4

/I,3--

_5o- I a o_

Report No. 957114-3

FINAL REPORT

UNIFIED CONTROL/STRUCTURE DESIGN AND MODELING RESEARCH

January 8, 1986

RI_C£1VF-D

FEB 11 1986

S£CT|Ol_626

D.L. Mingori

J.S. Gibson

P.A. Blelloch

A. Adamian

Mechanical, Aerospace and Nuclear Engineering Department

University of California

Los Angeles, CA 90024

[NASA-Ch-179948) UNI_IHD CON_OL/STR_CTURE

L_SIGN AND MODELING RESEARCH final Report

{California Univ.) 14_ p CSCL 22B

N87-13478

Unclas

G3/18 43637

This work was performed for the Jet Propulsion Laboratory, California Institute of Technology sponsored by the National Aeronautics and Space Administration under contract NAS 7-918.
TABLE OF CONTENTS

Io

2.

o

o

Introduct ion

Structure Models and Finite Element Approximation

2.1 Antenna Model

2.2 Finite Element Approximation of a Sector

2.3 Galerkin Approximation of a Sector

Statement and Solution of Control and Estimation Problem

3.1 Abstract Control System

3.2 Infinite Dimensional l_C Problem

3.3 Functional Gains

3.4 Approximation

3.5 Application to Space Antenna

Robustness

4.1 Overview

4.2 Loop Transfer Recovery Techniques

4.2.1 Robustness Measures for MIM0 Flexible Systems

4.2.2 Asymptotic Loop Transfer Recovery (LTR)

Control Design

4.2.3 Algebraic Loop Transfer Recovery

4.2.4 Robustness of LTR Designs

4.2.5 Examples of the LTR Approach

4.3 Sensitivity optimization

4.4 Optimized loop recovered designs

1

7

7

10

10

12

12

14

17

17

21

23

23

26

30

45

53

57

63

93

98

il
•

o

Control Driven Finite Elements

5.1 Comparison of finite elements and modes for

functional gain oomputat/on

$.2 Direct approximation of Hamiltonlan

Summary and Conclusions

References

Appendix A: Singular Values

Appendix B : Proof of Unstructured Uncertainly Theorems

Appendix C: Numerical Considerations in Algebraic LTR

101

i01

113

115

120

123

126

130

iii
List of Notation

A _

0

D -

0

T1,T 2 -

C r J C m -

1 -

r

Chapter 2

stiffness operator for the antenna

damping operator for the amtepma

tensions in the antezum mesh

damping coefficients

rib length 11

A .

0

B -

o

D -

o

M

o

C

0

Vl,Vo -

{d.

3

H -

Vg -

AJ -

Chapter 3

stiffness operator

input operator

damping operator

mass operator

measurement operator

noise processes

natural frequency

basic Hilbert space

kinetic energy space

strain energy space

total energy space

semigroup generator

performance index

12

12

12

12

12

12

12

12

13

13

14

14

14

iv
l

Page

K .

GIfA

]IA -

c

f,g -

e° -

1

V n ,HMn -

P(Ac) -

control gain matrix

estimator gain matrix

solution to the control Riccati equation

solution to the estimator Riccati equation

compensator semigroup generator

functional control gains

functional estimator gains

finite element basis vectors

approximation subspaces

spectrum of A

c

Abbreviations

1 .h.p. -

r.h.p. -

KBF (KB filter) -

Chapter 4

left half plane

right half plane

Kalman-Bucy Filter

LQR (LQ regulator)-Linear Quadratic Regulator

LSS

LTR -

MIMO -

RMSSISO -

Large Space Structure

Loop Transfer Recovery

Multiple-Input-Multiple-Output

Root Mean Square surface error with respect to

inertial space

Single-Input-Single-Output

General

1 ith closed-loop pole (ki=oi+Jwi)

15

15

15

15

16

17

17

17

18

21

51

51

26

26

36

23

30

24

30

95

v
Page

f

o

!

Af -

nominal plant frequency

true plant frequency

normalized variation in plant frequency

modal damping ratio

State Space System Representation

qc "

A .

Acl -

B -

C -

D -

K -

G -

M -

N -

e .

Q _

Qe "

Qr "

Rnumber of plant outputs (measurements)

large scalar parameter increased to achieve loop

recovery in the asymptotic (KBF) method

scalar parameter weighting the state cost matrix

in the LQR problem

plant system matrix

closed loop system matrix

plant input matrix

plant output (measurement) matrix

plant damping matrix in the modal representation

LQR control gain matrix

KBF estimator gain matrix

state noise covariance matrix in the KBF problem

measurement noise covariance matrix in the KBF

problem

solution to either the LQR or KBF Riccati Equation

state weighting matrix in the LQR cost functional

matrix weighting mechanical energy

matrix weightingRMS surface error

input weighting matrix in the LQR cost functional

4O

4O

4O

4O

53

48

68

47

93

47

47

63

47

47

49

49

49

68

73

69

68

vi
Page

U I

X -

X -

y-

@%

y-

¢-

A 2 .

plant input vector

plant state vector

estimate of x

plant output (measurement) vector

estimate of y

plant displacement state vector in the modal

representation

diagonal matrix of modal frequencies squared

Transfer Function (Polynomial) S_ystem Representation

deE{'} the degree of a polynomial function

s - complex frequency (Laplace transform variable)

w - imaginary part of s

_R " Nyquist D-contour

n - number of pole/zero cancellations

c

n I - number of uncancelled compensator poles

n - degree of the compensator numerator polynomials

n

(nc+n I)

nN - maximum degree of the plant numerator polynomials

n_ - degree of _(s)

(Upper case variables denote the Laplace Transform of their corresponding lower case time variables, with the following exceptions)

d(s) -

dc(s ) -

dl(S ) .-

compensator denominator polynomial

portion of d(s) corresponding to pole/zero cancellations

uncancelled portion of d(s)

47

47

47

47

47

63

63

55

32

37

36

55

55

55

55

55

51

51

vii
Page

ld(S ) -

1 (s) -

m

ni(s ) -

D(s) -

E(s) -

G(s) -

G'(s) -

Z(s) -

L.(s) -

1

L (s) -

o

R(s) -

Am(S ) -

Ad(S ) -

_(s) -

t(s) -

_(s) -

W D

X -

B -

C -

0

f,g -

upper bound on O[Ad(S) ]

upper bound on O[Am(S)]

ith compensator numerator polynomial

plant denominator polynomial

output error (R(s)-Y(s))

nominal plant transfer function matrix

true plant transfer function matrix

compensator transfer function matrix

input relative plant error

output relative plant error

control command signal

multiplicative unstructured uncertainty

divided unstructured uncertainty

numerator polynomial of K_B

(sl-A) "I

-i

(sI-A+BK)

Chapter 5

elastic deflection

generalized displacement

rigid-body rotation

damping coefficient

functional control gains

36

36

54

51

32

32

32

32

35

35

32

36

36

51

48

48

102

102

102

103

103

viii
1.0 INTRODUCTION

The purpose of this research project is to apply control theory for distributed systems to large flexible space structures, and to combine distributed system theory with finite dimensional control theory to produce

comprehensive control system design methods for large space structures. The

basic idea is to establish the existence and properties of an ideal infinite

dimensional compensator for the distributed model of the structure and then

approximate the ideal compensator with realizable finite dimensional compensators. For an analysis of the ideal compensator and the resulting performance

of the closed-loop system, distributed system theory, which is based on infinite dimensional analysis, is essential. Also, infinite dimensional analysis

is necessary for studying convergence of approximations and selecting

appropriate approximation schemes. For computing the approximating finite

dimensional compensators and determining efficient implementation, finite

dimensional control theory is used.

To demonstrate the applicability of the methods to real space structures, the research has focused on a model of a space antenna which consists

of a rigid hub, flexible ribs and a mesh reflecting surface. This model was

designed to include the most important characteristics of large flexible space

structures. In particular, the structure to be controlled has different types

of distributed components, as well as rigid-body modes, and, because of the

mesh surface, there are many closely packed structural frequencies. This project appears to be the first application of distributed system control theory

to a structure of such complexity.

1
It should be emphasized that, although the term "distributed system"

usually implies a system involving partial differential equations (as with the

model of the flexible components of the antenna here), the methods of this

research do n_ot require the solution of partial differential equations.

Rather, according to the approximation results, only finite dimensional Riccati and Liapunov matrix equations need be solved.

Chapter 2 describes the space antenna model used in this project and

discusses the finite element approximation of the distributed model. A Calerkin component mode approach is used to develop a finite dimensional approximation of the structure. This approach was found to converge rapidly to a model

of suitable accuracy.

The basic control problem is to design an optimal or near-optimal compensator to suppress the linear vibrations and rigid-body displacements of the

structure, which is modeled as a distributed system because of the flexible

components. As is common in finite dimensional optimal control of linear systems, a quadratic performance index is defined to penalize both the disturbance in the structure and the control effort used to eliminate the disturbance. Determining the optimal compensator for this problem mad analyzing the

response of the system require linear-quadratic-gaussian (LQC) optimal control

theory for distributed systems [BA1,CU1, CI1,2]. The application of this

infinite dimensional LQG control theory to flexible structure is discussed in

Chapter 3.

Distributed system theory yields an infinite dimensional compensator as

the optimal, or ideal solution to the control problem. As discussed in

Chapter 3 this compensator can be represented in terms of _ _ for

2
the feedback control and state estimator, and also in terms of an irrational

transfer function. Both the functional gains and the Bode plots of the

transfer function provide useful graphical-re-p_sentations 0f t_e ideal cornpen sat or.

Approximation theory is a central topic of this research. Because the

ideal compensator is infinite dimensional, it can be neither written down in

closed form nor built and implemented exactly. Thus, a finite dimensional

approximation of the ideal compensator is required. In this research, there

are two stages in the approximation of the ideal compensator. First, increasingly large-order finite dimensional LQG problems are solved until the resulting compensators converge to a finite dimensional compensator that is essentially equal to the ideal compensator. Next, the order of this large compensator is reduced as much as possible using balanced realizations. It is

important to note that, in both these stages of approximation, the convergence

and truncation criteria are tied to quantitative measures of control system

performance.

The approximation theory for constructing the large-order approximation

to the ideal compensator is discussed in Ref. [HRI]. At the heart of the

infinite dimensional I_G theory, are the infinite dimensional Riccati operator

equations whose solutions determine the optimal control and estimator gains

for the infinite dimensional compensator. The scheme for computing finite

dimensional approximations to the ideal compensator is based on the approximation of the infinite dimensional Riccati equations by finite dimensional aiccati matrix equations of the type found in finite dimensional control theory.

These finite dimensional equations are given in [HR1]. Also to be found in
[HRI] are typical functional control gains and the Bode plot for the ideal

compensator for the antenna.

While infinite dimensional, or distributed system control theory is the

main basis for this research, finite dimensional control theory plays an

important role. Indeed, the true richness of the project stems from the combination of a primary infinite dimensional perspective of control theory with

finite dimensional control theory and practice. Thus, the project really

represents an application of a finite dimensional methods in a infinite dimensional setting.

After the converged approximation to the ideal compensator has been

obtained, it must be verified that the lower order approximation to the ideal

compensator will still produce a satisfactory closed-loop system. Also,

robustness with respect to errors in plant parameters must be studied.

Robustness issues were a major focus of the work performed this year. Two

basic approaches for robustness enhancement were investigated: Loop Transfer

Recovery and Sensitivity Optimization. A third approach synthesized from elements of these two basic approaches is currently under development. The goal

of loop recovery is to produce a loop transfer matrix with the same frequency

response as full state feedback which is known to have very desirable robustness properties. Sensitivity optimization uses numerical optimization technlques to select regulator and/or estimator gains which reduce sensitivity of

the real parts of the closed loop poles to parameter errors. These methods

are presented in Sections 4. 9. and 4.S along with numerical examples which

illustrate their performance.
In general it was found that the methods worked well on relatively simple problems which were used to test and develop the software. However,

results were mixed when the methods were applied to the antenna model. The

light damping and complexity of the antenna apparently make this a challenging

system for robust controller design. The performance requirements were purposely chosen to ensure that several of the flexible modes were within the

controller bandwidth, and this contributed to the difficulty of the problem.

The form of the weighting matrix Q in the performance term of the cost function was found to have a strong effect on robustness. For example, Q's which

did not penalize velocity terms were found to be less robust than those which

did. Also, Q's in which the penalty on rigid body motion was much greater

than the penalty on flexible modes were less robust than Q's in which the

rigid body mode was not emphasized. These results are reported in Section

4.2.

Sensitivity optimization, a second method for designing robust compensators, is discussed in Section 4.3. This method uses nonlinear programming to

find control and estimator gains that minimize the sensitivity of the closedloop response with respect to uncertain plant parameters. Of the various

measures of closed-loop sensitivity that were tried, the sensitivity of the

real parts of the closed-loop eigenvalues proved to have the best correlation

to robustness. Simulation results are given. As with the loop-recovery

methods, the form of state weighting in the control problem affects the

robustness significantly.

Chapter 5 discusses control driven finite element approximation of flexible structures. This means choosing the finite element scheme that is most

5
efficient for approximating the solution to the optimal control problem for

the distributed model of the structure. Section 5.i compares three sets of

finite element basis vectors for computing functional control gains and demonstrates that some finite element schemes are much better than others. Section

5.2 discusses the possibility of constructing a finite elesent scheme to

approximate the infinite dimensional Halniltonian system directly, instead of

indirectly as in Chapter 3. The main application of this approach appears to

be for structures with history-dependent damping.
o STRUCTURE MODELS AND FINITE EL_4ENT APPROXIMATION

2.1 ANTENNA MODEL

The space antenna model that has served as the primary example of this

research is shown in Figure 2.1 .I. The flat antenna consists of a rigid hub,

eight ribs and a mesh reflecting surface. The ribs are modeled as beams cantilevered to the hub, and the mesh is modeled as o/rcular sectors of membrane

tied to the ribs and hub. The center of the hub is fixed but the hub can

rotate out of plane, so that there are two fiEld-body modes.

This model is based on the Lockheed wrap-rlb antenna [L01]. Since this

is the first complex structure to which the methods of this research have been

applied, the model has been simplified to a flat antenna with eight instead of

the actual 48 ribs, to allow concentration on fundamental issues while mainraining a complex structure with different types of components. Otherwise,

the parameters given in Figure 2.1.1 are based on the 4$-rlb antenna [EL1].

The two actuators apply torques to the hub, and the sensors measure the

rotation of the hub and the displacement of the tip of each rib. The compensator ks designed to control the out-of- plane motion of the antenna.

For small elastlc displacements, the symmetry of the antenna further

reduces the complexlty of the compensator because the motion of the antenna

decouples into two sets of orthogonal modes, with each set controlled independently by one actuator. Each of these sets consists of modes that are asymmetric about one torque axis. Although the actuators can control only the

controllable modes, these are the only modes excited by rotating the antenna
with the torques on the hub.

The equations of motion for rigid-body rotation and small elastic vibration have the form discussed in Chapter 3. See (3.1.1) - (3.1.10). Here, we

will discuss the stiffness and damping operators, which are the primary operators in the equations of motion. The stiffness operator Ao can be separated

into rib and mesh components as

where

Ao = Aor + Aom,

(2.1.1)

Act = EI d4(.)Idr 4,

(2.1.2)

Aom = T1[d2( .)Idr2 + (llr)d(.)Idr] - T2 d2(.)Id_ 2

(2.1.3)

and T1 and T2 are the tensions in the radial and transverse directions,

respectively. The term Aor operates on the elastlc deformation of each rib

and Aom operates on the elastic deformation of the mesh.

We assume that both the ribs and the mesh have viscoelastic damping, so

that the damping operator is

D

o = Cr Act + Cm Acre

(2.1.4)

where cr and cm are positive scalars. If cr and om are not equal, this structural damping couples the modes -- but not modes that are controllable by one

actuator with modes that are uncontrollable by the same actuator.
TogQuE£_

\

\

I

MODEL DATA (based on 48 rib Lockheed wrap rib antenna):

Hub Radius: 46 in. Hub Weight: 1000 lbs.

Hub Inertias: IN_ = Iv_ = 342 Ib-in-sec =

I== = 684 Ib-in-sec =

Rib Length: 86 ft. Rib Weight: 115 lb. per rib

Rib stiffness: El = 4.05x10 _ Ib-in =

Mesh Weight: b.94 Ib per sector

Fiqure 2.1.1
2.2 FINITE EL_T APPROXIMATION OF THE ANTENNA

The approximation of the antenna is a component modal synthesis for

which the component modes of the ribs and mesh were determined by consistentmass finite element approximations. The ribs were approximated by a standard

finite element scheme using Hermite cubic splines, and the mesh sectors were

approximated by a Galerkin scheme discussed in Section 2.3.

In an earlier project [HR1], the approximation scheme used in this proJect was compared to a NASTRAN finite element approximation of the antenna.

While both schemes converged to the same mode shapes and frequencies as the

order of approximation increased, the component modal synthesis converged faster and was more convenient and cheaper to use. An important advantage of the

conponent modal approximation for this project is the fact that the program

was written to make the order of approximation easy to vary. In design of

controllers for complex structures, it is impossible to determine in advance

what order of approximation will be necessary for designing an optimal, or

even acceptable compensator. Thus, in a project like this, it is essential to

have an approximation scheme of easily variable order.

2.S GALERKIN APPROXIMATION OF SECTOR

A Galerkin approximation of the antenna was used to compute the optimal

control law for the composite sector. The basis vectors have the form

(en' en" (2.S.1)

where (@n" er) are the natural modes of the hub and ribs without the mesh, and

_nare the natural modes of the mesh when the beams and hub are constrained

10
not to move. This Galerkin approximation is then a component modal synthesis

of the composite sector.

The mesh modes have the form

_n(6, r) = sinSn¢ fn, k(r), n = 1, 2, ...,

k - 1, 2, ...,

where the functions fn, k(r) are the eigenfunotions of the problem

Tl[P(64n21r2)f- if'' + f'Ir)) = te2 .f

(2.3.2)

(2.3.3)

f(r o) = f'(r o + i r) = 0

(2.3.4)

Here, the elgenvalues w

n,k are the squares of the ratural frequencies of the

mesh, T1 is the radial tension, T21T 1 = _ = 72 and Er is the rib length. This

value of p was selected so that the first mesh frequency is approximately half

way between the first two controllable beam-hub frequencies.

A standard Ritz-Galerkin approximation with llnear splines as basis vectors was used to solve (2.$.$) and (2.3.4) for the mesh modes. As many as 45

spllnes were used, but convergence of the first I0 frequencies and mode shapes

for n = 1 was obtained with 30 spllnes.

ii
3.0 STATEMENT AND SOLUTION OF CONTROL AND ESTIMATION PROBL_4S

3.1 ABSTRACT CONTROL SYSTEM

The space antenna model used here falls into the large class of flexible

structures represented by the abstract control system

Mo"x" + Dox + Aox - Bou + U1,

(3.1.1)

Y = Cox + (/o (3.1.2)

where the generalized displacement x(t) is in a real Hilbert space H, the control u(t)e Rm, the measurement y(t)e Rp. The disturbance _1 and the noise _o

are zero-mean Gaussian white noise processes, as in the finite dimensional LQG

problem. The mass operator M° is bounded, selfadJoint and coercive on H, Bo

is bounded from Rm to H and C is bounded from H to Rp. The unbounded stiffO

ness operator A° is selfadJoint and bounded below, with compact resolvent, and

the damping operator Do is symmetric, nonnegative and Ao - bounded. A similar

abstract description of the LOG control problem has been used by Balas in

[BA2] and [BA3] and Gibson in [GI1]. Also, see Balakrlshnan [BEI].

As usual, by natural modes we mean the etgenvectors xj of the probl_n

w_ M° xj = AoXj, J = 1, 9, ...,

where the mj, s are the natural frequencies of the structure.

problem, as in most structure control problems, the only nonposltlve eigenvalues of A° are zero elgenvalues corresponding to rigid body modes. When

these rigid body modes are controllable, they are contained in the positive

(3.1.3)

In the antenna

12
etgenspace of B "

oBo , so that the operator

Ao = Ao + BoBo I m __ O,

N

for some positive m; i.e. o A° Is coercive.

Next, we define the straln-energy space V - D(_ol/2)

(3.1.4)

with inner product

(3.1.5)

This is a natural space for the generalized displacement because usually. as in our example, <x,x> v is just the sum of twice the elastic strain

energy and the squares of the rigid body displacements.

To write (3.1.1) in first order form, we define the energy space g =

VxH M where HM is H with the equivalent kinetlc-energy inner product

(3.1.1) as

<hl,h2> M = <Mhloh2> H.

(3.1.6)

is twice the kinetic energy in the structure. Now we can write

Z = AZ + BU + (11,

(3.1.7)

y=Cz +(10"

(3.1.8)

where

Z" ()o (11 . 1(11 ] I °ol

(3.1.9)

and

13
A s

D(A o) x D(Ao)CD(A)

As in [GI1, Sec. 2], D(A) is chosen uniquely so that A generates a COsemlgroup T(t) on E.

(3.1.10)

3.2 INFINITE DIMENSIONAL LQG PROBL_4

As in the finite dimensional case, a separation prlnclple [BA1, CU1]

allows us to design the optimal compensator by solving the deterministic

optimal regulator problem and the stochastic state-estlmatlon problem

separately. The infinite dimensional optimal control problem is to choose u

to minimize

m

J-J

0

(<Ozo z>E + <Ru, u>Rm)dt ,

(3.2.1)

where z is the solution to (3.1.7), Q = Q* is a nonnegatlve bounded llnear

operator on E, and R - R°>O.

To slmpllDj certain technicalities about existence of solutlons to the

LQG problem, we will assume that DO + BoB o is coercive, whlch means that the

flexible components of the structure have coercive damping, and that Q is

coercive, whlch guarantees that the optimal closed-loop system is uniformly

exponentially stable.

For thls _termlnistlc problem, the optimal control has the feedback

form

U _ m KZ,

14
where

(3.2.2)

I " -R-1B_[

(3.2.3)

and II is the unique nonnegatlve selfadJoint element of L(E) which satisfies

the infinite dimensional Rieeatl equation

A'JI + I_ - IIBR-IS" II + O = o

(3.2.4)

See [BE1, C01, GI1, GI2].

The minimum variance estimator, or infinite dimensional Kalman filter,

is [BAt, CU2]

where

z = + Bu + G(y-C_z),

(3.2.5)

/% SA 4

G =]JC R-x

and ]][ satisfies the Riccatl equation

-He R--" cnr+o=o

(3.2.6)

(3.2.7)

A

The pzp matrix R and the bounded nonnegative operator _ = _* are the

covarlanoe operators for (/o and (/1, respectively. We assume that any undamped

modes are observable, so that the closed-loop estimator is uniformly exponentially stable.

15
A

We do not require alther Q or Q to be trace class, as is usually necessary for the inflnlte dimensional LQG problem [BEt, CU2]. This is not necessary for our definition of the infinite dimensional compensator° and it allows

more freedeom in designing the compensator to produce desired closed-loop

transient response. See also [012].

The optimal compensator consists of the infinite dimensional estimator

(S.2.$) and the control law

^

U = - KZ,

where K is given by (3.2.3) with (3.2.4), and (3.2.$) becomes

= oZ + Gyo

where

(3.2.8)

(S.2.9)

Ac - A- BK - GC .

(S .2.10)

Note that the optimal compensator has the irrational transfer function

K[sI - Ao]-IG

(3.2.11)

Even though this transfer function would require an infinite dimenslonal

state-space realization, the transfer function itself is only a finite dimensional matrix function of the complex variable s. For control of the antenna

quadrant, this is Just a 1 x 3 row matrix representing the three control chanreels from the displacement sensors at the rib tips to the torque actuator on

the hub.

16
3.3 FUNCTIONAL GAINS

When there is a single actuator (re=l), the operator B o is actually an

element of H and BeE. Hence

or

_Z I R--IH_ | _I_H--1 | Z)E |

(3.3.1)

where

Kz = <f,x>v + <g'X>M

(3.3.2)

with feV and gsH M . H.

(_) = I_R-1

We call f and E,_],U_,_aI control_ains.

(3.3.3)

Similarly, for a single sensor, the measurement y is a scalar and C*eE.

Then

G = *R-1 =

/% A

where fsV and gsH are _&_ _ _ains.

(3.3.4)

For the multi-input-multi-output case, there is a pair of functional

gains for each actuator and for each sensor.

3.4 APPROXIMATION

To obtain a sequence of approximating finite dimensional LQG problems,

we use a Ritz-Galerkin approximation scheme for (3.1.1). We assume a sequence

of linearly independent basis vectors el, which are complete in V. The nth

17
approximate solution to (3.1.1) is

n

i=l (3.4.1)

which is in Vn = span (e l, ..... ,e n). We will need both Vn and HMn, which are

the same set but have the V and HM inner products, respectively.

Initially, let us consider the ease U1 " 0 in (3.1.1). Then the coefficients an(t) satisfy

Mon_m + Don _n + Aon a n = Bon u,

where an(t) is the n-vector containing an(t), i = I .... , n.

tng and stiffness matrices are given respectively by

(3.4.2)

Tl_e mass, dampM

on, lJ = <el'ej>M = <Moel,ej>H, (3.4.3)

D

on, lJ = <Doei'ej>H" (3.4.4)

Aon, iJ " <Aoel 'ej>H (3.4.$)

In general, (3.4.4) and (3.4.5) are valid only if the basis vectors are

in D(Ao). Otherwise, we use

$ •

A°n'lJ " <ei'ej>v - <B°ei'B°ej>H (3.4.$)

(Recall (3.1.4) and (3.1.5)) and a slmllar expression far Don (in our model,

Do is essentially a scalar times Ao.) Also,

18
Bon,£ j - <ei,Boej>, t = 1, ..., n J - 1, ...o m

(3.4.6)

The convergence of such approximations and the corresponding finite

dimensional optimal control problems is discussed in [3.5] for eieD(Ao) , which

includes the cases where the basis vectors are either natural modes of the

structure or component modes. Here, we only outline the formulation of the

sequence of n-order LQG problems. The most efficient way to do this is to

note that, with (3.4.1), (3.4.2) is equivalent to the following differential

equation on HMn . span (el, ..... oen):

Mon'X'n+ DonX n + AonXn = BonU,

where Mon, Don , Aon, and Bon

matrices in (3.4.3) - (3.4.5) and the identification (3.4.1).

(3.4.7)

are the operators on HMn determined by the

Of course, we

can write (3.4.7) as

wlth zn = (Xn, Xn ).

zn = AnZ n + BnU

(3.4.8)

For (3.4.7) and (3.4.8), the optimal regulator problem leads to the finite dimensional Riccatl equation

AA + bAn- _BnR-1BA + Qn = O,

where the operator Q is defined as follows.

n

Pn the projection of E onto g n. Then

Let E n = Vn x HMn

(3.4.9)

and denote by

Qn = enQen[E "

n

(3.4.10)

With this Qn and our preceding hypotheses about damping and the completeness

19
of the ei, s in V, the I_ of (3.4.9) is guaranteed to converge strongly to the

solution II of the infinite dimensional Riccatl equation (3.2.4).

The functional control gains in (3.3.3) are approximated by

(3.4.11)

The strong convergence of _ and B n implies that fn converges in V to f and gn

converges in H to g.

The functional control gains fn and gn are associated with the nth order

control law

with

u = - Knz n = - <fn, Xn>V - <gn, Xn>M o

(3.4.12)

K n = R-1BA

(3.4.13)

To approximate the infinite dimensional compensator, we construct a finite dimensional state estimator

Zn " An_ n

A

+ BnU + Gn(Y- CnZ n),

(3.4.14)

where

Gn "_I n C:_ -1

A

and _ satlsfles the Rlccatl equation

A A $ A ,i,__. " A A

An_ + ]InA n -I[nC n R-1Cn_ + 0n - O.

(3.4.15)

(3.4.16)

2O
A

The operator Qn Is given by

Qn" PnQPn

for Q the covariance of the process noise U1 in (3.1.7).

covariance of the measurement noise _o"

onto En = VnX HM_.

(3.4.17)

As before, R is the

The operator Pn is the projection

We now have the components of the nth approximation to the infinite

dimensional compensator. The nth finite dimensional compensator consists of

(3.4.13) and the control law

U = - KnZn.

This compensator has the rational transfer function

Kn[SI- Acn] -I Gn

where

(3.4.18)

(3.4.19)

A

cn = An- BnKn - GnCn"

(3.4.20)

For each sep(Ac), this transfer function (see Section 9 of [GI2]) the value of

the transfer function in (3.2.11) as n increases.

3.$ APPLICATION TO SPACE ANTENNA

The generalized displacement vector x for this system has components

representing each of the two out-of-plane rigid body angles, the out-of-plane

elastic deformation of each rib and of each mesh. The basic space H is then

21
H- S2 x L2(O,1 ) x L2(O) x "-- x L2(0,1 ) x L2(9)o

(3.5.1)

for each rib and mesh

where 1 is the length of the ribs and 9 is the area of each mesh sector. The

straln-energy space V is llke H with L2(O,I) x L2(Q) replaced by _(0,i) x

HI(fl) .

In the numerical studies in [HRI], we used the following data. For the

performance index in (3.2.1), the state weighting operator Q and control

weighting matrix R are taken as

Q=I and R = .1 x I.

(3.5.2)

Hence <Qz,z> E is twice the total energy in the antenna, plus the sum of the

squares of the rigid body an_les.

The disturbance _I in (3.1.7) and the measurement noise _o in (3.1.8)

are assumed to have the respective covarlances

" and R = .01 x I.

The viscoelastic damplng coefficients in (2.1.2) are

(3.5.3)

cr - .001 and cm : .003.

(3.5.4)

The functlonal gains have the form

f = (al, a 2 , .. ,

functions for each rib and mesh

(3.5.5)

22
4.0 ROBUSTNESS

4.1 OVERVIEW

Robustness refers to the ability of a control system to perform satisfactorily even when the model used for design is an imperfect representation of the

actual system. Modeling difficulties are a major concern in designing control

systems for LSS. Much of the work on functional gains , described in the

Ref. [HR1], is aimed at one particular type of modeling problem: model truncation. Parameter errors are another important source of modeling error.

During the current year of research, we have worked on the problem of developing LSS controller designs which are relatively insensitive to parameter

errors, but still maintain a satisfactory level of system performance. For

antenna structures of the type considered, this is a difficult problem which

does not easily yield to standard techniques. Because of its importance, however, it is a problem which cannot be neglected.

Two different techniques for robustness enhancement have been explored during 1985: Loop Transfer Recovery (LTR) and Sensitivity Optimization. Sections 4.2 and 4.3, which follow, describe the basic features of these

approaches, and also give examples of their performance. Section 4.4 briefly

discusses a controller design approach based on elements of both Loop Transfer

Recovery and Sensitivity Optimization. At the present time this synthesized

approach is still under development.

As a general rule, an improvement in robustness can only be achieved at the

cost of a reduction in some measure of performance. In order to achieve a

proper balance between performance and robustness, some constraints must be

23
imposed. In the loop recovery approach, the open-loop and closed-loop

frequency responses are constrained to approximate those of the full state

feedback LQR design. Thus, the performance of the system is first determined

in the context of the LQR design, and then one tries to obtain an estimated

state feedback implementation which has similar performance and'also adequate

robustness. The sensitivity optimization approach also begins with the solution of an LQR problem and designs an estimated state feedback implementation.

In this case, however, there is no constraint to retain the LQR frequency response. Instead, the goal is to minimize the sensitivity of the closed-loop

eigenvalues to parameter errors. A constraint must be imposed to keep the

closed loop regulator and estimator eigenvalues sufficiently far in the lefthalf-plane, otherwise the optimization routine would generate a very low performance control system. Designs based on sensitivity optimization tend to be

more robust than LTR designs, but their loop gains are also lower.

Both of these approaches rely heavily on the statement of the LQR problem

which serves as a starting point for the designs. The state weighting operatot in the performance index for the LQR problem discussed in Chapter 3 is Q.

Generally, scaling up Q while holding the control weighting matrix R fixed

leads to larger control gains and a less robust compensator. The numerical

examples presented in Sections 4.2 and 4.3 indicate that in addition to the

maEnitude of Q, the relative penalization that Q specifies among the different

components of the state vector can affect robustness dramatically. For example_ Q's which penalize only the RMS I surface error of an antenna model were

Throughout this report, the term RMS surface error refers to the root mean

square displacement of the antenna surface relative to its nominal position

in inertial space. Thus RMS surface error includes the effects of rigid

24
found to lead to designs which were nor very robust. However, Q's which

penalize the energy or the RHS surface error plus the rare of change of RHS

error, were found to be quite robust. While the interpretation of these

results is nor entirely clear at this point, some observations can be made.

Penalizing RHS surface error alone leads to controllers which try to reduce

rigid body displacement much faster than elastic deformations. This means

that the closed loop eigenvalues for rigid body modes are substantially to the

left of those for the lowest flexible modes. In this case, large control torques are produced to control rigid body modes, and the system is nor robust

because these torques strongly excite the flexible modes, which may be inaccurately known. The other two Q's mentioned above penalize rigid body and flexible modes more equally. The real parrs of the closed-loop eigenvalues, and

hence the decay rares, for rigid body modes and lowest elastic modes are

approximately equal and the compensator is more robust. In our continuing

research effort, we plan ro pursue further the relationship between the selection of Q and robustness.

body rotation as well as elastic deformations of the antenna

25
4.2 LOOP TRANSFER RECOVERY TECHNIQUES

It is well known that linear quadratic--regulator-designs _[_el very attractive robustness properties, e.g., one half to infinite gain margin and 60 °

phase margin. This property is based on the use of full-state feedback. In

practice, full-state feedback is virtually never available (certainly not for

LSS) and the state must be estimated. The inclusion of an estimator frequently results in loss of the robustness promised by the regulator design.

Loop recovery techniques have been proposed as a way to recover a measure of

this robustness [DOI,2]. The Loop Transfer Recovery (LTR) approach was originally suggested by Kwakernaak [KW1] and later extended by Doyle and Stein

[DO1,2]. Kwakernaak first derived a method to asymptotically recover the loop

shape of a given Kalman-Bucy Filter (KBF) at the plant output, by allowing the

control weighting in a specific LQR problem to approach zero. This offers the

advantage of recovering the minimal sensitivity properties of the Kalman Filter. Doyle and Stein pointed out that the dual to Kwakernaak's approach,

allowing the noise covariance in a KBF problem to approach zero, asymptotically recovers the LQR loop shape at the plant input. This offers the advantage of recovering the gain and phase margins of the Full-State Feedback Quadratic Regulator. Both procedures will be referred to as Loop Transfer

Recovery (LTR), where Kwakernaak's approach [KWI] recovers a loop shape at the

plant output, and is only valid when the plant has at least as many inputs as

outputs, while the approach of Doyle and Stein [D01,2] recovers a loop shape

at the plant input, and is only valid when the plant has at least as many outputs as inputs. Both approaches require that the plant be non-minimum phase

and it is only in the case of square plants that the designer has a choice of

26
recoverinE the loop shape at either the input or output _ A recent paper

which discusses more fully the various aspects of examining the feedback loop

at various points is Ref. [FR1]

The present work examines the application of the LTR approach to designing

robust control systems for lightly dampedj flexible structures with an unequal

number of inputs and outputs. It is relatively common for a flexible structure to have more available measurements than available control inputs.

Hence, the case of an excess of outputs over inputs will be emphasized. In

particular_ the special case of 1-input and m-outputs_ which_ by reducin E the

loop gain at the plant input to a scalar function_ both simplifies the analysis and clarifies conceptual ideas_ will be used extensively. This is motivated by the study of the wrap-rib antenna.

The first implication of the excess of outputs over inputs is that the loop

shape can be recovered only at the plant input_ while a second implication is

that a number of extra degrees of freedom, required to produce a square system_ are available. The implication of these degrees of freedom will be carefully considered and two methods for takin E advantage of them will be offered.

In particular_ an algebraic method that will provide Loop Transfer Recovery

with arbitrary compensator pole placement and a reduced order compensator,

will be presented. An optimization method which chooses the extra degrees of

freedom so as to minimize the sensitivity of the closed-loop pole locations

with respect to plant frequency errors is discussed in Section 4.4.

Margins at both the input and the output can be guaranteed simultaneously

only when the inequalities _[I+K(s)G(s)]kl and _[I+G(s)K(s)]kl both hold

[_ll.

27
It should be noted that at least one paper on an application of LTR to a

flexible space antenna has appeared in the literature [SU1]. In this work the

design model was reduced to the three rigid body modes of the antenna, while

all the flexible modes were treated as uncertainties. Unstructured uncertainties are a Eood representation of the error due to neglected dynamics so this

is a valid application of LTR, but it will clearly result in a very low performance control law 2. In fact, the resulting control law indicates an openloop bandwidth below 10 -3 rad./sec., where the first flexible mode is not

encountered until approximately .75 rad./sec. The application studied in the

current work involves findin E hiEh performance control laws for flexible

structures with uncertain frequencies. This means that the control action

will actively control the uncertain frequencies, or equivalently, open-loop

bandwidths will fall beyond the uncertain frequencies. In this case the uncertainties are best described in terms of parameter errors rather than unmodeled (i.e. truncated ) dynamics. As illustrated in Section 4.2.1, unstructured uncertainties are not a good representation of parameter errors in the

modeled modes of a liEhtly damped flexible system and the difficulty in

achievinE a robust desiEn is Ereatly increased.

The organization of the next few Sections is as follows. First the question of robustness is discussed_ payin E particular attention to the modern

robustness Theorems of the past 10 years. The most important results relevant

to this study are the unstructured representations of plant uncertainty presented by Doyle and Stein [DOll and by Lehtomaki [LEI,2,3,4]. The unstruc2 The reason why this implies a very low performance control law is discussed

in Section 4.2.3.

28
tured uncertainty indicates that robustness can be guaranteed by makin E the

minimum sinEular value of either the return difference or inverse return difference transfer function matrices larEe enough. This leads to the idea of

definin E robustness in terms of loop shape, and provides the motivation for

loop shapin E as a control desiEn method, and LTR as a particular approach to

achievin E some desired loop shape. UsinE a simple one mode example, it is

shown that the unstructured uncertainty is far too conservative to treat

poorly modeled frequencies. It is concluded that in the case of a strict parameter uncertainty, such as an uncertain frequency, robustness should be

checked by varyin E the parameter within the expected ranEe , while examinin E

the closed loop eiEenvalues. A secondary, but crucial conclusion is that

robustness is not necessarily determined by loop shape, except in the special

case where the system component uncertainties are accurately described in

terms of unstructured uncertainties.

Next a description of the LTR method is riven, alone with a discussion of

some of its properties. AEain , particular attention is paid to the implications of some of these properties for liEhtly damped, flexible structures. It

is concluded that the LTR method does not always guarantee a robust system,

even when the correspondin E full-state feedback desiEn is extremely robust,

and should be used carefully in desiEning compensators for lightly damped,

flexible systems. This is a result of the previously noted conclusion, that

robustness is not necessarily a function of loop shape alone. The reasons for

lack of robustness in LTR desiEns is examined and methods for using the extra

degrees of freedom available in the non-square problem, to find a "better" LTR

design for a given LQR loop shape are suggested.

29
After presenting the desiF_n methods, they are applied to a wrap-rib antenna

model, in an attempt to achieve robustness in the face of frequency uncertainty, while maintaining a given loop shape. The antenna model is chosen

because it is much more sensitive to frequency errors than other models we

have tried. The results are compared with traditional LQG designs, traditional LTR designs and with a direct parameter optimization approach. In the

conclusions, the implications of this comparison, and a general approach to

using LTR to design robust control systems for non-square, lightly damped,

flexible structures are discussed.

Background information may be found in the appendices, including a review

of singular values, proofs of Theorems, and a discussion of numerical considerations in the application of the two LTR methods.

4.2.1 ROBUSTNESS MEASURES FOR HIM0 FLEXIBLE SYSTEMS

One of the major reasons for the application of feedback control, is to

minimize the effect of variations in plant dynamics (or equivalently_ plant

frequency response) on the system performance. Robustness is the study of

exactly how large of a variation in plant dynamics can be tolerated, before a

given feedback system goes unstable.

The basic work in single-input-single-output (SISO) feedback systems was

done more than forty years ago by Nyquist, Bode [BOI] and their colleagues.

The most important robustness result is the Nyquist Stability Criterion, and

the related concepts of gain margin and phase margin. These margins specify,

in an exact sense_ how much the gain and phase of the plant frequency response

30
can vary before the closed-loop system will go unstable. In particular the

gain margin measures the amount by which the open-loop gain can be increased

without causing instability, while the phase margin measures the amount by

which the open-loop phase lag can be increased without causing instability.

These essentially measure the closeness of the Nyquist plot to the -i point,

therefore generalizing the Nyquist Stability Criterion from a measure of absolute stability to one of relative stability.

The present work focuses on developing robust control methods for large,

flexible, lightly damped structures. There are two aspects of these systems

that cause some difficulty in analyzing robustness properties. The first is

their lightly damped, flexible nature, which implies a highly oscillatory frequency response, making concepts such as gain and phase margin much more difficult to interpret. This difficulty, however, can still be handled within

the context of classical control methods, though with some care. The second

difficulty proves to be more fundamental, and stems from the existence of a

number of inputs and outputs, all connected together. Since there is no longer one loop gain to be analyzed, classical methods cannot be applied

directly. Furthermore, it has been shown that analyzing each loop of a multiple loop system separately, does not always give results that are valid for

the overall system [DOll. Fortunately the robustness of multivariable systems

has been studied extensively in the last ten years. It is the purpose of this

Section, to give a quick overview of the state-of-the-art in robustness of

multiple-input-multiple-output (MIM0) feedback uontrol systems.

An extremely general approach to MIMO robustness can be found in the work

of Zames [ZAI,2] on cone bounded perturbations and Safonov [SAI,2] on even

31
more general perturbations. These results rely heavily on the mathematical

field of functional analysis and are applicable to both non-linear and timevaryin E plants. The special case of linear, time-invariant plants, however,

can be dealt with much more simply, and a number of more practical robustness

Theorems have been developed. These Theorems can be divided into _o groups,

paralleling the two modern approaches to control theory mentioned in the last

Section. The most popular approaches to describing the robustness of linear,

time-invariant MIM0 feedback control systems deal with a transfer function

description of the plant uncertainty. It's the transfer function approach

which leads to loop shaping as a method for obtaining robust control, and it

will be dealt with first.

Transfer Function Approaches to Robustness

Consider the simple feedback system illustrated in Fig. 4.1, where:

R(s) - Command Signal

K(s) - Compensator

U(s) - Control signal to the plant

G'(s)- True, possibly unknown plant, described by a nominal

plant, G(s), and some characterization of errors

Y(s) - Output variables (Measurements)

Dropping the dependence on s, some transfer functions describing the system

can then be evaluated as follows:

Y = [(I+GK)'IGK]R = [GK(I+GK)'I]R

Z = R - Y = (I+GK)'IR

U = [(I+KG)'IK]R = [K(I+KG)'I]R

(4.2.1)

(4.2.2)

(4.2.3)

Also define the following functions:

32
In

V

i

V

b

_mS

In

r.T.1

_uJ

.o

33
GK

KG

[I+GK]

[I+KG]

[I+(GK)

[I+(KG)

- Output Loop Transfer Function

- Input Loop Transfer Function .....

- Output Return Difference

- Input Return Difference

- Output Inverse Return Difference

- Input Inverse Return Difference

And make note of the following identities:

[I+(GK)'I] "I = GK[I+GK] "I = [I+GK]'IGK (&.2.4a)

[I+(KG)'I] "I = KG[I+KG] "I = [I+KG]'IKG (4.2.4b)

[I+(GK)'I] "1 + [I+GK] "1 = I (4.2.5a)

[I+(KG)'I] "1 + [I+KG] "I = I (4.2.5b)

The first two identities show that the inverse return difference is simply the

inverse of the closed loop transfer function. The second two identities show

that the sum of the inverses of the return difference and inverse return difference is equal to the identity. The implications of this fact, in terms of

fundamental limitations on design, are discussed in Ref. [SA3].

Note that the output loop and input loop transfer functions and related

return differences are not in general the same for HIM0 systems, though they

are for SIS0 systems. In fact, if the number of inputs is not equal to the

number of outputs, they will not even have the same dimensions. This property

affects the way in which an uncertainty is described. In particular the true

plant G'(s) might be described in terms of a nominal plant G(s) in one the

following two ways:

34
c' (s) = Lo(S)Q(s)

................... -or ....... G'(s) = G(s)Li(s)

(4.2.6a)

(4.2.6b)

Lo(S) will have the dimension of the plant output, and can be considered an

uncertainty acting at the output, while Li(s ) will have the dimension of the

plant input, and can be considered an uncertainty acting at the input. L (s)

o

would provide the best error description for the output loop since

G'(s)K(s)=Lo(S)G(s)K(s), while Li(s ) would provide the best error description

for the input loop, since K(s)G'(s)=K(s)G(s)Li(s ). The point at which the

loop is "opened" to examine robustness will therefore depend on the way in

which the error is described, and vice-versa. For simplicity, only errors

acting at the plant input will be considered, but the results for errors acting at the output can be easily found by replacing K(s)G(s) by G(s)K(s) in the

Theorems of the following Sections.

Unstructured Uncertainty

The most basic description of uncertainties in the form of (4.2.6a) or

(4.2.6b) is the unstructured uncertainty introduced by Doyle and Stein [DO1]

and by Lehtomaki [LEI,2,3,4]. An unstructured uncertainty is one that can be

given no more structure than a bound on a suitable measure of its size. For

matrices, singular values serve as such a measure. Singular values are discussed in Appendix A. It is assumed that the reader is familiar with the concept of maximum and minimum singular values as well as some of their properties.

35
Consider the following two _ possibilities for L(s) = Li(s):

L(s) = [I+Am(S)] (multiplicative uncertainty)

-I

or L(s) = [I+Ad(S) ] (divided uncertainty)

(4.2.7a)

(4.2.7b)

where in each case the perturbation can be bounded by frequency dependent

functions Im and Id:2

_[Am(S)] _ ira(s) (4.2.8a)

and _[Ad(S)] _ Id(S ) (4.2.8b)

THe function Im(S ) and Id(S) are real and positive on the Nyquist D-contour.

The closed loop stability of the true system (KG'[I+KG'] "I) can be determined by the multivariable generalization of the Nyquist Criterion [ROI],

which requires that the det[I+KG'], evaluated on the standard Nyquist D-contour (denoted by sz%), encircle the origin in a counter-clockwise direction,

as many times as there are unstable open-loop poles of KG' For most practical problems involving flexible structures, the only poles of G on the imaginary axis will be at the origin, and there will be an excess of poles over

zeros, implying that limK(s)G(s)=0. In this case the Nyquist D-contour reduces

, Two other unstructured uncertainty representations are the additive and subtracted representations, which are discussed in Ref. [LE1]. These, however,

are equivalent to the multiplicative and divided disturbances respectively

and will not be discussed here.

2 Another method for hounding the uncertainties is the cone-bounded perturbation [DO4]. This essentially corresponds to a multidimensional scaling of

the problem and is again equivalent to Eq. (4.2.8).

36
to the imaginary axis, with the possibility of an identation about the origin,

and siftR can be replaced by jw. Under the assumptions that G and G' have the

same number of open-loop unstable poles, and that KG' and KG have identical

open-loop poles on the jw-axis we have the following two results:

Theorem 4.1: (Multiplicative Disturbance) The closed loop system will remain

stable for all allowable perturbations iff:

"g[KG(s)(I+KG(s)) "I] < i/Im(S) _szfiR

(4.2.9a)

or equivalently if KG is invertible:

_[I+(KG(s)) "I] > lm(S)_szfl R

(4.2.9b)

Theorem 4.2: (divided disturbance) The closed loop system will remain stable

for all allowable perturbations iff both a) and b) are true:

a) L(s) has no zero or strictly negative real eigenvalues for any S¢_ R

b) _[(I+KG) "I] < 1/ld(S)

or equivalently:

_[I+KG] > ld(S) _szR R

_s¢_ (4.2.10a)

(4.2.10b)

A proof of the above results is given in Appendix B.

It should be noted that these are not conservative results, given the error

characterization of Eq. (4.2.8). In fact, if the above conditions are not

met, there exists a perturbation, Am(S) or Ad(S), whose maximum singular value

37
lies below Im(S) or id(s ) respectively, which will destabilize the system.

The unstructured uncertainty can however be unduly conservative in its characterization of error. If the error is known to have some structure it is quite

possible that the particular Am(S ) or Ad(S ) required to cause instability cannot occur.

One major difference between the above two results is the additional

requirement, in the case of a divided uncertainty, that L(s) have no zero or

strictly negative real eigenvalues for s_ R. This is needed to insure that

-1

the function [I+tAd] remains continuous as s varies from zero to one, and

can be interpreted as indicating that Theorem 4.2 cannot guarantee robustness

in the case where phase is completely arbitrary. See Appendix B and

Ref. [LE1] for further details. This places a limit on the situations in

which a divided disturbance can be used to study robustness. For most practical problems, however, the phase will be known to within ±180 °, and both Theorems 4.1 and 4.2 are applicable. In this case the system can be made robust

to multiplicative uncertainties by maintaining a large inverse return difference, or equivalently a small loop gain, while the same system can be made

robust to divided uncertainties by maintaining a large return difference, or

equivalently a large loop gain. The two representations therefore imply opposite requirements for achieving robustness, and together imply that a system

can be made robust either by maintaining very high loop gains, or by maintaining very low loop gains, while it will be most sensitive 'to errors in the

region of gain cross-over. This is a familiar result from classical control,

corresponding to the fact that perturbations of the Nyquist plot far away from

38
the -I point will not affect stability, and the unstructured uncertainties

provide its MIMO generalization. One final note is that Eq. (4.2.5b) indicates that the return difference and inverse return difference are not independent, implying that a system cannot be made robust to both multiplicative

and divided disturbances acting simultaneously at any fixed frequency.*

In applying the unstructured uncertainties to flexible structures, which

will have an infinite number of possibly lightly damped and poorly modeled

modes, the divided disturbance is not particularily useful. The reason for

this is that if some set of modes at "low frequencies" are robustly controlled

by maintaiming a high loop gain in that frequency range, there there will

necessarily exist a further set of modes at some "intermediate frequency"

which will lie near cross-over. If the low frequency modes are inaccurately

modeled, then the intermediate frequency modes will also be inaccurately

modeled, and the system will not be robust to errors in these modes. The

unstructured uncertainty, when applied to lightly damped flexible structures

therefore implies that the loop gain must be kept below some level, whenever

poorly modeled frequencies are present. To gainan appreciation for this

limit, consider a very simple SISO flexible model.

* The divided disturbance is often considered mostappropriate in dealing with

low frequency errors, while the multiplicative disturbance is most appropriate when dealing with high frequency errors. This is because the divided

disturbance indicates high loop gain, typically true at low frequencies,

while the multiplicative disturbance indicates low loop gain, typically true

at high frequencies. It should be emphasized that both are unstructured,

with the only additional structure available in the divided disturbance

being due to the fact that phase cannot be completely arbitrary.

39
Example of a Multiplicative Unstructured Uncertaint 7 Approach

Since the unstructured uncertainty approaches are generalizations of classical SISO theory, they will work for more complicated systems only if they

work for a simple SISO system. With this in mind consider a system consisting

of a single, underdamped, flexible mode whose gain and damping ratio are

known, but whose frequency is known only within given bounds. Such a system

might bedescribed as follows:

f'= fo 2

G'(s) = s2+2_f,s+f,= G(s) = s=+2_foS+fo = (4.2.11)

let Af = (f' - fo)/f 0

then Am(S) =

afs[(2 + Af)s + 2_f']

s 2 + 2_f's + f,2

(2+Af)' + (2{f') _

and lam(jW) I = Iaflw (f,2.w=)2 + (2_f,)2W,

(4.2.12)

(&.2.13)

This is the Im(W) for one particular error in frequency (Af), but Af is

actually only known to be within certain bounds. The correct im(_ ) would then

take the worst case of the above bound for every possible Af. An algorithm to

find Im(M ) for a set of bounds on Af would, for every w, calculate

Eq. (4.2.12) for a sufficient number of Af's between the bounds, and then plot

the worst case.

Fig. 4.2a illustrates the result of such an algorithm. In this case

fo = 1 rad/sec, _ = .01 and Af is allowed to vary between -.I0 and +.i0.

40
201ogl0(i/Im(W)) is plotted, taking the worst case of 200 different Af's between -.I0 and +.I0 at 400 frequency points between .75 rad/sec and 1.25 tad/

sec.

A _omputationally simpler approximation would be to consider only the two

worst cases of Af, (Af = -. 10 and Af = +.i0). The result of an algorithm that

did this for the same number of frequency points is plotted in Fig. 4.2b. The

plots are similar, though the shapes vary slightly between the limits on Af.

The second approach, might provide guidelines for a first cut design, though

it would not strictly guarantee stability by Theorem 4. i.

The function (i/im(W)) places strict limitations on the bandwidth of the

system, since the closed-loop gain must fall below it. This implies that the

bandwidth is limited by the first time that (i/im(W)) falls below 0 db. This

occurs when Im(W ) first rises above i, or when the errors in frequency response first rise above the nominal frequency response. It would be expected

that this would occur at some frequency for any realistic plant but will it

occur near a given uncertain natural frequency? In the cases plotted in Figs.

4.2a and 4.2b (I/Im(W)) actually dropped to about -20 db. Theorem 4.1 would

therefore imply that a SISO feedback system with with a i% damping ratio and

10% frequency error would be required to have a loop gain that fell below

-20 db at that frequency, sharply limiting bandwidth and therefore performance.

To get a rough idea of the extent of the limits imposed on such a system by

Theorem 4.1, consider an approximation of Eq. (4.2.13). Assume that _ and Af

41
3

m

3o

2O

10

C m m .

-20

-3o

I I I

• .9 1.0 1.1

Frequency (radlsec]

4.2____I/__Ic___for -o.I_ Aw _ +o.I

3G

2O

._ 10

.Q

0

3

W

-10

,-4

-20

-30

19 1 Io 1 _1

Frequency (rad/sec)

_4._._ l_m_O._ for _..0.1 _&[÷0.1

-...

42
are both small as compared to i, and that the function reaches a maximum at

......................... w_--__._f'.Then:

IACjw) lmax ffi Af/_ (4.2.14)

This indicates that relative frequency errors on the order of the damping

ratio will limit the system closed-loop bandwidth. This is a relatively

strict limitation, since if the damping ratio in a particular mode is 1%, a

reasonable value for flexible space structures, a I% frequency error might

destabilize a system whose bandwidth included that mode.

This limit seems overly restrictive, but to gain a more concrete appreciation of the conservativeness of the unstructured uncertainty for this particular system consider a simple constant gain controller in the feedback loop.

The closed-loop characteristic equation of the system is:

2

s + 2_f's + (l+k)£ '2 = 0 (4.2.15)

This is clearly stable for all k > -i, for all _ > 0 and for all {'. The

bandwidth of the closed-loop system can therefore be increased without bound,

for arbitrary error in frequency and arbitrary non-zero damping ratios, in

conflict with the requirements implied by Theorem 4.1. The reason that the

unstructured uncertainty is so conservative, even for this very simple, SIS0

system, is that it doesn't take into account any information on the phase of

the system. Essentially it defines limits on the gain error, while allowing

completely arbitrary phase error. For this system though, the phase error is

uniquely determined by the gain error, so that the exact phase shifts that

would be necessary to destabilize the system for a given gain shift can never

43
occur. The unstructured uncertainty is therefore not a good characterization

of error due to uncertain frequencies in the modeled modes of a lightly

damped, flexible structure. Next apply Theorem 4.2 to the same problem.

Application of Theorem 4.2 t_ooth__eeExample Problem

In applying Theorem 4.2, the first step is to ascertain that L(s) is never

zero, or strictly negative on SZ_R, or equivalently that the phase of L(jw)

never reaches ±1800 For the example:

f'2[(fo2-W2)(f'2-w2)+4_2fof'W2+2_W[fo(f'2-w2)-f'(foZ-W2)]j]

L(jw) = (4.2.16)

W2[(f'2-W2)2+4_'f'2w 2]

L(jw) is equal to zero or strictly negative only when the imaginary part of

eq. (4.2.16) is identically equal to zero. This in turn is true only when _=0

or when w2=-fof' , so unless _=0, there is no real frequency at which L(jw) is

either zero or negative real, indicating that Theorem 4.2 is applicable. _ For

the example the divided disturbance is given as follows:

.Afs[(2+Af)s+2_fn_

Ad(S) = s2+2_foS+foZ v,

where Af=(fo-f')/f' (4.2.17)

[Ad(jw) [ will reach a maximum near W=fo, so: IAd(JW) lmax = Afl_

Theorem 4.2 therefore implies that whenever the frequency error is on the

order of the damping ratio, loop gain at that frequency must lie above Odb, or

4 Actually, for small _ and large Af the phase of L(jw) asymptotically reaches

±180 °, but it is never exactly equal to ±180"

44
equivalently, that sufficiently high gain will stabilize the system. Theorems

4.I and 4.2 together indicate that a feedback control for this example can be

made robust by either introducing very high gains, or very low gains, while in

the case where Af>_ there will exist a zone of intermediate gains which will

not be robust, as was pointed out in the previous Section. This is a conservative result since it has been ascertained that the above example is robust

to arbitrary frequency errors for any positive feedback gain.

In conclusion, errors in the frequency of a lightly damped oscillator provide a highly structured parameter variation and the unstructured uncertainties of Doyle and Lehtomaki provide an overly conservative characterization of

these errors. This further implies that due to their conservativeness

unstructured uncertainties are not particularly useful for determining the

robustness of flexible systems with poorly modeled frequencies. An implication of this observation will be dicussed further in the context of LTR design

methods (Section 4.2.4).

4.2.2 ASYMPTOTIC LOOP TRANSFER RECOVERY (LTR) CONTROL DESIGN

LTR is an appealing approach to control design for two basic reasons. The

first is that it can be used to recover the gain and phase margins of a Linear

Quadratric Regulator (LQR) at the plant input, or of a Kalman Bucy Filter

(KBF) at the plant output. This is sometimes called robustness recovery [DO2]

or sensitivity minimization [KWl], but these are somewhat misleading terms

since they suggest that the loop recovered system will have the same robustness properties as the full state feedback LQ regulator or the KB filter.

This is not necessarily true as will be indicated more specifically later.

45
The second appeal of the LTR approach is its applicability to loop shaping as

a more general control design method. The idea of loop shaping is most

clearly spelled out in Ref. [DOll, where it is pointed out that a number of

system properties, including performance, sensitivity, noise rejection,

"robustness" and control effort, depend on the system loop shape (maximum and

minimum singular values of the system transfer function matrix). The LTR loop

shaping approach involves first designing an LQR loop or a KBF loop with

desired characteristics, and then recovering that loop shape by LTR methods.

Again there is an assumption in the loop shaping approach that all systems

with the same loop shape will behave identically. This is true for the nominal plant, but when the plant parameters are perturbed, the two loops may no

longer be identical and the system response (and stability) may vary considerably. In a later Section, a number of examples will be given of systems with

identical loop shape, but different robustness characteristics.

For simplicity, the input loop recovery procedure of Doyle and Stein

[DOI,2] will be considered, noting that output loop recovery [KWI] is simply

its dual. After describing the procedure it will be interpreted for SISO systems, and for 1-input_ m-output systems. This interpretation will lead to a

polynomial (pole/zero cancellation) approach to loop recovery. Methods for

using the extra degrees of freedom available in the loop recovery procedure to

find an "optimal" loop recovered system will be considered in Section 4.4.

Input Loop Recovery

The LQR loophas excellent robustness properties. In particular an LQ

regulator with diagonal input weighting matrix R is guaranteed to have at

46
least an infinite gain margin, a gain reduction margin of I/2 and a phase margin of ±60", simultaneously in all the feedback loops [LEI,2]. The idea of

input loop recovery is to design an LQ regulator with a desired loop shape and

then use an asymptotic procedure to design a KB filter that recovers that loop

shape. Since gain and phase margins are a function of loop shape, the output

feedback system, using a KB filter to estimate the plant states, will have the

identical gain and phase margins as the LQR loop. For this reason, input loop

recovery is sometimes called robustness recovery. This is the point of view

taken in Ref. [DO2]. Another advantage of the loop recovery approach is that

it is relatively simple to specify the LQR loop shape at low frequencies by

carefully choosing the weightings in the LQR cost functional [HAll. Loop

Transfer Recovery (LTR) can then be used to achieve the identical loop shape

for an output feedback system. This is the loop shaping control approach, and

is the point of view taken in Ref. [D01].

To see how loop transfer recovery works, consider a full-state feedback

control law, u=-Kx. The input loop transfer function for this system will be

K(sI-A)'IB. Now consider a state estimator of the form,

= A_ + Bu +G(y - C_)

and a control law based on the state estimate _, u=-K_.

pensator transfer function is,

K(s) = K(sI - A + BK + GC)'IG

and the input loop transfer function for the output feedback system is:

K(s)G(s) = K(sI - A + BK +GC)'IGC(sI - A)'IB

To get loop recovery choose an estimator gain matrix G, such that,

K(sI-A+BK+GC)'IGC(sI-A)'IB * K(sI-A)'IB

(4.2.18)

In this case the com-

(4.2.19)

(4.2.20)

(4.2.21)

47
In Refs. [DOI,2] it is indicated that any G matrix that asymptotically

satisfies

G _ qBW (4.2.22)

for some arbitrary symmetric, positive definite matrix W, will recover the

loop shape, q is a large scalar parameter which is increased to achieve "

asymptotic loop recovery. To illustrate this result, note that the transfer

function between y and y in the state estimator can be written as follows,

= (I + C_G)'IC_Gy (4.2.23)

where _ denotes (sI - A + BK) "I. This leads to the following alternate representation of K(s),

K(s) = K[_ - _(I + C@G) IC_]G (4.2.24)

which after some further manipulation can be rewritten as:

K(s) = K@--G(I + C_G) "1 (4.2.25)

Now let G _ qBW which means that,

K(s) -_ K_B (C_B) "l . (4.2.26)

Using the identity, _--B= _B(I + K_B) "1, where _ = (sI - A) "I Eq. (4.2.26)

becomes,

K(s) _ K_B (C_B) "I (4.2.27)

and,

K(s)G(s) _ K_B(C_B)'Ic_B = K_B (4.2.28)

This shows explicitly how loop recovery inverts the plant dynamics from the

left and replaces them by the full-state feedback dynamics.

So far there has been no requirement that K be derived from an LQR problem,

nor that the estimator be a KB-filter. In fact any appropriate method (e.g.,

48
pole placement) could be used to select the control gains K, and any estimator

(e.g., an observer) which both asymptotically satisfies Eq. (4.2.22) and stabilizes the plant, could be used to calculate the asymptotic estimator gains

G. However, the choice of an LQR approach for finding K has the advantage

that the-resulting loop will have desirable properties, and the use of a KBfilter to recover that loop shape has the advantage of guaranteeing a stable

plant at every stage in the asymptotic loop recovery procedure.

To use a KB filter for the LTR, the first step involves appending additional columns to the B matrix until the plant is square. The only requirement on these columns is that the resulting plant be minimum phase, and within

this constraint the extra columns can be viewed as free design variables.

Once the system is "squared up," a KB filter is found, where the measurement

noise covariance N is an arbitrary positive definite matrix, and the state

noise covariance has the following special form,

M = M + q2BwBT (4.2.29)

O

where M is some nominal noise covariance, q is the parameter which will be

O

increased to achieve recovery and W is an arbitrary positive definite matrix.

Note that the B matrix in Eq. (4.2.29) includes the appended columns, so the

rank of BWB T is equal to the number of plant outputs.

To show that this KB filter will asymptotically satisfy Eq. (4.2.22), consider the Riccati equation:

AP + PA T - pcTN'Icp + M + q2BwBT = 0

O

2

and divide by q :

(4.2.30)

49
A(P/q2) + (p/q2)AT - q2(G/q2)N(G/q2)T+ (Mo/q2) + BWBT = 0 (4.2.31)

Now let q _ -, under the assumption that the plant be minimum phase, to get:

-(G/q)N(G/q) T- + BWB T _ 0 (4.2.32)

The KBF estimator gains will therefore have the following asymptotic property:

G _ qBW1⁄2TN "1⁄2 (4.2.32)

Where T is an orthonormal matrix (ie. TTT = TTT = I). This of course satisfies Eq. (4.2.22) and the KB filter will asymptotically recover the loop shape

as q is made to approach infinity.

One quick note before continuing to an interpretation of loop recovery for

SISO systems, is that the estimator gain matrix G may become very large before

loop recovery is achieved. This may cause some computational problems and may

not provide the best solution for a robust compensator, as will be discussed

further in Section 4.2.3. A related problem concerns the convergence of estimator functional gains. As G increases in size, the performance of the estimator increases, implying that the number of modes necessary to achieve convergence of the estimator functional gains also increases. Asymptotic LTR

therefore implies a large system model if the estimator is expected to converge to the optimal infinite dimensional estimator, as measured by functional

gains. This issue still needs to be looked into. Next the SIS0 case will be

examimed to gain a more intuitive understanding of the LTR procedure.

SISO Interpretation of Loop Recovery

The SISO loop recovery problem is as follows. Given a SISO

50
G(s) = nCs)/d(s), find a proper (or strictly proper) K(s) = N(s)/D(s) such

..................... that K(s)G(s) = G(s)K(s) = _(s)/d(s) for some specified _(s). The obvious

solution is to let N(s) = _(s) and D(s) = n(S)Dl(S), where Dl(S) consists of

enough poles to make K(s) proper {or strictly proper), and these poles are

placed far enough into the l.h.p, that they do not significantly affect the

loop shape in the design region. This interpretation also clarifies the minimum phase condition, since if G(s) is non-minimum phase, this procedure will

result in a pole/zero cancellation in the r.h.p., that will destabilize the

closed loop system.

The asymptotic LTR procedure does not calculate an exact pole/zero cancellation, but achieves this cancellation asymptotically. It is clear, however,

that it is possible to achieve loop recovery by specifying a compensator that

does achieve exact cancellation, without going through an asymptotic process.

In both cases the cancellation will be close to exact before loop recovery is

achieved, so it will be assumed that it is exact.

This interpretation indicates the essential difference between the true LQR

loop and the LTR loop. While both loops may look identical over any given

frequency range, the LTR loop will contain a number of hidden pole/zero cancellations. These cancellations do not show up for the nominal plant, but as

soon as the plant changes, they will no longer be exact and the LQR and LTR

loop shapes may be considerably different. This is especially evident for

lightly damped systems, since the plant zeros in this case will lie close to

the imaginary axis. Very small errors in the plant zeros will therefore produce very large errors in the LTR loop shape. It is because of this property

that an LTR design will in general not have the same robustness characteris51
tics as the corresponding LQR design.

Some Comments on the MIMO Case

LTR is a more interesting and less obvious design procedure in the MIM0

case, but the same basic properties carry through. In this case the designer

would choose the following compensator,

K(s) = K#B(C@B)'I/DI(S)

where Dl(S) would have the same interpretation as the SIS0 case.

can be rewritten by dividing out the common factor det[sI - A] to get,

K(s) = [Kadj(sI-A)B][Cadj(sI-A)B]'I/DI(S)

where Kadj(sl-A)B is the MIMO generalization of _(s) and Cadj(sI-A)B is the

MIMO generalization of n(s).

(4.2.35)

For the special case of 1-input and m-outputs [Cadj(sI-A)B] will be a

square matrix where only the first column is fixed and the next m-I columns

are at the designers discretion (within the constraint of a non-minimum phase

plant). Thinking again in terms of the SISO problem, this is somewhat like

having an n(s) at the designer's discretion, corresponding to some freedom in

the placement of compensator poles. Once the poles are picked, however, loop

recovery will specify the compensator zeros. Keeping this in mind an algebraic, direct pole/zero cancellation, design of a LTR compensator for a

1-input, m-output plant will be considered.

52
4.2.3 ALGEBRAIC LOOP TRANSFER RECOVERY

As indicated in the previous Section, loop recovery can be achieved for a

SISO system by defining a compensator that exactly cancels the plant zeros and

replaces them by a set of "optimal" zeros corresponding to the LQR loop. For

the case of a 1-input, m-output plant, a similar procedure can be carried out,

but now the designer has a set of free design variables. This freedom can be

used in a variety of ways. Some possibilities include choosing a minimum

order loop recovery compensator or specifying the location of the compensator

poles. For the purposes of this discussion a minimal order compensator for

which the compensator poles can be chosen arbitrarily will be considered, and

an algebraic procedure to find the compensator numerator polynomials, or equivalently the compensator zeros will be used to achieve a given input loop

shape. It should be emphasized that this is not a requirement of the algebraic LTR approach. Lower order compensators might be found by removing the

freedom to specify pole location, while higher order compensators would allow

the designer a number of free parameters which could be adjusted so as to meet

some other criteria, possibly via an optimization process. The choice considered for this discussion, however, allows one to examine the important effect

of compensator pole location, without adding excess variables to the formulation.

A 1-input, m-output plant and compensator will have the following special

form:

53
"gl(S)l "nl(S 

g2(s) ] In2(s)

G(s) = . = . /d(s)

.gin(s)] nmCs).

(4.2.36a)

K(s) ffi[kl(S ) k2(s ) ... kin(s)] ffi[N1(s ) N2(s ) ... Nm(S)]/D(s )

(4.2.36b)

The input loop transfer function will be:

m

K(s)G(s) = Zni(s)Ni(s)/d(s)D(s)

ill

(4.2.37)

To achieve loop recovery it is necessary that:

m

K(s)G(s) = ___ni(s)Ni(s)/d(s)D(s ) = _(s)/d(S)Dl(S ) =

_(s)/d(s) (4.2.38)

ill

where _(s) is the numerator polynomial for the LQR input loop. Dl(S ) is again

a set of poles required to maintain a proper (or strictly proper) compensator,

where these poles are kept far enough out in the l.h.p, that they don't significantly affect the loop shape in the design region. Then D(s)=Dc(S)Dl(S),

where Dc(s ) must be cancelled out by the input loop numerator. Once the compensator poles are selected, the following equation must be solved to find the

compensator numerator polynomials:

m

E ni(s)Ni(s) = _(S)Dc(S)

iffil

(4.2.39)

Next consider the minimal order compensator that can satisfy Eq. (4.2.39).

54
Let n_ be deg(_(s)), nc be the deg(Dc(S)) , n I be deg(Dl(S) ) and nN be

deg(Ni(s)). The degree of the left side of Eq. (4.2.39) will be nN + nn,

while the degree of the right side of Eq. (4.2.39) will be n_ + nc, so the

first requirement on _ is that:

nN + nn = n_ + n c (4.2.40)

The second requirement on nN is that there must exist enough degrees of freedom to satisfy Eq. (4.2.39) for arbitrary _(S)Dc(S). The number of free parameters is m(nN+l), while the number of polynomial coefficients that must be

satisfied is ng+nc+l, this implies that:

= + i (4 2.41) m(nN+l) n_ + n c

Combining Eqs. (4.2.41) and (4.2.42) gives the following result for nN:

nN = (nn/(m-l)) - I (4.2.42)

For a strictly proper compensator, the order of the compensator is

n c + n I = nn/(m-l). So a 1-input, 2-output plant has a strictly proper loop

recovery compensator of the same order as the degree of the plant numerator

polynomials, while a 1-input, 3-output plant has a loop recovery compensator

of half that order, etc.. This approach therefore leads to lower order compensators than the asymptotic, observer based approach described in the previous Section. In fact, as m increases, the order of the compensator can become

very small, while still achieving loop recovery.

Once the order of the compensator and the location of the compensator poles

are chosen, the compensator numerator polynomials are found by the solution of

55
a set of m(nn+l) linear equations. Let the ith plant numerator polynomial be

written as follows:

n

ni(s ) = ni'nnS n + ... + ni,lS + ni,o (4.2.43a)

and similarily let the ith compensator numerator polynomial be:

n

Ni(s ) = Ni,nNS N+ ... + Ni,l s + Ni,0 (4.2.43b)

Then:

hl, 0 0 ...... 0 ..... nm, 0 0 ...... 0

nl, 1 nl, 0 ...... 0 ...... nm,l nm ,0 ..... 0

nl,nn nl,nn-I ....... nm,nn nm,nn-I ....

.,., ,,,..

0 nl,nn 0 n m_n n......

°,o. ,,°,. ,o.o. •

oooo °°_o, .°°°o °

,°°°°

0 0 .... nl,nn 0 0 ... nm,nn

NI,0

NI,I

Nl,n

Nm,0

N

m_n n

do

d I

n

= (4.2.44_

d

n +n

n nn

or symbolically, Nn = d, where di is the ith coefficienU of g(S)Dc(S ). The

above algorithm is easily implemented on digital computer whenever a unique

solution to Eq. (4.2.39) exists, or equivalently whenever the matrix N is

non-singular.

56
4.2.4 ROBUSTNESS OF LTR DESIGNS

The purpose of this Section is to present a discussion of particular

aspects of LTR designs which relate to robustness.

One observation concerns the relationship between loop shape and robustness

implied by Theorems 4.1 and 4.2. Specifically, if model errors are accurately

described by unstructured uncertainties, then LTR methods can be used to

ensure that the conditions of Theorems 4.1 and 4.2 are met, thereby guaranteeing robust stability. This constitutes the ideal case for applying Loop

Transfer Recovery_ since specifying constraints on the loop shape is a necessary and sufficient condition for robustness. On the other hand, some model

errors are not well described by unstructured uncertainties. In particular

frequency errors in the model of a lightly damped flexible structure fall into

this category (see Section 4.2.1)_ as do most cases of strict parameter variations. In this case Theorems 4.1 and 4.2 give sufficient but not necessary

conditions for robust stability, and a design based upon constraining loop

shape on the basis of unstructured uncertainties will be conservative, resulting in low performance. In the case of lightly damped structures the implied

constraints on loop shape would require that uncertain flexible modes not be

actively controlled (again see Section 4.2.1). Therefore, the robustness of

designs which actively control flexible modes cannot be analyzed by examining

loop shape alone. In fact, for a given loop shape, a whole family of LTR

designs may exist, where the robustness of these designs varies. In conclusion, the application of LTR methods to lightly damped structures with uncertain frequencies cannot be blindly approached in terms of conditions implied

on the loop shape. A major emphasis of the current study has been to identify

57
conditions which are important. These results will be illustrated by the

examples presented in Section 4.2.5.

As noted in Section 4.2.2, one variable among a family of LTR designs with

the same loop shape is the location of pole/zero cancellations. Whenever

these cancellations occur near the imaginary axis a small perturbation in the

plant, resulting in a small shift of the plant zeros, will cause a very large

variation in the open-loop frequency response. It would therefore be reasonable to expect that an LTR design which tries to avoid pole/zero cancellations

near the imaginary axis will be more robust than one that does not. One way

to reduce the number of plant zeros, and therefore the number of cancellations, is to add measurements. A second order flexible system, with m-inputs

and m-displacement outputs, can have at most (n-2m) transmission zeros [EDI] I

Remembering that in the asymptotic LTR approach, each additional measurement

entails an additional artificial input, this implies that every measurement

removes a pair of transmission zeros. In the version of the algebraic

approach used at this point, additional measurements are used to reduce the

compensator order, so it is unclear whether any robustness improvement might

result.

A method for shifting the cancellations away from the imaginary axis in the

asymptotic approach would be to use the extra degrees of freedom available in

the choice of artificial inputs to shift the plant transmission zeros to the

left. The best way to do this is still under investigation. This idea is

also the motivation behind the algebraic LTR approach, which allows the desigo..mm.oo_mo_o..o...

I For velocity outputs the maximum number of finite transmission zeros is

(n-2m+l)

58
ner to place the pole/zero cancellations as far into the left half plane as

desired. The general question of how to choose the most robust LTR compensator for a given LQR loop shape is certainly not closed, and one method which

we will continue to study is the optimal sensitivity approach described in

Sec. 4.4.

One final comment on the robustness of LTR designs is a short discussion of

Ref. [SH1], which deals specifically with the robustness problems associated

with any compensator design that incorporates very high estimator gains.

Asymptotic LTR designs can result in extremely high (on the order of 10 G or

higher) gains in the estimator Kalman gain matrix G. Two difficulties may

arise. The first involves the case where a plant perturbation changes the

structure of the plant transmission zeros at infinity. This is a perturbation

that affects the excess of poles over finite transmission zeros 2. An example

in which this occurs for a realistic flexible system is as follows: Consider

the system illustrated in Fig. 4.3, consisting of two disks (of inertia 1),

stacked one on top of the other, connected by a rod (of length 1) which acts

as a rotational spring (of spring constant 1/2). The input is a torque

applied to the lower disk. The nominal output is the rotation of the upper

disk. The nominal transfer function is:

G(s) = 1/2

s2(s2+l ) (4.2.45)

2 Note that this is not the same as the finite transmission zeros added when

neglected dynamics are appended to the design plant. In this case the number of poles increases along with the number of transmission zeros, indicating that the excess does not change, and the structure of the zeros at

infinity therefore does not change.

59
An LQR design which minimizes the mechanical energy, plus the rotation of the

lower disk squared, plus the control input squared, results in the following

control gain vector (in modal coordinates):

K = [ 1.0000 2.0555 1.6124 0.6325 ] (4.2.46)

An asymptotic LTR design can be achieved by setting the process noise covariance equal to (lxlO12)BB T, resulting in the following Kalman Filter gains:

G

98'255"11

1,000,000.

98,116.

995,170.

(4.2.47)

This design is relatively robust with respect to frequency variations (withstands ±30% errors in the flexible frequency), but consider an error corresponding to a manufacturing defect which places the rotation sensor at a point

¢ below the top disk. The true transfer function would then become:

= zs2+I/2

G'(s) s2(s2+l) (4.2.48)

Therefore, while the nominal plant has no finite zeros, the true plant has a

pair at ±_-i/2z, satisfying the conditions referred to in Ref. [SHI]. The

asymptotic LTR design defined by Eqs. (4.2.46) and (4.2.47) will be unstable

for z>0.04,* which produces finite zeros at ±3.54j or lower on the imaginary

axis. This might be a troublesome robustness problem in some applications,

but it can occur only under very special circumstances. For any square flexi-

* It will also be unstable for E<-0.01, but this corresponds to a measurement

location above the disk and is not physically feasible.

6O
ble structure with displacement outputs, the maximum number of finite transmission zeros is (n-2m) [EDI]. If the nominal plant has this maximum number

of transmission zeros, then no perturbation in plant parameters can result in

the conditions required by Ref. [SHI] for robustness problems.

In conclusion the robustness problems associated with LTR control exposed

in Ref. [SHI] depend on a nominal plant which is chosen such that there exists

a perturbation in plant parameters which will increase the number of finite

transmission zeros by at least 2. This possibility can always be ruled out by

ensuring that the nominal (design) plant has the maximum possible number of

finite transmission zeros, which will be the case in most practical problems.

In particular the antenna model studied in this report always has the maximum

number of transmission zeros and is not affected by the robustness problems

exposed in Ref. [SHI]. The extension of these ideas to infinite-dimensional

systems is not considered in Ref. [SHI] and has not been addressed in the present work. This would, however, provide an interestin E topic for future examination.

The second observation in [SHI] is that, due to the very high gains

involved, a small error in the realization of the compensator could destabilize the system. The authors suggest that a compensator be realized directly

from the transfer function K(sI-A+GC+BK)'IG. This issue also has not been

considered in detail, but it will be examined later.

61
I

FiEure 4.__.33Two Disk System

62
4.2.5 EXAMPLES OF THE LTR APPROACH

Antenna Model

The methods developed in the previous Section are applied to the wrap-rib

antenna described in Chapter 2. The system is modeled by a component mode

method, ie. mode shapes of the beams and the mesh Sections are found separately, and the "modal" representations are then combined to form a model for

the 90" sector. A subset of six modes are selected from this very large model

to constitute a design model for one quadrant of the antenna Frequencies,

modal input influence coefficients, and modal output influence coefficients

for the six modes included in the design model are listed in Table 4.2.1.

Damping is assumed to be visco-elastic (ie. the damping matrix is proportional to the stiffness matrix)*, but the model allows different damping coefficients _ for the beam and the mesh portions of the sector model. This

implies that damping is added at the component mode stage, and furthermore

that the damping matrix, after combining the beam and mesh component mode

models and calculating the overall modal representation, will include offdiagonal terms. The final antenna model, in a second-order modal reprensentation, will have the following form:

+ D_ + A2¢ = Bu, y = C¢ (4.2.49)

where A 2 is the diagonal matrix of the natural frequencies squared, while D is

i

* Visco-elastic damping implies that the damping ratio of a mode increases

proportionally with the frequency of that mode. Higher frequency modes will

therefore be more highly damped. This is a realistic situation for most

materials, particularily in the higher modes.

z The damping coefficient is a scalar number which multiplies the stiffness

matrix, resulting in the damping matrix.

63
symmetric, positive-semidefinite, but is diagonal only in the case when the

damping coefficients in both the beam and the mesh are identical• The particular model studied here has higher damping in the mesh, and the damping

matrix is therefore not diagonal, though the off-diagonal terms are in general

an order of magnitude smaller than the diagonal terms. The entire damping

matrix is listed in Table 4.2.1.

Table 4•2.1 Antenna Modal Data

Mode #

B

0

C 1

C 2

C 3

C4

C5

C6

O.

0.

•093

•093

.0

.0

.0

.0

.0

6•95

•0105

-•122

-. 122

•243

.002

.001

.000

•172

18.95

•0302

-. 285

-. 285

.183

•183

.129

-.019

•129

4

36.57

•0589

-.511

-.511

.617

•341

•241

.068

.437

51.89

•0519

-. 120

-. 120

•201

.099

•070

-.027

•142

55.32

•0537

-.Iii

- Iii

115

052

037

020

081

Damping Matrix

"0.0 0.0 0.0 0.0 0.0 0.0

O. 0 O. 146 -0. 030 -0. 050 -0. 016 -0.00_

0.0 -0.030 1.144 -0.151 -0.096 -0.049

0.0 -0.050 -0. 151 4.308 -0.009 -0.095

0.0 -0.016 -0.096 =0.009 5.385 0.099

0.0 -0.009 -0.049 -0.095 0.099 5. 940

m

Simpler models, including series of connected masses and springs, as well

as a beam/hub model have also been studied, but the difficulty in achieving

robust control designs has been much greater for the antenna sector model. It

is hypothesized that this is due largely to the addition of mesh modes to the

beam/hub model, resulting in a densely packed set of frequencies, and some

very slightly controllable/observable modes. From the point of view of con64
trol design, the important aspect of this particular model is that it is

difficult to control robustly, and therefore poses a challenging problem.

The number of available measurements (or plant outputs) is not fixed at

this time. There must be at least one measurement relative to inertial space,

to insure that the rigid body mode is observable. This measurement is provided by a rotation sensor colocated with the hub torque input. To effectively control the shape of the antenna, measurements on the antenna surface

must also be taken. Possible locations for these measurements are labeled and

numbered on Fig. 4.4, where all these measurements are relative to the central

hub position. The modal output influence coefficients are listed in

Table 4.2.1. In the compensator design stage, the measurements used will be

referred to by the corresponding number found on Fig. 4.4. For the examples

presented here, only one other measurement will be used. That measurement

corresponds to position #2, at the tip of the beam which lies perpendicular to

the axis of rotation.

One final note on the measurement locations is that all seven measurements

are not necessarily independent for a given mathematical model. This means

that, for a particular mathematical model studied, the value of one measurement may be directly related to that of another measurement. As an example,

consider measurement locations 2 and 6. The complete mathematical model of

the antenna sector includes symmetric beam modes (the two beams move together

in the same direction) and asymmetric beam modes (the motion of one beam

exactly opposes the motion of the other). With no mesh, the asymmetric modes

are uncontrollable, and even with the mesh they are only very slightly controllable. These modes were therefore neglected in the mathematical design

65
©

FiEure 4.4 Antenna quadrant Model

66
model, since they do not have much affect on the control problem. This,

however, implies that in terms of the mathematical design model, the two beams

will always move together, and the measurements at locations 2 and 6 will be

linearly dependent upon each other. This points to an important consideration

in determining the independence of measurements. The fact that measurements

are independent for a physical system or for some mathematical model does not

imply that they are independent for every possible, reduced order, mathematical model. Measurements 2 and 6 are clearly independent for the overall

mathematical (and the physical) antenna model, but due to the exclusion of

asymmetric modes, they are not independent for the reduced order mathematical

design model studied here, due to the exclusion of asymmetric modes. In terms

of this study, the implication is that adding measurement number 6 to number 2

will not result in any further information on the state of the system. This

is particularily important with respect to the algebraic design method which

must assume independence of all the measurements used.

LQRDesign Problem

The first step in a Loop Transfer Recovery (LTR) control design is the

choice of a full-state feedback control law, or equivalently the matrix K of

control gains. The matrix K can be found by a number of methods, including

pole placement, but, perhaps the most advantageous is by solving a Linear

Quadratic Regulator (LQR) problem. The statement of the LQR problem is as

follows:

Given _ = Ax + Bu (4.2.50)

67
Minimize the following cost functional:

J =f(xTQx + uTRu)dt

0

where Q is symmetric, positive semi-definite and R is symmetric, positive

definite. The solution to the problem, for full state feedback is:

u=-Kx

where K = R'IBTp and P satisfies the following Riccati equation:

PA + ATp - PBR'IBTp + Q = 0

(4.2.51)

(4.2.52)

(4.2.53)

The LQR approach has a number of positive features, including guaranteed

gain and phase margins [LE1,2]*, the ability to shape the loop gain at low

frequencies [HA1] and the simplicity of the approach. The designer "simply"

chooses the Q and R matrices and the computer does the rest.

For the antenna problem, the input is a scalar, so without loss of generality R can be replaced by I. The problem then reduces to finding a Q matrix in

the following form:

Q = qcQo (4.2.54)

where qc is a scalar parameter which is increased to increase system performance (and loop gain), and Qo is a matrix which specifies the form of the cost

weighting on the states. There are a number of bases for choosing Q. One

might wish to achieve a particular loop shape, or a particular closed loop

tim_m_mmommm_mm.m_m

* This constitutes a large part of the appeal of the LTR control design

approach

68
pole configuration, or there might be some considerations in terms of a

physically meaningful objective function to be minimized. For instance, one

might want to minimize the mechanical energy in the system, or a pointing

error, or perhaps the RMS error of a surface or signal.

Three LQR designs will be presented for the antenna. The first minimizes

the RMS surface error, since this provides a good meausure of antenna performance. The second minimizes the mechanical energy in the antenna, plus a term

corresponding to the rigid body mode (this is necessary since there is no

mechanical energy corresponding to rigid body rotation). The third minimizes

RMS error, plus the rate of change of RMS error. All three LQR designs remain

stable for combined frequency errors of -g9% or +100%. Any robustness difficulties in the LTR designs therefore are not due to similar difficulties in

the corresponding LQR design.

LQR Design #1

The first LQR design penalizes the RMS surface error of the antenna. This

was chosen because the antenna performance is expected to be closely related

to the surface error.

ing form:

The Q matrix which penalizes RMS error has the followQl = I Qr0 00]

The elements of Qr are given in Table 4.2.2. Notice that the weightings on

the rigid body mode and the first three flexible modes are approximately

equal, while the weightings on the last two modes are higher by approximately

(4.2.55)

69
a factor of 5. Qr therefore approximately weights the position of the rigid

body mode and the first three flexible modes equally, places a little more

emphasis on the last two modes, and places no weighting on the modal velocities. The resulting closed loop pole configuration, as qc is increased from

.01 to I00, is illustrated in Fig. 4.5. An asterisk is placed by the closed

loop poles for qc = I00, and the loop gain for this case is illustrated in

Fig. 4.6. This loop gain indicates flexible modes both above and below 0db.

This is a desirable, yet realistic situation, since if all modes were below

0db they would essentially be uncontrolled, while if an modes were above 0db,

there would have to be some modes (of the true system) below 0db.

Table 4.2.2 Matrix Qr

2166.16 385.25 -313.39 -754.52 -5980.06 -4880.05

385.25 1618.56 -632.98 42.80 -5976.23 -3724.72

-313.39 -632.98 1598.49 -771.90 6803.32 3031.96

-754.52 42.80 -771.90 2539.28 -5295.22 1537.05

-5980.06 -5976.23 6803.32 -5295.22 98137.61 -2965.06

-4880.05 -3724.72 3031.96 1537.05 -2965.06 102461.54

LQR _ #2

The second LQR design penalizes the mechanical energy (kinetic +

potential), plus a term corresponding to the rigid body mode. For a

mechanical system the kinetic energy is given by xTMx, where M is the mass

matrix, and the potential energy is given by JKx, where K is the stiffness

matrix. The appropriate Q matrix to weight the mechanical energy is

therefore:

7O
POLE C@NFIOURF1TION

-,d

L QR DES ION

X OPEN L@@P

[] @C = 0.01

0 QC = 0.10

A OC = 1.00

0 OC - 10.00

+ OC - 100.00

NUNBER i

L

O U

• Am

, v g

T

0.00 -8.00 I -6.00 I -U,. I O0 -2.0I

REFIL PFIRT

O

0

0'_

o

0

0'1

-0

c_

(_

139

-_c-)

oZ

o33

CO

._-_

• 2D

o--- t

P_

"o

(2)

o

-o

F__ure 4.5 Pole Configuration for LQR Design #_!1

71
72
Oe[MO]

If the rigid body coordinate is the first state vari_le, K will have a zero

first row and first column, indicating that there is no potential energy corresponding to rigid body motion, me resulting control law would therefore

fail to control rigid boo motion, so some weighting on the rigid body

motion must be added to the K marrY. The new K matrix will be identical

with the addition of a scalar patterer qb in the (1,1) position (qb can be

used to vary the relative wei_ting on the rigid body motion). In tr_sforming the system to modal coordinates both K and M will be premultiplied

by TT and postmultiplied by T,

2nd order problem. Since the

Table 4.2.1) is equal to TTb,

(4.2.56)

where T is the matrix of eigenvectors for the

input influence coefficient matrix Bs'o (see

where b is a vector with a 1 in the first

position and zeros in all other positions, the resulting Q matrix in

modal coordinates will be:

0 (4.2.57) [A'+qb,o,o 0 0]I

For the purposes of this report the relative rigid body weighting factor qb

has been set to 50.

Table 4.2.3.

The resulting A2 + qbBoB_ matrix is listed in

Table 4.2.3 Potential Energy + Rigid Body Weighting Matrix

I .43 -.57 -1.33 "2.38 -.93 -.52 .57 49.07 1.75 3.13 1.23 .68

.33 1.75 363.11 7.28 2.85 1.59

:-2.38 3.13 7.28 1350.40 5.10 2.84

-.93 1.23 2.85 5.10 2694.60 1.11

-'.52 .68 1.59 2.84 1.11 3061.3_

73
Note that this matrix places considerably higher emphasis on the higher

modes. This is because the matrix A 2 places a cost proportional to the square

of the modal frequency. The other important difference between this weighting

matrix and Qr is that a penalty is now placed on modal velocities. The closed

loop pole configuration, as qc varies from I00 to 10,000, is illustrated in

Fig. 4.7. An asterisk is placed next to the poles corresponding to

qc = 1,000, and the loop gain for this case is plotted in Fig. 4.8. As

expected, the control law resulting from Q2 places less emphasis on the rigid

body modes, and more emphasis on the flexible modes, than does that resulting

from QI" This is indicated, both in the closed loop pole configuration, and

in the loop gain. While LQR Design #i pushed the rigid body poles further to

the left than either of the first two sets of poles corresponding to flexible

modes, LQR Design #2 pushes all the flexible mode poles further to the left

than the rigid body poles. In comparin E the two loop gains, that resulting

from Design #I shows a continual decrease in loop gain throughout the frequency range where flexible modes are being actively controlled, while the

loop gain resulting from Design 2 indicates a non- decreasing loop gain

through the same frequency range (though the gain is decreasing for both

higher and lower frequencies). It is important to note though, that both

designs actively control the flexible modes and have approximately the same

bandwidth, though Design #i does have a higher gain at low frequencies. Later

in this Section it is found that the robustness properties of LTR designs

based on these two different LQR designs vary considerably. Thz real-reasons for this remain a topic for further investigation.

74
t,

POLE CONFIGURATION

LOR DES ION

X OPEN LOOP

OC = 100.00

0 OC = I000.00

A OC - 10000.00

NUMBER 2

X

-10.00

o

I I I I

-8.00 -6.00 -_.00 -2.00

REAL PAR T

0

0

0")

-(_

i

O

o

0'1

-(_

c_

O

l-,-,,,4

-_

oX_

co--_

-._

_ZD

X .

_ 0

_ ., .o

O.gO

Figure 4.7 Pole Configuaration for LQR Design #2

75
O

C)

CD

CD

II

__) 0

Ik

O_

_.1

C_

C_ 0

I.L.I

I.LJ

I.I.J

I---

C2_

I---

C_I

_..I

.._1 LJ_

00"01

75
There are two basic differences between Designs #i and #2. The first is

that Design #2 places much more emphasis on the higher frequency modes, while

the second is that Design #2 places a penalty on modal velocities as well as

modal displacements, while Design #1 does not. To separate these two effects,

Design #3 uses the same penalties on modal position as #1 while adding identical penalties to modal velocity. The resulting Q matrix is as follows:

Q3 = I Qro O"qr (4.2.58)

The corresponding closed loop pole configuration as qc varies from .01 to I0

is illustrated in Fig. 4.9. An asterix is placed next to she poles corresponding to qc=l, and the loop gain for this case is plotted in Fig. 4.10.

Next results for asympototic LTR designs, corresponding to each of the LQR

designs, will be presented.

Asymptotic LTR Designs

There are a number of design variables in the asymptotic LTR procedure.

These are the elements of the appended columns to the B-matrix, and a scalar

parameter q, which is increased until loop recovery is achieved. Since only

the case of two measurements is studied at this point there will be only one

appended column to the B matrix. This is chosen as a scalar parameter r

S

times a vector corresponding to an input collocated with the second output 4.

The input influence coefficients corresponding to an input collocated with

77
POLE CONFIGURATION

LOB DE S IGN

X OPEN LOOP

[] @C ,, 0.0!

• OC - 0.10

A OC ,, 1.00

NUMBER 3

l

0

C3

C_

"0

0

o

Ul

-o

0

c_

33 -@o

oZ

033

210

o0- 0

-°33

0-- H

-o

o

(:3

-0

oo

, I , i _lr'-I_t_" - ' <::::) •

O. O0 -8. O0 -6. O0 -I,_. O0 --_o O0 O.

REFIL PRRT

Figure 4.9 Pole Configuration for LQR Deslgn #3

78
0

II

0

(Z3

Q..

C)

._J

C_)

cn

0

LU

IJ_

O0"I

00

0_

00

,°

_0

Z[_E

00

L_

I

-J

J,

i

00"_ 0o'dz 0oJ0

(GO) NI_O

oo"0;_-

"b

r=,

.40 |

.in

Z

" ]

i

I

!

•,N !

f

N

1

I

II

It

ii

" S

I I

I I

I I

I I

I

"5I

!

I

i

_ L

i i

II_ I I

I I

I

'v

D

o

n

v

00"0_- 00'

_GiI"

- s.=,

-c_

-op

-if*

-r

' 1

i

I I i

I -m

! )

I !

I !

I

.io

;\t .io

i

0 "l I0 "91

(9_0) _igUXd

79
The reason that a collocated input location is chosen is that this guarantees

a minimum_hase-sys_emfor-r-_--i .....The case r = 0 is equivalent to not

s s

adding any columns to the B-matrix, and does not strictly satisfy the conditions for asymptotic loop recovery outlined in Refs. [D01,2]. However, loop

recovery is achieved for this case so it will be considered as a design possibility.

Asymptotic LTR compensators corresponding to each of the three LQR designs

are presented. In each case the loop gain for both the final step in loop

recovery and the preceding step are plotted.

The point at which loop recovery is achieved is somewhat of a relative measure. As the parameter q is increased, three effects take place simultaneously. First the loop shape begins to converge as pole/zero cancellations

line up, second the overall loop gain increases and third the uncancelled compensator poles move out into the l.h.p, on 45" rays. Loop recovery is

achieved when the overall shape and gain of the compensated plant loop match

the full-state feedback loop and the decrease in loop gain due to the uncancelled poles shifts past the point where loop gain has dropped below 0db.

This will be clearer when the loop gain plots are presented in

Figs. 4.11-4.17. The parameter q (listed as QEST on the figures) is actually

multiplied by 1010 to produce the results presented.

to produce loop recovery for reasonable values of q.

This scaling was found

q is increased by steps

of a factor of ten at a time, until it is judged, by visual inspection, that

the 2nd measurement wil be identical to the output influence coefficients

for the 2nd measurement, listed in Table 4.2.1.

8O
loop recovery has been achieved.

The tables describe robustness results. The first column gives the LQR

design #, the second column the LQR parameter qc' the third column the LTR

parameter q and the fourth column the 2rid input scaling factor r . The

s

robustness results indicate stability or instability of the closed loop system

when all the frequencies are varied by the percentage indicated. An S indicates a stable closed loop system, while a U indicates an unstable closed loop

system. This was chosen as a rough measure of robustness since the compensators we tested seemed to be most sensitive to a shift of all the frequencies

in the same direction. This is also a realistic possibility since a variation

in mass or stiffness properties would produce just such a result. In a final

analysis stage a more complex study of robustness (for all possible sets of

frequency errors) would have to be carried out, but the data presented in

Tables 4.2.5-4.2.7 give an easily read and meaningful measure of robustness.

Asymptotic LTR for LQR design #1

In this case loop recovery was achieved for q=lO0. The loop gains for the

case of q=10 and q=100 are plotted in Figs. 4.11 and 4.12 respectively. A

number of different values of the scalar parameter rs were tested and loop

recovery was achieved in every case. Robustness results are presented for two

cases: r =1 and r =0. Neither case proved to be robust, but while the case

s s

with rs=l indicated stability only for a 10% increase in frequency, the case

with rs=O indicated stability only for a 10% decrease in frequency. This does

81
suggest that the choice of the appended B-matrix columns will affect

robustness. Note that increasing q to get loop recovery does not seem to

improve robustness. In fact, the results do not change. However, increasing

q does increase bandwidth and low frequency loop gain, thereby increasing performance while maintaing robustness.

Table 4.2.4 Robustness of Asymptotic LTR Design for LQR Design #1

Q#

1

1

I

qc

i00

100

100

I00

q

I0

I00

i0

I00

r

s

1

1

-40% -30% -20% -10% +10% +20% +30% +40%

U U U U S U U U

U U U U S U U U

0 U U U S U U U U

lo u u u s u u u u

Asymptotic LTR for LQR Design #2

For the second LQR design, loop recovery was achieved for q=10. The loop

gain for q=l and q=10 are plotted in Figs. 4.13 and 4.14 respectively, for

this case different values of r were not tested since the LTR desing was

s

extremely robust for r =i. In fact Table 4.2.5 indicates stability for ±40%

S

variations in frequency at both steps in the loop recovery. This is an excellent robustness result, but we do not yet have a rigorous explanation for it.

Table 4.2.5 Robustness of Asymptotic LTR Design for LQR Design #2

I j:1,4i Q# qc rs [

2 1000 1 S S S S S S S

2 ]I000 I0 1 S S S S S S S

82
C_

C)

II

i,i

C_

O

C:)

O

w,,,,,-I

II

O

C3

Ii,

0._

(C)

E_

._I

LLJ

(3:

O")

I,i

Q_

(C)

(-.):

00"_

OO

_O

O_

OO

_00

_EZ_Z

E 00

O0"OZ(60J NIU_00 "0_- GO"Oh0

,01,,

'b

i I -ID

111,

-m

!-:_.

l---

I:

, F

i

-r,l

,i _

r

t-o

[

I

b

"D

-m

-r

_$elWd

0=I

_J

83
ii:+ ':/

m_ ILl :

.-.- /:

0

' -o

a.

a

OO'm oo'oo oo'ost oo'dz 13o"o oo'o _Io oo*o'h- oo'o oo'i'

(80| Nlkl9

I

I

I

I

I

I

I

I

I

I

l

, +

° 0

'D _

I

I

I

I

I

' il I

'

o ._

I

I

I

I

,

I

I

I

I

i

I

I

t

t

I

I

I

I

I

I

, !

I

I

i

t

I

o

I

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I

I

I

I

I

)

t

I

I

P

' !1 t

f

i01" 10_101]H;l:ll,td

84
As indicated earlier, one difference between the two designs is that design

#2 pushs all the closed loop poles corresponding to flexible modes further

into the l.h.p, than poles corresponding to the rigid body mode, while design

#i does not. This indicates that the second design does not attempt to move

the antenna rigid body mode too quickly, but it does pay a great deal of

attention to the flexible modes. The first design, on the other hand, has a

very high gain at low frequencies (indicating a fast response to errors in the

rigid body mode), but does not pay much attention to the flexible modes. The

performance of the second design, in terms of fast reduction in the RMS error

or of the rigid body displacement error, will be less than that of the first.

Though there is clearly a trade-off between performance and robustness, this

does not entirely explain the difference in robustness, since much lower performance designs based on RMS error were also tried and no large increase in

robustness was observed. This is still very much in the area of hand-waving,

but a more rigorous understanding of the great increase in robustness observed

for the second LQR loop shape, will be a major part of our effort in 1986.

Asymptotic LTR for LQR Design #3

LQR design #3 was concocted to separate the effect of the two major differences between designs #I and #2. That is the increased cost on the high frequency modal displacements and the cost on modal velocities. The modal displacement cost in design #3 is identical to that of design #i, while a rate of

change of RMS term is added to weight modal velocities. In this case, loop

recovery was achieved for q=100. Loop gains for q=10 and q=100 are plotted in

Figs. 4.15 and 4.16 respectively. Robustness results are presented in Table

85
II 00

I,MUJ

_J

86
0

C_

II

O

0

C)

II

C._)

0

Ik

O_

E3

C)

_J

r_

ILl

(_

O3

Z

b_

CL

E_

(J

_'©

_o

_0

0_

_o

_00

_oo

_XZ_Z

Z 00

(

D oo'_ 0o'_ _ _o _'o_- oo'os

(80) NIUO

.4,,,..

i ,

" \;

i

r

I

I

I

" \i

w

I

I

\ I

I*

tl

II

N 11

|

I

it

o,

"D il

I I

0 !

I

n !

I

I

r

I

I

I

n I

!

I

I

¥ I

!

!

I

!

I

I

- 06"0 00"91

'b

i _ i °cb

J -qD

/:

'b

,m

b_

,r,.

_LIp

-=r

I

I r

( i

I I

I I -f'v

, , _

i i

i i

i i

,I ,\: i

I I

-s,

4_

I

!

t

I

I

, _

I

[g]OJ_gi_d

87
4.2.6. This design is considerably more robust than #i, but not quite as

robust as #2,--Ag_,_.we do no_t hav_ a rigorous explanation of this fact, and

this will remain an area of active research. Also note that increasing q does

not improve robustness, but it does improve performance.

Table 4.2.6 Robustness of Asymptotic LTR Design for LQR Design #3

Q#

3

3

3

qc q

1 1

1 i0

1 I00

r

s

i

1

i

i

i

1

-40% -30% -20% -10% +10% +20% +30% +40%

S S S S S S S U

S S S S S S S U

S S S S S S S U

Algebraic LTRDesigns

The design variables in the Algebraic LTR approach are the locations of the

compensator poles. The compensator order is fixed by the maximum degree of

the plant numerator polynomials (n n) and the number of measurements (m). For

the 6-mode antenna model with two measurements this will be order ten, or two

orders less than an estimator based design. Of the ten poles available, eight

will be cancelled and two will not. The positions of the uncancelled poles is

set at -200±200j. This is far enough away from the controlled flexible frequencies that the loop shape in the design region is not significantly

affected. To examine the effect of varying pole locations, four sets of pole

locations were chosen. These are listed in Table 4.2.7.

88
E3

II

&

O_

E3

_J

1:3

bJ

hCE

O3

Z

LO

Q_

_--

E_

00"0

IN'

I / ''/

ii

I I I [

I I i

. i I -Ii,

I I i

I i

i , f

I I

I i

J J \

z-,

I I I .to

!

m'l i I

. i

I I

I I

I I

! I

t I

! !

t I

I I

I I

! I

all t I

I I

W I I

! I

! I

! I

IF i t

B I

! I

! I

I !

I i

I I

I I

I I

I I

I !

_. J ,

oo'1_ oo "l_l oo 'o oo'o_- oo'oh- oo"o oo "111- oo "1

lEO} NI I:lO

"b

_o

,ll,

b

-lid

.phi

U')

.r- -i_

!

-lid

*r.-

-ID

-In

,m

.hi

,Oi" iO]O) ._,5_Ha

B9
C)

C)

C)

q,,-I

II

i,i

(:3

C)

II

0

(:3

It,

_D

_D

_n

Z

Q_

Q

(J

O0

O0

_oQOlD

i.= oo

,Ill I

I

'41 I I I

I I I

'UI I I I

I I I

I I I

ql* I I I •

I I I

! I I

_11 I I I

I I I

i I I I

I I I

_ll I I I

I I I

I I I

I I I

I I I

I i I

:J , ,

411 I I I

'll I I I

I I I

4"" I I I

ql I I

I I I

qlP Ir

2

:ll ! ! I:=-

! -

b _

!

, _

oo'_ _" ol _.o_- ®-o'., -;ot.'_-I'_ol [ti_'_i

C_.-%

90
Table 4.2.7 Compensator Pole Locations

#

1

2

3

4

Pole #1

-l±lj

-lO±lOj

-.15±lOj

-1.9±9.8j

Pole #2

-2±2j

-20±20j

-1.s±2sj

-5.5±8.3j

Pole #3

-3±3j

-30±30j

-2.5±50j

-8.3.+.5.5j

Pole #4

-4±4j

-40±40j

-3.0¢55j

-9.8±1.9j

The first two sets of poles lie on 45" rays from the origin. The next set

are located close to the plant zeros in both channels. This is done in the

spirit of the asymptotic approach which cancels plant transmission zeros. The

fourth set of poles are placed in a Butterworth filter pattern with a cut-off

frequency of 10 rad./sec..

Loop recovery was satisfactorily achieved in every case, but robustness

results were not particularily positive. Table 4.2.8 lists those results for

all three LQR designs and all four sets of compensator poles. Only LQR design

#2 in conjuction with pole location #2 showed any respectable robustness

results. Again, a rigorous interpretation of these results has not yet been

made, but the number of possible cases for pole locations has not been

exhausted. Once an ideal LQR Loop shape is found an optimizing routine might

be applied to find the "best" pole locations.

91
Table 4.2.8 Algebraic LTR Designs

Q#

1

1

1

1

noo

2

2

2

2

3

3

3

3

qc

I00

I00

I00

i00

1000

1000

1000

PL

1

2

3

4

1

2

3

-40% -30% -20% -I0% +10% +20% +30% +40%

U U U U U U U U

U U U U S U U U

U U U U U U U U

U U U U U U U U

U U U U U U U U

S S S S S S U U

U U S S U U U U

i000 4

1 1

1 2

1 3

1 4

U U U U U U U U

U U U U U U U U

U U U U U U U U

U U U U U U U U

U U U U U U U U

92
4.3 SENSITIVITY OPTIMIZATION

Our second approach to robust compensazor design is to use parameter optimization (nonlinear programming) to minimize various measures of the sensitivity of the closed-loop system to parameter errors in the model. Initially, we

augmented the quadratic performance index in the optimal linear regulator

problem with terms penalyzing the gradient of the closed-loop response (sensitivity) with respect to uncertain parameters. While the resulting optimal

control problem could not be solved via the solution of a Riccati equation,

the function and gradient evaluations for parameter optimization involved

solution of Liapunov matrix equations, which can be solved rather efficiently

by well known methods. Unfortunately, this approach proved ineffective

because we were unable to find a quadratic performance index involving sensitivity vectors such that the compensator that minimized the performance index

was robust.

A much more successful approach has been to minimize an objective functional involving the gradients of the real parts of the closed-loop eigenvalues with respect to uncertain parameters; i.e., minimize the sensitivity of

the closed-loop eigenvalues. The closed-loop system has the form

where

93
Our typical sensitivity objective functional has the form

Js = Z[dRe(_i)Idaj]Z

where the ki's are the eigenvalues of Acl and the aj's are the uncertain parameters in A, B and C. The uncertain parameters usually are natural frequencies and damping ratios. The design variables for this optimization problem

are the control and estimator gains K and G.

Usually, we obtain the initial gains as the solution to an optimal LQG

problem, and then use sensitivity optimization to improve the robustness of

the closed-loop system. We use standard formulas [KA2,Ch. II.2] for computing

d_i/da j in terms of the eigenvectors and eigenvalues of Acl and dAcl/daj, and

we have found that the computation is much more efficient if done entirely in

real arithmetic.

For our beam-hub model, we have been able to take an initial LQG compensator that allows less than 10% variation in the natural frequencies before the

closed-loop system becomes unstable and obtain a sensitivity-optimized compensator that allows more than 20% variation in the frequencies while maintaining

closed-loop stability.

The following two tables list the dominant closed-loop eigenvalues of the

antenna and two compensators based on a six- mode antenna model. The first

compensator is optimal for an LQG problem in which state weighting Q penalizes

94
the r.m.s, surface error of the antenna and the disturbance is white noise

distributed uniformly over the antenna, along with white noise entering

through the actuator. This compensator allows less than 10% error in the

open-loop plant frequencies before the closed-loop system becomes unstable.

With the LQG compensator as the initial guess, our sensitivity optimization

program found the second compensator, which allows more than 20% error in the

open-loop frequencies while maintaining closed-loop stability.

95
Table 4.3.1. Dominant Closed-loop Eig envalues

with LQG Compensator

For correct fre_encies ......... For +-lO%err6f in

all frequencies

-1.2 ± .2j -2.0

-i.I

-3.3±4.8j -4.8± .Sj

-3.0 ± 6.7j - .4 ± 6.0j

-1.1 ± 7.0j -2.7 ± 7.0j

- .6 ± 19.j ÷ "----_& ± 17.j

-2.8 + 20.j - 3.5 ± 20.j

-3.8 + 30.j - .5 ± 33.j

For +20% error

-10.7

1.0

-1.6±1.3j

-.06f5.6j

-2.50 ± 7.0j

+.80±16.j

3.8±19.j

+0.2±36.j

Tabl_ 4.3.2.

Dominant Closed-loop Eigenvalues with

Sensitivity Optimized Compensator

For correct, frequencies

-1.3

- .6

-3.4 ± 4.83

- .6 ± 6.93

-3.0 -+ 6.7j

-2.8 ± 19.3

- .7 ± 20.3

-2.2 ± 36.3

Fo____r+10% error in Fo___rr+20% erro______r

all frequencies

---1.6 -2.2

- .6 .6

-4.1 ± 4.0j -4.5 +- 2.5j

- .6 +- 6.0j .6 ± 5.2j

-2.5 ± 6.7j -2.1 ± 6'.8j

- .3 ± 17.j -.05 ± 16.j

-3.2 +- 20.j -3.6 ± 20.j

-1.3 ± 33.j - .7 +- 29.j

96
We have found that the form of the operator Q in the optimal control problem affects robustness significantly, me Q for the preceding LQG compensator

penalized the rigid-body displacement primarily and did not penalize energy at

all. Table 4.3.3 gives the dominant closed-loop eigenvalues produced by the

LQG compensator for a Q which penalizes both the rigid-body angle and the

total energy in the antenna. We chose the Q-to-R ratio so that the real part

(-.65) of the least stable closed-loop eigenvalue with this compensator is

comparable to the real part (-.6) of the least stable eigenvalue with the previous LQG compensator (which resulted from the Q that penalized the antenna

surface error). In this case, sensitivity optimization did not improve the

robustness of the LQG compensator significantly.

Table 4.3.3.

For correct

-0.65 + O.13j

-0.70 ± 0.33j

-0.78 + 6.94j

-0.84 ± 6.96j

-1.13 + 18.9j

-2.05 + 18.9j

Dominant Closed-loop Eigenvalues

with Second LQG Compensator

For+__error in

-0.56 ± 0.173

-0.76 ± 0.423

-1.36 ± 6.833

-0.27 ± 7.813

-2.49 ± 18.83

-0.71 ± 21.13

For _ error

-0.53 ± 0.18j

-0.76 ± 0.48j

-1.50 ± 6.92j

-0.13 ± 8.46j

-2.61 ± 18.9j

-0.59 + 22.8j

97
4.4 OPTIMIZED LOOP RECOVERY DESIGNS

Another approach to improving robustness combines the Loop Transfer Recovery methods described in Section 4.2 with the sensitivity optimization methods

described in Section 4.3. This is the optimal Loop Transfer Recovery

approach, and is currently under investigation.

The idea of the approach is to take advantage of the robustness improvement

inherent in the sensitivity optimization methods, while maintaining a fixed

loop shape. This guarantees that robustness is not achieved at the cost of a

reduced loop gain, or equivalently reduced performance. As noted in Section

4.2.2, the loop recovery compensator for a given full state feedback loop

shape is not unique when a plant has more outputs than inputs. In the asymptotic LTR approach, this non-uniquenss corresponds to the choice of artificial

columns in the B-matrix. This allows the designer some freedom in the placement of finite plant transmission zeros, which in turn determine the location

of the compensator poles. In the algebraic LTR approach the degrees of freedom depend on a number of factors including the desired compensator order,*

but for the specific algorithm described in Section 4.2.3, the compensator

order is fixed and the degrees of freedom translate directly into the location

of the compensator poles.

Consequently two optimal LTR approaches are suggested. The first finds an

optimal set of elements for the artificial columns of the B-matrix in an

asymptotic LTR design. The second finds an optimal set of compensator pole

wm_m_omm_Dom_mm_

I As compensator order is increased, the number of degrees of freedom in the

algebraic loop recovery design also increases.

98
locations in an algebraic LTR design. In both cases the sensitivity of the

real parts of each closed loop pole to variations in plant frequency are evaluated. The objective function is equal to the sum of the squares of these

sensitivities. This objective function is used since it gave excellent

results in the more general sensitivity optimization approach described in

Section 4.3.

Since the data presented in Section 4.2.5 suggests that the asymptotic LTR

designs provide greater robustness than the algebraic designs, the optimal

approach is being applied first to the asymptotic design method. Currently

the software is written and preliminary results are expected early in 1986.

Two other applications of optimization to the LTR approach are also being

considered. The first is based upon the conjecture that the lack of robustness exhibited by LTR designs relative to the equivalent LQR designs, is a

function of pole/zero cancellations close to the imaginary axis. As noted in

Section 4.2.4, small variations in plant parameters can result in Very large

variations in loop shape whenever pole/zero cancellations occur close to the

imaginary axis. An optimization routine could therefore be used to choose

artificial columns of the B-matrix so as to shift finite plant transmission

zeros away from the imaginary axis. A second possibility depends on the fact

that the LTR estimator design is largely independent of the full-state feedback LQR control gain matrix K. An optimization routine could therefore leave

the LTR estimator design fixed while varying the elements of K so as to minimize closed loop pole sensitivity. Unlike the other optimal approaches

described in this Section, this approach would not maintain a given loop

shape, and performance might be reduced_ but it may produce some further

99
insight into the relationship betweeen full-state feedback loop shape and LTR

robustness.

These optimal LTR approaches to achieving robust control of the antenna

will be studied more extensively in 1986.

100
5.0 CONTROL DRIVEN FINITE ELEMENTS

5.1 A COMPARISON OF THREE FINITE ELEMENT SCHEMES FOR

APPROXIMATING THE SOLUTION TO THE OPTIMAL LINEAR

REGULATOR PROBLEM FOR A FLEXIBLE STRUCTURE

The Control S_stem

One end of the Euler-Bernoulli beam in Figure 5.1.1 is attached rigidly

(cantilevered) to a rigid disc which is free to rotate about its center, point

O, which is fixed. Also, a point mass m I is attached to the other end of the

beam. The control is a torque u applied to the disc, and all motion is in the

plane.

FiEure 5.1.1

!

i lJ_ I00 000

l _...... J

L------j _u. uuv

I01
Table 5.1.1. Structural Data

r = hub radius

1 = beam length

I0 = hub moment of inertia about axis

perpencilar to page through 0

m b = beam mass per unit lingth

m I = tip mass

EI = product of elastic modulus and

second moment of cross Section for beam

fundamental frequency of undamped structure

i0 in

i00 in

I00 slug in 2

.01 slug/in

1 s lug

13,333 slg in3/sec 2

.9672 rad/sec

The angle 8 represents the rotation of the disc (the rigid- body mode),

w(t,s) is the elastic deflection of the beam from the rigid-body position, and

Wl(t) is the displacement of m I from the rigid-body position. The control

problem is to stabilize rigid-body motions and linear (small) transverse elastic vibrations about the state e = 0 and w = 0. Our linear model assumes not

only that the elastic deflection of the beam is linear but also that the axial

inertial force produced by the rigid-body angular velocity has negligible

effect on the bending stiffness of the beam. The rigid-body angle need not be

small.

This system can be written in the form (3.1.1) with the generalized displacement vector

x = (8,w,w I) H = R X L2(0,1) X. R. (5.1.1)

102
We assume that the beam has Voigt-Kelvin viscoelastic damping, so that the

damping operator in (3.1.1) is

D O = c o A 0 (5.1.2)

where co = I0 "4. For the details of the energy-space formulation of this

problem and the operators in (3.1.1), see [GII].

The Optimal Control Problem

In the optimal control problem, we take Q = I in the performance index in

(3.2.1). This means that the state weighting term <Qz,z> E is twice the total

energy in the structure plus the square of the rigid-body rotation. Since

there is one input, the control weighting R is a scalar, and we take R = .05.

The functional control gains in (3.3.2) have the form

f = (af,¢f,_f), (5.1.3)

g-- (5.1.4)

where the o's and B's are scalar gains for the hub and tip mass and Cf and Cg

are functions which define the feedback law for the beam. Because the strain

energy in the beam involves the second spacial derivative of w, ¢f,, appears

in the the control law in (3.3.2), rather than el.

Approximation

To approximate the solution to the optimal control problem, we solve a

sequence of finite dimensional optimal control problems as described in Chap103
ter 3.4. To illustrate the convergence of the approximating feedback control

laws, we will plot _he_distributed func_t_!Qnal gain components _/' and _g.

We compare three approximation schemes; i.e., three different sets of basis

vectors e. for the approximation to the generalized displacement in (3.4.1).

1

The first approximation of the structure is based on a finite element approximation of the beam which uses cubic Hermite splines [ST1] as basis functions,

the second approximation scheme uses cubic B-splines to approximate the beam,

and the third approximation scheme uses the natural mode shapes of the structure as the basis vectors in the approximating optimal control problems.

Recall that cubic B- splines require the displacement, slope and second derivative to be continuous at the nodes, while cubic Hermite splines require only

the displacement and slope to be continuous at the nodes. To get the natural

modes of the structure, we used 24 elements in the Hermite spline approximation to get the first 12 modes of undamped, free vibration.

Figure 5.1.2 shows the distributed functional gain components computed

using the Hermite splines, Figure 5.1.3 shows the distributed functional gain

components computed using the B-splines, and Figure 5.1.4 shows the distributed functional gain components computed using the natural mode shapes. For

both spline schemes, we get convergence with ten elements and very near convergence with eight elements. Also, we get convergence with ten modes and

very near convergence with eight modes.

Now note that if n e is the number of elements, then the number of degrees

of freedom in the Hermite spline scheme is 2n + I, while the number of

e

104
degrees of freedom in the B-spline scheme is n + I. This means that the

e

B-splines and natural mode shapes are almost equally efficient for computing

the control gains and that the Hermite splines require almost twice as many

degrees of freedom. This is important because the dimension of the Riccati

matrix equation that must be solved in each approximate optimal control problem is twice the number of degrees of freedom in the approximation scheme.

We experimented with varying the location of the nodes for the Hermite

splines and B-splines. While we found that we could improve the rate of convergence slightly, we felt that the improvement was not sufficient to justify

the elaborate procedure. The improvement in convergence rate obtained by varying the node locations was insignificant compared to the improvement obtained

by using modes or B-splines instead of Hermite splines.

105
CONTROL

Figure 5.1.2a

FUNCTI ONRL GAINS

C@

CD

(%1

om

C_

0

Or)

O0

I

I

I

I

'I

C,,,0.00010

Re =0. 05000

=i =R- tL7213 (D

•,,,,R.tl7213 A

=,=_.t17213

ib.oo ab.oo 3b.oo _b.oo sb.oo sb.oo vb.oo eb.oo 9b.oo 1

Z06 X
Figure 5.1.2b

CONTROL FUNCTIONRL GRINS

o.

@@

lw'--@

I

6

GO

I.__

I

C)

r,:

I

¢0

¢0

I

W

I

If)

@r)

I

C)

W

It)

w--4

ZO

\

,oo zb.oo

6

C,=0.00010

R¢=0.05000

Bz=I-2_O0

Gz=-133.87

Bz=[.29728

G,=-139.69

B,=[.31063

Gm=-lql.15

B,=l.31ql3 0

G,=-Iql.5_

2b.oo ab.oo _b.oo sb.oo sb.oo 7b.oo eb.oo 9b.oo

io7 X
Figure 5.1.3a

CONTROL. FUNC T IONRL GRINS

CD

i

!

0

I

I

I

I

I

/

C.,,0. 00010

R, =0. 05000

" ,,.q,. P,'7213 +

_, =g- 1_'/213 O

,=,,=g. _'72i S A

", =tt. IA7E 1S D]

"re=q,. q'7213 _,

C)

0")

"_"0! O0

lb. O0 2b. O0 3b. O0 qb. O0 5_.00 6b. O0 "lb. O0 @b.O0 9b.O0

V
Figure 5.1.3b

CONTROL FUNCTIONRL GAINS

%

%

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

%,

\

C,=0.00010

R¢=0.05000

Ba'1.13657

Gz--121.B7

B,=1.2817_

G,=-137.98

B,=I.30785

GI--1_0.85

B1=I.313;6

Gm=-1_1.;6

B,=1.31515

G,=-I_1.65

',,,---i

i 0.00 ,_.oo _ _. oo _. oo s5.5_ _ _.oo _. oo _o-.oo

zo9 X

O0
Figure 5.1.4a

CD

O_

(D

m

c_

ou

(D

CD

CD

CONTROL FUNCTIONRL GAINS

C,=O.O0010

0m= I•00000

Rc=0. 05000

=a=tl-q7213 +

,===u,. u,7213

,===U=.u,7213 rn

====u,.u,7213 ._

/

/

/

!

I

iO.OO zb.oo 2b.oo sb.oo _b.oo sb.oo eb.oo ?b.oo eb.oo 9b.oo

].].o X
Figure 5oi.4b

coNTROL FUNCTiONRL GAINS

!

I

0

; i_"

I

CO

(X)

C_.

|

=

!

!

"3

Z 0.00

%

%

\

lb.oo 2b.oo 3b.oo

C,=O. O00IO

Re=O.05000

B1=1"11907 4

G==-119.78

B_=1.28170 0

G==-137.90

B,=1.30926 A

G,=-1_0.99

8,=1-31_ 3_

G,=-1_1.56

B==l.31566

G==-1_1.71

.b.oo sb.oo sb.oo 7b.oo eb,oo 9b.oo

"iil X

] )0. O0
5.2 DIRECT APPROXIMATION OF H_IILTONIAN SYSTEM

We have investigated approximation of the solution to the optimal control

problem for the distributed model of the space stucture by constructing direct

finite element approximations of the infinite dimensional Hamiltonian system.

To date, our numerical solution of the LQG optimal control problem for the

distributed model of the space antenna has been based on an approximation

scheme that is indirect in that we first approximate the open-loop system with

a sequence of finite dimensional systems and then solve a finite dimensional

optimal control problem for each approximating system. The solutions to these

finite dimensional problems converge to the solution to the infinite dimensional problem. However, for certain space structures, direct approximation

of the solution to the infinite dimensional problem may allow better choices

for the basis elements used in the approximation.

The Hamiltonian operator involved in the solution to the optimal regulator

problem is

(5.2.1)

where A is the open-loop semigroup generator, B the input operator, and Q the

state-weighting operator in the optimal control problem. Recently, we began

studying the extension of an idea used at JPL for solving static state estimation problems. As applied to the control problem, the idea is to choose a

finite element scheme to approximate the infinite dimensional Hamiltonian

operator directly with operators of the form

,_,,ECEDING PAGE ai_-'N_ _

113
An "C n

.l.

Qn "An

(s.2.2)

1

Here, A

n

is not necessarily the adjoint of A .

n

In particular, two different

sets of basis vectors may be used in approximating A and A

For the models of flexible structures and the finite element schemes that

we have used to date, direct approximation yields the same finite dimensional

Hamiltonian operator as does indirect approximation, but for more complicated

structure models, this should not be the case. Direct approximation of the

infinite dimensional Hamiltonian operator may be most useful for structures in

which some of the materials are viscoelastic with history-dependent damping.

For discussion of history-dependent viscoelastic effects see [COI,FUI]. Such

structures lack the basic selfadjointness of more common structures, and the

operators A and A have very different domains. Hence different finite element basis vectors are needed for approximating A and A . These basis vectors

should be chosen to produce appropriate convergence of the direct approximations to the Hamiltonian system and the corresponding direct approximations to

the solution of the infinite dimensional optimal control problem.

114
6.0 SUMMARY AND CONCLUSIONS

The work described in this report further develops aspects of modern control theory which make it better able to deal with issues of modeling and

implementation. The report represents an extension of previous work done for

JPL in which elements of distributed parameter control theory and finite

dimensional control theory have been synthesized into a design approach which

simultaneously produces both a model which is appropriate for the control

problem at hand, and an "optimal" controller based on that model. The interaction of modeling and controller design is central to the approach. The

techniques developed have been applied to a space antenna to demonstrate their

performance on space systems of realistic complexity.

A major focus of the work carried out during 1985 was robustness. Robustness refers to the ability of a control system to perform satisfactoril@ even

when the model used for the design is an imperfect representation of the

actual system. There are two types of modeling errors which are of major concern in designing control systems for large space structures (LSS). Much of

the work on functional gains is aimed at one particular type of modeling problem: model truncation. Parameter errors are another important source of

modeling error. Work during the current year emphasized the development of

LSS controller designs which are relatively insensitive to parameter errors

but still maintain a satisfactory level of system performance. For antenna

structures of the type considered, this is a difficult problem which does not

easily yield to standard techniques.

As a general rule, an improvement in robustness can only be achieved at the

cost of a reduction in some measure of performance. In order to achieve a

115
proper balance between performance and robustness, some constraints must be

imposed. In the loop recovery approach, the open loop and closed loop frequency responses are constrained to approximate those of the full state feedback LQR design. Thus, the performance of the system is first determined in

the context of the LQR design, and then one tries to obtain an estimated state

feedback implementation which has similar performance and also adequate

robustness. The sensitivity optimization approach also begins with the solution of an LQR problem and designs an estimated state feedback implementation.

In this case, however, there is no constraint to retain the LQR frequency response. Instead, the goal is to minimize the sensitivity of the closed loop

eigenvalues to parameter errors. A constraint must be imposed to keep the

closed loop regulator and estimator eigenvalues sufficiently far in the left

half plane otherwise the optimization routine would generate a very low performance control system. Designs based on sensitivity optimization tend to be

more robust than LTR designs, but their loop gains are also lower.

Both of these approaches rely heavily on the statement of the LQR problem

which serves as a starting point for the designs. The state weighting operator in the performance index for the LQR problem discussed in Chapter 3 is Q.

Generally, scaling up Q while holding the control weighting matrix R fixed

leads to larger control gains and a less robust compensator. The numerical

examples presented in Sections 4.2 and 4.3 indicate that in addition to the

magnitude of Q, the relative penalization that Q specifies among the different

components of the state vector can affect robustness dramatically. For example, Q's which penalize only the RMS surface error of an antenna model were

found to lead to designs which were not very robust. However, Q's which

penalize the energy or the RMS surface error plus the rate of change of RMS

116
error, were found to be quite robust. While the interpretation of these

results is not entirely clear at this point, some observations can be made•

Penalizing RMS surface error alone leads to controllers which try to reduce

rigid body displacement much faster than elastic deformations. This means

that the closed loop eigenvalues for the rigid body mode are substantially to

the left of those for the lowest flexible modes. In this case, large control

torques are produced to control the rigid body mode, and the system is not

robust because these torques strongly excite the flexible modes which may be

inaccurately known. The other two Q's mentioned above penalize the rigid body

and flexible modes more equally. The real parrs of of the closed loop eigenvalues, and hence the decay rates, for the rigid body mode and lowest elastic

modes are approximately equal and the compensator is more robust. In our continuing research effort, we plan to pursue further the relationship between

the selection of Q and robustness.

Another factor which has a strong effect on robustness in the number an

position of the measurements. In the case of the beam-hub model, control

designs based on two measurements were much more robust than those based on

one measurement. The implications of the number of measurements for the

_v_ w_ =_ .... more extensive •

Sensitivity optimization appears to be a successful approach for designing

robust compensators for complex structures, but we need more analysis and numerical experimentation to determine the best objective functionals and the

most effective methods of optimization. We have experimented with objective

functionals involving both first and second-order gradients of eigenvalues

with respect to uncertain parameters, and found that minimizing the second

117
gradients also does not improve robustness siEnificantly. Also, we are

comparin E performance of the optimization routines using analytic as opposed

to numerical Eradients with respect to the design variables.

The results in Section 5.1 demonstrate that a well chosen finite element

scheme can Ereatly reduce the order of approximation required to design an

optimal compensator for a flexible structure. It should not be surprisinE

that the natural mode shapes are at least amonE the best basis vectors for

approximating the optimal functional gains. It is more interestin E that the

cubic B-splines are almost as efficient in the example that we worked; they

are much better than the cubic Hermite splines, which are most common in enEineering finite element approximations of beams. This suggests that further

research on control driven finite elements could produce significantly more

efficient numerical methods for the design of controllers for large flexible

structures.

Since we know now that some approximation schemes are more efficient than

others for numerical solution of distributed optimal control problems, it

seems natural to try to develop a scheme especially for the approximate solution of the optimality conditions for the infinite dimensional LQG problem for

the distributed model of a flexible structure. These optimality conditions

are embodied in the infinite dimensional Hamiltonian operator discussed in

Section 5.2, where we propose the direct approximation of this operator

instead of the indirect approximation that usually is used. For the kinds of

material models and finite element schemes that we have used so far, the

direct and indirect approximations are identical, but for materials with

memory (particularly history-dependent damping) the direct approximation of

118
the Hamiltonian operator would be different and should be investigated.

119
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122
Appendix A: Singular Values

Singular values measure the "size" of a complex matrix, and are equivalent

to the concept of the modulus of a complex scalar. The singular values of a

matrix transfer function can therefore be thought of as the MIMO equivalents

of the gain of a scalar transfer function.

Any n x m complex matrix A can be decomposed in the following fashion:

k

A = U_V* = E O.u.v. *

i= 1 i I i

where:

k = min(n,m)

o i = = A],

o i k oi+ 1

U = [u I u 2 ... Un] ,

V = [v I v2 ... Vn] ,

UU =UU=I

VV =VV=l

The o.'s are the singular values of A, the vectors u. are the left singular

1 1

vectors of A and the vectors v. are the right singular vectors of A. The

l

largest singular value o I is denoted by _ and the smallest singular value ok

by _. _[A] is the 12 induced operator norm of A in the following sense:

123
_{A] = max I1_112= max IIAxII2

_0 ilxl[2 Ilxll2=l

while _[A] is a measure of the closeness of A to singularity in the followin E

sense:

_[A] -" rain IIAxII2= rain IIAxII2

x*0 ilxl12 Ilxl12-1

_[A] is the only computationally reliable tool for the determination of near

singularity, or rank of a matrix [KL1,LA1].

The following table includes a number of useful properties of singular

values:

I) E[A] > 0 _[A] = 1/_Y[A"1]

2) oi[aA ] =l=[oi[A]

3) R[A] S lli[A][ S _[A]

4) A=A _oi[A ] = Ill[A][

5) A=A _0_oi[A ] = li[A]

6) _[A + B] s _[A] +_[B]

7) _[AB] S_[A]_[B]

8) _[AB] _[A]_[B]

9) _[A] - 1S _[I + A] S o.IA ] +I

Io) _[E] < _[A] o[A + E] > 0

11) _[A] < 1 C[I + A] _ 1 -_[A]

124
Another slight variation on the singular value decompostion is the polar

decompostion_ used extensively by Postlethwaite [PO1,2,KOI]. Write the singular value decompostion as:

A = UZV - UV (VZV ) = (UZV )UV

where UYV is the usual singular value decomposition, UV (VYV) is the right

polar decomposition and (UZU)UV is the left polar decomposition. Postlethwaite defines the principal gains as the eigenvalues of VZV or UZU ' which

are of course just the singular values of A, and the principal phases as the

arguments of the eigenvalues of the unitary matrix UV . This decomposition is

useful in separating a transfer function into a gain part and a phase part and

will be used the proof of Theorems 2.1 and 2.2 in Appendix B.

125
Appendix B: Proof of Unstructured Uncertainty Theorems

Theorem 4.1 will be proved in detail. The changes that need to be made for

Theorem 4.2 will not be extensive and will be described in less detail.

Theorem 4.1:

The Multivariable Nyquist Criterion requires that det[I + KG'], evaluated

on the standard Nyquist D-contour, encircle the origin counterclockwise as

many times as KG' has unstable open-loop poles. If it is assumed that KG' has

the same number of unstable open-loop poles as KG, the encirclement count of

the origin must not change as G is warped continuously towards any allowable

G' For the Nyquist Contour to remain fixed as G is warped towards G', it is

also necessary to assume that any poles on the jw-axis be identical for G and

G' Any plant that has an uncertain frequency on the jw-axis will have an

infinite unstructured error in the neighborhood of that frequency, so it is

reasonable to assume that the only poles on the jw-axis are at the origin, or

equivalently, that the model have some positive damping. Furthermore, practical plants will be strictly proper, implying that lim K(s)G(s)=0. The Nyquist

S_

D-contour can therefore be replaced by the jw-axis, with the possibility of an

indentation about the origin, which will not have any effect on the results

for uncertain modal data.

Since KG(I+CAm) is a continuous function of E, requiring that the encirclement count of det[I + KG] not change as G is warped towards G' is equivalent

to requiring that:

det[I + KG(I+¢Am) ] # 0 for 0 _ t _ 1

126
or in terms of the minimumsingular value:

_[I + K(s)G(s)(I+ZAm(S)) ] > 0, for 0_z_l, _[A(s)] < lm(S), sz_ R (B.I)

Dropping the dependence on s, and noting that if _[A(s)]_l m, then_[zA(s)]_l m,

allows us to replace (B.I) by:

_[I + KG +KGAm] > 0 for _[A] < Im (B.2)

Assume that n[(I + KG)'I]=I/C_[I + KG]) > 0. This is true whenever

W[KG] < -, which in turn is true by the definition of the Nyquist D-contour.

Multiply (B.2) by _[(I + KG)'I]:

0 < _[(I+KG)'I]_[I + KG +KGAm] _ _[I + (I + KG)'IKGAm ] (B.3)

Now let (I + KG)-IKG have the singular value decompostion given by UZV or

equivalently the polar decompostion given by UV (VZV). Let A = -zl VU for

m

0_¢_I. =[A] =_[A] = E1 S 1 , so this is an uncertainty that falls within

m m

the specified bounds. Substitute this in:

0 < K[I + (I + KG) -IKGAm ] =A[I - E1VZV * ]

m

(B.4)

but (I-elmVZV) is hermitian, positive semi-definite, so the singular values

can be treated as eigenvalues to get:

0 < 1 - ¢imOi(VZV ) = 1 ¢imUi[(l + KG)'IKG] lB.5)

127
1 - Zlm_[(I + KG)'IKG] > 0

_[(I + KG)'IKG] > llCtl m) Z lI(Im)

(B.6)

(B.7)

Before continuing to prove sufficiency it is worth noting that the particular uncertainty that will destabilize the system must have a phase charactere

istic given by -VU . An uncertaintly with some other phase characteristic

might require that O[Am(jW)]>>Im(W ) before the system would go unstable. This

indicates that if there is a some specified relationship between gain uncertainty and phase uncertainty, the unstructured uncertainty representation of

Theorem 1 can be arbitrarily conservative. This point is discussed by Lehtomaki [LEI,3,4] in the context of most sensitive directions of the perturbation.

Now prove sufficiency by working backwards:

_[(I + KG)'IKG] < i/(Im) (B.8)

_i > im'6[(I + KG)'IKG] _Z[A]_[(I + KG)'IKG] (B.9)

_[I + (I + KG)'IKGAm] _ 1 -"6[(I + KG)'IKGAm] > 0 (B.IO)

Now assume that _[I + KG] > 0, which is true whenever the nominal feedback

system is stable. Multiply by _[I + KG]:

0 <_[I + (I + KG)'IKGAm].9.[I + KG] _[I + KG + KGAm] (B.11)

128
_[I + KG + KGAm] > 0

and Theorem 4.1 is proved I.

(B.12)

Theorem 4.2:

The major difference in the proof of Theorem 4.2 is in ascertaining the

continuity of L(_,s) as z varies from 0 to i. In particular KG(I+¢Ad)'I must

a continuous function of ¢ for O_E_I, or equivalently L(s)=(I+Ad )'1 must be

have no zero or strictly negative real eigenvalues. This is true since if

L(s) has no zero or negative eigenvalues, then neither does (I+Ad) , and thus

Ad can have no eigenvalues in the interval (--,-i], so sAd never has eigenvalues of -i and (I+¢Ad) is never singular. If the first requirement is met it

is then necessary that:

0 <_o[I + KG(I+Ad)'I ] VszRR, Vz¢[0,1] (B.13)

which in turn is equivalent to:

O<_[I + KG + Ad] Vsz_ R

The proof then continues as with Theorem 4.1.

(B. 14.)

* The only difference between our result and the result stated in Ref. [DO1]

is that we've replaced_[A(jw)]<l m by _[A(jw)]_l m. This is a minor variationj and it was corrected in later papers by Doyle [D03,4,5]. Furthermore,

for practical purposes, the results are identical.

129
Appendix C: Numerical Considerations in Algebraic LTR

There are essentially five steps in solving for an LTR compensator by the

algebraic method. This appendix will describe the numerical considerations

important in each step.

I) Calculation of the Plant Numerator Polynomials

The plant transfer function is as follows:

G(s) = C(sI-A)'IB (C.I)

where C is an mxn matrix, A is an nxn matrix, B is an nxl vector and G(s) is

an mxl vector function of s. To use the algebraic approach the numerator

polynomials of G(s) must be found. We have used two methods, the Fadeeva

method, which finds the coefficients directly based upon the Cayley Hamilton

Theorem [CHI], and a method based on the transfer function zeros. The Fadeeva.

method works well for small systems but not for large. The first reason for

this is that the number of steps is proportional to n 4, which can become computationally very expensive. The second reason is that the algorithm involves

raising the matrix A to the (n-l)th power. This works well for discrete time

systems, since the eigenvalues of A lie inside the unit circle, but it is very

poorly conditioned for continuous time problems, since the eigenvalues of A

may be very large.

The second approach is far more reliable. It is based on calculatin E the

zeros in each channel of the plant, calculating a constant that will premultiply the numerator polynomial and then multiplying out the zeros. The zeros of

a plant are defined as values of s which result in a zero output for a nonzero input. In terms of the state space representation of the plant for the

130
ith channel, this can be defined as follows:

(sl -A) (c.2)

where C. is the ith row of C and the plant zeros are the values of s which are

1

a solution to (C.2) for y = O. This can in turn be transformed into the following generalized eigenvalue problem:

;: 011:l [III

The generalized eigenvalue problem can be solved accurately and efficiently by

the QZ-algorithm_[ST2], where the computational cost is proportional to n _.

The only tricky step is the separation of finite zeros from infinite zeros.

The QZ-algorithm will result in one complex vector = and one real vector _.

The eigenvalues are found by dividing the elements of = by the corresponding

non-zero elements of _. The first step in separating finite zeros is to eliminate all elements where _ is zero. There are two possibilities for eliminating further infinite zeros. The first is to place a threshold on the elements of _, eliminating elements lying below this threshold. This method,

however, proves to be very sensitive to the threshold. A more reliable method

involves dividing all the remaining elements of = by the correspondin E nonzero elements of _ and then placing an upper bound on the magnitude of the

resulting zeros. The upper bound is easily estimated, with some knowledge of

the problem, and this method has proved to work very well. As a final check,

the number of zeros for a SISO flexible system will almost always be n-2.

The ith numerator polynomial will have the following form:

131
n i (s) = k i (s-z I) (s- z2). •.(S-Zn. 2) (C.4)

The next step therefore involves computing k. This is most easily done by not1

ing the following equality:

Ci(sI-A)'IB = ki(S'Zl) (s-z2)... (S-Zn. 2)

(s.Pl) (s.P2) ".. (S.Pn.2) (S_Pn. i) (S.Pn) (C.5)

where pj is the jth pole. (C.5) is valid for any s which is not a pole of the

system, k. can therefore be found by evaluating (C.5) for some s = s

1 O

(eg. so - i) and then solving for ki. Also note that only (Sol-A)'IB need be

evaluated_ using a linear equation solver, rather than (Sol-A)'l. The above

method was used in Ref. [EDI] to evaluate plant transfer functions for s = jw.

The final step in solving for the plant numerator polynomials is to multiply out (C.4). The approach based on plant zeros is more complex than the

Faddeva method and must be carried out for each channel separately, but it is

far more computationally reliable.

2) Solution to the LQR Problem -

Re_..major step in the _=_=_.-°_-"_"_^-v_^c_=_c"_...... =_=_= feedback control gains is

the solution of a Riccati equation for the LQR problem. This is solved by

Potter's method [KAI] which involves finding the eigenvalues and .eigenvectors

of the following Hamiltonian matrix:

H = (C. 6)

BR'IB T -A

Finding an accurate solution to the Riccati equation therefore reduces to

finding accurate eigenvalues and eigenvectors for (C.6). Inaccuracies can

132
develop when Q is made very large in comparison with BB T. This becomes especially important in the solution of the near singular Kalman filter problem

for the asymptotic LTR method, since Q may be very large before loop recovery

is achieved. This difficulty can be overcome by multiplying both Q and BB T by

the square root of qc' and then multiplying the final Riccati solution matrix

P by the same number, rather than simply multiplying Q by qc" This procedure

keeps H from becoming too "unbalanced" and results in accurate solutions to

the Riccati equation, even for very large qc"

The numerator polynomial for the optimal LQR transfer function K(sI-A)'IB

is found by the same methods used to find the plant numerator polynomials.

3) Compensator Order Determination -

The compensator order is found very simply by counting the maximum degree

of the plant numerator polynomials and applying equation (4.2.24). If a

proper compensator is required the compensator order will be identical to nN

in (4.2.24), while if a strictly proper compensator is required, the compensator order will be nN + I. The only problem in determining the compensator

order is dealing with cases in which the matrix N is (4.2.26) will be singular. This will occur when some plant numerators contain identical zeros.

This could be checked, but it essentially corresponds to a poorly posed problem.

4) Solution of Compensator Numerator Polynomials -

133
Once the plant numerator polynomials, the optimal loop gain polynomial, and

the compensator order are found; a set of compensator poles are chosen and the

compensator numerator polynomials are found by solving the m(nN+l) set of

linear equations defined by (4.2.26). The solution of these equations depends

on the condition of the matrix N, which is discussed in Section 4.4. Assuming

that a solution does exist, We use an iterative solution routine. This is

performed only once per design, and since the accuracy of the pole/zero cancellations depends heavily on an accurate solution, the extra computational

cost and storage required for the iterative solution is considered worthwhile.

5) State-Space Realization of the LTR Compensator -

Since the compensator is found in polynomial form it must be transformed to

a state space form• I use a observable-canonical representation [CHI] which

is as follows:

A

"0 ...... 0 -D 0"

I ...... 0

0 1 .... 0

• ° °°°• •

O0 .... I-D

nN

B _.

nl, 0 •.. nm, 0

nl,nN. •. nm, n

(c.7)

c--[co ...... o]

where D i is the ith coefficient of the compensator denominator polynomial,

hi, j is the jth coefficient of the ith compensator numerator polynomial, and c

is the product of the compensator poles which are not cancelled.

Since the polynomial coefficients are sometimes very large, and will possibly contribute to later numerical problems, the state-space representation is

134
Y

balanced so as to minimize the 1-norm of the matrix. This is a relatively

cheap operation (computational cost proportional to n_), and does result in

better numerical properties for the robustness calculations.

135