LMX' 

initial condftion (O,$o) with O0 < 

YUUX 

Let r denote the open curve along which the trajectory lies for all t 3 0. 

larly let r- denote the curve along which the solution lies for all t 3 0 when 

v(t) = v and y(t) = - Let r($o) = r+ u r-. 

two open regions D and D where (Os$) E D if I$ < $ Then all solutions of the 

differential equation with Iy (t)l & ymX and Iv<t>l s vmax with initial conditions m 

on r($ ) lie either in I.($ ) or enter D 

solutions are unbounded and lim g(t) = -0. 

Simi- + 

l-($ ) divides the plane into maX Ymax' 0 

C 

2 2 2 0' 

Since this is true for every 4 the 0 0 2' 0' 

A T b) Let x = [el,$l. 

- 

2 tLet D denote the region in R WxI lel/ < v-1 and let - m 

DC denote the complement of D. 

a trajectory is i(x> = -e * + e v < 0 for x E DC. 

can increase only in E. 

If W(x) = 1/2 x'x, the time derivative of W along 

C Hence IIx Ildecreases in D and 1 1 

Ue wish to show that .d constant c exists so that if 

Ilx(t,jll = c1 7 c over an interval [t ,t + T Is thenllx(tO + To)(I < cl. - 00 0 

If llx(toYI = cos integrating the equation for 6 in (17) it $an be shown that 

maX 

1 

C if x(tO) E D, then x(t ) E D for eome t E [tO,tO + To] if co > - , where 1 1 sin0 

cot e = 2[T + . Hence under the conditions specified in the theorem, the 

trajectory invariably enters D during every period T 

monScot.ially, the trajectory can be made to lie in a subdonain of DC for a finite 

time A with 0 < A < 6 

this subdomain, a constant c > c 

(Eo-Vmx)T 

C By increasing Ilx(tO)ll 0' 

over every period. 

0 

Sinceqx(t)fl decays exponentially in 0 exists satisfying the conditions of the theorem. 

-- Comments:l.The positive limit set of any solution x(t) lies in D. 

2. 

the solutions are bounded if 

To, and 6 

E~ will be referred to as the degree of persistent excitation. 

> vmax but the nature of the limit 

0' 

By the theorem, 

set depends on 

3. From the theorem it follows that for a given bound v on the perturbations, 

the system can be made robust by increasing the degree - of persistent excitation. Note that this is an example of practical stability. 

4. The conditions €or boundedness and unboundedness of solutions are given in 

this case in terms of y,(t). 

press them in terns of 

For design purposes it is more appropriatn to exthe reference input r(t). 

-5. HYBRID ADAPTIVE CONTROL 

In continuous adaptive systems of the type described in the previous sections, 

Recent advances in microprocessor and related digital computer techthe plant operates in continuous time and the cmtroller parameters are adjusted 

continuously. 

nology favor the use of discrete systems in which signals are defined at discrete 

187 
instants. 

tinuous elements. 

report [lo] the authors have developed analytical models of hybrid ystems in 

which control parameters are adjusted in discrete time even as the continuous 

plant signals are processed in real time. The same algorithms can also be extended to control disciete time plants so that the overall discrete system operates 

on two time scales - a fast time scale in which the system operates and a slow 

time scale in which the control parameters are updated. We shall refer to such a 

system as a discrete hybrid system. 

Practical systems on the other hand may contain both discrete and conSuch systems may be described as hybrid systens. In a recent 

In this section we describe briefly one of the hybrid adaptive algorithms and 

demonstrate global stability in the ideal case of an adaptive system which uses 

such an algorithm. 

when bounded external disturbances are present. 

seGtioc, arguments are put forward as to why hybrid schemes should result in more 

rcbust system and simulation results are presented to show that this is indeed 

the case. 

The behavior of a discrete hybrid system is then discussed 

Using the results of the previous 

a) Hybrid Error Model: 

error mcdels giver; in [lo] and discuss its properties. 

be derived in all the other cases. 

In this section we consider the first of several hybrid 

Similar results can also 

The error model is described by the equation ' 

kEN 

+m +1 where u: fi -P P , el: E! -+Ut 

red to as the input and output functions of the error model. 

ally increasing unbounded sequence i.n @t with 0 < TmLn & Tk 

where T = t 

od. $I:B +p is a piecewise constant function, referred to as the parameter 

error vector and assumes values $(t) - 4k, t r [tk,tk++l), where 4, is a constant 

vector. 

are piecewise continuous fuvztions which are referfor k E: N 

{t } is a monotonic- k + 

Tmax < 

- t When Tk = T, a constant, we shall call T the sampling peri- k + k+& k' 

It is assumed that 0 (and hence + ) is unknown, the values u(t) and tl(t) 

can be adjusted at 4,, - 'k 

The objective is to determine an adaptive law for choosing the sequence 

4 0 k 

can be observed at every instant t and AQk 

{A$k} using all available input-output data so that lim e (t) = C. 

Theorem 2:If in the error equation (19) the vector 0, is updated according to the 

adaptive law 

= tk+l' 

1 t- I 

I tk+l e,(.r)u(.c) 

- dT (20) -1 *" = % It, 1 + uT(r)u(r) 

then 

(i) 

(ii) 

+ if u(t) and G(t\ are sirormly bounded inP lim elit) = 0 

cif in add+.tion to the conditions in (i) vi ils persistently exciting over an 

interval T lim @k = 0 min' k188 
T 112 2 

(iii) If u E L: then e,(t) = p(t)[l + u u] , p E L . IT 1T _1_ Proof: If V(k) = y I)k Ok, using the adaptive law (20) we obtain AV(k) - - 7 I)k 

[21 - ppk 2 0 

where dr * 

Hgnce V(k) is a Lyapunov function and assures the boundedness of 4 

k=l 

Since k' 

E AV(k) < - it follows that lim AV(k) = 0. Henc? 

T 

el (TI 

dt = 0. 

2 

le,, k-m T lim +k \Ok = lim - 

k- k- Tk tk 1 + u (T)u(T) 

(i) If u is bounded, e is bounded and e E f . If is bounded lim e,(t) = 0. 

(ii) 

(iii) ,If u grows in an unbounded fashion with u E Le, el = pm where p E f2. 

Comments: In the three cases given in theorem 2 the first two assume that the 

input u is uniformly bounded and the corresponding results are applicable 

identification problem. 

able to the control problem. 

t-m 

1 1 

If u is persistently exciting \ is uniformly positive definite and hence 

-t 0 as k 3 a. 

m 

'k 

1 the 

The third case which treats unbounded inputs is applicThe fact that T need not be a constant is also worth noting. As shown in section 

6 a time-varying period may be used to improve the transient response of the system. 

k 

b) The hybrid adaptive algorithm 

described in the preceding section can be used to adjust the control parameters 

of a hybrid adaptive system. 

tion 3 for a continuoas time system the overall system can be shown to be globally 

stable, 

Stable Hybrid Adaptive Control - Ideal Case: 

Using an approach very similar to that used in secUsing the same notation as in section 3 we have for the adaptive law 

- rtk+l cl(?)S(r) 

A+k = -- dT 

Tk Jtk 1 + cT(~><(r) 

From the analysis in the previous section we conclude that 

(i) the parameter error vector is bounded k 

T 2 and (ii) E = pJI + g 5 where p B L , 1 

whi-h conditions are the same as those o ined for the continuous case. Condition (I) assures that: the signals y 

rate. 

vious assertion proving the boundednevs of all 

85v, Ilu(t)ll and 1 q(t)l grow at the same 

P). 

Condition (ii) results in Iy (t) I = o ~zrll w(r)ll which coiitradicts the pre- P 

the signals. 

The similarity between the continuous and hybrid syst?ms also extends to 

cases when external bounded disturbances are present and the methods described in 

I89 
sections 3 and 4 apply to the hybrid case as well, 

lowing section, the use of averaged values over an interval rather than instantaneous values, results in nore robust coutrol. 

However, as shown in the folc) Adaptive System with Two Time Scales: 

in section 5a and applied to hybrid adaptive systems in section 5b can also be 

SI' 

-- The hybrid adaptive algorithm developed 

. modified for discrete hybrid systems or discrete systems with two time 

' .es, shown below. 

1 Let the output error e (k) E B and the paranreter error 1 reldted by tne error equation 

T 

$ (k)w(a) = el(L) k,& E N, a ~[kT,(k+l)Tl 

where $(k) is a constant vector aver the interval [kT,(k+l)T], T E N and denotes 

the period of the interval and w(a) cBn is an input vector, 

collected over the entire interval, the parameter error vector $(k) is updated at 

time (k+l)T using the adaptive law 

Using information 

(k+l)T-1 e,(i)v(i) 

c A $(k+l) - $(k) = A$(k) - 1/T 

i=kT 1 + v(i)Tw(i) 

.. 

w( i>WWT (k+l)T-1 

where R(k) 1/T C I- . kT 1 + w(l)'w(i) 

T In [lo] it is shown that V(k) = 1/2 $ (k)+(k) is a Lyapunov function for the 

system (21) from which it follows that $(k) Is bounded if $(O) is bounded and 

ep) 

IEN (23) 1/2 = O i- [l+w(i)Tw(i) I 

1 im 

If the adaptive law (22) is used in a control. system to update the parameters, 

equation (23) can be used to demcnstrate global stability [lo]. 

When an external disturbance v is 

modified as 

T 0 (k)w(k) + v(a> = el(&) 

Using the same adaptive law as before, 

(k+l) T-1 

A$(k) -R(k)$(k) + Z 

i=kT 

present the error equation (23) have to be 

the error equation has the form 

w(i)v(i) 

1 + w(iITw(i) 

(k+l)T-l w(i)v(i) 

where s(k) = c ,F . A 

i=kT 1 + w(i)'w(i) 

The matrix R(k) and the vector 8(k) in algorithm (25) are averaged values 

over an interval rather than instantaneous values. 

may be considered to have more persistently exciting inputs in its homogeneous 

Hence the equivalent syscem 

190 
equation and a smaller magnitude of perturbation (if the mean value of the disturbance is small). 

tion results shown in Fig. 2 indicate the dramatic improvement in performance. 

Due to both reasons the outputs tend to be smaller. Simula6. NEW DIRECTIONS 

The criteria for judging the perforsance of an adaptive control system are 

In the preceding sections methods using 

no different from those used for any conventional control system and include stability speed and accuracy of response. 

persistent excitation of reference input, and nonlinear and hybrid adaptive algorithms were described which would make the overall system stable under perturbations. A judicious combination of these different methods may improve the robustness of the system substantially and result in schemes which are practically attractive. Some of these combinations as well as extensions of known methods which 

appear promising are given below. 

(i) A detailed analysis of a first order adaptive system containing a single control parameter was 

given in section 4. When a disturbance is present it was shown that a sufficiently 

large persistently exciting reference input would also result in bounded solutions. 

Further 

higher order systems and research is currently being done to determine the bounds 

on the sol-t ions 

(ii) Hybrid Adaptive Control: 

tion 5, it was shown that the sampling incerval T,. could itself be time-varying 

Robustness of nth Order System Using Perdstent Excitation: 

studies have revealed that siailar c-nclusions can be drawn regarding 

- In the adaptive control system described in sec- -. 

provided it lay in a bounded interval [Tmin,TmaxIKwith Tmin > 0. In practical 

systems it Pppears possible to adjust T on line to improve the transient response 

of the 7 n. 

k 

(iii) tmd-Zone, Persistent Excitation and Plant Identification: A sufiiciently 

large deLd-zone in the adaptive algorithm was shown to result in bounded solutions 

in section 3. The results in section 4 indicated that boundednesa of solutions 

could also be achieved by increasing the PE of the reference input. 

appears likely that the same results can be achieved using a combinatioq of a 

smaller dead-zone and a smaller degree of persist, .: excitation. 

dies have shrwn that this le indeed the case and a:,empts are being made to demnstrate this theoretically. 

It therefore 

Simulation stuWhen the reference input is persiatently exciting and the adaptive loop is 

stable, the plant paraineters can be estimated on-line and used in second level 

adaptation to reduce the dead-zone further. 

of reference inputs appears to be of boLh theoretical and practical interest. 

Hence combining a dead-zone with PE 

Ile*II maX and Persistent Excitation: As in (iii) a persistently exciting input enables e* to be estimated and hence an attempt could be made to use tha information to decrease the region of search. 

(v) a-mcdification and Persistent Excitation: The a-modification sclleme, in its 

basic form, described in sectiou 3 is unappealing, since the parameter error can 

be large if IIe*Jl is large. Using identification methods as in (iii) and (iv) and 

estimating t3* on line, second level adaptive procedures may result in a smaller 

bias 

19 1 
The second level adaptation problems stated in (ii)-(v) while practically attractive, lead to stability questions in more complex nonlinear systems. Further, 

it is worth pointing out that all of them consider external disturbances rather 

than perturbations in plant dynamics. 

the design of a low order controller to adaptively control a higher order plant is 

generally agreed to be the single most important theoretical question in the field 

of adaptive cmtrol. 

area, it is acknowledged that even a proper formulation of this problem is a formidable one. 

section will contribute significantly towards this end. 

The reduced order problem which deals with 

While considerable research is being carried out in this 

It is felt that the answers to some of the questions raised in this 

Acknowled pen t 

The work reported here was supported by the National Science Foundation under 

grant ECS-8300223. 

REFERENCES .. 

1. K.S. Narendra, Y.H. Lin and L.S. 

- Part 11: Proof of Stability," 

448, Tune 1980. 

2. A.S. Horse, "Global- Stability of 

Trans. on Autom. Control, AC-25, 

Valavani, "Stable Adaptive Controller Design 

IEEE Trans. on Autom. Control, AC-25, pp. 440- 

Parameter Adaptive Control Systems," - IEEE 

pp. 433-439, June 1980. 

3. K.S. .Narendra and Y.H. Lin, "Stable Discrete Adaptive Control," IEEE Trans. 

on Autom. Control, - AC-25, pp. 456-461, June 1980. 

G.C. Goodwin, P.J. Ramadge and P.E. Caines, "Discrete Time Multivariable Adaptive Control," IEEE Trans. on Autom. Control, AC-25, pp. 449-456, June 1980. 

B.B. Peterson and K.S. Narendra, "Bounded Error Adaptive Control," IEEE Trans. 

on Autom. Control, AC-27, pp. 1161-1168, Dec. 1582. 

G. Kreisselmeier and K.S. Narendra, "Stable Model Reference Adaptive Control 

In the Presence of Bounded Disturbances," IEEE Tranu. on Autom. Control, 

AC-27, pp. 1169-1175, Dec. 1982. 

4. 

5. -- 

6. 

7. P.A. Ioannou and P V. Kokotovlc, "Robust Redesign of Adaptive Control," IEEE 

Trans. on Autom. Con:rol, AC-29, pp. 202-211, March 1984. -. 

8. B.D.O. Anderson and R.M. Johnstone, "Adaptive System and Time Varying Plants," 

Int. Journal of Control, vol. 37, No. 2, pp. 367-377. 1983. 

., 

9. B.D.O. Andei son and R.M. Johnstone, "Robust Lyapunov Stability Results and 

Adaptive Systems," Pro:. 20th IEEE CDC, San Diego, pp. 510-515, Dec. 1981. 

K.S. Narendra, I.H. Khalifa and A.M. Annaewamy, "Error Mcidelr; for Stable Hybrid Adaptive. Systems - Part 11," Tech. Report No. 8404, Center for Systems 

Science, Yale Univemity, New Haven, Ct., Fcb. 1984 (To be published in IEEE 

10. . 

192 
Transactions on Automatic Control). 

11. W. Hahn, Theory and Application of Lyapunov's Direct Metnod with Applications, Academic Press, New York, 1961 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

J.P. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method with 

Applications, Academic Press, New York, 1961. 

Krasovskii, Stability of Motion, Stanford University Press, Stanford, 1963. 

C.A. Desoer, R. Liu and L.V. Auth. Jr.. "Linearizy VS. Nonlinearity and 

Asymptotic Stability in the Large," IEEE Trans. on Circuit Theory,-pp. 117- 

118, Harch 1965. 

K.S. Narendra and A.M. Annaswamy, "Persistent Excitation and Robust Adaptive 

Systems," Tech. Report No. 8302, Center for Systems Science, Yale University, 

New Haven, Ct., Aug. 1983. *. 

K.S. Narendra and Y.H. Lin. "Design - of Stable Model Reference Adaptive Controllers," Applications of Adapti8e Control, pp. 69-130, Academic Press, New 

York, 1980. 

K.S. Narendra and L.S. Valavani, "Stable Adaptive Controller Design-Direct 

Control," IEEE Trans. on Autom. Control, AC-23, pp. 570-583, Aug. 1978. 

K.S. Narendra, A.M. Annaswamy and R.P. Singh, "A General Approach to the Stability Analysis of Adaptive Systems,'' Tech. Report No. 8402, Center for Sys- 

+ems Science, Yale University, New Haven, Ct., Jan. 1984 (To be published in 

International Journal of Control). 

A.P. Morgan and K.S. Narendra, "On the Uniform Asymptotic Stability of Certain Linear Nonactonomous Differential Equations," SIAM Journal of Control 

and Opti-ization, vol. 15, pp. 5-24, Jalz. 1977. 

20. B.D.O. Anderson and C. Richard Johnson, Jr., "Exponential Convergence of Adaptive Identifi. ation and Control Algorithms," Automatica, - vole 18, pp. 1-13, 

1982. 

21. S. Dasgupta, B.D.O. Anderson and A.C. Tsoi, "Input Conditions for Continuous 

Time Adaptive Systems Problem," Proc. of the 22nd IEEE CDC, San Antonio, 

Dec. 1983. 

22. S. Boyd and S.S. Sastry, "On Parameter Convergence in Adaptive Control," Tech. 

Report, June 1983. 

23. A.P. Morgaq and K.S. Narendra, "On the Stability of Nonautonomous Differentia; 

Equations x - [A + B(t)]x with Skew-Symmetric Matti:= B(t)," SIAn Journal of 

Control and Optimization, vol. 15, pp. 163-176, Jan. 1971. 

193 
Y, * 2.0 v--6.0 ? 

2 4 

'1 

6 8 10 12 

* 

( c) 

g1 - -el + 4Cel + y,) + v 

i = -el(el + YJ 

Fig. 1: Persistent Excitation 

and Xobustnees. 

(d) 

(a) autonomous case I ym I c Iv I unstable 

(b) autonomous case I yml > I v 1 stable 

(c) non-autonomous case y < v unstable 

(d) non-autonormus case E > v stable 

majc m&x 

3 max 

194 
2.00 - 

e1 k 

-1.00- 

2.m} 

’p I 

Jtr) - I.W.W).ds(.WI 

rig. L 

Fig. 2: Improved Robustness Using Hybrid Control 

i95 
MAXIMUM LIKELIHOOD ESTIMATION WITH 

EMPHASIS ON AIRCRAFI' FLIGHT DATA 

K.W.IliiraadR.E.W 

NASA Ames Rtscarctr Center 

Dryden Flight Rtswch Facility 

Edwuds. CA 93523 

Accurate rodeling of flexible space structures is an important field that is 

currently under investigation. Parameter estimation, using rethods such aa MXimm likelihood, is one of the ways that the aodel can be inpzm?ed. The ~dau 

likelihood estimator has been used to extract stability and control derivatives 

from flight data fox many years. Hoet of the literature on aircraft estimation 

concentrates on new developments and applicatitma, assuring familiarity with 

basic estimation concepts. This paper presents sow of these basic concep+s. 

me pawr briefly discusses the maxinu likelihood estimator and the aircraft 

equations of motion thut the estimator uses. 

and estimation are exadned for a simple computed aircraft example. The coat 

functions that are to be riniaized during estimation are defined and discussed. 

Graphic representations of the coat functions are given to help illustrate the 

minimization prxess. Finally, the basic concepts are generalized, and estimation from flight data is discusaed. Specific examples of estimation of structural dynamics are included. 

example are also developed for the analysis of flight data. 

'Iha basic concepts of dnidzation 

Sou of the major conclu8iona for the computed 

INTRODUCTION 

Accurate modeling of flexible space structures is an inportant area that is 

Such techniques have been succe88- 

currently under investigation. 

can be improved using parameter estimation. 

fully used to estinate aircraft stability and control derivatives and refine 

aircraft mathematical models. Sore of the experience gained in the aircraft 

problem can be applied directly to analysis of flexible space structures. 

The mathematical waling of these structures 

The maximum likelihood estimator has been used to &thin stability and control estimates from flight data for nearly 20 yearcr. 

cations have been reported worldwide. Reference 1 contains u representative 

list of some of these reports. Several good texts (includifig Refs, 2 and 3) 

contain thorough treatments of the theory of .axinurn likelihood estimation. 

Experience reports (Refs. 1, 4, and 5) poicting out practical considerations for 

The results of many appli- 

*Senior Staff scientist 

*Aerospace Engineer . ?REC:EDWG PAGE BLAXK b?OT F!LIV,n 

197 
applying the maximum likelihood estimator ham also been published. 

and control derivatives eatimted from flight data are currently required for 

correlation studies with predictive techniques, handling qualities ductmentation, 

design compliance, aircraft sirulatx enhancement and refinement, and control 

system design. Correlation, sinulation, and control sy~ta~ design epplicatiom 

(including the space shuttle) are dircusaed in Ref. 6. Current studies have 

concentrated on estimation model structure deternination (Refs. 7 and 81, equation error with state reconstruction (Refs. 9 to 111, and maximum likelihood 

estimation in the frequency domain (Refs. 12 and 13). 

Stability 

Most of the reporta in the estimation area concentrate on new developments 

and applications, assuming familiarity with the basic concepts of maxinu. likelihood estimation. In this paper s’- of these basic concepts are reviewed, 

concentrating on simple, idealized models. These simple models provide insights 

applicable to a wide variety OE real problems. 

This paper also presents sone of the basics of maximu likelihood estimation 

as applied to the aircraft probltr. It briefly discusses the maximar likelihood 

estimator and the aircraft equations of rotion that the sstimtor uses. 

basic aspects of minimization and twithation are then examined in detail for a 

simple computed aircraft example. Finally, the discussion is expanded to the 

general aircraft estimation probler including specific examples of estimation of 

structural dynamics. 

me 

system matrices 

normal acceleration positive upward, g 

longitudinal acceleration, g 

lateral acceleration, g 

normal acceleration poritive upward, g 

reference span, ft 

coefficient of rolling mmnt 

coefficient of yawing moment 

coefficient of axial force 

coefficient of side force 

coefficient of normal force 

general functions 
GG* 

g 

H 

I, Iyr I,, I,, 

i 

J 

KP 

L 

L' 

LYJ ' 

n 

m 

N 

n 

fig 

P 

9 

9 

R 

r 

- 

S 

1 

t 

U 

V 

measurement noise covariance matrix 

acceleration due to gravity, f t/sec2 

approximation to the information matrix 

moment of inertia but subscripted aria, slug-ft2 

general index 

cost function 

sidewash factcr 

rolling moment divided by I,, deg/mec2 

rolling momeat, ft-lb 

rolling moment due to yaw jet, ft-lb 

pitching moment divided by fy, deg/(1ec2 

Inass, slug 

number of time points or cases or yawing moment divided 

by IZ, deg/sac2 

state noise vector or nunber of unknowns 

estimated roll rate due to turbulence, d*g/sec 

roll rate, deg/sec 

pitch rate, deg/sec 

dynamic pressure, lb/ft2 

innovat! rn covariance matrix 

yaw : te, -g/sec 

reference area, ft2 

time increment, aac 

time, sec 

control input vector 

forward velocity, ft/sec 

199 
X state vector 

z 

a 

A 

6 

5 

Q 

T 

9 

w 

Subscripts: 

0 

m 

distance between lateral accelerometer and the 

center ol gravity along the appropriate axis, ft 

observation vector 

predicted Kalman-filtered estimate 

angle of attack, deg 

angle af sideslip, deg 

estimated angle of sideslip due to turbulence, t2eg 

time sample interval, sec.. 

contrGi deflection, deg 

aileron deflection, deg 

elevon deflection, deg 

rudder deflection, deg 

measurement noise vector 

pitch angle, deg 

mean 

vector of unknown8 

standard deviation 

time, sec 

transition matrix or bank angle, dag 

integral of transition matrix, or heading 

f requency, rad/sec 

angle, dag 

partial derivativs with respect to subscripted quantity 

bias or at time zero 

measured TGantity 

200 
Other nomenclature: 

.., predicted estimate 

* estimate 

transpose 

I indicates moment in ft-lb 

MAXIMUM LIKELIHOOD ESTIMATION 

The concept of maximum likelihood is discussed in this section. First the 

In 

general heuristic problem is dh~~~saed, and then the specific equations for 

obtaining maximum 1 ikelihood estimates for the 'aircraft problem are given. 

the following sections, both the concepts and the computations involved in a 

simple tut realistic example are discussed in detail. 

The aircraft parameter estimation problem can be defined quite simply in 

general terms. The system investigated is assvmed to be modeled by a set of 

dynamic equations containing unknown parameters. To determine the values of the 

unknown parameters, the system is excited by a suitable input, and the input and 

actual system response are measured. The valuas of the unknown parameters are 

then inferred based on the requirement that the model response to the given 

input match the actual system response. 

problem of identifying the unknown parameters can be easily solved by many 

methods; however, complicating factors arise when application to a real system 

is considered. 

When formulated in this manner, the 

The first complication results from the impossibility of obtaining perfect 

measurements of the response of any real mystem. The inevitable sensor errors 

are usually included as additive measurement noise in the dynamic del. Once 

this noise is introduced, the theoretical nature of the problem changes drastically. It is no longer possible tc exactly identify the values of the unknam 

parameters) instead, the values rust be estimated by some statistical criterion. 

The theory of estimation in the presence of waeurentent noise im relatively 

straightforward for a system with discrete time ob~ervations, requiring only 

basic probability. 

The second complication of real syster~r is the presence of state noire. 

State noise is random excitation of the system from unmeasured sources, the 

standard example for the aircraft stability and control problem being 

atmospheric turbulence. If state noise io prerent and measurement noise is 

neglected, the analyeir results in the regreemion algorithm. 

When both state and measurement noise are considered, the problem is more 

complex than in the cases that have only state noise or oirly measurement noise. 

Reference 14 develops a mixed continuoue/diecrete maximum likelihood formulation 

that allows for both etate and measurement noise. This formulation has a continuous system model with discrete aampled observations. 

20 1 
The final problem for real systems is modeling. It has been assumed throughout the above discussion that for so10 value (called the "correct" value) of the 

unknown parameter vector, the system is correctly described by the dynamic model. 

Physical systems are seldom described exactly by simple dynamic models, so the 

question of modeling error arises. 

available. The most common approach is to ignore it: Any modeling error is 

simply treated as state noise or measurement noise, or both8 in spite of the 

fact that the modeling error may be deterministic rather than random. 

assumed noise statistics can then be adjusted to include the contribution of 

the modeling error. This procedure is not rigorously justifiable, but, combined 

with a carefully chosen model, it is probably the best approach available. 

No comprehensive theory of modeling error .is 

The 

With the above discussion in mind, it is psible to make a -re precise, 

mathematically probabilistic statement of the parameter estimation problem. The 

first step is to define the general system node1 (aircraft equations of motion). 

This model can be written in the continuous/discrete form as 

where x is the state vector, z is the observation vector, f an3 g are system 

state and observation functions, u is the known coritrol input vector, 4 is the 

unknown parameter vector, n is the state noiae vector, and q is the measurement 

noise vector. The state noiae vector is assumed to be zero-mean white Gaussian 

and stationary, and the measurement noise mctor is assumed to be a sequence of 

independent Gaussian randop variables with zero mean and identity covariance. 

For each possible estimate of the unknown parameters, a probability that the 

aircraft response time histories attain values near the observed values can then 

be defined. The maximum likelihood estimates are defined as thoee that maximize 

this probability. 

characteristics; for example, it yields asymptotically unbiased, consistent, and 

efficient estimates (Ref. 15). 

Maximum likelihood eatimaMon has many desirable Statistical 

If there is no state noise and the matrix G is known, then the maximum 

likelihood estimator minimizes the cost function 

where GGf is the measurement noise covariance matrix, ad z~(ti) is the computed 

response estimate of z at ti for a given value of the unknown parameter vector 

6. The cost function is a function of the difference between the measured and 

computed time histories . 

202 
If Eqs. (2) and (3) are linearized (as ie the case for the stability and 

control derivatives in the aircraft problem), 

N 

For the no-state-noise case, the zE(tj.1 term of 4. (4) can be approximated by 

where 

When state noise is important, the nonlinear form of Eqs. (1 1 to (3) is 

intractable. For the linear model defined by Eqs. (5) to (71, the cost function 

that accounts for state mise is 

where R is the innovation covariance matrix. 

is the Kalman-filtered estimate of z, which, if the state noise covariance 

is zero, reduces to the form of Eq. (4). If there is no state noise, 

the second term of Eq. (11) is of no consequence (unless one wiuhes to include 

elementa of the G matrix as unknms), and R can be replaced by GGf which makes 

Eq. (11) the same as 4. (4). 

The ;((ti) term in Eq. (119 

To minimize the cost function J(F)# we can apply the Newton-Raphaon 

algorithm which chooses succeesive estisttes of the vector of unknown coefficients, i. 

obtained from the L estimate as follows: 

Let L be the iteration number. The L + i estimate of is then 

203 
The first and second gradients are defined as 

The Gauss-Newton approximation to the second gradient is 

The Gauss-Newton approximation, which is sometimes referred to as modified 

Newton-Raphson, is computationally much easier than the Newton-Raphson approximation because the second gradient of the innovation never needs to be calculated. In addition, it can have the advantage of speeding the convergence of 

the algorithm, as is discussed in the SIMPLE AIRCW EXAHPLE section. 

Figure 1 illustrates the maximum likelihood estimation concept. The measured response of the aircraft is compared with the estimated response, and the 

~ifference between these responses is called the response error. The cost functions of 4s. (4) and (11) include this response error. The Gauss-Newton computational algorithm is used to find the coefficient values that maximize the 

cost function. Each iteration of this algorithm provides a new estimate of the 

unknown coefficients on the basis of the responae error. These new estimates of 

the coefficients are then used to update the mathematical mode1 of the aircraft, 

providing a new estimated response and, therefore, a new response error. The 

updating of the mathematical de1 continues iterati aly until a convergence 

criterion is satisfied. The estimates resulting from this procedure are the 

maximum likelihood estimates. 

The maximum likelihoc? estimator also provides a measure of the reliability 

of each estimate based on the information obtained from each dynamic maneuver. 

This measure of the reliability, ana!.~o\m to the standarq deviation, is called 

the Cram&-Rao bound (Ref. 16) or the uwerrtainty level. The Crambr-Rao bound 

as computed by current programs should generally be used as a measure of relative accuracy ra4&er than absolute accuracy. Tha bound is obtained from the 

approximation of the information matrix, H. This matrix equals the approximation to the second gradient given by Eq. (14b). The bound for etch unknown is 

the square root of the corresponding diagonal element of H. That is, for 

the ith unknown, the Crambr-Rao bound is m. 

204 
The Maine-Iliff formulation (Ref. 14) and minimization algorithm discussed 

The program and computational algorithms are described fully 

above are implemented with the Iliff-Maine code (MMLE3 maximum likelihood estimation program). 

in Ref. 17. All the computations shown and described in the remainder of the 

paper use the algo-rithm exactly as described in Ref. 17. 

AIRCRAFT EQUATIONS OF MOTION 

For the disr:ussion tnat follows in later sections of this paper, some ' 7wledge of the airxaft equations of motion is assumed. To clarify am of tr~a. 

discussion, the aircraft equations are discussed briefly in this section. 

rirst, tire axis system on which the aircraft equatims of motion are baaed 

is discuas:d. Figure 2(a) shows th? aircraft reference .&dy-axia system and the 

conventio.ia1 control rvfaces. The origin of the body-axis system is at the 

center of gravity. 

right-hand rule with the x-axis defined a8 positive forward on the aircraft. 

The longitudinal accel.eration (a,) and nondimensional axial force coefficient. 

(Cx) are defined along this axis, and the roll rate (p) and rolling moment (L') 

are defined about this axis. The y-axis is defined as positive out the right 

wing. 

(Cy) are defined along this axis, and the pitch rate (9) and pitching moment 

(H') are defined about this axis. The z-axis is defined as poeitive out the 

bottom of the aircraft. 

force coefficient (Cz) are r'afined along this axis, and the yaw rate (r) and 

yawing moment (N'j are defined about this axis* '=fie normal acceleration is 

sometimes defined 3s positive upward but is then referred to as aN. The three 

,omenta (L', M', and N') are usually divided by the corresponding moments of 

inertia (Ix, Iy, and Iz!, cnd are then referred to without the prime as L, M, 

and N. The83 qSAantities are nondimensionalized ((24, G, arld Cn, respectively) 

for use in the eq.:ations of motion soon to be discussed. The primary control 

about th. roll aiis (x-axis) is the aileron (da), about the pitch axis (y-axis) 

is the ele rator (tSe), and about the yaw axis (z-axis) is the rudder (dr). Some 

aircraft have other controls, but in this paper these vi11 only be defined where 

they are discussed (the reac'iion control jets on the space shuttls, for example). 

The sign convention for'this axis system is defined by the 

The lateral acceleration (a,) and nondimerlbional side force coeffichnt 

The normal acceleration (a,) and nondimensionai normal 

The Euler angles +, 8, and $ define +he aircraft attitude with respect to 

the earth. 

to the aircraft reference body-axis system of Pig. 2(a). . The order of rotation 

must be atout the z-axis (jr), then the y-axis (81, and finally the x-axis (+I 

for the aircraft equations of motion that will be written mbrequently. 

These angles define the rotations which transform earth-fixed axes 

For etabilit-* aad control analysis, the velocity of the aircraft with 

respect to the air (not with respect to the earth) is of primary interest. 

Figure 2(b) shows We relationship between the aircraft axis system pnd the flow 

angles. 

flow anqle in the x-y plane is the angle of sideslip (8). 

The flow angle in the x-z plane is the angle of attack (a), and the 

A more rigorous and 

20 5 
detailed definition is required for the derivation of the equations of motion, 

but the above definitions are sufficient to define the following equation of 

motion. 

Generalized nonlinear equations of motion are given in detail in Ref. 17, 

which fully describes the Iliff-Mine code (MtUE3 program). All computations 

an(’ aircraft examples in this wper use the linearized form €or the lateraldirectional equations. 

remainder of the paper. 

These equations are given below and referred to in the 

- 

21 = (cy + io) + 8 cos e sin + + p sin a - r cos a (15) 

where 

where the 6 term is summed over all controls. 

The observation equations are 

PmIP 

r,,, - 1: 

9m * 0 

. .. Em = P + PO .. r, = r + ro 

206 
The state, control, and observation vectors for the lateral-directional lode 

can then be defined as 

SIMPLE AIRCRAFT EXAMPLE 

The basic concepts involved in a parameter estimation problor can be illustrated by usir,g a simple example representative sf a realistic drcraft problem. 

The example chosen here is representative of 9n aircraft thet exhibits pure 

rolling motion from an aileron input. This example, although simplified, typifies the motion exhibited by many aircraft in particular flight regimes, such as 

the F-14 aircraft flying at 9igh dynamic pressure, *.e P-111 aircraft at derate speeds with the winq in the forward position, and the T-37 circraft at low 

speed. 

Derivation of an equation describing this motion is atraightforuard. 

Figure 2(c) shows a sketch of an aircraft with the x-axia perpendicular to the 

plane LE the figure (positive forward on the aircraft). "he rolling moment (I.@), 

roll rate (p), and aileron deflection (gal are positive as shown. 

exaaple, the only state is p and the only control is 6ae 

moments is 

For this, 

Ihe result of summing 

The first-order Taylor expansion then becomes 

where 

Since the aileron is the only control, it is notationally eirpler to use 6 

i.nstead of 6, for the discussion of this example. Bquation (33) taxi then be 

written as 

. 

~=*++6 (34) 

;.- ?Iternate approach that results in the 8aae equation ia to coablne Eq. (16) 

wit3 dq. (20), ELbstit-uting for C.:# snd then eliminate the terms that are zero 

for our example. This yields 

20 7 
where p is the roll rate and 6 is the aileron &flection. 

the equation can be put into the dlmnsional derivative form of lh.. 134). 

Rearranging terrs, 

Equation (34) is a simple aircraft equation &ere the forcing function is 

provided by the aileron and the damping by the dappping-in-roll term, 5. 

subsequent sections we examine in detail the parameter estimation problem where 

Bq. (34) describes the system- For this single-degree-of-freedom proble~n, the 

maw- likelihood estimator is used to entirate either Ip or L6 OX both for a 

given computed ti- history. 

In 

We will ass- that the system has measuremeat naise, but no state noise as 

in Bqs. (11, (21, and (3). Equation (4) then gives the cost function for mxbu 

likelihaod esthation. 

let it equal 1. For our exarple, Qt3- (2) and (3) btcm xi = pi and ti = Xi. 

Therefore, Eq. (4) becares 

The weigf?ci,xg a;* is unimportant for this probla, so 

where pi is the value of the peasured response p at tire ti and fii($8&3) is the 

cquted time history of 

rest of the paper, where coaputed data (not iiight data) are used, the msured 

time history refers to pi, and the computed tire history refers to &($#La)* 

The computed time history is a function of the current estirutes of Lp and Q, 

but the measured tine history is not. 

. at tire ti for Lp = $ and Lg = ;a. Throughout the 

The mst straightforvard method of obtaining pi is with -8. (3 and (8). 

In terms of the notation stated above8 

where 

4 = exp ($A) 

ar.9 A is the length of the sample interval (ti+l - ti). Simplifying the 

notat ion 

then 

208 
The maxima likelihood estimate is obtained by minimizing Eq. (36). The 

Gauss-Newton method described earlier is used for this minimization. 

tion (12) is used to determine successive values oL the estimtes of the 

unknoms during the minimization. 

quaA 6.. a 

For this simple problem, 5 = fLp ql* and successive estimates of Lp ud 

are determined by updating Bq. (12). The first and second gradients of Ipq. (12) 

are defined by 4s. (13) and (14). The complete set of equations is given in 

Ref. 17. 

The entire procedure can noy be written for obtaining the mxirur likelihood 

estimates for this simple example. 'Po start the algorithm, an initial estimate 

of Lp and Lg is needed. This is the value of €0. With m. (121, (1 and sub 

sequently 5~ are defined by using the first and second gradients of J(+,L&) 

from Eq. (36). The gradients for this particular example from Eq. (13) and 

'14b) are 

A 

a 

With the specific equations defined in this section for this simple example, 

we can noy proceed in the next section to the computational details of a specific example. 

Computational Details of ninimizatios 

In the previous section w specified the equations for a simple example and 

described the procedure for obtainins estimates of the unknowns from a dynamic 

maneuver. In this section we give the corprtatioril details for obtaining the 

estimates. Some of the basic concepts of parameter estimation are beat shown 

with computed data vhere the correct answers are known. Therefore, in this section we study two examples involving computed ti- histories. me first e-le 

is based on data that have no measurement noise, which results in e8l.iwtes that 

are the same as the correct value. The socond example contains sigdficant 

measurement noise; consequently, the estimates are not the same as the correct 

values. Throughout the rest of the paper, diere computed dau are used, the 

term "no-noise case" is used for the case with no noise added and 'noisy case' 

for the case vherc noise has been added. 

Since ve are studying a simple conputed example, it is desirable to keep it 

simple enocgh to complete soma or all of the calculatiora on a home computer or, 

wit some labor, on a calculator. With this in mind, the number of data points 

needs rn be kept snzll. Por thitr Fquted example, 10 points (tine samples) are 

2 09 
used. T1.e simulated data8 which we refer to as the measured data8 are based on 

Eq. (34). 

respectively) for both examples. In addition, the same input (6) is used for 

both e, mles, the sample interval (A) is 0.2 sec, and the initial conditions 

are terc 

examp?i. These values ire given to four significant digits, although to obtain 

- exac:2 t; 

signizicin: digits, as in the corputatica of these tables. If the four-digit 

numert are used in the carputation, the answers will be a few tenths of a 

percent off, but will still serve to illusLrate the minimization accuracy. 

both exarples, the initial values of Lp and 

respect ively 

We use the 8- correct values of Lp and L6 (-0.2500 and 10.0, 

Tables of all the significant internediate values are giwn with each 

8- values with a computer or calculator requires the use of 13 

In . (or €9) are -0.5 and 15-08 

-ample With No kkasurerent Wise 

The easurement time history for no measurant noise (no-noise case) is 

shown in Fig. 3. The aileron input star<& at zero, goes to a fixed value, and 

then returns to zero. The resulting rod-Fate time history is also shown. The 

values of the measured roll rate to 13 sigrificant digits are given in Table 1 

along with the aileron input. 

ATable 2 shows the values for $8 Lg, and J for each iteration, along with 

the values of 0 a..d J, needed for calculations of pi. In three iterations the 

aigorithm c;nverges to the correct values to four significant digits for both $ 

and L(j. Q overshoots slightly on the first iteration and then comes quickly to 

the correct answer. 

- 

A 

A 

$ overshoots slightly on the second iteration. 

Figure 4 shows the natch between the measured data ar?d the computed data for 

each of the first three ttrations. The match is very good after two iterations. 

The ratch is nearly enact after three iterations. 

Although the zlgoritha has converged to four-digit accrrlracy in % and La, 

the value of the coat function, J, continues to decrease rapidly between iterations 3 and 4. This is a consequence of using the WM~BU~ likelihood estilrator 

on data with no measurement noise. 

value of J at the minimum should Lo zero. RQVCver, with flnitt accuracy the 

value of J becomes small but never quite zero. 

nuabet of significant digits that are being used. For the 134igit accuracy 

used here, the cost eventually decreases to approximately 0.3 x 

Theoretically, using infinite accuracy the 

'Ibis value is a function of the 

Example With MeasurePent Noise 

The data used ,n this &le (noisy case) are the same as those used in the 

previa8 sectiun, except that pseudo-Gaussian noise has been added to the roll 

rate. The time history ik 8.10~-n in Fig. 5. The signal-to-noise ratio is quite 

low in thi- example, as is readily apparent by comparing Figs. 3 and 5. The 

exact values of the time history to 134igit accuracy are shown in "able 3. 

values of Lp, Lb, p, J,, and J are sham far each iteration in Table 4. The 

The AA 

2 10 
algorithm converges in four iterations. The behavior of the coefficients as 

they approach convergence is much like the no-noise case. The most notable 

results of this case are the converged values of 5 and L6, which are somewhat 

different from the correct values. 

time h.' tc-y if+ shown in Pig. 6 for each iteration. No change in the latch is 

apparent for the last two iterations. The match is very good considering the 

amount of measurement noise. 

L L 

The match between the measured and computed 

In Fig. 7, +he computed time history :or the correct values of $ and 

compared to that for the noisy-case estimates of T , and L6. 

algorithm converged to values sorevhat different an the correct values, the 

tvo compute. time histories are similar but not identical. 

is 

Because the 

The accuracy of the converged elements can be assessed by looking at the 

CraGr-Rao inequality (Refs. 16 and 17) discussed earlier. 

can be obtained from ae following approximation to the infomation matrix. 

The Cram&-Rao bound 

The Cram&-Rao bounds for 5 and Q are the square roots of the diagonal elements of the H matrix, or (H(1,l) and /H(2,2), respectively. 

bounds are 0.1593 and 1.116 for Lp and u, respectf-;-,ly. 

L6 are less than the bounds. 

The Cram&-Rao A A 

Ihe errors in L,, and 

Cost Functions 

In the previous section we obtained the maximum likelihood estimtes for 

computed time histories by minimizing the values of the cost functioa. To fully 

understand what occurs in this minimization, we lust study in more detail the 

form of the cost functions amid some of their more important characteristics. In 

this section, the cost function for the no-noise case is discussed briefly. me 

cost function of the noisy case is then discussed in more detail. The same two 

time histories studied in the previous sectlon are z-amined here. The noisy 

case is more interesting because it has a meaningful Cram&-Rao bound and is 

more representative of a i r c r a f t f l i g h t data. 

First ve will look at the one-dimensional case where L6 is fixed at the 

correct value, because it is easier to grasp some of the characteristics of the 

cost function in one dimension. Then ;*e will look at the'two-dimensional case, 

where both Lp and Lb are varying. It i3 important to remember that everything 

shown in this paper on cost functions is based on computed time histories that 

are defined by Eq. (36). For every time history we might choose (computed or 

flight data), a complete cost function is defined. 

the cost tunction defines a hypersurface of n + 1 dimensions. 

us that we could just construct this surface and look for the minimum, avoiding 

the need to bother with the minimization algorithm. This is not a reasonable 

For the case of n variables, 

It might occur to 

211 
approach because, in general, the number of variables is greater than two. 

Therefore, the cost function can be described mathematically but not pictured 

graphically. 

To illustrate the many interesting aspects of cost functions, it is easiest to first look at cost functions having one variable. 

tion, the cost function of tp and Lg was minimized. 

interesting in the Lp direction. Ther=:ore, the one-variable cost function 

studied here is J(Lp). 

to the carrect value of 10. Figure 8 shows the cost function plotted as a function of Lp for the case where thsre is no measurement noise (no-noise case). As 

expected for this case, the minimum cost is zero and occurs at the correct value 

of Lp = -0.2500. 

more negative Lp than for a positive TT. 

to become less negative where $ is more negative than -1.0. 

makes sense since the more negative values of 5 represent cases of high damping, 

and the positive Lp represents an unstable system. Therefore, the pi for positive Lp becomes increasingly different from the measured time history for small 

positive increments in 5. 

would show essentially no response. Therefore, large increases in damping 

result in relatively small changes in the value of J($). 

In an earlier secThat cost function is most 

All subsequent discussions are for J(Lp) with Lg equal 

It is apparent tirat the coat increases much more slowly for a 

In'fact, the slope of the curve tends 

Physically this 

For very large damping (very negative $1 the system 

In Fig. 9, the cost function based on the time history with measurement 

noise (noisy case) is plotted as a function of Lp. 

Lp (-0.2SOO) and the value of Lp (-0.3218) at the minimum of the cost (3.335) 

are both indicated on the figure. 

Pig. 9 is similar to that shown in Fig. 8. Figure 10 shows the comparison 

between the cost functions based on the time histories with and without masurement noise. The comments relating to the coat funotion of the no-noist zase 

also apply to the cost function based on the noisy case. Figure 1@ shows 

clearly that the two coat functions are shifted by the difference 1:; the value 

of Lp at the minimum and increased by the difference in the minimum cost. One 

would expect only a small difference in the value of the cost when far from the 

minimum. This is because the "estimated" time history is so far from tho 

measured time history that it becomes irrelevent as to whether the measured time 

history has noise added. Therefore, for large values of cost, the difference in 

the two cost functions should be small in comparison to the total cost. 

The correct value of 

The general shape of the cost function in 

Figure 11 shows the gradient of J(LP) plotted as a function of Lp for the 

noisy case. 

equivalently, the minimum of tht cost function) using the Gauss-Newton method of 

a previous section. 

the value of the minimum of J(Lp). 

This is the funrtion for which we were trying to find the zero (or 

The qradient is zero at Lp P -0.3218, which corresponds to 

The difference between the Newton-Paphson method (4. ( 14a) ) and the 

Gauss-Newton method (Eq. (14b) ) of minimization has been mentioned previously. 

21 2 
For this simple one-dimensional case, we can easily compute the second gradient 

both with the second term of Eq. (14a) (Newton-Raphson), and without the second 

term (Gauss-Newton, 4. (la) 1. Figure 12 shows a comparison between the 

Newton-Raphson and the Gauss-Newton approximation second gradients. The 

Gauss-Newton second gradient (dashed line) always remains positive because it 

is the sum of quadratic terms (squared for the one-dimensional example). Ths 

Newton-Rapnson second gradient can be positive or negative, depending upon the 

value of the second partial with reapct to Lp. Other than the difference in 

sign for the more negative $, the two curves have similar shapes. 

As stated earlier, the Gaiiss-Newton method can be shown to be superior to 

Newton-Raphson in certain cases. We can demonstrate obvious cases of this 

with our example. 

Raphson method will occur is to look for places where the second gradient (slope 

of the graditnt) is near zero oL aegative. Figure 11 has such a region near 

Lp = -1.0. 

are forced to divide by zero in E’. (12) with the Newton-Raphson metr,A. 

point is at Lp = -1.13 in Fig. 12. If the value of the slope of the gradient is 

negative, then the Newton-Raphson method wiil go to very negative values of Lp. 

For very negative values of $, the cost becomes asymptotically constant and the 

gradient becomes nearly zero. In that region, the Newton-Raphson algorithm 

would diverge towards negative infinity. If the slope of the gradient is 

positive but small, we still have a problem with the Newton-Rapheon method. 

Figure 13 shows the first iteration starting from $ = -0.95 for both GaussNewton and Newton-Raphaon. The Newton-Raphson method selects a point where the 

tangent of the gradient at $ = -0.95 intersects the zero line. This reaulta in 

the selection of an Lp of approximately 2.6 in the first iteration. 

value it requires many iterations to return to the actual minimum. On the other 

hand, the Gauss-Newton method selects a value for Lp of approximately -0.09 and 

converges to the minimum to four-digit accuracy in two more iterations. With 

more complex examples a comparison of the convergence properties of the two 

algorithms becomes more difficult to visualize, but the problems are gerieralizations of the sitaation we have observed with the one-dimensional example. 

An easy way to select a spot where problems with the HewtonIf we choose a point where the gradient slope is exactly zero, we 

This 

From that 

The usefulness of the Cra&r-Rao bound was discussed in the Example With 

Measurement Noise section. At this point it is useful to digress briefly to 

discuss some of the ramifications of the CramhRao bound for the one-dimensional 

case. 

example, the estimate of Lp is -0.3218 and the Cram&-Rao bound is 0.0579. 

celculation of the Cramk-Rao bound was defined in the previous section for both 

one-dimensional and two-dimensional examples. 

mate of the standard deviation of the esti.mate. One would expect the scatter in 

the estimates of $ to be of about the same magnitude as the estimate of the 

standard deviation. For the one-dimemioaal case discussed here, the range 

(Lp (-0.3218) plus or minus the Cram&-Rao bound (0.0579)) nearly includes the 

correct value of $ (-D.2500!. If noiay cases are generated for many time hiatories (adding different measurement noise to each time history), then the sample mean and sample standard deviation of the estimates for these cases can be 

calculate?. Table 5 gives the sample mean, eample standard deviation, and the 

The Cramdr-Rao bound only has meaning for the noisy case. In the noisy 

The 

The CramhRao bound is an esti2 13 
standard deviation of the sample mean (standard deviation divided by the square 

root of the number of cases) for 5, 10, and 20 cases. ?!he sample mean, as 

expected, gets closer to the correct value of -0.2500 as the number of cases 

increases. This is also reflected by the decreasing values in column 4 of 

Table 5, which are estimates of the error in the sample mean. Column 3 of 

Table 5 shows the sample standard deviations, which indicate the approximate 

accuracy of the individual estimates. This standard deviation, which stays mor3 

or less constant, is approximately equal to tne Cram&-l?ao bound for the noisy 

case being studied here. 

noisy cases used here (not shown in the table) do not change much from the 

values found for the noisy case being studied. Both of these results are in 

good agreement with the theoretical characteristics (Ref. 16) of the Cram&-Rao 

bounds and maximum likelih. a3 estimators in general. 

In fact, the Cram&-Rao bounds for each of the 20 

The examples shown here indicate the value of obtaining more sample time 

histories (maneuvers). More samples improve confidence in the estimate of the 

unknowns. The same result holds true in analyzing actual flight time histories 

(maneuvers); thus it is always advisable to obtain several maneuvers at a given 

flight condition to improve the best estimate of each derivative. 

The size of the Cram&-Rao bounds and of the error between the correct value 

and the estimated value of + is determined to a large extent by the length of 

the time history and the amount of noise added to the correct time history. For 

the example being studied here, it is apparent from Fig. 5 that the amount of 

noise being added to the time history is large. The effect of the power of the 

measurement noise (GG*, Eqs. (3) and (4)) on the estimate of Lp (that is, Lp) 

for the time history is given in Table 6. The estimate of Lp is much improved 

by decreasing the measurement noise power. A reduction in the value of G to 

cine-tenth of the valu.? in the noisy example being studied yields an acceptable 

estimate of Lp. 

the accuracy of the output of the measurement sensors. 

A 

For Lilght data, the ueasurement noise is reduced by improving 

Two-Dimensional Case 

In this section the cost function (which is deiendent on both Lp and t&) is 

studied. The no-noise case is examined first, followed by the noiey case. 

No-noise case. Even though the cost function is a function of mly rwo 

unknowns, it is much more difficult to visualize than the one-unknown case. The 

cost function over a reasonable range of Lp and L& is shown in Fig. 14. The 

cost increases very rapidly in the region of positive $ qnd large values of 

Lg. The reason it3 just an extension of the argument. fcr poaitive giver, in 

the previous section. The shape of the surface can be depicted in greater 

detail if we examine only the values of the cost function less than 200 f@r 5 

less than 1.0. Figure 15 shows a view of this restricted surface from the upper 

end of the surface. The minimum must lie in the curving valley that gets 

broader as we go to the far side of the surface. Now that we have a picture of 

the surface, we can look at the isoclines of conatant cost orr the Lp-VerSUS-L6 

plane. These isoclines are shown in Pig. 16. The minimum of the cost function 

2 14 
is inside the closed isocline. 

positive-$ direction is once again apparent. Inside the closed isocline the 

shape is more nearly elliptical, indicating that the coat is nearly quadratic 

here, so fairly rapid convergence in chis region would be expected. 

becomes an asymptote in cost as L6 approachea zero. The cost is constant for 

L6 = 0 because no response would result from any aileron input. 

response is zero for all values of $, resulting in constant eoat. 

The st.eepness of the cost function ir, the 

The $ axia 

The astimated 

Figure 16 shows the region of the minimum value of the cost function, which, 

as seen in the earlier example (Table l), occure at the correct valuea for Lp 

and L6 of -0.2500 and 10, respectively. 

cost function surface shown in Pig. 17. The surface has ita minimum 3t the 

correct value. As expected, the value of the cost function at the minimum is 

zero. 

This is also evident by looking at the 

Noisy case. As shwn before in the one-himemional case? the primary difference between the cost functions for the no-noise and noisy cases was a shift 

in the cost function. In that imtance? the noisy case was shifted so that the 

minimum was at a higher cost and a more -mgative value of 

dimensional case, the no-noise and noisy cost functions exhibit a similar shift. 

For two dimensions the shift is in both the Lp and L6 directions. 

small enough that the difference between the two cost functions is not visible 

at the scale shown in Fig. 14 or from the perspective of Fig. 15. Figure 18 

shows the isoclines of constant cost for the noisy case. The figure looks much 

like the isoclines for the no-noise case shown in Pig. 16. The difference 

between Figs. 16 and 18 is a shift in $ of about 0.1. 

in the value of Lp at the minimum for the no-noise and noisy cases. 

cally, one can see that the same would be true for cases with more than two 

unknowns. 

mini mum. 

In the twoThe shift ie 

This is the difference 

HeuristiThe primary difference between the two cost functions is near the 

The next logical part of the cost function to edne is near the minimum. 

Figure 19 shows the same view of the cost function for the noisy case as was 

shown in Fig. 17 for the no-noise case. The aha-w is roughly the same as that 

shown in Pig. 17, but t)ie surface is shifted such that ita minimurn lies over 

Lp = -0.3540 and L6 = 10.24, and is shifted upward to a cost function value of 

approxinu te ly 3.3 . To get a more precise idea of the cost of the noisy case near the minimum, 

we once again need to examine the iaoclines. The isoclines (Fig. 20) in this 

region are much more like ollipsse than they are ir Figs. .16 aad 18. We can 

follow the path of the minimization exaiaple used before by including the results 

from Table 4 on FiQ. 20. The first iteration (5 = 1) broughc the values of Lp 

and I,6 very close to the values at the minimum. 

selected the valuea at the minimum when v-:smd at this scale. One of the reasons the convergence is so rapid in thie region is that the isoclines are nearly 

%lliptical, demonstrating that the cost is very nearly quadratic in this region. 

If we had started the Gauss-Newton algorithm at a point where the isoclines 

are mch less elliptical (as In rome of the border regions in Fig. 181, the 

The next iteration essentially 

21s 
convergence vould have been much slower initially, but much the same as it 

entered the nearly quadratic region of the cost function. 

Before concluding our examination of the two-dimensional case, we need to 

exanhe the Cram&-Rao bound. 

is based on the Cram&-Rao bounds defined in an earlier section. 

ships between the Cram&-Rao bound and the uncertainty ellipsoid are discuss,:d 

in Ref . 16. The Uncertainty ellipsoid almost includes the cxrect value of Lp 

and Lg. The CramGr-Rao bound for Lp and L6 can be determined from the prJjection of the uncertainty ellipsoid onto the Lp and Lg axes, and compared with the 

values given earlier, which were 0.1593 and 1.1 16 for Lp and Lg, respectively. 

Figure 21 shows the uncertainty ellipsoid, which 

The relatiorESTIMATION USING FLIGHT DATA 

In the previous several sections we examined the basic mechanics of obtain-. 

ing maximum likelihood estimates from computed examples with one or two unknown 

parameters. Now that we have a grasp of these basics, we can explore the estimation of stability and control derivatives from actual flight data. For the 

computationally much more difficult situatio? usually encountered using actual. 

flight data, we will obtain the maximum 1ikeli.hood estimttes with the ISrfMaine code (WE3 program) described in Ref. 17. The equations of motion that 

are of interest are given in the AIRCRAFT EQUATIONS OF MYTION section of this 

paper; the remainder of the equations are given in Ref. 17. 

In general, flight data estimation is fairly complex, and codes sxh as the 

Iliff-Maine cade must usually be used to assist in the analysie. However, one 

must still be cautious about accepting the results; tkac :s, +&e eatimates must 

fit the phenomenology, and the match between the m;sured and computed time histories must be acceptable. This is true in all flight regimes. ?xt one must be 

particularly careful in potential prQblem situations such as (1) in separated 

flow at high Mach numbers or high angle of attack, (2) with unusual aircraft 

configurations such as the oblique wing (Ref. 181, or (3) with modern highperformance aircraft with high-gain feedback loops. 

one should be particiilsrly careful where there are even small anomalies in the 

match. These anomalies may indicate ignored terms in the equations of motion, 

separated flow, nonlinearities, sensor problems, insufficient resolution 

(Ref. 1 1, sensor location (Ref . 1 1, time or phase lags (Refs. 1 and '91, or aiIy 

of a long list of ocher problems. 

In any of the above cases, 

The following brief examples are intended to show how the above caveats and 

the computed examples of previous sections can be used to'assist in the analysis. 11. the computed example, the desirability of low-noise sensors, an adequate model, and several maneuvers at a given flight condition is shown. 

Hacd Calculation Example 

Sometimes evaluation of a Fairly comp:.ex flight maneuver can be augmented 

with a simple hand calculation. One example of this can be found for the space 

216 
shuttle. 

enter the atmosphere from space and land horitontally. The entry control system 

consists of 12 vertical reaction-control-system (RCS) jets (six up-firing and 

six down-firing), 8 horizontal RCS jete (four left-firing and four right-firing) 

4 elevon surfaces, a body flap, and a split rudder surface. The locations of 

these devices are shown in Fig. 22. The vertical jets and the elevons are used 

for both pitch and roll control. The jets and elevom are used syrawtrically 

for pitch control and asymmetrically for roll control. The space shuttle control system is described briefly in Ref. 6. 

The space shuttle is a large double-delta-winged vehicle designed to 

The shuttle example used here is from a maneuver obtained at a Mach number 

of approximately 21 and an angle of attack of approximately 40°. The controls 

being used for this lateral-directional maneuver are the differential elevons 

and the side-firing jets (yaw jets). The maneuver is shown in Fig. 23. 

Equations (15) to (31) describe the equations of motion. A simplified approach 

can be used to determine some of the derivatives by hand. 

that has ben used since the beginning of dynamic analyaie of flight maneuvers. 

In particular, for this mneuver the slope of the rates can be used to determine 

the yaw jet control derivatives. This is possible for this example, even with a 

high-gain feedback system, because the yaw jets arc essentially step functions, 

and the elope of the rates p and r can be determined before the vehicle and the 

differential elevon (aileron) responses become significant. The rolling moment 

due to yaw jet (LYJ) is particularly important for the shuttle (Ref. 6 discusses 

the essential nature of flight-determined LYJ in the redefinition of entry 

maneuvers) and is, in general, more difficult to obtain than the more dominant 

yawing moment due to yaw jet. Therefore, as an illustrative example, LYJ is 

determined by hand. Figure 24 shows yaw jet activity and smoothed roll rate 

plotted at expanded scales. 

The approach is one 

The equation for LYJ is given by 

LYJ = ~Ix/(Number of yaw jete) (42) 

Therefore, given that I, B 900,OOG elug-ft2, and the nuaber of yaw jets is 4, 

LYJ f 2750 ft-lb. 

The same maneuver was analyzed with MEUIE3, and the resulting match * shmn 

in Fig. 25. The match is very good except for a small miaxnatch in p at about 

6 sec. This smal.1 mismatch was studied separately with MMLE3 and found to be 

caused by a nonlinearity in the aileron derivative. The value from =E3 for 

LYJ is 2690 ft-lb, which for the accuracy used here is eeeentially the same 

value as obtained by the simplified method. The aileron derivatives would be 

difficult to determine as accurately as the yaw jet derivatives. 

estimates can seldom be obtained with the elope method discussed here, rough 

estimates can usually be obtairied to gain some insight into values obtained with 

WE3 (or any other maximum likelihood program). These rough estimtes can then 

be used to help explain unexpected values of estimate6 from an estimation 

program 

Although good 

217 
Sometimes a flight example becomes too complex to ollow anything other than 

qualitative estimates to be determined by hand. The example shown in Fig. 26 tr 

the determination of the rudder derivative for tke F-8 aircraft with the yaw 

sugmentation syetem on. "hie example, taken from Ref. 20, include8 an aileron 

pulee and a rudder pulse. Although an indepmdent pilot rudder pulse ie input 

during the mnauver, the rudder is largely reeponding to the lateral acceleration feedback. 

moving, thus making it difficult to use the eimplifinll qproach jurlt diecussed. 

However, Cn6 

1.7 sec fron the start of the maneuver. Most of the slope of yaw rate probably caused by the rudder, but a poor estimate would be obtained using the hand 

ca 1 cu lat ion. 

When the rudder ia moving, eeveral other variables are also 

can be roughly detcrmined when the rudder moves, approximately 

r 

Cost Function for Full Aircraft Problem 

The analysis of a lateral-directional maneuver obtained in flight typically 

has from 15 to 25 unknown parameters (a8 shown in -8. (15) and (31)), in 

contrast to thc one or two in the simple aircraft example. This make8 detailed 

examples unwieldy and any graphic presentation of the co8t function impoeeiblr,. 

Therefore, in thie section we are primarily examining tLe estimation procedure 

and the process of the mipimization. 

For our flight example, we have chosen a lateral-direction61 manevver, with 

both aileron and rudder inpute, that has 97 unknown parmeetere. The data arc 

from the oblique wing aircraft (Ref. 18) with the wing unskewed during the 

maneuver. This example was chosen because it is a typical maneuver. 

history of the data and the eubeequent olzcput of MMLE3 have been published in 

Ref. 21. Some reeults of the a.ia7ysis are shown in Table 7. The match between 

the measured time hietciry (eolid Anea) and the estimated (calculated) time 

history (dashed lines) ie shown as a function of iteration in Pig. 27. Figures 27(a) to (e) are for iterations 0 to 4, respectively. Table 7 ehows that 

the cost remains unchanged after four iterations. 

for the two-dimensional simple aircraft example in Fig. 6 and Table 4. 

The time 

A similar result was obtained 

Of the mny thing8 the analyst met consider in obtaining estimates, the two 

nost impaxtant on88 are how good is the match and how good is the convergence. 

A satisfactory match and monotor 

cient, conditions for a succesr! A analysie. Figure 27(e), althczgh not perfect, is a very good match. 

the normalized cost TI the laet row of Table 7. The coet has converged rapidly 

and monotonically in four iteratione, and it rewaine at the converged cost. 

These factors are convincing evidence that the convergence ie complete. Therefore, the criteria of match and convergence are satisfied in our example. In 

some cases we might encoi?nter coet that doe8 not converge rapidly (in four to 

six iterations) or monotonically, or stay "exactly" at the dnhum value. TheRe 

situations usually indicate at least a small problem in the analysis. 

probieme, if found, are usually traced to an instrumentation or data aquieition 

problem, an inadequate mathematical model, or a mneuver that contain8 a marginal amount of information. 

convergence are necessary, but not suffiThe convergence can best be eva!uated by looking at 

These 

218 
Table 7 also show8 that the startup values of all the coefficients are zero 

for the control and bias variables. Wind turnel estimates could have been deed 

for starting value;? but the convergen-e of the algorithm is not very dependent 

on the startup vslues. As part of the startcp algorithm, the MMLE3 program normally holds the derivatives of the state variables consta:-t until after the 

first iteration, ad is evident in Table 7. 

Figure 27(a) shows the mztch between the measured and computud data for the 

startup values. The match is very poor because the startup values for the control derivatives are all zero, YO the only motion is in response to +he initial 

conditions. The control derivatives and biasc. are determined on the first: 

iteration, resulting in the much improved match shown in Fig. 27(b). The match 

after two iterations, shown in Fig. 27!c), is improved as the Srogram furthor 

modifics the control derivatives and, fcr the first time, adjusts the deri7,atives affezting the natural frequency (CnB and CQ~). By the third iteration 

(Fig. 27(d)), the improvement in the match is.a'lmo8t complete, because minor 

adjustments to the frequency are made and the damping derivatives are c5anged. 

Fig. 27(e) shows the match when all but the aost minor derivatives have ceased 

to change. 

Several general observations can be made based on this well behaved example. 

The strong or most important coeff icionts have essentially converged in three 

iterations. 

verged faster than Lp (Table 4). 

coefficients have only converged to two plitces after three iterations and are 

still changing by one digit in the fourth place at the end of six iterations. 

Another observation is that for some coefficients (C.tr, Cnda, and Cab 1 even 

r 

though the sign is wrong after the first iteration, the algorithm-quickly 

selects their correct values once the important derivatives have stabilized. 

The same effect was s8en in the simple example - that is, Le conSome of tke less iuportant or eecond-order 

In general, if the analysis of a maneuver has gone well, we do not need to 

spend much tine inspecring a table analogous to Table 7, emever, it there have 

been problems in cozz~2rgsiiea or in the qilality of the fit, a detailed inspection 

of such a table may be necessary. The data may show an important coefficient 

going unstable at an early iteration, which could caase problems later. If the 

starting values ate grossly in error, the algorithm is driven a long way from 

reasonable values and then for many reasons does net behave well. Occaeiorally 

the alqorithm alternately select8 from two diverse se% of values of two ~t more 

csefficients on successive iterations, behaving as if the shape of the cost 

function were a narrow multidimensional valhy analogous to but more extreme 

than the two-cYnensIona1 valley shown in Figs. 18 and 20. 

Cram& -Rao Bounde 

The earlier sections regarding the computed example have shown that the 

Cram&-Rao bound is a good indicator of the accuracy of an estimated parameter. 

The Cramir-Rao bounds can be used in a similar, but eomewhat more qualitative, 

fashioi on flight data. 

well as many other maximum likelihood estimation programs) have been useful in 

The Cram&-Rao bounds that are included in MULE3 (as 

219 
determining whether estimates are good or bad. 

here has bca reporced previously (for example, in Refs. ? and 16). However, 

this exmple of the use of the Cram&-Rao bound in the assessment of flightdprived estimates is pertinent to the thrust of this paper. Figure 28 shows 

estimates of C as a function of angle of attack for the ?A-30 twin-engine 

general aviation aircraft (Ref. 22) at three flap settings. There is a significant. amount of scatter, which makes the reliability of the information on C 

questionable. The data showr. are the estimates from the MMLE3 program, which 

also provides the Cra&r-Rao bounds for each estinute, 

has shown that if the Cram&-Rao bound is multiplied by a scale factor (the 

result soinetimes being called the uncertainty levre1 (Refs. 1 and 16)) and plotted as a vzrtical bar with the associated estimate, it hel- in the interpre+ation of flight-cietermined results. Figure 29 shows the same data as Fig. 28, 

with the uncertainty le--1s now included as vertical bars. The estimates with 

small uncertainty levels (Cra&r-Rao bounds) are the best estimates, as was 

discussed earlier in the section on Cram&-Rao bounds for the one-dimensional 

case. The fairing shown in Pig. 29 goes through the estimates with 8-11 

Cramir-Rao bounds and igmres the estimates with large bunds. One can have 

great confidence in the fairing of the estimates, because the fairing is well 

defined and consistent when the Cram&-Rao bound information is included. In 

this particular instance, the estimates with small bounds were from maneuvers 

where the aileron forced the motion, and the large bounds were from maneuvers 

where the rudder forced the motion. Therefore, in addition to aiding in the 

fairing of the sstimates, the Cram&-Rao bounds help show that the tileronforced mtineuwrs are superior for estimating Cn 

The aircraft example discussed 

"P 

"P 

Past experience (Ref. 1 ) 

for the PA-30 aircraft. P 

This example illuacrates that the Cram&-Rao bounds are a useful tool in 

assessing flight-determined estimates, just as they were found useful for the 

simple aircraft example with computed datae 

Atmospheric Turbulence (Stzste Noise) 

Atmosphtric turbulmce (state mise) cannot always be avoided in blight; 

therefore, it is desirrrble to be able to obtain s*dbility and control derivatives in the presence of turbulence. In addition, an estimnte of the turhlence 

time history can be of interest, particularly in the implementation of turbulence svK>pressioc systems. 

Many years ago it was demonstrated that the stability and control derivatives Cdil be adequately determined with maximum likelihood estimation techniques 

fo.- maneuvers performed iil smooth air. 

account for turbulence, are a2plied to data obtained in' turbulence, nst only are 

the resulting matches of the time histories unsatisfac tory but the estiiaated 

coefficients are unacceptable (Refs. 23 to 25). The tecnnique deacribd in 

Refs. 14, 23, and 25 can rrccouilt for the effect of turbulence. With this technique, maximum likelihood astimates of the stability and control derivatives as 

well as esthatas of the turbulence time histories are jttained by zrinimieing 

the cost function given by m. (11). ReRults of the application of the technique to lonqi tudinal maneuvers obtained in turbulenco have been resorted 

previously (Refs. 23 to 25). 

If the;e techniqies, which do not 
Trre lateral-directional equations (Eqs. (151, (161, (17;, (181, and (29) 1 

can be modified in a manner similar to that used to dify the longitudinal 

equations in Refs. 23 to 25. The turbulence (state noise) -1 is the Dryden 

expression, which is described in Ref . 26. The Iliff-&ne code (Ref . 17) c.m 

be used to obtain the maxirum likelihood estimates where state noise fs present. 

Thirty-eight semnds of data from the PA-30 aircraft flying in turbulence 

was analyzed at 50 samples/sec. The best match that could be obtained with the 

maximum likelihood estiration cethod that does not account fcr turbulencz ia 

shown i.? Fig. 30. 

the stability and control derivatives. Figure 31 shars the ratch obtained with 

the maximum likelihood estimation technique t!at accountcl for turbulence 

(Re’ , 14 and 17). The laatch is excellent and the maneuver provided acceptable 

es .mated stability and control derivati7es. ft is also of interest to compare 

the power spectra of the estimated turbulence time histories. 

spectrum of the turbulence component affecting-angle of sideslip, B,, is shown 

in Fig. 32. Figure 33 pre,sents the power spectnu of t!e turbulence corponent 

affecting roll ?ate, pg. The slopes of the asymptotes shwc in Pigs. 32 and 33 

are those defined by the Dryden expression given in Ref. 26. Good agreement is 

ShoWir becdeen the power spectra and the asymptotes for 8, and pg. 

The ratch is unacceptable and resulted in poor estimates of 

The powr 

A 

A 

A A 

The algorithm used here is based on a linearized system described by 

Eqs. (5) to (7) and solved by minimizing the cost function giver. by Bq. (11). 

The system need not resemble that for the aircraft stability and control problem 

other than in the requirerent for linearity. Therefore, many formulations for 

the structural problem are wrizten in the form of 480 (5) to (71, and the 

algorit.hm under discussion can be directly applied with these formulations. 

ESTIMATION FOR SIMPLE STWCTUUL . PRoBLEn 

The problas of t.?e Llexible space structure is mast fully ctaracteziied as a 

distribuced parameter system with its associated distribu-d system control 

laws. 

enaronesnt, such as szlar heating. rAs in most cases, the preferred solution is 

the simplest successful approach, 

and computationally far more efficient chan the fully distributed paraneter system approach. 

the-art approaches has proved very successful. Admittedly, the aircraft structure is heavier than most spacecraft, but Pany c i - r c r a f t s t r u c t u r e s are highly 

complex, consisting of many subetrxturos within the main structure. 

novice, many of the sp-ce structures xrretitly king i.ivestigated appear sieqlsr 

thar. modern, large aircraft. If the luqed parameter system approach used for 

the aircraft pro’ sm is found to be inadequate, it s?em likely that distributed 

parameter estimatiorl codes wiil evolve tu whatever complexity in zecessary to 

solv? the f Leuible space structure problea. 

The model will vary depend~;.3 upon changes in its configration or its 

\%e lumped svstem approach is much simpler 

For example, structural rode control base? on earrent state-ofTo the 

22 1 
This paper has discussed some of the experience gained from the applicatitn. 

of aircraft sability and control analysis to flight data. The codes used for 

this analysis are for lumped parmter systems in the time domain. 

have been used successfully for structural problears and are fully adaptable to 

the frequency domain Ff that is found to be preferable. 

The codes 

Although feu results have been obtained fer tire-domain structural analysis 

at the Ames Dryden Flight Research Facility, sose superficial experience in 

structural tire-dcrain analysis has been obtained. 

show how the techniques being used for stability and control analysis can be 

applied to simple structural problems. The preceeding section discussed the 

incorporation of state noise in the del. The following examples do not 

include the use of state noise, but state noise, if warranted, could easily be 

incorporated in the types of examples to be discussed. 

The folloving tvo examples 

Estimation of Structural Characteristics 

All aircraft have observable structural mdes. These lodes usually cause no 

difficulty in estimating atability and control derivatives because the structural frequencies are higher than the aerodynamic frequencies. In general, if 

the structural frequencies are higher than the highest aerodynamic frequency by 

more than a factor of 5 to 10, they can be neglected unless their amplitude is 

so large as to mask lreasurementr desired for the aerodynamic analysis. However, 

if one or more structural nodes are affecting the aerodynamic modes, as MY 

occur in Large aircraft, these structural des must be included in the aathematical model being analyzed. 

Even though no completely satisfactory practical results are available that 

account for structcral modes and their interactions with the aerodynamics, it is 

interesting to assess the time-domain aax~mum likelihood analysis of the structural modes independent of any interaction. This can be done where a s"crurtura1 

mode is observed and no significant coupling is apparent. 

Figure 34 shws a structural rode on the lateral acceleration of an aircraft 

where little effect was observed for structural-aerodynamic coupling. The frequency of the rode is high enough that the mode does not interact with the aerodYMuC modes. Therefore, the stability and control derivatives were obtained 

separately and held constant for the succeeding analysis, !l'hc analysis consisted of using the raxintm likelihod estimation program lOlLE 3 (Ref. 17) with 

a sixth-order model that included the latexaldirectional aerodynaaic unies plus 

one structural Me. The dynamic presaure and the velocity were allowed to -rsry 

in the analysis. The structural mode frequency and damping were estimuted as 

lifiear functions of dynamic pressure. The initial conditione were also estirated. A structural mode frequency of 7.S4 Hz was chosen to start the estieatidr process. The comparison between the original data and the match obtained 

wid. the maximum likelihood estimation method ie shown in Fig. 35. The two time 

histories are in gd agreeaent at the beginnin5 of the maneuver and at the end 

of the maneuver, hit they aye 180° out of phase at a time of approximately 

222 
0.3 8ec. The match shown in Pig. 35 suggests Lkt the mmcisu likelihood estimator has reached a local minimum but ,?ot %he global dnirur. 

aze not normally a problea when obtaininq the stability and control derivatives 

of aircraft with the maximum likelihood estimation method. 

Multiple minima 

The reason for the rmltiple dnira is dtronstrated by the following simple 

scalar example. Let the noiseiess measured response be t(t) - sin (wet) and the 

estimated response be zc = sin (ut), where o io the only unknown coefficient. 

Then, by Eq. (41, the cost function becoares 

If T is chosen to represent 10 cycles, as shown in Pig. 35, then for an 00 of 

1 radlsec, T equals 20r. In Fig. 36, the cost function J(w,2Or) is shown as a 

function of w. The global minimum is at an o of 1 rud/scc, aa it should be, but 

there are many local minim at increments of approximtely 0.05 rad/st?c. If a 

value of less than 0.97 or greater than 1.03 were choaec for a starting estimate 

of w, the algorithm would converge to a local minimum. If a value of betveen 

0.98 and 1.02 were chosen, it would converge to the global minllun. Therefore, 

for this example where 10 cycles were observed, the starting vahe of w mist be 

less than 3 percent from the correct anwner to converge to the global minimum. 

Figure 37 shows a sine Wave for the global minimum along With a sine wave 

with a frequency that varies 10 percent from the qlobal minimum. The sine waves 

are in phase at the beginning and em!, ami 180" aut of phase in the riddle. 

These data appear similar to those shown for flight dam in Pig. 35. 

one or two cycles were used for the aMlySi5, the problem illustrated in Pig. 37 

would be minimized. This is apparent in Pig. 38 where only the first cycle of 

Fig. 37 is shown. 

If only 

If T is chosen to represent only one cycle and 00 remains 

(as in Fig. 381, then T equals 2rr. The cost function J(w,2*) 

fanction or w in Pig. 39. The glabal minimum is correctly at 

but now the algorithm converge& t~ the global minimum if 0 is 

approximately 25 percent of the correct value. 

equal to 1 rad/sec 

io preser.tad as a 

an w of 1 rad/sec, 

started within 

Knowing the sensitivity of the a1goriV.m when a record with many lightly 

damped cycles is being analyzed, the data of ?ig. 34 can be reanalyze6 stzrting 

closer to the observed frequency. 

method with an w of 3.0 results in the fit shown in Fig. 10. This is an acceptable fit of the data. 

Starting the maximum likelihood estimation 

223 
Based on the preceding results, if data are to be analyzed where many cycles 

of a structural mode are present, the structural lode frequency, w, must be clcsely approximated before starting the estimation process. 

Structural nodes in Space - 

In the process nf analyzing aircraft flight data, the authors haw frequently observed results that clearly exhibit unmodeled dynamics. fie udeled 

dynamics could be caused by many phenomena, such as higher-order aerodynamic 

modes or structural modes. These mdes can usually be ignored and left UMOdeled because they have no effect on the results of primary interest in tne analysis. If the unrodeled modes cannot be ignored, then the system equations must 

be revised to include the deled modes. 

The authors have not yet found it necessary.to model structural modes for 

data obtained in space in the process of obtaining control derivatives for the 

space shuttle. However, the structural modes have been observed. Figure 41 

shows the response of the space shuttle to the firir.9 of a roll jet and a yaw 

jet at an altitude of 430,000 ft. The space shuttle configuration and the location of the RCS jets are shown in Pig. 22. 'Ihe changes in the rigid-body rates 

and lateral. acceleration caused by the jet firings are apparent in Fig. 41. The 

structural modes are also excited by *.e jets, as evidenced by the increased 

ringing in each signal at the time of the jet firings. The roll jet firing has 

little effect on the rigid-body responae for the yaw rate and lateral acceleration; nowever, the yasr jet results in a rigid-body responre for all the signals 

chosen. This maneuver waa analyzed to obtain control derivatives for the rigidbody response described by -8. (1s) to (31 ). The resulting match between the 

measured and computed response is shown in Pig. 42. The estimated coatrol derivatives are in good agreement with those obtained from the maneuvers. The unmdeled structural dynamic modes are evident, but it is apparent that the des 

vi11 have little effect on the rigid-body control derivatives. The differences 

between the measured and computed rigid-body responses (the residuals) for the 

time close to when the jets were fired are ahawn in Pig. 43. The data shown 

here are for a sample interval of 0.006 8ec. Same persistent structural 

ringing is shown for the two rates and the latemal acceleration. Havever, when 

a jet is fired, the increased structural response is evident, The structural 

coefficients can be extracted directly from the residual as they were for the 

example in the previous section. It appears that there my be some contamination caused by the rigid-body response at the instant the jets fire. If so, 

this contamination can be eliminated in one of two ways: either analyze the portion of the maneuver a tenth of a secocd after the jet fires, or adapt the equations of motion to include the structrlral dynamics in addition to the rigid 

dynamics. 

but the procedure is straightforward. 

same as that used on the example in the preceeding section. It is apparent, 

howe-rer, that more than one structural atode wqwld need to be included in the 

mode 1. 

The structural dynamics depictea in Fig. 43 have not been analyzed, 

The procedure used on this case was the 

All the analysis techniques discussed in this paper apply to the analysis of 

this space shuttle example. 

model, then the linear form of 3s. (5) to (7) would he required. In geaeral, 

If state mise is included in the mathematical 

224 
if the structural partial differential equation can be expressed in the linear 

for= of 4s. (5) to (7) (with or without state noise), the structural modes can 

be analyzed readi:, With the lQlLE 3 program (Ref. 17) in the time domain. If 

the analyst prefers, the problem can be expressed in the linear constant c-wfficient form and analyzed in the frequency domain, as described in Ref. 12. The 

relative advantages and disadventagee of tiredomain analysis as compared with 

frequency-dorain analysis are also diacuseed in that reference. 

tions are nonlinear, but in th~ form of Dqs. (1) to (31, then ~Xim likelihood 

estimates can be obtained in the time domain. 

If the equaThe corputed simple aircraft example shawed the basics of minimization and 

t-.d general concepts of cost functions therselws. In addition, the example 

demonstrated the advantage of low measureant .noise, rultiple estimates at a 

given condition, and the Cram&-Rao bounds, and the quality of the match between 

the measured and computed data. The flight data shcned that many of these concepts still hold true even though the di.ensionality of the cost function makes 

it impossible to plot ar visualize. In addition, the techniques used fx the 

aircraft problem were shown to be applicable to the flexible structure problem. 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

Iliff, K. W., "Aircraft Identificatioc Experience," AGARD-LS-104, 1979, 

pp- 6-1 to 6-35. 

Goadwin, Graham C., and Payne, Robert L., Dymaic System Identification: 

Experiment Design and Data Analysis, Academic Press, New York, 1977. 

Sorenson, Harold W., Parameter Estimation; Principles and Problems, 

Marcel Dekker, Inc., New York, 1980. 

Kl.ein, V., "On the Adequate We1 for Aircraft Parameter EstimEtiCfi," CIT, 

Cranfield Report Aero No. 28, Mar. 1975. 

Hamel, P., "Determination of Stability and Control Parameters Prom Flight 

Testing," AGAFtD-LS-114, 1981, pp. 10-1 to 10-42. 

Iliff, Kenneth W., and Maine, Richard E., "NASA Dryden's Experience in 

Parameter Estimation and Its Use in Plight Teat," AIAA Paper 82-1373, 

Atmospheric Flight Mechanics Conference, San Diego, Calif., Aug. 1982. 

Gupta, N. K., Hall, Earl W., and Trankle, T. L., "Advanced Methods of Model 

Structure Determination from Test Data," AIAA Paper 77-1170, 1977. 

Trakkle, T. L., ViLcent, 3. H., and Franklin, S. U., "System Identification 

of Nonlinear Aerodynamic Models:" The Techniques and Technology of 

Nonlinear Filtering and Kalrnan Filterins, AGARD-AG-256, 1g82. 

2 25 
9. Klsin, V., and Schiess, J. R., "Compatibility Check of Ueasured Aircraft 

Responses Using Kinematic Equations and Extended Kalman Filter, ' NPSA 

TK D-8514, 1977. 

10. 

!I. 

12. 

13. 

14. 

15. 

1 e. 

17. 

18. 

19. 

20. 

21 

22. 

Bach, R, E., and Wingrove, R. C., "AppliCatiOn8 of State Estimation in 

Aircraft Flight Data Analysis," AIM Paper 83-2087, 1983. 

Mulder, J. A., Jonkers, H. Lo, Horsten, J. J., Breemn, J. H., and 

Simons, J. H., 'Analysis of Aircraft Performance, Stability and Control 

Measurements," AGARD-LS-104, 1979, pp. 5-1 to 5-87. 

Klein, Yladislav, "Maximum Likelihood Method for %stimating Airplane Stability and Control Parameters From Flight Data in Frequency Domain," 

NASA TP-1637, 1980. 

Fu, K. H., and krchand, M., "Helicopter System Identification in the 

Frequency Domain,* Ninth European Rotorcraft Forum, Paper 96, Stresa, 

Italy, Sept. 13-15, 1983. 

Haine, Richard E., ad Iliff, Kenneth Wmt "Formulation and Inpleaentation 

of a Practical Algorithm for Parameter Estimation with Process and 

Keasureaent Noise," SIAM J. Appl. Math,, vol. 41, 1981, pp. 558-579. 

Balakrishnaz, A. V., ColarrurAcation Theory, HcGraw-Hill Book Co., c.1963. 

Haine, Richard E., and Iliff, Kenneth W., *The Theory and Practice of 

Estimating the Accuracy of Dynamic Flight-Determined Coefficients," NASA 

RP-; 073 + I981 

Maim?, Rjchard, E., and Iliff, Kenneth W., "User's Manual for HMLE3, A 

General FORM Program for Maximum Likelihood Parameter Ektimation," 

NASA TP-1563, NOVO 1980. 

Maine, Richard E., Aerodynamic Derivatives for an Oblique Wing Aircraft 

Estimated From Flight Data by Usiqg a Naximum Likelihood mchnique," NASA 

TP-1336, 1978. 

Steers, Sandra Thornberry and Iliff, Kenneth W.; "Effects of Time-Shifted 

Data on Flight-Determined Stability and Control Derivatives," NASA 

TN D-7830, 1975. 

Shafer, M. F., "Flight-Dererrined Correction Terms for Angle of Attack and 

Sideslip, " AIM Paper 82-1 374, 1982. 

Maine, Richard E., "Fryrammer'a Manual for MMLE3, A General FORTRAN 

Program far Maximum Likelihood Parametor Estimation, " NASA TP-1690, 1981 

Fink, Marvin P., and Freeman, Delma C., Jr., "Full-Scale Wind-Tunnel 

Investigation of Static Longitudinal and Lateral Characteristics of a 

Light Twin-Engine Airplane," NASA TN D-4983, 1969. 

226 
23. Iliff, K. W.; *Identification and Stochastic Control With Application to 

Flight Control in TUrbulance,* UCLA-QJG-7340, School of Engineering and 

applied Science, Univ. Calif., Loa Angeler, Calif., Hay 1933. 

24, Iliff, Kenneth W., 'An Aircraft Application of Systen Identification in the 

Presence of State Noise,' Pew Directions in Signal Processing in 

Communication and Control, J. K. Skwirzyneki, ed., Nwrdhoff International Publishing (Leyden, The Netherlands), c.1975, pp. 529-541. 

25. Iliff, Kenneth W.; *Estimation of Characteristics and Stochastic Control of 

an Aircraft Flying in Atmospheric Turbulence,' Proceedings of AIM 3rd 

Atmospheric Flight Mechanics Conference, c.1976, pp. 26-38. 

26. Chalk, C. R., Neal, T. P., Harris, T. M., Pritchard, F. E., and 

Woodcock, R. J., 'Backgrounu Information and User Guide for 

MIL-T-878SB(ASG), *Military Specification - Flying Qualities of Piloted 

Airplanes,' AFFDLTR-69-72, Air Force Flight Dynamics Lab., WrightPatterson Air Force &Be, Aug. 1969. 

227 
Table 1 Values of computed tme 

history with no maasureaent noise 

i 6, deg P, deg/sec 

1 0 0 

2 1 0.9754115099857 

3 1 2.878663149266 

4 1 4.689092110779 

5 1 6.411225409939 

6 1 8.049369277012 

7 1 9.607619924937 

8 0 10.11446228200 

9 0 9.621174135646 

10 0 9.151943936071 

Table 2 Pertxnent values as a function of iteration 

- 

L ip(L) (L) 4(L) S(L) JL 

0 -0.5000 15.00 0.9048 2.855 21.21 

1 -0 30C5 9.888 0.9417 1.919 0.5191 

2 -0.2475 9.996 0.9517 1.951 5.083 x 10-4 

3 -0.2500 10.00 0.9512 1.951 1.540 x 

4 -0.2500 10.00 0.9512 1.951 1.060 X 

Table 3 Values of compueedtime history with added measurement noise 

1 0 0 

2 1 C.4875521781881 

3 1 3.230763570696 

4 1 3.429117357944 

5 1 6.286297353361 

6 1 6.953798550097 

7 1 10.80572930119 

8 0 9.739367269447 

9 0 9.788844525450 

13 0 7.382568353168 

228 
Table 4 Fertment values as a function of iterrtron 

- 

- L fp(L) &j (Li O(L) WL) JL 

0 -0.5000 15.00 0.9048 2.855 30.22 

1 -3.3842 10.16 0.9260 1.956 3.497 

2 -0.3518 10.23 0.9321 1.976 3.316 

3 -0.3543 10.25 0.9316 1.978 3.3 16 

4 -0.3542 10.24 009316 1.978 3.316 

5 -0.3542 10.24 0.9316 1.978 3.316 

Table 5 Mean and standard deviations €or estimateo OF Ip 

Sample standard 

Number of Sample mean, Sample standard derivation of the 

cases, N IJ (4) deviation, uiLp) mean, a&)/& A 

5 -0 2668 0.0739 0.0336 

10 -0.2511 0 0620 0.0196 

20 -0 34.52 C 0578 0.0129 

Tble 6 Est-te of 5 and Cram&-Rao bound as 

a function of the square root of noire powst 

Square root of Estimate Cr&r-Rao 

noise power of Lp bound 

U.0 

0.01 

0.05 

0.10 

0.2 

0.4 

0.8 

1.0 

2.0 

5.0 

10.0 

-0.2500 

-0 2507 

-0 2535 

-0.2570 

-0 264 1 

-0.2783 

-0.3071 

-0.3218 

-0.3975 

-0.65 19 

-1.195 

--- ---- 

0.00054 

0.00271 

0.00543 

0.0109 

0.0220 

0.0457 

0.0579 

0.1248 

0.3980 

1.279 

2 29 
Fig. 1 Waxfaurn likelihood -timat ion concept. 

X--- 

(a) neasuremcnt and control surface 

definitions. 

(b) Flow angle definitions. 

Fig. 2 Aircraft axis system. 

230 
Roll roto @) 

Rdllng momrnt (1') 

Y 

(c) simp1 ified aircraft mmezlature. 

Fig. 2 Concluded. 

ltrrrtion 3 r 

0 1 2 

Tlm, aM 

Fig. 4 Comparison of Ipeasured ani 

computed data for each af the first 

three iteratiom . 

0 

L 

1 2 

nm, MC 

231 
20 

Roll rrtr, 

dSg1-c 

P? 10 

0 

20 

10 

It.rrtlon 0 

MWmm - 

-0.25 

- 

I I I 1 

4 

(I 

L 1 2 

flnw. wc 

ltrrtlon 1 

0 1 2 

Tkm, 8.c 

Fig. 6 Comparfson of measured ani 

computed data for each iteration. 

coat 

tunctkn, 

(LP) 

I 5 .5 

Fig. 8 

function If Lp 

Cast fu&ion (J(Lp)) as a 

for m-mise case. 

rig. 7 Comparison 

of tmtimted roll 

rate f roa no-noise 

arrd noisy caaesa 

lwr 

CaTm 

I I 

/!--Q@, 

-50 -1.0 0 1.0 

pig. 9 Coat function as a fuHion 

of L~ for noisy cade. 

ormplno In dl, 5 

232 
la0 

W80- 

70- 

a60- 

40- 

30- 

10 

0 

-10 

-20 

mormpina In roll. Lp DImplno In rolb 5 

Fig. 10 Comparison a€ the cost pig. 11 Gradient af J(LpJ a3 

functiono for the no-noise alrd noby 

cases. 

function of L~ for misq cam. 

- 

- 

- 

- 

- 

-20 -1.5 -1.0 - .S 0 

D.mplng In roll, Lp 

Fig. 12 Cornparisen of NewtonRaphson ad Gauss-NI values of 

the second gradient the noisy 

case b 

lon - 

/- -- T~rntm-~rphI I 1 

-2 -1 0 1 2 3 

Damprn~ In rdl, Lp 

Fig. 13 

ation step slzo for the Ipewtoe 

Raph80n ad Gauss-Neprton algorithm 

for the noisy ,car&. 

Carparison of flrut iter233 


Ad 

onbd 

porr. 

Ld 

16 

15 

14 

13 

?2- 

onw 

m. 'lL* 10 

O& 

15 - 

14 /- 

lo 

't 6 

t- 

- 

- 

- 

- 

Se--- (Cmnir-RUJkud T1 I I I i 

Pig. 20 Isoclirres a€ constaat axpt 

for region mt minium for misy 

case. 

L I 9 A 

Pig. 22 Isoclines ad uncertainty 

ellipsoid of the cost i%mztion for 

the noisy case. 

236 
Pig. 22 space shuttle cold'guration. 

2r 

4r 

?ig. 23 laterrl-directiorral space 

a mch NVlVlr of 21. 

Fig. 24 

slmple calculatfom for the shuttle 

data from rig. 23. 

ExaqpleS of obtaining Lyj by 

237 
- 10 

10 

1 

r- 

- 10 1 i I I I 1 

Q 

- .1 I I J 

R 

-nh, d.01- -4 l!?kIzz 

0 4 81216aD24 

-, UC 

Fig. 25 

shown in Fig. 23. 

MLE3 nmtch of maneuver 

I I 1 I ! I 

-4 I 

-2 

r 

- 10 v 

Pig. 26 Latsral-directional mneuver from P-8 aircraft with augmentation on. 

238 
-40 kzz!kzY 

.1 r 

0 

- .1 

A 

10 -. 

A 

"I'I' 

uv 

0 2 4 6 8 10 12 14 16 18 

Tirrywc 

(a) Zero iteration. 

-40 :& 

0 2 4 8 8 10 12 14 16 18 

M, ne 

(b) One iteration. 

Fig. 27 Hatch between measured and coqpted ti#, histories as a fullction a€ 

iteration. 

239 
-20 Ot!szi&z 

': F*V--- 

- 10 IIU 

0 2 4 6 8 10 12 14 16 18 

Timr, - 

(c) Two iteratiom. 

Fig. 27 

- ': 10 0 bu 2 4 6 8 10 12 14 16 16 

-. ea 

(d) Three iterations. 

contirued. 

240 
Bank 8ngI0, 

cp. 

d.Q 

Yaw rat.. 

1, 

.1 r- 

-2Q :w 

0 2 4 6 8 10 12 14 16 10 

Tlm, Sec 

(e) Four iteratiom. 

Fig. 27 CO~lUdede 

.8 

pw=J - 8 .4 

-d 

- 12 

-2 

Pig. 28 Variations of C, with an3le 

of attack without uncertainty levels. 

P 

24 1 
&r I 

-41 R, 

1 -T I 

Fig. 29 Variatiorrr a€ Cn ulth angle of attack 

P 

with umrtainty levels. 

Fig. 30 Match of flight data obtained Pig. 31 Hatch of flight data obtained 

in turbulence [state mise) and con+ in turbulence (state noise) ami comput& data obtained from .MXiWS puted Eata obtained fron a nmxiaum 

1ikslihaad estimtor that does not 1ikelihM estimator that accounts 

account for turbulences. for turbuleme. 

24 2 
lo’ r 

I 

10-5 I I I \ 

10-2 lo-’ 100 lo1 102 

Fnqurcry. HZ A 

Fig. 32 Power spectral density af 8g 

obtair- from maneuver shown in Pig. 

31. 

lo1 

loo 

10-l 

Fig. 33 Power spectral demity of $9 

obtained fivm maneuver shown in Fig. 

31. 

Q -.2 1 I I I I 1-1 

0 2 A 4 d 1.0 1.2 1.4 

-9 - 

Fig. 34 Structural mode oscillation 

observed on the lateral axeleration. 

, 0 ;I; -A 

0 9 A b .8 1.0 1.2 1.4 

-. BaC 

Pig. 35 ELatch af measured an3 computed lateral acceleration obtained 

when maximum likelihood estimator 

converged to lxal minimum. 

1.11 +ln 1.0t 

1 

abl wt, 

ah oot O 

0 1 2 3 4 5 

W, radlsec 

Fig. 36 Cost functioml for 

10 cycles of data as function 

of frequency, showing close 

proximity of iwal minima 

to global minimum. 

Fig. 37 Simple scalar example illustrating a local minimum similar to 

that shown for flight data in Fig. 35. 

24 3 
$In ut. 

rln wot 

Pig. 38 Simple scalar example showing 

only the first cycle. 

Fig. 29 Cost function for one cycle 

of data as function of f,requency, 

showing wide region of mnvergence 

for global minilaurn. 

Fig. 40 

a lld computed 1 at era1 a ccelera t ion. 

Acceptable match of measured 

244 
4r 'r 

4 

= -4 OF,T7n-; 

*04 i- I 

Lateral 

Q 

$*>; Roll ratr, 

R 

Wl* - .2 

0 .0 1.8 24 3.2 4.0 4.8 

Th, erc 

Fig. 41 Dynamic response of space 

shuttle to firing of roll and yaw 

jets at an altitude of 430,000 ft. 

Roll jets, 

numkr 

Yaw job, 

number 

Fig. 42 

0 4L-J-- 

-4 - 

I 

OW 

-.04 1 I I 1 

- .2 LII 

Maximum IiJrelihOod match 

of rigid-body response of the space 

shuttla. 

245 
.1 #-- .I IYaw rate I Jdual, 0 dw- 

(a) Roll jet. (b) Yaw jet. 

Fig. 43 Differ?. ..? _* en measured am- computed rigid-body response 

(residual) for 8. 5: 3 shuttle. Altitude = 43Or000 ftz dytnmdc 

pressure - 0. 

24 6 
i N85-31209 

OPTIMAL SENSOR LOCATIONS FOR 

STRUCTURAL IDENTIFICATION 

F. E. Udwadia 

University of Southern California 

Los Angela, CA yooS94242 

Jet Propulsion Laboratory 

California Institute of Technology 

Pasadena, CA 9! 109 

J. A. Garha 

SUMMARY 

The opthiurn sensor 1-cation problem, OSLP, may be thought of in terms of the 

set of systems, S, the class of input time functiocs, 1,. and the identificatior. 

algorithm (estimator) used, E. Thus, for a giyen time history of input, the 

technique of determining the OSL requlres, in general, the solution of the optimization and the identification problems simultaneously.. However, this paper 

introduces a technique which uncmples the two problems. 

of the concept of an efficient estimator fgr which the covariarce of the parameter 

estimates is inversely proportional to the Fisher Informatlon Matrix. 

This is done by means 

INTRODUCTION 

The problem of Ftructural identification in structaral engineering is one 

which has received considerable attention from several researr-sers in the recent 

past (Refs. 1-4). Though various methods have been developed for identifying the 

difierat parameters that characterize a structure from records obtained in them 

under various loading conditions, few investigators, if any, have looked at the 

question of where to locate sensors in a structure to -iequA ? data for "best" 

parametric identiffcation (Ref. 5). The problem of optimally locating sensors in 

a scructursl system arises from considerations of. 

instrumentation; ana (2) efficiently detecting structural changes in the system 

with a view to acquiring improved assessnent of structural integrity. 

(1) minimizing the cost of 

The problem addressed in this paper can be stated as fdlows: Given m senscrs, ..here should they be located in a strucfxre so that records obtained from 

those 1oca:ions yield the "best" estimates of the unknown parameters? 

In the past, the optimal sensor location problem (OSLP) was solved by 

positioning the g2.m number of sensors in the system, using the records obtained 

At those lccations with a speciric ezfIm.tor, and repeating the procedure for 

different sensor locations. 

estimates would then be selected as op'cimal. The essimates obtained, of course, 

would rraturally depend upor. the type of estimator uied. 

tions are estimtor dependent, and .an exhastive search needs to be performed for 

each sp:c+.fic estimator. Such a procedure, besides being highly computation2 *r 

intensive, suffers from the rnajr,r drawb'qck of not yielding any physic21 inri 

into why certain locatlons are preferable to others. 

The set of locations which yield the "best" parameter 

Thus the optimal locaRecently, work c the solution of the OSIP was done by Shah and Udwadia 

(Re:. 5). In brief, .hey used a linear relatimship between small percurbarions 

24 7 
in a flnite dimensional re9resentation of the system parameters and a finile 

sample of observations of the system the response. 

estimates are minimized, yielding the optimal locations. In this paper,vedevelop 

a more direct approach to the problem which is both computationally superior, and 

throws c..qsiGerable light on the rationale behind the optimal selection process. 

The error in the parameter 

We uncoupia the optimization problem from the identificatton problem using 

the concept of ari cfiiciect estimator (e.g., the maxFmum likelihood estimator as 

tine becomes very large!. thtor the covariance of the parameter 

estima es is a minimum. 

ment;, a rigorous for= lation ad solutim of the OSLP is presented. The method 

is applied to a building structure mcdelled as a general linear dynamic system. 

For the N +egree of fre2dom system consida-ed, the methodology for selecrkig 

m(m<N) of the nodai displacements for purpost: of measuramt is presented. 

For such an 

Using this technique and motivated by heurlstic arguSample calculations are made for a simple building c;rTu?ture modelled as a 

two-degree-of-freedom system subjected to base excitations. The ~~?-rll sensor 

location for the identification of: 

ratio is investigated. 

(a) the mass ratia; and (b) the stiffness 

The results indicate Khat the OSLP depends on: 

i) the class of systems, S, to which the structure belongs; 

2) the type of excitation; 

3) the actual system para-Aers involved; and 

4) the parameters to be identified. 

THEORY 

Consider a system modelled by the equation 

.. 

MX + Ci + KX = F(t) 

where M, C, and K are the (NXN) mss, damping and stiffness %trices, F(t) is 

an (Fix11 vector containing inertial forces and extei.dly applied loads and X 

is thz N-vector of nDdal displacements. Let BM, Bc and 8K be vectors containing 

the various parameters related to the mass matrix, the. damping ma-rix an8 the 

stiffness matrix, respectively, which need tc, be identified. For convenience, 

we collect these quantities in the parameter vector, e, defined as 

% 

where the superscript T indicate matrix transpose. If the M, C and K are 

symmetric each of the three scbvectors has a maximum dimension of N(N+1)/7. 

Given m sensors (m<N), we $hen need to find where to locate them so that 

the covariance of the estimate, 6, is a minimum. 

measurement vector Z(t) can be expressed as 

Assutz further that the 

24 8 
where Zi is the ith component of Z(t), and tte functionals gi represent the 

"measurement process". 

vector 8 is explicitly noted The aeasurement noise Ni(t) is taken as nonstationary Gaussian White noise with a variance of J12(t). Therefore, 

The dependence of the response X on the parameter 

where 5K and 6~ stand for the kroneker and the dirac-delta functions, 

respectively. 

they contain the nost inforuation abcut the system parameters and are maximally sensitive to any changes in the parameter values. 

process can be represented by an m-dimensional vector Y such that 

A total of m out of N responses need to be selected so that 

This "selection" 

vhere S is the (m x N) upper triangular selection matrix witk each row containing null elements except for one which is unity. The m different components of Z selected to be aeasured sre so ordered in vector Y, that if the 

element in the i-th row and k-th column of S is unity, the (i+l)-ith row has 

unity in its E-th column with E > k. 

P = STS in an (NXN) diagonal matrix with unity in its i-th row if, and only 

if, Zi is selrctea t3 be measured. 

Hence, me can write 

The matrix S has the property that 

The elements of P are otherwise zero. 

If gi is linearly related to ;he response X in general, then 1' 

r,ix(e,t)l = SRX (6) 

where Rit) car, be thought of as a dynamic gain matrix. In the case that gi 

is relatee t? :he resporss Xi oAy, then matrix Rwill reduce to a diagonal 

matrix, r3j J 

The prcblem cjf locating sensors in an optimal mamer then reduces to 

~letrrminin~ t\e se1ic:ion matrix S, or alternatively, finding the m locations 

iri P that Thould be unity. These locations must be so chosen as to obtain the 

bc s t 'I pa rame t e r es t ima t e s . 

5GME MOTIVATING THOUGHTS AND THE FISHER INFORMATION MATRIX 

Consider a Lase in whi-b one tries to estimate only one parameter, 91 (to 

be Identified) involved in a dynamic system model with only one sensor provided. 

Therefore, one dants to ideally choose a location i (out of N possible such 

locations) such that the ineasurernent yi(t), ie[l,N], te(0,T) at location i yields 

the best. estimate of the parameter 81. Heuristically, one shculd place the 

249 
sensor at such a location that the time history of aeasurements obtained at that 

locatim is nost sensitive to any changes in the parameter 81. He-ce, in equation (5B) it is really the slope of H[X(el,t)] with respect to 81 that needs to 

be maximized. 

interest, it is logical to wanr i find i (or equivalently determine the selection matrix S described previousl?) such as to maximlze (aH/a81)2 over the interval (0,T) duing which the response is to be measured. This leads to naximizing 

the following integral: 

However, since onlv the absolute magnitude of this slop* is of 

When there is more than one parameter to be estimated, and the number of 

sensors is greater tban unity, this intuitive approach aeeds to be extended i? a 

more rigorous aanner. In such cases recourse to mathematical treatment is 

necessary, and Le shall see that. such treatment will be in agreement with our 

heuristic solution outlined above. 

To further understand the problem, let us loo': at it from another angle, 

namely, the concept of an efficient unbiased estimator. 

che covariance of the estimates is a minimum. Furthemre, it can be shown that 

for any unbFased estimator of 8, 

For such as estimator 

where 6 is the estimate of 8 and the matrix [aH/ae],j p aHi/Wj. 

=tor is "efficient", the above inequality becomes an equality. 

:he left-hand side of inequality (8) takes its lowest value (minimum covariance). 

Hencs , 

If the estiThis means that 

The term inside the bracket on the right-hand side of the equation (9) is known 

as the Fisher Information Matrix, Q(T). 

to a ainjmization of the covariance of the estimate, 8. ' 

Thus, maximi:ing Q(T) would indeed lead 

sj2 note then that the m sensor locations need to be so chosen that a suitable 

nom L thc matrix !?(TI given by 

250 
is maximized. 

tically derived earlier for the scalar case, to the vector situation. 

equation (6) in equation (10) one may write 

This constitutes an extension of equation (7), which we heurisIntroducing 

where the ij element of X can be written as: e 

where X = xi)^ and 8 = {ei),. We note that the Fisher Matrix is symmetric and 

is dependent on ihe length of the record avaiiable, 2s well as the locations of 

the sensors as determined by the matrix P. 

If the m locations where the sensors are to be placed are denoted by 

sk, k = 1,2,. . ,m, then 

P=C I 

k=l 'k 

where the (N x N) diagonal matrix' Isk has all Its elements equal to zero excepr: 

the element of the Sk row, which is unity. Noting that P is a diagonal matrix, 

equation (11) can be simplified to yield 

where rSk is the Sk row of the matrix R. 

made of the dependence of the Fisher Matrix on the time length T of the available 

data, the sysiem S, the parameter vector 8, and the time-variant input I. If the 

matrix R is diagonal, with diagonal elements PI,. . . ,pN, then the ij element of 

the matrix Q, after some manipulation, reduces to 

Also in eq. (13) explicit mention is 

Each element of Qij represents the cross-sensitivity of measurement with respect 

to the response x of node sk. 

k S 

25 1 
The cptimal sensor locations are then obtained by 

k = 1,2, ..., m, out of a possible N, so that a suitable 

maximized (e.g., the trace norm, etc ...). This may be 

cond it ion 

picking m locations sk, 

norm of the matrix Q is 

spec if ied by the 

Although there are several matrix norms which could be used, perhaps the 

most useful and physically meaningful in this context Is the trace norm. 

order not to detract the reader from the basic methodology we defer an exhaustive 

treatment of suitable matrix noms to a later cormnunication. 

In 

The methodology presented up to this point is valid for both linear and nonlinear systems since the criterion developed in equation (13) was derived using 

only equations (5) an? (9). We will now indicate its application to linear 

multi-degree-of-freedom systems. 

APPLICATION TO LINEAR DYNAMIC SYSTEMS 

Consider the N-degree-of-freedom dynamic system whose governing diffe5ential 

equation of m2tion is given by eq. (11, together with X(to) = Xo, X(to) = Xo, 

where Xo and Xo are the given initial conditions for the system. 

system to be classically damped. 

Assume the 

Introducing 

where 9 is the (N x N) weighted modal matrix and n(t) Is the N-vector qf 

generalized coordinates we ge: 

(17) .. T T 

q+2SNwN;HA'l = 4 F(t), dtO) = +TMXo, (to) = 4 Go, 

where the (N x N) diagonal matrix A is given by 

The so1utic.i of equation (17) is given as 

t 

'0 

where n and no are initial conditions and 

Oi i 

252 
ui(t> = ED(-Siwit) 

v.(t) = - 

i "d 1 

Sinwd t , i 

cui 

di 

Coswd t + - 

w i 

pi(i> = oTF(t), i = 1,2 ,..., N. 

Also, differentiating equation (1) with respect to 8, yields 

/\. /\. A f.di. e +Cie+KXe = Fe(t) - (MeX+CeX+KeX ; Xe(0) = 0, ie(0) = 0 

where 

i = 1, ..., N, and j * 1, ..., L. 

Introducing 

x = 4z e 

yields 

z+2FN%;+hz = G(t) 

where 

G(t) = @T[F6-(MeX+CeX+KeX)] A. A. P . 

Equation (21) can further be simplified to give 

253 
where i and can be obtained by differentiation of eq. (18). This may be shown 

as follgws 

where 

t 

rli(t) = rl w ( t - t o) +;I oi T i( t-t ()) + j i+)Pi(T)dT 

Oi e 

where 

254 
T pi(t) = F(t), i = 1,2 ,..., N. 

Therefore, substituting equations (23) and (24) into equation (22) gives G(t). 

Consequently the solution of equation (20) can be written as: 

ij 

where hi(t) is the same as that of eq. (18). Notice that the initial conditions 

in eq. (20) are zero. This is due to the fact that the initial conditions of 

(18) are known constants. 

If we assume that [C] is expressed as a linear combination of [K] and [MI, 

then eq. (22) can further be simplified. Namely, 

C = 2aK-t2$M, (26) 

where a and $ are known constants. Hence in equation (17j, the percentage of 

damping, C, can be expressed as: 

, i = 1,2 ,..., N B 5, = aw + - i wi 

To further simplify equation (22) under this assumption, let us consider the 

Following three cases: 

1) The vector 8 contains only 8M, i.e., only estimation of mass parameters is undertaken. Then 

2) The vector 8 contains only the subvector eK. Then 

255 
3) Finally if th? vector e = [a B] T , 

(28B) 

Lf the input F(t) is not a function of 6, thon Fe would be omitted all through 

this discussion. Once the solution of equation (25) is obtained, the Fisher 

&trices may be obtained as in equation (13). Hence 

TTT T z @ rs rs @z 

dt 

q2W Q=2 f 

L= 1 0 

Je note that the summation form of relation (30) is particularly amenable to the 

maximization of the trace nOrm of Q. 

256 
EXAMFLE 

To illustrate some of the ideas of the previous section, consider the prchlem 

of finding the optimal sensor location (OSL) in a stractural system modelled by the 

two-degree-of-freedom system (shown in Figure 1) which is subjected to the base 

excization of f(t). 

The governing differential equation of motion can oe expressed as 

.. 

M + Ci + KX = -W f(t) 

where X = <x x s~, C= cxK, W = <Am and the matrices M and K are 12 

A case study for locating sensors to best identify (1) the mass ratio, A, of the 

first to the second floor and (2) the stiffness ratio, B, ;If the first to the 

second flcor, will be presented. 

Let si denote the lower mass location and s2 the upper mass location. 

selection between the locations can be equated to determining the one non-zero 

element of the [1x2] selection matrix, S, with the measurement H(t) defined by 

The 

H(t) = SX + V(t) , 

where, Vlt) is Stationary Gaussian White Noise (S G W N) with Jl(t)=JIo. 

If S = E1 01 the lower mass is selected for measurement; if S = [G 11 the 

clpper mass is selected. 

tion 92 for identifyixg the parameter A, if Q[T,sl] > Q[T,s2], where T is the 

time that the measurement is taken, 

The location si would then be preferred aver the loca2 

=L 2 [ (2) dt, 

d'O 

(32A) 

2.57 
and 

Since only one parameter is being estimated the Fisher matrices reduce to 

scalars. 

The dependence of the OSL on various types of the base excitations can be 

studied now. Let us for this presentation consider ground acceleration ir? the 

form of a delta function, i.e., f(t) = 6(t). 

In this case, closed form solutions foi 0' and Q, can be obtained. 

For the OSL problem for the "best" (minimum covariance) identification of 

1 

the parameter A (given the parameters B and u) using an impulsive base excitation, 

Figure 2-A shows the plots of th;. raKio of the information matrices Ql(T)/Qz(T), 

for T = 50 secs, for various values of the parameters A (which is to be idsntified) and a* 0 awo, where uo Points on the graph with ordinates greater 

than unity indicate the opth-1 location to be the lowx mass level and vice 

versa. The graphs indicate that the optimal location in most cases, for the 

range of A Considered, is the upper mass level. However, we obse -ve :bat for 

some small values of A and a* the OSL is the lower level. 

enough, that the optimal sensor lccation for identification of 4 actually depends 

not only on the actual values of B and a which are presumably known, 'nut also on 

the value of the parameter A itself which is to be identifier!! 

to ascertain the optimal sensor location some a priori assessment of A is 

necessary. 

m. 

We note, interestingly 

Thus to be able 

Figure 2-B shows that the optimal location for identification of the parameter B (given A and a), using an impuisive base input, is again the upper mass 

level for the range of B values considered. For larger B values, however, and 

a*>0.05, the ;rend appears to be more and more in favor of the upper mass. This 

seems intuitively correct, for as B becomes larger, the lower part of the system 

becomes stiffer and the OSL would be the upper mass level. 

Figure 2-C is associated with the OSLP for estirating the parameter B iisir~: 

a sinuscLa1 base excitation, f(t) = a sin ut. The figure shows that as the 

normalized driving frequency y = u/wo varies, the OSL changes. For this example 

the Fisher Matrices can be computed in closed form. For the estimation of B, 

(g'ven A and a* = 0) the dimensiLn3.eso driving crequency y = yields no 

information on B from records at either of the two mass lzvels. The responses at 

the two mass levels yield identical amour,'^ of information on B at y = 0 and 

y = fi for A. # 1, as indicated by the values of 41/42 = 1 at ttese frequencies. 

The value of 41/42 = 0 at y = 1 is indicative of the fact that the upper mass 

level is ,. far better location for a senpor when estimating B with a* = 0. 

Figure 2-LI shows the mean value of the rLtio Ql/Q2 for a random Gaussian white 

noise base excitation together with the 1-CI band. 

upper mass level for identification of A. 

The OSL appears to be at the 

258 
CCINCLCSIONS AND DISCUSSION 

This paper presents a general methodology for determining the optimal sensor 

locations in dynamic systems for obtaining records vhich would enable the "best" 

(minimum covariance) identification of a given set of unknown parzmeters in the 

system. The technique utilizes the concept of an efficient estimator to uncouple 

the identification €rom the optimization problem. 

?.de& in as clear a fashion as possible, we have restricted the discussion in this 

sequel tc linear systems. 

In order to present the basic 

The method has been illustrated by application to a two degree of freedom 

system. Though the results presented here for the simple system chosen form only 

a first step towards acquiring a detailed understanding of the OSL problem, the 

follQwing conclusions appear t3 be relevat at this time: 

(1) The OSL for a given sys'tem heavily depends on the class of fx-cing 

functions used for obtaining response data. In this study, an impulsive base mtion is considered. 

(2) The OSL for linear dynamic systems is independent of the amplitude of 

the forcing function. 

(3) The OSL depends in gweral on all the values of system parameters. For 

instance, the OSL for estimating A with minimum covariance depends not 

only on the actual parameter values B P.? a but on the vaJ.ue of A itself for the system! 

identifying a given parameter (or a set of parameters) in a dynamic 

system necessitates the Lnovledge of soae a priori estimates of the 

unknown parameter (s) . 

This implies that the OSL problem associated with 

(4) ihe results of OUT simple example show that ihe OSL problem may yield 

solutions which may be dirficult to predict on purely heuristic 

grounds. 

problem, in a rather cmplex manner on the actual parameter values of 

the system and the nature of the base excitation. 

The OSL appears to depend, even iur this relatively simple 

REFERENCES 

Evkhoff, P., "Process Parameter and State Identification", Pc-i. IFAC Congress, 

LL'ndon, 1966. 

Hart, G.C. ed., "Dynamic Response of Structures: Instrumentation, Testing 

and System Ident if rcat ion", ASCEIEMD Specialty Conf erknce Proceedings, 

U.C.L.A., 1976. 

Hudson, D.E., "Dynamic Testing of Full Scale Structures", Firthquake Engineering, Prentice Hall, pp. 127-149. 

Udwadia, F.E. and Sharr, P.C., "Identification of Structures Through Records Obtained During Strong Gro-.ind Shaking", ASME, Journal of Engineering for Industry, 

Vol. 98, NO. 4, 1976, pp. 1347-1362. 

259 
5) Shah, P.C. and Udwadia, F.E., "A Methodology for Optimal Sensor Locations for 

Identification of Dynamic Systems", Journal of Applied Mechaics, Vol. 45, 

March 1978. 

Figure 1. 2-degree-of -f reedom 

generic structural 

systep. 

t 

1.5 c 

0.00 I I I I i I I, 

0.00 1.5 3.00 

A 

Figure 24, Variation of Q1/Q2 for V&-,OUS 

values of the parameter A. 

Ql /Q2 greater than unity 

indicates that optimal is 

at 1c.ver raass. 

I 0" 0- _L *Irn 1.0 

Q2m 

0. I 

0.00 

[LL 

L 

L-U- 0.00 1.5 3.00 

Figure 2-9. Variation of Ql/Q2 for various 

values of B. 

260 
Flgure 2-C. Variation of 41/22 with y 4 iu/wo for differc7t a* gicen 4=&=m=l, 

k=100 and f(t) = Smut. 

0.30 "F 0 - 

A 

F:%u. z 2-3. Variari.or\s in Lt,e mean v.. .ie of Qi(T, !Q2(T) and the 1-c band with 

li'fererit values of A, w%w R=m=l, ic=lOO, a*=C.l. LP-O, ir.puc is 

Saussian knite h'oise. I.ntegrstian was done over , c.ec second 

period I 

26 1 
COMBINED STATE AND PASAMETER ESTIMATION 

FOR A STATIC MODEL OF THE MAYPOLE 

(HOOP/COLUMN) ANTENNA SURFACE 

H. T. Banks 

Lefschetz Center for Dynamical Sy.,tems 

Brown University 

Providence, RI 02912 

P. K. Lunm 

Southern Methodist University 

Dallas, TX 75275 

E. S. Armstroag 

NASA Langley ResePlrch Center 

Hampton, VA 23665 

Parameter and state estimation techniques are discussed for an elliptic system arising in a developmental model for the antenna surface of the 

Maypole Hoop/Column antenna. A computational algorithm based 311 spline 

approximations for the state ad elastic parameters is given and numerical 

results obtained using this algorithm are summarized. 

Results are presented from a Langley program directed towards 

developing computattonally efficient identification techniques for flexible 

systems modeled by partial differential equations with an emphasis on large 

space structures. Initial efforts have been directed towards extendidg the 

spline-based theory and computational techniques used by the first tw 

authors [11-[6] in solving identification problem with delay and partial 

differential equation eels in one spatial variable to solve distributed 

problems in several spatial variables. Additionally, ir. order to support 

Langley's technology development program [7] in large space antennae, a 

narameter aad state estimation algorith has been derived for a prototype 

distributed model of the Maypole (Hoop/Column). antenna reflector 

surface [8]. Thz next section describes the Hoop/Colum antenna and presents the identificotion problem being considered. The state and pazawter 

estimation approach is rhen outlined and rliscuased in tne :.antext of the 

Hoop/Column application. Subsequent sections include mathematical details 

of the antenna application and numerical results. 
For the purpose of technology development, the NASA Large Space 

S:rstems Technology (LSST) program off ice has pinpointed focus missions and 

identified future requirements for large space antennas for communications, 

earth sensing, and radio astronom; [7]. In this study, particular emphasis 

is placed on mesh deployable antennas in the 50-120 meter diameter category. One such antenna is the Maypole (Hoop/Column) antenna shown for the 

lOOm point-design in Mgures 1 and 2. This antenna concept Is being developed by the Harris Corporation, Melbourne, Florida, under contract to the 

Langley Research enter [8]. 

The Hoop/Column antenna consists of a knitted gold-plated molybdenum 

wire reflective mesh stretched over a collapsible hoop that supplies the 

rigidity necessary to maintain a circular outer shape. The annular 

membrane-like reflector surface surrovnds a.telescoping mast which provides 

anchoring locati-ns for the mesh center section (Pig. 1). Thq mast also 

provides anchoring for cables that connect the top end of the mast to the 

outer hoop and the bottom end of the mast to 48 equally spaced radial 

graphite cord truss systems woven through the mesh surface [8]. Tensions 

on the upper (quartz) cables and outer lower (graphite epoxy) cables are 

counter balanced to provide stiffness to the hoop structure. The Inner 

lower cables produce, through the truss systems, distributed surf ace loading to control the shape of four circular reflective dishes (Pigs. 1 and 2) 

on tLle UBL,.~ surfacer 

After deployment or after a long period of operation, the reflector 

surface may require adjustment. Optical sensors are to be locatc3 on the 

upper mast which aeasure angles of retroreflective targets plactd on the 

truss radial cord edges on the antenna surface. This information can then 

be processed using a ground-based computer to determlne a data set of values of mesh surface location at selected targeb points. If necessary, a 

new set of shaping (control) cord tendione can be fed back to the antenna 

for adjustment. 

It is desirable to have an identification procedure which allows one 

to estimate the antenna mesh shape at arbitrary surface points and the 

distributed loading from data set observations. It can also be rinticipated 

that environmental stresses and the effects of aging will alter the mesh 

material properties. The identification procedure must also allow one to 

address this issue. 

Considering the antenna to be fully deployed and’in static equilibrium, a distributed mathematical model whick deocribes the antema surface 

deviption from a cbrved equilibrium configuration is under investigation 

(for preliminary findings, see [9]). Using a cylindrical coordinate system 

with the z-axis along the aast, it is expected that the resulting model 

will entail a system 02 coupled second-order linear partial differential 

26 4 
equations in two spatial variables. The coefficients of these eqwrtions 

are functions of the material properties of the stretched mesh. The derivation and computer software for this model are still under dPvelopment. 

In the meantime, a simpler developmental (prototype) problem has been 

solved which is descriptive of the original problem. 

For the developmental problem, the loading is assumed to be normal to 

the horizontal plane coataining the hoop rim, and tbe mesh surface is 

assumed to be described by the static two-dimensional stretcbd membrane 

equation [ lo] with variable stiffness (elastic) coefficients and appropriate boundary conditions for th2 Hoop/Coluun geometry. Hathematically, in 

polar coordinates, we have 

where u(r,O) is the vertical displacement of the mesh froa the hoop 

plane, f(r,Q) is the distributed loading force per unit area, and 

E(r,O) > 0 is the distributed stiffness (elastic) coefficient of the meoh 

surface (force/unit length). Efvation (1) is to be solved over the annular 

region Q = [E,R] x [0,2a]. Appropriate boundary conditions are 

along with the periodicity requirement 

where R is the radius from the mast center to the circular outer hoop, 

c is the radius fraa ti= mast to the beginning of the mesh surface (see 

Fig. 21, and uo ia the coordinate at r = E of the meah surface below 

the outer hoep plane. 

We further assume that the Jistributed loading slo.ng with a data set 

of vertical displacements, u (r 8 ), at selected points (r,,e,) on the 

mesh surface is known. Given this ‘-nform*+ion, the developmental problem 

is to estimate the material properties of the mesh as represented by 

E(r,O) anti produce state estimates of the eurcace represented by u(r,e) 

at arbitrary (r,O) points within n. The procedure applied to solve this 

problem is discussed in the next section. 

m i’ j 

26 5 
The first two authors and their colleagues have derived techniquec for 

approximating the solutions to systems identification and control problem 

involving delay equation models and partial differential equation models in 

one spatial variable and have used them in a variety of applications 

[ll], (121. The Hoop/Column application requires an extension of the theory 

and numerical algorithms to elliptic distributed systems in several spatial 

variables. The approach, when specialized to the system identification 

problem, may be 6' marized as follows: (1) select a distributed parameter 

formulation containing unknown parameters for a specific system; (2) mathematically "project" the formulation down onto a finite dimensional subspace 

through some approximation procedure such as finite differences, finite 

elements, etc.; (3) solve the identification problem within the finite 

dimensional subspace obtaining an estimate .dependent upon the order o€ the 

approximation embodied in the subspace; (4) successively increase the order 

of the approxima+.ion and, in each case, solve the identification problem so 

as to construct a uequence oi parameter and state estimates ordered with 

increasing refinement of the approxiaatiun scheme; (5) seek a mathematical 

theory which providcs conditions under which t!e sequence of approximate 

solutions approaches the distributed solution as the subspace dimension 

increases with a convergent underlying sequence of parameter estimates. 

In applying this approach to the developmental problem, the stiffness 

function is parametrized in terms of cubic splines of fixed order; thir 

converts the estimation of E(r,6) into a finite dimensional parameter 

estimation problem. After writing the energy functional generic to the 

membrane equation, the Galerkin procedure is used to project the dtsttibuted formulatian onto a finite dimensional state subspace spanned by tensor products of linear spline functions defined over n. The approximate 

displacement (state estimate: thus obtained is expressible in terns of the 

spline basis fuwtions. The Galerkln procedure in this case yields algebraic equations which def :ne the displacement approximation coordinates in 

tei- of the unknown E(r,8) parameters. In order to solve the approximating parameter estimation problem, the parameters defining E(r, ct) are 

chosen so that Y -east squares measure of the fit error between the 

observed and predicttd (by the estimated state) data 3et is minimized. 

Finally, following steps (4) ad (5) an algorith Is constructed to 

determine the order of the linear spline approximation above which little 

or no further improvement is obtained in the unknown quantities 88 one 

increases the dimension of the subspaces. Details of this system 

identification approach 3re ?resented in the following sections. 

266 
Prior 

dimensional 

cor?diticns 

dependent v 

to applying the Galerkin procedure [13,14] to perform the finite 

approximation for the developmental problem, the boundary 

(2) are converted to homogeneous form by introducing the new 

fariable 

Equation (1 ) then becomes 

with boundary conditions 

Iollowing the standard formulation (nee 113,141) for the weak or LI 

variational for- of (51, the energy functional E associated with (5) is 

where 0 is the gradient in polar coordinates which, in the form used 

here, I, equivalent to 

H 

The function f is giber by 

h, rE( r ,e )u, 

f(r,e) - f(r,e) +-- r lar (E-R) 

and the vertical displacement z(r,e) of the mesh surface awq from the 

hoop equilibrium plane is a function satisfying the boundary conditions (6) 

and posseszing first derivatives on Sl in the distributional sense (we 

denote thie by (Sl) f 2). The first variation 6; of ; about 

the function y(r.0) is given by 

%,per 
= i2'lR {E(r,B)Vy V-r - [f(r,e)v + E(r,e)Z Vv]) rdrde (10) 

where 

k= (;I ($) (11) 

(0) and v is an arbitrary function in Z - .Ho,per 1. 

A 

Given a finite dilaen810d subspace 2 of 2, the Galerkin procedure A A 

defines the approximation y as the solution in Z of 

*a 

for all VEZ. 

For computational efficiency, the baels functlons used for the 

representations of y in (12) are taken as tensor products of linear 

B-splines ([13], p. 27; [14:, p. 100). Thus v and y are in the space 

spanned by 

a 

A A 

- aNw, , (j 5 j . 

.*,N - 11, M where aM - ai(r), (i = 1 ,..., M - 11, and i 

are standard linear B-aplines with knots uniformly spaced over and 

!0,2n], respectively, modified to satisfy the appropriate bocndsry 

condi t ions. The elements {a:} %re modified to satisfy homogeneous 

boo ' iry conditions while has been altered to satisfy- periodic 

boundary conditions [ 151. 

f&,R] 

8: 

26 8 
For yMsN(r,O) within the subspace spanned by v ’pN we can write 

il 

Replacing j(r.0) in (12) by yMSN(r,Q) from (14) and successively settiaig v(r,O) = vMBN(r,Q) for i = l,...,M - 1 and j = l,...,!l leads to 

a set of high-order linear algebraic equations for ths 

We avoid sparse matrix methods in s.olving the 

ij 

M*N coordinates. 

MsN eqdation by wij imposing a separability condition: 

As shown in [15], condition (15) reduces the 

~0.1uti~i of the matrix equation 

wid MsN calculaticn to the 

with 

269 
and 

where, in (17)-(22), i,p = ls.oosH - 1 and j,q * 1 ,..., N. 

Equation (i6) is rewritten in the equivalent form 

and solved by the Bartels-Stewart algorithm [161. 

In order to estimate, via a numerical scheme, the functional 

coefficients E1 and E2, we parametrize these functions so that 

idertification is performed over a finite-dimenaional parameter set. To 

this end, let 

where vk and are scalar parameters and 

cubic B-spline funct on8 defined 113, p. 611 over 

respectively, whose orders are independent of M and 

2 & and lJ are 

(€,R] and f0,2nl, 

N. The basic spline 

functions are modified so that 

boundary conditions. uj and its derivatives satisfy periodic 

We turn next to the computer implementation of the identification 

scheme. 

2 90 
Appealing to the ideas found in previous sections, we now detail an 

algorithm for estimating thc coefficients Vk, k - I,..., MI and 

j = 1, ..., NI, for E(r,8) that provide the "best fit" between 

estimations of the state u and observed data obtained from various 

sample poince on the surface. We may equivale3tly consider data for y by 

making the transformation 

for i = l,..., L, and j = I,.-., LQ. 

A parameter estimation algorittrm may 'k organized into the following 

steps. 

1. Select an order of approximation for the cubic spline elements 

N1, used to ";is j = lB***s 

Ah, k = I,.. ., M1 and 

represent E1 and E2. Set n = . 2. Select M and N, .r: number of the linear spline basis elements 

used to represent UNSN (and PsN). 

3. Assume a nominal set of values for 

v - (vl, V*,...s 'MI) (27) 

and 

6 - pl, 6* ,..., 6 . (28) \ N1) 

4. Calculate the ioefficient matrices in (23) and solve for 

#sK( V, 6). 

5- Calculate, from (141, f19N(r4L,@j; \:,a) and evaluate 

6. Proceed to step 8 if J'*N(u,6) €8 sufficiently small. 

Otherwise, through an optimization procedure, determine a net pair 

(v,6) which decreases the value of J . If no euch pair can M,N AI 

be found, go to step 8. 

271 
A7. Set (v,&) - (V,6) and return to step 4. 

8. Preserve the current values of JHsN and the corresponding 

(v,6) yair as the hith entry in a sequence of these pairs, 

ordered with increasing N and N. 

9. Proceed to step 10 if sufficient data has been obtained to analyze 

the sequences. Otherwise, set G = n 4 1 and return to step 2 

with increased M and N. The current values of (V,6) Will bt 

used ig initial values for the next optimizatdcm process. 

10. From analysis of the numerical sequences, eelect the (M,N) entry 

which indicates the beet numerical results. The correspoading 

parameter estimate (u, 6) pair yields E(r,B) which determines the 

material properties of the antenna'.mesh. The matrix drN( V, 61, 

when used in conjunction with (141, determines a state 

approximation yMrN for the shape of the antenna surface. 

A convergence thiory for the identification algorith may be found 

in [15]. Numerical results are described in the next eecrion. 

Experimental data for the Hoop/Column antenna is not available at this 

time. Therefore, synthetic data is constructed to deaonetteee the 

preceding algorithm. 

As shown in Figure 2, the parent reflector has four separate area8 of 

Illumination on its surface. Each area is assumed to have tk same 

parabolic shape. For 0 < 0 < - and E 5 r 5 B- - -2 
where 

ql(0) = stnO + cos0 . (31) 

The functions q2!0) and q3(0) are cubic poljaomial fits used to 

ensure smoothness in regions of 0 near (3 = 2 , f, 2 3n , 2n. Formulae for 

qZ(0) and c3(0) may be found in 1151. The parameter k > 0, a 

stretch factor used to perturb the surface below the conic (k = 0) shape is 

taken as 0.25. 

f 

For the complete surface, we defice, fnr E L r R, 

It is expected that the mesh will be stiffest near the outer hoop 

(r = R) and around the Fnr\er radius (t = E), For this reavn we choose a 

known value of El(r) as 

where T is a constant dependent on the mesh material. The stfffneee In 

the angular dtrecion 18 exrectod to oe uniform with 

From data rrovld :8] for the IO’l-meter point design, a reasonable 

value for (given in units vGG is 

A 

T = 3.391 ; (35) 

similarly, other parameters are calculated to be uo = -7.501, E u P.2351~ 

and R - 50m. 

273 
A 10 x 24 grid of data points u+ri,€)j) is calculated by 

- 

e.-aluating u(r,O) at points (ri,ej) wtth 

(37) 

n 0 0 [7.5" + (j - !) 15'1 E (j = l,Z, ... ,Le = 24) 0 

j 

Values of Qj correspond to data taken along every other radial cord 

truss system with reflectors assumed located on the gore edges. 

Distributed loads are obtained bv 8ubstitutj.q (32)-!J4) int- (1) and 

evaluating f ( : ,O ). .. 

In the examples of the imctlficatlon pracess to oe presented, an 

equal number of linear spline basis functions are ueed in boLh r ~.nd 8 

directions. That is, M = N + 1 for an increabiq sequence of N-values. 

The cubic spline appruximrtions (24) and (25) are ueed with fixed 

M1 - N1 - 4 to xepresent E1(r) and E-;e). The L%L version 

( WSSQ) of the Levenberg-Harqusrdt dgorithm [ 17f is employed to minimi ee 

JMsN given by (29). For tte first choice of N; sdnal (v,6) parameter 

values to Initialize the Levenherg-brquardt slgorithm are obtained by 

finding those (ut&) coordinateb which came (24) and (25) to beet approximate assumed ftli.ctions E1(r) and E2(2) chosen as guessed fom far 

E (ri 83d t2(e), respectively. P,,L Iorger N, the latest previoueLy 

obtained set of converged coorlfnatee Is used as norn1-d parameters. 

PI. rical calculations are performed on a CDC Cyber 170-series digits1 

. ;z.puta: using default valuee 02 the IHSL convergence paramcters. 

0 0 

- 

1 

Two measurzs of identification scherae performance are employed. The 

quantity 

is uaed as a measure 02 state estlmatim accuracy. Aaditimally, 

2 74 
measures the relative error between the true 

and the estimatec. E(r,O) denoted by 

which is calculated from (24) and (25) using the (M,NIth level of state 

approxic-tion obtained at step 8 of the computational procedure. In (391, 

1.1 ,c‘.~otes the L2 norm on [6,R] x [0,2+1. RMPN provides a 

lneasure of parameter estimation accuracy. 

Convergence in the sense that 

#,N + 0 

and 

as 

depends on the ability of the cubic spline approximates (24) and (25) to 

accurately represent El(r) and E2(8). An exact pointwise fit can be 

obtained for E (e) by choice of the 4 6-coefficients in (25). However, 

E1(t) 

- - 

- 

- 2 

can at best be approximated to 

relative error by (24) and (27) with M = 4, Consequently, entries in the 

(RM’N, jMgN) (M,N) 

value. Less realistic examples in which (24) acd (25) exactly fit simpler 

1 

sequence can be expected to cease decreasing past some 
- 

E (r) and E (0) functions, and and RnsN monotonically decrease 

with increasing (M,N) can be found in [15]. Also, using the best cubic 

spline fits to E1(r) and E2(8) obtained fra (24) and (25) to 

define E(r,e), along with the exact f(r,e) data, we observed that 

1 2 

- - 

jnSN - 0.087 

unifomly in (l4,N). Tk folloving nurerical remlts shcw that the 

parameter estimates from the ldentlflcatlon procedure tend to Improve 

( reduce) this J An*N value at the expetme of RwJ 

Example 1: Estimate E2(8) holiing El(’) fixed at the best 

cubic sp!i.re estimate of (r) using (24). Mnd Fsraneterb tor the 

N = 4 

1 

starting value are obtained by fitting (25) to 

3 1 

2 E2(e) - 1 +- COS e . 

Four 6-parameters are estimated and results scrltcd below. 

N RnBN, % 8 the, see 

4 0 . 0390 5.13 

6 0.0384 5.57 

8 0.0322 5.69 

10 0.0347 6.01 

12 0 0330 5.83 

8 

23 

86 

105 

132 

EssentiaLLy no improvement in state estirate vu obtained past N = 8. 

E;’N(6) tepded to 3.591 instead of 

attribdted to the Inability Jf (24) to exactly fit xl(r). 

The 

The - 0.20 bias is 

- 

E2(8) E 3-39?. 

276 
Example 2: Estimate El(r) holding E2(8) fixed at the best 

cubic spline estimate of E2(B) using (25). Nominal para~eters for the 

N = 4 starting value are obtained by fitting (24) to 

- 

Four +parameters are estimated aod results surmkarized below. 

N ;'~N, m RMsN, % 2 time, sec 

4 

6 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

a 

0.0355 

0.0270 

0.0293 

0.0275 

0.0273 

0.0271 

0.0267 

0.0264 

0 -0260 

0.0267 

0.0250 

0 A203 

0.0259 

0.0343 

32.25 

24.5 

4.39 

13.17 

7.44 

7.59 

7.68 

7.91 

8.11 

7 -49 

7.58 

7.71 

8-08 

8.03 

22 

41 

75 

103 

130 

168 

222 

292 

370 

460 

578 

751 

847 

1050 

Prom a state estimation viewpoint, N = 28 provides the best 

accurrcy. Overall, considering state, parameter and ease of co8putatioa, 

N = 8 is best. Figure 3 shows the character of EYBN(r) for selected 

values of N. 

27 7 
Example 3: EstiMte both E1(r) and Ez(e). Ndnal parameters 

are obtained aa before for N = 4 from 

Ep) = 1 - - sin e 0 

For each N, the first coefficient, 61, is bcld fixed at its iaitial 

value. Seven parameters are estimated. 

N ItnBN, 6 the, Bcc 

4 

6 

8 

10 

12 

14 

16 

13 

20 

22 

24 

26 

0.0356 

0.0341 

0.0270 

0.0293 

0.0275 

0.0273 

0.0271 

0.0264 

0.0262 

0.0260 

0.0260 

0.0267 

32 . 24 

28.71 

4.42 

13.18 

8.09 

7.45 

7 059 

7.69 

8.04 

7.90 

8.12 

7.47 

40 

67 

168 

209 

236 

337 

411 

490 

567 

65 1 

768 

945 

Again, fra overall consideratioxm, N * 8 giver, the beet results. 

278 
In all examples we have been able to succeesfully estimate the surface 

shape of the model antenna. Similr.. results have been obtaiced where 

random noise (approximately 5% noise level) has been added to the data. 

These and other findings may be fod la Section VI of [ 151. 

Research reported here was supported in part by NASA Grant NAG-1-258 

for the first and second authors, in part by WSP Gram MCS-8205355 and in 

part by AFOSR Grant 81-0198 for the first author, and NSP Grant M3-8200883 

for the second author. Parts of the efforts reported uere carried out 

while th. first two authors *are in residence at the Institute for Corputsr 

Applications in Science and Engineering, NU Langley Research Center, 

Hampton, VA, which is operated under NASA Contracts No. NAS1-15810 and 

NO. NAS1-16394. 

Banks, H. T., "Algorithms for Estimation in Mstributed lbdels With 

Application8 to Large Space Structures." Proc. Workshop on Applications of Distributed System Theory to the Control of Large Space 

Structures, JPL-Calif. Inst. Tech., Pasadena, cb, July 14-16, 1982. 

Banks, H. T., and Crowley, J. M., "Parameter Estimation in Timoshenko 

Beam Wels," Lefschett Center for Dynarical Systems Report 

No. 82-14, Brown University, Providence, RI, June 1982; 

J. Astr. Sci., Vol. 31, pp. 381-397, 1983. 

Banks, H. T., Crowley, J. M., and Kunisch, K., "Cubic Spline 

Approximation Techniques for Parameter Estimation in Distributed 

Systems," IEEE Trans. Auto. Control, Vol. AC-28, No. 7, pp. 773-786, 

July 1983. 

Banks, H. T., and Daniel, P. L., "Parameter Estimatim of Nonlinear 

Nonautonomous Distributed Systems," Proc. 20th IEEE Conf. on Decision 

and Control, San Diego, CA, pp. 228-232, 1981. . . 

Banks, H. T., and Kunisch, K., "An ApproximatLon Theory for Nonlinear 

Partial Differential Equations With Applications to Identification 

and Control," Lefscnett Center for Dynamical Systems Rept. 81-7, 

Brown University, Providence, RI, 1981; SIAW J. Ccntrol and 

Optimization, Vol. 20, pp. 815-849, 1982. 

279 
I81 

I91 

[ 161 

Banks, H. T., and Daniel, P. L., "Estimation of Variable Coefficients 

in Parabolic Distributed Systems," LOS Rept. 182-22, Sept. 1982, 

Brown University, IEEE Trans. Auto. Control, to appear. 

Russell, R. A., Campbell, T. G., and Reeland, R. E., "A Technology 

Development Program for Large Space Antennae," Paper No. IAF-8OA33, 

Presented at Thirty-First International Astronautical Congress of the 

International Astronautical Federation, Tokyo, Japan, Sept. 21-28, 1980. 

Sullivan, M. R., "LSST (Hoop/Colurn) Maypole Antenna Developent 

Program," Parts I and 11, NASA CR-3558, June 1982. 

Banks, H. T., and Majda, G., "Modeling of Flexible Surfaces: A 

Preliminary Study," ICASE bpt. No. 83-19, Berpton, VA, iJay 1983; 

J. Math Modeling, to appear. 

Sagan, H., Boundary and Eigenvalue Problems in ~therat~cd Phyeics, 

John Uiley 6 Sons , Inc., 1966. 

Banks, H. T., "Distributed Systea Optimal Control and Parameter 

Estimation: Computational Techniques Using Spline Approxiutioas," 

Lefschett Center for Dynamical Systems kpt. 82-6, Brown University, 

Providence, RI, 1982; in Proc. 3rd IFAC Sgrposiu on Control of 

Distributed Parameter Systems, Toulouse, Prance, S.P. 21-S.P. 27, 

JUIW 29-July 2, 1982. 

Banks, H. T., "A Survey of Sow Problem and Recent Results for 

Parameter Estimation and Optimal Control in Delay end Mstributed 

Parameter Systems," In Volterra and Functional Different a1 Equatione 

(K. B. Hannsgen, et al., de.), Mrcel Dekker, New York, pp. 3-24, 

1982. 

Strang, Go, and Pix, G. J., An Analysis of the Finite Element Hetirod, 

Prentice-Hall, Inc., 1973. 

Schultt, M. H., Spline Analysis, Prentice-Hall, Englewood Cliffs, KT, 

1973. 

Banks, H. T., Daniel, P. L., ad Armstrong, E. S., "A Spli=-Based 

Parameter and State Estimation Technique for Static Me18 of Elastic 

Surfaces," ICASE Rept. No. 83-25, Haapton, VAS June 1983. 

Bartel8, R. H., and Stewart, G. U., "Algoritta 432 - A Solution of 

the Matrlx Equation AX + XB - C," Corun. AQI, Vol. 15, pp. 820-826, 

Sept. 1972. 

More, J. J., "The Levenberg-Marquardt Algorit'm," Implementation and 

Theory Iir Nuplerical Analysis (G. A. Watson, ed.), Ucture Notes in 

Mathematics 630, Springer-Verlag, 1977. 

280 
FEED ASSEMBLY 

(4 REQU I RED) 

FEED MAST 

UPPER MAST HOOP SUPPORT CABLE 

hOOP SUPPORT CABLE 

Figure 1 .- Side View of Heypole (Hoop/Column) Antenna. 

Figure 2 .- Maypole (Hoop/Coltimn) Antenna Reflector Surface. 

28 1 
N=4 

RMsN = 4.39% 2- FIN = 0.0270m 

11 I I I I 1 I I I 

2 RM*N = 7.58% jMIN = 0.0203m 

0 

I@ 15202530 35 40 45 50 

r, 

Figure 3.- Estimate E1(r) with E2(8) fixed. 

282 
EXPERIlMlENTAL VERIFICATION OF IDENTIFICATION 

ALGORITHMS FOZCONTROLOF 

FLFXIBLE STRUCTURES 

B. Sridhnr, J.-N Aubm, and K. R. Lod 

Lockheed Palo Alto Research Laboratory 

Palo Alto, CA 94304 

ABSTRACT 

This paper describes an on-going simple laboratory experiment, referred to 

as the Beam Control Experiment (BCE), which has the essential features of a 

large fisxible structure. The experiment is used to develop and evaluate identification and control algorithms which 1ook.promising in the active control of 

high performance large space structures. 

hood identification of the parameters of the beam-actuator-sensor assembly from 

experimental data is presented in the paper. 

Some results on the maximum likeli- 

'I. INTRODUCTION 

One of the major problems in the design of control systems which operate 

in the presence of a flexible structure is obtaining accurate information about 

the plant dynamics. In particular, knowledge of the frequencies, damping ratios 

and mode shapes c the flexible modes is critical to the successful design of 

a high performance system. 

success of which depends upon the choice of algorithm and system model, the 

choice of inputs to excite the system, and the quality of output data. A cereful integration of these items is especially critical in the case of large 

flexi )le struztures. 

Systen! identification is an iterative process, the 

Ln this paper we describe the development and performa,ice testing of a 

simple laborat2ry model of a jitter control sys !m designed to provide a stable 

image with optical components mounted on a flexible structure. 

be carried out in three stages: (a) identification with simulated data, (b) 

identification with real data, and (c) comparison of closed loop performance 

with simulated results. Results from the first two stages are reported in 

this paper. 

The study will 

This paper is organized as follows: Section 2 describes tne experimental 

set-up and a msthematical model for the BCE is developed in Section 3. A brief 

description of the maximum likelihood estimation (MLE) algorithm is presented 

in Section 4. Results on the identification of the parameters of the BCE using 

both simulated and experimental data are discussed in Section 5. 

and future work is described in Section 6. 

A summary 

11. BEAM CONTROL EXPERIMENT 

The idea behind this experiment (Fig. 1) is to demonstrate the interaction between the control of an optical system, symbolized by a laser beam, 

and control of a flexible structure, represented by a flexible aluminum beam 

to which passive and active mirrors are attached. These mirrors bounce the 

laser beam toward a desired targtt. The intezesting control problem stems 

283 
from the fact that the active mirror is in fact part of a proof-mass actuator. 

Thus, any attempt to control the laser beam will tend to disturb the aluminum 

beam, thereby also disturbing the laser beam. This intricate coupling is 

quite a challenge for a classical design but more amenable to modern techniques. 

The other aspect of the experiment is the use of a commercially available 

cated microprocessor (IS1 MCP-1001 capable of handling at maximum a 32--, e 

Kalman filter at 3 2000-E2 sampling rate. Such implementation, aside f 1 its 

laboratory usefulness, bring control technology one step further toward r?al 

space implementation, and the experience gained is valuable. 

The schematic of the experiment is shown in Figs. 2 and 3. The laser beam 

bounces first on a mirror situated near the middle point of the vertical aluminum beam, then on a mirror attached to the tip of a pivoted proof-mass actuator (PPM). The laser beam finally reaches a photodetector, which measures the 

laser beam position. A beam splitter provides a visual display of the jitter 

by projecting the spot on a remote screen. ; 

Two sensors are used in the experiment: the photodetector measuring the 

lue-of-sight (LO'S) error, and the I'PM rate sensor measuring the relative velocity of the proof-mass. 

A preliminary experiment had been performed earlier (Ref. 1) using commercially available software for identification and control synthesis. Only 

one sensor was used (position) end the control system was able to significantly damp out the beam vibrations; thus stabilizing the line-of-sight. 

However, in order to eliminate the static error and achieve a higher performance, a better model is needed and thus more sophisticated identification 

techniques are sought to that end. 

For purposes of identification, a known control force is applied to the 

beam ueing the proof-mass actuator. The control force time-history and the 

beam position and relative rate outputs are recorded on a Nicolet 4094 digital 

oscilloscope. 

Nicolet 4094 to a Harris 800 computer where the identificatim algorithms 

were exercized. Thus an efficient link between hardware ter'-s and eophisticated 

computer analyses (rig. 4) was established. 

Special software transfers the input and output data from the 

111. MATHEMATICAL MODEL 

In this Section, a state space model of the system is developed. The 

mathematical form of this model will be used both for simulation and identification of the parameters of the BCE. 

The angular displacement, ea, of the proof-mass actuator is limited to 

to a few degrees. 

tor are given by the equations (Reference 2) 

For small angles the force and torque applied by the actuaT = INa 

284 
where m is the mass of the proof-mass actuator, b is the distance from the center of mass to the proof-mass pivot and I is the centroid inertia of the proofmass actuator. The dynamics of tte aluminum beam will be defined in terms of 

the principal modes and mode shapes. 

mode and define: 

Let qi be the modal amplitude of the ith 

w 

Ti = damping of the ith mode 

$i = translational mode shape of the ith mode at the 

= angular velocity of the ith mode i 

beam tip (where the PPM is mounted) 

and +i R = rotational mode shape of the ith mode'at this tip. Let $im and $im R 

be the corresponding values for che mode shapes where the mirror is mounted 

on the flexible beam. The modal equations for che beam are: 

where M is the number of modes represented in the model. Due to the actuator 

dynamics, the control force fc applied to the actuator is related to fa by the 

equation 

where r = lever arm of the actuator 

g = acceleration due to gravity 

K = spring constant of the actuator 

D = damping constant of the actuator 

and €It = rotation of the beam tip 

The rotation of the beam tip can be expressed in terms of the modal amplitudes 

by 

285 
Equations (1) - (5) can be reduced to the set of equations 

(6) .. R R r ea = - (K + mgb!/I ea - D!I ba + tU1 Z qi + DII C 9, ii + i fc 

and 

+ Bi fc (7) 

where 

Bi = Ai r/K (9) 

and ‘i = Ai D/K (10) 

Let y1 be the displacement of the laser beam from the reference point \%.e., 

this is a measurement of the L.O.S. error). 

rate between the actuator and the beam tip. 

Let y2 be the relative angular 

Then, 

and 

where El is the distance between the mirror on the beam and the mirror on the 

proof-mass actuator and a2 is the distance between the photodetector and the 

mirror on the proof-mass actuator (See Fig. 3). 

Equations 6, 7, 11, and 12 give a state space representation of the 

input/output behavior of the BCE with [ea ea q1 q1 . . . qH 

state vector. I 

as the 

For a single mode model the equations are given by 

x = ‘rx + Gu 

y = hrx 

286 
whert 

c.- 

2 I 4~1O-~ Kg-m 

m 0.07895 Kg 

b 0.06985 m 

r 0.021 m 

0.165 m 

0.217 m 

5 

$2 

- 

F= 

and 

0 

[; 1 

1 

-D/ I 

0 

-C 1 

0 

2R 

@? 61 

0 

2R 

-wl+AIC1 

[O r/I o B1 '1 (16) 

where uz = (K + mgb)/I . a 

R The F 

$lrn,+tm) and the Irn-cm parameters 

known parameters are tabulated in Table 1. In this aodel the number of unknown 

parameters is equal to (6Vt2) where M is the nuiiber of modes. 

G and H matri2es depend on the unknown parameters (Ua,D,w1,51,41,@1, 

(I,m,b,r,Rl and a2 1. The values of the 

Table 1. Kqown Parameters of the BCE 

287 
IV. MAXIMUM LIKELIHOOD ESTMATIClN (MLE) 

There are several methods available for the estimation/identlfication of 

parameters Figure 6 shows the main components of an identification method. 

The system to be idenzified and a mathematical model, W(p), of the system 

is excited by a known input u. An error function, L(p,e), is fonned from the 

outputs of the system and the model. 

adjusting the model parameters p to minimize the error furictior. 

of M(p), L(p,e) and the adjusting mechanism for p lead to different identification algorithms. In this paper, we shall restrict our attention to the 

maximum likelihood estimation of parameters. 

Identification Is the process of 

The choice 

The MLE can be applied to a large class of problems and has good statistical convergence and accuracy properties. In addition, HLE is well suCted 

for identifying physical parameters of thc system. This is a drawback with 

most recursive algorithms. The main disadvantage of MLE is the amount of 

computation. However, the amount of computation can be reduced significantly 

by taking into account the special features of the dynamics of the large space 

structures. 

The flow of computation in the MLE is sham in Fig. 7. The mathematical 

model for the system is assumed to be 

x = F(p) x + G(p)u + w (20) 

where x is the n-dimensional state vector, y the p-dimensional output vector 

and u the m-dimensional input vector. ‘1) and represent the process and 

measurement noise respectively. 

of unknom system parameters. 

applied t3 the system and the output y of the system has been observed at discrete times to, tl, . . . tN. 

zero-mean gaussian random variables with 

The matrices F, G and H depend on p,the vector 

~n input signal 

Further, it is assumed that x(o) 

[u(t), o < t < tN] has been 

and w are 

cov (x(0)) = P(0) (22) 

cov (0) =Q (23) 

and cov (v) =R , (24) 

The identif icatlon problem consists of estimating the parameters p from the 

experimental data u(ti), y(ti) , i - 1,2, ... N. Let be the state estimte, 

the output estimate, and e(tR) be thc outDut error 

where eq = Y(tk) - &t,L (25) 
The negative log likelinood function, v(p,e), can be written as 

N- -1 

i= 1 

L'(p,e) = - log L(p,e) = C b(i) E(i) e(i) + log jB(i)l (26) 

The maximum likelihood estimate of the parameters p is obtained by maximizillg the ?ikelinoJd function L(p,e) (or by dn'imizing V(p,e)). This nonlinear minimization protlem has to be solved by numerical methods and makes the 

XE cornputationally intensive. The computational aspects of MLE are discussed 

in Reference 4. 

V. IDENTIFICATION RESULTS 

Numerical results on the identitication of the parameters will be presented in two steps. First results from the identification of simulated data 

will be snown. This will be followed by results from experirental data. 

a. Simulation Results: Identificaticn with simulated dcta was done 

to get a tetter understanding of the dynamics of the BCE, to provide guidelines to set up the experiment and to test the MLE 

software. The system was simulated using 4 modes and was excited 

by a "bang-bang" type input with ar. amplitude of tO.1 Newton. 

Figure 8 shows the laser beam position and relative velocity 

output from the Simulation. This i?put/output simulated data 

was used to identify a single mode beam model of the system 

(see equations 13-17). The negative log likelihood function 

V(p,e) was probed at a few points to see its variation with paramrter p. Figure 9 shows the variation of the likelihood surface 

with wa ar,d w,. 

The damping terms Ga and G1 were set to thfi simulatior values and 

vary. Table 2 shows the results based on simulated daca. There 

is good agreement between simulated and estimated values. 

are ready to try the identificationwith experimental data. 

The surface is well-beh&ved in these two variables. 

only parameters (wa. w1, $1, $1, R $i, and +in) were allowed to 

Now we 

b. Experimental Results: The aluminum beam was excited by applying 

to the proof-mass actuator a sinusoidal force with a lizearly 

varving frequency (so-called "chirp" excitation) . Figure 10 shows 

the control force fc. The position and rate measurements are shown 

in Figure 11. As before, the single mode model will be used as a 

starting point €or the identification of paraaeters (ua, D, w1, GI, 

91, I$1, $im, $im). 

These were related to one or more of the following causes: 1) large 

differences between the values of some of the actual BCE parameters 

and those of the original simulation, 2) bias in the input force 

and position measurements, and 3) error in rate measurement calibrat ion. 

R R Initially, the MLE had convergence problems. 

269 
The bias was accounted for simply by subtracting a constant from 

the force input and position output. The error in rate calibration was taken into account by defining a scale factor a. This 

results in a new H matrix where 

R 

- ($im - ("+") 4J 

H= [:. 

The scale faztcters. The esth 

parameters show 

0 

a 

was 

0 

estimated along with the other 

] (27' 

R 

-01 a 

8 parametea v lues are shown-in Table 3.. 

in Table 3 for "simulation valties" were obtained 

Th- model 

from an extremely simplified model of the aluminm beam (cantilevered with a point mass at tip). Thus, it is not surprising 

that some of the values obtained fro- the identification are very 

different. In particular, values of r3tational mode shapes are 

quite sensitive to local inertias and masses. 

w 

a 

1 w 

'a 

';1 

41 

01 

0 im 

* irn 

R 

- a 

Simulated 

Data 

46.36 

54.63 

0.005 

0.005 

3.328 

-13.51 

1.068 

-10.6 

1 .o 

Estimate 

(Simulation 

Data) 

45.5 

54.58 

.005* 

.005* 

3.49 

-13.61 

0.98 

-10.3 

1.0* 

Est hat e 

(Experimental 

Datc) 

37.18 

40.32 

25E-4 

0.01 

7.48 

-110.0 

21 .o 

2.49 

3.80 

'These parameters were $et to their simulation values. 

Table 3 Results with Experimental Data 

290 
i.'igure llsl ".IS a comparison between the measured i:nd estimated values of the 

or:t.puts. The estimated values were generated for the prameter set which 

rejulted from identification using experimental data (Table 3). Figure lla 

.:.k,ows the measured and estimated values of the position. Figure llbis a 

t .ow-up of the same curve to show the difference between measured and estimated 

?lues. Figures Ilcandlld shou curvessimilar tolla and llb for the rate 

casurement. There is very good agreement between the model output and the 

:xperimental data. 

This model will be validated by taking the direct approach. In the third 

stage of the experiment, a control system will be designed using the identified model. 

with its experimental. behavior. 

ThE predicted behavior of the control system will be compared 

VI. CONCLUSIONS AND SUMMARY 

In this paper, we have described a laboratory experhect which has the 

.ialient features of controlling an optical system located on a flexible structure. The experiment will be used as a test bed for designing control and 

i-dentification algorithms for large space structures. The plrameters of a 

:lode1 suitable for designing a control system were identified using maximum 

jikelihood estimation. The real test of a model is of iourse how well ic 

satisfies the goal of modelling. Currently, we are designing a control 

 stern based on this node1 and the results of this final stage will be 

reported in another paper. 

VI;. REFERENCES 

1. S. C. Shah, R. A. Walker and D. B. Varvell, A Control Design Wurkstaiion, 

IEEE Control Systems Society Symposium on Computer-Aided Control System 

Desigr sept. 28-30, 1983, Cambridge, Mass. 

.1. J.-N. Aubrun and G. Margulies, Low-Authority Control Synthesi, for Large 

Space Structures, NASA Contractor Report NASl-14887-Task 11 Fay 1982. 

3. K. J .4strom, Maximum Likelihwd and Prediction Error Methods, Autornatica 

Vol. 16, pp 551-574, 1380. 

4. N. K. CI;.~~ and R. K. Mehra, Compucational Aspects of Maximmi Likoiihood 

Estinati and Reduction in Sensitivity Function Calculations, IEEE 

Trai lctions on Automatic Control, AC-19, No. 6, 1974. 

29 1 

0 

STRVCNML 

rnMIISE AllD 

AcnJAIol 

SlCluL 

GMEIATOI. ' 

FIG, 5 BEAM EXPERIMENT GEOMETRY 

LIGHT 

?Am ' e LINEAR - 

PIIOTODCTtCmR 

293 

* , 
ELKRIPTI(III Y 

OF THE loOEL 

)I (PI 

INPUT U 

Y 

REM 

SYSTM 

t 

. NOISE 

FIG. 6 COMPONENTS OF AN IDENTIFICATION SCHEWE 

I MODEL #scRIPTIoII 

IlITIM Pm 

Am COVARIMQ I 

I I PERTURBATIOllS 1 

NOWLINEAR NllCTIffl 

nINI~IuTIo(r 

FIG. 7 COMPUTATIONS IN THE MLE 

294 



N85-312 IZ 

AN EIGENSYSTEM REALIZA"I0N ALGORITHM (ERA) 

FOR MODAL PARAMETER IDENTIFICATION 

AND MODEL REDUCTION 

j. N. Jupns and R. S. Pappn 

NASA Langley Research Center 

Hammn, VA 23665 

ABSTRACT 

A method, called the Eigensystem Realizatjon Algorithm (E! 5 developed 

for modal parameter identification and model reduction of dynamic systems frtm 

test data. A new approach is introduced in conjunction with the singular value 

decomposition technique to derive the basic fornrulatlon of mlnimun order 

realization which is an extended version of the Ho-Kalman algorithn. The basic 

?ormulation is then transformed into modal space for modal parameter idmtification. Two accuracy indicators are developed to quantitatively identify the 

system modes and noise modes. Fbr illustration of the algorithm, examples are 

shown using simulation data and experimental 'data for a rectangular grid structure. 

I. INXCWCTION 

The state space model has rece 

analyses and design in recent control 

these areas, in particular, is control 

design controls for a dynamic system 

model which wi 11 adequately describe 

constructing a state space represent 

system realization. 

ved considerable attention for system 

and systems research programs One of 

of large space structures. In order to 

it is nebessary to have a mathematical 

the system's motion. The process of 

tion from experimental data is called 

bring the past two decades, numerws algorittmns for the construction of 

state space representations of Ifnear systems have appeared in the control 

literature. Among the first were the works of Gilbert [l] and Kalman [Z], 

introducing the important principles of realization theory in terms of the 

concepts of control labi 1 i ty and observabi 1 ty . Both techniques use the transfer 

function matr'x to solve the realization problem. Ho and Kalman [3] approached 

this problem from a new viewpoint. They showed that the minimum realization 

prcblem is equivalent to a representation problem Invclving a sequence of real 

matrices known as Markov parameters (pulse response functions). By ninimum 

realization is meant the smallest state-space dimension among systems realized 

that has the same input-output relations within a specified degree of ac.curacy. 

Questions regarding the minimum realization from various types of input-output 

data and the generation of minimum partial realization are studied by Tether 

[4), Silverman [5], and Rossen and Lapidus [6] using Markov i---dwters. Rossen 

and Lapidus [7) successfully applied Ho-Kalman [3] and Tether [4] methods to 

chemical engineering systems. A cmon weakness of the above schemes is that 

effects of noise on the data analysis were not evaluated. Zeiger and McEwen 

299 I'KECEDING PACE I3LAh;I-, NOT FILMED 
IS] proposed a combinition of the Ho-Kalman algorithm [3] wit4 the singular 

value decomposition technique for. the tmatment of noisy dato However, no 

theoretical or numerical studies were reported in Reference [P1 unony followup developments along similar Itnes, Kung [9] presented anc a 1 gor i t hm .i n 

conjunction with the singular value deconposition techniqG:o ti' Incorporate the 

presence of the noi se. Note that the singular value decorr!::#ai tion techniqLe 

[lo-113 has been widely reqnized as beiclg very effect'vd and numerically 

stable. Although several techniques of niinimun realization are irvailable in the 

literature, formal direct application to the modal parameter identification for 

flexible structures was not yet addressed. 

In the structures fieiu, the finite-element technique is used almost 

exc1usivt:ly for constructin: analytical models. This approach is we'll established and normally provides a model accurate efiough for structural design 

purposes. Once the structure is built, static and dynamic tests are performed. 

These tes- results are used to refine the fi.nite-element model, which is then 

use4 for::final analyses. This traditional approach to analytical model development may not be accurate enough for use in designing a vibration control 

system for flexible structures. Another approach is to realize 3 model directly 

from the experimental results, This requires the construction of a minimumorder model from the test data that characterizes the dynamics of the system 

at the selected control and measurement positions. lhe present state-of-the-art 

in structural modal testing and data analysis is me of controversy about the 

best technique to use. Classical test techniques, which may provide only good 

frequency and moderate mode shape accuracy, are often considered adequate for 

fini te-element model verification purpoces. On the other hand, advanced data 

ar,alysis techniques which offer significant reductions in test time and improved 

accuracy , have been available [12-163 but are not yet fully used. For example, 

Ibrahim [13] presented a method based on state space for the direct identification of modal parameters from free responses. Recently, Void and Russell [161 

presented a method using frequency response functions and time domain analysis 

which can also identify repeated eigenvalues. A comparison of Contemporary 

methods usinq data from tne Galileo spacecraft test is provided by Chen r171. 

A1 though structural dynamics tethniques are generally successful for ground 

data, further incorporation with work from the controls discipline is needed t3 

solve modal parameter identification/control problem. For example, it is Known 

from control theory [18] that a system with repeated eigenvalues and Indepsndent 

mode shapes is not identifiable by single input and single ouput. Methods 

which aliows only one initial condition (input) at a time [13], will miss 

repeated eigenvalues. Also, if the rea:ited system is not of a minimum order 

and matrix inversion is used for constructfng an ovcrbfzed state matrix, 

numerical errors may become dominant. 

Under the interaction of structure and control disciplines, the objective 

of this paper is to introduce an Eigensystem Realization Algorithm (ERP,) for 

modal parameter identifica* and model reduction for dynamtcal systems from 

test data. lhe algorithn consists of two major parts, nemely, basic formulation 

of the minimum order realization and modal parameter identification. In the 

section of bas'lc formulation, the Hankel matrix which represents the dFtS 

structure for Ho-Kalman algorith is generalized to allow random distribution 

300 
of Markov pdrameters generated by free decay reponses. A unique approach 

based on this generalized Hankel matrix is developed to extend the do-Kalmn 

a1 ortthn in combination with the sinylar value decoaposition technique [lo113. Through the use of the generalized Hankel matrix, a linear model is realized 

for dynamical system matching the input and output relationship. lhe realized 

system model is then transformed into modal space for modal parameter identifications. As part of ERA, two accuracy indicators, namely, the modal amplitude 

coherence and the modal phase collinearity, are developed to quantify the 

system modes and noise modes. The degree of modal dxcitation and observation 

are evaluated. The ERA method thus forms the basis for a rationa: choice of 

model size determined by the singular values and accuracy indicators. 

Two examples are given to illustrate the ERA method. The first example 

uses simulated data from an assuned structure. The effect of repeated eigenvalues 

on the parameter identification is shown. The second example uses experimental 

data fra I simple grid structure. Conparison of the ERA results with a finite 

element mdel of the grid is performed. Experimental results for a more complex 

structure--the Gal i \eo spscecraft--are sham in Ref. [19]. 

11. BASIC FORMULATIONS 

A finite-dimensional, discrete time, linear, time-invariant dynmical 

systein has the state-variable equations 

x(k+l) = Ar(k) t Bu(k) (1 1 

where x is an n dimensional state vector, u is an m dimensional control input, 

and y is an p dimensional output or measwernent vector. The integer k is the 

sample indicator. The transition matrix A characterizes the dynamics of the 

system. For flexible structures, it is a representation of mass, stiffness 

and damping properties. The problem of system realization is then the following. Given the measurement functions y(k), construct constant matrices [A, B, 

C] such that the functions y are reproduced by the state-variable equations. 

With different sets of inputs and outputs, several C?SBS can be obtained. lhe 

simplest case, namely, si3gle input and single output, is treated first to 

allow the reader familiar wlth notations for the treatment of multi-input and 

mu1 ti-output cases. 

Single input and single output 

For the system (1 ) with free pul se-response (or i qi ti al-state-response) , the time-domain description is given by the function know1 as Markov parameter 

y(k) - CAk-16 [or y(k) - CAkx(O)] (3) 

where x(0) is the system initiz' -0nditions and k is an integer. Note that the 

matrix 6 is a collrmn vector (si..%ie input) whereas the matrix C is a row vector 

30 1 
(single output). br free initial-stateresponse, the matrix B only represents 

the infomtion of initial conditions rather than the control influence matrix 

as shown in Eq.(l). The probles of system realization is to construct matrices 

[A, B, C] in term of the measurement function y(k) such that the identities of 

Eq.(3) hold. Now observe that 

Assume that this nth order system has no repeated eigenvalues. mere exists 

a row vector C from observability theory (Ref. 18) such that V has rank n. 

Consequently , rearrangi ng Eq . (4) becomes 

- y( k+l ) = VAkB = VAV-ly( k) (6) 

Given the sequence of measurenent vectors T(ktl), the generalized Hankel 

matrix H(k) is defined as 

: I ('I 

,y(k+l) ,..., y(k+n-1) 

H(k-1) = G(k) I y(k+l) ,..., y(k+n)l = ,y(k+2) ,..., y(k+n) 

) ,y( k+n) , . . ,y( k+2n-2) 

It inrnediately follows from Eq.(6) that 

H(k) = VAV'lH(k-l) - VAkV-ltl(0) (8) 

or from Eq. (4) that 

H(k) = VAk[B, AB,. Ak-161 - VAkW (9) 

*here W is a conrrollability matrix (Ref. 18). Again if the systf!m with order 

.I has no repeated eigenvalues, there exists a colun vector 8 such that Y has 

rank n. This means that H(k) is invertible if the system is controllable and 

observable. Letting k = 1, Eq. (8) will then detemfne the state matrix A in 

the following way 

VAV-1 = H(I)H-~(o) (lo) 

To rigorously prove this result, define E as the column vector ET=[l,O,...,O]. 

The measurement function y(kt1) can then be written by 

y(k+l) = ETH(k)E = ETH(k)H-I(O)H(O)E - ET[H(l)H'1(O)]kH(O)E (11) 

with the aid of Eqs. (8) and (10). Hence by €q. (3), the triple [H(l)H'1(0), 

H(O)E, ET] is a realization in the sense that if the triple [A, 6, C] in the 

30 2 
system equations (1) an? (2) is replaced by the [H(l)H'I(O), H(O)E, ET], the 

measurement functions y(k) are reproduced as proved in Eq. (11). In other 

words, state variable equations (1) and (2) are transformed to the following 

quat i ons 

- 

x(k+1) = H(l)W1(0)F(k) + H(O)Eu(k) (1 2) 

y(k) - EG(k) (1 3) 

where y( k) = V-l x( k) . (14) 

Let us summarize the case as follars. 

A finite-dimensional, discrete tine, 1 inear time invariant dynmical system 

with a single input and a single output is realizable if the state matrix A has 

no repeated eigenvalues, and the system is controllable and observable. 

Mu1 ti -i nput and 9ulti-output 

(or initial-state-reponse) function knorm as Markov parameter 

The time-donrain description for this case is given by the pulse-response 

Y(k) = CAK'lS (or Y(k) - CAk[xl(0),x2(0),...,x,(0)]) (1 5) 

where xi(0) represents the 9th set of initial condition and k is an integer. 

Note that 8 is a nxm matrix and C is a pxn matrix. The problem of system 

realization is that, given the functions Y(k), construct constant matrices [A, 

B, C] in terms of Y(k) such that the identities of Eq. (15) hold. The algorithm 

begins by forming the rxs block matrix (generalized Hankel matrix) 

where ji(i=l,...,r-l) and ti(i-l,...,s-l) are arbitrary integers. For the 

system with initial-state-response measurements, slatply . replace Hrs(k-1) by 

Hrs(k) It is easy to prove from Eq. (15) that Eq. (9) also holds for 

this multi-input and mrlti-output case, 

where Vr and Ws are respectively the observabi 1 i ty and control 1 abi 11 ty mat -ices 

in a yeneral sense. Note that Vp and US are rectangular matrices with dimensions 

rp x n and n x ms respectively. Assme that there exist a matrix HI satisfying 

the relation 

303 
Ys#Vr = In (31 

where I,, is an identity matrix of order n. 

order p. E: CS. o~.".. and [h. %..... o,,,I. In view of 4s. 

(16) and (18), the measurement function b(k+l) can be obtdned through either 

of two algorithms A1 and A2. lhe algorithm A1 is 

&fine Op as a null matrig 

Y(k+l) = EpTHrs(k)C = EpVpA Tki YsH V#sh 

.. and the algorithm A2 is 

Y (k+l) = E$,,( k) €,, = E$PsH'VPkYsQ 

EpTVrY,CH'VrMS Ik 

= EpTH,S(O)CH'HrS(l )Ik% (20) 

Hps(O)~Hrs(O) * Vp#V$s Vrus Hrs(0) (21) 

Cience, by Eq.(15)D CHr (1)~'s H (0) +, $1 or CH~H 01, E, $~(o)I is a realization. There fs no dm62 tha t matrix #plays a 'or role in 

solutions (19) and (20). mat is M? Observe that, fror Eqs. (17) and (18). 

Hx is a pseudo-inverse of Hrs(0) in a general sense. When the rank of Hrs(0) 

equals to the colmn nmber of Hrr(0) , then H~=~[Hrs(0)]THrs(O)J'l~Hrs(O)]T. 

If the rank equal to the row nunber, then ~l[H~s(0)]TIHrs(O)[Hr~(O)J 1 1 -1 . Ihe matrix Hrs(l)H 5) has been used in structural dynmlcs area to identify 

system modes and frequencies.13 Both are special cases representing either 

single input or single output which can not realize a system that has repeated 

eigenvalues, or a noise-free system unless the system order is a priori known. 

A general solution for ## is given belm. 

For an nth order system, find the nonsingular matrices P and Q such 

that10,ll .. Hrs(O) = pwT (22) 

where th.? rpxn matrix P and the nxrm matrix QT are isometric matrices (all the 

columns are orthonormal), leaving the stngular values of HrS(0) in the diagonal 

matrix D with positive elements [dl,d2,...,d 3. The rank n of Hrs(0) 4s 

determined by testing the singular values !or zero (relative to deslred 

accuracy)l2 uhlch will be described In the next section. kfh 

Hrs(O)= pOpf CPDICQTI * PdQT (23) 

304 
Each of the four 

Eq.{1?) with k=O, 

- 

VrWs - 

Multiplying on the 

matrices [P&QT,Us,V:] has rank and row number n. By 

T is nonsingular because if U = USQ(QTQ)”= WsQ, then TU= I by Eq. (25). Since 

TU = I = UT far nonsingular T ar,d ‘.I then 

Hy= [()][(P~Pd)-lP~] = [Q][o“PT] = QP$ i27) 

The dimension of matrices Q and Pa with rank n are respectively msxn and 

nxrp. To this end, sunmrize the case as follws. 

A finite-dimensional , discrete time, linear time-invariant dynamical system 

with multi-input and multi-output is realizable in terms of the measurement 

function if the system is controllable and observable. 

Note that no restrictions on system eigenvalues are given for this case. 

In other words, this technique can realize a system with repeated eigesvalues. 

The system (1) with this realization will be transformed into the following 

x(k+l) = Hrs(l)&(k) + Hrs(0)E,,,u (28) 

(29) 

(30) 

(31 1 

equation - 

F Y(k) = EpX (k) 

where x(k) = Vs#x(m. [lr 

~(k) EpTHrs(O)T(k) (32) 

- x(k+l) = HtHr,(l)z(k) + E,,u 

where xfk) = Wsx(k) (33) 

The realizations (28)-(33) are not of minunrun order, since the dimension of ‘51 

is the number of either columns or rows of the matrix Hrs(0) which is larger 

than the order n of the state matrix A for nultt-input and multi-output cases. 

With the aid of Eqs.(l7), (18) and (27), a minimum order of rea!ization 

can De obtained from 

305 
where Eq.(23) has been used to obtain the last equality. This is the basic 

formulation of realization for ERA. 

The triple [PxHrs(l)V, Q is a minimum realization, since the 

of the state vector x. he sane 

solution in a different form for the case where ji = ti = i (i=l,...,r-l) can 

be obtained by conpletely different approach as shown in Refs. [3 8 20). he 

system (1) with this realization is written as 

order n of P$lrS(l$Q equals 

%(k+l) = PdHrs(l)Uz(k) + QT$p (35) 

y(k) = ETP&k) 

where 

X(k) = U,QY(k) 

(36) 

(37) 

A simple exercise such as replacing Y(k+l) by Y(k) In Eqs.(lS), (20) and (34) 

shows that all the algorithms developed above are also true for the realization 

of a system with initial-state-response. 

Examination of Eqs. (19), (20) and (34) reveals that algoriths (Al) and 

(A2) are special cases of ERA. A1 is formulated by inserting the identity 

matrix (18) into the right hand side of the state matrix A as shown in Eq. 

(19). On the other hand, A2 is obtained by inserting the identify matr!x (18) 

into the left hand side of the state matrix A as sham in Eq. (20). However 

the aigorithn ERA is forned by inserting the identity matrix (18) into both 

sides of the state matrix A as shown in Eq. (34). Because of the different 

insertion, A1 and A2 do not minimize the order of the state transition matrix. 

Mathematically, if the singular Val: 2 deconposition technique is not included 

in the cqutational procedures, A1 and A2 can not be numerically implemented, 

unless a certain degree of artificial noise and/or system noise are present. 

Noises tend to make up the rank deficiency of the generalized Hankel matrix 

Hrs(0) for algorithns A1 and A2. Since the degree of noise presence is generally 

unknown, algorith-r A1 and A2 are not recommended. 

I I I. MOWL PAR.WETER IDENTIFICATION AND MOOEL REDUCTION 

he presence of almost unavoidable noise and structural nonlinearity 

introduces uncertainty about the rank of the generalized Hankel matrix and, 

306 
hence, about the dimension of resulting realization. By employing the singular 

value decomposition (SVO) technique, the rank structure of the Hankel matrix 

can be quantitatively displayed. The Set of singular values can be used to 

judge the distance of the matrix with determined order to a lower-order one. 

Therefore, the structure of the generalized Hankel matrix can be properly 

exploited to efficiently solve the realization problem. These include an 

excellent numerical performance, stability of the realization and flexibility 

in determining order-error tradeoff . Assume that, by Eq. (22) 

If the matrix Hrs(0) has a rank n then all the singular values di(i=n+l,...,N) 

should be zero. When singular values d (1-n+l,...,N) are not exactly zero but 

away from a n-rank matrix. However, there can be real difficulties in determining a sap between the computed last n@nzero singular value and what should 

be effectively considered zero, when measurement noise is present. Possible 

sources of the noise can be attributed to the measurement signal, computer 

round-off and instrument imperfections. 

based on measurement errors incurred in estimating the elements of HrS(0) and/or 

round-off errors incurred in a previous computation to get them. If 6 is 

chosen as "zero threshold" such that 6 < dn, then the matrix Hrs(0) is 

considered to have rank n. Unless information about the certainty of the 

measurement data are given, the nunber 6 is defined as a function of the 

precision limit in the computer machine. For example, 6 'dn/dl cannot exceed 

the precision limit. Further details are found in Ref. [ll]. 

After the test of sir rlar values, assune that the matrix [P$HrS(k)Q] has 

rank n. 

very.smal1, then one can easily recogn 1 ze that the matrix Hrs(0) is not far 

Look at the singular value dn of the matrix HrS(0). Choose a number 6 

Find the eigenvahs L and eigenvectors II, such that 

The modal damping rates and aanged natural frequencies are simply the real and 

imaginary parts of s, after transformation from the z- to the s- plane using 

the relationship 

where AT is the data sampling interval and j is an integer. Note that k is 

generally chosen as 1 for simpiicity. Although z and $ are in complex domain, 

con@utatlon of Eq.(40) can be performed in the rsal domain (Ref. 21) sirce the 

state matrix real i zed for most flexible structures has independent eigenvectors. 

S = [(lnz) - + 2j11 ]/(kA.t) (41) 1. 

-1 T The triple z, g Q Em, ]js obviously a minimum order of realization s m ly by observing Eq. ( E P J, is called Sensor modal displacements and g' t't Q E,,, initial modal amplitudes.'T: quantify the system modes and noise 

modes, two indicators are developed as follows. 

30 7 
Modal Amplitude Coherence y 

If the infornetion about the uncertainties of the measurement is minimum, 

the rank thus determined by the SVD beccnnes larger than the nmber of excited 

and observed system modes to represent the presence of noises in mal space. 

In modal parameter identification, the indicator referred to as nodal amplitude 

coherence is developed to quantitatively distinquish the systm and noise 

modes. Based on the accuracy parmeter, the degree of the modal excitation 

(controllability) is estimated. 

The modal amplitude coherence is done by calculating the coherence between 

each modal amplitude history and an idea one formed by extrapolating the initial 

value of the history to latter points using the identified eigenvalue. Let 

the control input matrix (initial condition) be expressed 

where * means transFclse and coaplex conjugate, and the 1 x m colmn vector bj 

corresponds to the system eigenvalue s~(J=A,...,~) Consider the sequence 

which represents the ideal modal amplitude in conplex domain Containing 

informations of the magnitude and phase angle with time step AT . W, define 

vectors qj such that 

The complex vector qj represents the modal implltude time history from the real 

measurement data obtained by the decmposition of ,ne Hankel matrix. Let yj 

be defined as the coherence parameter for the jth mode, satisfying the relation 

where I I represents the absolute value. lhe parmeter yj takes only the values 

between 0 and 1. Y- + 1 as 1- + qj indicates that the reallzed systm 

eigenvalue Sj and the jnitial modal awlitude b. are very close to the true 

values for the jth mode of the system. On tie other.. hand, if yj fs far 

awdy from the value 1, the jth mode is a noise mode. However, to make a clear 

cut between the system modes and noises requires further studies. obviously, 

the parameter y* quantifies the degree to which the modes were excited by 

a specific input, i.e. the degree of controllability. c 

Modal Phase Collinearity p 

Ti? lightly damped structure, normal mode behavior should be observed, An 

indicator referred to as the modal phase collinearity is develo ed to measure 

imtginary part of the sensor nodal displacement (mode shape) for each mode. 

Bwed OR the accuracy indicator, the degree of the modal observation Is 

estimated. &fine 

the strength of linear functional relationship between the rea P part and the 

308 
where c . (j=1,2,. . . ,n) i s the sensor modal displacement corresponding to the 

jth realized mode. Let the column vector of order p be 

LI 1T = [1,1,.. .,1] (47 1 

in which p is the number of sensors. Now compute the following quantities 

for the jth mode shape. 

- cj = CJyP 

'rr = [Rea 1 ( c -Fj - 1 ) ] T[ Rea 1 ( cj -zjJ ) 1 

'ri = CReal(~~-~~J)]~[Imag(c~-~~J)] 

e = (cii - ~rr)/z~rj (52) 

0 = arctance + sgn(e)(l + e2)1/2] (53) 

where Real( ) and hag( ) respectively ar5 the real part and imaginary part of 

the complex vector ( ), and sgn( ) is me sign of the scalar ( ). The modal 

phase collinearity uj for the jth mode is then defined as (Ref.22) 

pj = I crp + crj C2(e*+i)sinZ(e)-13/e l/(Crr+cii) ; j=l,?, ..., n (54) 

This indicator checks the deviation from 00 - 180" behavior for components of 

jth identified sensor modal displacement. The parameter pj takes only the 

values between 0 and 1. Dj + 1 indicates that the accuracy of the modal 

displacement is high. (31 the other hand, if p- is away from 1, the jth mode 

is either a noise mode or high damping is preseni. 

Model Reduction 

The dynamiczl system is composed of an interconnection of all the ERA 

identified modes. lhe accuracy indicators allow one to determine the degree of 

individual mode participation. Model reduction can then be made by truncating 

all the modes with low accuracy indicators. The accuracy of the complete modal 

decomposition process can be examined by conparing a reconstruction of Y(k) 

formed by Eq.(35) with the orginal free decay response$, using the reduced 

model . 

IV. SUMMARY OF ERA 

A flowchart of the procedures to be followed to use ERA in system model 

identification is presented in figure 1. The computational steps are sunmarized 

309 
as follows: 

1. Construct a block-Hankel matrix HTS(0) by arranging the masurement 

data into its rows with given r, s, ti (i = 1, 2,..., s-1) and ji (1 = 1, Z,..., 

r-1), (Eq. 16). 

2. kconpose Hrs(0) using singular value decomposition (Eq. 23). 

3. 

4. 

ktermine the order of the system by examining the singular values of 

Construct a minimum-order realization (A, 6, C) using a shifted blockthe Hankel matrix Hrs(0) (Eq. 38). 

Hankel matrix (Eq. 34;. 

5. Find eigensolutions of the realized state matrix (Eq. 40) and conpute 

. . 6. Calculate the coherence parameter (Eq. 45) and the collinearlity 

the modal damping rates and frequencies. 

parameter (Eq. 54) to quantify system modes and noise modes. 

(Eq. 41). 

7. Dtermine the reduced system model based on accuracy indicators, 

reconstruct function Y(k) (Eq. 35) and conpare with measurement data. 

Note that the determination of r, s, ti and ji in Step 1 above requires 

further developnent. This determination is related to the choice of the 

measurement data to minimize the size of the Hankel matrix Hrs(0) with the rank 

unchanged. 

V. EXAMPLES: SlMUlATNN AND EXPERIMENT RESULTS. 

To illustrate the ERA method, two examples are given. First, a numerical 

problem will be posed and solved for an assuned structure with distinct and 

repeated frequencies. Second, experimental data for a simple, tbtO-dimenSiORal, 

grid structure as shown in Fig. 2 is used and realited..in terms of a linear 

system. Experimental results are compared with those predicted by a finite 

element model. 

Numerical Simulation 

Figure 2 shows a representation of a typical flexible structure. The 

dynamical equation for this typical structure with initial-state-response in 

terms of system modes in moda? space can be written as: 

dg/dt A g (55) 

Y=b (56) 

where A is a canonical matrix with the diagonal blocks {Ai, ..., Ak). 

3 10 
g is the generalized modal anplitude and C is the generalized sensor influence 

matrix. The quasi-diagonal matrix Aj (j=l,---,k) has the matrix form 

The complex values 6j - + Iwj are the eigenvalue of the frame structure. 

Given a model described as in Eq. (55), results of some numerical simulation 

using the ERA scheme can be surmarized in the sequel. Two cases will be given 

including systems with and without repeated eigenvalues. The numerical test is 

performed by taking as “data” y the output values of the solution of a model 

with the form (55) whose parameters A, C and initial condition g(to) are 

known. In the analysi s of physical systems, experimental methods generate the 

measurement data y. It is then desired to realdze a system by using the data y 

and compare with the known model. 

Case I: A model with distinct eigenvalues 

coeffiLient of the assuned structure are adjusted to give 

Assume that parameters such as bending rigidity, mass density and damping 

; j - 1, 2, 3, 4, 5 

-0.Olxj JI r-0.01 x j 

Aj = -j 

To illustrate applications of ERA in a single input and single output case, A 

sensor is chosen and located to give 

(59) 

- 

c = [l,0,l,O,l,O,l,O,l,U3 

Let the initial condition for free decay responses be 

y’(to\, = ~0,1,0.1.0.1.0.1.0.11 

Then the functions y with a sample time interval 0.05 second generated 

from the model (55) with known parameters (58), (59) and (60) are used as 

measurement data for the ERA procedure. 

Using jj = ti= i and r=s=90 in Eq. 16, the ERA realization of a dynamical 

system is 

.’ 

C = [U.709,2.529,-0.347,-1.706,0.814,-1 .183,-l.382,-0.276,\ .129,1.257] (61) 

gT( to)=[0.103 ,O .Xi7 ,-0 .I 14,-0 .563,0.395 ,-0.574 ,-0.696 ,-0.139,0.396 ,O .44O] (62) 

and A ts identical to that shown in Zq. (58) with the accuracy close to the 

precision limit of the computer. In the process f realization, the number 

6 =d,/dl as defined in Eq. (38) is set to be IO-’’. The singular valaes of 

the generalized Hankel matrix HrS(0) are 

31 1 
0 = [49.86,44.84,33.69,27.64,23.69,21.04.13 . 57 .I 0.95.6.374.5 . 5083 (63) 

All the values d- (i =11,..., 90) which has the number d-/d less than loo1* 

are considered td be zero. The rank of the Hankel matriJHr:(0) is obviously 

ten which is identical to the order a priori given in Eq.(58). lhe realized 

state matrix is a minimum order of 10 and the eigensolutions are obtained from 

this 10 x 10 matrix. All the parameters for modal amplitude coherence (Eq.45) 

and modal phase collinearity (Eq. 54) are 100%. Although Eqs. (61) and (62) 

are a different realization from the system (59) and (60), they are equivalent 

in the sense that a unitary transformation and normalization will make them 

equal . 

By forming the matrices V in Eq. (5) and W in Eq. (9) with the aids of 

Eqs.(58)-(60). the reader can see that this realization is controllable and 

observable. 

Case 11: A model with repeated eiqenvalues and independent eigenvec+a 

.. 

Assume now that the system model is represented by 

and 

-0.01 1.01 

- 0.01 1 

-0.Olxj 1 

A1- A2 = 

L 

w.0lxj j 1 

% j= 3, 4, 5 = 

Using the same process as last case, the ERA realizatlon simply miss the repeated 

eigenvalue Ai. The result is expected since, by control theory for a linear 

system, single input or single output does not make a system with repeated 

eigenvalues and independent eigenvectors control l ab1 e or observable. It can 

be verified that the matrices V in Eq. (5) and W in Eq. (9) formed by Eqs. 

(59). (60). (64) and (65) have rank 8. Multi-input and mu\ti-output nust be 

used to realize such a system. Let two sensors be chosen and located such 

that 

and two initial conditions far free decay responses 

Note that the rows in Eqs. (66) and (67) are independent. For each initial 

condition, a series of "measurement" funtLion y with a sample time interval 

0.05 second can be generated from the rmd,?l (55) where each y In this case is a 

vector with two elements for two different sensors. The free decay function 

Y in Eq. j15) is then a 2 x 2 matrix. Using that ji ti = 1 and r - s = 45 

for Eq. (16). the ERA realization for a dynamical system is then 

312 
C = 0.1 35,-1~686s~~155,-0,172,0.111 s-0.032,0.099,0.035,0.195s0.177 

L-O . 004 ,Ow 1 07 ,O. 1 42 s -O,d 1 36 s O .1 1 1, -0.032, O .099 ,O .035, O. 1 95 ,O. 1 77 

I- ] (68) 

] (69) 

-0.014,-0~457s3.840,-3.692,8.338,-2~4~~,8.9~.,3.1~1,2.81~,2~5~~ [ -0.051 , 0 092,3 -605, -3.508,8.338, -2.406,8 . 956 3 . 181 ,2.818 ,2 554 

T 9 (to)= 

where A is identical to that shown in Eqs.(64-65). The singular values D are 

D = [7O.16,44.32,37.97,25.~5,11.18,9.050,7.950,3.873,0.127,0.026] (70) 

The sne error window All the parameters for modal 

amplitude coherence and modal phase colinearity are 100%. Again, Eqs. (68-69) 

and Eqs. (66-67) are equivalent in the sense that a unitary transformation and 

normalization will make them equal. The reader can easily verify that this 

realization is controllable and observable. . . 

6=dn/dl as last is used. 

Sample Experiment61 Results 

A sample set of modal 1dent;fication results that have been obtained from 

laboratory ..est data using ERA are included in tbis section. The test article, 

shown in Fig. 2, is a 7 ft by 10 ft flexible grid structure that will be used at 

NASA Langl ey for vi brati on con: rol experimentatf on. It 1 s constructed of overlapping aluminum Cars of 1/4 in, by 2 in. cross section, riveted together at 

one-foot intorvals. Four rive?s are ussd at each joint to provide a tight 

connection. The structure is %-?ended from a stiff overhead bean using two 

short cables attached to the top horizontal member. The results to be shown 

are from a preliminary dynamics test of the grid. It was conducted by exciting 

the structure with an airjet and measurlng the free vibration response using 

nine non-contacti ng proximity sensors. The response sensors were attached to 

a stiff frare located adjacent to the grid for the measurement of out-of-plane 

motions. Eight different excitation frequencies corresponding to resonant 

responses were used. The sampling rate was 32 samples par second. 

The ERA analysis was performed using a single matrix of data from all 

nine response measurements and eight initial conditions, Each response functian Y as shown in Eq. (16) was thLs a 9 x 8 matrix. The hankel matrix Hrs 

of 72 rows by 400 columns was formed to perform the analysis. Table 1 provides 

5 comparison cf the ERA results with analytical prediction from a NASTRRAN 

finite-element wodel. The entries in the center of table are Correlation 

coefficients in percent between each ERA-identified mode shape and each NASTRAN 

mode shape. High correlation values indiccte good agreement between the two 

shapes. The results show reasonable agreement in both.frequencies and mode 

shapes, except for the damping result of the first mode. The main redson for 

the ffrst mode discrepancy is inadequate data length. Only 50 data points were 

used which corresponds to less than one cycle of data for the first mode. The 

results can be improved by using more data points. Note that few high correlations occur for some modes with significantly different frequencies. This is 

because only 9 sensors were used in comparison. More detailed experimental 

results for a complex structure are shcwn in Ref.[19]. 

313 
CONCLUDING REMARKS 

An Eiyensystem Realizat.ion 4lgorithm (ERA) is developed for parameter 

identification and model redvction for djnamical sjstems. Twc developments 

are given in this paper. First, a new approach is developed to derive the 

basic ERA formulation of minimum realization for dynamical systems. As byproducts of this approach, two alternative less powerful algarithms, identified as A1 aid A2, are derived. A special case of A1 Os shown to be equivalent to an approach currently in use in structural dynamics. Second, acccracy indicators are developed to quantify the partipatlop of Systbll mcdes 

and noise modes in the realized system model. In other words, degrec of 

controllability and observabjlity for each participate0 mode is determined. 

A model reduction can then be made for controller deslcjn. 

important features of the ERA algorithm are summarized as follows. 

(1) From the computational staradpoint, the algorithm is attrxtive, since 

only simple numericdl operations are needed; (2) the cm3tational procedure 

is numerically stable; (3) the structural dynamics requirements for Mwlal 

parameter identification and the control design requirements for a reduced 

state space model are satisfied; (4) data from more than me test can be used 

simultaneously ta efficiently identify the closely spzccsd eigcmvalues; (5) no 

restricti6;;s on nunber of measurements are imposed. 

REFERENCES 

E. G. Gilbert, "Controllability and (bSerVdbi1;ty !n Multi-variable Copt - 1 

Systems," SIAM 3. Control, Vol. 2, No. 1, E' I' 21, 1963. .- 

R. E. Kalman, "Mathematical kscription of Linedr Qyt;?ii?+rz; Systsis,' - SIAM 

J. Contr., Vol. 1, pp. 152-192, 1963. 

B. L. Ho dd R. E. Kalman, "Effective Constructim of Cnew State-variable 

Models From InFut/O.itput Diita," Proc. 3rd Arm. Allerton Conf. Lircuit and 

System Thzory, 1955, pp. 449-459; also Reyelungstech, Vol. 14, pp. 545-548, 

1966 . A. Tether. "Constructions of Minimum Linear State Variable Models From 

Input/output Data, "IEEE Trans. on Auto. Contr., Vol. AC-15, No.4, Aug. 

L. M. Silverman, "Realization of Linear Qynamical Systems." IEEE Trans. on 

Auto. Contr., 16, 554(1971) 

1970, pp. 427-436. 

R. H. Rossen and L. Lapidus, "Minimum Realizations and System Modeling. 

I. kndamental Theory and Algorithms," 

R. H. Rossen and L. Lapidus, "Minimum 

Theoretical and Numerical Extensions," 

1972, pp. 673-684 

;972, pv. 881-8920 

AIChE Journal, Vol.18, No.4, July, 

Realizations and System Modeling: I1 

AICHE Journal, Vol. 18, No, 5, Sept. 

314 
La] H. P. Zeiger and A. J. McEwen, "Approximate Linear Realizations of Given 

Dimension Via HO'S Algorithm," IEEE Trans. Automatic Control, Vol.19,(1974), 

153. 

[9] S. Kung,"A New Identificatioic and Model Reduction Algorithn Via Singular 

Value kcomposition," 22th 4si?mar Conf., Circ!iitr, System and Gqut.e**s, 

Novo 6-8, 1978. 

G. F, Gaiub and C. Seinsch,"Singul3r Yalue kconposition and Least Sqirare 

klutions," Numer. Math., - Vol . 14, pp. 403-420. 

V. C. Klema and A. J. hub, 'line Singular Value kconposition: Bs W utation and Some Application," IEEE Transdctions on Automatic Contr6 Vol. 

AC-25, No.2, Apri1, 1980, pp. 164 - 176. 

R. C. Stroud. S. Smith and G. Hama."MOIWUB--A New system for Structural 

@namits Testing," Shock and bibratyon I Bulletin, No. -4E, Part 5, August 

1976, pp. 153-175. 

S. R. Ibrahiin and E. C. Mikulcik, "A Method for Direct ?dcntification of 

Vibration Parameters from the Free Response." The Shock and VibratirBulletin, No. 47, Sctpt. 1977, pp. 183-198. 

D. L Brown, R. 3. Allmang, R. Zimerman aod M. Mergeay,"l)armeter Btimati c.n Tecnniques for Modal .!hlysi s ." SAE paper 790221, Feblmary 1939. 

R. N. Coppoiino,"A simultaneaus Frequeficy ibrrain Technique for Estimtion 

of Modal Parameters from Measured &ita,"SAE paper 811C46, October, 1931, 

H. Vo!d and R. !luU;sr?ll,"Advaficed Ana:ysis kthods Improve Nodal Test Results," bund md Vibration, March, 1983, pp 36-43. 

J. C. Chen,"Ev3luation ot koddl Testipg kthoas," AIAA paper 84-1071, 

Presented at A XAA Qnami cs 5peziali st Conference, My 1984. 

C. T. Chen, Introductioc to Lipear System Theory, Holt, Rierthart ana 

Winstsn, Inc. New Ywk, 1970. 

R. S. Pappa and J. N. Judng,"Galileo Spacecraft Modal Identification Using 

an Eigensystm Realization Algorithm," AIAA paper 84-ir)70-Cp, Presented at 

the AI4A jjnmics Specialists Conference, Palm Spring, Caw, May, 1584. 

H. E. itaiman 2nd R. S. EnyIer,"Computation of A M'ninal. Realization," A User 

kaoual for ASPC, Ch, XV, 1966. 

F. R. Gdntmscher, 'The Theor of Matrices, Vol. I, Ohelsea Publishing CO., 

C. R. Q30,'lR;e Use dnd Interpretation of Principal Component Anilysis 

in Appl red Research,"Sankhva - Ser. A, 26, 329-338. 

New York, Yew York + 

315 
a 

d 

eq 

ruooooooo 

QI 

W 

N 

0 

h 

OI 

Q\ 

d 

d 

0 

0 

r( 

ui 

a3 

0 

316 
c I 

e 

c 

317 

A RESIDUALS APPROACH TO FILTERING, 

SMOOTHING AND IDENTIFICATION FOR 

STATIC DISTRIBUTED SYSTEMS 

G. Rodriguez 

Jet Ropulsion Laboratory 

California Institute of Technology 

Mna, CA 91 109 

ABSTRACT 

This Frpcr rdvarxes an approach for stat-e estimation and identification of spatinlly 

distributed parameters embed4ed in static distributed (elliptic) system models. 

The method of maximum likelihood 5 used to .find parameter values that muimize 

a likelihood functional for the qstem model, or equivalentiy, that minimize the 

negative logarithm of this functional. To fiad the minimum, a Newton-Raphron search 

is conducted that from an hitid estimate generates a convergent sequence of 

parameter estimates. Central to the numerical search are a gradient functional and a 

Hessian operstor, which are respectively the first and second function-space 

derivatives of the negative-log likelihood fpactionrl with respect to the parameter 

distributions being identified. For simplicityc a Gauss-Mar'rov approach irr used to 

approximate the Hessian in term of prodoctt of first derivatives. The gradient and 

approximate Hessian are compnted by first arraDdlzLg the negative-log likelihood 

functional into a form based on the -re-root factorization of the predicted 

covariance of the measurement process. The resulting data-processtisg approach, 

referred to here by the new term of predicted-data-corrrhnce rarun-root filterin& 

maker the gradient and rpproximete Hessian calculations very simple. Since the 

parameter estimates are only approximations to the actual parameter vaLOCs, there is A 

parameter estimation emr inherent in ohe estimation process. Ciamer-Rro bounds are 

obtained for the covariance of the estimation error in term8 of the informrtion 

operator associated with the likelihood functional. There error covariance boards are 

then used to outline methods for optimal input design. 

A closely related set of state estimates is also produced by the maximum likelihood 

method: rmocthed estbtes that are cptinul in a conditional mean tense and filtered 

estimates that emerge from the predicted-data-covrrhnce square-root filter. The 

terms 'smoothed' and "filtered" are used because the fonnrrlu which generate these 

estimates, when expressed in operator notation, are symbolically very similar to those 

used in filtering and smoothing for linear dynmnical rgstemr. A key sirrd.hrity is the 

presence of a predictor-corrector rtrocture conuinlag estimator gains that, as in a 

Kahn filter, can be expressed in terms of the state estimation error covari.nces. In 

addition, a residual procem can be definedc in the usual way, as .the difference between 

the actual dru and the predicted data obtained from the filtered state estimate. The 

residu~b have properties nearty idattical to those of an hovationr process: the 

residuals are whits with a unit covariance; and the residuals and measurements can be 

obtained from each other by means of reciprocal linear transformrtionr. Because these 

tranrfonrutionr &re not Volterra (causaU, the residuala are not a boar fide bovrtions 

process. Even though they are not a true irpnovaUonr procem, the residuals are very 

useful- because they lead to itate ad parameter estimathn scheme8 for elliptic 

systems that retain conceptually the rimpllcity of those obtained by the innovatiom 

approach to fllteriry, smoothing and identification for hear dparmical systems. 

319 
1. INTRODtJCTION AND SUMMARY 

The elliptic models considered in thi8 paper can be cast a8 

where A is a formally 8elf-8djOht elliptic differential operator defined over the 8pati.l 

domain Q; B and C are approprhtely dimclruioned operaton that model the influence of 

the process emr o and the input f on the state w H is an operator that characterher 

the SU%~-~O-Q~~~~V~~.~OXU nupi o and n are nhtte-nobe model errors formlag the model 

error vector t=[o,n]; ad f is a deterministic iapPt. Bumpl6r of the application of rrrch 

models tc the problem of static ahape determirution of &me -ace renrcturer are 

contained in Ref. 111. 

The central aim here is *to develop n maximum-likelihood approacb to the 

estimation of the parameters 8 (there parameters could in general be spatially 

distributed) by using the data 9 and the qstem del itdelf. It is awumed thrr the vpe 

vrhre eo of +he parameter 8 is a determinlrtic but poorly known quantity. The input f 

can be ralected to optimize the data generated for estkzation. A related but somewhat 

seconderg ah it fo develop a metbdolo~ for compz#rtfon of the correm 8tAt.e 

estinuter. 

h Formula for the Negative-LoR LbUhood Ram 

lt will be shown in Sec. 3 that thc neaative-lo8 likelihood functional is specified by 

when 

where 

Predicted-Drta-Cov~rhnce Sawre-Root Form of the Likelihood Ratio 

idcater an innat product in the fixaction space to wbich the data belopllr. 

A number of alternative formal.? ior ths nS8atiVS-lOg likelihood functional are 

developed in Sec. 4. To wlve the above minimization problem, the mort convdenlt 

fonmP1. L: 

320 
(1.4) 

where 

Equation (1.5) can be viewed as specifying a filter, characterized by the operator L(8), 

that processes the data y and the suspected mean m(8) to provide a ‘filtered’ state 

estimate de,. Thi8 filter L(8) will hereafter be referred to a8 the predicied 

data -covariance square-root filter because the key calculation required to tptcify L(81, 

as in (1.61, is the evaluation of the square-root of the predicted-data-covariance 

operator [I+B(6)]. The equivalence between (1.3) mnd (1.4) can be esublished by 

substitution of (1.5) and (1.6) into (1.4). 

Mcte for later reference that the definitions in (1.6) imply tbst K(8) and L(8) are 

related by 

[I+K(e)]-’ = I-L(8). (1.7) 

Furthermore, (1.7) implies that K(8) = Lle)+K(e)L48) = L(8)tL@)K(€3i. 

Gradient of the Likelihood Functional 

The gradient functional aJ/a, to be defined more completely in Secs. 5 and 6, is 

specified by 

where 

with 9 = y-m, and aL/a, am/* be@ the function-space Frechet derivatives of L and 

m. These equations can be obuined from (1.4) by faction-space differentiation with 

respect to 0. 

The gradient functional aJ(B;y)/a in (1.8) Ls the Frechet derivative [2) of the functional 

J with respect to the parameter 8. The derivative is a linear transformation (a8rumed 

to be bounded) that map8 an adnxbaible parameter perturbation be into the 

corresponding perturbation 6J(0,be;y) of the likelihood functional by means of the 

equation 6 Jte,M;y) = [aJte;y)/a] be. Detailed computation of the function-space 

derivatives above is condncted in Sec. 6 arw a perturbation analysis of the 

eigemyatem of the covariance operator II - H@BB*O*H* obtained in Sec. 5. 

Note that in Sec. 7 it will be established that 

32 1 

(1.10) 
so that the expected value of the gradient vanishes at the optim~l parameter value 8,. 

Hessian of the Likelihood Functional 

Similarly, differentiation of (1.8) leads to 

tWae*taa/ ae), (1.11) 

and to its expected vrb at &eo of M(0$ = B[a' J/a02]lexe , Le., 

0 

Furthermore, substitntion of (1.9) in the last term of (1.13) leads to 

Note that the expected valw of ?.he Hessian operator a'J/W2 evaluated at O=ec is a 

sum of two terms each of which is positive definite. Consequently, in 4 prababillrtic 

sense nude precbe by (1.131, the likelihood ftmctifmal is strictly convex in ths vfctnfto 

of the optimal value &eo. Note that by definit.bn Weo) in (1.13) is also the 

information operator arrocirted with the likelihuod functional. 

Newtori-R8~hson Search for the O~timrl Parameter BStiuute8 

Since the problem of minimhation of Jt0sy) in (1.4) has no clored-form solutio& it is 

nacesmry to consider iterative numerical search techuiques for optid~~tion. The 

following constitutes a function-upace Neuton-Paphtan iteratioa: 

(1.14) 

where 8n = aJ(#;y)/a is the gradient functional (1.8) evaluated at e=& and where 

is an approximation to the Hessian operator aaJ/a2 in (1.11). Thit approximation is 

obtained from (1.12) by tsphcina the second term B[(aZ*/ae)@z/W)J with the rctrul 

value [CaZ*/a)tae/a)J obtained in a ri4sle realization. Under certain coaditiom, to be 

322 
examined in more detail in future work, the sequence en specified by (1.14) converges 

to A 10Cd InhbUXI of J(8;y), If the initid esthte used to start the search 

sufficiently close to the optimal value. 

Cramer- RAO Bounds and 0Ptimal Input Desim 

The above nametical search results in an estimate 8 of the actual parameter value eo. 

In Sec. 7, A C-R bound for the covariance B(8 8 *) of the estimation error €3 = 8-e0 is 

obtained from the inequality. 

PP P 

where the information opera 31: M(80) is specified in (1.131. The related mca-square 

estimation error is bounded by B(8 *8 1 L TrfM-'(e0)]. PP 

The informrtion operator M(eo) can a&o be wed to specify CiiteriA for optimal input 

design. Perhaps the simplest optimal selection method to imglement is that which 

seeks to maximize Tt [M(e,)] with respect to f, subject to the constraint that f sati8fy 

Ihe normalization condition of f*f=l. Thh method results in an optimrl f which is the 

eigenvector correspsnding to the largest eigenvaiue of a positive-definite matrix 

described in detail in Sec. 7. Other criteria for optimal selection based on calculation 

of M-'(80) may be more diffidt to implement but tmully lead to superior 

performance. 

mhe Corresponding State EIIthUteS 

Related to the parameter estSmsth approach are the followiry two distinct state 

estimates (denoted by uo AM zo1: 

where C and g are Krlman-like gab2 (see Sec. 8) specified by 

G = 1 8in2%\@k*, g = (1-C08\)\@k*. (1.18) 

h these equtionr, Ot. are the eigenvectors of the operator EL - H@BB*QH*, so that 

Ai@k, with A; being the related eigenvalws. Also, \ and \ are defined by 

The #Ute estimate uo s MWy) is defined as the conditional expectation of the state 

given the data y. Since uo is an optimal esttmrte of u base6 on the entire data set (as 

323 
opposed to a mbret), uo can be vieued a8 a best pnoothed ertimate. The other 

ertiuute, 2Q in (1.171, will be referred to u filtarea rute eatbate. Th3 filtered 

ertim~te has no known probabilbtic intefpreution rimilrr to uo - B(W) above. 

However, in spite of the apparent lack of probaMllrtic mmlplt, thir ertirrute is rueful 

in Sec. 8 that zo in (1.17) .ob I, the estimate emerging from 

the predacted-data-covrrhnce 8~1~re-mot filter, are related by 3 - €bo. Hencel xo is 

a born fide estfnute of the entire state, whereas % - &o ir 8 parti.1 ertimate defined 

8@U9h# tb 8r-t Ad Hetrira Crk&tfW (1.8) md (1.11). It dl bo 8honly at the observation locations. 

Kahaa-Ue Gains rad Error Covarhnces 

The 8riPr G ad g in (1.17) can altemtively be specified in term8 of the 

covariance of the state ertirrstion error inherent in-uo and to, Le., 

G = PH*, 

where 

The corretponding mean-rqurre state ertiuution error is 

B[(u-uo~*(u-uo)] = Tr [PI, B[(u-xo)~u-aol] I Tr @pel. (1.211 

Pufthcrmore, Y ad p are related by 

Since the term pHfHp io rum-negative, the mean-s~~rrs estimation emr auocirted 

with the smoothed erthte uo is 069er hreer than thrt of the ffltared estimate to. 

Pilterina ad Smoothiq 

While uo ad p0 have been defined sow that independently in (1.171, they are 

related by: 

u =Z +pH*e, (1.23, 00 

where 

k the residual process defined a8 the difference between the data 9 and the 

obreroad-state eotimrta fro. The symbol 9 in (1.34) denotes the^ mean-centered data 

324 
process 9 = y - HWf. It will be shown in Sec. 8 that (1.22) and (1.23) constitute a 

generalization to elliptic rystemr of the forwrrd/backward weep meW for solution 

of smoothLng problem in linear dpnrmical rystemr. 

The Residuals as b Pseudo-humvations Process 

The residuals in Bq. (1.24) have two propexties that are rimil~r (but not identical) 

to those of an innovations process: 

E(ee*) = I, (1.25) 

e = (I - L) jjc f = (I + E)e. (1.26) 

Eq. (1.25) reflects WB?- *I8 Of the ~8hdS. Bq. (1.26) rate8 th.t th8 rS3idUal .nd 

mean-centered data pmccrser e and 9 can be obtained from each othbr by mea01 of 

reciDroca1 transfomtions, i.e., (I + ~1-l = (E. - L) as in (1.71. ~~eness of the 

innovations and reciprocal rektionships between hnovatFonr and measurements are the 

two central features of the innovations approach to least-rqumrer estimation for linear 

dpLlmica1 systems. Bqs. (1,251 and (1.26) are similrr to these conditions. Howeverc 

there is a key difference: the transformation8 (I t XU md CI - L) in (1.26) are Predholm 

operators whose domain is the entire measurement space. Thlr is in contrast to the 

Volterra (causal) operators in the hnovatiorrr approach for hear dpPImtca1 systems. 

The notion of causality ft not even used in this paper, although such a notion can be 

defined for certain classes of elliptic systems [L]. Becaore of thir difference the 

residual process it not a bo- fide inmovationr proceu. Emever, the resislul process is 

stili useful in 0bt.W the relatively simple foz'xdat in (1.8) - (1.26) for filter& 

smoothing and identification. 

Pwer Outlinq 

This section has at a su.nriury level addressed miry of the fundamental iur;er involved 

in the mrximum likeltaod approach *to est,imation. The mbsequent section€ of the 

prpar contai a more coraglete dercr'rption of the above mdts. 

Sectiong. Development of the mrthemrtical framework -- inclnding integral 

operator models, a priori covrrbnce analysis with white-noise ?nodel erron, Fredholm 

resolvenu, and eigenfuaction expaarionr -- required to arrive at formrrlr (1.3) fw tho 

likelihood functional and to evaluate the corre8podxug ftmction+prce gradient in (1.8) 

bd the rpproxim8te Htrrkn Ln (1.15). 

Section3. Deriv&oioa of the negative-log likelihood functional in (1.31. Thir 

functional is the negative logrrithn of the Ukem9d ratioc urociated with the 

detection of a Gaussian s-1 in additive Gaturkn mise, trditiok.lly encountered in 

the theory for cornmuaication and signal detection. 

Sectiou4. Development of alternative forxxmlas for the likelihod ratio, some of 

which are more convenient to use than (1.3) in implement4 the numerical search for 

optirnizatioa -- in patticSr, development of the predicted-dau-covrti*nce 

)auare-root filter form i1.4) upon which tha NsWton-Rapbr~ search L brsed. 

Add~tiltrul farm8 of the likelihood ratio which mre of interest in their own right 

325 
(although not subsequently used in the paper) are: a smoothing form exprerrred in terms 

of the best man-squats state estimrter an eigearprtem erpans!.n form bared on the 

eignenvalues and eigenvectors of the operator B-H@BB*@*H* in (1.3); a trigonometric 

operator form with which most of &e mrnipuhtionrr involved in the mrximum likelihood 

approach can be visualized using their similarities to simple trigonometric formulas foscalars. 

Sections. Development of a first-order perturbation annl9sis to evaluate the 

infinitesimal changes in the eigensystem of the operator R=H@BB*WH* in (1.3) due kr 

similarly small changes be in the parameter distributions bekrg identified. This is the 

central calculation required to compute the function-space gradients aJ/a, Wa, 

aL/a and am/a ‘JI (1.8) and (1.9). 

Section 6. Calc~~lation ofthe gradient functioml ad approdmrte Hessian of the 

likelihood functional based on the pcrturbstion analysis of SCC. 5. There are the two 

calculations whlch are central to Smplcmentatiod of the Nawton-Paphton search ad 

which have been used as a bash for computer programs to implement the mucmtUa 

likelihood approach. 

Section 7. Parameter estimtion error covariance ana&& ad Cramer-Rao boundr 

based on explicit formulas for Zhe Hesrhn (informrtlon) operator in (1.13). Outline of 

un optimal input design approach based on us@ tbc! Crrmer-Rro l?ousid as &n OptiaUrUty 

criterion. 

Section8. Aarlyrir of the filtered and smoothed state enfmrter embedded h the 

parrrmeter estimation approach. Amlysir of the predicted-dau -covaiirncc square-rtmt 

filter resulting in Kalmm-IlLe formula9 for the filter grin, esrhuticm of the state 

estimation error covariance, and rehtdoXUhip8 betwoen filtensd md smoothed es~t8~. 

Section9. 

the n\\a.efical search for the optimal esthates. 

Swmary ad explanution of ca1culBtfonr required for hnp1exnentatiOn of 

Section 10. Conclusions and directions for future work in tba rxaa of development of 

as: iptotk propeflies of the erthatts ad of optimal @Ut de8igL 

The aim of this section is to dcvemp L set cf onircehneous results that will be useful in 

subsequent sections in conductw detailed derivation of: Zhe negative-log likelihood 

functional in (1.3) eo be minimiaed, the corresponding function-space #radiant in (1,8h 

and the approximate Hesrirn operator in (1.15). The main result8 of the section can be 

summarized as follows: 

e conversion of the pareid differential operator model in (1.1) to an 

equivalent integral operator formulation. TLnir in’cegrrl operator 

formulation is introduced because it simplifies the statement and solution 

of the estimation problem in (1.1) - (1.3). 

326 
0 evaluation of the observed state covariance operator B=H@BB*@*H* In 

(1.31, under the assumption that E = [c,n] is a spatially distributed 

white-noise process with a unit covariance operator. Related to evalurtim 

of this covrficrce operator B is the similar evaluati- of the mpacted 

mean m = d@Cf in (1.3). 

0 evaluation of the dual observed-state covariance operator Q=B*@*H*H@B 

- which can be vie;.;ad as &he covariance of the outrut of a system model 

dual to (1.11, unde the rrrmmption that thi8 dual apstem is driven by a 

white-noise process. 

e def*inition of two sets of eigenvectors qk and qk of R and Q above, with ).; 

being a set of common eigenvalues. 'ihese two sets of vecwrn can be used 

to expand functions in the input space H1 and the outpm space Hg. 

* 

definition of iYe vectors \ and pi in the state rpace ti2 and its dual IT7., 

related to tJrk and 9, above by 5 = Xi14bBWk rile pk = XiL@*H%k. 

two sets of vectors \ and pt satisfy a boundorg-value problem 4hLar to 

those traditionally encountered as necessdlry and sufficient coxLdJ.ions for 

optimality in quat atic optimal control rad estimrtion problem subject GO 

linear constrair .I. 

8 analysis cf the basic relrti.onship betwsen R and Q above and thslr 

corresponding Fredholm resolvents 'P ad S defined a8 P = I-(I+R1-A and 

s = I -(1+~)-'. BxpArrton of the operators EL, Q, P and s in terns of ae 

eigenfunctions et qk defined above. 

0 development of trigonometric operator fom for B and P. There 

trigonometric forms allow dsvelopment of ir' westing trigonometric 

alternatives ta (1.3) in evaluating the likelihood IP' ,tmul. 

Wde the section concentrRtes on the development of a mathematical framework to be 

used kn subsequent sections, uuny of tine above result8 (such as tht trigommettic 

operrtor fannulns for the covarbnce operaton) are of interest in their OWD aht, 

sornewbt indepenlent!j cf thetr wbmquent epplication. 

Hilbert Space Notation 

There are three Hilbert aprcer of gximary interest: Lhe'lqFut llpscc HI to wMc% 

the process error o ~nd the aeterdnbtic hput f b5long; the state space H2 conutning 

the state u; end the measurement space H3 where the data y ad tho obsewrti~n error 

n belong. The her ptoduct between two arbitrary elements u ad v Ln the space Hi is 

denoted by <u,v>i or by tho simpler notation u*v = <u,v>~. Similarly, uv* denotes a 

Hilbert space outer product. 

327 
Conversion to htenral Operator Mdcd 

It is convenient for subsequent developments to convert (1.1) to an equivalent 

integral operator formuhtion. 'Po this end, define the Green'r function #WE) of A at 

the sdution of 

wher.: 6 is the impulsi-!e deita function, and where the mbscript x in Ax donotes that 

the spatiril differentiations embedded in A are performed witb respect to x (a8 opposed 

to being performed with respect to I). Define then the mcegral operator 9 whore 

kernel is the Green' 8 function, i.e., 

for all admissible functions v. Note that 9 is the integrrl operator such that A@ = I, 

-where I is the appropriately dimensioned identity. 

With these definitions at h.&, it is possible to recut 11.1) and (1.2) as 

where m(8) is the *suspected mean" 

Equation (2.3) can be c&st inro tte €cll~-~ing even more compact no*c;stion 

Predicted-Data Mun unl Covariance 

The evalartion of the predicted mean rrd covariance of J, needed as a preliminary 

step to arrive at (1.3), k based on the key asmmption that the mode1 error vector t .I 

[o,n) is a zero-mean spatially distributed white-mise process whose covariance 

operator E(ct*) is the identity, i.e., 

where I is the appropriately clmsnrioned identity. Note tht tW dnunption tr not at 

all rertrictiwe, because the more ~eneral cue where the model emn t = [o,n] are 

correlated (with a nonidentity covfirirnce operator) can be hadled within the came 

formulation hv rebetion of the operator B in (1.1). It is assumed here that BB* is 

bounded ad trace-clast, with kernel bWt) satbfy~ I, b(.hM. <-. 

328 
Remark 2.1 The process J is a random field with mean and covarhnce specified as 

E(y) = mW, E((p-m(8)] I(y-m(8)]*1 = I+ R(8), (2.7) 

where R(8) = H(8)~8)B(8)B*(8)9Y8)H*(8). Thrt E(y1 = m follows from (2.5) and the 

fact that E is zero-mean. The second of Bqs. (2.2) follows from the following sequence 

of operaticns: E[(y-m)(y-m)*] = B[hcc*b*] = hE(cclh* = hh* = I+B. A more detailed 

development of the above resuk is contained in Ref. 11). 

Remark 2.2 

field with mean and covariance tptcifiea by 

The process u in (1.11, r3preeenting the sute of the system, ir a ndom 

Note that the ttrte covariance 

(2.8) are related by 

and the 'observed-state" covariance B in (2.7) and 

Remark 2.2 The state covariance ii tatitfie8 the partial differential equation 

ARA* = BB*, (2.11) 

A result which can be established by pre-multiplication of # in f2.9) by A and 

subseqwtlt post-multiplication by A*. 

Bemark2.4 

integral operator 

The state covariance operator @ can be represented as the following 

Biv = I, rCx/f) V(Ehg, (2.12) 

where the kernel rW[) satirfies 

and where b(x/E) is the kernel of BB*. This result cm be established by means of the 

following sequence of operations. Consider an admissible function P (adminible in the 

seme that i; can be operated on by the operaton A&A* and BB* in (2.11) 80 that 

ARA% = BB*V makes sense). In tern of the corregonding kernels r and b, this last 

equation be comes 

(2.14) 

I' where the first eqwlity is valid because by definition A * is the formal adjoint cf A 

Since 8q. (2.14) must be valid for all rdmirsible v, then (2.14) implies (2.13). 

c 

329 
Remark 2.5 The state covariance kernel r in (2.12) CUL be expremed u 

* 

w PROCESS c 

where @ t the Green's fwmioa of A, d b is the kernel ol BB*. Tbia redt can be 

established By -re- (2.9) in term of the operator kernat 9 amt b of 0 .ad BB* 

rrd by robsequent reverul of the order of integration. 

Hu SERVED 

(2.16) 

8 ERROR 

where the spmnution is taken over the dbmrbace locationt. 

Q, &E , i L 

5 DUM '. 'STATE '' r! OBSERVATION, ,,. ERROR L - 

8. h BSERWD 

!UALsiAlE 

330 
The primal system model is based 00 (2.31, with o, u and €Io dmting the process eftor, 

the system state, and the observed atate respectively. For this IILD4e1, = E(uuq is the 

covariance of the strte, ui& R = HgH* is the correspodhq~ covariance of tht 

observed state. It is assumed €or the sake of this dircpuion that the determfnirtic inpat 

f in (1.1) has been set to aero, SQ that the swpectd mbm rn in (2.3) i8 zero also. With 

this assumption, it is not necessary to show m in the block -am in PQ. 2.1, .ab tbe 

rehtionthig between the priMl and b1 models is ilhwrated more easily. Ths bp.i 

system model is chancterittd by the dud operators If+, W and B*, by the &as1 or 

adjoint state A, rad by the observed dud state B*L It is usumed that the &A model 

is driven by a tmit-corarbnce nhite-noire process so that B(llpq = I. Thir lnppt process 

n drivhq :he dual model can be thou@a of u being the obrervation error process in 

(2.3:. For this dual model, 0 = B(hA*) ad Q = B(B*XA*B) are respectireiy the 

covariances of the state h and the o'oreroed state BfL Upon dtiplication of A in 

Fie. 2.1 by A*, the following pawl differential equation8 remlt to describe the dud 

model 

A*k = H%. (2.16) 

Note that the dual state covariance 0 = .ad the dnal obrerred-state 

covariance Q = BW*H*H@B are nkted by 

Q = B*QB. (2.19) 

In the same spirit used to rmve at (2.11)-(2.16), it is now possible t3 develop the 

folloning properties of the operator Q rod its corrtspodiq~ kernel q. 

Remark 2.7 The dual-state cotrrhnce oper8tot 0 ratbfier 

A*QA = H*Hl (2.20) 

a result which can be obtained from 0 = @*H'H@ in (2.17) upon maltiplicatioa 

by A*( *)A. 

Remark 2.8 In termt of its kernel q, the opentor 6 can be expressed 88 

& = I, cl(w'fMt'Mtr, (2.21) 

where q satisfies the differcn*&l equation 

and uhere h(f/x) is the kernel of H*H. Thsr nsolt can be established by an amroach 

quite 8imil.r to that used in arriviq at (2.13). Thc 8-1 v denotes a~ah an 

admissible function defined to be admissible if (2.21) makes sense. 

EenurB 2.9 The kernel q(EM of can be expressed as 

331 
(2.24) 

where the muaxnation is t.&m met the ret of tsnrorlocatiom. 

with % being the con- e&smectort. Note *ht, In cuas where H hu a 

finite-- * rat, the operator R is an N-by-N xmtzix with a finite marrhrr of 

euenvectorr. In the more @merd cub u&re obs me of H k Mfnftalbimsarf0ll.L 

thm R is pmrlly compact ad s+O u b-. In both of these cues, the fom 

Mercer expansions bold for E and its kernel r 

Furthermore, the normalized eigemecton + form an ortbmrzml buL for the 

observation space %. Tbir impbt that C %%* - I, where I ia the identity in %. 

Closeiy related to the buit @k abme us the dud racton vk defined u 

~euurk3.11 

obraned-state corerlance Q = BW%W@B, Le., 

n;C vectors qL defined by (2.27) are +.&e s~mvectoit of the dual 

Wk (2.28) 

332 
This result can be established by premultiplication of (2.27) by H@B and use of the 

condition R@k = '.:ek. Note that, if the dimentian of the 4mt space HI ir mater 

than that of the output space Hg, then the qk do not span the input space. They do, 

however, span the ranee subspace of the operator BWrH*. Conrcqucntly, they curnot 

be used to expad vectors in the null space of HaB. 

Remark 2.12 The vectors *k are ato related to ak by the equ8t.i- 

(2.29) 

a nsalt that can be obtaiped from (2.27) upon pl'eomltipllc8tion the operrrtor H@B 

use of the condition R% = 

Remark2.13 

can be expressed as 

The dual-state corariurce operator Q .rd its corn- kernel q 

a set of equations which are 8rulog;Ou to (2.26). Thb redt can be obtained from the 

observation that Q = B*a*H*H@B = BWH* (C *kqk*. Use has 

been nude of the condition E ek%* = L 

HaB = C 

The vectors +k span the ohsenration space %. while the vectors vk do not 

span the input space H1, they do span the range of BW. So far, no at- hu 

been made to obtain vectors that can be used to erpd functions in the state space 5 

or in its dual space Ha*. To this d, define 

The Vector \ b b the $Ut& space thr, 8d)Obt V8fbbh8 pk UC b th!! dorl 

space. In general, neither one of these two vectors however spuu the state space 'fa. 

EL-& 2.14 Tb vectors xL d p am oitborro-1 with m&Ct to H*H 8Ild BB* 

respectively, Le., 

%*H*Hx, = 0, 

= 1, 

- 0, 

- 1. 

(2.32) 

(2.33) 

333 
Then, substitution of (3.34) in (2.36) implies (2.35) 

Note the similarity between this problem a.3 those tnditFonrllp encmterud u 

necessary (ad at times sufficient) corditirJIpr for optimaliw in -tic optimal 

control and estimation problemr for lfneu ryrtams. 

In terms of the zorrespcding kernels r ad p, tbase equation8 become 

(2.38) 

for the case with continuously dinributed data. In cases with discrete data, P and P. 

are matrices whose general elements P and Ptm are related by k.m 

(2.39) 

334 
In both cf there equations (2.38) ad (2.39). the unknown is the Predholna rerolvent P, 

whereac the obrerved-rkte covariance kernel EL i.8 known. 

Remark 2.17 

result, which can be stated a8 

The integral operator EL and iu Predholm rarolvent P commute. This 

RP = PR, (2.40) 

is a direct consequence of (2.37). 

Remark 2.18 Equations (2.37) also imply that 

P = (I+R)-~R = R(I+R)-', R = (I-P)-'P = PGPI-'. (2.41) 

Remark2.19 (2.411, it is portible to define the 

resolvent S of the dual-state covariance operator Q by the rehtiomhip (I+Q)-' = LS 

which implies 

In a manner analogou8 to (2.37) 

Q = S + QS, Q = S + SQ, SQ = QS, (2.42) 

and 

s = ~I+Q)-'Q = Q~I+Q)-', Q (I-SI-~S = sa-si-'. (2.43) 

Remark 2.20 The Fredholm rerolvent.8 P and S can be expressed at 

These expansion8 can be ertablbhed by 8ubrtb~ (2.26) and (2.30) into (2.37) and 

(2.4 1). 

TriRonometric Operator Fonnr 

Remark 2.21 The predicted-data-covariance operator (I+E) can Le exprerred br 

I + R = I+ TAN'a = SBC'a, 

where TAN'a and SBC2a are the operators 

(2.46) 

(2.4?) 

335 
and tan ~1: is defined by tan % - kk. Note aho for later teference that 

Proof: A # 8 c = & kn 'a @ @ < thereby ertrblir~ (2.47). Use 

of the formal expresrion I = & ekek*for the identity I implie8 tibat cf+JL) = & (l+flln2\) 

+k+k*, which leads to (2.46). Bqutiw (2.48) are obkhed from (2.46) and (2.47) by 

performance of the square-root operation. 

Lecall that R = 

Pemvrk 2.22 

operator R can be expressed as 

;he Predholm resolvent defined as P = I - (I+R) -'of the covariance 

where SIN2a is the operator defined by the expaasion 

- Proof: This redt can be established by substitution of tan2% = k: in (2.44). 

Remark 2.23 BqtmUoar (2.47) and (2.48) togethar im~iy that 

P = B(I+R)-' = TAN2a[I+TAN2a]-' = TAN2= [SBC'a]-' I SIN'a, (3.51) 

a trigonometric operator identity that can be viewed as a 8eneraUation of a hilar 

identity involving scakrr. 

3. DBPIVATION OF THB LIKBLXHOOD PUN cmow 

Based on the ndt8 of the preview section, it is now pouible to derive the 

likelihood functioarl in (1.3) to be minimind. Since the development required to 

achieve this is fairly lengthy, it k convenient to SrUrpMrjre in advance the pivotal steps 

involved in the derivation: 

0 the integral operator mode! 7 = m + HBBo + n in (2.8) is first coverted into an 

equivalent 'spectral" foxm yk - 5 + Akak + 'fr( ivhere yk - ek9, oL = 

wk+o, 5 = +k*' n are the correspaidirq wectral coefficients. 

the spectral coefficient8 pk of the dru y are a reqwnce of independent 

Gaursirn raudom vaxiables with muan B(yk) - a+ covariance CY; - 1 + A; and 

probability denrity pk(yk;8) - T 

-'/BCY-1 

8- [-(yk-tnQ"/2a;] 

336 
0 a "finite-dimensional' Ukelkhood ratio is then defined 88 the prduct Of 8 

finite number N of terms involving the probability dsmities p (y ;e) above. kk 

an 'infinite-dimensional" likdihood ratio is Gbtained by let- the number N 

of spectral coefficients approach infinity. The related negative-lo8 

likelihood functieM1 in (1.3) is obuined by taking the negative logarithm of 

the functional that. results from the limiting process. Of come, 18 cases 

where the data b finite-dimensional (obtoitad by means of a finite number of 

discretely located measurements), the limiting procerr involved in this krt 

step is not necessary. In tBi, case, the 'finite-dimensional' likeUhood 

function obtained tE the previous step is the fumxion to be minirniaed to 

obtain the parameter estimates. 

The remaindc 

result& 

of this rection contains a more detailed derivation of the foregoing 

Recall that 

y=mtH@Bo+n, (3.1) 

where m = H@Cf and f ir tbe input. A8 outlined sbove, the fint stpp toward evsfartfng 

the likelihood function ir to convert (3.1) into an eqnivalent 'spectral' form by wing .he 

eigenvectors +k ad Wk, Le., 

Substitution of (3.2) and (3.1) and premultiplication of (3.1) by +k* lead8 to 

Result 3.1 s , % and % are independent Gau88hn random varlrbler with mean ad 

covriu*icc given by 

MEAN COVARIANCE 

where f = yk - 9 Hence, yk is a sequence of independent Gawrira random vafbbler 

with mea= p4: and covariance 1 t 1;. 

337 
N Let 9 J [yl, ...,yN] be an N-dimearionrl vector conriot&g of the fht N rpeetral 

coefficients yi ob the data 9. Becalue yk are independent Go an random variables 

with mean % and covariance a; = 1 + Lk, their corretpoodiqj probcbiUty dsaritSar 

pk(yk;B) = wJ%iAexp (-7;?2ai) can be dtip!icd to obtain the pnobaMlltg density 

p(y ;6) of tbe composite Ebdimemfonrl vector ]I , Le., 

a 

N N 

In order to obtain a likelihood funetionrl for ,ta4 identification problem with the 

would be in the Unit a probability density ftmcdoxul (PDF) for tha mcew p. 

Unfortuartely, thia limit -9 not erirt bccarrro the ri@it ride of (3.4) ma9 not comcqa 

as N*, and con8equently s PDF for tha procerr y csrmot be defined !a tbir nunnet. 

function-rpace ptoCe88 9 a8 the d8t.v it WOUld bt derirrble to &M W 8ad Oh& nzIt 

However, this can be circumvented by divibn# by 

This results in 

N which can be viewed 88 a likelihood ratio cOnrirtin8 of the PDF of the proceu 9 with 

the “8k3tu1BU ok and 5 nonzero, divided by the rimhr PDF of yN Vrith ?he 8-l8 ok 

and 5 set to zero. The term likelihood ratfa rrred to dercribs (3.6) i8 COarbtant with 

terminology common in the thecry for detection of Caorrtm rl~pult in additive 

Caurrirn noise [SI. 

bl N Uth0W.h the limit8 of p(9 ;e> mnd p0(9 ;a) appsrriry rerpectivsiy in (3.4) and (3.5) -7 

not erirt when taken inde;.endcntly, the limit of their ratio in (8.6) ia a well-defined 

quantity npecified by 

338 
This Is the desired erpression for the likebod ratio that the nwtmum-k hod 

method seeb to maximize. It can be interpreted as the likelihood ratio far the 

detection of the "slgnrl' m t H@Bo in (3.11, in the preeance of the noisy Gaussian signal 

n. Insteed of maximizing h(y;8) directly, it k more convenient to minimize the 

negrtive-log likelihood factional defined as J(8;y) - -log [h(yi(31], or, more a@licltly, 

Note thvt for the special case with no deternjnirtic bp&ti .=m=O in (3.11, and the 

negrtive-log likelihood in (3.8) r4ucer to 

where P(8) = I - [ItR(t))]-' is the previauly defmed (in Sec. 2) Fredholm resolvent uf 

the prcdicted-drta-covariance operator B. 

Tbe first tern. ib both of these hst two equations can be cart into an equivalent and 

somewhat more convenietlr form 'q -=e of the identity [4] 

Substitution of (3.10) in (3.9) leads to 

which has been recorded previourly a8 (1.3) ad conatitutbr the central aim of thi8 

section. 

B eorientrtion 

The method of mximurn lihlihood, a8 defined here, results in cstinrtes that 

minimize J(8;y) in (3.11). Thb minimization problem can be viewed (IS A function-space 

nonlinear programming problem subject to the 8ystern model conatraint.8 that 

soiurion tG this problenl erlrt.8, it i8 nece8sary to u8e .numerical method8 for 

optimization. However, &ere out alterartivs formulrs for the Illelbod retio tlut 

are more convenient to we in the imgiementation of the wmericrl znethods. Such 

formulas are developed in the following section. 

~(8) = M(e)9(e)*(e)B*(e)~*(e)~*(e) and mW = ~(e)@(e)c(e)f. Since no closed- form 

3 39 
SPECTRAL 

SQUAPB-ROOT 

PILTBP 

TPIGONO&X3TPIC 

OPERATOR 

In the above table, the bas?c tonrmlr t expressed in termr of the suspected mean m ad 

covariance ItB = I t H@BB*@*H* of the data 9. The smmtthg form is specified in 

terms of the optimal smoothed estimate uo = E(P/y), representing the coditbad mean 

of the statc u given the data y. The twctral fonmrlr ir obtdaed by mjbstitation in (1.3) 

of the eigensywm expamioar P = %ek*, y - zyk+ umi m = E%+ where 1; 

and % are the eigenvalues and ebeavactors of the obseired-state covariance operator 

B. The -re-root filter formalr, prew recordsd in (l.4?, t bud on the 

frctorizrtion of the predicted-data-covariance operator u (I+B) - (I+PI%+R;’ and 

o^u the definitions z = Lp t (I-Lhn and (I+K)=(I-L)-’ - Q+P<”. PhUy, the 

trinmmetric werator formula ir obkiped from the square-noot filter expression by 

use of the identities ItR - SBC2a and L = I-COSa develogjd in Sec. 2. 

Although the derivation of the above erpreaioru ha& to 8ienific.nt insigh &Soat the 

structure of the likelihood functional, it ir not with3.n the $oops of the paper to 

inver’ig8te all of there alternrtivet to the same level of detail. The formula imow 

he predicted-data-covariance smare-mt filter appears to ba the most comuabnt to 

implement the numerical rearch for the mtimrl estimates. ’Llhir section, however, rimt 

to first develop the results suxmuriaed ahve. 

Formula8 Bued on the Optimal Smoothec %ate BsUmaea 

Bcdt 4.1 The negrtiue-log likelihood function81 can be 4 premed 8s 

340 
where uo = E(e) is tbe ccOGitioru1 erpectrtion of the state u dven the data 9, and G is 

the estimator gain. 

Proof: It will be rrhoun in Sec. 8 that uo in (4.2) is the corditiot.1 meen .nd that G in 

(4.3) is the cornrpoxuling estimator gain. Therefore, for the sake of the discussion 

here, amme that (4.2) and (4.3) valid. Multiply uo in (4.3) by H and use (4.3) in the 

resulting equation u) obtain 

= HGy t (I-HGh, (4.4) H'O 

where m = HWf is as before the suspected mean of the &A 9. However, recall the 

identity HG = HgHWtHEH*)-' = 1 - atH@H*)-' so that I-HG = (ItI3EHq-l - (XtR1-l. 

Hence, substitution of this lrst identity in 14.5) le& to 

Thit is the central result required to establish the equivalence between (4.1) ad 23.11). 

To thir id, substitute (4.6) into the sccoad term on the at 8ide of (4.11, and observe 

the equivalence with (3.11) by irispeetion. 

where B(d I 9Cf and B(o/p) are respectively the ~pacondiw and coadiuarvl expected 

values of the state u. 

Proof: 

the last two term cm the tight side of (4.1) .ob use of the eqaritioa m = HWf. 

This re8ult can be establirh4d u a corobry to the Result 4.1 by combhhg 

Both of these results express the likelihood fum%.iorul in term of a q.rrmuty uo in (4.2) 

which ia the conditioar- arpectation B(Wy) of the state given the data yo This quantity 

L also known to be the best linear mean-square ert!xtuts as well as the optimal 

least-squarer stinute. The coincidence of the best mean-square est!mte and the 

optinul least-squares est&te, both of which can be computed by the conditional 

expectation formu& (4.21, is explored at length in Ref. 111. 

34 1 
-- Result 4.9 The negative-log likelihood functional cm be expressed u 

where yk = tpk% and 5 = tpk% are the spectral coefficients of the data .rd thc 

suspected mean m, uui Ai an the eigenvduts of P. By substitution of \ - in 

(4.81, this equation can be cut u 

Proof: 

(3.6) ad letting N-. Use of the identity Lk = y in (4.8) le& to (4.9). 

Bquation (4.8) can be established by taking the megatire log of MyN@) in 

Redt 4.4 The negative-log likelihood ratio cm be expressed u 

(4.10) 

(4.11) 

(4.12) 

as the difference between the data yk ad the 'filtered' eaimate 9 Observe that ek = 

CO~QL(~~-~Q by substitution of (4.11) k (4.12). sllb+titpte tbfr lut eqmtion into the 

second term on the right side of (4.9) to obtain (4.10). 

The formdr for the likeellhood fpnctionrl in (4.10) can be viewed u the 'spectral' 

version of the predicted-data-corate quare-root formula described below. 

342 
where use has been made of the fact that (I+R)-' t (I-Lq (I-L). 

Result4.6The operators L .Ild K can be represented in tern of the following 

eigensystem expansions: 

where "+ = tm-')ck. and ek are the eigenvectors of EL. 

(4.18) 

-'/a in and then evaluate the as yet undetermined coefficients \ from L = I - U+R) 

(4.15). To this end, premultiply L in (4.15) by %* and p0mnukipl.y by $bk to obtain 

L, = 1 - (1t)LL) 2 -va = l-co- which is the desired remlt. A 

Similarly, to obtain the desired expansion for K, seek to determine thc coefficients 

in 

K= 1 $ek@k* vrfth \*$bk*K@k. (4.19) 

(4.20) 

where 

34 3 
Proof: Recognize that (4.17) implies that 

and use these identities in (4.13) ami (4.14) to obuin (4.20) .Id (4.21) respectively. 

(4.24) 

and as before uL = tan-’$ with $ be- the eigemdue8 of R. 

Selection of Preferred Formark for NIDDlhllc.1 Sd- tiom 

In principle, all of the above forrrmlu for the likellbood fimcthd J(8v) can be OICd u 

a point of departure to compute the g.diant aJ/ae .ad the correspomlbg Hessian 

a’~/a’ - .lpd to thereby obtain the necessary iugtedients to implemunt the 

Newton-Rapbon search for optimtutiap. Ths c.lcol.tionr finohed in the mrzlperical 

search can vary significantly, however, dapeadfry on nMch of the form8 is d u a 

%tam point. It it therefore of hterest to cht adet.ileb tmssc@tion cf the 

relative advant.8es ad diudvurta8e8 of the vuioor method8 to implement the search 

that arise from the vrricms form of the likelihood functimd. Such an invertbation it 

currently in pro8fi~~ ad will be mporttd on in future work. In thir paper hanever, the 

formula selected to compute the gradient .ab Hessian is that bued on the 

predicted-data-covariance sQ\ure-tOot filter in (4.13). 

5. COVARIANCE BIGBNSYSTBM SBNSITlvrrr TO SMALL PA-R CHANGBS 

As a prelim(nrfy to the evaluation of aJ/ae .od a’ J/ae2 involved in the uumerical 

search for optimization, it is necessary to conduct an aztdyrir of the perturbationt 6kk 

~nd Mk of the eigenvalue8 and eigenvectors of R = H@BBWcH*, with rerpcct to 

344 
variations de of the parameter dirtribption 8. Such an uulysis will provide tbe 

mathematical tools that will be used in subsequent sections to evehate all26 .ob 

a‘J/a’. 

where the dependence on 8 of EL, % rod Ak hu been explicit. The uUxutc objective 

of this section is to develop udytical formnlu for c.lcpl.tiqj the first-order 

parameter distributions 8. 

wflarb8tiOllS 6Ak .pd % Of )ck d $ with reSpt?Ct t0 tnUU Cue8 68 fn the 

It is assumed here that the Prechet differential [Z] of \ at 8 and that it can 

be computed by 

nhen y it 8 8cal.r .ab 60 it admissible pertprbation of e. ~q~rtioo (5.2) it .cthe formul. typically used for comppt.rion of the Gateaux differential. However, it ir 

assumed here that both of thew derivative8 exist .nd coincide .od that thcrefore (5.1) 

can be used to calculate the Prechet derivative. 

Since Ak is Prechet differentiable (admittedly by assumption, u an investigation of 

the technical conditions required for differentiability it not within the scope of this 

paper), its differential &hk(e;&O1 can be expremed u 

where ckk(81/aB is a Wed hear functional ref$& to a8 the Prechet derivative of 

)Ct at e. The tranrfo~.ion aA,/ae can ah be viewed u a function space gradient of 

Lk at 8. Similarly, the eigenvector differential @p;E”) L defined a8 

where [a~$~(e)/aBJ is the Prechet derivative, assumed to be linear ad med. 

Calculation of 6A. and aFL. 136 

Recall that the ek in (5.1) are orthononarl 80 that 

345 
which can te takeu u the point of departure fzr calculation of bAt uui a\-. 

Result 5.1 The Prechet differential 61 (8;60) can be expressed u 

where 6A(9;68) is the differential of A defined u ._ 

- Proof: Performance of first-order perturbation on (5.61, ard use of the colditFon 

f$k*& k= 0 leads to 

It can be observed from (5.10) th*t evrhuth of 6@ ia the central calculation 

required to determine 69. In ofder to simplify aotatiapI without loss of generality, it 

has been assumed in ami- at (5.10) that B .nb H do not dvperd on 8. In mort 

practical cases, this assumption ia satirPied because the poorly known parameters occur 

in the operator A. 

TO compott ti@, as nguited (s.10)~ mcau tht A(eme) = I, thet AN + 

A(6cD) = 0, .ad 

Substitution of (5.11) fa (5.10) leerds to 

346 
Finally, use of the def'nitions p = AiA and 5 = '.;'@13qk in (5.19), and 

substitution in (5.91, implies (5.7). In performiag this last step, it has been assumed that 

A = A* is formally self-aajoint, a cdtion that is valid on most problems of practical 

interest. 

k 

Discussion and Additional Assurrr~tions on A 

The above result, although a step in the right direction, is still ioII1cwh.t 

intermediate because the differential &Ak in (5.7) is expressed in term of the yet to be 

determined differential 6A. To proceed f-er, it is convenient tc make two additional 

assumptions (typicaliy satisfied in practice): 

0 A(8) is linear in 8 so that A(B1 + = Ate,, + for two admissible 

distributions e ad e2. 1 

0 A(8) can be factored as Ate) = DW)D, where D .ad its correspondir@ formal 

adjoint D* ma9 in general be matrix differential operators. 

Based on these assumptions, it is now possible to derive the following more explicit 

formulas for &Ak and aAk/%. 

Result 5.2 The Prechet derivative a\ (e)/= of \ is 

Proof: Since A has been assumed to be linear and factoriz8ble 

where the last equality is a consequence of a process aarl0g0~8 to integration by parts. 

where a)Ck(8; *)/at3 is an element of X*(Q). Futhetmore, @Ak@; *)/a01 can be evaluated 

from 

34 7 
Proof: The -orour derivation of this reat is not as yet available. The remlt is 

accepted somewhat formally on the bash Uut a bounded hear frmctioxul can be 

represented by an element in the dual *%o the qace in which the functional is defined. 

Calculation of 4%. and *e 

Result 5.4 The Prechet differential % (8;68) of $ can be expressed as 

(5.18) 

m#k 

where 6P is the differenthi of the observed-state covariance operator B. 

Proof: Since Pak = Ai+k, 

(62j~$~ + R- = 2Ak(6Ak% + (5.19) 

Now, seek an expansion for &Pk in term8 of the orthononarl barb @m, Le., 

(5.20) 

where cLm are scalar coefficients to be determined. Note that the orthononarlity of 

@k implies that cLL = 0, so that hk does not have a component in the direction of % 

To evaluate cb, premultiply (5.20) by em* to obtain 

Use of the conditions @m*~ - \:em* and chn = +m *&k and rearrmement of terms 

leads to 

Substitution of (5.22) in (5.20) leads to (5.181, thereby establlrhiry the remlt. 

Equation (5.18) is similar in nature to (5.9) in that it expresses the desired 

differential in term8 of the yet to be determined quantity 6B. 

Pedt 5.5 The Prechet differential %(&be) of $ can be expre888d as 

34 a 
- Proof: Substitute (5.12) in (5.18) and use the definitionr for pk and 1~. 

Bquation (5.23) is valid witbt -Ling the rdditionrl urpmptioa that A(8) is 

linear in 8 and factorizable as A(8) = D*(8)D. If there two assumption8 are now made, 

the following result can be obtained. 

Result 5.6 The Prechet derivative aQI (e)/= is specified by 

(5.24) 

- Prodfi This result follows by rubstitution of bA(0) t DY6e)D in :5.23). 

Closely related to Mk is the differential 

of the outer product akak*. 'The corresponding Prechet derivative a(~@~@~*)/a is 

evaluated in the following result. 

Result 5.7 The Prechet derivative [a($$ *)/a] L specified by 

Proof: Use (5.24) to evaluate the r&ht ride of (5.25) md recall that 6(+k@k*l = 

[a@kfpk*)/aelae. 

Discussion 

The results obtained above provide the key tools required to evaluate the 

function-space gradient aJ/M and Hessian a2J/# of the likelihood functional. The 

most useful formulas are (5.17) for the derivative aAk/a of the eigenvalue by, (5.24) 

for th,: derivative &#k/a of the eieenvector ek, and (5.26) for the derivrtire 

a(@k@k*)/a of the outer product (@ 9 *I. There formula8 will be used repert.edly 2c 

kk 

the following rection. 

349 
6. SPECTRAL REPRESENTATIONS FOR THE GRADIBNT, AP PROXIMA TB HBSSIANL 

AND NEWTON-RAPHSON SEARCH 

Implementation of the modified Ne~t~i~-R~phr~n rearch for the optimal parameter 

estim~ter requires calculatiom of the aJ/ae ad of an apprordmrtion to the 

H~SS I operator a2~/a2. These calculaticnr are be8t achieved pra the 

predicted-data-covariance rm~are-mt filter in Remlt 4.5 that expreuer the likellhoal 

functional AS 

where z(8) = L(&y t [I-L(8)] m(8). Function space differentiation of (6.1) with respect 

to 8 leads to the gradient functional 

and to the approximate Hersibn operator 

upon which the Newton-Raphson numerical rearch is to be bared. An updated estimate 

en+ 1 = en - is obtained by specification of the parameter change an defined as 

The min objective of this section is to replace the ophrator .:i'a+bzu (6.2) aml (6.3) 

with a set of equivalent tmtrfx eqp.tiaru mora cmenht i:r cahhtlanr. Th4 

fundamental approach to bc used cmirU of reprerentin# the .cg~tion space derivatives 

aL/a, am/= ad az/a - which have oply been derived in tern of operator symbols in 

of the obsented-state covariance operator R. 

(6.2) and (6.3) - h tr?m Of specific O-ml ba defined by the abem?Cton @k 

SPectral Reorerentation for the @- 

Result 6.1 The Prechet derivative &/a of the predicted-6.u-covariance 

square-root filter L can be represented ss 

km 

350 
Note that ab defhes a matrix whore diagonal elements are provided by (6.6) and 

whose corresponding nodiagonal elements are given by (6.7!. 

- Proof: Observe L = (l-co~a~h$~@~* implies 

Substitution of thb equation in ah = +k*taL/Mwm and use of orthonormality of t$k 

lead to 

where aA,/a and *k/= are the function-mace derivatives evaluated in (5.18) ad 

(5.25). Substitution of there two equations from ‘sec. 5 in (6.9) leads to (6.6) and (6.7) 

thereby establishing the result. 

Result 6.2 The Frechet derivative am/* of the suspected mean m(8) is reprerented by 

(6.10) 

ad @Cf in (6.11) denotiq the suspected value of the rate u. 

- Proof: Since m = HWf, then 6m = Hb@Cf = -H@A(:Q) @Cf, where the trt equality 

follows from the condition 5Q = -@A(68)@. Defhe now ‘6mIk a8 the k* rpectral 

coefficient of 6m, Le., 

where as before p = AiL@*H*@r Use of the identity pk*A(68)@Cf = -Dpk *D(@CW 

in (6.12) results in (t,mIk = (am/aelk 66. with (Walk given by (6.11). 

k 

Result 6.3 In the 8peckl :&IC in which the deterministic irrlprrt f ir asrmaed to be a 

kctor f = [f .,..., f,] of M inputs applied at the dircrete locatbm E,, ALL rltenutive to 

(6.11) in evaluating (&I&@), id 

m= 1 

35 1 
where @(x/[I is the Green's function of the system model operator A. 

Result 6.4 The gradient &/a = (aL/S)pt (I-L) tam/ael of the filtered state estkute z 

can bc represented as 

where the spectr.1 coefficients (&/a), = ak* (&/a) are given by 

(6.15) 

with ab specified in (6.6) md (6.7) ad 

Proof: Substitute aL/a and WS lrom (6.5) ad (6.10) into &/SI = (aL/aq t 

(1-L) (am/%) and then compute the spectrd coefficients (&/a), in 'C.14) from 

taz/ae,, = 9k*(az/ae). 

Result 6.5 The gradient g(e;y) in (6.2) can be represented AS 

where ex: = ak*e &re the spectral coefficients of the residual process e = y-z, ad 

(aZ/a), ate given by (6.15!. 

Prod: Substitute aL/a in (6.51, &/a in (6.141, e u & ek$bk And I t = 

E secak+k@k* into (6.2) ad use orthonormality of ak. 

Equation (6.17) provides ?.he means to ewluate the likelihood fuacticnrl gradient, one of 

the key ingredients bf the Newton-Paphson iteration. Tb4 Approximate Hersha 

operator M(8;y), which . the other major element required to implement the search, 5s 

evaluated below. 

xedt 6.6 The &pptuxiotc Herrhn M(e;y) in (6.3) is an integral operator whose kerne! 

M(rJ[) i8 specified by 

(6.18) 

35 2 
th where zlk = = @k*taz/a) is the k spectre1 coefficient nf &/a. 

Prcof: Substitute (2.26) and (6.14) into (6.3) aad use ?&e orthonomadty of e,. 

Implementation cf an iteration step in the Newton- 2aphson search requires calculktion 

of 68" = M-'d;y)g(Bn;y), representing the incremcntal change in the parameter 

estimate. Inversion of M(On;y) is therefore required at every steF of the -eerch. Thi3 

iinersion is achieved by solving an integral equation as outfined in the follcwing result. 

Result 6.7. 

of the following integral equation 

The incremental parameter change 60" can be computed as the so1ut;on 

where M is the approximate Hessian :.ernel in (6.181, and g (XI is Lhe value of the 

eradient at the spatial location x. The subscript n in Mn and gn denotes that the 

corresponding quantities are evaluated at the n 

n L 

th n parameter estimate 8=8 . 

Proof: Observe that 

in cermq of the kernel M to obrain (6.19). 

= M.ilgn implies Mnb# 1 gn, and express this last equation 

n 

7. PARAMETEX ESTIMATIOK ERROR, CBAMBB-RAO BOUNDS AND OP-L 

INPUT DESIGN 

The objectives here are to obtain a C-I2 boa for the covariance of t?e pavmeter 

estimation error and to Segh an investigation of the problem of optimrl input detign by 

using the C-R bound as a criterion fgr optimrl input selection. 

Recall that the covariance of an unbiased estimate 9 satkfier the inequlitv 

whert M(Oo) t, the information opextor utiinrd as 

(7.2) 

The corn>sponding mean-square estimation error Et0 '@@ 1 sc.tu'*.;? i:4: related 

inequality 

PP 

35 3 
It can be observed that the key calculation required to obtain the C-B bound is the 

computation of E[a'J/a'; as outlined below. 

-- Craxm,-Rao B~und for the Bstmtiw Bmr 

Result 7.1 The incormation operator M(e0 1 is specified by 

(7.4) 

where h = H@EE*@*H+ is the data-covariance operator, @LBO) is the derivative of 

L = I - (I+R)-''', and tam/a0) is the derivative of the datn mtan m=H@Cf. 

Proof: Differen-tiate g(@;y) in (6.2) to obtain 

Take the expecte+ '*E in (7.5) .bo-.+ evaluate at 0 = eo, rod simplify to obkin 

Finally, use 

Result 7.2 In spectral form, the inform~tion operator M(e0 1 is specified by 

i) in (7.6) to amve at (7.4). 

(7.7) 

- PrcDf: 

Inspection of (7.4) reveals Jut the informatior operator Weo) consists of the sum of 

two terms bct5 of which are positive definite. In the fint term, the daU-COV8il.nce 

operator (I+R) appears a8 m %ei&bting" that is multiplmd by the tenritiivty filter 

aL/=. Note parenthetically that in fact L b 8elf-adjokt 80 that L o C*. The recond 

tern-. on the other hand, will be shwm to be a quadretic ftmztion of the input f. 

"esult 7.3 Assume that f = [i ,...&I is a vector Gf M inputs applied at the 1VI discrete 

locations Cm. Tie inlormation c3eratsr M(Bo) b an integral operator whose kernel 

M(IC/€) can be e-mzased as 

Use an approach similar to tbrt used :3 arrive at (6.18). 
(7.8) 

where 

T ad where bk(E) is the M-dimensional vector 

(7.9) 

(7.10) 

(7.11) 

with being the Green's function of A in (1.11. 

Proof: 

taL/a) in (6.51, and for am/a in (6.10) into (7.4) to obtain (7.3) and (7.13). 

Sutstitc?e the eigeruystern expansions for R in (2.261, for L in (4.171, for 

The second term in (7.8) b a qurrdratic form in the input sign~l f. This property can be 

used as a basis for optimal input derOptimal Input Design 

The La'ormcrtion operator can be used to state criteria for optimal input design. 

While several posrible criteria exist, the one that it easiest to use is perhaps the 

maximization of Tr M(e0): 

(7.12) 

where 

The optimal input fo. which is the solution to the above optimbrtLn pioblem, is the 

eigenvector corresponding to the largest eigenvalue of the M-by-M matrix V. 

Other criteria for optimrl input selcctim include: minimiaatim of Tr @I-'), which 

woulE, correspond to minimia hg tht Cramer-Rao bd; and minimization of 

M-'), where A- fs &e m~ximum eigenvalue of M-'. While these kst two 

cnterir could be superior to (7.121, they both have the dimdvantrrge of requirixlg 

inversion of the operator M( However, the requirement for such an b.rt.rsion may 

not be a reriou, additicorl dzawback because a similar calculation is required to 

implcment the Newton-Raphsoa search outlined in the previous sections. 

Lax 

355 
VanisWng Bias of the Gradient 

Closely related to the rbove analysis is an investigation of the bias in the 

parameter estimate 8. The central result is as follows. 

Result 7.4 The expected value of the gradient functional g(8;y) vanishes at 8 = eo, Le., 

(7.14) 

Proof: Observ- that &/EX3 = taL/X3* + (I-L) tWXN, .nd recall that $ = (I+K)e. 

Substitute this in (6.2) and take the expected value. Finally, use the whiteness of the 

residual process, to be established in (8.46). 

8. FILTERING, SMOOTHING AND THE RESIDUAL PROCESS 

The central aim of this section is to con-uct an analysis of the smoothed estimate 

u and of the filtered state estimate zo that emerges from the 

predicted-data-covariance square-root filter. This analysis leads to the fobwing 

major results: 

0 

0 The smoothed estimate uo is opthml in a condition41 mean sense. 

0 The formulas that generate uo aud zo have a predictor-corrector ~tnrctpte in 

which the fbl state estimate is the mm of: a prediction term-bued on 

application of hcwn inputs to the system model; and a comctiua term based 

on the difference between the actwl and predictud data. The key ehment in 

these formulas k an estimator gain th.l provides the relative weighbetween the two terms. 

0 The covariance of the state estimation error inherent in both esthates can 

be evaluated by mean8 of equations which, if written in operator notation. 

resemble those encountered in tilte- ad smoothin& for linear dprumicnl 

systems. 

0 Investeation of a residual process associated with the filtered state esUmste 

zb that has properties nearly identical to those of an innovations process: the 

residuals are a white noise process with a unit covariance; the residuals and 

the measurements can be obtained from each other. by means of recQrocal 

linear transformations. BecaLe these transfomtio~~ are not the 

residuals are not a bonr fide innovaticma process. However, they are as 

useful in cierivinq filtering, smoothLng and identification sohtions for elliptic 

syst .ms e$ the innovatioar process is in deriving simL1.r solution8 for hear 

dynamical systems. 

356 
Development of relationships between the filtering ad smoothiag estimates 

thrt can be thought of as extensions to elliptic systems of the 

forwrrdmackwrrd sweep method for solution of filtering ad smoothing 

problems in linear dynrmical systems. 

Development of spectral representations for the predicted-data-covarirnce 

square-root filter rad the optimal smoother in terms of the eigenrpstem of 

the state covariance P = QBBW*. This leads to sixuple ways to implement 

filtering and smoothing solutions on a computer. 

Smoothed and Filtered Bstimates 

The smoothed and filtered state estimates uo &xd go have been defined in (1.17) as 

2 = @Cf.+ g(y-H@Cfl, (8.1) 0- u = OCf t G(y-HQCfl, 0 

where G and g are Kalmm-like gab specified by 

The estimate uo it referred to as a smoothed estimate because it is the 

minimum-variance estimate of the state given the entire data ret. This is established 

by the followinp resalt. 

Result 8.1 The smoothed estimate uo in (8.1) is the ccmiitional mean u t B(&y) of the 

state given the data. Purthermore, the estimator grin G in (8.2) can be expressed 

altermtively as 

G = RH*(I + H&H*)-', (8.3) 

in tern of the state covarirace P = OBB*@'. 

- Pro3f: Recall the general form& 

derived in 141 for the conditioprl expected valae of a 2ero-mern random process D @wen 

the related zero-mean random procerr v. Note that thir formula require8 crlculatim of 

the *cross-covariance* operator B(m9 and the auto-cottlirfrace operator B(w*). 

Define now the meru-centered state 5 = u-@Cf = @Bo ard the mean-centered data f = 

Hii+n. By this definition, 6 and f are zero-mean. Therefore (8.4) can be used directlp 

to coapute u' = B(5/& Le., 0 

35 7 
which indicates that to evaluate $, it is necesstiry to first evaluate the covariance 

operators Bfii*) and BCm*:. These calcal.tions are: E(*-*) = B(QBoo*BW*) = 

@BB*@* and E(W) = E [(Hiitn) (Htitn)*] = I + HgH*. Use of this in (6.5) lea& to 

I . 5G ' lH L 

This together with the definition of 0 .ad 9 in terms of P ad y implies (8.1). The 

equivalence between the two different exprerrionr for G in (8.2) ad (8.3) is established 

by use of the spectral expansion= in Sec. 2. In particular, use expuuLoar (2.46) - (2.47) 

for I + B and the definition for 3 in (2.311. 

As established by this result, the estimate no h.8 a very well defined probabilistic 

interpretation. It is not presently known if the Atered estimate a. has a similar 

interpretation. Nonethelex, this estimate phys a very si@ficant role ia the filte-, 

smoothing and identification methodology for elliptic t]lrtemr der develcpmeat here. 

Its role is analogous to that of the filtered ertinute erne- from a Xabn filter in 

the case of dynamical system. This is further invertipted below. 

Predictor- Corrector Structure 

To examine this structure, c-mtider the equation for uo in (8.1) am3 Wustrated in 

Fig. 8.1. Use of the deterministic inpat f"] and the system model @C"' leads to a 

predicted estimate "I. The difference procem y-HWf '41 ir then formed d operated 

on by the estimator grin C Is] to obtain the correction term G(y-H@Cfl 16'. F~PA~, 

the correction term is added to the predicted estimate to obtain the Opumrl estimate 

u The equation for the filtered ertinutt zo in (8.1) also has a predictor-comctor 

struciure. The key difference between the twc equations in (8.1) is thst the e8thutor 

gains are different. A relatimh@ be'ween there two different 8.h C and 8 

explored later in this section. 

0' 

SYSTEM MODEL 

MEASUREMENT DATA DIFFERENCE 

PROCESS Y 

Fig. 8.1 Predfctor-Corrector Form of the Smoothed State Brtinutor 

358 
Estimation Error Covariance and Kalmm-lh Gains: Smoothiq 

Since uo rad zo are only estimates of the actual state u, it ir of interest to 

investigate the inherent estimation error u = u-u and z = u-iO. In particular, the 

aim is to determine the estimation error covariance. der the rsgumptim that the 

actual model errors o and n in (1.1) and (1.2) are white-noire processes. 

Pedt 8.2 

u-uo is specified by the folloning alternative formulrs: 

P 0 P 

The covariance 9 = P* - Mu u *) of the state estimation error u I 

PP P 

T; = (1 - GHIPa - GH)* t GG*, (8.7) 

- Proof: To show (8.71, observe that u = OCf t @Bo and u0 = OCf + Gty - HWE) 

implythatu =u-u i8 P 0 

u = (I - GH)OBo - Sn. (8.11) 

Hence E(u u *) = B[T[-GHk#Boo*B~* (I-GH)* t Gnn*G*] = CI-GHI&(I-C,H)* t GG*, 

where we h~s been made of the fact that t = [o,n] irr a white-noise process with 

covariance Nee*) = I. To show (8.81, observe that (8.7) implies 

I; = i$ - GHS - ~H+W t cc t H~H~G*. 

P 

PP 

f8.1:) 

Substitution of G = n'H* (I t HgH*)-' in (8.12) leads to (8.8). To show (8.91, observe 

that (8.8) can be expressed as P' = 6 (I-GHP = CI-GEnE by prbq G - RHWI t HPHq-' 

in the last two tern of (8.12). To establish (8.101, substitute P = @BB*Q* in (8.8) and 

ut the identities Bq*H* atH@BB*O*H+)-'H@B = CI+B~*H*HQB)-'B*O*H*HQB = I - 

(I+ B*Q*H*HQB)'-'. 

Result 8.3 The operator HPH* is the Predhlm re80lVe~t of HgH* 80 thrt 

Proof: Compute HPH* from P' in (7.8) to obtain H@H* = HaH* [I - (I+HgH*)-'J. Use 

th? identity (ItHilH*)-'H&H* P I - (ItZIeH*)-' twice in thh kst equation to obtain 

(8.13). 

The rim now is to use (6.13) in (8.2) to obtain an alternative expression for the 

estimrtor gain. 

359 
Result 8.4 The estimator gain G = EH*(I+HgH*r' can also be expressed 8s 

G = FH*, (8.14) 

where P = E(u u *) is the covariance of the smoothed state estimation error u 

PP P' 

Result 8.5 The mean-square smoothed state esthtion error is given by 

E(u *u 1 = Tr [F]. PP (8.15) 

Proof: This follows from the definition of P' as P = B(u u 4). - PP 

Note that many of the above formulas are very similar in form to the ones 

traditionally encountered in K8hn filtering for dyxumical systems. For instance, Bqs. 

(8.3) and (8.14) are very timilar to those used to Compute the grin G for a Kahn filter 

in which R and are the covariances of the estimation error associateb_ with the 

predicted and corrected state estimates. Note also that (8.8) implies that P is always 

smaller than E, which implies that the covariance of the estimation error after the 

observation y has been accounted for is smaller than the emr covariance before the 

estimate correction occurs. 

Estimation Error Covariance and Kahn-like Gains: Filteriq 

The aim here is to obtain results similar to result8 (8.2) - (8.5) above, but that are 

applicable to the filtered estimate zo. 

0 

Result- 8.6 

is given by 

The covarhnce E(z z *I af the filtered sktc estimation error z = 0-2 

PP P 

E(z Z *I = (I-gH) Qia-gH)* + Bg*, (8.16) PP 

where R = @BB*@* is the state covariance, and 8 is the filter cain in (8.2). 

Proof: Note that u = 9Cf + @Bo. This and (8.1) imply that 

z = (I-gH) @Bo-gn, (8.17) P .. 

whwe use has been made of y-H@Cf = HQBo + n in (8.1). Calcul-tion of B(z z *I, PP ..,, rb.17) and the conditions B(oo*) = 1 and B(nn*) = I, leads to (8.16). 

TIUS result applicable to the filtered estimate is AIAA~O~OUS to (8.7) of the smoothed 

estimates. To obtcrin results that are analogous to (8.8) - (8.10) requires, howzver, a 

f+w preliminary definitions and results. The need for these preliminaries arises fzom 

altimate desire to find A spectral decomposition for the state covariance R = 

'-, *a*. It is straightEorward to obtain the spectral representation for the 

360 
observ’d-state covariance H6H*. However, f- a 8im’l.r decompo8iUon of is not 

as simple. The primary reason for thir lack of simplicity ir that the vecton = I 

A?B*@*H*~P. may not necessarily span tbe entire space H. TMS is particularly true in 

cases in which the dimension of the input space HI is greater than the dimension of the 

obsenration space Hg. In order to consider this case, assame that the operator H@B has 

finite-dimensional range. Thi8 comtpondr to the situation where there are only a 

finite number hJ of se’3jors and the observed-state covariance P = HQBBW*H* is an 

N-by-N matrix. Asswne also that the input space is either infinite-dimensional or 

finite-dimensional with dimemion bl greater than N. This second assumption 

corresponds tc cases where the uncertainty is distributed at M discrete locations or 

throughout the entire spatial domain Q. 

1 1 

Result 8.7 The identity operator I mapping HI into itself can be dtiompored as 

1=1 tII, (8.18) 0 

where 

= I - B*@*H+R-~H@B, I, = -J @*H*P-’H~B, (8.19) Io 

and B = H@BB*@*H* is the obsemed-state covariance. In addition, Io is in the 

null-space of the operator 

R (-1 = H@B(.)B*@*H*, (8.50) 

mappine the space of bounded linear traatfonnrtioat on H1x H1 into the space of 

N-by-N matricca. Furthermore, Io and I, are orthoroar1 comlements so that 

I *II=Tr[I I ]PO. (6.21) 0 01 

- Proof: This result and ita comrpolding proof are illustrated graphically in Fig. 8.2. 

Bq (8.18) follows from (8.19). Subrtitutb of Io in (8.19) into (8.20) shows that P (Iq) = 0 

so that Io is in the null space of EL(* 1. That Io and II are orthogonal complement8 

follows from substitution of (8.19) in (8.21) by calculation of Tr [Io I*] wing (8.19). 

-- Bed 8.8 The i - . y operator I mapping 5 into itrelf cab be expressed as 

36 1 

(8.22) 
SPACB OF BOUNDBD LINBAP TRANSFORMATIONS 

FPOY INPUT SPACB INTO ITSBLP 

n 

.'NULL SPACE OF I\ R( 1- HcDB ( 1 BW H* 

Pig. 8.2 Orthogonal Complement Decomposition of the Identity in H1 x H1. 

- 

The above result simply reflects the fact that the do not span HI, because (by 

assumption) there are only a fiaite mznber of them, and this number is smaller than the 

dimension of the hput space. 

Result 8.9 The state covariance E = @BB*@* can be decomposed 80 

i 

where - 

uo = 9131~~4~ 6 

(8.23) 

(8.24) 

Furthermore, 

HB~H* = 0, ~5~ = 0, iio~* = 0. (8.25) 

Proof: 

A-'@Bt) . To show (8.251, mbstitute I fmm (8.19) hto (8.24) ab (8.25). 

To show (8.291, substitute I from (8.23) into P(1) = @B(DB*9*, ad use x - -- j 

1 j 0 

36 2 
-- Besuit 8.10 The dual state covariance 0 = @*H*H@ can be expressed as 

N 

6 = 1 A2 p.p.* 1 I1 

j= 1 

where p. = AyLQ*H*@.. II J 

N Proof: Since the cp. span the observation space Hg = B , th-;l 1 

(8.27) 

v:here IN denotes the identity in RN x RN. To obtain (8.261, substitute (8.27) in Q = 

@*H*I €I@ and use definition of p.. 

1 

N 

Define now the quantities 

N N 

j= 1 j= 1 

r = i? t (seca.-l) x.x.*, q = 1 (seca.-l) p.p;* 

0 1 11 11 (8.28) 

and note the following key identities. 

Result 8.11 The state covadaoce a = @BB*@* and r defined in (8.28) are related by 

= r t r*trH*Hr. (8.29) 

Furthermore, 

I t HsH* = (ItHrH*) (I+Hr*H*). (8.30) 

- Proof: Substitute r from (8.28) and H from (8.231 into (8.29). Use the orthonormality 

of x. with respect to H*H. This establishes (8.29). Equation (8.30) follows from (8.29) 

by forming ItHBH* from (8.29) and rearranging terms. 

Result 8.12 

1 

The dual state covariance 3 in (8.26) and q ’.? ,0.28) satisfy the identity 

6 = qtq* t qBB*q*. (8.311 

Furthermore, 

I t B*QE = (ItB*qB) (I+B*qB). (8.32) 

36 3 
- f Proof: 

with respect to BB*. This establishes (8.31). To establish (8.321, form I t B*QB using 

(8.31, and rearrange terns b the resulting equation. 

Substitute 6 in 13.26) and q in (8.28) into (8.31). Use the orthonormcrlity of p 

These are the preliminary resclts needed to evaluate the covariaace of the estimation 

error associated with the filtered state estimate zo. 

Result 8.13 Tk filter gain g defined in (8.2) can be expresssed alternatively as 

where r is defined in (8.28). 

Proof: Substitute r from (8.28) into (8.33) an$ use goH* = 0 .ad ej* -- rj*H*. Thi~ 

recovers g in (8.2). 

Note the simil~rity between (8.3) ad (8.33). ThAe equation in (8.3) expresses the 

smoother gain G in tenns of the rta%e covariance R = (DBB*O. Bq. (8.33) is a similar 

equation for the filter gain in tern of r. The operator n in G CAO be interpreted a8 the 

state covariance. No 8imll.r probaListic interpretation for r L known. However, its 

introduction L- very useful because it allows development of fotmulrs fo? the eatimotion 

error covariance AC; for the filter gain that very Cl08eiJ7 resemble those obtained for 

LiiS suloothing solutions. 

Result 8.14 

is 

The covariance Bfz z *) of the filtered state estim~tion crrw z = IS-z PP P 

E(z 2 *) = p + p*, (8.34) PP 

where p = p* is s3acified by the 8lternrtive formul.8 

p = (i-g~)r(~-gm* + gg* (8.35) 

p = r-rH* (ItHriW-k, (8.36) 

p = (1-gH)r = r (I-gH)*, (8.37) 

p = (~2) a' t (l-cosa x x *. (8.38) 

To establish (8.34) A& (8.351, substitute (8.29) in (8.16) and use the identity 

0 i )i 

- Proof: 

(I-gH) rH* = tH* ,i t HrH*)-'= g. (8.39) 

To establish (8.361, observe that (8.35) implies that 

p = r-gHr - rH*g* t g(1 + HrHr)8*. ,d.40) 

364 
Substitute g = rH* (I 

the second term of 

(8.38), substitute t in 

+ HrH*)-& in (8.40) to obtain (8.36). To obtain (8.37) observe that 

'8.36) cat be exprerscd rltemtivdy a8 6th and rH'g*. To obtain 

(8.38) into (8.36) and we orthonorrmlity of 9 

j* 

Result 8.15 

estimate is given by 

The mean-square estj~.~tioa emi asrocirted with the filtered state 

N 

j= 1 

E(z *z 1 = Tr Ip t p*] = Tr 1s J + 2 1 (1-cora 1 x *x.. PP 0 jjl (8.41) 

- Proof: This result follows from (8.34) azid (8.38). 

Result 8.16 The filter grin 8 can be e~tprersed a8 

where p is related to the filtered state estim~tion error covariance by B(z z - p + p*. PP 

This equation is analogous tc (8.14) in that it exprewer an ertimrtor gain in tern of 

the covariance of the state esthtion error. 

Result 8.17 The operators 1 + HrH? and I -HpH* are reciprocal, i. e., 

- Proof: 

8 = pH*. 

Recall (1 + €IrH*)-' = I-Xg = I-HpH*, where the lwt eQpIUty bldr becrtue 

Note that thfs redt impUer that the operator HpH* is the Predhotnr resolvent of 

MrH*. The identify also Lmmedp.tely inrpltsr whitenerr of the reridwb process a8 

investigated in more detail below. 

Pseudo-Innovatiom Pro~ertios of the Re EidWl8 

Define the reridwl procsaa in the -1 WAY, AI the dffforsrr between the actual 

measurements and the predicted data emerging from the prsh-eted-dru-covrrirnce 

square-root filter, Le., 

= Y-Hz~ (8.44) 

Thfs ptocerr tunrs out to have two key propertSm8 that 8n marly identical to those of 

an inaovatixu pmcerrr: the reridub are white mtB a pnit covrri.ncei the tesidub 

365 
and the measurements can be obtained from each other by m01m of reciprocal 

relationships. Theac twa properties are estabbhed in the following rerults. 

Result 8.18 

i.e., 

The residual process defined in (8.44) k white with A unit covariance, 

-. Proof: Observe from (8.1) that e = CI-Hg) (y-H@cfl. Hence, Nee*) = CI-Hg) (1; HqH*) 

(I-HgP = I. This last equality follow8 from B[(y-H@cf) (y-H@Zf)*] = I + HEH* and from 

(8.42; and (8.43). 

Result 8.19 The residuals e - y-Hzo and the tnern-centered measurement process 

ji =y-H9Cf can be obtained from each other by me- of mcigroc~! linear 

transformmiom, i.c., 

where 

(I t HrH*) -' = (I-HpH*). (8.47) 

- Proof: Eq. (8.47) has been ertrblbhed in (8.43) A& is restated hem only to 

emphasize its relatiomhip to the poperties of .;he nsidud procers. Bq. (8.1) WUes 

e = (1-Hg)y. 

We the smoothed rad Cutered ertimrter have bean defined s0medA.t independently 

of tach ather, these ebtim~ter &re h fact very Cl08eb related. ft is possible to 'Wn 

one in terms of the other, a8 outlined io the followiry remlt. 

Bemlt 8.20 The smoothed and filtereo e8tbnAtes to and so are related by 

u = zo+ ee, 0 18.48) 

where 

e -y-Hzo (8.49) 

is the residual process, and 8 Ls the predicted-data-c~varirnce rqture--.obt filter gain. 

- Proof: Observe that (8.1) ab (8.3) imply lao= @Cf + CH* (I + HPH*)' (y-HOCtI. 

Use of (8.46) leads to uo = OCP + aH* (I*HrH*)-'e SUrly, zo in (8.1) ad n in (8.33) 

lead to zo I <bCf + rH*e. Hence, ~~-2~ = [&H* (I+LK*)- -rH*]e. Use of the identity 

(8.29) in 'is bznpUe8 that uU,-ao I le, which ir the ds;ired result. Note that (8.481 can 

be r:* *: r n In the altenutive form 
Clcsaly related to the above reltkonship between filtered and smoothed stote estimates 

is a relationship between the ccrresponding covariances of the state esthution errors. 

'This is developed below. 

Bemt 6.3& The filtered state estimation c-or z = u-z ad the residsal process 

e = y-Hz are related by 

P 0 

0 

P' 

e=ntHz (8.51) 

where n is the measurement error. 

Proof: 

P5sult 8.22 

u = u-uo can be expressed as 

Note ttdt e = y-Hzo = Hu+n-Nzo = Htu-zo) *n = Hz + n 

P -- 

The covafiam-e 2 = B(u u *) of the smootbeci stqt? csti;nrUon error PP 

P 

where p + p* = E(z z *) ?? the covariance of the filtered state estimabon zrror z = 

u-z Furthermore, 

PU P 

0' 

Proaf: Use (8.52) to obtain 

E(ez *) = B(nz *) t !iE(z, 7. *) = Hhz *) + H(p+g*). P P 3P P 

Now use (8.17) to compute B(nz *I, Le., F 

Nnz *) = -g*, 9 

(8.54) 

(8.5:' 

since !Uno*) = 8 bj assumption. Substitution of (8.55! 1& (8.54) ad use of 2 I 1 iP leads 

to 

Since u - u-u the2 3 = z - ge from (8.48). Hence, P 0 PP 

Now use ,8341, (8.43, (8 45) and (8.C6) to r%ta,itr (8.52). Eqiution (8.53) follow^ 

immedirte!v from (8.52) by fohming I - HPH* am. rearranging tenas {TI %li-: temlting 

expressior. Nota that (8.52) implies that the gab C . ld 8 sre related by 

36 7 
The kst three resulk can be viewed as a generalization to elliptic systems of 

relationships encouatered in filterino, ax& smootbing for dylumical systems. For 

example, Bqmtion (8.481 is a generalization to elliptic system$ of the foxwardmrckward 

sweep methd for tolution of two-point boutxhrg-value problems. This method in 

general terms states that the smoothed states estimates can be obtained as a remlt of a 

two-stage process: fornard filteriq by means of a Kalmrs filter to obtain a filtered 

state estimate and a residua1 ptocew ad backward smoothif16 to process the re8iduah 

and obtain a smoothed state estimate. This two-stage data procetricg approach has 

been extensively stadied for bear dyn~mical systems. Bqs. (8.48) and (8.491 have 

exactly ?.he same structure. This structure is illustrated in Pig. 8.3. 

The ok~all diagram illustrates how the data yl" and the detenninirtic inpnt +2J are 

processed to arrive at a smoothed estimate IS'". The estimation procem contirtr of two 

stages: a FILTERIN". stage that remlts in a filtered estimate z!' rtd a residual 

pro~ess'~]. This filtem stage ir characterized by a predictor-corrector structure 

where a predicted is first produced and then corrected by a correctLon 

ne results of the film- stage are then proced by the SMOOTHING 

stage. Central to both the filum and smoothing sugar is the lain J81. ~h4 

foregoing structure b nearly idendcal to that of the forward/b.ckuud sweep method in 

linear dyrumical systems. There are, however, some key differences. One of the 

differences is that the filtem sta~e in the cue of dyn8dC.l tystemr L bued on the 

Balmrn filter, whereas in the elliptic case under consideration here, thir filter is 

replaced by the predicted-data-covariance -re-root filter. Another key difference 

is that the Kahn filter b causal whereas the predicted-data covariance square-mot 

filter is not, i.e., the filter gain g ir a Fredholm operator as opposed to be- a Qolter~a 

operator. In the same vein, the smooe stage for dynamical ryrtemr is backward (in 

time) or anticaural. In the elliptic system case, however, the smoo- stale L also 

characterized by Predholm operators. The notion of causality ir not even introduced 

here althowh it is possible to do this for cerufr chws of elliptic rprtemr 111. 

-. 0 

Fig. 8.3 Filtering and Smoothing 

36 8 
Spectral B epresentatiaat: Smoothing, Piherim, and the Reriduh 

The ah here are: to obtain spectral representations for the filtered and tnnoothed 

estimates uo and a. and the corresponding error covariances P and pi to explore the 

predictor-corrector structure of the spectral represenktiont of the filter ud 

smoother; ad to investigate the pseudo-innovations properties of the spectral 

representation of the residul process. The term "spectral representation' meuu the 

use of an expansion in term of the eigeaspstem + of R and of the related function8 

j 

= AF'B*@*HWi, xi= AY'QBw. and p. = A?@*H*.. ? If I 

Result 8.23 The smoothed state estimate uo can be represented u 

u = @Cf + sin'a. (9. - m.) x., (8.59) 0 11 If 

where 9. = a.9 and m. = @.*m are the spectral componentr of the &t. y .nd the 

.I I I1 

sup-cted mean m = HWf. The related observed-state estixxute Huo is specified by 

= m + HG (y-d, Huo = (I-HG) m + HGy. (8.60) HuO 

i In spectral form, Hu = 1 P @. where 

0 01 

2 nj = m. + sin ai (yj-?), uo i = cor$j 9 + sinhy (8.61) 01 i J' 

P = 3t + 1 sin'a x xi* HPH* = sim'~~q~p~*. (8.62) 

Furthermore, the corn- mean-square ertimrticm erron B(u So 1 = TdF] and 

E'u *H*Hu 1 = Tr [HPH*] are 

0 ij4 

PP 

P P 

B!u *u 1 t Tr [go] + sin'aj xj+Xj, Bh %*Hu 1 = rinaai (8.63) PP P P 

Proof: To establish (8.591, substitute y = Zy Cp and m t Cmpj in (8.1). To show 

(8.60) mulitply uo in (8.1) by H and recall that m = H9Cf. To ert~bli3h 8.61, multiply 

(8.60) by tbj*. The equation for P in (8.621 follows by substitution of (8.23) iu (8.8) and 

use of the conditions Hio I CoH* = HgoH* = 0, The equation for HFH* in (8.62) 

follows trr J: P axxi we of . Bq (8.69) follow8 from (8,621 ad the orthoaorpuUty 

- if 

i-9 

of 4j. 

369 
Result 8.24 The filtered state estimate a. can be represented by 

z = Wf + 1 (1-cosa.) (y.-m.) x., 

0 t Ift 

The related observed state estimate z = Hzo is 

z = m t Hg(y-m), z = a-H@ m + Hgy. 

(8.64) 

(8.65) 

In spectral form, z = 1 2.4. 

11 

ai = cosai 7 + (1-Cora v (8.66) f ,j* 

2. = m. + (1-cosa.) (y.-mJ, 11 111 

Let z = 2-zo denote the estimation error. The e&timrticn error covari.ncet B(2 z +) = 

p +, p* ad MHz 2 *H*) = H(p + p*)H* can be represented as 

P PP 

PP 

p = (%) io + (l-cos~r 1 X.X *, HpH* = (~-COM 1 + *. (8.67) i li i ii 

Furthermore, the corn- mean-rquan estimation errors an 

E(z *z 1 = Tdp + p*), B(2 *H*& 1 = Tr [H(p + p*)H+J, PP P P 

where 

(8.68) 

Tr [p] = ('/a) Tr [go] + 1 (1-corn 1 x , Tr [HpH*] = 1 (I-cosa 1. (8.69) i 17 i 

- Proof: (8.1). Bq. (8.65) 

follows from multiplication of (8.64) by H and use of m = HWf. Bq. (8.66) is obtained 

from (8.65) upon multiplication by 4* and use of the orthonormrrlity of 4 The 

equation for p in (8.67) has be- tstabbhed in (6.38) and is repeated hem only for 

convenience. The recd of Bq. tu) follonr from PIC of the identity +j - 7. Bq. 

(8.68) follows from the definition of p = p* in (6.381. Bq. (8.69) is establish& by 

perfonaing the trace operation on (8.67). 

To show (MI, substitute 9 = &y rn = Cm p )into 2 

j I' 

Result 8.25 The residual process e = y-Hz can be represented as 0 

e = 1 ej+j, ei = ej*e. (8.70) 

The spectral components e. are independent random vrriibler with zero-mean and unit 

covariance, Le., 

f 

370 
Furthermore, the spectral components ei and yi of the residual and difference processes 

e = y-Hzo and J = 7-m are related by the reciprocal rel~tionsbips 

- e = cosa yi = seca. e.. (8.72) i i i' 11 

- Proof: To show (8.711, observe 

that Eke.) = @i* %(e& @. and then use (8.45) and the orthono~lity of @ Equations 11 1 j- (8.72) are the spectral representations of the reciprocal relationships (8.47). Note that 

(8.72) can also be established by the simple trigonometric identity (l/cosai) = secai. 

Eq. (8.70) is valid because @ iare orthonoxmal in H 

9. NUMBRICAL SBAPCH CALCULATION SUMMARY 

Since the development of the estimation. approach is rather lee, it is 

convenient to summ~rize the steps that are requited to implement the rmmerical search 

It is assumed that the process starts with a Lnown input f, a set of data y rad an 

initial parameter estimate en. TO conduct an itentiom in the nnmerical search requires 

that the following steps be performed: 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

Compute the suspected mean and covariance m = H@Cf ukd R = H@BBW*H*. 

Compute the eigenvalues 1; and ebenvecton +k of R. 

Conduct a spectral a~lysir of the data and of the suspected mean to obtain the 

spectral coefficients yk = @k- and % = ak* 

Use Result 6.5 to evahaau the gradient aJ/a of the likelihood functiolul. 

Use Results 6.6 ad 6.7 to compute the Hessian Mn .Id to determine the 

incremental cbwe wn of the parameter estimates. 

Obtain o new parameter estimate 8 n+l = en - 6e'L, mhvn to 8t8p 1 above, ard 

iterate throw steps 1 to 6 =til conver8ence is achieved. 

If Cramer-Rao bounds and/or an optimrl input are deshedhre (7.6) - (7.13). If the 

covariance of the state ertimatiom em; L de&d we Pemlt 8.2 and/or 8.13. 

The calculations involved in condwting a single iteration in the nuximum-likelihood 

parameter estimation approach are summarized in block diagram form in Fig. 9.1 A 

single iteration consbu of all of the compuktiml step8 required to obtain an updated 

pwameter estimate @+'by procersing the avribrble data, the known deterministic 

input, ad the curreat parameter estimate en. 

371 
Fig. 9.1 Calculations Required for Sue Iteration in Modified Newton-Raphson Search 

To simplify the description of these computations, the steps performed in a single 

iteration have been grouped into the following four major blocks (delineated by the 

broken lines in the diagram): 

a SQUARE-ROOT FILTER block that processes the mearurernea data 9 and 

the external input f to obtain a filtered estimate 2 and a correrpoading 

residual process e, defined as the difference between the data and the 

filtered state estimate. The square-root filter implements the equations a = 

Ly + (I-L)m and e = 9-2. The central compnution in the square-root filter 

block is that provided by the operator L = I - CI+R)-'h defined in terms of the 

square-root of the predicted-data-ccuvariurce (f + R). ThcI operator appears 

in two disthct places in the diagram: in the data filter, wbae primrig 

function is to process the m%amrementr 9; and in the mean filter, whore 

main function Is to process the suspected me- m. The suspected mean is in 

turn obtained from the known extenul input by meam of the input-output 

model. 

372 
0 a SQUARE-ROOT FILTER SBNSITMTY block that processes the 

measurement data y and the detenainistic input f to obtain the filtered 

estimate sensitivity &/a. Thit block implements the equation &/a = 

(aL/a)(y-m) + (I-LHWa). The computation of the sensitivity aL/a is the 

main calculation performed in this block. 

a GRADIENT-HESSIAN SYNTHESIS block that forms: the function-space 

gradient aJ/a of the likelihood hctiorul by mean8 of the equation aJ/W = 

Tr((aL/aXI+K) - tWS)e*J; ad the function-space approximate Heen by 

means of the equation M = Tr [(aL/aeWI+BWaL*/a)] + (&&Woa). Note 

that the qu~ntity that is actually evaluated in this block is the kernel MWt) 

of the Hessian operator. This kernel is a fmrctiot of two spatial variables x 

and f defined over a "square" domain (r/f)cQ I Q, where Q is u before the 

spatial domain of definition of the system model. 

a NEWTON-RAPHSON ITERATION block-whose input is the gradleat and the 

approximate Hessian and that generates as an output the updated parameter 

distribution 8'' for the next iteration. The central calculation in this block 

is the solution of the integral equation M~& = that rernltr in the 

parameter estimate Wte ti#. 

After specification of the parameter estimate en+', the square-root filter L& .pd its 

sensitivity aLt&/a are redesigned by lettin# 8 4 , and the steps outlined above 

are repeated in order to conduct the next step in the iterative process for optimization. 

n n+l 

The predicted-data-covariance smare-root filter processes the data y and the 

suspected mean m to produce a filtered state estimate 2 and a set of re-idruls e - y-2. 

This is done by mean8 of the equation a = Ly + (I-Lh, where L = A-(I+R)-'~. Thir 

equation, while providing a very succinct symbolic description of the splure-root filter, 

does not by itself provide a recipe to conduct comzmutfonr. In order to provide such a 

recipe, it is convenient to use the corretpordky gpectral form 5 = (1-co-l~~ + 

COSQLPLL, which expresses the spectral amplitudes \ - ek% of the filtered state 

estimate z as a linear combbtiW of the &ta and suspected mean rpectral amplitudes 

yk and Such a spectral form of the predicted-6.u-covariance square-root filter is 

illustrated in Fig. 9.2. 

The diagram in the figure illustrates the main calculation8 involved in the square-root 

filter. On the upper branch of the diagram, a set of data"] y =.[yl,...,y J is assumed to 

be avahble at N discrete location8. A spectral -lyrir[21 ir conducted on this data to 

obtaia the data spectral amplitudes'3J Iy' ,...a 1. These spectral amplitudes are 

then multiplied by the coefficients (1-cosq) in the data filteJ4], red- in the term 

(l-cosa$. On tho lower branch of the diagram, the detenainirtic inputs ft6] are 

N 

N 

373 
processed by e inputJoutput qstern to obtain the suspected mean 

m = [m ,...,%] A The spectral amglitudet mk * m[g]of the suspected mean are 

then computed and subsequently multiplied by the coefficients CO~ in the mean 

filter[''' to produce the terms cos(qmL Ill1. This last term is then added to 

(1-cosap ' in"] red- in the filtered state spectral amplitude8 #21 and the 

residuals e:']. Note that the physical state ertiaute a d the residual e can be 

recovered from at. and ek by means of the sumfmt.ion8 2 = &2 + and e = &eht$k, 

1 * *k 

kk 

akhough for SimpUCitJ? this h8t trUh8fO~t.h b nOt 8h- Oa the diagram. 

112 13 

Fig. 9.2 Spectral Form of Predicted-Data-Covariance Square-Boot Filter 

The foregoing remark8 have scmtinized the spectral form of the rqprre-root filter 

ct,wtion z = Ly t (I-Lhn. The immediate aim z~oyo Sr to conduct a 8imft.r detailed 

analysis of the spectral representation of the -re-root filter rmitivity equasiou 

WaS 5 (i3L/20b t ;I-L) Cam/ae,. The spectral form of thi8 eqtution ir mted in Bq. 

(6.15) and illutrated in the block diagram in Pi& 9.3. The overall phry 

374 
function of the square-root filter sensitivity is to process the N mean-centered data 

spectral amplitudes"1 rad the M determinirtic inputs[21 in order to obtain the spectral 

amplitudes oft3] of the filtered state estimate sensitivity ad*. An intermediate 

calculation embedded within this overall process involves processing of the 

mean-centered &U spectral amgUtuder y"] by means of the N-by-N matrix, with 

general elements ah, representing the data filtsr sensitivity ~L/w[']. Other 

intermediate steps involve: processing of the deterministic trrpatr fm [21 by the 

filter 17] to obtain the terms CO~ tam/aeIk [el . 

input/output model sensitivity muir bb "1 to generate the suspectad mean spectral 

amplitudes (am/ae), I61; ad dseqrrent processing of these coefficients by the mean 

Fig. 9.3 Spectral Form of Sqarre-Root Filter Sqitivity 

IO. CONCLUDING REMARKS AND FUTURE DIRECTIONS 

The 'a of erthmtion for elliptic qsthrmr is so full of interests research 

problem that, in spite of all that this paper has covered, much more remains to be 

done. There are some of the problem that lie ahead: 

0 Conduct of an asymptotic statistical property analysis that explorer the 

convergence of the parameter estimates as the uumber of Obrtnratioar 

increases. 

3 75 
Development of approximation approaches thrt rQomusly urive at 

f inite-dimensional approximations to the infinite-dimensional solutiorvl 

advanced here. 

More complete investigation of the optimal input design problem. In 

particular, development of "spectral' domain design approaches which would 

do for elliptic systems what the frequency domain methods achieve for linear 

time - invariant dynamical systems. 

Development of more precise mrthemrtical arguments to justify 

function-space differentiation, eigensystem expansions, covariance 

calculations, likelihood-ratio derivations, etc. 

Investigation of alternative (to the square-root) factorization of the 

pradicted-data-covariance that could result in easier calculation of the 

function-space derivatives necessary for the Newton-Raphson search. 

Numerical experimentation with the filtering, S~OO- and identification 

algorithms to gain further insight into the state and parameter estimation 

approaches and solutions [SI. 

As a final remark, this paper is a concrete example of the power of the functional 

analysis approach to estimation advanced in Ref. [4]. Because of the conceptual 

simplicity of the method, it has been possible to solve in this paper problem that would 

have defied solution by any other method. It has also made it possible to conceive areas 

for future research that would otherwise have been left unidentified. 

RBPBRBNCBS 

1. 

1. 

3. 

4. 

5. 

Rodriguez, G., "A Function Space Approach to State and Model Bmr Bstimrtion 

for Elliptic Systems,' Proceedings of the Workshop on Application8 of Distributed 

System Theory to the Control of Large Space Structures, JPL Publication 83-46, 

July 1983. 

Luenberg%r, D. G., Optimization by Vector SDace Method$, Prentice-Hall, 1971. 

Balakishnsn, A. V., Communication Theom, McGraw-Hill, 1968. 

Balakrishnan, A. V., Applied Functional Analysis, Springer-Verlag, 1977. 

Scheid, R. E. and C. Rodriguez, "Numerical Experimentation with Maximum 

Likelihood Identification in Static Distributed Sy8tem8*, Proceeding8 of 

NASA/ J PL/LEIC Workshop on Idenrification and Control of Flexible Space 

Structures, San Diego, CA., June 1984. 

AC KNOWLBDGBMBNT 

The research described in this paper was carried out at the Jet Propalriw 

Laboratory, Califorair Institute of Technology, under contract with the National 

Aeronautics and Space Administration. 

3 76 
NUMERICAL EXPERIMENTATIQN 'WITH MAXIMUM 

LIKELIHOOD IDENTIFICATION IN STATIC 

DISTRIBUTED SYSTEMS 

R. E. scheid, Jr., and C. Rodriguez 

Jet Propulsion Laboratory 

California Institute of Technology 

Pasadena, CA 91 109 

1. INTRODUCTION 

Many important issuet in the control of large space stmctotcs are intimately 

related to the fundament.1 problem of parameter identification. Very often, a 

complicated structure can be adequately modeled for certain operationr by the 

fitting of a rather rimple model with a number of free parameten. Tblr simple 

model then can be referenced for necestary control operations. Important 

applications include the many tpace station design8 which are baaed on the amembly 

rrnd joining of discrete module* by crew mewem. Thiz crew-misted coartruction 

will result in a configurrtion -v nich t a Irrge-scale composite of wuy structural 

element8 and whore static and dyrumic chrracterlrtics cannot be abeqortely 

modeled in advance. In fact, any mode- will require periodic upflat- as more 

modules are added to the system and as the rtn;rctural properties of the element8 

rlowly change over the Wetime of the station. 

One might nl8o ask how well thi8 identification process can be carried out h the 

presence of noisy data since no sensor qstem is perfsct. With these conrideratiom 

in mind our algorithm are designed to treat both the case of uncertainties in the 

modeling and uncertaintier in the data. 

Thi8 paper aerves as a companion to [a] where the analptical arpect.8 of 

mrximum likelihood identification are considered in wme detail. Here we foctu on 

the questions relevant to the implemenution of these schemes, p3rticularI9 as ttray 

apply to models of lwge space structures. OPT emphriis will be ,-a the influence of 

the infinite-dimenrim1 character of the problem on finite-dimenrioPrl 

implementation8 of the 8lgorithm8. We highlight those areas of currev'- ad futm 

an8lgsL which indicate the interplay between error analysis and possible truncation8 

of the rtate and parameter spacer. 

2. MODELS 

A8 in (61, we conrider the syrtemr of the form 

Here A it 8 formally self-adjoint elliptic dif€crenti&l operator defined over the 

spatial donuin n; the integral operator @ it related to A by 

377 
where I is the identity. B and C are appropriately dimeariooll operaton that model 

the influence of the procesr error o and the input f on the state u. H is an 

operator that chrracteriaer the 8t8te-tO-Ob8eW@th mapi o ad 9 are model 

erron that form the model error vector 

and f is a deterministic input. Conceptually the error vector E representa spatial 

white note ad is characterized by the covarhnce operator 

uo and uq are non-negative scalar wemting parametem that respectively 

meam the relative importance of the modem error ab the meammm40t error. 

Thus, the limit uo + 0 corresponds to the caw of perfect modeling while the limit 

u + 0 correqoads to the case of yerfect mearruemsnts. v 

8 is the porribty infidte-dimen8ional parameter which must be ettimated. For 

simplicity we shall generally contider cases where the parameter dependence is 

restricted to the operator A. potthhrmore, we astom4 a8 in [6) that the parametsr 

enters linearly into the expression for the potentirl energy of the system. Thru we 

assume 

where D* denotes the form81 rdioint of Di the corre8pod.q~ potential eXl6rm 

given in tern of the appropriate state-rprce inner-product: 

And finally, the detenninfrtic ab 8tochartic forcings will be localized to 

discrete point8 which might correrpoad to actuator locrtkr. Similarly, the 

obrenration mrp retimu a vector of obrervations at dircrete poinu which might 

correrpond to sensor locations. We armwd that there are Ns point-rearom at 

locations {[Q and N, point-actuations at locationr {TJ. 

Became of there last assumption8, many of the relevant calculation8 mtbd h 

[6] reduce to matrix and vector mraipulrtiofu. In thir paper the mution 3 will 

(k) refer to a finite-dimension4 vector whore k-th component 14 given by g . Similarly 

relevut dimemiom of vector AC matrix qurntitier will rlwayr be clear from the 

context. 

(hj). me is the notation for a matrix whom (i,j)-component .is given by G 

After taw formal limit8 in the system (2.1) we bye: 

AU = uo B3+ Cf 

(2.6) 

378 
where 

s 

w 

(2.76) 

(2.7e) 

(2.7D 

in the cue of more general senrin4 and rcturtin$j ryrtem, the oaodeliq 

requii,mentr for the system can be reduced to rolvin~~ equations OE the form 

AU = f. (2.8) 

Thus the discussion in tu section will facut on how the infiniie-dimena!oml 

stm-ture of the ryrtem (2.8) influencer the choice of finite-dimensional 

approximations which ern be mde. In this paper we consider two specific 

structural rnodeb: a strirq under tefuion and a wrap-rib ataxia. 

Let be a distance coordinrt: ernved in meten along A of Length L which 

is aho given in meters. Let u(x 7 be the displacement in meters and let &&I Re the 

tension parameter given in upla of newtons. The forcirq dcnsity f&) in 

unit8 of newtondmetez. Then the energy potential [3! la @en by 

given 

(2.9) 

3 79 
Of come the energy potential is given in units of newton-meten. The equations of 

motion can be derived immediately bared on the principbt of the calculus of 

varhpon8 but it will be convenient to first transform to dimenrionlerr coordinates. 

Let a* be some chara~terirtlc value of the tension parameter. We introduce the 

dimemionless variables: 

A 

X 

.x = - 

L 

A at3 

a(x) = -fi- 

"* 

U(X) 

L 

Ah 

&XI= - 

%)L 

t4 

$(& 

%L 

f(x) = -- 

V(u) = --- 

and the potential expremion becomes 

(2.10) 

(2.11) 

For simplicity we prercribe boundary condition8 ,ontpolding to fixed end points: 

u(0) = dl) = 0 (2.12) 

Then arguments based on the calculua of varhtim give the mystem 

(a(xlu'(x))' - f (x) 

u(0) = ull) = 0 O<x<l 

(2.13) 

This example has been studied many timer in the clardcal literature but m 

analogous approach @ver comparable upresrim for much moh complex systems. 

We consider now a planar model for I wrap-rib antem which is wed to study 

out-of-plane vibrations (sce PWe 1). The antenna model comprber N gores 

(subsections) modeled by *uterconne:ted rib8 and mesh. Since the tramformrtions 

are similar to thore wed in the care of the string, we imm66i.tely write the 

potential expression with dimenrionlasr coodinater. Let the vector of rib 

dirplacemenu be ar) where the k-th component of zi8 u~', the di8placament of the 

k-th rib (0 < r < 1). Let the vector of merh dirplacementd bs%r,8) where the k-th 

component of ais v(') the displacement of the k-th merh rector (0 < r < 1,0 < 8 < 1). 

3 80 
Pig. 1 Simplified model for wrap-rib antenna 

Based on malyrk of actul antenna dew, our model consists of N identical 

beams fixed at a central hub. Stretched between the beams are N identical 

anisotropic membranes. The potential eqyation k given by 

d'z d'u' dr 1 1 

2 V=- I C1 7 - 

0 dr' 

(2.14) 

Here the coefficients {GQ are related to the phyeical parameters of the beam 

8d memtranea thudy: 

Gl= --- 

UL 

Tr eo L' 

Ga = 

U 

T~L~ 

Gg = 

OB0 

(3.15) 

381 
E and Io are respectively the Youzy'~ modulus ud the moment of inertia of the 

beams. Tr and Te are rewctively the radial .nd circumferential tensions of the 

membrane. L is the radiw of the antenna d eo it the .Iytllar width of a rector, 

that is, we have 

2n 

N 

g =- 0 8 (2.16) 

where N is the number of gores. Pixully, u it lome convenient scaling parameter 

with the dimensions of energy (nt-m). We note that the phydcd forca densities rR and FM having retpective dimensions nt/m ami nUm3 were rescaled accozding to 

(2.17) 

Appropriate geometrical boundary cozditioar follow from fixing the center .nd 

attaching each of the ribs to its adjoining membranes: 

(2.18) 

Here is an N x N periodic matrix: 

C= 

N 

382 
As in the case of the s-, the equations now follow from .rgumeatr based on 

the CdCdUS Of V8fhtiOlU: 

la 

r& 

with the additiond natural boi..d.ry conditions: 

(2.20b) 

(2.31) 

One of the focuses of this study is the conrideration of nonm0d.l approaches in 

the finite-dimensional approximation schemes. In practice, tbir gene- will mean 

directiy sol- a hear system of equations rather tlun proceeding fron 30- 

finite modal synthesis. But the infhW-dimen$ional 8tructure of the 8y8tem (2.8) 

also can influence the particular finite-dimensional approrinution schemes d. 

Our approach is sufficiently generd so that any adeqgate finite el- aent model of 

the system (2.8) should yield adequate Eumerical approdnutim. Br me can often 

do much better for a particular model or a particular chrr of modelr. 

We use the antenna model to illortratr the point md make some observations 

that should influence the approximation scheolts regardless of which finite element 

or finite difference scheme is employed. We exkphmixe that these coaridentions 

also apply to much more complicated antenna model8 whkh rbrre 88lient features 

with the system (2.18) - (2.21). First we note that the rtnrcfurt is periodic in the 

9-direction. This cyclic symmetry leads to conriderable saving8 in the compukth 

of solutions to (2.8). This can be deduced from either the differentid equation or 

the energy expression (2.14). The periodic mrvix can be di.g&ed bY means of 

a finite Fourier transform [l]. That tr, let U be the N r N matrix whore (f,k) 

component has the form IC) 

(2.32) 

383 
We then h8ve: 

(2.23) 

This tr8nsfoxm8tion decouple8 the 8ystem 8ince the potentid exprc88ion (2.10) 

with 

= 

u =wu u 

(2.24) 

The brlurc@ of terrm in the equation also can influence the choice of 

discretization. Based on A report by Lockbed on the 8pecific8tionr for a SS-meter 

wnp-rib antenna with 48 ribs [2], the follo*riry nomid parameter me8 were 

derive : 

- 1.31 10*m4 

B - 9.72 10'ont/nr' 

IO 

TR - 1.75 lO-'nt/m 

Te - 3.50 lo-' nt/m 

(2.25) 

384 
G - 2.29 

1 

G - 8.58 lo-' (2.26) 1 

G -1 

8 

This -ne- that the ndi.l terms of the mesh potantid ue comparatively I11D.u 

ucepr nhen the rrdi.l derivatives ue large. How thir affects the stnrcture of the 

system is demonstrated in the follo- example (see Figure 2). 

RIB 2, 

Fb. 2 Si@e antemu gore 

h thir example we rtudy the equationr for a rector of membrane where the 

prescribed boundary conditions depend on the adjoining rib ~kcexnents (fx(r) uxl 

fa(r)). For simplicity we take the forcing on the mesh to be zero although the more 

385 
general case could be handled in a similu fashion. We are of course interested in 

the case where 

0<€<<1 (3.28) 

which corresponds to the parameter ranges (2.26) in (2.30). Physically one expects 

that the radial terms contribute little to the sutic bch.vior except perhaps at the 

boundary when the gradients my become we. One is also interested in the 

behavior near the cornen (r,8) = (LO) and Cr,e) = (1,1) since some rinsplu behavior 

may be possible. Using the techniques of pcrturb.tion$, (see for example [a]), one can thou thrt as € approaches zero we have 

This expansicm could be continued to higher orbera, .od, u noted before, a more 

complicated expression wo-dd redt from forcing8 on the membrane. One possible 

approach to &e numerical 801Uth of the 8y8tem (2.20) k-dd by the eUminltion of 

the mesh behavior entirely be mbstitu- an expre88ion'rimilu to (2.29) into the 

beam equations (2.90.). Then one would have only equations alone the beams to 

solve. If higher-order accuracy on the mesh is required, one could then apply 

finite-element techniques to the system obtained after linearbation about the 

asymptotic expamion for the mesh behavior. Finally we note the appearance of 

logarithmic singularities in the mesh gradients (&re, ar tole cornem (r,8) = 

(1,O) and (r.8) = (1.1) are approached from the interior of the mesh. This 

consideration should also influence my finite element approximation of the me&. 

We emphasize that this uulyris applies not only to the simplified antenna 

model we have considered but would hold for more elaborate configur8tion8 when 8 

similar structural balance of term8 govern the system. Thus, m8ny modeliYuJ 

options can be considered for parameter identification in important clwes of 

structures if one does not insist on a traditional 11106.1 churrcterization of the 

system. 

3. THE LIKELIHOOD FUNCTIONAL 

A detailed discussion of the likelihood pxinciple is given in (61. The functiond 

we consider is the negative logarithm of the likelihood ratio associated with the 

detection of a Ga-asian signal in additive Gaussian nobe; thir framework is 

traditional in the theory of communication and 8-1 detecticm. 

In accordmce with the diacusrion @veri in (61, md the notation discussed in 

Section 3 ,the log-likelihood functional is given by, 

386 
(3.1) 

Here Fl*q indicates the Euclidian inner product in the N*dtmbnriOo.l wee to 

which the obsemators belong. 

Prom the assumptions of Section 2, it is easy to see t&t 5 El is an N N lnutrix 

whose (i, j) component is given b9 

where g(xl<) is the point-source solution of the underlying elliptic -tern 

Au~~(x- E) (3.4) 

with the appropriate boundary conditions. And likewise the expected observation 

has the form 

?a;,= g f- 

- (1; iN IT f = (f ,..., f a ) (3.5) 

We note that (3.1) also differs from (1.3) of [a] in accordance with the 

rntraductioa of the poritive weigh- parameters, uo and u,,, into the system (2.1). 

An equivalent form for the likelihood functional follow from a rearrangement of 

terms. 

387 
where 

(3.6) 

This form is useful since one can amve at the fuactiod given in [a] directly by 

the substitutions 

This correspondence rllows one to we the algorithms derived in [a] directly on the 

functional 

The goal is to find the parameter value 0 which minimizes the log-likelihood 

Eunctional; that is, we wish to solve -c 

min Jt0,$) , (3.10) 

e 

where 0 ranges over some appropriate infinite-bimenriorul qace. As- that 

the functional hr.s a Prechea; derivative rrad rati8fier UI appropriate convexity 

condition, one can restate the problem (3.10) as 3 .I 

aJ/S (e,$) = 0 (3.11) 

Both problems (5.10) and (3.11) have been studied in a variety of contexts (See, for 

example, [SI). 

Since both th~, parameter space and the state space are infinite-dimensional, 

one mwi make dual approximations in order to achieve problem that are 

finite-dimensional and therefore computationally tractable. Thus in practice one 

solves a sequence of problems of the dorni 

388 
(3.12) 

or 

(3.13) 

where the state-space and parameter-space have been replaced by finite 

dimensiolul spaces. Then the problem reduces to a finite minimixation problem 

which can be treated numerically by a variety of techniques (see, for example, [1,5)). 

The state-space Can be approximated by a finite-element space which is 

appropriate for approximating solutions to (9.41, and the parameter-space can be 

conveniently represented by a spline-based space. Let Nx be the dimeruion of the 

finite-ctimensiorul approximation to the state-space and let Ne be the dimensions of 

the finite-dimensional parameter space. This leads to the natural substitutions 

b1 

k =1 

(3.14) 

(1) RJ )T u = [u I...) u x ] s 

where the sets {tpd and give the basis elements for the skte and parameter 

spaces respectively. 

It will also be convenient to consider the strte-space 

weighting given by the basis elements of the parameter -ace. 

< u,v >) = <u, rp> 

(i {la..., - 

inner-product with a 

Thus we define 

(3.15) 

Jn the following we restrict our attention to these finite-dimensional problem, 

and, when the context irr clear, we suppress the ^-notation. Questions concerning 

the convergence of the numerical schemes and the general relationship between the 

infinite-dimensional and firlite-dimensional problems will be discussed more fully in 

a future report. 

3 89 
Most nO&ear Opthhatiou teChniqlM8 reqube 80lVhg Uearfred 8y8tetXM 

iteratively, ad conuequently one must solve system of the form (3.41, where the 

dimension Nx may be quite large. Since the complexity of 801- an m-dimenrionrl 

linear system is O(ma), the speed of convergence of the iterates is an impoxtant 

consideration. With this in mind, we emphasize the use of quasi-Newton methods 

for the solution of (3.13). Consequently much of the resulting effort is directed 

towards deriving adequate approximations for the Ne- dimemional Jacobian vector 

aJ/M and the Ne x Ne Hessian matrix aa J/a2. 

We briefly outline the procedure here; as noted previously, a more complete 

description is given in 16). In general for the finite problems, the dimension of the 

state space (Nx) is much larger than the dimension of the parameter space Ne), the 

number of sensors Ws) or the number of actuator8 RJ.1, and so it is preferable to 

carry out. the necessary manipulations in spaces whose dimemions do not depend on 

the dimension of the state space. 

Therefore, as in [a] we represent calcrrlatim in term8 of the eigen-structure of 

the Ns x Ns matrix R. 

N 

(3.16) 

n Ak = tan ak (0 S a <-) ri 

From the spectral components of R we define useful quantitks a8 given in [a]. N 

From (3.2) we have the expected observation 

(3.175 

and also we define the filtered observation 

where the Ns x NS matrix L is given by N 

(3.18) 

(3.19) 

aad tl;e related matrix K is given by N 

390 
(3.20) 

Par algebraic convenience we also define the residual of the process: 

--ce=y-z 

ek = Tk* 'i' 

The gradient of L is represented by 

(3.21) 

(3.22) 

k m#k 

(3.23) 

i Pot later convenience we derive another form for the coefficient8 {ah]. usstandard trigonometric identities one can easily verify the relation 

cosak - cosa (cosa 3 '(cosa m) a 

m =- (3.24) 

tuna k) a - (tanam la COSUm + Cos% 

This leads to an alternate form for the coefficients 

- ( Cos\)' [(kk)' <mk, % >j] k-m 

(9.25) 

- (cosak cosam)'/ (COS% + cosam) . 

[Am%.; Wp,, % >i + Ak ILL <Dpk, Dx > 1 mi , k # m 

39 1 
The point of this last derivation is that the bracketed terms reduce to rimpler 

expressions. Prom (2.2)’ (2.71, and (3.4) one can easily rhow: 

and also from (2.7) we have: 

x m -h;@B*p, 

By this we have: 

where the Ns x Ns matrix A has the form -1 

And similarly we have the useful relation 

’k<-k’ D3j =q*B k -j f 

u = @Cf - 

where the Ns x N, matrix B is given by -i 

(3.26) 

(5.27) 

BPSrn) = <Dg (xJEk), Dg (X,F~)>~ (3.311 1 

We now give expressions for the gradient and the Hessian fn terms of the 

quantities given above. As in [a] the gradient can be represented as 

35 2 
(3.33) 

Buct expression8 for the He8dU are givca in [6]j in general, \owever, all 

terms need not be estimated to @ve an adequate approximation. In par'Jculrr, the 

crlculatfonr are much simpler if the terma With second-order derivative* CUI be 

ignored. The simplest approximation comes from only keeping thore terms which 

contribute to the expected VL of the Hessian. Thua, from (1.12) of (41 the 

(i,j) - corzrponent of the N, x Ns Hearian approximation e i8 given by 

(3.34) 

where by (3.20) ad (3.22) we have Vi ( m*k) aFsec% 

This esti~natc is justified when the covariance 3s amall as one might expect if the 

number of measurements is large. This point will be inveatigated more rigorously i~ 

a future paper. 

We now ~utnmarh? the search procedure for the 8yrtem (2.6) where the 

Ns-dimensional obsenratian vector 7 is even and an WYa1 N e-dlm8tUid 

parameter estimate go is rvauable. 

Pint the expected observation ad the covarhnct matrix E are dete&b from 

(3.3) snd (3.5). The spectral decomposition of E a9 well as the qpantkiear given by (b.17) 

- (3.33) then can be determined by standard matrix abebrr routher. Aab therefore 

from (3.921, (3.33) and (9.941 one obtains an Nc-dimruhi grdient spptoXimat3m 8 

an3 an Ns x N8 Het5au approximation ?$. 

The parameter eatinlate eo can then be updated by making the quasi-Newton 

correction: 

-- -1 -. e* =eo- Yo8 g (3.35) 

Here' yo i.8 an appropriate scalar choreo to improve the updated parameter 

ertimrle. In accordance wl '1 the geueral ?.&eory of Ne.rton iteratiom in function 

spacer IS], one can repeat this procedure until the 8olntLoru of the Ilnowri2ed problems 

converge to the rolutlon of the underlyw nonllnerr problem. 

This analyrir completes our outhe of the marinrum Ueubood identification 

prOse88. h SSCtiOn 4 we give eXaf&~ple!J Which ub8w8U the 8UCCgEifd 

implementation of there schemes in useful applications. 

333 
+. 

previously discussed aborithms. 

BXAXPLBS: In this section we give examples of successful implementations of the 

We first consider the string (cf (2.13) 

Au = - (ak) u'(1t))' 

do) = u(1) = 0 

(4.11 

For the case where the unknown tension parameter is constant, the point-source 

solution can be explicitly given: 

x< = min{x,El . 

And thus, as outlined in Section 2, all calculations could be given in terms of these 

quantities, without any uvncation of the state space or the parameter space. 

In general, however, truncat'ons in both spaces are necessary. For the string 

problem we consider an Nx - dimensional state space of linear splines; the state 

variable then beccmes the vector of nodal values on the corresponding grid. For 

simplicity we take the grid to be uniform; thus, since the endpoints x = 0 and x = 1 

are fixed, we have: 

1 

Ax= -- 

Nx t 1 

The state-space elements are then given bv 

(i) NX 

U = U K.(X) 

1 

i= 1 

wherc, as illustrsted in Figure 3, the basis elements {r{x)) have the form 

x -(i-llAx 

- , (i-1) A x < x < (i) Llur 

Ax 

x -ti)Ax 

, (i) Ax <x < (i+l)LLt 

Ax 

0 , otherwise 

KCI(X) = 

(4.3) 

(4.4) 

(4.52 
(i-l)AX (i)AX (i+l)AX 

0 AX 

Fig. 3 Linear spline elements 

A rimL1.r dircretizrtion of the parameter rprce k pO88ible. Pint we consider the 

augmented upline space 

where, ar illustrated by Figure 3, the endpoint-elements K~ and ‘c~ are given by 

X 

39 5 
(&I - x 

Ax 

0 , Ax< x < 1 

------ ,o< x <Ax 

IC (XI = 0 

Thus we have a corresponding parameter element 

(4.7) 

(4.8) 

i =o 

d 

which would give a parameter space {a with dimension N,t2. 

However, as previously noted, the resolutim of the pairmeter space often does 

not need to be as fine as the resolution of the state-space. We consider then the use 

of I) piecewise linear parameter space of lower dimension where the only 

requirement is that the nodal points must be a subset of the nodal points of the 

state-space. The new parameter space is then a subset of the (Nxt2)-dimemional 

space given by (4.6). Let a be an No -dimentional parameter element 'Ne 5 Nxt2). 

Then a identifies with an element Z of the larger (Nx+Z)-dimensioarl space ad the 

relationship is given by 

--* hr 

a = &a (4.9) 

where E is an (rt2) x Ne matrix. And correqmndingly, we have 

&==B - . (4.10) 

This relationship simplifies the algorithm as described below since B is easy to 

construct, ad the more cumbersome calculations which are needed to determine 

partial deriV8tiVe8 with respect to the parameter space are *en specified in terms 

of the grid associated with the state space. Thus we have: 

a/s: a/s (4.11) 

39 6 
We illustrate these points with a sample calculation (see Pig. 4). We consider 

the case where there are seven sensors at the locations 

A A A 

S S S S S S S 

I I I I I I 1 

0.0 0.25 0.5 0.75 1 

S: SENSOR LOCATION 

A: ACTUATOR LOCATION 

Fie. 4 String tension identific-tion: sensor and actuator locations 

and three actuators at the locations 

h 

E E (-258 -5, -75). 

The data vector was derived from a phnt with specifications 

a (x)=3 t x plant 

C F= 6(x-.25) +6(x-.S) + Mx-.75) 

u = .001 

w 

(4.13) 

(4.14) 

u = .QO1 rl 

For the state space we take the seven-dimensid space of linear splines fix)} 

with nodes corretporading to the sensor locations (4.121, and for the parameter space 

we take the five-dimensional subset of lheu sppllnes with nodes corre$pondbg to 

the set 

The relaxation parameter yo in (3.35) was chosen to speed up to the convergence of 

the iteration; these issues will be discussed more fully in a future report but we give 

the results nf the calculations Ln Fig. 5. There numerical experiment8 appear Lo be 

very encouraging although with a crude approximation to the Hessian the 

canvergence can be very slow. 

39 7 
Fig. 5 Distributed parameter identification via spline analysis 

In a similar way, the algorithm wa8 successfay applied to the wrap-rib antenna 

model (2.4). To simplify the calculation8, we assumed here that the stiffness 

parameters were scaler8 although one could introduce a spline-baed @ace ad in the 

previous example. 

Again for simplicity we consider the case where there are six gores (N = 61, 

where a sensor k placed on the oute~ endpoht of each rib (r = 11, and where an 

actuator is placed at the midpoint of each rib (r =.SI. This scheme is outlined in 

Flgute 6. 

We introduce the set of N-dimeasional dt vectora 

(4.16) 

- 

where the components of each ak are determined by 

398 
POiNT - SENSOR LOCATION 

POINT - ACTUATOR LOCATION 

Pig. 6 Rib stiffness identification: sensor and actuator locations 

(4.17) 

The parameters of the likelihood functional are then given by 

CYa = .001 

u = ,001 (4.18) 

cT= 1 < 6 (r - .SI 

rl 

And the stiffness parameters of the plb..,. are given by 

E1 = 1.25 .lo’ nt-m’ 

TR = 

(4.19.) 

1 ’5 .IO-’ at/m (4.19b) 

L = 2.75 .IO-’ m (4.196) 

We applied the algorithm then to the case where the unknown parameter was E1 

while the other stiffness parameters were asawned to be kxiown. 

399 
To discntize the state-space eight equal mbdivisionr were made in the radial 

direction on each rib and in each mesh sector; in the circumferential direction five 

equal subdivision8 were xnade in each ae8h sector. The shape functions on the ribs 

were given by Hermite cubics while on the mesh the shape functiom were given by 

splines linear in r and 9. In test cases this discretization produced at least three 

digits of accuracy in solving problems of the form (3.4). In all calculations the 

principle of cyclic symmetry (cf. (2.24)) was exploited to reduce the number of 

calculations. 

4 

5 

Convergence of the likelihood algorithm was very fast (see Fig. 7) when the 

relaxation parameter was taken to be 

* 

1.13866 105 

1.23889 l@ 

yo = 2.5 . 

LIKELIHOOD FUNCTIONAL 

-0.0048 - 

- 

- 

-0.0052 - - 

i I 

0.1m 0.126 0.1 w 0.175 

RIB STIFFNESS X IO-‘ nwn2 

IT€ RIT~N f STlMATED 

Nr .18€9 PAR AM ET ER 

I1 I 1.75000- l@ 4.- 10-1 I 

2. 10-1 I 

3. 1oA2 ---I 2. 10-4 

Fig. 7 Distributed parameter identification of beam stif fnesa parameter 

400 
Numerical experimenu rho demonstrated an improvement in the sensitivity of the 

identification rrchemes a8 the number of rnerruremenu was increased. Thus, Figure 

8 illustrates how, for the antenna problem considered, an increase in the number of 

sensors led to a stew of the likelihood functional. Here the curves mre 

shifted transversely for illu8trative purposes. We note that no corretpondhg 

improvement in the parameter estinute occurrcd in these trials, possit$ because of 

the lea8 favorable &pS-U-noire ratio which corresponds to renting in the interior 

of the ribs. 

LIKELIHOOD FUNCrlONAL 

I I I I 

0.125 0.150 0.175 0.100 

Fig. 8 

(a) One sensor per rib at r - 1.0 

(b) Two aensors per rib at r - 0.5, 1.0 

(c) 

Sensitivity of antenna etlffaese - phsameter ldtantiflcation according to number of k,eneors 

Three eensors per rib at r - 0.5, 0.75, 1.3 

More detailed numtricrl experiments with distributed antenna rtiffneu 

parameters will be given in a future report. But the resuits outlined in t&i8 report 

dernorutrrte already the great potential for these algorithmr. 

RBPBRBNCBS 

1. Dahlqubt, G. and Bjork, A., &gerical Metha, Pwntice Hall, 1974. 

2. El-Rahab, M. and Wapt, P., "Static ad Dpnrmic Chrractcrirticr of Large 

Deployable Space Reflecton" AUA Conference Paper 81-0503-PC, AIAA 

Dpnrmicr Speciubtr Confetduce, Atlanta, Geor@a, 1981. 

3. Calfand, X. NI., and Pom& S. V., CrlcuSur of Vrriatim, Prentice-Hall, 1963. 

601 
4. Kevorkian, J. and Cole, Jej Perturbation Methods in Amlied Mathematics, 

Springer-Verlag, New York, 1980. 

5. Luenburger, D. G., OPtimization by Vector Space Methods, Prentice-Hall, 1971. 

6. Rodriguez, G., "A Residuals Approach to Pilte-j Smoothing and Identification 

for Elliptic Systems", Proceedings of NASA/ACC Worbhop on Iden*ification 

and Control of Large Flexible Structuteir 1984. 

402 
DISCUSSION-FUTURE RESEARCH DIIWCTIONS 

Moderator: Heman A. Rediess, H. R. Textron, Inc. 

Panel Members: A. V. BaWrbbm, University of California, Los Angeles, 

R. F. Crviisk, NASA Headquarters, J. L. Lions, College de France 

and Institut National de Recherche en Informatique et en Automatique, 

R. E. Welton, Purdue University, and W, E. Vrrnder Velde, 

Massachusetts Institute of Technology 

SYNOPSIS 

The intent in thi8 session was to present several viewpoints on the direction of 

future research in identification and control of flexible space 8uUCtnres. The penel 

members were given the option of prepam formal papers or presenting informal 

comments. Professor Lions was unable to attend the meeting becau8e of a last Ininate 

change in plans but did submit a formal written paper which is printed in Session M of 

these proceedings, This synopsis attempts to capture the main points diacarred but has 

not been reviewed or endorsed by the speakers. 

Professor Vander Velde di3cussed uncertainty management methodology for large 

space structures. He identified six major types of ~x~certamties that are likely to exist 

in control system design and operation. After assessiag the present methodology for 

handling each type, he wested the additional technology developments needed. 

Altogether, he su~~gerted specific research topics covering: remot nobe ir nonlinear 

situations; external or internal diaturbancer; modeling approrirrution8; model. parameter 

errors; component operational status ad reliability; and, spacecraft c~-afiaurati.on 

changes. He introduced, and discasred to some extent, the concept of an on-line 

system diagnostic modtar for detecting certain types of fault8 or uncertainties, mch at 

model parameter errors, sewor failures or actuator failures. 

Professor Vander VeLde concluded that the dynamic cbracter of flexG'le qace 

structures and the likely complexity of their control rystema nuke them erpsqially 

likely to have uncertain component operatiag sktur. The problem of detec+iry 

component failures is coupled with the effects of modem error whether we ?LLe it or 

not. Perhaps this couplirq can be utilkad, or at least controlled, in the procerr of 

monitoring the health of the system by use of an integrated 8yprtem dhgnortic monitor. 

Professor BalaLrIshnan discussed the important research area of uncertainty 

modeling from a perspective of estimation and identification problems in random 

fields. Currant estimation and system, identification methods only consider noise with 

rational spectra. We need to look at signals with non-rational spectra. One area iu 

which non-rational spectra appear naturally is in the area of random fields. Random 

fieldr are random processes in which the parameter is no longer time and arise on space 

structures in describing, for example, deformation in an antema or mirror. Yrofersor 

Btlakri8hn used a geophysical example of gravity anomalies to illustrate the 

effectiveness of random field theory in estimation problems when dealing with 

non-rational spectra. Although there extat.8 rubstantial research on the theory, there 

are only a few practical applicstim in the literature. Techniques are available that 

can be applied to practical large space structures problems. He suggested that more 

effort be made to apply these techniques. 

403 
Professor Skelton presented his views on the inseparability of the control and 

modeling problems. When a model is used to derive a control law, then :ne cannot say 

how good the model is without hawing what it will be used for and what the control 

will be. That means that one cannot evaluate the impact, of modeling errors until the 

control is specified. There is a challenge to the community to develop techniques that 

do the complete model error estimation and compensation problem. 

Mr. Carlisle of NASA Headquarters discussed the challenges in controls for the 

Space Station. He views the problem of des- the S?ace Station as involving trades 

among "subsystems". For example, in trades between controls and structures, one uses 

all of the technology options in both disciplines to optimize the overall design. NASA's 

plan is for an evolutionary Space Station with the initial operating configuration to be 

placed in orbit in 1992. Block changes in configuration and maybe in performance will 

be made subsequently. There is a challenge to the controls community to develop the 

technology that will provide the flexibility for changes and not be the limiting element 

in future expansion of the Space Station. There is a five year period to mature 

technology for the initial system. Periodic block changes will provide opportunities for 

some later technology to be considered. 

Although the Space Station configuration has not yet been determined, it appears 

likely that it will involve large flexible structures because of the solar arrays, radiators, 

or even the bnsic structure as it evolves and grows over many years. Mr. Carlisle 

described several possible configurations and pointed out the controls challenges. One 

of the major challenges identified was to devise an affordable scheme for developing 

and validating the technology for control of flexible structures in view of the high cost 

of full scale flight experiments. 

Dr. Rediess presented his thoughts on the role of txperim-nts ia the development 

of control technology for flexible space structures. The three main points covered 

were: why experiments are important for developing the technology; the need for a 

coordinated program from analysis to experiments; and the need for a cooperative 

program among the various participants to make effective use of the relatively rare 

experimental data. It will become increasiagly more difficult if not impossible to 

perform ground vibration testing on large flexible space structures. Designers will have 

to rely more on analytical techniques to predict controVstructure interactions. At the 

same time, controlhtructure interactions are becoming mare important because of 

increased overlapping of the required controller bandwidth and the structural modes. 

Damping characteristics, important in the controller design, are probably the most 

difficult to predict. Even the properties of new materials, such as metal matrix 

composites, can islfluence the cantrol/structures design optimization. His conclusion 

was that experiments on the grow and on-orbit will be necessary to develop and 

validate the analysis and design techniques. A well coordinated program is needed to 

bring together the promising analytical methods and the experiments 

Dr. Bediess' final point and recommendation was that NASA, the Air Force, and 

other organizations conducting an<i/or sponsoring major experimer'ts either in 

laboratories or on-orbit should establisL a cooperative data base with broad access to 

the technical community. Such experiments are costly and often difficult to get 

supported. It is imperative that the most effective use be made of these very limited 

experimental data. 

404 
Before opening the discussion to the general audience, Dr. Rediess invited Dr. R. V. 

Ramnath from Raytheon and an Adjunct Professor of Aeronautics and Astronautics at 

MIT to present some remarks on the application of asymptotic techniques to controi of 

flexible space structures (to expand on some of the ideas in Professor Lions' paper). 

Asymptotic analysis is defined as a study of applied mathematical systems in 

limithg cases. The following are some of the benefits of usu it. In model reduction 

techniques, an asymptotic ordering of a complicated system into a hierarchical 

structure can be used in selecting the order of the model. The principle of minimal 

simplication gives a stopping rule for this process. In solving a complex linear 

differential equation, the domain of the independent parameter can be extended into a 

higher dimension. An ordinary differential equation is thus converted into a partial 

differential equation and solved asymptotically in the new space. The solution is made 

to coincide with the solution of the original system along certain trajectories. 

Asymptotic analysis yields a quasianalytic solution which can give insight into stability 

and controllability of the system. Approximation errors are generally smaller. Dr. 

Ramath gave two examples where asymptotic analysis was successfully used to 

generate the controls. He stressed the need of more applications in controi problems of 

large flexible space structures. These techniques have been used very effectively in 

other fields such as celestial mechanics and fluid dynamics, among others. 

The following are what appeared to be the most important comments and 

recommendations made during the open discussion period. 

Some people have suggested that Skylab controls techuolon will be adequate 

for Space Station. If so, there is no need for controls research to support the 

Space Station. 

0 If we rely on Skylab controls technology, we my seriously limit the 

evolutionary growth of the Space Station. 

0 It was mentioned earlier in the workshop that NASA cannot afford f1d.l scale 

on-orbit flight experiments. What evidence do we have to say that we can 

afford not to conduct full scale flight experiments? 

If NASA cannot afford full scale flight experiments, then the project 

managers will not use the advanced controls terhology and will take a more 

comxvative approach. 

There seems to be A change in emphasis in NASA's program away from 

fundamezltal research on controls and towards large experiiiental programs. 

If that is tme, we msy be in danger of cutting off new innovative concepts. 

0 From the perspective of A user of advanced technology, it is necessary to 

periodically take advanced theory snd apply it to practical problerns in order 

to develop an adequate confidence for application to res1 systems. Without 

tht confidence, the theory would never be used, and we would be reluctant 

to support more fundamental work. There needs to be a balance of 

theoretical and applied/experimental effort Ln NASA's program. 

405 
0 NASA is to be complimented on the excellent program plrn for control of 

flexible structures that was presented by Mr. Russell at t*& workshop. It 

presents a balanced program and a truly integrated controls and structures 

technology plan. 

EDITOR'S NOTE: There follows a synopsis of the prepared remarks given by the 

panel members. 

H. A. REDIESS: I have a particular personal interest in this workshop. About three 

years ago, when 1 was at NASF Headquarters and rozponsible for this technical area of 

NASA's RhD program, 1 encouraged G. Rodricaez of JPL and L. W. Taylor, Jr. of 

Lnnglel Research Center to organize a workshop on control of flexible space 

structures. The first one was held about two years ago at JPL and was very effective. 

I'his, the second one, has been equally as successful, and I am pleased to chair this final 

plenary session. One objective of this workshop is to provide some feedback to NASA 

and JPL on research opportunities in controls technology for flexible space structures. 

We have invited several noted technologists to lead this discussion with you, the 

workshop participants, to identify these reseaxh opportunities. After thc prepared 

remarks by the panelists, the audience will be invited to join in the discussion. 

406 
W. le. VANDBB VRLDE. Unceruinty Mananemeat Methodology for Larne Space 

Structwes I think it is safe to cay that the majority of the papm we have lirrt.ened to 

in the: last couple of days have deal; with one aspect or another *L: handling 

uncertamtie; in control systemss bazause, after all, it is one of tne f adamintal 

purposes of a feedback control system to accommodate disturbances, noise, and plant 

uncertainties in certain bandwidth ranges. That is really fundammtal in what we are 

doing. Our pupose here, in this pr' ent tat ion this mom, b to summarize the nature 

of the uncertaintics that we need to deal with, to summarize the technologies that we 

have availabk to us to deal with them, and to iddentif9 where there are gaps. 

Now, suppose you were working onboard a space 8tation, and you gradueily became 

vaguely aware that there WAS some persistent oscillation going on. In fact, it WAS 

betting bigger and tigger, and very soon it was cleat to you that you had a problem. 

The question is, wht would you do? Naybe more '~3 the point, would you have designed 

your control system in such c. wv that it aht have monitored itcl om Sehrvior and 

taken some action when it observed tlut it had a problem? if ea. what would that 

action be? Well. this is one evidence of uncertsintfr in action, becrwc, of cow :, based 

upon our understanding of the dynamics of the system and environment, the tystcta 

should not behave in this frstrion. It L only because we are uncertain atout somethat we might conceivably get this Liud of behavior. But here we are oscillating along 

in the space station. WbAt are we go@ to do next? We need to tuve some fall-buck 

position which is as robust and rehbie as p088ible. It is certrirly that the most 

robust system ycru can build is one that does not do anything. Altho-ph, in the context 

of spacecraft control, it may very Well be important to at least keep me rteoar 

pointed toward earth to mdntain wideband commuaication8. Certainly, +he loop thnt 

accomphhes that pupose should be extremely l@w bandwit3. It should not attempt to 

control any flexible mode actively. It should just try to do +it oa= Ctwction. Well, 

suppose we have indeed disengaged our uaakble spacs ,*C* . .em, or at m..t gone 

back to a truly robust feedback system. That would tm,, x ., little brecr'chizg 

room, and we wouid have time to sit back ad conts?n$ate whrr cuht bsve gone 

wrong. What are some of the uncertainties? What are the tools tb+, we ned in crrdcr 

to handle these different types of uncertainties? whk? arc some of the research 

directions that we ought to recommend? I am gohg to addr-8s these qws?,iorL* in my 

discussion. The list of uncertaintier in control system deafgn ad Gperation thAt 1 will 

discuss hludes: sensor noise, noire in other components; external or intern1 

disturbances; modeling approxLnzrtion8; model parameter erron; component operational 

8tAtUS; ad cOnfigWat&On CbIL@S. 

sensor noise is A source of uncerkinty that has been addresred for a long time. As 

you well know, if you have A full9 linear situation, linear filtering in esttmation theory 

is complete. Even for sbhtly nonlinear mea8urernent.8 or slightly noPUILdar dynamics, 

approximate extensions of the linear thewy, mch as various forms of extended Kihn 

filtering, work well. It is true that if you have II si&nificantJy nonlinear problem, A 

nonlinzar set of measurements or dynamics, there ir a bod9 of theory which L 

AppkAble. Rut this i8 very cumbersome to implement. If you cannot assume from thc 

start that you have Gaussian distributions, you have to, in effect, estimate the entire 

probability density function for the state, given the measurements, and that is 

inherently an infinite dimensional problem. In the context of spacecraft contral, I do 

not think thrt the nonlinear situation will be important, unless there are certain types 

of sensors with nonlinear properties. The only additional technology required are some 

spcciahed techniques for handling state estimation for particular types of nonlinear 

situatioas. I do not think that it is fruitful to pursue general nonlinear estimation as 

much AS specific techniques that might be applicable to specific problems. 

40 7 
The second category of uncertainties 1 will discuss includes wise in other 

components of the system. This is a very important source of noise. There are may 

instances in which autopilots pick up electronic mise that, results in noisy actua+.oz 

operat,ion. Analytically, this appears AS an internal system disturbance. I have included 

that in my nexL category, which is internal ad external disturbacces. Here again, if 

the system is fully linear, and if these disturbances can be modeled as randoin processes 

with rational spectra, then linear sysLem theory IS complete. I mention rational 

spectra, because, if one were to design filters in the frcquen-;y domain, one would 

probably use spectral factorizrtim. This is only easy to do for rational spectre. In any 

case, for a significantly nonlinear system, the situation is just 4s it. is for sensor noise. 

Again, I would say it would be more fruitfd to punwe specialized aarlysis tc aiques 

that would apply to particular situaVons. 

I have mentioned one other topic Mer additiorul technology requizements: 

intemal disturbances that are generated in oilc part of a spacecraft ad propagated to 

orher parts. In fact, just vesterday, we head one presentrtion in which it was 

emphasized that, for multipurpose sprcecraft,' very often the most significant 

disturbances are those that one payload imparts to the other payloads. Fcr example, 

you might have a scanner, with an oscillating mirror or something similar, which is 

creating a disturbance to other pcrts of the spacecraft. So. why not try to isohta tho-,. 

disturbances unthin the modules where *ey originate. I am (NTC Chat this is riot a new 

idea. I list it tere and emphasize the fact that active disturbance isolation my make it 

possible to do this even better then purely passive stnactim design can do. 

This next category is one that has received a great deal cf attenticn in this 

workshop. This is warrreted because we art 8csured in advance that, in dealing with 

flexible sprcecraft godelixu ~~ncertahties, we are going to be working witb an 

inaccurate model of r$e true dynamics. There a number of methodologies that we are 

all familicrr with to deal with the situation. One such method involves designing 1 

control system on tne brsis of a reduced-order model and then evaluating it with a 

higher fidelity madel. We know several approaches to the problem of madel order 

reduction. Absolute stability theory is a way of dealing with stability of systems with a 

ntAzlinear operator confined ody to one sector. And finaL9, there is P very useful 

approrch ustag singular values for evaluation of robustness properties of multi-variable 

systems. There are seveirl additional techniques in model- approximatiom that 

would be very useful. The fht one has to do with 8-U but arbitrary damping 

characteristics cf space svstems. The characteristics of flexible structures wtrlch will 

be least uuderstood in advance will be their dampine ptoperties. But, if the damp@ is 

small, you have the impression that you should be able to design control system which 

does not realiy depend on what is the specific damping. I have not attempted to do 

this. It just seems to me that it might be possible. In a similar spirit, sright 

nonhearities and uncertrin stiffness properties in bigher order system may not be 

important under steady regulation. They might be important, for example, under slew 

maneuver;. The Mea of design procedures for control systems, in which modeling 

inaccwacies of an unspecified form are incorporated directly, is a highly desirable 

objective bit a difficult one to be sure. 

Model I arameter err- are a somewhat more structured Lsad of uncertainty in the 

plant. Our conception is that wc. have a model which is adequate in some scnse. We 

just do not know quite what the values of the parameters are. The standard approaches 

available to w include yrosivvely robust controllers and adaptive systems. There are two 

basic types of adaptive system: Ln one you explicitly identify parameters; in the other 
you do not. We have already heard some interesting thiags about adaptive systems rt 

this workshop. Under the heading of robust controllers, it is always my impression that 

robustness normally is bousht at the cost of some aspect of performance. It would be 

well tc work on that trade-off and attempt to improve it. Under parameter 

identification, the isrruc cf how to efficiently model a system is important from the 

standpoint of the origh! design, as well AS from the standpoint of identifying the 

system. One of the thoughts in an earlier paper rt this workshop appeals very much io 

me. The idea is to model cert.in modules of the system as distributed parameter 

models and then piece them together at the bomdaries. It strikes me that this might be 

an efficient way to handle some configurations at least. The idea of optimizw inputs 

for identification is nat a new thought by any meam. It is true that, in order to identify 

all of the properties of the system, you have to push on the system with potu own 

actuators ad measure the effect on th9 output v8riables. You would *&e to do it in a 

way that affords yau the best identification. In the context of flexible spacecrafts that 

has to be done carefully. I mention the problem of adjusting the controller foUowing 

identification of the plant simply because of the.fact that the design of the controller 

in the first place is such a difficult chore in tht large space structure applications. If 

we indeed identify the plant, and find it to be slightly different tha the model we have 

in the origlasl design process, then the controller h8s to be adjusted to account for the 

differences. That may not be a trivial thing to do. In our present state of knowledge 

about the design of these systems, control design seems to be very tricky. The designs 

have to be tuned jut about =ight in order to perform well. 

Finally, under adaptive control we actually heard some interesting result8 on 

stability theory yesterday. It CS still true that it is somewhat of a risky business. The 

conditions for stability generally depend on the relative magnitude of things like 

disturbances relative to knowxi inputs. It would be very well if we could use and depend 

upon adaptive controllers. Any research that can be done to clarify strbility properties 

of asaptive controllers would be helpful. 

Component operational status has to do with the fa- status of the components 

of the system. We do have a number of metha for failure detection and isclation 

(FL':'. Less work hrs been done on ae problem of reconfw the system following 

the detection of a failure. h the area of computer8 and 8-1 tran8mission networks, 

there is ¶??rite a substrnzial background of work. Under additional technology, I mention 

the fact that PDI system concepts are needed that are less sentitive to munodeled 

dpmics. It is true that, if your PDI depends upon system dyn8mics modeling, then it 

suffers from unmodeled dynamics, jut like the controller design problem suffer8 from 

unmodeled dynamics - and maybe in an even more sensitive manner. Methods for 

reliable reconfiguration are cert.inly not settled at thix point. Ln effect, we have to do 

redesign of the system using om less component once we discover a failure. The 

redesign process is not all that simple because the design process is difficult in this 

application. 

i nentioned fad: tolerant atrembbr of 1ooseh coupled COmpUteg! or& to suggest 

tbt the fault tolerant computer assemblies that we are der- with now, for the most 

part, are based upon vary- degrees of syncbronirm - very tight synchroni8m Ln some 

cases and looser in other cases. In a largo spacecraft - particularly one that has been 

assembled in space out of Y number of moduks, with each module controlled separBtely 

prior to its mssembly with the rest - y0-a nrfli have control 8ystems With a8ynchronous 

computers. You would like to be able to monitor those for their failure performance as 

well. And ffnally, a very important item is the validation of the operational software 

which is executed in the system. 

409 
The last category of uncertakti 't rmfkuration changes. This is very importm? 

for the space station, which is to 3c a .owing system. It does seem to me that we will 

need to use pre-programmed changes in the controllers to coincide with confQpr8tim 

changes. 1 suspect that we will use a low bandwidth robust controller d- the perid 

of a change. Someone observed yesterday that it would be nice to avoid *he need for 

that by us- adaptive controUers that were adequate to hand12 coafigur.tion changes. 

You certainly would have to suppress transients in the control system when a change in 

a controller is made, hat, I thixsk that is not hard to do. Another problem will be the 

isolation of the disturbance due to m~ting a new module with the edtu assembly. 

One should isolate the effect of that to the local region, if possible, to prevent the 

disturbance from propagating throughout the atmcture. 

These are some of the different types of uncertainties and the technology that we 

have to deal with. But, what about the unstable spacecraft I mentioned at the 

beginning? We still bve it h the back14 mode and have mt yet figured out what is 

wrong. Do the methodologies that we taLked about answer this question? Are they 

adequate to help us discwer what bppened? Well, I wauld ray not really. Nothiag of 

what we spoke of k direct* applic8ble to the dirgnotk of a problem We W. So, it 

would be very nice to have an additional tool, which I am calling here a spttem 

Diagnostic Monitor. A System Diagnostic Monitor has the propcfly that it monitors the 

health of the system. Not only does it monitor the health and give a go/no-go 

indication. When it is no-go, it isolates the frh. This is helpful in fisnrtag out what 

has happened. In particular, you might be able to monitor parameter values in the 

model to detect changes 8s well as monitoring for failures in components. A desirable 

property of such a monitor would be that it can be reconfigured from time to ?he so 

that it can monitor different characteristics of the system. A suggested possible 

structure for such a system monitor is shown in Figure 1. The controller can either be 

in operation or not. It does not make any difference as far as the monitor is 

concerned. The lower part of the figure shows a frihxe detection filter which hrs the 

same structure as a Kahn filter or an9 other Linear filter. In fact, if the gab matrix 

D which operates on the measurement residual were chosen to be the Xalm8n gain 

P-c 1. A Possible Structure for the System Diamostic Monitor 
matrix, this would be a Kahn filter. But it is possible to design the gain matrix in 

different ways. In particular, it is possible to design that ~ain matrix such that, for 

certain selected events, the residual can be constrained at the output to lie in a fixed 

direction. This idea was first proposed for the purpose of doing ccmponent failure 

detection, but it can A~SO be used to detect iudiyidul parameters of a .nodal that have 

been mismatched I will just give you a quick illustration of this in Figure 2. 

1 

la 

3 

- 

c) 

0 

0 

- 

I 

Figure 2. Simulation of the System Diagnostic Monitor 

(Actuator 2 Paw off at 1 Sec) 

Figure 2 shows a dynamic system modeled with this type of fUter or monitor 

designed to monitor two events: a failure is actuator mmber 2 (there are four 

actuators in this model); and a mismatch in the frequency of the first kendm mode. 

The graph is a time history of two output indicator8 for the two events. Initially, 

everything is nominal, and the outputs are indiaingpio bble from zero. But after one 

second, we simulate a failure in actuator 2. At that point, the output indicator of the 

first event bzcomes non-zero, while the output indicator of the second event, which is 

related to the frequency, still remaim zero. If the system were configured to monitor 

these two events, you could clearly say what was wrong. The other case would be if the 

value model of that first bending mode frequency were wrong, as is shown in Figure 3. 

In that case, the monitoring system is being excited right from the start. The second 

event indicator is wnzero, where a8 the first-event indicator, corresponding to the 

actuator failure, stays zero. Again in this case, tha monitor would be helpful in 

discovering what had gone wrong. I do not claim that this is a closed book. In 

particular, the effect of modem bccurrcy on this had of a monitor has not been 

resolved and is very important. We have made other fuzls for the CASC where some 

additional bending modes, bcgond those that were involved in the design of the filter, 

411 
were simulated. That thoroughty confases the monitor became of the leakage of the 

unmode1e.i dynamics through the filter to the indicator outputs. 

In conclusion, the management of uncertainty is certainly nothing new in control 

system work. BaSlCallp, controlling mceruinties is a fuxiamenttl reason for fred5ack 

in cootrollen. The large space structure problem is ewecia* sensitive to some of 

these uncertainties. It is especially susceptible to component failures. Now, whether 

we like it or not, the problem of trying to detect component. failures is coupled with the 

problems of modeling itaccuracies or model error parameter errors. I would suggest 

that if the couplixq cannot be used to our advantage, then at least it could be used in 

monitoring of the health of the system. 

EVEKT I. ACTUATOR 2 PAILIIRE 

1 

EVENT 1 A 

p. 

0 

0 

I 

I 

r I 1 I 1 I 1 

-..0 .3 .6 .9 1.2 1.5 1.8 2.1 

TfWE (SEC) 

Figure 3. Simulation of the System Diagnostic Monitor 

(First Bending Mod6 Frequency 10 Percent Low) 

412 
A. V. BALAKPISHNAN: Some Estimation a& Identification Problem in R.adoar 

Fields In this lecture, the urefdneerr of random field theory in the estimation ad 

control of large space rtrpctnres is outllned. Random fiald theory tan be wed to 

cbaracterize the deformation of &aces of antems or of other flexible itnrctnres. 

Professor BalaLrisharn has developed a filtering technique for rmdom fields which 

pmcesses dit. to obtain estimates of the shape and model parameters. Most of the 

current filtea techniques exploit time dependency and the rational power density 

spectra of the signal. Random fields often generate a nonrationrl power spectral 

density of the signal ad they do not have a time parameter. They have 2 or more 

parameters. Therefore, current filtering techniques cannot be used. Batch estimation 

is cumbersome to apply because it requires all the data to be processed at the s8me 

time. Hence, sca1ppip8 schemes are superior. In cwo-dimensions, if the data is scanned 

in certtrirr directions, the problem is converted to a one dtmensioacl problem, a~ 

Kahan filtering can be applied. Professor Balakrishnan has suggested an improved 

method. Instead of consideting one scan line at a time, consider scan lines 

sispultaneousfy. Thus, in a direction orthogoarl.to the scan line, a vector can be 

constructed consisting of states at points on thk scanring lines. Thut, the problem is 

converted into a vector filtering problen. If the spectral model sarifies certain 

sufficiency conditions, infinite-dimensicrd Katmm filters can be used. Necessary 

conditions are not known to date. However, the resultinjj Kahn filter8 can be used to 

handle two dimensional data very efficiently. 

41 3 
R. E. SKELTON: Model Error Structure ~'Ird the ImeParobility of the Control and 

Identiiication Problems This moe Iwould like to share with you my view; on some 

of tbe problems we all face in the modeling and control of large space :;tructures. 

Specifically, we will be reminded, in several different ways, that the mdeling and 

control problems are not indepedtzt, and that this fact has consequences i? the field 

of identification, a m]or subject of this workshop. 

A Simple Emcrime& 

A graphic dzmcnstratioa of the inseparability of the model- and control problem 

is presented in [l]. The essence of this expetimmnt is as folhws. Let S1 deEote a model 

of the system S, but S1 # S due to modem errors. Let S2 be another model of the 

system S, but also S2 # S. In the experiment of 111, S was represented as the first 44 

elastic modes of A flexible spacecraft, S1 WAS on?: subset of these modes (1, 2, 5, 9, 

171, and S2 WAS anothez subset of these modes (1;2, 17, 4 io}. Now, let Scl denote the 

controller which is optimal with respect to the model SL, ad let Sc2 denote the 

controller which is optimal with respect to model S2. Let the performance of these two 

controllers be evaluated when driving the 'real' ayttem S. Which model should be 

better for control design S or S2? Pig. 1 illustrates that the answer is that neither 

model is always best. Por a particular range of con*ml gains S is best, and for another 1 

range S is best. In other words, ate cannot say which model is best iadependeatly of 

some statement of the control l~w. This reinforces the notion that 'one cannot say 

what is cb good model without swing (precisely) what the model is goinn to be used for". 

1 

2 

Pigure 1. Any given model not always best for control design 

414 
llow, if there exists a model S2 for which the control der@ cannot be improved 

(Fy. 21, then the modeling and control problem can be called separable with resptct to 

that model S2. Of course, such notions involve all possible (infinite in number) models, 

rud the question makes sense only with respect to 4 given control design methodology 

(LQC in the case of Pi& I). 

Figure 2. Modem and control problems separable with respect to S2. 

Nonetheless, this inseparability of the modeling and conuol problems; holds without 

regard to the method of producing the model, whether thrt method be modeline from 

first principles (known physical hwr, P = MA, etc.), or whether tbt method be 

modeling from on-line data, coxnfnonly called the identification problem. Our 

experience at PurBue has always produced the inseparability described in Fig. 1 and 

never the separability phenomenon of Fig. 2. 

The theorg of Root Locus requires the model to be the for all control gains. 

Hence, the Boor Locus method seems useful onlp when the rnodeli- ad control 

problems are separable. Ln the orighl Root Lonu work of Bvuu [a], he tmrgfned that 

the most appropriate repreienution of the plant model remainub the same for all 

controls (as it tuma out, this is equivalent to the assumption of no error in the plant 

representation). Since modem error creates such hvoc with our textbook theories, it 

is worthwhile to look more closely at the nature of modeling errors. 

Modal Error Structure 

Considering only linear system, let us label the state of our finite dimensional 

representation of th2 plant as x and the interconnection with the remaining states as 

5. Hence, the system rcturlly obeys equation8 (1) 

R 

415 
15J 

bzt the representation available to the control designer is 

P x =AXptBu y = cx B 

(19 

AB=AtAA, BR=B+AB, CE=C+AC 

subject to pararr,eter errors in AA, AB, AC. But, rewri- (1) using these definitions 

t 

e.=S 1 A A RTAkle ~R~R(uMo = PixB 

0 

allows 5 to be eliminated in !l), yielding, 

x = A x t B u+ etW t e (x 1 + fl(u) i61 

y=c x (7) 

B RB B 1B 

RB 

(8) A i -IA 

'iXAB~AT TB e=P.x+e. i=l, ..., oo, i i B i+1 

(9) i -1 

RT T BT fi = Qiu t fitl i = 1, ..., 00, Qi = A A 

where we have fixed the cocrdinates in (1) so that CT = 0. Such coordinates olways 

416 
edst if dim y 5 dim xR. It is also possible to imagine coordinates for which ART = 0, 

CT f 0. la this case the error term of the sort (et t e + fl) appears in the output cq. 

(7) instead of the state eq. (6). But, these variations do not alter the story, so I shall 

stick to the form (6) - (91, which is now a representation of the euct modal errm 

svst2m. Comparing (2) and (6) - (71, it is clear that any model (2) which we may write 

for a Linear system always has m associated model error vector of the form e = etW t 

e !x ) + f (u) t e , which is the sum of four types of modeling errors: et(t) = errors 

which are functions of time (we usurll9 call these external and internal disturbances), 

e (x 1 = errors in model order (an integral operator P on x R1 1, f (u) = errors in model 

order (an integral operator Q on u), eA(M, AB) = errors in parameters = (AAxp + 

ABu). Note especially that due to fl(u), the imact of *he model error e = e + e + f tll 

cacnot be determinL i indePendcntlp of the control inwt u The integral operator 

fl = Qu is an explanation of the inseparability &f the modeling and contro! problem. 

Model reduction theories concern themselves with the term el+ fl. Disturbance 

accommodation techniques concern themselves with e& There is 110 control theory 

which promises the simultaneous accommodation of all four types of mode- error. 

1 

1R 1 a 

1R 

+ =ti 

More importantly, note that the phrase 'parameter error' haz no precise meaning, 

since the sabmatrix Ag -- A t AA k coordinate dependent (even though we have a 

spcci.f,Flc A in mind) and, in fact, this coordinate choice affecta &l the four model error 

terms. Therefore, each term of the mcael error vector is nonunique. The challenge 

here is this: If we szlcceed (by c2nice of coordinates or by compensation) in reducing or 

eliminating one t-lrpe uf exmr, rno: ~nr type may get worse. 

In the future, I think we mut fin8 ways to estimate the eJtLre model error vector 

and not jlist certain terms in it. Because the subject of model error estimation 

embraces i. broader class of errorsc it has no substantfrl progress yet to report. One 

attempt 131 writes the model error systems (6-9) in the form < 

;=Ax 

y = cx 

(10) 

T'I T where x = ,e ,e ,..., fT,fT ,..., y 1. (The time dependent enor e has for convenience 

been written in the form et = P (t), where the equality is only in a mean-squared sense 

and holds when the independent bark functions y !t) = (y,(t),y (t),...) form a complete 

set.) The matrix A has a specific form dictated by the agplregation of equations (6-9) to 

form (101. The model error vector is estimrted if x is, since (xi, e: f; 7's = xTC 

for some C1. The parameter A ad *.he state x can be estimated under certain 

conditions 13 1. 

(xR 1 

Y 

T 

a 

417 
The dpamical system described by 

AA A 

x = Ax + PCy - CX) 

(1 1) 

A -1 T A, *,'2 A = Q C Q(y - CX,X 

drives this Liapunov function downhill & 5 01, 

provided 

TT T C~QC(A - FC) + (A - PC) c QC + L L = o 

is satisfied for some L. 

l'he good news is that estbtors of the form (111, when wed in the above spirit, 

embrace 8 broader class of model errors. The bad news is that condition (13) is 

unverifiable even when it is satisfied (A is unknown). Paititionad parts of (13) yield 

certain specialized results, reported in [4]. Most adapthe control and estimation 

algorithms can be explained in terms of simplicatioas of problem (11-13). Adaptive 

control ignores (ei, fi, y) in the x of (10). Orthogonal filter8 [4] ignore (ei, fi) in the x 

of (10). 

A more direct approach to model error estimation was posed by Rodriguez [5,6]. 

By lumping e + fl t et into one term ex = e + f + et, ad allowing error terms also in 

the output equation, y = CxR + e 

1 11 

the model error system (6-9) can be written 

Y' 

xB = AxR + Bu + ex 

y=Cx +e 

RY 

or in operator notation (as- zero initial conditions) 

[;] y = K u + K e + e = Klu + [KzI] 

where the operhtors are defined by 

1 2x Y (15) 

418 
Hence (15) may be written 

&L A 

y = y - K1u = Ke , K = [Kz, I! 

TA T T 

e =(ex e 1 

Y (17) 

and has the minimum nom solution 

e = K*y. (18) 

The good news is that the model error estimation (18) requires no decomposition of the 

model error vector as in (6). (This is an advantage due to the nonuniqueness of the 

decomposition.) 'ihe bad -1ews is that the pseudo inverse of an integral operator E* is 

required, and this is not an on-line calculation. Some numerical examples are given in 

161. 

Identifying Structures Under Control 

We now look at the parameter identification problem as a modeling method. 

Having already reminded ourselves that the modem and control problems are not 

independent, we certainly expect that the identified model will be dependent upon the 

control inputs. Suppose that the closed loon system of P&. 3 is "identified' as G'W. 

&en, since the controller H(s) is known, the identified plant is recovered from 

knowledge of G'W and H(d, 

Now, the interestiag ob86 -4on hem is the "identi€kd plant' G'(0 is a function of the 

controller HW. Hence, if the controller H(r) is clmged, the plant as the controiler 3ecs 

& is different. The fmct that the plant look8 different with erch controller (hence with 

and without control) has not penetrated the identification research, ]udgu from a 

sparsity of papers on the subject of identification under feedback. 

I 

I 

Pigut* 3. Identifying stnrctures under control 

4 13 
Since the identified closed loop system G'(si is obviously a function of the feedback 

law H(s), let us write C'(H(s),s). Then, it is clear that the :den ification and control 

problems are separable iff CYs) h invariant under HW; that is, iff 

But this can happen only if there are no modeling errors (G'(s) is Zxsctly correct). 

Hence, the inevit;ble modeling errors force the inseparability of the modeling and 

control problems. 

The conclusion her6 is that it is verg difficult to define a meaningful identificrtiun 

experiment, one that propides improved knowledge with which to design a control kw. 

One logic81 approach to this dilemma is to reject any identif!.cotion result which does 

not produce the correct Hh). That is, apply an identification method to the CIO~C~ loop 

system treating both plant and controller as unknowns. Then reject the "ident;fied' 

plant unless the "identified" controller matches the known zovtroller This is not a 

sufficient condition for a successful identification experiment, however. The problems 

involved in such a unified identilication plan include: 

1. One identification approach my satisfy these conditions and anotiicr may not, 

even though both methods may hrve convergence *proofs* (which are eased upon 

the assumption that parameter emrs are present in the rr;odel error vector). 

2. Even sfter identification, the math-mtical techniques used to separate the 

identification of the plat ad cortroller subsystems will cffect the results rlad 

conclusions. 

3. Convergence criteric need to be established to determine when identification under 

feedback is successful. 

_.- 

Conclusions 

Dotb dynamics and control textbooks are wfkten as thowh the modern problexE 

and the control problem are separable. However, due to inevikblc ztodellng errors, 

they are not separable. Moreover, since the modeb and control problems are not 

separable, neither are the identificrtion and control problems. Beserrcb is needed to 

identify stnactures under feedback control. Otherwise, it is not clear that open loop 

identification experiments will be uaeful for proVLdian model8 €or conlr~l design. EL. A. 

Frosch put it well this morning in hi8 ACC plenary talk when he said, "We are not me 

of what we are doing when we abstract the rerl world". 

[!] B. E. Skelton, J. Davis, and Y. Yousuff. "A Compute; Aided Design for Suboptimal 

LQG Contro?", IFAC S~mposiurn-on Computer-Aided Ds.sW, West Lafayctte, IN., 

Sept. 1982. 

[2] W. 8. Evans, "Graphical Analysis of Control Systems", AlEE Trm Part H 67, 1948, 

pp. 542-551. 

4 20 
[31 B. E. Skelton, Course Notc I AAB 664, Pdue University, West Lafapette, IN., 1983. 

(41 B. E. Skelton, "Adaptive Orthogonal Filters for Compensation of Model Errors in 

Matrix Second Order Systems", J. Guid. & Control, VoZ. 4, I~L. 2, March, 1961. 

[SI G. Rodriguez, "Model Brror E: :irnation for Large Space Systems", JACC, 19110, San 

Francisco. 

[6] C. Rodriguez, "A Function Space Approach to Smoothing with Applications to 

Model Error Estimation for Flexible Spacecraft Control', IEEE CDC, 1981. 

421 
R. F. CABLISLB: Future Research TechnoIoror Directions for Space State What we 

are attempting to do, using the Available literature dnli expertise, is to get a soiutioa 

that can optimize the station by further application of controls. In the world of 

spacecraft systems, when the power subsystem fellows tell me to bring on-line 

improvements in power, I tcU them to forgst it. Thiz is an expense that I do not tBinsc 

t\e system can afford. We will go back to the user and reduce the power requirements 

by applying other techniques. We are dealing with trades in the integration of 

subsystems. Thus, when a control system aesigner says I Cannot solve your problem. I 

go back 50 structures. We just have to beef it up and fly it anyway. When I cannot 

afford the structure and the weight to do that, then I go back to the cor trol people and 

say how can YOU help me. That h the world tht We airc dealing with. 

On the space station, AS you how, we have A planned approval for the program. 

We are all ~nxio~lg waiting for the approved budget this year. If successful this year, 

we are looking for A flight date approximately in the 1992 time frame. In the world of 

research and technology and advanced development and des@, that means we can 

expedite some promisizq hings in the piseke. This will be done by increasing funding 

for certain technologies and bringing them to itate of mtwity and readiness that 

perhaps can improve the initial space station design. Let me share with you some of 

the things that I see in the design profile. 

First, we have a ground de that say3 the space station is going to be an 

evolutionary design. That means it is going to grow and change with time. There will 

be block change= in configuration, and maybe block changes in perfomncz. Therefore, 

it behooves us to do the best we can in the initial design, so that we can evolve 

gracefully and economically to meet those challenges. If we lock ourselves in with a 

design that does not hve flexibility and growth potential, then the control system may 

be one of the things that tourids the extent of evolution that we could hope to achieve. 

AU of the sub-systems are trying to bring robustness, flexibility, and evolution of 

growth, so that they itre DCt the ones to bound the potential for space station 

evolution. Another interesting thing in space, as opposed :.o A~I-OIUU~~CS, is that the 

spacc station is essentially the first time (really the shuttle was) we can repair 

on-orbit. We are asking for 20 years continuous life with on-orbit mrintearnce. That 

gets into the practical world of hardware and failure analyais. It also gets into the 

question of how we accept fahurer in tke most ecouomical way, and whether we want 

to have built-in redundancy and fault tolerance or repair on-orbit. Repairing on-orbit 

means we need test2 ad analysis in order to locate failures, and a means of 

accomplis- the repairs. Thus, we are loow at extremely interest- challenges 

with the space station. Bssentially, we have 5 years to mature promising technolop 

and applications. But in an evolut.iowry senre, promising rerearch that could be made 

into a block change is also an opportunity. Overall, it is a world of opportunity that I do 

not think I have ever seen before. Thus, I want to illrstrate just 8 little more what I 

think the problem is, and review with you whmt I understund of where NASA has gone 

with their fundirq that is applicable to space station. Pin~Uy, want to leave with vou 

our definition of what we are looking for that, is in the spirit of .ppomm.ities for future 

research. 

*The 8pCAker refers to slides used in the oral presentation that were not ~vail~hle lor 

inclusion ih this wtfttcn report; however, the major points are evident even without the 

Slides. 

422 
If this were our initial space station-and it is not, it is just typical-each one of 

these would be a payload f'Sght for the shuttle. So you place one of these ia orbit ad 

leave it there, and then you bring the shuttle up and bring another module to it. Now, I 

think this is fairly complex for the initial opera- configuration, but let us just armme 

that for now and walk through the control problems. You Lave to be able to ttab*&e 

something like this. You h~ve to be eble to hold it well enough in attitude SO you can 

mate to it, and mate to it with only the rerolution of what the shuttle can do, or what 

this can do. As that changes, there are obviourly major shifts in m~ss diftribution. 

Maybe two modules hooked together can still be considered a rigid body. But, as it 

grows in size, as the whole stmcture gets Ilrger, then you begin to wonder whrt the 

relative rigidity of these joints is. The initial space st~tion has a requirement for 75 kw 

useful on-board power. lhat is A solar array of about 20,000 square feet, half an acre 

and about 20,000 pounds. So, it is not trivial. There are significant distributed 

low-frequency appendages on this structure. This is the radiator, which damps the 

waste heat. Because of the efficiency of a closed cycle like this, then is almost 75 kw 

being radiated out. So, this is not A trivi~l sppendage either. 

This is another illustration of a larger space station. Now lGok what we have done 

to ourselves. If these were rigid bodies with cl~ssical control, and then romehow this 

whole matrix grew to this configuration with these appendages, then I wonder if that 

original classical control design would stand some graceful economic evolution to 

control that cluster as it is shown there. I would say that is reasonably raalistic as to 

what the evolution would be. Gcing A bit further, this is a collection of conceptPrl 

drawings of what it my grow to eventually. I cannot define anyone of those as being 

realistic, but an9 of them could be. We would like to drive in that direction. The 

challenge to you is that control systems of the oxi@ml configuration should not 

constrain us, if that is 8 reasonable rewst. 1 do not know whether it is or Pat. 

Let me go back rhea to a very quick review of where we have been in NASA 

funding for control of large flexibk structures. ID rbout 1978, we started worLing on a 

large structures program. Initially, we homed it on antennas. There are two antenna 

configurations: the hoop column and the map rib, representing Harris Corporation .ad 

Lockheed designs. In the control %rei, we dealt with the problem of maintainhqg the 

relationship between critical points of the feed and the reflector. We also dealt with 

the probEem of srziace coatrol. This program started as a structures program. As we 

got into the progism, we recotmimd *ht we were not der- with o system tml~sr we 

addressee iae contrn! ;:cLiem and the BP characteristics. We had a grouxtrl test 

program that built up fuil-size segment8 of there antennas. Here is a section of four 

gores of the hcop column. This shows it built up on rhe around and spreading the mesh. 

These are srdall reflectors that are nut on the mesh for hmg up the mcrh. They can 

also be uszd to close the loop around to measure and control the surface conditions. 

This is P. full scale model of the wrap rib, with the mesh deployed. The ribs cannot 

support themselves in one G. They are hung from spring rpsemblies attached to tbe 

ceiling and to the floor. We hoped when we started that we could measure some 

dynamic ChArACtCriStiCS. Ir: retrospect, 1 hove to say that it was succersful research 

since we learned that we failed. We could not get significant resolution of any 

importance to measure dynamics. We did not spend enough money or time to get the 

precision of realistic measurenients for dynamics. This led ua to conclude that since we 

did not know what to do on tht ground anymore, we were soing to have to fly. We 

proposed a flight program planning exercise in-house. We recognized the expense in 

Undertaw such ambitious planning, but we felt it had to be done. The 

423 
plan grew to the point that we killed it with economics. W2 ended up with an in-house 

esthate of approximately 350 million dollars to nm perhaps a flight that would 

only learn something about control dymmicr. Pm *-he system no-dd not stuxl that 

mucn expense, we decided try to get a paylord to share expenses with us. We couldn't 

find 8 payload wllliry to take the risk of committing to an nnproven stmcture. Stace 

nobody wanted to go with us, ud we could not afford it alme, we stopped that program 

and faced the fact *that we do not know how to go full size in flight. We are still 

working on the problem. I think, and several others agree, that we are developixq 

powerful technology. We want to achieve that plateau of technology where we CIS 

handle control of low frqacncy interactive modes. We are now pka an exhaustive 

ground test on a much suuller article. The westion is how do we proceed to validate 

that design. Can we get sufficient meamrements on the gromd, with inatnuaentation 

and fixturing errom to have the confidence to fly full scale, kqer vehicles? Or do we 

need a flight test to validate the ground test? If we have to have a mht test, how do 

we scale? If we use a small grooDd sample and fly that ume sample, to get the 

relationship between the on-orbit ad gf~~~d. characteristics, do we gain enough 

confiience to fly fa-size structure:? hose are the things thrt we are rea* worrying 

The real configuration has nat been decided. The eight contlrctorr that have been 

worLing with us all recome that controllability is a critical isme fer confidence in the 

origin~l design. Thit configuration, called thz delta, has a ri,gid structure. It L UI 

inherently stiff design, with the modules bed up at the ends of the of a 

pyramid. It is the most rigid design. It is also probabiy the me with the least growth 

potential. These two are quite similar, with the modales gathered in 8 fairlp stiff 

arrangement and with a flexible appendage. This design offers the 0pportPrrity aad 

advantage that the sbttle can come in am3 berth to it away from the solar arrays; but 

it can, if it grows in evolution, significant!.! charge and complicate the control 

problem. The question that I pore is: how can you help IU with the tough decisions that 

we have to make in *he next five yeus? wh.t is the or&inrl cont=o1 configuration that 

can meet most of the desirable features that I mentioned earller? Our problem mry be 

summarized as evolutionary growth, size and complexity, possible need for on-orbit 

test, and c long-life r-quirement. We have the opportunity for solving it with control 

technology or wirh structures technology or by some combirutioc of each. No matter 

what that combination is, we still have chalhge of how the hithi design shonld give US 

the most options. 

about. h the CUtteilt phw Of the Space there SeVCd COd-tim. 

424 
H. A. REDIBSS: The Role of BltPertments in the Develmment of Control Technolo=* 

Experiments, both h Irmratories on the gruund and flight experiments on-orbit, an 

necessary in the development end validation of controls technology for large flexible 

space structures. I was pleased to see that it was recognized in thir workshop, and that 

there were so many papers that touched on experiments. I have not prepared a formal 

paper on this subject, .od for the most part I am ringing to the chcir. However, there 

are three points that I would like to discuss: 

Why experiments are important for developing the technoioE 

0 The need for a coordinated program to validate techniques from analysis to 

simple experiments to complex experiments. 

The need for a coopentive pmgr8m among govexnment, UdVersity uui 

induttry to m~ke the most effecthe use of experimental data. 

Why Broeriments are Inrso runt. We an belrinnidp a new en with hrge space systems 

that will tu our cmt method. of developing spacecraft- We an coatiderirb space 

systems that are sufficiently Irqe that they must be developed in space in sections ad 

assembled. The systems will be designed for the aero-g vacuum envltopmc1Dt of space, 

and we will not be able to fully assemble .ad test the entire system on the ground 

before deployment. We will not be able to do complete grad vibntion testing. 

Ground vibration testing has been an important upect of developing control 

systems for aerospace vehicles nhen there are significant stlOCtures and controls 

interactions. We curmntly do the best job we can in tnodellng the stmctunl dyncmics 

of a vehicle, but we have 8n opportnnity to fine-trme oot madeis d, in tUrXA, the 

control systems with grod vibration data. 

If we cannot test the complete space symm, we ueed to have a higher degree of 

confidence in our modeling and design methods than we currently have. We may have 

to do on-orbit systems iden+Xicatloa of the complete space system, or even as it is 

being constmctcd in the case of a space sutlon, before the attitode control system can 

be operated. We must have confidence in our systems identification method8 before we 

get to operational sys-.ems. 

There are other factors thrt further complicate the problem of having confidence 

in our an~lysis and der- methods. Three significant met are: 

1. Structural modes will be overlapping the required controller bandwidth. 

2. Uncertainties in predicting damping. 

3. Introduction of new composite arterials. 

Figure 1 points out the trend we expect to see in the control ad structure 

Lnterrctions of future spacecraft, a8 they get much larger and hence more flexible. In 

today's spacecraft, the rtmctural modes are sufficiently separated from the controller 

bandwidth tht relatively simple filterin# L adeqarite to deal with the interaction. In 

future flexible spacecraft, the structural modes are expected to overlap the required 

*Several of the charts used by H. A. Rediess are from a NASA presentation on Can :?I 

of Flexible Structures ad were used with NASA permission. 

4 25 
cotxroller bandwidth. Requirements for rapid slewing, precise pointing, and very 

accurate shape control will force the designer to coder the control rad structure 

interactions. Active stmctural mode suppression -9 be necessary to achieve effective 

and affordable designs. We need a higher degree of confidence in our structural 

modeling and control design technic, for active controls. 

Figure 1. Trends in spacecraft control and structures interactions 

(Prom R. A. Russell of NASA HeAdqrurters). 

The second factor that complicates the problem is the mcertainty in predicfJng the 

inherent damping of space structures. ProZessor Vandcr Velde mentimed earlier that 

damp- characteristics are probably the most difficult to predict and yet are very 

important in control design problems. Damp- charactertStic8 of complicated 

structures, such as a large tetrahedral mass auteap. with a long offset feed, can be 

very different in the space envircmment than on the groud. Figure 3 shows 

experiments1 data on simple-joint damping obtained by Proferror Meul of South 

Hampton University. Test results are shown for a no-joint specimen .ad for a joint 

specimen in air and in a vacuum. Note that joint damping in vacuum is about me-third 

that in air. These data are for one simple type of joint, for one type of material 

(aluminum), and for the fundamental vibration mode in osclltrting tension and 

compression. Considering the multitude of types of joints, matel-hls, ad vibration 

modes, one can see the difficulty in predictinq damping of large comlp'ex space 

structures. There Ls a need for A series of experiments on small and full scale 

specimens. If possible, these experimenu shonld be conducted in the space environment 

to develop an adequate data base and prediction techniques. 

The third factor is due to the introduction of new composite materials, such AS 

meal matrix composites (MMC). MMC have several characteristics which make them 

particularly well suited for spacecraft applications. They have a high strength to 

4 26 
Figure 2. Joint damping &t. for oscillating tensidcompreasion tests 

(Prom Prof - Mead of South Hampton University). 

weight ratio, excellent dimensional stability, and low coefficients of expansion. 

Cert.in important characteristics, such as stiffness and damping, can be chrnged ad 

triiored by the way the MMC is made. The effect of material characteristics could 

have on spacecraft ,oerfornunce is illustrated in Figure 3. It depicts the elements of a 

study being conducted by 98 Tearon for the Naval Ser Systems Comm~nd. The 

spacecraft is a hypothetical ~-eill.nce system that is required to Lp.ILcover for 

survivability and yet sect stringent pointing and shape control requirements evenwhile 

maneuvering. The chart in the center illustrates the effect of one material parameter, 

stiffness, on line-of-right error at various maneuver acceleration levels. It is necessary 

to characterize there new materials in laboratory tests in order to assess the effects on 

spacecraft perfonnmce. That would, in turn, improve the ability to predict the 

structural dynamics characteristics. Control and structural interactions will become 

damping characteristics is imponant for effective conmller design. Chracterization 

of MBlC properties could lead to optimization of new materials, structures, and cGntrols 

for flexible space structures. These and other factors will nq.uire a V~I. :y of 

experiments in 1aboratories in space to develop the technology with confidence. 

Need for a Coordinated Program The second major point I wanted to make was the 

need for a well coordiarred program to validate techniques from ArAlysis to simple 

experiments to complex experiments. The final report of the NASA Space Systems 

Technology Advisory Committee, ad hoc Subcommittee on Controls/Structures 

kltCfACtiOIlS, dated Tune 8, 1983, strongly supported the need for ground and on-orbit 

testing for developing control/rtructures interaction (CSn technologp. Figure 4 is from 

that report and shows the ad hoc Subcommittee's recommendation for a coordinated 

more SignifiCAZt in fnture large space Stmct-S. The abut9 to predict ACClXAtely the 

427 


ad the Air Force develop a national cooperative data base for these major ground and 

on-orbit experiments. The coatrolr/strrrctures technical commmity - includh 

universities, industry, ad government laboratories - should be involved in establishing 

the experiment ad dak requiremenu. A data base should be established so that many 

of the different techniques for modeliq and controm of itNcture& could be tested 

and validated. There must bve been at least 50 different one8 discussed at thh 

workshop alone. The data base should be devised in such a manner that it can be easily 

accessed and used without extra progrsmming by each user. It should be available at a 

low enough cost that universities could use the &k in relatively sm4U research grants. 

If these requirements are conridered and incMed at the outset. there should be 

minimll impact on cost and schedule of the programs. 

I Ill rfZ3?3Et I 

Figure 6. Test model descriptions for NASA's control of flexible 

structures propsm (Prom R. H. Russell of NASA Headquarters). 

Et will not be 1possibLe to rem there experiments with each new analysis technique 

or cxm! concept. It also gets very expensive to re-amlyxe elpartmanu1 data with a 

scceak 0. third researcher, i€ the &ta are not alreuly k a 8Oob user format. With 

cppmpri :-:e pre-planning and setting trp of a rnulti-user data base, the cost for 

r.d&iourl andysir by alternate researchers should be qnite small. 

In t:onclusion, my three min points were thrt: ground ad flight test experiment8 

will be necesuarp to have bigh confidence in our systems identification and 

controldstructures interaction desipa techniques; a well coordinated program L needed 

that hOlVe8 atulysis, ground testa ad flight testingi and, l.st!y, an interagency 

cooperative data base should be set up for all major ground and €light test programs to 

achieve the most effective use of the very rare experimental data. I would like to 

coinmhod NASA on constructing an excellent program plan for developu and 

vdidatiug the techno&gy for control of flexible rtnretures a8 presented in Mr. Ru18cU's 

paper. 

BDITOq'S NOTE: Thir concMe8 the formal remarks made by the panel members. 

There follows 4 discusion period between the panel members and audience participants. 

4 30 
8. V. RAMNATH: I would like to share with 90u some of my experiences, Fnterests, and 

aspirations in the area of asymptotic anrlysk. Asymptotic methods have been wed 

celestial mechacics, mecbnics, etc. It is my feeling that asymptotic metha have not 

been used as well as the9 rhoald have been in the areas of large space structures, 

structural dynamics, and control. There bve been a rprinklhg of papen using 

perturbation methods and asymptotic amlysis in the last ten or twenty yean. Professor 

Lions’ paper highlights some of the issuer. We have been working in this area for 

sometime. I would like to tell 9012 about our experience. 

With grert =Cess in m8ny branches of the phy8iC.l IcieLLCCI, XXUthtXtl8tiC8, phydC& 

Asymptotic hbsis. We all agree that real physical problems arc very hard to pose 

exactly, mud, even if the9 can be pored, they will result in highl9 POPUDCar coupled 

equations, for which closed €om analytical solutions are hard to obtain. For example, 

large flexible space stnrctates are described by partial differential eptions with 

variable parameters. Bven worse, these panmeterr may be nodhear functions. For 

such problems, it is very hard to obtain closed f0.m solutions. So what we do is try to 

get some kind of an approxix~tion. Hirtorically, approdnutionr have been tried as 

approximating the problems, ad we have used it fairly successfully. 

long a8 people have been a-bahg real problem. AIpmpfotic aXl&ly8b OOc Way of 

I would like to define asymptotic ~ufysis as a study of applied mathematical 

systems in limiting cares. There seem to be a little bit of confusion in the literature 

about the convergence ad the asymptoticity of eqprtionr. These two wordr are wet! 

interchangeably. We can discuss the difference in the context of a swle example. 

Consider a slowly decaying exponential function exp (-at), where L is a small 

parameter. The series representation of this function is convergent within an infinite 

radius of convergence. But, in order to get the approximation by uPIlcatiag the series, 

one ius to be careful. If the argument is very largec while the series converges, it 

converges very, very slowiy. Thus, if one taker a few terms to represent the series, me 

gets v~ry poor results. Thus, this is an eumple of a ccwergent series that is not 

asymptotic in certain domains. Also, there are other function8 which diverge 

everywhere, but for certain valuer are known to be asymptotic. 

Asymptotic analysis is considered a8 an art ami science by practitlonen. The 

science is what mathematicians worry about. For example, they worry about how to 

calcuhte these faxlctioar, etc. On the other had, engineer8 are faced with UuknOWn 

parameters, so that there is some kid of wement requjred. Thb is the art aide of 

the problem. 

Asymptotic amlyrb csn be used in simplifying complicated problem. The 

principle of simplification states that in the asymptotic limit the system simplifies. 

Professor Liont talks about the need for rnom asymptotic onrlpsir in his paper, mainly 

because it lead8 to simple dercri@tims of very complicated ryrtem8. 

The principle of miainnrm 8impUficatFon or maximum balance rtrter that there 

exis28 a representation of the problem which ratah mort of the prominent features of 

the problem, while it is simple enough to obtnia a closed form solution to it. The real 

problem may not be amenable to a clorod form solution. There two prfnciples are not 

theorem but give a guideline for approrimrting problem which I have found very 

tueful. 

431 
We have Applied this to model the gace rhuttle. It har A bO-rt~te differential 

equation dercrib~q~ rigid body motion, fledble body motion, control actuator dynadcr, 

ctc. The hierarchical rtnrcture coold be obtained becaw we could differentiate by 

time scaling. Thio approach is called the multiple scale appro~ch. 

Multiple Scale Awroac4 This approacb. consists of taw an independent vrrhble in 

the differential equation and extandinp, it8 domain into a @ace of higher bsmsruion. 

When the problem is rest~ted in this new 8pacer the problem 8ek rimplitied 

considerably. 

For eumple, a dynamic phaaotneaon which exhibits qualitatively different 

behavior, such as fast xnotion combined with rlow motion, can be converted into two 

problem. The independent variable, tirne, ni?l have two rcale8, one for the fast motion 

and the other for the slow motion. With there two new fobepeadent vatiabler, two 

partial differential eqpatlonr can be dtten h the p&ce of one differential eqtutbn. 

Phy8iCAw~ it Of htUm8t to thlnL Of tb f0- Cnrge-t. hUgh8 them 

observer8 with clock8 obrerdq a phapomeasn. Back clock count8 time oil a different 

be different from the othen. The obrerrer with the fat clock obrerper tbs fart 

chnget of the dynamics whcle the observer with the rlovr clock obremer only 8lou 

together to get a composite type of rapmrsntrtion. Now, ow 801Ve8 thi8 new problem 

in higher dimensions rrymptotically and reqphr the wlution to match wi?h the 80hth 

of the original problem along certain he8 wMh are cabd trajectorier. Thor, we are 

able to get back to the origin81 problem. 

8CAk. Thus, each obrerver Obren788 e ceruin A8peCU Of phSnOmmol0 nhich nUy 

ChAge8. ThU, it h pO88ibh t0 reparat8 th4 ftWuOQ8 aod than Cod- ai0 

AmUcatioru_. The multiple 8C.k approach war rued to rolve the problem of 

electrortaticai.€y controlling the membrane mirror ob a apace antenna which would be 

used in radio artnnromy expeitmsau. A meznbrane &e of m&r is preferable bscm~88 

it ir light and can be folded. Thar, it ia convenient to crny in rprce. Tha rbpe of thi8 

membrane it to be controlled by an etctrort~tic field #enerrtub by elecuobes. Bosn 

though the electroder we finite, the electrostatic field fr contirnrotu. Thus, A 

contboua membrane is to be coatrolled by a bircontktUOU8 force. If the rhrpe of the 

membrane it a perfect circle, the mode rhrpsr are #ive by Bema1 .mctioDI. But when 

eSper&nce tell8 UI that thir fr whom rrt comer into pby. The independent variable can 

Beriel functions can be obtained, whore argument8 are there nodhear functions. Thw, 

one obtains an asymptotic 80lUtbn for the mode rhaper af the membrane. 

the 8hApe But, out 

be 8put hr0 two VAdbb8 tht are nonlinear fUZlCthU. Then, A 8OlUtiOn h tern Of 

M)t CirculAr, 80lVbg fOt th8 mode 8hApe8 b Vr?- difficult. 

432 
"Le payoff of thir asymptotic solution b that we can understand the dylumicr of 

the model asymptotically. "he error is not aero, but it it not large. We are able to 

think of a feedback sohtion for the control problem The 8OlUtions have a Bersel 

function of A different Liad with the feedback gain embedded in ita arguments. Also, 

we can study its subility rad controKabUty oud come up with a conrtructive procedure 

to calculate feedback @ab and rtrbLUty. 

Conchwiom: It is my fee- tht more and more studies would indicate the usefulness 

of approximations using argmptotic methods for problem8 which are very difficult to 

solve analytically. The advantages ~f ssymptotic methodr are that one obufnt 

anrlytical or quasi-analytical solutions to these problems. Thir helps in studying the 

properties of the system, mch as 8tabWty ad controllability. Also, the methods can be 

used to solve control problems. Even though the erron are not aero, they are rather 

small compared to other methods. So I want to reiterate Professor Lions' suggestion of 

looking at asymptotic solutions of distributed parameter systemr. 

QUESTION: There are a range of opinions on Space-Station controls, some of which say 

that it k just another Skylab. If that L true, why can't we merely use th. control hwr 

from that program? 

RBMNATH: I want to paraphrase what R. A. b08Ch said, 'We are not ran of what we 

are doing when we abstract the real world.' When we develop models for these things, 

we cannot know the importance of modem error8 until we fly. T).erein lies the rub. It 

depends oa the performance requirement8 of the Space Station. If the performcnce 

requirement8 are benign, we can u8e Skylpb technology. If you are as- for 

complicated requirementt, that require control system8 bandwidth8 less than the order 

of hours, then we should worry about flexibility. 

CARLISLE: I think it it reasonable to assume that experiments that require very 

precise pointing would have some control system just as they did in Skylab. What 

concerns me the most L that we have to be rble to hold a hrgs station stable enough to 

rendezvous and dock with birtuaLt)r aero energy transfer. Then we want to be able to 

evolve. The evolution L 'to be determined'. We jwt do not kww what ita nature L. 

The question is: what can we put bn the first one as A control ryrtem architecture that 

will give us flexibility to evolve in the future? I think if we 80 with Skylab technology, 

the control system is goiiy to bound the evolutionary growth prematurr'v. 

RBDiBSS: I think that the availrble controls technology hac got to be self-fiplf(llinn in 

terms of the type of space station one might have. For example, if we relv on the 

technology that presently exists, which was adequate for Skylab, we are gou to be 

limited to a type of space station that k compatible with that or suffer very expemive 

modifications in space at some later time when we want to expand that caprbuty. 

.'KELTON: It WAS mentioned earlier that we cannot afford to do full scale testing in 

space. What evidence do we have that we can afford not to -- in other wozds, it my 

turn out to be 2esr e%'pcxuiVe in the long run. 

CARLISLB: The problem is: if we have to have full scale testing in space to validate 

new technology, and the system says we cannot afford it, then we are bound to live with 

something less than the potential tech.ology that control analyrk &nd research shows i8 

very promiring. 

433 
QUESTION: In various meetings in which I have participated recently, I hrve sensed an 

opinion that we have had a lot of effort on theoretical development directed toward 

large space structures. There reems to be an unjustified hnytti . with theoretical 

developments, and a concern that researchers are going oPf in *,ne wild blue yonder 

trying to address problems that may not exist. The impression is that it is more 

appropriate to develop an extensive experimental plan and defer further ,~mphasis 031 

theoretical activities. I would like to get the panel to comment ca this. 

CARLISLE: I think research is verj Fromising. However, in the world of Frogram 

management, unless we validate the research, it witi never get used. The question is, 

how do you get your kicks? In pure research, you can get your kicb by doins; analysis 

and identifying a promising application. Then, you get bored with it and wvf to take 

on another new idea. If nobody ebe picb it up, develops feasibility, a.ui validates your 

first idea, that idea will never get usad. I think there is a necessity far all phases of 

research. I think it is terrific to do fundrmenul work. X think it more imporant that 

research be develcped to the point that there is enough confidence in it to be used. 

That is A tough decision. I have had some experience with pmgrrm management. 1 can 

tell you that, when I was nmnitq a major program, I was not interested in rsssarcb 

unless I had a critical problem. When I ha4 a critical problem, I needed all tue help 1 

could get. I would take as many chancer as I had to, to meet my obligation s,c program 

manager. Unlerq I had that critical need, I did not have any incentive to add complexity 

in maturing an experimental technology. A program mmager has no hcent've to do 

that. It has to be done in the pre-development phase. Thb relat=.r to my COIACCIQ m 

Space Station. We have five years to mature pmmfsing ideas. I am not sa*&f i that 

we are pa7ing enough attention to sort out which are the u ost prombizrg view ... With 

an evolutionary approach to the Space Station, we should not gc with last year's 

~echnology, we should 80 with next year's technology, with a plan to evolvz to the 

future -- do something even better. I feel quite strongly that there is a necersity for 

several views and that we need to coordinate all those in some complementary way. I 

also believe that there is a xsonrlity clash between the types of people with original 

ideas and those required to carry oat the ~pplicatiom. We have to do something as a 

community to improve the communicraon and cooperation, so that we can sort out the 

really promising idear ad get them uaed. The more new ideas get used, the more the 

whole discipline will proceed. Soon the rest of the Space Station will be doing things 

with structures that we never did before, because *be control system technology is 

needed. I do not think we are really up to rpeed now in that area, and I would like to 

encourage it. 

QUESTION: As program manager of a Space Station proBram, would you be satisfied to 

incorporate methodology in control systems or operations Ut you tu6 not 

demonstrated in some kind of space experiment? 

CARLISLE: It depends. T&t is a point I Vied to touch on earlier today. If the 

technology is too expensive to mrture full-scale in orbit, theq the problem is to gain 

confidence to use it the first time full rcale. We flew the shuttle, end we did not do as 

much flight testirq on the shuttle as we do on aircraft. When the first Shuttle flew, it 

WAS a brand new thing in space. We had enough maturity, confidence, and experience, 

so that was a good decision. 1 also think that the decision to use new technology varies 

with the experience of the decision maker. You think of your own field. If somebody 

suggests a new technology to you, you get comfortable based on your own expsrience. 

If somebody aiiggests something in a held foreign to you, and you are a good researcht: 

or engineer, you Will have all kinds of doubts, without the experience to back them up, 

and you wiil never buy the proposal. 

434 
SKELTc"V_: I would just like to ake a point about the problem with waiting until there 

is a need for a new method tefor.. we start, worrying about it. We dc uot have a 

continuity that sewes as a baris for & continuow flcw of ideas into the sys~em. Om of 

the things LhaL striker me a8 odd is at in thir country, snd in the western world, there 

is no Lmstit9te of significant rize that focuses on control and system science irraes, PA 

there is ~il the Soviet Union. The Soviets have more trun a dosea iust~tutes, spotuored 

by the Academy of Sciences, each one employing 3 to 4 tbGuand eogineert with 

advanced degrees. They have the resppoasibility to carry 012 research all the way from 

thc very theoretical to the applied ad experimental in -stem topics such 88 control, 

identification, artificial intelligence, ad all of their intar*ctioxu. It seem to me, if 

you look at our organization charts, we find guidmce ad control bloc& sosewhere 

along the line. That narrow vlew of it my have covered the subject 30 yews ago. I do 

not think it does today nor tt will tomorrow. I am wondering, where in the wester-r 

world it should be covered, whether it should in the Air Force, in NASA, or somewhere 

else. I think evcrybody I, ellpectu the other guy to do it. Pme ir ILO consirtent 

substantial support of the size ad focus required. to addr-ss *Le i. me8 tbut we deal 

with. When we launched the Sc sp V vehicle, we had interr Ation problem between 

propul;ion systems and control. The?, we had hteraction problems between the 

structural dynamics rad controls. We can name r number of instances and eumgler 

where we defined the control concept too naowl.; We ,iought of it a8 a little black 

box to be added tu the system rfter eveim ehe wa3 3one. I think the cuntroh 

community has a much brooder view today than it did 10 Yean ago. However, in tenus 

of the organization8 that sponsor rerecrch, there i8 still only a very 8-U se~ment that 

understands this brcrder view. 

SKBLTON: One concern is this. We should determine whether or not to create an 

institute on systems science or control syst :ms itrues, or whatever you waat to crU k. 

There my be good reasons for not creating such an iastitute, but there should be a 

study to decide. The decision not to do it should be a conmiour effc-t. Let us not lose 

the ,ethnological leadership by default. 

VANDBB VELDB: I would like to comment on your concern about combining conti01 

theory with other discipliner. We see &is pbnomerum at MIT (I asrurne you do rho at 

Purdue): the graosrte 8tJdents today very commoal.)i 8tudy quite 8 good deal in control 

and structiual dynamics. This interest has been rthuhted by the problem of hrgc 

spacecraft control, which they look upon as an interesting challclge. It was very 

uncommon a number of years ago for out student8 to sttdy those two areas veq' much. 

CABLISLB: I would like to comment that in the &st, four or five years, within the 

sphere of NASA we have made a significant change. Funding for controls has gone up 

by a factor of 10 Gr mc;e. We have thia visibility that wa8 talked about -- the 

aecessity of addressing the problem of strxtures and control dynamics interactions. 

We holve visibility in the badget md are loow for furthbr expansion. I think that we 

have a thing going now because of this recognition. This is a powerful tool and can 

make a big difference b Space Station. We should try to pay rttrqtior, to it and take 

advant, qe of that exposure. 

-- SKBLTGN: I agree there ir progress. 

4 35 
BALAKRISHNAN: I just wafri to make a comment on an earlier statement mahtainiq~ 

that structures students have discovered control theory. Actually, the number of 

students going into controls is much more .;mailer than it used to be. At UCLA, many 

more are going into communications. I have a question regarding the evolutiomry 

design concept. Xs this somethilg new for NASA, or bs NASA had any expc .ience doirr 

something like that before? What kind of comments from university, industry and 

scientific coaamuOities are taken before such decisions are de? Is there any formal 

mechani-m, or does somebody just make the decision? 

CARLISLE: I think the opportunity for permanent presence in space just spontaneonrk, 

developed the evolutionary idea among those of us who have worked on it. The only 

way to achtwe permanent presence is by service and maintenance. The Shuttle @vet us 

the opportunity to evolve. I think we are in the infancy of even undent.nding what we 

mean when we say that. I do not know of any formal wa9 that we consulted with 

anybody. 1% was just handled spo~taneously, with a very preliminary plan which could go 

in any direction. The system engineering ad integration task on the Space Station has 

been planned to be done within the agency with contractor support. Though the 

program is very new, we hwe 3een planning technology Tor a couple of years. But we 

are really in an infancy state. I meant what I said. I do not think we know what we 

mean when we say evolutionary design, except th8t we know we do not want to be 

bode4 with last year's ;echnologp. We want to open it up to take advantage of the 

Shuttle and logistics end have it grow. 

QUESTION: Is the project pretty definite? 

CARLISLE: It is definite that we did not yet win a vote in Congress, but we did win the 

vote of the President. In bb speech of last January, it was one of the thbgr that he 

quoted as bc'q one of the future commitment in space for thir country. 

-- BBDIESS: I would like to go back to the point R. E. Skelton raised about the institute in 

control. In part, it relates to my experience within NASA, particularly in the controls 

area. I think there is a long term fnab.ment.l outlock within NASA of not really having 

treated controls as a science and discipline. It has been treated more as an engineerhg 

area. It takes quite a bit of effort for an orguriaatloa like NASA to turn that am. 

There are times where we have received rabttanthl &reare8 Lo support in the area. It 

is due largely to program managers mch as R. S. Carlizle, who udantands cokmb and 

its -361 difficulties and conccrn8, and who provides perronrl mpport. When WjOr 

projects come along that clearly require some new technoloSg in controls, there is tin 

OppOrtUdty to increase the SUppOrt for developing the teChOlOg)t, PArtiCUhrlY tOWrfd8 

the mote costly end of it. I think there still is a difficulty in really recognizing controls 

AS an equal disciptinrry area with some 2 the more well established disciphes, which 

have had decades of support with NASA. There are a number of people who are aware 

of that within NASA. It k just a duficult institutional thing to t&n around. 

QUESTION: Do you think it ** a reflection of th, t evaluation of the relative 

importance of the rubjcct? 

BBDIESS: No, I think :.he problem i8 that it is often treated as an engineering problem 

which can be solved thnugh a good engineelring approach-as compared with developing 

the fundamental concept), methods, and took a8 a discipline, such as we do in 

aercdynamics, matetirk and other disciplines. 

4 36 
LANDER VBLDE: La aeronautics, we are all frrmlirr with the term control-conf*e 

aircraft. In fict, N4SA has done work in-house and has sponsored a certain amouat of 

work in that area. You never hear the term control-configsred spacecraft, even though 

you mentioned the requirement this morning for integrated des-. It seems as if there 

may be perhaps more to be gained in the lam6 spacecraft area through the integrated 

design of the configuration, assuming from the met t&t ttzre be tome form of 

control. That would really be a control confipred spacecraft. 

CARLISLE: I think aeronautics is ahead of space technology in the integration of 

structures ad controls. 

SKBLTON: I think the universities should take some blau for this problem. In the 

past, students would take dysmks cm~e, 4 they were told: this is whrt p3p want 

to model, ad this is how you model it. They take a contr~h course, and they hear: if 

that is a model of a system, then this Q wlut yo3 do with it. However, there is a big 

world in-between for uhich we are here utembltd. I think the universities have to 

raise a new breed that understands integration of the dbciplines. At Purdne, ne are 

really concerned abaut the integration of discipliner, ami we are now reviring our 

curriculum accordingly. Structures and dpnrmics have had a 150-pear history of 

development of very sqhisticated method8 to answer specific needs. Even though 

controls is in its infancy be- only a 50-year old t-Aject), it has grown very rapidly 

and has achieved some sophisticated level. On the other hmd, the premise8 upon which 

each of these two disciplines arc based fall apart when put together. I do not think it is 

possible to take formally trained structural designers and rtk them to understand 

enough about "controlsm to do the job. Conversely, it i8 not porrPle to ask formally 

trained control engineers to learn enouh about structures to do the job. I think you 

need the universities to take it upon themselves to merge thote disciplines at the 

fundamental educational level. 

BBDIESS: I would like to come back to one other point. There was a presentation by 8. 

A. Russell from NASA Headquarters on a program plan for the control of dedble 

structures. I do not know how many of you hul an opportunity to hear him, but it is in 

part responding to P special ad hrc subcommittee that was esubliahed by NASA to look 

at the control of flexible space rtrrrctures. That particalrr mbcoxnmittee was n 

combination of controls and rtructurez experts. I personally feel they did a very good 

job of reviewing NASA's program and -Ling some tecommeab.tlonr. I al8o fird tJut 

the particuL- plan put together in retponding to that it an excellent plan. I have had 

involvement for two pean it trying to put together them types of plans for NASA. I 

think the people who did th:; should be coIILmended, because they present an ap3roach 

that really couples the dtscipher of control8 and 8tructurer. Often, when we have 

tried to put together such a program, we ended up pack~gbg Ln one plan, a controls 

program ad separate structura: dylrrrmics program. I think this plan b-8 the tw3 

discipline8 together. I would like to encourage thbt NASA implement 8s mpch of it. as 

possible. I undentaad that it hat been favorably received up to thir point. I fully 

understand the problems of funding these program, since they get to be extremtly 

expensive. I would encourage them to proczed and accomF1~h as much as pottiblc. 

_I- CARCISLB: I do not kn,~ if you are aware, but that was one of two specific new 

thrusts for next year's budget. So, we in NASA are tryu to proceed. I qgest that 

the urrtversitier lelp. If the government and univerritiet can get together to recognize 

the problem, it might help. 

437 
BBDIBSS: The plan proposed was Weed a combination of &~lytical rod theoreticd 

work up through significant groard and flight testing. I think it real€y is responsive to 

the community's needs for technology. 

BALAKRISHNAN: I just wanted to sa9 that we must not forget the scale of things. In 

microelectronics, we have projects at the level of $20 million in the univentities. As 

you mentioned, you- NASA project is CO- along. However, if each group is given 

only $1OK to carry on and participate, yon will not get major results. That kixxi of thg 

h8s to change if you want hrge scale involvement. 

SKELTON: I would like to turn the question around to the audience. I am on the 

Aeronautics and Space Engineering Board (ASBS) of the National Research Council. A 

ccmmittee of the ASBB is loo- at the rektioasbips between faculty, WUSW, *nd 

universities. I would appreciate those of yon nho have specific concerns or idea 00 

how those relationships can be improped to drop me a note, or somehow register those 

thoughts, so that I might take them into accuunt. The concern is that in the 

universities a lot of our potential graduate students ue not goin& on to higher 

educ8tion. If you have concerns thar would help tc broaden my own perspective in 

formulating ideas and getting pohts of WW, I wopld certainly appreci~te hearing them. 

438 
TECHNICAL EVALUATION REPORT OF THE 

WORKSHOP ON IDENTIFICATION AND CONTROL 

OF FLEXIBLE SPACE STRUCTURES 

H. A. Rcdicss an4 N. Nay& 

H. R. Textron, Inc. 

Imine, CA 92714 

INTRODUCTION 

The main objectives of the workshop were: 

- to provide a forum for exchnging ideas and thoughts on how to effectively 

control large flexible multibody spacecraft; .9d 

- to identify the important unsolved problems of cmt ad future interest 

leading to possible future collaborative NASA/University/Industq efforts. 

The workshop was organized with several sessions addre88aing the major technical 

ad theoretical issues through invited and contributed papers, three panel &CU%~~OPS, 

and a final wrap-up session. The workshop age- is presented in AppenGix A. The List 

of attendees is presented ia AppeediX B. The purpo8e of thb report is to present 8 

technical evaluation of the workshop that -theshes the most important results, 

couclusions, and recommendations for future research. 

The workshop covered a wide spectrum of issues involved in tbc dentification and 

control of flexible space st~~~tures. Many different concepts, ideas, and novel solutions 

were presented in this workshop. A new tread in the workshop was visible. There was a 

consensus that an integrated approach be adopted to solve the complex and challenging 

problem of control of large flexible space stiucttues. A mt - -g of control theory, 

structural design, ad materials among others can produce k znt solution to the 

problem. 

Since NASA is planning for the deployment of krge space structures, such as 8pace 

station ad antennas, it was felt that much of the theory that has been developed h.s to 

be validated by doing laboratory experiments both on the ground and anboard in 8pace. 

Of course, tk-9 is 8 continuing need for more basic research in many areas to refine 

the ideas and concepts and to develop new solutions. 

4 39 
The recommendations outlined below are believed to represent a consem view 

from the workshop and draw heavily on the results of the panel discussions and the 

papels presented. One should not conclude, nor is there any intent to imply, that all 

participants of the workshop or the wrap-up panel discussion endorsed these 

re commendations. 

MAJOR RECOMMENDATION FOR AREAS OF FUTURE RESEARCH 

1. 

structures 

An integrated approach to modeling, control law design, and optimization of space 

R8tiOMb: Future space stnrctures will be verp. large and complex with stringent 

performance requirements. Tim, the control requirements for slewing and rubilizing 

the structure, and for providing shape control .nd vibration rpgpreuion, necessitate 

careful review. To minimiae control problems, these control requirements must be an 

integral part of the structure design process. Since a fa--proof control strategy 

satisfy4 these requirements does not exist, it is important that .ddiLional research be 

focused on structural modeling and de- problems from a controls viewpoint. 

2. Model reduction ad robustness 

Rational: Because of the large size and light weight, the space structure will exhibit a 

very large number of flexible mode shapes with low frequencies that will interact with 

the controllers. To control such stnrctures, it is impesative to model the stmcture very 

carefully. To keep the model reasonably tractable from a compautioml point of view, 

A reduced order model has to be obtained. Some new results are now available to give 

guidelines in obtaining reduced order models, but good definitions and techniques are 

needed to measure robustness. 

3. Interdisciplinary approach in the control of flexible structures 

Rationale: The control of a large flexible space stnrctare is a complex .rd challenging 

problem. In order to keep the control simple, fmrctional, and tractable, all the 

available adjustable parameten mrmt be exploited. For example, by charyiiy certain 

material properties, the amount of power required for control can be decred. Ttmr, 

by considering options offered through the material and structural ptoptdes involved, 

the control problem can be ~llevirted coaribcrably. Researchen and engineen are 

recognizing the potentid of this interbirciphary approach. 

4. Adaptive control, estimation and identificatbn 

Rationale: Some large space structures, such as space station, will &ergo changes in 

their be dynamic characteristics. Adaptive control is potentially suitable under tach 

conditions, but there i8 a need for extensive research. In addition, for adaptive coattol 

to be effective, a good estimation and i.lentL=i:ation scheme is needed. 

440 
5. Control of distributed systems 

Rrtiorule: Since &e proposed space structure will be large, it can \e viewed as a 

distributed parameter system. Results obtained in a distributed parameter setting can 

give A good understanding and insight into choosu a proper control strategy. Control 

of distributed systems can be viewed as a limiw case of rPmped parameter rystem 

control. Thus, one can exploit the strong theoretical results of distributed parameter 

systems theory to study the limiting behavior of lnmped parameter systems. 

Rationale: There wen a considerable number of laboratory experimental results 

presented at the workshop. These experiments are necruarg to valid~te theory. There 

should be me or more benchmark tea articles ret up when various 

contmYestimatiodidentification algorithm can be tested .ab compand to provide a 

better understanding of the overall control problem. Carefully selected and well 

designed on-orbit experiments should be coldpc&l to further validate identification 

and control techniques. Particular emphasis should be placed on validating grad 

teste techniqozs and techniques for estrapohting grmd test data to the space 

environment. The 2rogram should be well coordiruted 80hg from arulysb, to simple 

experiments, and to full-blown realistic experiments. 

7. Sensor/Actartor placement ad development 

Ratiolule: A proper sensor/actutor set on a fleale space svuctute can be hidhly 

effective in controlling the structure. Large and unique space structures will probabiy 

need different types of sensors ad actuators. 

8. Control system fault detection and tolerance research 

Rationale: It k img0it.n: to detect the faults and failures of sensors and actuators in 

time to control flexible space structures properly. This is particularly critical for 

stnngent performance criteri.. After fault detection, the coxtrol must be able to 

recjnfigure to fie new and smaller set of senson actuators. 

9. Norlinear control 

Bhtionde: Because of th2 complex nature of the flexible stnrcture, nonlinearity may 

be a pirt of designing A good control Irw. More attention nee& to be paid to this area 

to develop a more mature and practical theory. 

OTHER BBCOMMENDATIONS 

1. Real-time processor control Irwr 

High speed of computation (10 to 100 million opevtions per second) 

High reliability and durability 

Software langruger, verification, validation, And fault tolerance 

2. Nonlinear state estimation 

44 1 
3. Active disturbance isolation for Space Station 

4. Analysis of high order structural systems with slightly nodinear stiffness properties 

5. Theory of asymptotic properties of system 

6. Applications of random field theory to estimation 8nd system identification 

7. Combined treatment of identification and control problem accouuu for modeling 

errors 

8. h cooperative program among NASA, DoD, dversitiet and industry to make the 

most effective use of limited data from -)or gmpad and flight experiments 

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L, 
WORKSHOP OH IDEUTIPICATIO% AWD COWTROL OF FLKXIBLB SPACE STRUCTURES 

A. Adamian 

UiLA 

4731 Boelter Hall 

Los Angeles, CA 90024 

A. Alexandridis 

Gn Researr'. Labs 

Dept. 15 

Warren, HI $8090 

*G. Andersen 

L~RC n/s 364 

R. Araya 

ucw 

BYH 4531 

Los Angeles, CA 90024 

*E. S. Armstrong 

UKC n/s 161 

a. Aswani 

The Aerospace Corp. 

n/s n41921 

P. 0. Box 92957 

Los Angeles, CA 50274 

C. 8. Auchter 

United Space Boosters 

P. 0. Box 1626 

Huntsville, AL 35807 

tU. E. Bachman 

JPL n/s 198-ii2c 

T. Baker 

U.C. Berkeley 

262 Cory Hall 

Berkeley, CA 90026 

A. V. Baldkrishnan 

University of ralifornia 

Los Angel 3 

Los llngeles , CA 90024 

H. 3. Balas 

BSCE Department 

Rensselaer Polytechnic Institute 

Troy, MY 12181 

S. S. Banda 

Plight Dynamics Lab 

APWAL/FIGC 

Wright Patterson AFB 

OH 45433 

H. T. Banks 

Brown University 

Providence, RI 02912 

Applied Uath Dept. 

H. Barrett 

Honeywell, Inc. 

2600 Ridgway Parkway 

Box 312 

Hinneapolis, 55440 

D. Barrows 

UDAC 

5301 Bolsa Avenue 

Huntington Beach, CA 9264? 

L. Bindin 

ISA 

P.T. Arun EL 

Singapore 

P. Blelloch 

UCLA 

4731 Boelter Hall 

Los Angeles, CA 90024 

J. A. Bossi 

University of Washingte? 

Dept. of Aero. & Astro. FS-10 

Seattle, UA' 98195 

J. Breakwell 

Lockheed Research Labs 

3251 Hanover Street 

Palo Alto, CA 94304 

+Affiliated with Yet Propulsion Laboratory, California Institute of 

Technology, 4800 fat Grove Drive, Pasadena, CA 91109 
T. Brennan 

Aerospace Corporation 

H4/971 

P. 0. Box 92957 

Los Angeles, CA 90009 

?. E. Brey 

Wayne State University 

Dept. of Hech. Engineering 

667 Herrick 

Detroit, HI 48128 

Toronto, Ontario, Canada LLsSlA4 

+J. Cameron 

JPL HIS 19P-326 

R. Carlisle 

NASA Headquarters 

Code S, 8-536 

Washireton, D.C. 20546 

D. Carlstrom 

Prito-Lay Research 

Irving, TX 75206 

900 1. Loop 12 

H. J. Chang 

Goodyear Aerospace Corp. 

1210 Hassillon Road 

Akron, OH 44315 

B. Chang 

TRU Systems 8212024 

One Space Park 

Redondo Beach, CA 90278 

+J. C. Chen 

JPL HIS 157-316 

W. W. Chiang 

Stanford University 

hrand 017, Dept. of Aero/&istro 

Stanford, CA 94305 

E. H. Cohen 

The Poxboro Co. 

D835, Bristol Park 

Poxboro, 1u 02035 

A. Campito 

Aerospace Corparation 

2350 E. El Setundo Blvd. 

El Segundo, CA 90250 

+J. J. Cucchfssi 

JPL n/s 511-202 

+J. 8. Dahlgren 

JPL n/s 108-326 

A. Das 

AFRPL 

AFiWL/DYS/Stop 24 

Edwards Air Force Base 

CA 93534 

E. J. Davison 

Dept. of Electrical Engineering 

Toronto, Ontario, Canada L15SlA4 

I?. De Canio 

C. S. Draper Laboratories 

555 Technology Square 

Cambridge, 1u 02139 

R. DeCarlo 

Purdue University 

School of Elec. Engineering 

Uest La:ayette, XU 47906 

S. M. DeCaro 

A.T. 6 T. Bell Labs 

7 Hamilton Rd., Apt. 3C 

Morristown, 1y. J. 

+L. Demsetz 

JPL n/s 157-316 

J. A. Donaldson 

Howard Univers Lty 

Uathematics apt. 

Washington, D.C. 20059 

A. Doran 

UCLA 

57r3 Boeltor Hall 

Los Angeles, CA 90024 

Y. L. Eastman 

McDonnell-Douglas 

16441 Space Center Blvd, E-2 

Houston, TX 77062 

D. L. Bdberg 

Stanford University 

uapt. of Aero 6 Astro 

Stanford, CA 94305 

+Af.'iliated with Jet Propulsion Laboratory, California Institut of 

Technology, 4800 oak Grove Drive, Pasadena, CA 91109 

454 
+D. 8. Bldred 

JPL MIS 198-326 

J. Garibotti 

H. R. Textron, Inc. 

2485 1cCatm Yay 

Imine, CA 92714 

+ S. Graff 

JPL M/S ‘i1201 

R. Gran 

Grunman Aerospace 

Bethpage, MY 11714 

c. L. Gustafson 

Aerospace Corporation 

P. 0. Box 92957 

Us weles, CA 90003 

A. L. Hale 

University of Illifiais, Allg hpt. 

104 S. ’I .hews Avenue 

Urbana, IL 61801 

D. R. Haley 

BITS, Inc. 

10210 Greenbelt Road 

Suite 440 

Seabrook, 20706 

v. n. HHarris COT. (CUD) 

P. 0. BOX 94000 

nelbourne, FL 32902 

1/S 15-SOOA 

R. Harding 

General Electric 

Valley Forge, PA 19481 

D. R. Hegg 

C. S. Draper Laboratorius 

55: TechnoLogy Square 

Cambridge, NA 02139 

n/s 60 

D. Herrick 

R 6r D Associates 

International Airport, AT0 9377 

AlbuqueFque, lpLl 87119 

V. 0. Hoehne 

mAL/PIGC 

Yright Patterson AFB 

OH 45433 

+C. Ih 

JPL MIS 198-326 

Y. Iliff 

UASA hs/Dryden 

WS D-OFD 

P. 6. Box 273 

Bdwards, CA 93523 

.D. J. fnmm 

University of Buffalo 

Dept. of llech. & Aerospace Bng. 

suyycs 

Buffalo, Ipy 14260 

P . Jabbari 

UCU 

Los males, CA 90024 

S . Jacobs 

P. 0. Box 1626 

Huntsville, At 35801 

US81 

U. J. Jasper 

mhes Aircraft Co . 8143. S41, WS 8323 

F. 0. Box 92919, Airport Station 

Lo8 Aiieles, CA 90009 

*S. 1. JoshL 

taBc MIS 161 

*J. 1. Juang 

LaEC HIS 230 

c. Karl 

HIT 

77 ~ssachusetts Avenue 

Cambridge, M 00213 

10-025 

+Affiliated with Jet Propulsion Laboratory, Colifornja Institute of 

Technology, 4800 Oak Grove Drive, Pa88dolLo, CA 91109 

*Af€Fliated with UASA Laryley Research Center, Hampton, VA 23665 

455 
J. K0lly 

McDonne11 Douglas Astronautics 

5301 Bolsa Avenue 

Huntiryton Beach, CA 92647 

+R. Key 

JPL ws 180-101 

U. 3. Klepl 

Rockwell International 

201 Y. Douglas Street 

P. 0. Box 92098 

Los An&eles, CA 90009 

D. Klinger 

Lnsc 

7538 Uaterford Dr. 

Cupertino, CA 95014 

I. Landau 

C.Y.R.S. "Systemes Adaptatifis" 

Ensieg BP 46 

38402 St. Martin r)*Heres, Prance 

Tt,. Lange 

DFVLR, UT Rn 

8031 Oberpfaffenhoten 

Post Webling, PRG 

W. J. Lange 

Kirtland AFB 

7640 Arroyo del 380 YE 

Albuquerque, MI 87109 

J. H. Lilly 

Electrical Engineering Dept. 

University of Kentucky 

Lexington, KY 40506 

J. G. tin 

Control Research Corp. 

6 Churchill Lane 

Lexington, HA 02173 

+Y H. tin 

JPL U/S 198-326 

K. 8. Lorell 

Lockheed Research Labs 

3251 Hanaver Street 

Palo blto, CA 94306 

II. Lukich 

TIM Electronics & Defense 

One Space Park 

Bld8. R4, RE 1189 

Redondo Beach, CA 90278 

P. Uak 

Aerospace Corp. 

2350 E. El Segundo Blvd. 

Ul "egmdo, CA 90245 

n/s n4-971 

D. B. Malkoff, M.D. 

Uavy Personnel & Develop. Center 

Code 71 

San Diego, CA 92152 

L. Markus 

University of Minnesota 

Mathematics Dept. 

'Minneapolis, nI 55455 

+Is. Harmell 

JPL WS 198-326 

G. Y. Matsuntoto 

TRU 

One Spacc. Park, B4/1082 

Redondo Beach, CA 90278 

M. llasubuchi 

Osaka University 

Dept. of Hech. Engineering 

Suita, Osaka, Japan 

W. A. Hatthews 

Rockwell Int * 1. 

16323 Parksley Drive 

Houston, TI 77059 

+E. LIettler 

JPL ?-1201 

C. J. Meyers 

U.S. Air Force 

Kirtland APB 

Albuquerque, .1pn 87109 

It. K. Miller 

University of So. California 

Dept. of Civil Engineering 

LO8 AngGleS, CA 90089-0242 

+AfEiliated with Jet Propulsion Laboratory, California Institute of 

Technolcgy, 4800 Oak Grove Drive, Pasedena, CA 91109 

456 
T. 8. Hiller 

IBM Corporation, Dept. 235 

:.101 North Street, Bldg. 250-2 

gndicott, UY 13760 

D. i. ningori 

UCLA 

4731 Boelter Hall 

Los Angeles, CA 90024 

V. J. Modi 

Dept . of Hechanical Rngineering 

University of British Columbia 

Vancouver, B.C., Canada 

*R. C. bntgomery 

URC nis 161 

F. Moon 

Cornell University 

Dept. of Theoretical & Appl. llech. 

Thurston Hall 

Ithaca, W 14853 

A. P. Uayak 

ii. 8. Textrorr 

2485 McCabe Way 

Irvine, CA 92714 

Y. Ohkami 

Uational Aerospace Laboratory 

1880 Jindaij i Chotu 

Tokyo, Japan 

H. Okubo 

University of Osaka Prefecture 

Dept. of Aeronautical Engineering 

Sakai, Osaka 591 

Japan 

H. H. Ottesen 

IBM 

5905 Chateau Bd. U.U. 

Rochester, m 35901 

C. Partin 

Lockheed Engineericy & 

Ilonageeoen t services 

1830 Uasa Road One 

Houston, TX 77223 

n. Paluszek 

C. S. Draper Laboratories 

555 Technology Square, n/S 60 

Cambridge, HA 02139 

L. 11. Passeron 

Aerospat iale 

100 Boulevard du nidi 

06322 Cannes La Bocca Cedex 

B.P. 99, Prmce 

A. U. Payne 

Laurence Livermore Laboratory 

Livermore, CA 94550 

P. 0. BOX 5540, 6153 

J. Peebles 

McDonnell Dollglas Astronautics 

5301 Bolaa Avenue 

Huntington Beach, CA 92647 

A.3-222-13-3 

k4. P. Polis 

Uational Science Foundation 

1800 G. Street, U.U. 

Uashington, D. C. 20550 

T. A. Posbergh 

University of naryiand 

College Park, HD 20742 

G. A. Price 

101 228th Avenue, S. E. 

Eedmond, UA 98053 

R. V. Raumath 

Ray theon 

12 Pessenden Uay 

Lexington, HA 02173 

A. S. S. 8. Baddy 

Efoward University 

Dept. of Uach. mineering 

Uaahington, D.C. 20059 

H. A. Redieos 

H. 8. Textron, fnc. 

2485 WcCabe Yay 

Irvine, CA 92114 

D. B. Ridgely 

P 1 igh t D;mamic s Lab0 ra t rrry 

AFUAL /PIGC 

Uright Patterson AFB, Ohio 45433 

‘a‘LAted with Jet Propvlsion Laboratory, California Institute of 

*r !,ogy, 4800 Oak Grove Dciwe, Pasadena, CA 91‘09 

‘,‘if. -* ?;44 with UASA Langley Rssearch Center, HPmpton, Virginia 23665 

45 7 
+G. Rodriguez 

JPL n/s 198-326 

G. Rosen 

C. S. Draper Lattoratories 

555 Technology Square, W/S 60 

Cambridge, UA 02139 

D. L. Russell 

University of Wisconsin 

480 Lincoln Driva 

Nadison, UI 53706 

*R. Russell 

L~RC n/s 162 

+U. Salama 

JPL n/s 157-326 

R. L. Sanders 

United Technologies 

Huntsville, AL 35804 

C. H. Satter 

C. S. Draper Laboratories 

555 Technology Square 

Cambridge. MA 02139 

D. Schaechter 

Lockheed Palo Alto Research Labs 

3251 Hanover Street. B205,O 92-30 

Palo Alto, CA 94304 

+R. E. Scheid, Jr. 

JPL n/s 198-326 

J. R. Sesak 

General Dynamics Convair 

P. 0. Box 85357, WS 21-9530 

San Diego, CA 92138 

l4. Shafer 

UASA AmesIDryden 

P 0. Box 273 

Edwards, CA 935; 

HIS D--OVDD 

D. C. Schuab 

LLlsC 

1111 Lockheed Way, Bldg. 104 

Dept. 62-11 

Sunnyvale, CA 34089 

R. P Singh 

Honeywell, Inc. 

13350 U. S. Highway 19 South 

Clearwater, PL 33563 

+S. U. Sirlin 

JPL n/s 198-326 

it. E. Skelton 

lhrrdue University 

Uorth Lafayette, II 47907 

G. Smit 

Aerospace Corporation 

2350 E. El Segundo Blvd. 

El Segundo, CA 90245 

B. SridhRr 

Lockheed Research Labs 

3251 Hanover Street 

Palo Alto, CA 34304 

D. S. Stauffer 

MTI 

Bradford Drive 

Huntsville, AL 35805 

M. Stieber 

Communcations Research Centre 

P. 0. Box 11490, Station H 

3701 Carling Avenue 

Ottawa, Ontario. Canada U2H852 

+s. 2. szicmay 

JPL MIS 198-326 

*L. Y. Taylor, Jr. 

taRC HIS 161 

+Affiliated with Jet Propulsion Laboratory, Califomia Institute of 

Technology. 4800 Oak Grove Drive, Pasadena, CA 91109 

*Affiliated with UASA Langley Research Center, Hampton, VA 23665 

458 
P. E. ‘phsu 

City College of the City 

University of leu York 

Let York, UY 10031 

G. Thieme 

Dornier Systea GMBH 

Dept. ERHR 

Postfach 1360 

KJ-7990 Friedrichshafen 1 

West Germany 

+A. F. Tolivar 

JPL t4/S 198-326 

S. S. Tripathi 

Perkin-Elmer Corporation 

100 Uooster Heights Road 

ganbury, CT 06810 

C. T. Tseng 

The Aerospace Corporation 

P. 0. BOX 92957, n4/97i 

Los Angeles, CA 90009 

r. Tung 

One Space Park 

Redondo Beach, CA 90278 

J. D. Turner 

Cambridge Research 

Belmont, M 02178 

P. E. Uwadia 

USC 

University Park, 364 DUB 

Los Angeles, CA 90007 

S. Utku 

Duke University 

122 Engineering 

Durham, UC 27706 

8. Venkayya 

AFWAL /FIBK 

Wrigbt Patterson MB 

OH 45433 

W. E. Vander Velde 

bss. Institute of Technology 

Room 33-109 

Cambridge, MA 02139 

D. Vervaerde 

Dornier System CWRH 

BP 27,94190 Villeneuve 

St. Georges, France 

L. Q. Vu 

University of California 

Davis Hall #516 

SBSt4 Division 

Berkeley, CA 94720 

tB . Wada 

JPL n/s 157-507 

P.K.C. Uary 

UCLA 

4531 Boelter Hall 

Loa Angeles, CA 90024 

+S. J. Wang 

’ JPL H/S 198-326 

D . Waahburn 

R&D Associates 

105 E. Vennijo, Suite 450 

Colorado Springs, CO 80903 

T. Weber 

General notors Research Labs 

Gll Techrdcal Center 

Power Systems Research Dept. 

barren, MI 48090-9057 

M. Webster 

Hr’onnell Douglas Astronautics 

Huntington, BLach, CA 92647 

P. Wei 

Lockheed-Georgia Cmpany 

Dept. 72-42, Zone 41& 

South Cobb Drive 

Marietta, GA 30063 

U. Wright 

TRU 

One Space Park, 82/2754 

Redondo Beach, CA 90248 

E. woo 

Rockwell, Int’ 1 

12214 Lakewood Blvd. 

Douney, CA 90241 

tAffiliateQ with Jet Propulsion Laboratory, California Institute of 

Technology, 4800 Oak Grove Drive, Pasadena, CA 91109 

459 
APWAL/FIGC 

Wright Patterson APB 

OH 45433 

Y . Yeung 

1926 Federal Avenue, Apt. 3 

West Los Angeles, CA 93025 

Y. Yez-Pien 

UCLA 

500 Landfair Avenue, 717 

Los Angeles, CA 90024 

K. Yong 

The Aerospace Corporat Lon 

2350 E. EL Segundo Blvd. 

El Segundo, CA 90245 

J. W. Young 

SDRC, Inc. 

11055 Roselle Street 

San Diego, CA 92121 

tM. Zak 

JPL MIS 157-102 

tAf€iliated with Jct Propulsion Laboratoiry, California Institute of 

Technology, 4800 Oak Grove Drive, Pabadena, CA 91139 

460 ?L&!!--JpL--corm, LA, u