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DEPARTMENT" OF MECHANICAL ENGINEERING AND MECHANICS

SCHOOL OF ENGINEERING

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA 23508

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ADAPTIVE CONTROL OF LARGE SPACE STRUCTURES j' RMSnFA=,

USING RECURSIVE LATTICE FILTERS ,^*

By

I Gene L. Goglia, Principal Investigator

INASA-CR-176402) ADAPT:rVE CONTP.OL OF LARGE N86-133$8

SPACE STRUCTURES USING RECURSIVE LATTICE

FILTERS Final &eport, period ending 31 Dec.

1985 (Old Dominica Univ. , Norfolk, Va. ). Uncl.as

80 p HC A05/MF A01 CSCL 22B G3/18 04980

Final Report r

For the period ending December, 31, 1985

Prepared for the

National Aeronautics and Space Administration

Langley Research Center

Hampton, Virginia 23665

Under ,

Research Grant NAG-1-429

Dr. Raymond C. Montgomery, Technical Monitor

FDCD-Spacecraft Controls Branch

AIEW

December 1985
DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS

SCHOOL OF ENGINEERING

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA 23508

ADAPTIVE CONTROL OF LARGE SPACE STRUCTURES

U0,41a RECURSIVE LATTTCE FILTERS

By

Gene L. Goglia, Principal Investigator

Final Report

For the period ending December 31, 1985

119d

Prepared for the

National Aeronautics and Space Administration

Langley Research Center

Hampton, Virginia 23665

Under

Research Grant NAG-1-429

Dr. Raymond C. Montgomery, Technical Monitor

FDCD-Spacecraft Controls Branch

Submitted by the

Old Dominion University Research Foundation

P.O. Box 6369

Norfolk, Virginia 23508

December 1985

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TABLE OF CONTENTS

PAGE

SUMMARY............................. ............................... 1

I' REFERENCES .......................... P., ..................., ...... 3

APPENDIX A: IDENTIFICATION OF THE DYNAMICS OF A TWO-DIMENSIONAL

GRID STRUCTURE USING LEAST SQUARES LATTICE FILTERS.... 4

APPENDIX B: EXPERIMENTAL EVALUATION OF FLEXIBLE STRUCTURE

IDENTIFICATION USING LATTICE FILTERS .................. :18

APPENDIX C: PROGRESS IN ADAPTIVE CONTROL OF FLEXIBLE

SPACECRAFT USING LATTICE FILTERS ...................... 26

APPENDIX D: ROBUST CONTROLLER SYNTHESIS FOR A LARGE FLEXIBLE

SPACE ANTENNA ......................................... 34
/Whr I'IVC CUNTROL OF LARGE SPACE STRUL I LIKES

USING RECURSIVE LATTICE FILTERS

By

Gene L. Gogl i a*

SUMMARY

This report summarizes the research activities performed under grant

NAG-1-429. The objective of the research has been to study the use of recursive lattice filters for identification and adaptive control of large

space structures. Lattice filters are Used widely in the areas of speech

and signal processing. Herein, they are used to identify the structural

dynamics model of tree flexible structures,. This identified model is then

used for adaptive control. Before the identified model and control laws are

integrated, the identified model is passed through a series of validation

procedures and only when the model passes these validation procedures control is engaged. This type of validation scheme prevents instability when

i

the overall loop is closed.

One of the main aims of the research has been to compare the results 

obtained from simulation to those obtained from experiments. In this regard, the flexible beam and grid apparatus at the Aerospace Control Research

Lab (ACRL) of NASA Langley Research Center were used as the principal candidates for carrying out the above tasks. Another important area of research

namely that of robust controller synthesis was investigated using frequency

domain multivariable controller synthesis methods. The method uses the

*Eminent Professor, Department of Mechanical Engineering and Mechanics, Old 

Dominion University, Norfolk, Virginia 23508. 

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Linear Quadratic Gaussian/Loop Transfer Recovery (LQG/LTR) approach to ensure stability against unmodeled higher frequency modes and achieves the
desired performance. Such a controller was designed for the 122 mr. HoopColumn antenna using a single 3-axis torque actuator and attitude sensors.

References 1 and 2 present the detailed analysis of identification

results for the flexible grid apparatus using lattice filters. The scheme

provides on-line identification of number of modes, mode shapes, modal damping and natures frequencies. The results indicate that the lattice identi

fication scheme is a viable scheme for identifying the structural dynamics

of flexible structures. The experimental results also indicate differences

between those predicted by finite element analysIF and obtained by experiments. The difficulties are not as such in finite element analysis but in

modeling the apparatus for finite element analysis. This fact empnasises

the need for on-orbit identification of large space structures before control is attempted. A summary of the experimental results obtained using

lattice filters is described in reference 3.

An adaptive control scheme using lattice filter identification and

modal description has been developed in reference 4. Alternate schemes of

using input-output models instead of modal form from lattice filters is 

described therein. The problem in this approach is to obtain efficient 

control schemes as the identified model of the system becomes coupled and to

calculate the pole placement control law on-line is computationally complex.

Presently, the identification scheme using lattice filters for obtaining the

input-output model is under development in the Charles River Data Systers in

the ACRL.

A new approach of v•'signing robust controller for a large flexible

space antenna using the LQGOLTR approach was developed in reference 5. The

method was used in designing robust controller for the 122 mr. Hoop-Column

antenna using only a 3 axis torque actuator and attitude sensor. The objec2
tive is to design the controller based on a lower order model to achieve the

desired bandwidth and at the same time ensuring stability against unmodeled

higher frequency modes. The results in reference 5 indicated that if one

uses only a rigid body model for design stability against unmodeled modes

can be obtained but not the performance. Based on detailed studies in references 6 and 7 it was concluded that with the fiat three flexible modes

(corresponding to the 3 axes) included in the design model both stability

and performance can be ensured.

REFERENCES

1. R. C. Montgomery and N. Sundararajan, "Identification of the Dynamics

of a Two-Dimensional Grid Structure Using Least Square Lattice Filters," Proc. of American Control 'Conference, San Diego, June 1984, pp.

704-709.

E Also, appears in J. of Astronautical Sciences, Vol. 33, No. l;, JanuaryMarch, 1985, pp.,35-47.

2. N. Sundararajan, "Experimental Evaluation of Flexible Structure Identi- 

fication Using lattice Filters," Proc. of 7th I(: AC Symposium on Identification and System Parameter Estimation, York, England, July, 1985, 

pp. 687-692.

3. N. Sundararajan and R. C. Montgomery, "Progres <,; in Adaptive Control of

Flexible Spacecraft Using Lattice Filters," Proc, of Workshop on Applications of Adaptive Control, Yale University, New Haven, CT, May 1985,

pp. 162-167.

Also, to appear 'in a book on "Adaptive and Learning Systems: Theory

and Applications," Ed. Prof. K. S. Narendra, Plenum Press.

4. N. Sundararajan, S. M. Joshi, and E. S. Armstrong, "Robust Controller

Synthesis for a Large Flexible Antenna," Proc. of 23 IEEE Conference on

Decision and Control, Las Vegas, Nevada, Dec. 1984, pp.

Also, revised version has been submitted for publication in AIAA J. on

Guidance, Control and Dynamics.

5. N. Sundararajan, S. M. Joshi, and E. S. Armstrong, "Application of the

LQG/LTR Technique to Robust Controller Synthesis for a Lar e Flexible

Space Antenna." To appear as a'NASA - Technical Paper (TP^-

3
I

APPENDIX A

IDENTIFICATION OF THE DYNAMICS OF A TWO-DIMENSIONAL GRID

STRUCTURE USING LEAST SQUARES LATTICE FILTERS
t

t

f

Thai Journal of the Astronautical Sciencos, Vol 33, No. 1, 14nuary-March 1065, pp 35-47

i

Identification of the Dynamics of a

Two-Dimensional Grid Structure using

Least wares Lattice f=ilters'

1

R, C. Montgomery' and N. Sundatrarajana

Abstract

The basic theory of least squares lattice filters and their use in identification of structural

dynamics systems is summarized. Thereafter, this theory is applied to a two ,diimensional grid

structure made of overlapping bars. Previously, this theory has been applied to an integral beam.

System identification results are presented for both simulated and experimental tests and they are

compared with those predicted by means of finite clen% at modeling. The lattice filtering approach works well for simulated data based on finite element modeling. However. considerable

discrepancy exists between estimates obtained from experimental data and the finite element

analysis. It is believed that this discrepancy is the result of inadequacies in the finite element

modeling to represent the damped motion of the laboratory apparatus,

Introduction

The ability to predict the dynamic behavior of large space structures (LSS) adequately for control system design is doubtful because of their expected size, appreciable

flexibility, and on-orbit assembly anomalies, Hence, dynamical modeling from onorbit measurements, followed by modifying the control system as dictated by the

identified control system design model (adaptive control), is of interest, The goal of this

paper is to determine, using a generic grid structure, whether a priori modeling of the

structure is adequate for a high authority control system design oe whether on-orbit

identification is needed.

An approach for identifying the dynamic behavior of LSS that estimates model order

in addition to model parameters is presented in [ 11. It uses lattice filters which provide

an order as well as a time recursive algorithm for linear least squares signal estimation.

[2) provides a tutorhil on lattice filter theory and applications. The outputs of the theory

of [I) are the least `square estimate of the measurement sequence, the model order

required to fit the measurements, the associated lattice model (this includes mode shape

'This paper was originally presented at the American Control Conference. San Diego, California, June 6.8.

1984.

'NASA Langley Research Center. ,Hampton, VA 23665.

'Old Dominioth University Research Foundation. Hampton, VA 23666.

35

5 
36 Montgomery and Sundoraralon

estimates that arc orthonormal in the measurement +pare), and the us,ociated auto.

regressive moving average (ARMAX) model of the measurement sequence.

Some distributed adaptise control strategies require identification of the natural

modes of a structure 13. 41. l nfortunattl). the lattice filter provides mode shapes that

are orthonormal in the measurement Spare and, hence, are not the natural modes.

Natural modes can be obtained, however, either through an eigenvector anal) sis of the

identified ARMAX model or through a transformation that provides spectral decomposition of the lattice filter modal amplitudes 151, The latter.method is used herein to

obtain the natural modes. Using spectral decoupling to determine mode Shapes. one can

obtain mode frequency and damping using an equation error parameter identification

method (3) that employs a second-order ARMAX model to represent the natural mode

amplitudes, T)te procedure of 131 tracks frequency and damping coefficients required

for the modal amplitude input sequence to tit the second order ARMAX model.

Because the input sequence may have an unfavorable signal to noise ratio, the parameters so derived must be tested for validity before use in control system design. This

parameter testing is treated in detail in (6[ but is not employed herein.

The foregoing procedure has been used to identify the dynamic characteristics of an

integral free-free beam in [1. 71. 151 describes the test apparatus used in those studies.

In this paper, the'theory is applied to a more complex, two-dimensional grid structure

made of overlapping bars. First, a brief overview of the theory used is presented, Next,

system identification results are presented using both simulated and experimental

data, Finally, the experimental results are compared. with those predicted using finite

element modeling,

Summary of the Method Used to Identify Structural Dynamics Systems

For the application considered here v,e assume that the kth measurement sample is

of the form

where NS represents the number of sensors. It is assumed that Y is generated from a

model system such that

►•k. chew + n t (2)

Here, (P is a made shape matrix, `)'k is the modal amplitude vector. and n k is a Gaussian

random variable with a zero mean and a covariance matrix R. Typically, for structural

dynamics applications, each component of 111k is the output of an uncoupled secondorder process, The task here is to estimate the order and obtain the least square estimate

of IPk. from N + I measurement samples ?b through y, . [ I ► presents a derivation of the

equations that relate order n, and time i. recursions for the normalized forward and

backward residuals (e and r, respectively) as well as the least squares estimate of the

measurement vector y. These equations are listed below;

e,.,,.i = (I — k ,-i)-1:(e,.,, — k,.,.ir,-j,) (3)

rj.,,- i — (I — k2^.^)-i:(r^-i.^ — k,.,-ie,.,) (4)

N-1

w Z E ( e.v.,, I rx-1,J (S)

n-o

a

':..1 r 
Identification of the Dynamics of a Two•Dimensional Orld Structurs 37

with

k,,OQ (e, n, t',>1 „) (6)

acid tr'(.rjy) is the orthogonal projection operator of the vector .Y onto the vector y. The

symmetry of the recursion ;formulae is apparent. The equations are coupled by the term

All ",, which is customarilycalled the "reflection coefficient."

Clearly, in this approach one may "`fit the noise" by continually increasing the order

of the system; however, once the order of the estimator has increased sufficiently, the

residual errors should lie within a noise band which can tae predicted based on assumed

noise characteristics. A threshold value can be selected based on this predicted noise

band and order determined by a test of whether or not the residuals have been reduced

to lie within the noise band. Also, the test can be made considering several samples of

data; that is, using a data window. (1) documents experience in order determination

based on this threshold test,

Having defined the order required to fit the data using a linear model, we seek a fixed

set of basis functions that are spectrally decoupled for modal control, Therefore, u fixed

orthonormal basis is used during intervals when the order estimate is constant. (However, the order estimate is checked at each measurement sample based on the threshold

test.) The lattice filter uses the current measurement as the first mode shape and, using

a modified Cram-Schmidt orthonormalization procedure, generates additional basis

functions from estimation residuals. Consequently, the output of the lattice filter produces coupled mode shapes and corresponding modal amplitudes wherein the first

coupled modal amplitude will contain all significant natural moues. Since the order

estimate it been determined. the first coupled mode digital Fourier transform (DFT)

amplitude spectrum is searched for the it significant peaks and corresponding

frequencies, Because the spectrum contains rt peaks for the it separate modes, a

transformation matrix can be obtained that decouples the spectrum. This transformation

matrix is the inverse of the matrix whose elements are the real part of the transform of

the it coupled modal amplitude channels (rows) evaluated at the it peak frequencies

(columns), It effectively transforms the lattice filter modes into spectrally decoupled

natural. modes. These decoupled modes are not orthogonal. This procedure is described

in (5).

Thus, the decoupled modal amplitude time series, w(k), is obtained by applying the

transformation to the direct output of the lattice filter. ) his time series is then analyzed,

for each mode, to identify the parameters of its autoregressive moving average

ARMAX model. The inputs to each ARMAX modal model are the generalized forces

and hence, each model takes on the form

w(k) = A i x'(k — 1) + Azw(k — 2) + B l f(A — 1) + B: f(k — 2) (7)

wheref represents generalized forces. The parametersp' _ (A i , A , , B,) are the ones

which are identified and which are required for the control law design process. Thus,

the ARMAX model output error is

e(k — 1) = w(k — 1) (A,(k — I)w(k — 2) Ir A 2 (k — Ihj"(k — 2)

+ B# — i)f(k — 2) + Ba(k — 1)f(k — 3)} (g)

The method of [3] is used to identify the parameters (p) using the iteration sequence

7

a
31 OAonta4mery and sundsrsml4n

p(k) - p(k — 1) + e(f. — 1)

[Wl w(k — 2),W2w(k — 3), Wif(k — 2),W41(k — 3)] (9)

As Indicated in [3], the weights W, Y - 1, . , , , 4) must be selected so that they are

consistent with the relation

Wt w2 (k — 2) + Wiw2(k — 3) + W2 f2(k — 2) + W,.f2 (k — 3) 6 2 (10)

and the inputs to the algorithm (w and f) must be sufficiently varying and large if the

parameters are to converge to their correct value.

Damping ratios and natural frequencies of the movies can be obtained from A i and

A2 of equation (7). However, the solution is not unique, due to the foldover phenome•

non of sampling. By finding the roots, s„ of the characteristic equation ( 7) and using

the relation z, - e" in the primary strip, where r is the sampling period, the following 

equations for damping ratio and natural frequency are obtained for a typical root,,

say xt MA

w ¢/(21rr) 01)

c(c 2 + 02) 112 (fir

,^ herein

d - tan"(b/a) (13)

c - — I In(a2 .+. b = ) (14)

and a = Re(z) and b - lm(:). The behavior of this overall system identification methodology with both simulated and experimental data is discussed in the subsequent sections.

Description of the Flexible Grid Facility

Figure I shows the flexible grid experimental appartus currently being built at the

NASA Langley Research Center. The grid is a 7 ft by 10 ft planar structure made by

overlaying aluminum bars of rectangular cross section. The bars are centered every foot

so that there are 8 vertical and 11 horizontal bars, As shown in Fig. 1, the grid is

suspended by a cable at two locations on the top horizontal bar. The motions of the grid

perpendicular to the plane of Fig. 2 are the ones of interest in this study. There are nine

noncontacting deflection sensors mounted on a back frame which give a 9 x I

measurement vector. The sensor data are linked to the main CYBER 175 Real-Time

Computer System at NASA Langley Research tenter so that the identification can be

carried out in real time. For the experimental tests, the locations of the sensors are

indicated in Fig. 2.

Simulation Studies

A Finite element anaiysis of the grid was performed which included the suspension

cables. Nodes were place1 at each overlapping joint on the grid. the ceiling attachment

points of the cable, and every one-half foot along the cable. The grid elements connecting the nodes were modeled as bending elements, whereas the cable elements were

rrIf

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FIG. 2. A Schematic of the Grid Apparatus Indicating Locations Referred to in the Text and

Subsequent Figures

modeled as two-force members. Thus, a total of 165 elements were included in thel

model, Four degrees of freedom zppropriate for motion normal to the plane of the grid

were considered. No damping was included in the model. Thirty modes were obtained

from this analysis. The frequencies of the first ten modes are listed in Table 1. The first

three modes are the pendulum modes, the fourth is the first bending mode and the fifth

is the first torsional mode, The finite element analysis uses an iterative method to

calculate mode frequencies. The frequencies used in simulation are believed to be

numerically accurate since the change in eigenvalue iterate of the highest frequency

mode used in the simulation is l0" ' on the final iteration, The corresponding eigenvalue iterate was 1565, which corresponds to the mode 8 frequency in Table I -

A simulation was developed that accommodates the first 15 modes of the analysis,

but only four mod°s were used herein. Modes 4, 6. 7, and 8 were used. A sampling

rate of 32 Hz was simulated with a standard deviation for the measurement noise of

0.005 in, which was, based on actual sensor characteristics, Modes were simulated with

modal amplitude initial conditions of CIA. The data window for order determination

included eight samples. In this work, the sensor locations were chosen based on several

simulations. These (locations differ from those of the experimental apparatus in that they

were selected to maximize the ; Yect of simulated modes on the sensors. This was

accomplished by visual examination of the simulated sensor outputs. The selected

locations are indicated in Fig, 2, An asterisk is used to distinguish simulation sensors

locations from experimental ones. One may expect that location 5 would be preferablej

• 10
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Identlilatlon of the Dynamlee of a Two.Dlmenatonal arld Structure 41

TABLE 1. Modal Frequencles Obtained

from the Finite Element Analysis or

the Grid

Mode Number Frequency iNz ►

1 0 364

2 0.625

3 1.398

4 2.29

3.07

6 4.791

7 5.933

8 6.297

9 7.337

10 10.352

to location P; but, since some simulated modes had little input to a sensor at location 5, location 5* proved to be better location.

Based on the entire measurement vector, the lattice filter o;rdcr estimate is shown in

Fig. 3. Also, sensor 5* data, typical of those of the other sensors, is shown in Fig. 3.

After estimating the order, we carried out a transformation based on the discrete Fourier

.^ 2 ,Irl 1 ► ^' ' '

*^h

0 11V V p V . V" 0 VV q V V q VVII

-2 _VWWPWV07*1

1. 6

d

0 3

0

e .10

.0S

< < < 1 1 i

0 2 4 6 i3 10

^ N

t

^I

<_^

i

i F

]

z

TIME, SEC.

FIG. 3. Simulation Time Histories of Sensor 5", Lattice Filter Order Estimate, and the rorrn of the

Estimation Error for the Entire Measurement Vector.

. 

11

4

I
cl

- l

A "

42 Montgomory and Sundararslan

1

y l , in,0 ^ ^^ 4 1^4

-1

Y4 0

 in. 1

0

4

Order

A

I I l I I I

0 1 2 3 4 5

Time, see,

FIG, 4. Sensors I and 4 from Experimental Tape 5 and the Estimate of Signal Order Obtained by the

Lattice Filter.

transform (DFr) using 128 samples in order to obtain the natural modes, and used the

equation error method to identify associated modal frequencies and damping. The

resulting modal frequencies, damping and mode shapes are compared with those

predicted by finite element analysis in Table 2. The identification of frequencies and

damping are close for all four simulated modes, However, the mode shape estimates

agree with simulation for only three modes, One possible explanation for this is the

limitation imposed by sampling rate and the number of samples used to decouple the

lattice filter modes. Sampling at 32 Hz and including 128 data points in the DFC, a

TABLE 2. Camp2rison Between Simulated and Identified Results )

^. Mode 4 Mode 6 Mode 7 Mode 8

e

j Simu- Identi- Simu- Identi• Simu- Identi• Simu- Identi•

lated fled lated fled lated tied lated tied

Frequency

-% ! (Hz) 2.29 2.4 4.79 4.8 5.93 6,0 6.3 6.4 {`'

Sensor I

1 • 0.29 0.30 -0.16 -0.17 0.31 0,32 0.59 0.45

2" -0.40 -0,41 -0,12 -0.08 -0.43 -0.43 0 -0.43

3' 0,29 0,30 -0,16 -0.18 0.31 0.32 -0.16 0.21

40, 0.30 0.31 0.38 0.37 -0.06 -0,06 0.01 -0,09

5' -0.39 -036 -0,72 -0,74 0.49 0.49 0,01 0,49

6 0 0,30 0.30 0.37 0,36 -0.06 -0,06 -0.01 -0.06

7 0 0.31 0.31 -0.20 -0.15 -0,32 -0.31 0.39 -0,19

8' -0.39 -0,38 -0.26 -0.29 0.41 0.41 0 0.37

K ' .

I 9' 0,31 0.31 -0.20 -0,14 -0,32 -0.31 -0,39 -0.38

a

I

4

{

MEMO"

12`

j
kfsnttficstlon of the Dynamics of s TwoDinsnslonsl ck{d.$tmchxs 43

frequency resolution of only 0 . 25 Hz is obtained. Since the expected frequency separation between modes 7 and 8 is only 0,4 Hz, good decoupling cannot be achieved.

To summarize the lessons learned from the simulation studies:

1. The least squares lattice filter gives good identification of simulated modal

frequencies, damping ratios, and mode shapes in the presence of sensor noise

expected in the experimental apparatus.

2. The DFT method of obtaining natural modes from the lattice modes is inaccurate

if the modes are closely spaced in frequency. This may be improved by adding

more samples to the DFT,

3. Sensor locations should be properly selected to insure good identification of

simulated mode shapes.

The next section will discuss results obtained from the experimental apparatus.

Experimental Results

Experiments were conducted using the grid apparatus previously described, The grid

was excited using an air shaker which periodically exhausted a jet of air that impinged

' on the grid at sensor location 1. The frequency of the jet was adjustable from 0 to

P 50 Hz. Although the resulting grid excitation was periodic, it was not purely sinusoidal

but was rich in harmonics. Because of the range limits of the deflection sensors - 0 to

approximately 2 in. - the maximum peak-to-peak deflections of the grid wei-e limited

to about I in, When the peak-to-peak deflection neared this limit, the air shaker was

turned off and the grid was allowed to vibrate freely with only air and material damping.

w' A CYBER 175 Real-Time Computer System sampled the deflection sensor data at

r ;' 32 Hz for 5 seconds. The data were stored on a system data file for further analysis.

Since only free-decay response data were recorded, the B, and B 2 parameters of

equation (7) were not identifiable, Figure 4 presents data from file 5. Here, the order

estimate is seen to converge to an oscillation between 2 and 3 at about 0,5 s. At about

0.8 s, the order estimate was fixed at 3 and data collection (at 32 Hz) for the 64 time

samples required for the DFT was started. The DFT was accomplished at about 2.8 s

and the decoupling transformation matrix was calculated. The modal amplitudes after

this time should contain a single frequency and the transformed mode shapes should

correspond to the excited natural modes of the structure. Thus, three modes were

extracted from the experimental data tape. These have frequencies near 0,5 Hz.

2,5 Hz, and 5 Hz. Table 3 presents the mode shape estimates obtained from the

experiment, Also presented are selected mode shape predictions taken from finite

element analyses. The modes selected were those whose frequencies bracket the experimentally derived ones. The following discussion deals with the Table 3 data in order 4

of increasing frequency.

A good comparison does not exist between the first or third experimental modes and

either of their bracketing finite element analysis modes. Additionally, there is some

bending in the first experimental mode as evidenr..ed by sensors 4, 5, and 6. The

amplitude of this mode is shown in figure 5 along with its ARMAX parameters A, and

A 2 and their primary strip equivalents of damping and frequency. Figures 6 and 7 show

the same information for the second and third modes. respectively. For the second

Y °" t mode, a good comparison does exist between it and the 3.07 Hz finite element analysis

mode. Note that, however, the output of sensor 4 is opposite in sign and reduced in

j

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44 Montgomory and Sundwaralan

TABLE 3, Comparlson Dowten the finite Element Predictlons (P) and Fxperimentol

Identification (E)

Mode I Mode 2 %lade 3

Cumpansons Compansons Comparisons

Origin P E P P E P P E P

Frequency

tliz) 0.364 0.5 0 635 3 29 ZJ 3 07 4.79 5 5.93

Sensor

1 -0.51 0.26 0.45 0.35 -0.45 -0.47 -0.36 -0.04 0.43

2 -0.51 0.46 0 -0.25 0.11 0 -0.34 -0.38 -0.35

3 -11151 0.66 -0.45 0.35 0.44 0.48 -0,36 -0.14 0.43

4 -0-16 0.24 0.41 0 38 -0.08 0.13 0.22 O.Is -0,07

5 -U.:IV -011 0 -0.25 0,05 0 0.34 0.10 0.07

6 -0.36 0.41 -0.41 0.38 -0.03 -0.13 0.22 -001 -007

7 -0,08 0.10 0.36 0.38 0.54 0.51 -0.38 -0,65 -0.45

8 -0.08 0.20 0 -0.24 -0.07 O -0.43 0.39 0,32

9 -0.08 0.10 -0.36 038 -0.54 -0.51 -0.38 -0.54 -0.44

^ 1

1

0

-1 P3.

Al

0

0

A2

----

-2

1.

w, Hz.

0

.z

0

.2

I L i I I I

0 1 2 3 4 5

Time, sec.

FIG. S. Characteristics of the First Mode Identified from Experimental Tape 5,

4

14
I'l

t

Identification of•the Dynamics of a Two-Dlmrnslonal Grid Strut,Kure 45

1.

0

-1

3.

A

1

0

v

A2

—2

3.

.2

—.2

0 1 2 3 4 5

Time, sec.

FIG. 6. Characteristics of the Second Mode Identified from Experimental Tape 5,

amplitude from the finite element prediction. This means that a feedback on that sensor t'

based on the finite element analysts will be destabilizing near the 2,5 Hz frequency.

The validity of this deduction can be established by examining the outputs of sensors 1

and 4 (4). According to the finite element analysis, the 2.5 Hz content of the sensors

^E

!f

should be opposite in sign. However, they are in please in agreement with the identification results.

Conclusion

The application of the least squares lattice filter in system identification has been

extended to a non-integral, two-dimensional grid structure made of overlapping bars.

Previous experience has been limited to an integral free-free beam. Both simulation and

experimental data were used to evaluate the system identification capabilities of the

method, In the simulations, the least squares lattice filter gave good identification of {

simulated modal frequencies, damping, and mode shapes in the presence of sensor

noise expected in the experimental apparatus. However. the spectral decoupling method

of obtaining natural modes from lattice filter triodes required a large number of data

points in the discrete Fourier transform to get adequate ;requency resolution when the

modal frequencies were closely spaced. This problem can be overcome by an eigenvector analysis of the lattice filter's associated Al2MAX model. When the lattice filter'.

I s i l ^

15

,a ,
f i t

r

i'

M{ Montgomoty and Sundarrarslan

>R . 5

tl)3 0^

3

XL

A I 0

1 0 ...^

A2^. }

—2

Yy

0

^.. .2 4# i

o

t'V j I {

.2 bi

0 1 2 3 4 5

Time, s*c, i

FIG. 7. Characteristics of the Third Mode identified from Expcnmentai Tape 5.

was used for system identification with experimental data, the mode shapes identified

t differed significantly from those of the finite element analysis. This has been corroborated by examination of the sensor data and indicates that on-line identification of large

structural dynamic systems may be absolutely necessary to get acceptable performance

f

in a high gain system that requires knowledge of mode shapes. t ,,

References

(I] SUNDARARAJAN, N, and MONTGOMERY, R. C. "Adaptive Identification for the Dynamics of

a 1

Large Space Structures," Proceedings of the 1982 AIAA Guidance and Control Conference, Paper

No. 82-1565, August 1982, pp. 379-381 u+

121 FRIEDLANDER, B. "Lattice Filters For Adaptive Processing," Proceedings of the IEEE, Vol. 70,

No, 8. August 1982, pp. 829-867.

(31 JOL NSON, C. R., JR. and MONTGOMERY. R. C. "A Distributed System Adaptive Control Strat• I

egy," IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-15, No, 5, 1979.

PP,

 601-612.

(41 THAU, F. E., MONTGOMERY. R. C., and HORNER, G. C. "On-Line Structural Parameter Identification," Proceedings of the A/AA Guidance and Control Conference, Albuquerque. New Mexico,

August 19.21, 1981, pp. 530-539.

(5) SUNDARARAJAN, N. and MONTGOMERY, R. C, "Decoupling the Structural Mode Estimated

Using Recursive Lattice Filters," Proceedings of the 21st IEEE Conference on Decision and Control,

December 1982, pp. 998-999,

4

i 16 t

i
I

f

Identification of the Dynernks of a Tw"Imenslonal Grid Structure 47 II

(6) SUNDARARAJAN, N.. WILLIAMS, ).P., and MONTGOMERY, R.C. "Parameter Testing for

Lattice Filter Based Adaptive Modal Control Systems," Proceedings of the AIAA Guidance and Control

Conference, August 1983, pp. 399-605,

(7) SUNDARARAJAN, N. and MONTGOMERY, R.C. "Exp:riments Using Lattice Filters for the

Identification of Structural Dynamics," Proceedings of the 24th AIMtASME/ASCElAHS Structural

Dynamics and Materials Conference, Lake Tahoe. Nevada, Paper No. 83-0880-cp, May 2 .4, 1983.

(8) MONTGOMERY. R.C., HORNER, G.C., and COLE, S.R. "Experimental Research on Structural

Dynp mi,.s and Control," Proceedings of the Third VPI do SU/AIAA Symposium on Dynamics and

Control of Large Flexible Spacecraft, June 1981, pp, 365-377.

17
e-1

i.

FY I

APPENDIX B

EXPERIMENTAL EVALUATION OF FLEXIBLE STRUCTURE IDENTIFICATION

USING LATTICE FILTERS

ii
rt

xmimunAL wALuATion OF nsxIALE SMCTURE

IDENTIFICATION USING LATTICE FILTERS

By

III

VI 

, tOundararai '

A

-an

Ad Dominion University Research Foundation

NASA Langley Research Center

Hampton, VA USA

To be presented at the

7th IFAC Symposium on Identification

and System Parameter Estimation

York, England

July 2-8, 1985

.19

+).!'I
a(

i

^} l

F

t

e

ORIGINAL f-on: "o ­14

NF, POOR QUALITY

EXPERIMENTAL EVALUATION OF FLEXIALB STRUCTURE IDENTIFICATION USING LATTICE FILTERS

N. Sundararejan

old Dominion University Research Foundation, NASA Lsngltiy Research Center, Hampton, VA

Abstract. This paper presents the use of least square lattice filters in

idescation of the dynamics of highly flexible structures. Lattice filters have

been used extensively in the areas of adaptive signal procesing and speech synthesis.

Herein, they are used for on-line identification of the number of codes, made shapes

and modal amplitude time series from the measurement data. The theory is illustrated

using experimental data for a simple free-free base and a sore complex, flexible,

two-dimensional grid apparatus. Results presented Indicate that t1W lattice filter

approach produces effective identification of structural dynamics for the class of

structures studied to this time.

Keywordat Identification, Lattice Filters, Space Vehicles, Least Squares

Approximation, Vibration Maasurements.

i

i;

i}

INTRODUCTION

With the site of the structures currently

contemplated for building in space Mcoming

larger, identification of the dynamic characteristics of these structures is an important area of

research. Accurate on-orbit identification

becomes anecessity as these structuraa cannot be

assembled fully on the ground because of its size,

and also it is difficult to predict an accurate

model on the ground. An the "rfore-- nct rcquira=

ments for these structures in space become

stringent, however, it becomes imperative to

identify their characteristics on<. orbit and modify

the control system as dictated by the identified

control system design model (adaptive cont rol).

This paper highlights the model determination

phase of the adaptive control problem. This phase

involves not only determination of parameter

estimates for an assumed linear form, but also the

order of the linear model form.

An approach for identifying the dynamics of Large

Space Structures (LSS) thdt estimates model order

In addition to model parameters is presented in

Sundararajan and Montgomery (1983). It uses

lattice filters which provide an order as well as

time recursive algorithm for linear least square

signal estimation. A comprehensive tutorial on

the theory and applications of lattice filters has

been given by Freidlander ( 1980). The main

resulte from the paper of Sundnrarajan and

Montgomery (1983) arcs the least square estimate

of the measurement sequence; the model order

required to fit the measurements; the associated

lattice model ( this includes mode shape estimates

that are orthonormol in the measurement space) and

the associated auta-regressive moving average

(ARMA) model of the measurement sequence. The

mode shapes obtained by the lattice, filter are not

the "natural" modes but a linear combination of

them. In order to compare the identified mode

shapes to those predicted by finite element analysis, a decoupling method to obtain natural mode

shapes from the lattice mode shapes have been

developed in Sundararajan and Montgomery (1982).

Using the above spectral decoupling method to

obtain natural mode shapes, mode frequency and

damping can be obtained using an equation error

parameter identification method (Johmaon and

Montgomery (1979)) that employs a second order

ARMA model to represent the natural mode amplitudes. This procedure is followed herein.

The objectf.ve of this paper is to present the

experience in using lattic.- filter theory for

Identification of structural dynamics of two flexible structures. They cons.st of a one-dimensional free-free beam and a two-dimensional

flexible grid apparatus. The structures are part

of an experimental facility at the Aerospace

Control Laboratory at NASA Langley Research Center

for studying advanced control concepts for large

space structures. The beam apparatus provides a

simple structure to test the basic concapte first,

and the grid apparatus provides a more complex

structure close to the real spacecraft. Before

presenting the results of lattice filter identification for these structures, a brief outline of

the basic theory is given. Results are presented

for the identification of the dynamics of the beam

using experimental data. Next, the same is

repeated for the grid apparatuu. Conclusions

based on the above study results are then

summarized.

Summary of Lattice Filter

Identificarion Theory

For the application considered here, we assume

that the kth measurement sample is of the form

Yk ° l

y l (k), y2 (k), . . ., YNS(k) I

where NS represents the number of sensors. It is

assumed that y is generated from a model system

wherein

y k - aq k + n (1)

Here, 4 is an NSxNM mode shape matrix, q k is the

NMx1 mortal amplitude vector, and nk is a NSxI

gauseian random variable with zero mean and b

covariance matrix R. Typically, for structural

dynamite applications, each component of q k fa

the output of an uncoupled second order process.

The task here is to estimate the order (NM) and

20

r

,.
w

a

t

obtain the least squats estimate of qk free the

N+I measurement oamples yo through yN.

SundararaJan and Montgomery (198'•) present a derivation of the equations that re.ste order, n, and

time, i, recursions for the normal!;ad forward and

backward residuals as well so the least square

estimate of the measurement rector. These equations are listed belowt

2 -1/1

-i,n+l (1 ` k i,n+1 ) ( 4 ,n kl,n+l-i-1,n)

r - (I - k

2 )

-1/2 (c k e )

—t,n+l i0n+1 ^i-1 ,n "' l , n+i^t,n

N-I

yN - I E(4,n JE1- i ' n ) (2)

n-0

wherein

k i,n+l - <.,i ^ n , T-i-1 ^n>

and k(x1y) is the othogonal proje.tion operator of

the vector x onto the vector y. The symmetry of

the recursion formulae are apparent. The equations are coupled by the tens ki,n+1 which is

custowarily called the "reflection coefftctenc.'

The dtructure of this equatton is depicted In

Fi g . I where we have used the symbol t' i to

reprosent the time shift operator, i.e.

- 1

Z 

r i,n - ri-I,n

it should be noted at this point that the lattice

fitter is a modified Gram-Schmidt procedure

involving both forward and backward residuals

wherein the backward residuals form an orthogonal

basis for the entire observation sequence. Hence,

any least square estimate 0 the orthogonal

projection onto this basis. Assuming at this

point the order NN has been obtained (vhich is

explained below), the lattice filter has

decomposed the estimation of y into the form of

equation (1),

wherein

4-Irl1,rl,,.)

and

1

i.NM - q(1)

i.e., the backward residuals r i, n form the

orthonormal basis, or the mode shape matrix 0 and

the forward residual Ei,n represents the modal

amplitude time series.

The lattice filter has the following advantageci

1. Given a basis for order N, a basis for

order N+l caei be abtained using the

recursion formulae.

2. Because of the modified Cram-Schmidt

procedure, the baste for all orders n

between 0 and N are the first n elements

of the basis of order N.

3. The estimate assuming any order n

etween 0 and N+I can be computed using

e.:atiorl (2).

Tl^ tat, ice filter provides the information

nee.:ea letermine the residual sequence for any

model order between 0 rnd N+1 Inclusive, This

information provide$ the basis for the model order

dttermtwtan method described next,

Clearly, in this approach one MAY "fit the noise"

by continually Increasing the order of the system;

however, once the order of the estimator has

Increased beyond the correct order, then the

residual errors should its withi p a noise band

which can be predicted based on assumed noise

characteristics. A threshold value can be

selected based on this predicted noise band, and

order can be determined by a test of whether or

not the residual# have been reduced to lie within

the noise band. Also, the test can be made

considering several data samples when using a data

window. SundararaJan and Montgomery (1983) document the experience in choosing the data window

site Nw and the threshold level based on # imulations. Having defined the order required to fit

the data using a linear model, for comparison with

finite element analysis predictions, we seek a

fixed set of bads functions that are spectrally

decoupled. A method to obtain the decoupled modes

from the lattice filter modes using digital

Fourier transform ( OFT) has been presented in

SundararaJan and Montgomery ( 1982). Essentially,

at this point we have estimates for order NM, mode

shapes 0 and modal amplitude time series q(k) from

the lattice filter.

Since the ultimate objective of identification is

for control system design, an ARMA model is

identified using the modal amplitude time series

q(k). The method is b--:: an art equation error

method described In Johnson and Montgomery

(1979). For each mode, the model to described by

the equations

q(k-l) n A l q(k-2) + A2 q(k-3) + B I u(k-2) + B2u(k-3)'

(3)

The equation error is given byi

e(k-I) - q(k- U - q(k-1)

- q(k-1) - ( AI q( k-2 ) + A2q(k-3)

+ e I u(k-l) + 12u(k-3)1

where q is the model amplitude estimated by the

lattice filter, u is the modal control force, k is

the sample number and Al, A2 , B 11 8 2 are

the ARMA coefficient,. The term in brackets is

the model equation. The ARMA coefficients are

then updated byt

A I (k) AI(k-1) q(k-2)

A 2 (k) - A2(k-1)

+ o(k-00

q(k-3)

s I (k) BI(k-l)` q(k-2)

^2 (k) 82(k-Id P(k-3)

The weight a assures stability if

0 < u < 2/(q 2

 N-2) + q2 (k-3) + U2 (k-2) + u2(k-3)(

C3

s

i F

This Identifier performs well to a low noise

environment, but when the information content of

the signal is small, it attempts to fit the noise

(Thou, et. al. (1982). Also, the ideal ARNA model

for the beam has input parameters (B's) which are

three orders of magnitude smaller than the (A's).

This causes a very high sensitivity to noise in

the identification of the B's, and when the input

force Is applied, it tends to cause the identifier { `

gain on the A's to decrease significantly.

21

^r

^..a...a....^^A— •i[lll.l 4'' .__^:^^._^-_..-.__. _. :__..^^
AIM igh these effects are evident in the results

prescoced here, they did not prevent successful

Identification.

It one to Interested In determining the damping

ratios and natural frequencies of the modes, they

can be obtained in a straightforward nannor from

the equation (7), However, It should be noted

that this is not unique due to the foldover

phenomenon due to sampling. by finding the toots

at the equation (9) and using that relation

z n sat In the primary strip, where Y is the

asapling period, the damping ratio and natural

frequency can be obtained.

The behavior of this overall system identification

methodology with experimental data as discussed in

the subsequent sections.

uparimental Studies for the

Identification of a Free-free Dean

In this section, the lattice filter theory

developed earlier is illustrated for the idencification of a one-dimensional tree-free beam. The

identitication scheme yields the strdctural

dynamic characteristics of the basis. The experimental. apparatus for the frae-free beam is shown

in Fig , 2. It consists of a 12-foot beam of

rectangular e^.oas-section which is suspended from

the ceiling 'ay two cables and to attached to tour

electromagnetic (area actuators, There are nine

noneontact.ng deflection sensors that measure the

translational deflection of the beam. The actuators are compensated to eliminate the effects of

friction as much as possible. This cumpancation

Is nonlinear. producing a force in the direction

of the beam motion at the actuator attachment

points which is designed to equslite the effect of

frictions Testing Liao done ^,, manually exciting

the beam approximately in its first flexible mode

and sampling the nine sensors at 64 samples per

second. A total of 5 seconds of data ties stored

on a tape which was post processed with the

algorithm. Figure 7 shows a time history of some

of the maoourdment data processed by the algorithm. The innovations sequence for sensor 4,

INOV 40 Is shown dust below its time history,

Also shown to the norm of the forward estimation

residual, ENORM, which includes all components of

the measurement vector. Below the norm is the

estimate of model order. This was obtained using

a data window of eight samples. Initially, the

urdur estimator fills the data window, and hence,

the Indicated order estimate increases to 6.

After this the order estimator settles to 2

indicating that, even though we attempted to

excite only one mode, there were, in fact, two

significant modes excited. Note also that the

norm of the forward estimation error is small

compared with the value at the start of the

process when the order estimate was settling.

The modal amplitudes obtained from the lattice

filter are spectrally decoupled, tieing the procedure discussed earlier, after enough data are

taken to accurately take the OFT (64 time samplei,

about t second). This occurs at about 1.75

seconds, the first .75 seconds being used for the

identification of mode shapes and model order (see

Fig. 4). Figure 4 shows the modal amplitudes for

both of the identified modes. These are the

signals that are inputs to the parameter identification scheme used to identify the parameters of

the ARMA model of the modes. The identified ARMA

parameters are ohown on Fig. 4 for each of the two

modes identified. The a priori parameter estimates are initially offset from the values

predicted by a finite element analysis which are

also Indicated in Fig. 4. These parameters track

the instantaneous value re q uired to mintease the

output error. one possible smtplanstion of the

oscillatory behavior of the Mods 2 parameter

estimates is the nonlinearity of the actuator

compensation. Nonlinearity is apparent in the

sensor 6 dsta an Fig. ), Note that lattice filter

produces o linear least square fit of the data to

the msssurestnts, and in so doing, produces a

predominantly linear first made estimate and lumps

the nonlinear dynamics into the higher modes,

Thus, the parameter tracking Is more stable in

Mode I and produces estimates of an undamped

(A2­1) oscillation at neatly 2.7 Nt. It the

algorithm is constrained to an order estimate of

one, the predominant response to linear, however,

the fit error to Increased by an order of

magnitude.

The node shapes estimates obtained from the

lattice filter are shown in Fig, $. In this

figure we compare the estimates obtained by three

methods, one analytic, and two experimental.The

4n+0ytic result Is the primary mode oHpa of the

btam using Euler-Darnoulli theory. The two

experimental results which are in substantial

agreement are the nonlinear least squares

algonthe of Thou. at al. (1962) and the lattice

filter algorithm of this paper. Agsin note that

there is apparently an effect of th. four attached

actuators on the dynamics of the test article.

The lattice filter products two modes, one near

the mode of Thou at al. (1982) and another that is

shown on Fig. 5. This other estimated made does

not resemble any mode analytically predicted using

linear Fuler-Bernoulli theory, rather, is required

to modal the effect of nonllnearities in the

apparatus,

Experimental Studies for the

Identification of a Flexible and

iVaxt, the lattice filter identification scheme is

tested in a more complex structure compared to

that of the boom. The candidate structure

considered is that of a two-dimensional tlaxible

grid. Identification results are given using Ow

experimental data obtained from the laboratory

apparatus.

Figure 6 shows the flexible grid experimental

apparatus in the Aerospace Control Laboratory, at

NASA Langley Research Center, The grid to a

y it x 10 ft planar structure made by overlaying

aluminum bare of rectangular cross section. The

bars are centered every foot to that there are

7 vertical and 11 horizontal bare. An shown in

Fig. 6, the grid is suspended by a cable at two

locations on the top horizontal bar. The motion#

of the grid perpendicular to the plane of Fig. 6

are the once of interest in this study, There are

nine noncontaeting deflection sensors mounted on a

back frarme which give a 9x1 measurement vector.

The sensor data is linked to the main Cyber 174

Real Tint Computer System at NASA Langley Research

Center so that the identification can be carried

out in real time.

In order to compare the experimental results of

lattice filter Identification of the flexible grid

facility with predicted values, a finite element

anaaysis of the grid was made which included the

suspension cables. Four degrees of freedom appropriate for motion normal co the plane of the grid

were considered. No damping was included in the

model.

Experiments were conducted using the grid

apparatus descrihed above. The procedure for

conducting the experiments was to excite the grid

using an air shaker. The shaker was capable of

periodicall y exhausting a jet of air that impinged

ORIGINAL P,%, -

"

a 1-

F O

POOR QUALITY

22

E

'sE c

- r-
I'M Cho grid at ucRt,er loaatlom I which wee be the

bet tam loft hand corner of the grid. The

frequency of the jot was d)uotabio from U to

NI me. Tile ro*wltlng geld •xeltatlall Woo Mt

p.to 1 slauseldal but was 11th In he rmoa lc*.

got &.00 of the tango limit* of the deflection

•*-onto - 0 to ppro.tlmrely 2 In. - tho mmstmwe

peek to pea. defloclt*ee of the grid were limited

to about I lo. when ten pedal to peak deflection

moored this limit, (fix air shaker wee turned off

and the grid was allowed to vibrate freely with

only air ad material damping. A Cyber 171 Foal

film computer system .ampiod the deflectiom "wear

data o . 12 • Na. fee 5 soc. The data was stored on

a arotoo date file for further analys is.

This

toot peateduro was repealed for ooveral shaker

frequencies In the rfRge of 1 to IU 48. This

range hem an uspot limit because of the asepling

frequency M w.) The lower limit to "lotted to

Include the predicted lowest vibration node

(2.2 Mr.) tight 4494 sets cerrseponding to

different Shaker eacttation froquencloo were

c f a a 9*4 and Stored on tape$. The following

discuseion pertains to result• •.trotted from dots

met five.

Flgute f presents data from "Moore I and . as

veil me the lattice filter order estimate. •.r

this ca" the order oettmato was based on a 4894

.1ndow, of 4 sample• and spectral decoupllng wee

done With 64 ties samples. From the figure, the

older estimate 1s seen to convergo to an ooclllstlom between 2 and 1 at about .$ sec. at about .4

see, the order estimate was Vnmd at ) and data

collectlnn of the " Claw samples revulred for the

uR was started at )2 Na. The DR as accor

pushed at 41out 2.4 "c and then the Jwcoupling

transformation motrta was calculated. The modal

amplitude• after t41s Claw should contain a single

frequency and Cho transformed mode *hopes should

:urre.pond to the natural modes of the structure

which Were excited. In that mannot, three modes

Wore extracted from seporlawnial data tape 5.

These hew* frog onctea no 4f .5Nr, 2.5 me., and 5

No. Table 1 presents the mode •hope estimates

ont41ned from the exporlment. Also presented ere

selected •ode shape predictions taken from finite

olaMnt anal

y ses. The xo-" "lotted were those

Whose frequencies bracket the experimentally

dwrtced ono@. The followtntg discussion deals with

the TableI data In order of intrees l ng

frequency. A good comparison doeo not exist with

wit her 6racketlng finite element anal

y sts mode and

the first esperlmental mode. Additionally, there

is sees bonding In the o.p*ttmental mode as to

evidenced by sensors 4, 5, a" 6. The medal

amplitude for this mode to shown In Fig. a along

.9th the AAf1A parsawtefe A, and A

Z for ten

mods and their primary strip counerparts of

d4mptng : and frequency m. For the second mode, a

mood comparison does •miff betwen It and the 1.07

Na finite element analysis mode. Note that,

however, the output of the sensor 4 to opposite in

sign and reduced In amplitude from the ftnito

element predict/ , .,. This mean* that a feedback on

that soneor based on the finite element analysis

will be daatabillsing neat the 2.5 Na frequency.

The validity of this deduction can be setabllshed

by sustaining the outputs of " nsora I and 4

(Fig. 7). According to the finite oleduat analyst•, the 2.5 Na content of the "nsure should be

opposite In sign. However, they are In phase In

agrrement With the identification results.

CONCLUSION

The application of least square lattice filter* In

Identifying the dynamic characteristics of highly

flevlble structures has boon presented. The

theory has been used to identify the structural

characteristics of tom experimental hardware,

Mammly that of a fro*-fee* Mee and a limitable

grid structure. The results indicate that the

lactic* filter con be elfe.ctively weed for on-tine

I dent I I a c at ton of the ',..must of modes, code

shapes, medal looping and wool frequencies from

the meaeurelmnt data. The ospmrlM ntal reawlte

also Indicate that there to considerable disagreesent between them and 410141vtical predictions.

Dash M thee* osporteental studies, the main

conclusion :hat can be drawn to that for large

apace structures, em-erblt testing and Identifiesties ss"ntlal before central to attempted.

44Ft RA NCI s

Friedlander, 4. (19110). Lattice Filters Cur

Adoptive 

p rocessing. proceedings of the ![it. 1 0. pp. f27-467.

Johnson, C. A., law. and A. C. Mantgoawry (1979). A Distributed System Auapttvo Control

strategy. 1199 TfansectiOM$ on A4lfo*f4c* and

Electronic System*, ALS-15, pp. 601-612.

Sondarors;ar., N., and woncgomerv, R. C. (1942).

Decoupling The Structural Mode Estimated Using

Recursive Lattice Filters, proceedings of the

21et Ittt Conference on Decision and Control,

pp. 494-41

Sunderare)ea, N., and NontdoMry, R- C- 1194)).

Identification of Structural Dynamics system

Using Least Square Lactice Filters. Journal

of Guidance, Control end _Dynamic pp.

)74 )4; .

Them, F. 1., 111smov, T., A., and Montgomery,

A. C. (1v42). Least Square wqu*ntlal and

Parameter and State Estimation for Large Space

Structure$. Procee4ln a of the 1 1062 American

Control Conference, pp. le - 1.

Table 1.- Comparison Eotwoon the Finite tloawnt

r* ctTOtl^ aT 'IF

r Mn tom - --

Id entification

Node I Mode 2 Mad* )

Caparisons Comparisons Comparison./

originP

E

pt

EF

Ptp

q.

) .625 2.2i 2.5 1.07 6.79

5 5.91

Na

sur 1

6 .45 .)5 -.45 -.4) -.3n -.04 .43

2

r

6

0 -.25 .11 -.00 -.3 1. -.34 -.)5

3

1

6 -.45 .35 .44 .44 -,3h -.14 .43

4 14 .41 .3R -.O4 .1) .22 .14 -.07 5 .1 0 -.25 .05 0 .2 1. .10 .07 6 1 -.41 .)4 -.03 -.1) .2! -.01 -.07 7 10 .)6 .34 .54 .51 -.31 -.65 -.45 4 20 0 -.24 -.07 0 -.4) .24 .32 9 10 -.]6 .36 -.54 -.51 -.l4 -.54 -.44

F - Prediction based on finite element analysis

4 - Wculatlon based on Identification from

experimental data

23

t

C.Ji
t. tt ..f1^,

i, .. ^ NI

4.1.

/1•. i Flow dt.drae of lattice e.ttion

Ir

.1 . M•

. 1 vvvvvvvvvvvv

.1

•

.1

M•

IIM," 1. d^Gd^

as 1_

l• r

M- .. r

o .Mti•wwll4Aw^.^MMM...r

' fT a

n. w..

o ^^

A

rw....

A^

II{. ) Data tole -ant to the foal ties proC••e1,tR

of the •lborithe

AI

AZ

^I

I

; I

AX

l^

r

VIS. 1 A photograph of t11e beam dt•p•r.tu.

P^

As

ORIGINAL PA' t^

OF POOR QUALITY

i

..

mods

1 DP! data

0

fv Decoupled trod..

Order .etioation

0

0

lbde Z

.e

0 vvvvvvvvvvvv-v

-.e

.

0 1

0

rr^J^`M

0 I ) 5

TLao. sec

Fig. A tatt" ted •odei amplitude. And Identified

ALUMpar.•etera ret.ultlru from the

1 •ISOr I he

24
*.ea r

a2

r, N*

C

V

1

c

r

0

-1L

O

3r

lLI^

2

«^^;;Trr

}

ORIGINAL V.— 7 1

OF POOR QUALITY

1

1

, 4 . la. O a1

Ord*'. a 1

O

1 1 1 f^^

O 7 S S

T law. sec

Fig. 2 Sensor 1 and 4 from experimental tape S

andthe estloste of "gr.&I order obtained

ey the lattice (liter

Fig. $ Mode *hope estimate* obtained from the

algorlthaa and a comparison with an

&nely[!c pr*dlctlan and the estlmates

obtained using the nonllnear lout squares

t*chnlqus of Thau.

O 1 2 ] 4 S

Item. ..c

fig. 8 Characteristic* of the ••cord mode

identified from the experimental tape S.

Fig. 6 A photo8:aph of the flexlbie drld

apparatus.

25

0
|

^

|! »

°

.

. .

°

APPENDIX C

PROGRESS IN ADAPATIVE CONTROL OF FLEXIBLE SPACECRAFT USING

LATTICE FILTERS

}

`

|

^

}.

i

i

' '"
PROGRESS IN ADAPTIVE CONTROL OF FLEXIBLE

SPACECRAFT USING LATTICE FILTERS

by

N. Sundararajan

Old Dominion University Research Foundation

Hampton, VA

and

Raymond C. Montgomery ;F

NASA Langley Research Center

Hampton, VA

Workshop on Applications of Adaptive Control

Yale University, New Haven, Cr

May 29-31, 1985

27

nay
OKK

P J I

FROC".SS IN ADAPTIVE CONTROL OF FLEXIBLE VAQtEAFT US I1C LATTI C X FI LTUS

N. Sunderara)an

Old Dominion University Research Foundation

Raymond C. Montgomery

NASA Langley Research Center

ABSTRACT

This paper reviews the use of the least square

lattice filter in adaptive control systems.

Lattice filters have been used primarily in speech

and signal processing, but they have utility in

adaptive control because of their order-recursive

nature. They are especially useful in dealing with

structural dynamics systems wherein the order of a

controller required to damp a vibration is variable

depending on the number of modes significantly

excited. Applications art presented for adaptive

control of a flexible beam. Also, difficulties in

the practical implementation of the lattice filter

In adoptive cc.ntrol are discussed.

1 N'rRODIICTl ON

V- or large flexible spacecraft, design models will

probably not be adequate. Wence, an adaptive

control s y stems is highly desirable. Earl) research

into adaptive vibration control of large flexible

structures is reported in reference I. Therein,

adaptive control of a spinning annular most ntun

control device (AMCD) was studied. That scheme

consisted of simultaneous Identification and

control with the objective of regulating the outof-plane deflections of the spinning A140). Some of

the disadvantages of the method were the requirement of selecting t he number of modes to be used

for controller design, the use of aualytI ally

predicted mode shapes, and the coupling between

modes due to ir.homogenities In the system. Lattice

filter adaptive control is a new method which

attempts to overcome these problems. It Is, hence,

well suited for t;.e adaptivr ccntrol of flex'.ble

spacecraft.

The least square lattice filter has been used

extensively in the field of speech and signal

processing (reference 2). In there applications

the filter is desig:,ed based on a predetermined

estimate of system order. Reference 3 1s a comprehensive tutorial on this su bject. Concerning

adaptive control, reference 4 proposes a selftuning controller configuration using lattice

ftlter4. This scheme requires cumputing the polynomial coefficients for the plant end controller at

each Iteration and enforcing a known feedback

structure for the controller. Reference 5 proposes

inverting the transfer function of the plant for

general adaptive control. This idea, with the

least mean squares ( LMS) algorithm, was utilized In

reference A to obtain adaptive control.

Reference 7 proposes a similar approach using

lattice forms instead of the LMS algorithm.

Reference R takes this approach but uses a lattice

model instead of an autogressive, moving average

with exogenous variables (ARMAX) model where

familiar controller techniques could be used. All

of these scheate attempt simultaneous tdentificatlon and control or direct adaptive control. For

each cost stability questions are not resolved

analytically; neither are simulation results

available in the open literature.

As opposed to simultaneous identification and

control, the scheme discussed herein consists of

conducting tests to obtain A design model, valldating the model, designing a controller based on

the validated model, and finally, engaging the

control system. This approach is Ideally suited to

the control of large flexible spacecraft because of

the passive environment of outer space and the

potential of relaxation to a controller that is

known to be stable - that of collocated rate feedback. It was originally presented in reference 9

and represented the first use of a recursive variable order struc-,rs for adaptive control.

Therein, the lattice filter was used to provide an

on-line estimate of the system order, mode shapes,

and modal amplitudes to provide a validated medal

control design model. After the identified model

parameters are validated through a series of test

pr.)cedures, they are used In a modal pole-placement

ontrol law design. Figure I shove the adaptive

control scheme using lattice filters.

P ,I

Js

a^

a^ r .^...^ aawea•^ ^ a,..^

Figure 1.- Adaptive Control with Lattice Filters.

28
The purpose of this paper is to assess progress in

using lattice filtera in adaptive control of

flexible spacecraft and to highlight problem areas

for further researeh. First, lattice filter theory

and order determination is summarized following the

original development of reference 10. Then, their

use in adaptive control is discussed along with

applications to the vibration control of a bees„

Finally, difficulties arising in the practical

implementation are discussed.

SUMMARY OF LATTICE FILTER THEORY

AND ORDER DMOM MATION

For application considered herein, we assume that

the 1th measurement sample is of the form

YTi n (y 1 (i), y2 (i), " " YNS(i))

.here NS represents the number of sensors. It is

assumed that y is generated from a model syste+a

wherein

yi - ^. T i + v i

(1)

Here, I is an NS x NM mode shape matrix, ^i is

the NM x i modal amplitude vector, v i is a

Gaussian-random variable with zero mean and

covariance matrix R. NM represents the number of

modes in the system or order of the system.

Reference 10 presents a derivation of the equations

that relate any order. n, and time, i, recursions

for the normalized forward and backward residuals

as well as the least square estimate of the

measurement vector. These equations are listed

below:

e - (1 —k 2

)-1/2 (e —k r )

—1,n+1 i,n+l —i,n i,n+l—i— 1 ,n

Z

- (1 — k2

)-1/2(r —k e )

—i,n+l i'n+l —i-1 ,n i,n+1—i,n

wherein

ki,n+i - Cei,n' ri—1,nl

and C ? represents an inner product. The symmetry

of the recursion formulae is apparent. The equa -

tions are coupled by the term ki,n+l which Is

customarily called the "reflection ccefficient."

The estimate of the measurement (reference 10) at

sample 1 for a model of order n is

n-1.

'^04 (e i,iI ri—i,j)

where represents an orthogonal projection

operator. Hence,

k

yi,n - (r n-1,0' rn-1,1' ** " rn—l,n-11

i,l

k1,2

k'i ,n

so that

2

yi - 4 L' Y i + ei,n (2)

where 1L is an orthonormal NS x n matrix

I rn-1 ,0' " ', %—l,n-1I generated from the

lattice filter, and i'i is the n dimensional

vector of reflection coefficients and ei,n is the

NS dimensional estimation error vector.

Clearly, in this approach one may "fit the noise"

by continually increasing the order of the system;

however, once the order of the estimator has

increased beyond the correct order, then the

residual errors should lie within a noise band

which can be predicted a rp^ior i based on assumed

noine characteristics. A threshold value can be

selected based on this predicted noise band and

order determined by a test of whether or not the

residuals have been reduced to lie within the noise

band. The residuals will generally consist of

signal and noise parts — the signal part being

reduced as the correct order is reached until the

residuals essentially consist only of noise. This

teat is carried out based on a data window of NW

saspL^ts. Thus, assuming that the data can fit a

linear model and that the noise process is

Gaussian, for i large enough,

E(

NW

ei, ci n ) NW E(vivT

n

i-1 '

NS

- NW tr E(v ivi) - NW a23)(

n 1

where E in the expectation operator. This can be

used as the one sigma threshold for the order

determination test. In the last equation a j is

the sta0dard deviation of the noise process

for the nth sensor. Reference 10 documents experience in choosing the data window size NW and the

threshold level based on simulations.

ADAPTIVE CONTROL USING LATTICE FILTERS

Independent Modal Space Control (IMSC)

(reference 11) is a control scheme specifically

designed to deal with flexible spacecraft in a.

modal form amenable to control law design.

Unfortunately, it requires natural modes and not

the orrhonormal basis provided by the lattice

filter. Consequently, in order to interface the

lattice filter outputs with the target adaptive

control scheme (figure 1) and to make comparisons

with finite element analysis predictions of natural

modes, a method is needed to obtain natural mode

shape estimates from the lattice filter basis. The

filter updates tha NM basis vectors at every sampl::

instant. While the order estimate NM remains

constant, the updated basis vectors are related by

a mere rotational transformation. The assumption

of the target adaptive control scheme is that the

system motions can be modelled by a constant and

finite set of natural modes and their associated

modal amplitudes over a reasonably long time

interval. Hence, when the estimated system order

Is constant, the basis elements used to derive the

29

r

a4

kf

if

i

,

.. ..^..4^^.rrwr—..ru^r^.^.nY^r^l^rurf •. ^u ^.. _. ^._^ _.• _._ __—_r^^^^...J^^____._._
3

30

model amplitude time series required by the target

` adaptive control scheme are not changed.

The transformation from the lattice filter to a

natural made basis should satisfy

y - # L Y 

L - # N 

Y 

N

wherein the subscript L refers to the lattice

filter and N refers to the natural modes. A nonsingular matrix T, will satisfy this condition

provided

i,

# L T - I N T N - TYL

(4)

Since the order estimate is assumed constant, this

matrix can be approximately determined on-line

using the digital Fourier transform OFT). Herein,

this is accomplished as follows, Since the lattice

filter uses the current measurement sample as its

first basis element, the corresponding modal amplitude time series contains NM frequencies. Hence,

the DFT spect um of this series will contain NN

peaks care" ponding to these frequencies. The

frequencies (w l , w2 , ..., wNM) can thus be

identified by searching this spectrum for these

peaks. Assuming that the motion is comprised of

undamped structural vibrations, the matrix T, which

pro4tAces the desired transformation can be calculated as

[Re(Yl(wl )j . . . . . . . RefY'L(wNN))

T -

Re[YNM(w1)) . . . . . . . Re(YNM( wNM)^I

wherein, (*L(w), .., (w)) is an NM dimensional vector of •the modal amplitude transform.

Using this matrix, the digits! Fourier rranafo m of

each component of *N wit, be zero at the discrete

frequencies, w i , jsi. ,;a item which degrades

this approximation is cite error in using DFT

Instead of the true Fourier transform. still

another is the assumption that the motion is made

up of undamped structural oecillati^ns. In spite

of these items, reference 12 shows that this

approach produces good estimates of the natural

modes for the beam used herein.

The decoupled modal amplitude time series, N(i),

as obtained above in equation (4), is then analyzed, for each mode, to identify the parameters of

its autoregressive, moving average (ARMA) model.

The inputs to each ARMA modal model are the

generalized forces and hence, each model takes on

the form:

7 N (i) - AI T N(1-1) + A2YN(1-2)

+ B l f(i-1) + B 2 f(i-2) (5)

where the f represents the generalized forces.

Given the ^N and f I a, the parameters A and B

above are identified and used in the control law

design process. Thus, the ARMA model output error

is

e(i-1) :N(i-1) - JA I (i-1)Y N(1-2) + A 2 (1-01 N(1-3)

+ B l (i-1)f(i-2) + 62(i-1)f(i-3)) (6)

The gradient technique of reference 1 1s used to

identify the parameters p - (A i , Bi ) using the

iteration sequence

PM - P(i-1) + e(i-1) [W I Y N(1-2), W2 TN(1-3),

W3 0i-2), W4 f(i-3)) (7)

As indicated in reference 1, the weights W must be

selected consistent with the relation

W

1 '1

2

N

(1-2)+ W2YN(1-3) + W3f2(i-2)

+ W4 f 2 (1-3) < 2

and the inputs to the algorithm, *N and f, must

be sufficiently varying and large if the parameters

are to converge to their correct value.

For the identification and control scheme explained

above to work satisfactorily in a closed loop

environment, it is necessary to validate the design

model. Three teats are suggested herein which

check the following; 1) model fit error; 2)

parameter convergence; and, 3) signal information. These tests have been used successfully in

simulation and experimental work. The fit error

test uses a fixed parameter set to calculate an

estimated modal displacement for the past NT

samples.

NT

°fit ; ;r N(1=n) - tn1 WN (i-n-1) + YN(i-n-2)

n-0

+ B 1 f N(i-n-1) + B2f(i-n-2)), k 5 NT

If the absolute sum of the error between the

modelled displacement and the displacement calculated by the lattice filter exceeds a given threshold, the fixed parameter set is updated with the

present identified parameter set. This process is

repeated until the parameter set fits the data.

The convergence test runs concurrently with the fit

test. It simply checks the magnitude of the

changes in successive estimated parameters.

ogonv )n

UT

-O IPin - pn-lI for pT - [11

2, 91 0 B2]

If the absolute sum of ten successive parameter

estimates changes is above a specified level, a

logical switch is set to indicate failure. The

third and final test is on information content of

the estimated modal amplitude signals from the

lattice filters. The purpose of this test is to

check whether enough information is present in the

signal for proper identification of the

parameters. If this test fails, the controller
11 .

^f

gains are not updated based on the identified

parameters, but are frozen at the last values

before the test failed. Here, the estimated modal

amplitudes and velocities from the lattice filter

are checked for sufficient excitation by summing

over can samples.

NT

oinf 

< n 7 oN) 

+ I ON - *N-1I

The second term in the above equation represents a

measure of velocity estimates. If the sum is below

a threshold, o inf , the updating of the control

gains based on the identified parameters is

stopped. The information and fit error taste

constitute one test for each mode and the convergence and reasonability tests constitute four tests

for each mode. Thus, six tests must be passed

beforecontrol is applied to a given mode. The

actual stability and performance of the controller

is directly affected by the criteria chosen for

passing a test. If the test criteria are too

stringent, system noise and nonlinearities may

preclude initiation of control. However, if the

tests are not adequate, it Is possible that an

error in the estimated parameters could result in

gain calculations which produce an unstable

system.

Now, consider the philosophy to be used when the

tests described above pass or fail. When all the

nests for parameters of a given mode have passed,

control gains are calculated according to a

previously devoloped pole placement scheme

(reference 1). The control force commands are then

calculated using these control gains. Considering

the philosophy used when the tests fail, two cases

were studied. In the first case, Then the taws

failed, control was turned off and the control

forces were made zero. In the second case, when

the tests failed, updating of the control gains was

stopped and they were frozen at their values prior

to the test failure. In this case, the control

forces were not made zero and were computed using

the frozen control gains. From a detailed study of

both cases ; it was found that the performance of

the adaptive control system in the first case was

superior to that of the second case.

APPLICATION TO A V=1BL6 WEAN

The closed-loop adaptive control scheme of figure 1

has been tested in the digital simulation for the

12-foot, flexible free-free beam located at NASA

Langley Research Center. The simulation contains

the mathematical model of the beam apparatus in

modal form. For this study, the simulation

contains one rigid-body mode, the first three

flexible modes, nine deflection sensors, and four

actuators for control purposes. The initial

conditions on the modal displacements were set to

.05 in. and the modal velocities were set to zero.

The modal damping was also set to zero. A digital

sampling rate of 32 Hz was selected for the simulation, and the standard deviation for all measurement noise was assumed to be .005 based on observed

noise in the available hardware. The lattice

filter estimates were based otz a data window size 4

(reference 10). The testing procedures were all

carried out based on data window (NT) of ten

samples. Initial parameters estimates were offset

from the mathematically correct values to test and

verity the rapid convergence of the identification

algorithm, An arbitrary delay of 2 seconds was

added between the time identification starts and

when the control would be applied to show the

behavior of the identification scheme.

At the start of the simulation, the lattice filter

determines the number of modes in the simulation

along With the mode shapes. Modal amplitude time

histories are then generated. From the lattice

filter mode shapes and modal amplitudes, natural

modes and modal amplitudes are obtained through a

linear transformation explained in the earlier

section. The application of the transformation is

delayed for 2 seconds because the online

transformation technique of reference 12 requires

2 seconds of data for a digital Fourier transform

data base to obtain the required transformation.

The natural modal amplitudes are input to the

equation-error parameter identifier which

identifies the ARHA parameters. The identification

results are then tested using the test procedures

described above. When the tests are passed, the

control is turned on. Results of the simulations

are presented in figures 2-4.

Figure 2 shows the estimated modal displacement for

the ftrst lattice filter mode. The order estimate

plot shows that the correct order of 4 is obtained

in .3 seconds. After the parameter identification,

when all the testa are passed, th6 control is

turned on at 5.5 seconds and the modes are damped.

The result of the adaptive control or the natural

modes is shown in figure 3. It is evident that

when the identification is validated by passing the

tests and control turned on, the vibration

suppression is achieved. When the modes are damped

out the lattice filter order estimate drops from 4

to l indicating the flexible modes are damped out.

Although the lattice filter order decreased, the

control design order was maintained at 4 throughout

the time interval when control was on. Allowing

the order to vary in real time and updating the

control order is a topic for further studies.

11 r4•f doror•Inotton

f^ I.attk• lllt•r d.co.p1hq

arnwar Id.od(leseIOA AM t••tlnt

Ilk I In

/ ^^,--^/+-..` antrpl rrd.atn and epp lle.t ton

5

LATTICE

FILTER p

NODE 1

-.0

B •

CONTROL

OE5164

ORDER

0

0 2 4 6 8 IO

TIME, SEC.

Figure 2.- Typical time history of an adaptive

control run using identification,

testing, and control design.

I

k

31

j' 

4
n

0:1E

0

•.a

a ^^

s ^ II1^ ^ 11

1OU S t

_ I

o•

• rr,

Larrttt

I^.

0 i tta

ADl n

0

I I I 1 I I

0 2 4 6 E 10

TIME. SEC.

Figure 7.- Tins histories of three natural modes

with the lattice filter order Indicated.

The main results of the Identification and the test

procedures are summarized in figure 4. for the

first flexible mode, the fig to shows the time

histories of identified free,uency parameter AI

the fit error, a parameter that Indicates algorithm

• uwrn ^`^ ^^- I L

e

(, fI

.75 

r 1

<om.r •cn st* 50

25 I/.... rlPP 1.4.

C

,•rotil•

am_ _

.02 I, r

111, 14,1100 d •. 4,.I• .... ,n.....0

0 

IMON—ION,50

.25 N ..........•,......• ,.

r - Vesrti nu r

—J 0 2 V 6 !1 !0

TIME. SEC.

figure 4.- Time histories of the test variables for

one mode with the test thresholds and

logic sum of the tests Indicated.

convergence, and a parameter that Indlestes

Information content of the measurements. When all

the tests are passed, the corresponding pass

parameter (plotted as a binary logical varalble) is

set to one. The various thresholds for the testa

are /Iso marked to indicate when the tests pass.

Theca thresholds were determined based on detailed

sensitivity studies of the modal control schema for

the been (reference 1]). An error was intentionlly

put on the Initial estimate of A I so that the

convergence of the estimates to the correct value

could be observed. When the Identifier to turned

on, the •attest• converges to the true value of 1.8

from 7. The thresholds indicate that the fit error

test to passed first and than the convergence

test. With enough signal In the messur@menre the

Information test Is always passed. When all the

tests are passed of 5.5 sec, the control Is turned

on. When control is fully effective, that is when

the modal are damped out, the measurement date will

contain only the noise and the information test

will fall. This to immediately seen from the

history of At

 as it starts oscillating with large

amplitude Indicatint; that the nodal amplitude

signal contain/ mainly note@. Also, If the

parameter excursions are large, the convergence

tests will also fail indicating a failure for the

binary variable pass. Once this happens, the

control gain updating is stopped, and control

forcer jar@ made :.aro.

PRO g mts IM PRACTICAL IMPIJJI/NTATIOM

The adaptive control scheme of figure 1 is good

from the engineering point of view since only validated models are used for control oysters desi g n. A

natural Question arises as to the course of action

when validation tests fail. The operat17 , g environwent for large flexible spacecraft is, fortunately,

benign and a ayetme designed to suppress vibrations

can be shut down at the expense of having to

conduct relatively lor.g tern maneuvers. Another

saving feature of large flexible spacecraft is that

collocated rate feedback Is stable and relaxation

of the system to this mode of operatlon is also

possible, again, with corresponding degradation in

performance. Therefore, two options that can be

evoked are; one, to shut down the control system

and the other, to revert to a robust control system

design which Insures stability.

At first glance one say wish to use the ARMA model

generated by the lattice filter directly in the

design process rather then using IMSC with its

requirement of generating natural modes. Ilnfortunately , the current online design ca p ability for

controllers of vector ARMA processes is not adequate. Having selected IMSC, one must obtain

natural modes from the vector ARMA model or from

the measurement time series. Here the same problem

arises, that to, the current capability of

eigenvalue / vector analysis for vector ARMA

processes is :.adequate for online implementation.

Hence, a time aeries analysis using a DFT has been

selected. The accuracy of the process of

extracting natural modes is directly affected by

the number of data points processed. Hence, there

is a tradeoff to be made between the higher

t 

32

FW D
complexity in control computations versus the error

in the natural modes using the OFT approach. Also,

significant computational saving results if the

approximation of zero damping can be evoked. If

this approximation cannot be made, then one must

work with complex modes.

Since several approximations are required by the

system, a method of validating the models used In

the online controller design is essential.

Analytic methods of validating models based on

statistical error analysis (e.g. Cramer-Rao bounds)

are not adequate. Currently, tests on fit error,

algorithm convergence, information content of the

measurements, and reasonability have 'seen u"d.

The thresholds and design constants for these tests

can be determined only by exhaustive simulation

and/or hardware testa and is not an online

procedure.

CONCLUSION

This paper reviews the use of the least square

lattice filter in adaptive control systems.

Emphasis is placed on the integration of the

lattice filter into a practical parameter adaptive

control system. One novel feature of the

recommended system is the inclusion of a design

model validation scheme based on model fit error,

algorithm convergence, and signal information

content. An application is presented for adaptive

control of a flexible beam. These results indicate

that the lattice filter adaptive scheme 49

practical for vibration control of large ft:-ible

spacecraft. Difficulties in the practical

implementation of the lattice filter in adaptive

control are also discussed. These centered around

the computational burden of transforming lattice

filter modes into natural modes and the selection

of the thresholds for online validation tests.

7. Olcer, S. and Morf, M.: Adaptive Control by

Ladder Forma. Proc. of American Control

Conference, June 1984, San Diego, pp.

1265-1270.

8. Shah, S. C., Walker, R. A. and Saberi, H. A.:

Multivariable Adaptive Control Algorithms and

Their Mechanizations for Aerospace Applicstione,. Proc. of IEEE Conference on Decision

and Control, Las Vegss, Nevada, December 1984,

pp. 381-386.

9. Sundararajan, N., Williams, J. P. and

Montgomery, R. C.: Adaptive Nodal Control of

Structural Dynamics Systems Using Recursive

Lattice Filters. J. of Guidance, Control and

Dynamics, Vol. 8, No. 2, 1985, pp. 223-229.

10. Sundararajan, N., and Montgomery, Re C.:

Identification of Structural Dynamics System

Using Least Square Lattice Filters. Journal of

Guidance, Control and Dynamics, Vol. 6, No. 5,

September-October 1983, pp. 374-381.

11. Msirovitch, L. and Oz, H. Modal Space Can rol

of Distributed Gyroscopic Systems. Journal of

Guidance and Control, Vol. 3, No. 2, JanuaryFebruary 1980, pp.140-150.

l2. Sundararajan, N., Montgomery, R. C.: Decoupling the Structural Mode Estimated Using

Recursive Lattice Filters. Proceedings of the

21st IEEE, Conference on Decision and Control,

December 032, pp.99i;=9979.

13. Williams, J. P. and Montgomery, R. C.: Simulation and Testing of Digital Control on a Flexible Beam. Proceedings of the AIM Guidance and

Control Conference, August 1982, pp. 403-409.

, t

'i

7 f 1

F V^}

t

I

REFERENCES

1, Johnson, C. Re and Montgomery, R. Co: A

Distributed System Adaptive Control Strategy.

IEEE Trans. on Aerospace and Electronic

Systems, Vol.-AES-15, No. ^, 1979, pp. 601-612.

2. IEEE Trans. on Acoustics, Speech and Signal

Processing, Special Issue on Adaptive Signal

Processing, Vol. ASSP-29, No. 3, June 1981.

3, Friedlander, Be: lattice Filters for Adaptive

Processing. Proc. of IEEE, Vol. 70, No. 8,

August 1982, pp. 829-867.

4. Friedlander, B.: Recursive Lattice Forms for

Adaptive Control. Proc. JACC, San Francisco,

California, 1980.

5. Martin-Sanchez. J.; A New Solution to Adaptive

Control. Proc. IEEE, Vol. 64, No. 8, August

1976, pp. 1209-1218.

6. Widrow, Be, McCool, J. M. and Medoff, Be P.:

Adaptive Control by Inverse Modelling. Proc.

of 12 ASILOMAR Conference on Circuits, Systems

and Signal Processing, 1978, pp. 90-94.

6 33
4

^a

f

APPENDIX D

ROBUST CONTROLLER SYNTHESIS FOR A LARGE FLEXIBLE SPACE ANTENNA
.

ROBUST COWROLLER SYNTHESIS FOR A LARGE

FMIRLE SPACE ANTENNA

`. N. Sundarara an*p S. M. Joshi, and E. S. Armstrong

NASA-Langley Research Center

Hampton, Virginia 23665

SUMMARY

This paper investigates the application of the linear-quadraticGaussian (LQG)/ loop transfer recovery (LTR) method to the problem of

synthesizing a fine-pointing control system for a large flexible space

antenna. The LQG/LTR approach of synthesizing a multivariable controller

in the frequency domain is selected because large flexible structures can

be mod.41led with elastic mode transfer functions as additive perturbations

on the rigid body model and the LQG/LTR approach uses this formulation

naturally for robust control design. The study is based on a finite

element model of the 122 m Hoop/Column antenna, which consists of three

rigid-body rotational modes and the first ten elastic modes. A robust

compensator design for achieving the required pointing performance in the

presence of modeling uncertainties is obtained using the LQG/LTR method.

For the Hoop/Column antenna, a satisfactory controller design meeting the

desired bandwith of .1 rad/sec and ensuring stability with unmodelled high

frequency modes was obtained using only a colocated pair of 3-axis

attitude sensors and torque actuators. This study also indicates that to

achieve the desired performance bandwidth of 0.1 rad/sec. and to ensure

stability against higher frequency elastic modes, the design model should

include the first three flexible modes together with the rigid body modes.

* Old Dominion University Research Foundation, Norfolk, VA

,j

j:.

35

w
IWMODUCTION

One of the planned activities of the NASA's Space Transportation

System is the placement in earth orbit of a variety of large space

antennas. Potential space missions will require antennas and structures

ranging from 30m to 20km in size. Applications include communications

(mobile), remote sensing (soil moisture, salinity, etc.), deep space

network (orbital relays), astronomy (x-ray, observatory, optical array,

radio telescope, very long baseline interferometry, etc.), energy and

space platforms. Specific missions have been pinpointed and future

requirements have been identified for urge space antennas for

communications, earth sensing and radio astronomy (1). 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

schematically in Figure 1, basically consisting of a deployable central

mast attached to a deployable hoop by cables held in tension (2). The

deployable mast consists of a number of telescoping sections, and the hoop

consists of 48 rigid segments. The reflective mesh, which is made of knit

gold-plated molybdenum wire, is attached to the hoop by graphite fibers.

The mesh is shaped using a network of stringers and ties to form the radio

frequency (RF) reflective surface. In order to achieve required RF

performance, the antenna must be controlled to specified precision in

attitude and shape. For example, for missions such as land mobile

satellite system (LMSS), which is a communication concept for providing

mobile telephone service to users in the continental United States, it is

necessary to achieve a pointing accuracy of 0.03 degree RMS (root mean

square) and a surface accuracy of 6 mm RMS. It is also necessary to

I

36

i 
is

^• R f

2 

M

' r

have stringent control on the motion of the feed (located near one and of

r;

the masi) relative to the mesh. Because of its large size and relatively

light weight, the antenna is highly flexible, with a large number of

significant elastic modes. Its dynamics can be represented by partial

R differential equations, or by very large systems of ordinary differential

equations. The resulting equations have many resonant frequencies, some

of which may be very law, and possibly closely spaced. The natural

damping is usually very small. For these reasons, control of large space 

structures is a challenging task (3). Since the system is inherently of 

high order, a practical controller has to be based on a reduced—order

"design" model. Furt%ermore, the parameters (i.e., frequencies, mode 

shapes, and damping ratios) of the" system are known imprecisely. This

introduces additional modeling errors. Typically, the modeling errors for

finite element models increase substantially with increasing modal

frequency. j

Reduced—order control synthesis for the Hoop/Column antenna using the

» standard LQG theory was investigated in (4,5). The standard LQG procedure {

yielded satisfactory control, i.e., rigid—body bandwidth of up to 0.25 jJ

rad/sec, satisfactory time constants for the elastic modes, and acceptable

- root mean square (RMS) pointing errors in the presence of sensor noise.

'w It should be noted that the LQG approach in reference 4 used a large

&. number of actuators and sensors ( four 3—axis torque actuators and four

3—axis attitude and rate sensors). It was found in (4) that the first

three flexible modes had to be included in the "design" model (in addition

^ R

37
t k

3

k

to the three rigid modes) to obtain satisfactory performance. The main

problem with the LQG method was that a large number of weighting

parameters had to be simultaneously adjusted to obtain a good design. In

addition, the stability robustness property with respect to inaccuracies

in the modal parameters could not be properly evaluated because it was

difficult to effectively characterize the bounds c:n modeling errors in a

time-domair. setting. In order to reduce these difficulties, normally one

checks the control design for robustness after the control design is

completed using LQG or any other method. Such an approach in the

frequency domain using singular value measures was presented in (6) for a

large space structure using different control design

integ ral ftedback ace aiu ^.n, , I,rn uenn nhnnnvi I AG etc' a+^ ww....^u -P . 4 f 

Unlike

the LQG/LTR approach provides a means of including

uncertainities, in the control design process itself*

frequency domain, it extends the basic frequency domai

methods like LQC,

the above methods]

the robustness-toSince it is in the

n design guidelines

like bandwith, cross-over frequency, etc. from a scalar system to a

multivariable system.

The newly emerging robust control synthesis methodology which uses

frequency domain matrix norm bounds (i.e., singular values) has received

considerable attention in the recent literature [7-9). The basic

framework for frequency domain synthesis using the LQG/LTR methodology was

developed in (7-9). It has been applied to diverse oystems such as power

systems [10) and aircraft engine control (11). The LQG/LTR design

philosophy uses a low-frequency "design model" of the plant and a

38
A.

high — frequency characterization of the modeling errors. This method,

which characterizes unstructured uncertainty with singular value bounds,

appears to be particularly well suited for the control of large flexible

spacecraft due to the considerable uncertainty that inherently exists in

the mathematical models.

The purpose of this paper is to investigate the use of LQG/LTR

multivariable frequency domain methodology in the design of an attitude

control system for the Hoop/Column antenna. A low order compensator is

obtained by treating a sequence of finite element design models ordered

with increasing modal frequency and choo;iing the final design model as the

first one which allows the performance/robustness objectives to be met.

In this sequence of design models, the first one consists of the rigid

body modes only. Subsequent design models are obtained by the successive

t

addition of flexible modes. The designs use 3—axis torque actuators,

colocated attitude sensors, and attitude feedback.

The organization of this paper is as follows: The mathematical model

of the system is described in section 2. The control objective is briefly

discussed in section 3, followed by a brief description of the LQG/1,TR

technique in section, 4. The reduced order (low frequency) design model

and the high frequency model uncertainty barrier are also discussed in

this section. Section 5 presents the results of synthesizing the

controller based on the above procedure using only attitude feedback.

Some of the problems and limitations observed are also highlighted. Based 

on the study results, the conclusions are summarized in section 6.

r

39

•,^'m.

t^
S

Z. MATEOKMATIQL MDEL

As a consequence of its large size and light weight, the Hoop/Column 

antenna is a highly flexible system having a Large number of significant

structural modes. A finite element model of the antenna [Ref. 2] is used

in this paper. The mathematical model considered consists of rotational

rigid-body dynamics (about the three axes) and the elastic motion. We

assume that the control will be accomplished by using n T three-taxis

torque actuators. The linearized equations of motion are:

n

I sT^T (1)

j=1

q+Dq+Aq t u (2)

where Is is the 3 x 3 inertia matrix, T^ is the 3-axis torque applied

by the nth actuator, a s

=(^s, 860 ^s) T

denotes the rigid-body ;E

attitude, q is the nq

x 1 modal amplitude vector (for nq

structural

modes), D 2 diag(p l w l , P 2 w2 , •••+ pn wn) is the inherent :? 

r^

t F

damping matrix, (where p i is the damping ratio for the ith mode). 0 is

F

the m x nq

mode-slope" matrix (where m = 3nT ), u - (T1, T2, ...,'

t

T

T )T is T the m x 1 vector of actuator torques, and A = diag(wi )

w2, ..., wnq) where wi

is the frequency of the ith elastic mode.

The rigid-body parameters and the first ten elastic frequencies are given

in Table 1. Each value of p i is assumed to be 0.01 for i=1,2, ...,n .

i

R^

R^

ti

40

Ir•+^ - ale.. .. .. _ _, ._^-_.. _._' __ _^.^__.^._^__^__--
6

Normally, the sensors used include attitude and rate sensors. A

3-axis attitude ya at a sensor ( e.g. a star tracker) output is given

by:

y a - a s +*q+w (3)

where * is the 3 x nq mode-slope matrix at the sensor location, and w

is the sensor noise. If an attitude rate sensor ( e.g. a rate gyro) is

used, the sensor output yr is , given by an equation similar t o (3),

except that as and q are replaced by a s and q, respectively. Torque

actuators and attitude sensors are assumed to be located near the top of

the mast at the antenna feed ( Fig. 1.) TT

Defining x = (a s , a s , qT^ qT) T an n x 1 vector, the state

space model can be written in the form:

11 AF X + BF u (4)

Y = CF x + w (5)

The sensor noise w is not used in the design process in this paper;

however, it will have to be included when computing the RMS pointing

errors. Ignoring the noise, the transfer matrix between the input ( 3-axis

torque) and the output ( 3-axis attitude) is given by:

G(s) - G 1 (s) + G 2 (s) (6)

Ir

{ 

t

41
^t

7

a;

i^

where

G1 (s) - Ig1 /s 2(7)

n

G2 (s) - I (V► i ^TMs2

 + 2pi wi s + wi ) (8)

iml

(* i and ^i represent the mode-slope matrices at the sensor and

actuator locations corresponding to the ith mode).

3. DESIGN OBJECTIVES

f

The basic design objectives for the control systems are: (1) To

obtain sufficiently high bandwidth ( i.e. closed loop frequencies

corresponding to the rigid body modes) and satisfactory closed loop

damping ratios for the rigid body and structural modes; and (2) To obtain

satisfactor= y RMS pointing errors, feed motion errors and surface errors.

The first design objective arises from the need to obtain sufficiently

fast error delay when a step disturbanct(such as sudden thermal distortion

caused by entering or leaving Earth's shadow) occurs. The second design

objective arises from the RF performance requirements. These two

objectives may not necessarily be compatible, and may even be

conflicting For example, the use of increased feedback gains for

obtaining higher bandwith and damping ratios will, in general, result in

higher r.m.s errors (because of the amplified effect of sensor noise)

beyond a certain point. Therefore, it is necessary to carefully consider

the trade-offs between the speed of response and lower RMS error. In this

study, the main control system specification is that a minimum bandwith of

42

1
3

0.1 rad/sec for the closed loop system is to be ensured. The upper limit

on the low frequency gain is not specified, but it is desired that it

should be as high as possible. Also, for this r.4udy no specification on

RMS errors was made and this aspect along with measurement noise will be

considered in the future.

h. THE DESIGN PROCEDURE

The LQG/LTR method has been described in detail in (7-9). Here, the

main steps are summarized first and then each step is discussed in detail.

(1) Define a "design" model of the nominal plant which is an acceptable

low frequency representation. Define the high frequency uncertainty

(robustness) barrier, and the low frequency performance barrier.

(2) Design a full state feedback compensator based on the steady state

Kalman-Busy filter (KDF)b This assumes that the * loop is broken at

the output. Adjust the weighting , matrices in the KBF design until

it;,, :Frequency response meets the robustness specifications at high

frequencies and bandwidth specification at low frequencies.

(3) Design a LQ regulator to asymptotically "recover" the frequency

response obtained in step 2.

(4) Verify stability, robustness, and performance for the entire

closed-loop system.

The first step, which consists of the definition of the plant and the

uncertainty (robustness) barrier, is often the most important one. The

basic problem in controlling a flexible structure is the presence of a

43
a 

9

large number of lightly damped structural modes. Practical limitations

` necessitate the use of reduced-order controllers. Therefore, the

uncontrolled modes, as well as the error in the knowledge of the

f

controlled modes, represent uncertainty. Since the number of structural

modes is usually large and finite element modeling accuracy typically 

decreases with increasing model frequency, the design model should consist

of the rigid-body plus the first few elastic nodes. The remaining

structural modes then (partly) constitute the plant uncertainty. In order

to obtain an acceptable low-frequency representation, the design model

must include at least the three rigid body modes. The uncertain

barrier is a measure of the plant uncertainty at high frequencies. Tne

plant uncertainty can be represented as either multiplicative or additi

uncertainty (Fig. 2). Additive uncertainties are of the form

^.• 

` G' a G + AG

while multiplicative uncertainties are of the form 

G' ,• (I+A) G

r^

Multiplicative uncertainty form is the preferred form in the literature on

robustness studies as the compensated transfer function has the same

uncertainty representation as the raw model. However, since flexible

structure models exhibit naturally the additive uncertainty form of the

transfer function matrix, this will be used in the following studies. The

LQG/LTR approach requires the characterization of the uncertainty in terms 

44
10

of a frequency-dependent upper bound. Frequency domain sufficient

conditions are used to test the robustness in the presence of

uncertainties within that bound.

For the case of multiplicative uncertainty LP (s) of figure 2a, the

closed-loop system is stable if

i ;[I,p00-1] < o [I + ( Gp (J W ) GC OW)) -1 ] (9)

where Gp(s) and Gc (s) are the design model (plant) and compensator

transfer matrices, and a and o denote the largest and the smallest

singular values of the argument matrix, respectively. At high

frequencies, assuming I[Lp( j w)]l » 1 and 1[Gp(Jw)Gc(jw)]M << 1, (9)

approximately yields

( Gp Gc ) < 1 (10)

o (L p)

The "uncertainty (or robustness) barrier" is an upper bound Im(w) on

;

a(Lp ). The system is stable in the presence of such unstructured i

uncertainties if v[GpGc I < lml (w) at high frequencies.x

When the additive uncertainty formulation (Fig. 2b) is used, a

sufficient condition for stability robustness is given by [12] 

v (I+GG )

w p c 7 v (AG) (11)

f1.

_

Q(Gc)

l

Sri 1 

f 

45

e..._,.. -._._.._ _

_._.-.....e._,.^.`.^.".—^•^§. ^'^.^.'1^_^ r^6y'F^,PC.--. -a^-z-+z -ruwcucv.^.....
r ^ f

I

77^

^w 

rt

^^ I

-

r

1F

r

At high frequencies, assuming IGpGcl << 1, ( 11) (approximately) yields

c(Gc) < 1/3wo (12)

That is, the compensator must roll off sufficiently rapidly at high

k frequencies. The main objective of the LQG/ LTR approach is to first

I ^`

design a full state compensator ( based on KBp) which has the behavior of

the desired loop transfer matrix (i.e., the loop gain GpGc).

Therefore, (from step 2) any loop shaping should involve the product

G G rather than GC

as in ( 11) and ( 12). Assuming that G p c c p

is a square matrix,

GC = Gp l (GP

 Gc ) (13)

v (G) < v (G1) o (G G

C

p

n c)

or

v (G c ) < v (Gpl) v (Gp Gc ) (14)

Using (12) and (11), the following sufficient condition for stability

robustness is obtained:

v (I + G G c) a (G p)

v (AG) (1S)

v (GpGc)

The second step in the design procedure is to design a full state 

feedback compensator having desirable singular value properties. The

46

11
I

performance of the closed-loop system depends on the low frequency gain

and the crossover frequency of the loop transfer matrix G pGc; that is,

on the behavior of c [GpGc] • Larger low frequency gain and crossover

frequency indicates better tracking performance. Thus, o[GpGc] should

lie above the performance specification as shown in Fig. 3a. The other

requirement is the stability robustness in the presence of model

uncertainties. If the multiplicative uncertainty formulation is used,

according to (10), the a [G pGc ] plot should pass under the robustness

barrier o -1(Lp) at high frequencies ( Fig. 3a). On the other hand, if

the additive formulation is used, the robustness condition ( 15) should be

satisfied ( Fig. 3b). The advantage of an LQG-based full state design is

that it has excellent classical properties, and its frequency response can

be shaped in the desired manner by varying the weighting matrices [8].

As discussed in [7], this design can be accomplished using the LQR

Riccati equation if the loop is broken at the plant input, or the KBF

Riccati equation if it is broken at the point where the residual signal

enters the KBF. Herein we select the latter because the objective is to

control the attitude output. This selection is also consistent with

(9-11]. The KBF equations are:

12

AE + EAT + LL - u EC C T E 0

H = 1 ECT

P

(16)

(17)

j 

47
13

Y.

where L and u are the design parameters, L being an u x m matrix, and u a

scalar. The matrix H is the KBF gain and E is the corresponding Riecati

matrix. The KBF loop transfer matrix is given by:

G

KF

(s) - C (SI - A) -1 H (18)

Generally, the trequency response o(G KF (jw)) would shift higher as u

decreases, and the crossover frequency can be adjusted by changing L (6).

Having obtained satisfactory singular value behavior of KBF, the next

step is to design a LQR to "recover" the desired frequency response. This

is accomplished by solving the algebraic Riccati•equation

ATP + PAT- PBB TP + q CTC - 0 . (19)

where P is the Riecati matrix and q is a positive scalar. The control

gain matrix G is given by

G - R-1BTP

It has been proven in references 7 and 8 that the loop transfer matrix

GpG c for the overall system (consisting of the plant, the KBF and the

LQR) tends to GICF(s) as q+co, provided that the open-loop plant has no

transmission zeros in the right half plane. The compensator G c (s) after

recovery is given by:

Gc (s) - G (sI - A + BG + HC)-1H

F ,

s

t

r..

48
I 

k

I

49

4

I ^'

f t

i

1 t

f

14

k

Since the compensation obtained has no guaranteed robustness

properties, the last step will consist of testing the eigenvalues of the

entire closed-loop system to ensure stability and robustness. If

instability is discovered, it will be necessary to return to step 2 and

redesign the KBF for lower bandwidth and the LQR for robustness recovery.

If this does not produce satisfactory results, it would then be necessary

to return to step 1 and include more elastic modes in the design model.

Application of the foregoing LQG/LTR procedure for the Hoop/Column antenna

is described in the following, section.

5. CONTROLLER DESIGN BY LQG/LTR MMOD USING ATTITUDE FEEDBACK

The foregoing procedure has been'applled to the Hoop / Column antenna

model. The computations of singular values of various matrices (e.g.

loop transfer, return difference, inverse return difference matrices) were

carried out using a recently developed multivarable frequency domain

analysis software package (FREQ), and the LQG designs were carried out

using ORACLS [ 13]. The nominal plant includes three rotational rigid-body

modes and the first ten elastic modes. We assume three torque actuators;

hence, the order of B matrix is 260. Assuming three attitude sensors

(one for each axis) at the same location as the actuators, C is a 3 x 26

matrix. The plant, input, and output matrices were obtained from a finite

element analysis of the antenna.

t

Y 

E

{ 
15

Before starting the controller design, the maximum and minimum

singular ve,lues (F and a) of the full, nominal, open-loop plant transfer

matrix were obtained and are shown in figure 4. The g plot clearly shows

the peaks at the elastic mode frequencies (i.e. the poles), the most

prominent being the first mode near .75 rad/sec. The dips in a indicate

the presence of transmission zeros for the multivariable plant at those

frequencies. The controller synthesis studies are performed using the

design model consisting of:

a) rigid-body model (n . 6, nq - 0)

b) rigid-body and the first flexible mode (n - 8, nq R 1)

c) rigid-body and the first three f1vible modes (n 12, nq =3)

The measurements available are the three attitude angles at the feed

location. One 3-axis torque actuator is used at the same location. The

compensator is designed based on these sensors and actuators.

5.1 Rigid Body Model:

► :

i.

W ,t

In this section the controller design is carried out based only on

the rigid body design model. The largest and the smallest singular values

of the rigid-body transfer matrix (n - 6) are of the form 11s 2 . The

corresponding additive uncertainty AG, which consists of the (20th order)

flexible dynamics, is plotted in figure 5. Figure 5 indicates the

presence of poles at the undamped flexible mode frequencies of 0.75

rad/sec, 1.35 rad/sec, etc. Also, the pole of the first mode frequency of

0.75 rad/sec produces the highest peak since it is most lightly damped.

50
16

(The importance of this fact will be seen later when the stability

condition is violated at this point).

For this sixth order design model, a compensator design was carried

out using the Kalman ,filter design methodology to achieve satisfactory

performance (i.e., large gain and bandwidth) at low frequencies, and

robustness at high frequencies. This design was carried out using the

Kalman filter Riccati equation ( 16). The Kalman-Bucy filter (KBF)

transfer matrix GKF( s) is given in equation (18). Appropriate

loop-shaping can be accomplished by proper choice of the weights u and L

in equation ( 16). Since the controller design model is of the form

1/s 2 , one can analytically evaluate the singular values of I+Gvv

using equations (16) and (17). Assuming u - 1 and L - (L 1 ,L2 ) 'r the

left hand side of (15) can be solved. For L1 - 0 and L2 - k2 I, it

can be whown that equation (15) is satisfied by:

k2 < 10-7

This implies that the Kalman filter gain computed using (17) will be very

low. Figure 5 shows plots for condition (15) with two L matrices, with

L 1 - 0 and k2 - 10-6 and 10-7 . The right hand side of (15) is

also plotted in figure 5. It is evident; that condition (15) is satisfied

for k2 -10-7 . As k2 is decreased further, the curve shifts upward

thus increasing the margin.

51
t.

E

i

17

M F

A

W

k

The next step consists of LQ regulator design. Having obtained an

acceptable compensator through Kalman — Bucy filter equations, the LQ

regulator is realized via the loop transfer recovery method 18). Figure 6

presents the aingular value plots of the complete loop transfer matrix

GpGc (s) (which consists of the plant, the KBF and the LQR) for

different wetighting parameter q (Eq. 19). The q selected was q-10 6 and

1.0 7 . It is easy to check condition (11) in this case. As q is

increased, the plots approach those of the compensator obtained from the

Kalman filter design approach. The LQ design for q • 106 was considered

to be satisfactory.

'The standard LQ^/ LTD procedure requires the definition of the

"desired" loop transfer characteristics (see step two in section three.)

That is, Q,(GKF) must satisfy the low—frequency performance

specifications, and v (G KF ) must satisfy the high—frequency robustness

specifications. Thus, in the presence of additive uncertainty AG, the

procedure states that the robustness condition

Q (I + GKF ) o (Gp)

> v (AG)

v (GKF)

should be satisfied. However, in the case described above, it was found

that the above condition makes the "desired" design (G KF ) extremely

conservative. From Ue,F we 6, it is seen that the closed loop bandwith is

quite low and nowhere near the desired value of .1 rad/sec. Therefore,

52 r 
r ^

18

recovering this conservative loop gain yields a compensator with poor

performance. This fact led to a modification of the LQG/LTR procedure.

In particular, the above robustness test on % is omitted in the

modified procedure. Instead, the recovz , is carried out first, and then

the (lens conservative) stability test (11) is applied directly for the

compensator Gc . The Kalman filter transfer matrix GKF is based only

on the desired performance and not on satisfying the stability test of

equation (15).

With the revised Qest on Cc , the following choices on L and u

matrices were made.

0

L ; ^^ R1 -

10-2I

Using the recovery procedure, the compensator is obtained for this

case wth q - 104 . The resulting stability test (Eq. 11) is shown in

figure 7. It is seen that the stability margin is lowest at the first

mode frequency (0.75 rad/sec). Any increase in the gain (obtained by

q > 104 ) resulted in violation of stability condition at that point.

The overall loop bandwidth 'is obtained from the singular values of the

loop transfer function GpG. shown in figure 8. It is seen that the

bandwidth (i.e;., the frequency at which o(G p

C

c ) = 1) is far short of

the required 0.1 rad/sec. In order to increase the bandwidth, the gain

53
19

has to be increased by increasing q. However, this results in the

violation of the stability condition (11). Thus it is evident that, with

a rigid—body design model, it is not possible to meet the performance

specifications.

5.2 1 Flexible Mode Design Model:

To overcome the above problems, the next alternative that was

considered was whether the inclusion of the first flexible mode (0.75

rad/sec) in the design model would improve the performance. The inclusion

of the first flexible mode, which is predominantly a torsion mode, results

in a design model of order 8. The singular values of AG shown in figure 9

are an order of magnitude lower` than those in figure 5 (wherein AG

consistEd of all the flexible modes). The first peak of a(AG) occurs at

1.35 rad/sec, which is the frequency of the: second mode. This is the

critical point in the stability test (Eq. 11). After a number of trials,

the following choice of L and u was made to obtain the desired performance

GKF•

0

L 10-113 u 1

10 -1 1 2

0

The recovery is obtained for q = 10 5 and the stability test is shown in

figure 9. Fig. 9 indicates the critical point to be at about 0.28

54
20

rad/sec. There ib a good margin at the peaks of AG due to upward sloping

of the upper curve. The resulting loop transfer function (GpGc) plots

are shown in figure 10. The plots indicate that the required 0.1 rad/sec

bandwidth is not obtained (although it is much higher than. the rigid—model

case). Any increase in the gain (for q > 10 5 ) was found to result in

the violation of the stability condition (11). Figure 10 indicates the

presence of an open—loop invariant zero near 0.082 rad/sec, which was also

confirmed by independent computations. This zero is almost on the

imaginary axis (i.e., the transfer matrix is close to being nonminimum I

phase). Therefore, (as would be expected) the recovery procedure is not

very effective for making GpGc approximate GK.

5.3 3 Flexible Mode Model:

In order to improve the performance further, the next step was to

include the first three flexible modes in the design model. It is logical

to do this because they represent the first modes about each axis, i.e.,

the first torsion mode, and the first h nding modes in the XZ and YZ

planes. Thus, the order of the design model was 12. The singular value

plots for G and AG are shown in figures 11 and 12, respectively. It is

seen from figure 11 that Gp has zeros near 0.082 and 0.22 rad/sec, and

poles near 0.75, 1.35, and 1.7 rad/sec. It is seen from the AG plot

(Fig. 12) that v is considerably lower than that in figures 7 and 9.

After numerous trials, the following choice of the L matrix and the scalar

V was made for a suitable Gam,:

i

55

D
21

f

4 •

0

10-113

-10-412 i 0 -

1

10 -4 1 2

0

10-4 1 2

0

; V 0 1

F

t

I

C

s

t

L

The recovery was accomplished with q - 10 10 . The stability test is

shown in figure 12. It can be seen that condition (11) is satisfied with

a wide margin. Also, at the peak for AG (at 8 rad/sec) the upper curve

slopes upward, indicating good toletrance of high-frequency uncertainty.

The limit for increasing the gain (indicated by the lowest point in the

upper curve in figure (12) occurs at about, 0.3 rad/sec. The resulting

compensator Gc is shown in f:tgure 13. The gain of i G is much hi c higher

than that obtained in the previous cases. Generally, the LQG/LTR

technique attempts to choose Gc in such a way that the product GpGc

is replaced by GKF (i.e. Gc is attempting to invert G in the

frequency range of interest). The 3-mode design plant shown in figure 11

has elastic mode eigenvalues at -.0075 ± 3.75, -.0135 ± J1.35, and -.0170

i 31.70. Figure 13 shows that Gc has zeros with frequencies near these

locations. The design plant also has transmission zeros at -.9 x 10 -4 ±

3.082, -.37 x 10-3 ± 3.22, and -.29 x 10-3 t 3.22. Ideally, Gc

should have poles with frequencies near .082 and .22. However, the design

56
w..

i

22

I plant zeros are too near the JW-axis and tend to numerically behave as

!

f?

r

nonminimum phase. Some attenuation is obtained by the compensator pole 

near 94 rad/sec. The plots for the loop transfer matrix GpGc are

given in figure 14. It is seen that a bandwidth of 0.1 rad/sec. is

v

j

obtained except for the presence of the invariant zero near 0.082

' rad/sec. which causes some deterioration of performance. At frequencies

past .4 rad/sec., GpGc behaves like G and eventually rolls off at 

i

60db/decade. Also, v and a are closely spaced, indicating good system

' behavior. Thus it is seen that the inclusion of the first three modes in

the design model yields a robust compensator which also meets the

bandwidth specifications.

1 The final step is to check -the suability of the complete nominal

system when the compensator Gc(s) designed above is used. The overall 

closed-loop system is:

x AF -B

F

G x

X

HCF A-BG-HC x

}

where the subscript F is used to denote the full-order nominal plant, and

x denotes the state estimate for the design model. The eigenvalues of the 

overall closed-loop system using the 3-mode controller are given in Table

II. It can be seen from Table II that the overall closed-loop syste-i is

stable.
23+

6. CONCLUDING REKAM

The LQG/LTR multivariable frequency domain technique was employed in

the design of an attitude control system for a large flexible space

antenna. The LQG/LTR method was noted to be especially attractive for

overcoming spillover effects common to large space structures control

problem modelled from finite element data. The design objective of

avoiding excitation of higher order modes while satisfying performance

criteria was met by including these modes in the robustness uncertainity

barrier.

Design was based on a reduced order model chosen as the rigid body

dynamics plus the fewest number of low frequency vibrational modes

necessary to meet a desired closed loop bandwith. Inclusion of the first

three vibrational modes (corresponding to the three axes) was found to be

necssary to meet a 0.1 rad/sec bandwith. For wider bandwidths, design

models width greater than three modes may be needed. A satisfactory

control design was obtained using only a colocated single pair of 3—axis

attitude sensor and torque actuator for the Hoop/Column antenna problem.

Performance degradation was observed due to the presence of invariant

zeros within, the design bandwith. These zeros were unavo,dable given the

prescribed sensor/actuator locations and emphasized the fact that

consideration should be given to control aspects when building large space

structures.

A modification of the standard LQG/LTR procedure was introduced in

which the robustness test was performed with the full LQG compensator

58

i
i

24

b

instead of. the intermediate Kalman filter design. This approach was found

to produce higher gain compensators and helped overcome the basic

conservativeness shortcoming of the LQG/LTR approach.

tSP UNCES

1. Russel, R. A., Campbell, T. G., and Freeland, R. Be: A Technology

Development Program For Large Space Antennas, Paper No IAF-8OA33,

31st International Astronautical Congress of International

Astronautical Federation, Tokyo, Japan Sept. 21-28, 1980.

2. Sullivan, M. R.: LSST (Hoop/Column) Maypole Antenna Development

Program, Parts I and II. NASA CR-3558, June 1982.

3. Balas, M. J. Trends in La7rge Space Structure Control Theory:

Fondest Hopes, Wildest Dreams. IEEE Trans. Auto. Contr., Vol. AC-27,

June 1982.

4. Joshi, S. M.: Control Systems Synthesis for a Large Flexible Space

Antenna. Acta Astronautca, Vol. 10, No. 5-6, May 1983.

5. Wang, S. J. and Cameron, J. M.: Dynamics and Control Of A Large

Space Antenna. AIAA J. Guidance, Control And Dynamics, Vol. 7, No-1,

January-February, 1984, pp 69-76.

6. Kosut, R. L., et al.: Robust Control of Flexible Spacecraft. AIAA

J. Guidance, Control and Dynamics, Vol. 6, No. 2, March-April 1983.

7. Doyle, J. C., and Stein, G.: Multivariable Feedback Design:

Concepts for a Classical/Modern Synthesis. IEEE Trans. on Auto.

Contr., Vol. AC-26, No. 1, Feb. 1981.

59
f

25

1

P

kh

89 Stein, G.: LQG—based Multivariable Design: Frequency Domain

Interpretation. AGARD Lecture Series LS-111 on "Multivariable

Analysis and Design Techniques," 1981.

9. Athans, M.: "Multivariable Control System Design Using the

LQG/LTR Methodology". Unpublished lecture notes (lecture given at

NASA Langley Research Center, 1984).

10. Chan, S. M., and Athans, M. Applications of Robustness Theory to

Power System Models. IEEE Trans. Auto. Contr., Vol. AC-29, No. 1,

Jan. 1984.

11. Kappos, F..: Robust Multivariable Control of the F-100 Engine. MIT

Rept. No. LIDS— TH-1328, Sept. 1983.

12. Cruz, J. B., et al.: A Relationship Between Sensitivity and

Stability of Multivariable Feedback Systems. IEEE Trans. Auto.

Contr., Vol. AC-26, No. 1, Feb. 1981.

13. Armstrong, E. S.: ORACLS — A Design System for Linear Multivariable

Control. Marcel & Dekker Control and System Theory Series, Vol. 10,

1980.

60
0

26

1 1.

LIST OF FIGURES

Figure 1. Hoop/C61umn antenna concept

Figure 2. (a) Multiplicative uncertainty

Figure (b) Additive uncerIzainty

Figure 3. (a) and (b) Performance and robustness barriers

Figure 4. Singular values of full nominal plant

Figure 5. Stability robustness test (Eq. 15) for KBF—based loop shaping

Figure 6. Singular values of GKF and "recovered" G 

p 

GC*

Figure 7. Stabilty robustness test (Eq. 11) based on the recovered

compensator.

Figure 8. Singular values of recovered loop transfer matrix G 

p 

GCO

Figure 9. Stability robustness test (Eq. 11) for 1 mode model.

Figure 10. Singular values of Gp G. for 1 mode model.

Figure 11. Singular values of G p for 3 mode model.

Figure 12. Stability robustness test (Eq. 11) for 3 mode model.

Figure 13. Singular values of the compensator Gc for the 3 mode model.

Figure 14. Singular values of G

p G c

 for 3 mode model.

61
w

!able I. Antenna Paraa►etera

Rigid-body parameters

Mass-4544.3 Kg.

lertia about axes through center of mass (Kg-m2)

I xx M 5.724 x 106

I zz M 

4.383 x 106

I yy • 5.747 x 106

27

Iyz IXz Ixy 0

Structural Mode Frequencies (rad/aec)

1.35, 1.7, 3.18, 4.53, 5.59, 5.78, 6.84, 7.4 0 8.78

4 1 

;-

62
:X. Eigenvalues of the Full Closed-Loop system

Real part Imaginary aginary Part

-8.535 (10

3

) 8.054 (102)

-8.535 (10-3

) -8.054 (10-2)

-7.557 (10-2 ) 1.250 (10-1)

-7.557 (10

2) -1.250 (101)

-7.604 (10-2

) 1.248 (10-1)

-7.604 (10-2 ) -1.248 (10-1)

-2.237 (10 

1) 2.236 (10-1)

-2.237 (10-1 ) -2.236 (10-1)

-2.330 (10-1 ) 2.154 (10-1)

-2.330 (10-1 ) -2.154 (10-1)

-2.379 (10-1 ) 2.113 (10-1)

-'t.379 (10-1) -2.113 (10-1)

-7.466 (10-3 ) 7.466 (10-1)

-•7.466 (10-3

) -7.466 (10-1)

-1.346 (10-2 ) 1.346

-1.346 (10

2 ) -1.346

-3.076 (10-1 ) 1.373

-3.076 (10-1 ) -1.373

-1.016 1.267

-1.016 -1.267

-1.702 (10-2 ) 1.702

-1.702 (10-2 ) -1.702

-4.028 (10

1

) 1.737

=4.028 (10-1 ) -1.737

-3.181 (10 -2 ) 3.181

-3.181 (10-2 ) -3.181

-4.422 (10-2 ) 4.529

-4.422 (10-2 ) -4.529

-5.579 (10-2

) 5.590

-5.579 (10-2 ) -5.590

-5.731 (10-2 ) 5.776

-5.731 (10-2 -5.776

-6.685 (10

2 ) 6.841

f. l -6.685 (10-2 -6.841

-6.390 (10-2 ) 7.401

l

-6.390 (10

-2

) -7.401

-8.326 (10-2 ) 8.782

-8.326 (102) -8.782 63 ^

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