N90-23024

Modern CACSD using the Robust-ControlToolbox

Richard Y. Chiang and Michael G. Safonov

Department of Electrical Engineering - Systems

University of Southern California

Los Angeles, CA. 90089-0781

Abstract

The Robust-Control Toolboz [1] is a collection of 40 "M-files" which ex-

tend the capability of PC/PRO-MATLAB to do modern multivariable robust

control system design. Included are robust analysis tools like singular values

and structured singular values, robust synthesis tools like continuous/discrete

H2/H o° synthesis and LQG Loop Transfer Recovery methods and a variety of

robust model reduction tools such as Hankel approximation, balanced trunca-

tion and balanced stochastic truncation, etc.

In this paper, we will describe the capabilities of our toolbox and illustrate

them with examples to show how easily they can be used in practice. Examples

include structured singular value analysis, H _° loop-shaping and large spacestructure model reduction.

1 Introduction

The fundamental issue in robust control theory - to find a stabilizing controller that

achieves feedback performance despite the plant uncertainty, is still the same issue

addressed by the classical 1930's feedback theory of Black, Bode and Nyquist (ref.

Fig. 1.1). Modern robust control theory has resolved many of the issues concerning

the "gap" between the theory and practice that had grown to troublesome proportions

in the 1970's. One key to bridging the "gap" has been the singular value Bode plat.

Recent progress in Structured Singular Value (SSV), H °° optimal control theory and

the model reduction techniques utilizing singular values have made the modern robust

control theory highly practical

The inavailability of quality software implementing the techniques of robust con-

trol theory has, until very recently, significantly limited the access of both researchers

and engineering practitioners to these techniques.

294

https://ntrs.nasa.gov/search.jsp?R=19900013708 2020-03-14T08:59:44+00:00Z

y

. TYU.............................

s_ .................................................................

Fig. 1.1 Robust Control Problem.

Our Robust-Control Toolbox implements the most up-to-date robust control theory

like Perron SSV, optimal descriptor 2-Riccati H °° formulae, LQG/LTR and singular

value based model reduction techniques, ..., etc. The toolbox consists of a library

of 40 functions which extend the capabilities of PC�PRO - MATLAB TM and the

PC�PRO - MATLAB Control Toolboz. The toolbox itself represents four man yearsresearch work done at USC.

These 40-functions can be catalogued into 3 major areas:

• Robust Analysis

- Singular Values

- Characteristic Gain Loci

- Structured Singular Values

• Robust Synthesis

- LQG/LTR, Frequency-Weighted LQG

- H 2, H °°

• Robust Model Reduction

- Optimal Descriptor Hankel (with Additive Error Bound)

- Schur Balanced Truncation (with Additive Error Bound)

295

- Schur Balanced Stochastic Truncation (with Multiplicative Error Bound)

In this paper, we will highlight the most important functions in the toolbox,

demonstrate how easily they can be used and show what kind of results can be

achieved with practical problems. As for the details of the modern robust control

theory, they can be found in [7], [2] and the references therein.

2 Robust Analysis

The objective of robust analysis is to find a proper measure of the multivariable

stability margin (MSM) against uncertainty. Uncertainty may take many forms,

but among the most significant ones are noise/disturbance signals, transfer function

modeling errors and unmodeled nonlinear dynamics, etc. Uncertainty in any form is

no doubt the major issue in most control system designs.

Several tools to measure MSM are available: [1], [8]

• Singular values (Safonov, 1977; Doyle, 1978)

• Perron eigenvalues (Safonov, 1982)

• Diagonal scaling via nonlinear programming (Doyle, 1982; Tekawy et al., 1989)

Let's define the MSM first:

Definition 1 Multivariable Stability Margin (MSM)K,,,, #(.)-1

K,,(Tt_) A=#(Tt_)_l = inf{_(A)ldet(I - T_A) = 0}.

In other words, it's the smallest _(A) that can make the determinant (I - TtmA )

singular (or the closed-loop system unstable). See Fig. 1.1.A theorem summarizes the whole MSM idea:

Theorem 1 The system is stable for all stable Ai with ]lAilloo < 1, if the MSM

g,,,(Tv,,)> 1.

Unfortunately, exact computation of K,n (or/,-1) would require solution of a non-

convex optimization problem and is therefore impractical. Fortunately, computable

upper bounds on K,_ are available, viz.,

K_I(Tt_) = #(Tt_ ) < inf IIhTt_D-111o_ < inf IIDabs(Tt,,)D-111oo << IITtmlloo-- DED -- DED

where D := {diag(dlI,... ,d,I)ldi > 0}.Then using these upper bounds (some may be more conservative than others), one

can assure that the system is stable against the norm-bounded uncertainty ]lAIIoo <

gm.

296

A comparison of the available upper bounds reveals that some are much easier to

compute than others. See table 2.1.

Table 2.1

Method Prop erty Computation

Optimal Diagonal n = 3, exact K,n demanding

Scaling n > 3, 3 15 % gap

easyPerron Eigenvector

Diagonal Scaling

very close to optimal

diagonal scaling

Reference

Doyle, 1982

Tekawy et al., 1989

Safonov, 1982

Singular Value can be very easy Safonov, 1977

conservative Doyle, 1978

Let's see the following example.

Example: Given a system G(s) with multiplicative uncertianty at its input. Findthe MSM.

4: -I- 8s 8•-_- _'s oYs

Theorem1 implies IIAII < (lla(I + a)-'ll.o)-x.

3O

20

10

-20

-30

40

%

_ v-'_--",,m.Paros E{immm_

\

\

lO s If 10 s

Fig. 2.2 Singular Value vs. K,,, (upper bound).

This example reveals that the singular value upper bound is too conservative in

"robust analysis". Whereas, Perron SSV is much simpler to compute than diagonally

scaled nonlinear programming p.

3 Robust Synthesis

Classical control system designers often do Uloop-shaping_ to meet design specifi-

cations. So do modern robust control system designers. "Loop-shaping" for mul-

297

tivariable systems is done via the singular-value Bode plot. However, to shape the

loop transfer function L(s) is nothing but to shape the sensitivity function S(s) :

(I + L(s)) -x and the complementary sensitivity function T(s) = L(s)(I + L(s)) -z.

See Fig. 3.1

_Yr

L(s)Lt )

}, V Bo d

Fig. 3.1 SISO & MIMO Loop-Shaping.

There are several loop-shaping methods available in the Robust-Control Toolbox

(see Table 3.1), but H** is one of our favorites.Table 3.1

Methods

LQR

(Iqr.m)

LQG

(Iqg.m)

LQG/LTR

(Itru.m,ltry.m)

H 2

(h21qg.m)

H oo

(hinf.m)

Advantages

• guaranteed stability margin

• pure gain controller

• use available noise data

• guaranteed stability margin

• systematic design procedure

• address stability and sensitivity

• almost exact loop shaping

• closed-loop always stable

• address stability and sensitivity

• exact loop shaping

• direct one-step procedure

Disadvantageso need full-state feedback

o need accurate model

o possibly many iterations

o no stability margin guaranteedo need accurate modal

o possibly many iterations

o high gain controller

o possibly many iterations

o design focus on one point

o possibly many iterations

Example: Classical loop shaping vs. Ho* for 2nd order low-damped system.

Given a plant G(s) which is 2nd order with damping 0.05 at 20 rad/sec, find a

controller to meet frequency response Bode plot (see Fig. 3.2)

298

40

2O

ca -20

-4O

-6O

-8O10o

PLANT OPEN LOOP & SPECIFICATION

_f/_..: i i ! i i i iii ! i i i ] ! iiI

............... i ......... i---.-.4..---i.--i...i.- _-.i.._ ..... i ......... i.----.i----i.-..i.,.i., i..i. _................. _ ........ i......._....i...i..i..+.i.

: : : : : : ::: : ; $_. : : : ::: : : . : : :

........................._......_....b.._..._.._._.-".-.........................._......_...._....i.._..b._._.............'---_......_-..._....b..._.._.._

i ill iilil i i i iilil i"'-i, i i ilii i i ii!ii i i ! iii_i i i"_.,i i

.........................::....._....!.._.._.-._.!.._.........................._.-....i....!....i.._.i.4÷..............._........_....÷...i.-_,_,._-

i ! ! i iiii :: i i i iiii i i i i i i::: : : : : ::: : : : : : ::: : : : : : : :

........ : : : : : ::: : : : : : : :, i i iiliii . i i i iiiii i i i i i ii

I0_ IOs lOs

Fn_qumcy -Rad/Sec

Fig. 3.2 2nd order plant open loop (¢ : 0.05,0.5) and the L(s) spec.

A classical design might be decomposed into the following: (see Fig. 3.3)

Step 1: Rate feedback to improve damping.

Step 2: Design high frequency (phase margin, BW, roll-off..).

Step 3: Design low frequency (DC gain, disturbance rejection..).

The classical result is shown in Fig. 3.4. Now, let's see how H °° approaches the

problem.

H _ Problem Formulation

We axe solving the so-called H °° Small-Gain Problem ([3]) using the numerically

robust "optimal" descriptor 2-Riccati formulae of Safonov, Limebeer and Chiang [4]

[5].

H °° Small-Gain Problem:

Given a plant V(s) (re]. Fig. 8.5), find a stabaizing controller F(s) such that

the closed-loop transfer function Ttau I is internally stable and its infinity-norm is less

than or equal to one.

But what makes H °° work is its unique and remarkable "all-pass" property:

At H °° optimal, the frequency response of T_aul is all-pass and equal to one (i.e.,

IIT,, .IIThis means that designers can achieve EXACT frequency domain loop-shaping via

suitable weighting strategies. For example, one may augment the plant with frequency

29_ ¸

lead-lag

235(s+20)(s+40)

(s+2)(s+599)

plant

(0.05 @ 20r/s)

rate feedback

(k = 28)

Fig. 3.3 Classical loop-shaping block diagram.

t_

60

40

20

-20

-4O

-6O

-8010 o

O_ASSICAL LOOP-SHAPING

! ! ! ! !!!! ! ! ! ! ! !!!! , ! ! ! ! !!!

i_ _7 i !i iliiii i i i ii....... i i i i iiii - i i i i iiii i i i i !il

_n_[_'')l< ........ _l'l --_'.i ......................... i • "" • "'i" ll'';'' ''_ "''_ "'i "'i' *_ ............................ _'JJll" Jlllllljl_}l II I "_'' _1

_[Li :_i_',',i',i - ',i ii!_............. i ....... .:-,...:...:: ..... : ..............._.'..'.,,_ i _ _ _ _ i i i i ! ii

J....... L....!.....! ,,, .".._........... :....':'::b.................... _............... _......... _.....L .._.._..L..._

•iliilii.i.lO t 10 2 10 3

Frequmcy - R_ec

Fig. 3.4 Classical loop-shaping.

3OO

u I

u2 -_IP(s) Y2

Yl

F(s) [_

Fig. 3.5 H _ Small-Gain Problem.

u 1

u 2

AUG MI_I'I_D _ANT P(s)

p ............. °.°..°o_o_..°o°°ooo°o°ooo°o..°°°.o°oo.°°°o°°°o.oo_

Fig. 3.6 Weighting strategy block diagram.

= Yla

= Y2a

= Y3a

Y2

3o_

dependent weights W1, W2 and Ws as shown in Fig. 3.6. Then, the Robust-Gontrol

Toolboz functions augtf.m and augss.m will perform the augmentation and create a

state-space for function hinf.m to find an H _ controller. Of course, these frequency

weighting functions have to be chosen so that a stabilizing solution satisfying the H _

norm constraint exists.

In a typical application, either W2(s) or Wn(n) would be absent, leading to weighted

H _ costs of the forms

F(,) W3T < 1 or _(_ W_.FS < 1.oO

In our example, the frequency domain spec. can be split into W1 and W3:

P (0.2s + 1) 2 40000W1-1 = 100(0.0058 + 1) 2; W_-I - s 2

as shown in Fig. 3.7.

SPECIFICATION ---> WEIGHTING STRATEGY

]00 i i r i i ill! ! l i i i i _ii J i l ! i i ir

...............i i il il i i i!iiiii i!iiill80 ............ :::'J ......... i.....-.!...-.i.--J-.-i-.i.. J..i ................ i ......... i......i....._....i..._.., i-.i.i ......................... _......._....i....i...i.. _"•J-

_ . i ]7::ii [i ..........! ::-tL! i _i!::...............!........! _ i i ii i- i i i i i_iii __/i_ i i iiiii

i i i illili o i _: i i!iiii _ _ i ii_

...............i......... ...............0 -_'_!!! ........!_.,_.....i....!-.._.._.._.i

/ i [._.i-_i _-_ i i i iiiii i i i_40 _ ....... _..... _-'. _ _ i iii Y i i i i i i iii i i i i i iii

10 e lot 10_ 10_

Frequency - R_I/Sec

Fig. 3.7 Weighting strategy for 2nd order problem spec.

The results are shown in Fig. 3.8 for different p's. Clearly, in the limit (as p

goes to 3.16) the cost function becomes "all-pass". The parameter p of W1 is the

only parameter on which we iterate for design; the Robust-Control Toolboz script-file

hinfgama.m automates this iteration.

302

H-inf Design A (rho = 1)

5o. i i i I ITITI T i i T IIIII _ _ i I_I_iI

!,o_....i:}:i:i. ...............i-!iii

.,o...............i........._......_....!.._.._.._._.._................_........._Tr_!.... i_:...:i_ij_; ...............i.........i..............ii11................i.........i..........f-i-ii--ii...............__-_ ...............i.........i......i4..._..i..i.i..i.i.........i......i....i..i.._..i..i.i...............i........L..-....i..]:_i.l.i.5o _ _ _!ii!il i i i li!!ii i i i iiii"i

10 o 10 n 10 2 10 3

Rad/S_

H-infDeqn B (rhoffi3.16)

. ':[i i iiii : ii::i:i:i:il:[ i iii:ii i li iif i iiiill i iii iiiiiiii

,_ -10

-15

I0 e 10 n 10 2 I0 3

P.ad/Sec

H-iaf C.mt F,um:tica (Daisn A & B )

D_/6A/ 8 (opr/_1-)o 6"

-20

• ,%

(x¢__Ma_ .). ',

....... io, ....... i_ ....... _'o,

Fig. 3.8 I'I _ resultsfor 2nd order system

303

3.1 H _¢ Software Execution and Sample Run

To do an H _ control design with the Robust-Control Toolboz is relatively simple.

Table 3.2.1 shows the complete user inputs for our sample problem.

Table 3.2.1

>> hUg----[0 0 4001; dng = [1 2 400];

>> [ag, bg, cg, eg] = tf2ss(nug, dng);

>> sysg=[ag bg;cg t/g]; zg=2;

>> wl=[2.5e-5 1.e-2 1;

0.01 • [4.e - 2 4.e - 1 1]];

>> w2 = [ 1;

>> w3 = [1 0 0;0 0 40000];

> > [A,B1,B2, C1, C2,DII,D12,D21,D_2] = augtf(sysg, zg, wl,w2,w3);

>> hi./

Table 3.2.2 shows the output which appears on the screen for a successful run of

hinf.m.

Table 3.2.2

<< H-inf Optimal Control Synthesis >>

Computing the 4-block H-inf optimal controller

using the S-L-C loop-shifting/descriptor formulae

Solving for the H-inf controller F(s) using U(s) = 0 (default)

Solving ricoati equations and performing H-infinity

existence tests:

i. Is D11 small enough? OK

2. Solving state-feedback (P) ricoati ...

a. No Hamiltonian jg-axis roots? OK

b. A-B2*F stable (P >= 0)? OK

3. Solving output-injection (S) riccati ...

a. No Hamiltonian j,-axis roots? OK

b. A-G,C2 stable (S >= 0)? OK

4. max eig(P*S) < 1 ? OK

all tests passed -- computing H-inf controller ...

DONE!!!

3O4

4 Robust Model Reduction via Basis-Free Tech-

niques

In the design of controllers for complicated systems, model reduction arises in several

places: 1). Plant model reduction, 2). Controller model reduction, 3). Simulation oflarge size problem.

However, naive implementations of model reduction methods such as Rosenbrock's

stair-case algorithm, Moore's balanced truncation, optimal Hankel approximation and

balanced stochastic truncation, etc., can fail on even relatively simple problems due

to numerical instability.

The Robust- Control Toolboz implements the "basis-free" version of the latter three

of the model reduction techniques, which are not only numerically robust but also

tend to achieve the ultimate result for practical problems. In particular, they all

possess the following special features:

1. They bypass the ill-conditioned balancing transformation, so that they can eas-

ily deal with the "non-minimal" systems.

2. They employ Schur decomposition to robustly compute the orthogonal bases

for eigenspaces required in intermediate steps.

These methods all enjoy attractive L _ error bounds - either an additive error

bound or a multiplicative error bound.

• Additive Methods:

- Optimal descriptor Hankel MDA (ohklnw.m).

- Schur balanced truncation (balmr.m, schmr.m).

• Multiplicative Method:

- Schur balanced stochastic truncation (rc_mr.m).

4.1 Robust Model Reduction Theorems

For robust control system design, it is desirable that the reduced order model satisfies

the conditions of one of the following two theorems (ref. [1] [2]). Otherwise, the

controller design based on the "blind"reduced order plant model can be unstable !

Theorem 2 Additive Robustness Theorem: (see F_. _.1)

x/_(_'A) <___(_) for _, <_,0, (_ith _A ..d 0 open toop a,_te), t_. t_ do, ea-loop system will be stable provided that the control bandwidth is less than w., where

o.,,. := rnaz{oJ [ o-(O(jw)) _>_,(_A(jw))}.

305

TRUE PLANT G(s)................................ "l

|

b .............:1Fig. 4.1 Additive modeling error.

Theorem 3 Multiplieative Robustness Theorem: (see Fig. 4.2)

If'_(_u) <_1 for _ <_o,, (with _M and _ oven loop stable), then the closed-loopsystem will be stable provided that the control bandwidth is less than w,, where w, :--

max{_ I _(_(j_)) _<1).

TRUE PLANT G(s).............................................. i

i

.....Fig. 4.2 Multiplicative modeling error.

4.2 Examples of Model Reduction

Example 1: Find a 3-state reduced order model for the transfer function

0.05(87 + 801s s + 1024s s + 599s 4 + 451s 3 q- 11982 + 498 + 5.55)

G(s) = 87 + 12.6ss + 53.48s s + 90.94s 4 + 71.83s s + 27.228 z + 4.75s + 0.3

with Hankel singular values of the phase matrix

0.1 0.2 0.3 0"4

0.9959 0.9972 0.9734 0.7166

0.5 0.6 0.7

0.5635 0.0021 0

3O6

2OSchur BST-REM vs. Schur BT and Des. Hmkel ('/-state -> 3-staw)

10

0

-10-20

-30

-40

-5010-_

...................... + ....................... ) ....................... ,.............................. o ....................... ) ......................

...................... £ ....................... _........ "+....Ltd. ................... _......................

......................_.......................i............t._ ......_......................

_ o_._ _o..____4_lMOA !

......................i.......................i..............l .......................i......................l

.... 610-2 104 10o l0 t 1 1 104

Frequency - Rad/Sec

20O

100

._ -100i

el

-200

-30O

-4O0

-50G10-3

Schur BST-REM vs. Schur BT and Des. Hmlml ('/-state --> 3-state)

...................... ,_ ....................... j ................. ::._ ..................... z.............................................. _......................

• i _-_-_..:_;.....................+.....................-;.......................+,.......................i.....................

:'.\i :......................'+".......................'..................:"i".......................'.......................' .......................' ......................

solid : oii_ moll i "_. _ i . i .............;_..i._i_.-ii,i+ ........._................................-. ....................................................................

..... : O#mal Des. Htnkea MDA_ "\ :,-.-.-.: Schur BST-RTaM i X_ i.._;_, ".....

..................... : ....................... :........................ _....................... _._ ................... _ ................. _._" .-'.....................

i i i i i i

lO-1 10-i 10 o 101 10a 10 _ 104

_m_-Rad/Sec

Fig. 4.3 Schur BST-REM vs. Schur BT and Descriptor Hankel.

307

The results produced by the Robust-Control Toolbozfunctions - ohklmr.m, schmr.m,

reschmr.m, are shown in Fig. 4.3. Clearly, the Schur BST-REM method that keeps

the reduced model staying inside a prescribed relative error bound produces the ul-

timate result in model reduction. Note that or7 = 0 indicates only the "basis-free"

methods can handel the problem without numerical difficulty.

Example 2: Model reduction for a large space structure [6] (see Fig. 4.4).

Our design reequirement is to find a controller to track LOS loops in 300 Hz BW

and to reduce plant disturbance response by a factor of 100.

The Hankel singular values after the inner loops are closed indicate that the system

is non-miuim_l. Therfore, only the "basis-free" methods such as - Schur BT (schmr.m)

or Schur BST-REM (reschmr.m) can be used. Again, only the Schur BST-REM can

match the originxl model up to a "robust frequency" which is high enough so that

the required BW of 300 H_. can be satisfied (see Fig. 4.5 & Fig. 4.6).

Secondm'y

Mirror

Structral [

Member _

(.13-18)

Primm'y Mirror ////

Disturbances

(25-30)

)Disturbances

(19-24)

1

Fig. 4.4 Large space structure.

3O8

L5,5 44_e SChur-BI mode/vs. 116-su_ oc_imd

-2o..................i....................i..................i...................i...................i....................i....................i...................

40.................i..................ii....................i...................i..................i...................ii..................._{64)i...................

"i.................i...................i...........................................................""16_1q 10 4 10-1 10 o l0 t 102 10s 104 10S

2O

0

-20

-.4¢

.-6C

.._

-I00

-120I0 a

L.SS 44ta_ Sdmr-BT mode.l v_ mml m.,,m.

[

iiii i......................................iI0 _ 104 I0o l01 I02 I03 I04 I0s

Fig. 4.5 Model reduction using Schur Balanced Truncation.

°LSS 4-mine Schta"BST l_t,/model v,t. 116-mine oa,NI.

c .i....................!........................................_......................................' _ i i i i

-2o.................._....................,....................i...................i...................!.....................i....................!..................

-ioo...................i..................i....................i....................i...................._...................i.............-m .................._....................i....................i....................i...................._...................i..................._...............

_,6o,ii'l_...................;..................._...................iiii....................i!!!!!!!!!iiiiiiiiii!i!!!!!!ii!i!i!iiii!!!!!!!i!i_I...................!...................i....................i...............

1o-_ io-2 to-, ioo lo, lo_ _o_ _c_ t_

Rzd/S_

LSS 4-sm_ Sdmt BST-I_ mo_l vs. _tl __0

o.................._....................!-._...................i....._..(.__..i........................................

! _(_,,,-_)'i'_"-'L:.............i_ i

_ ..................T...................i................................................................."..................._..........._:":.::................

i i i _ i_i i.,................-2_.... _ /0

-25010 "_ 10 "a 10 4 I0 ° 10' 102 103 104 105

Fig. 4.6 Model reduction using Schur Balanced Stochastic Truncation.

3O9

5 Importance to the Control Community

If the ultimate goal of research is to apply the theory to reality, then the contributionof Robust-Control Toolbox is dear:

It provides a vital bridge between modern robust control theo_ and real control appli-cations.

The following diagram (Fig. 5.1) shows how different groups of people with dif-

ferent backgrounds can utilize the Robust-Control Toolbox to achieve their personal

goals. For example classical control designers can use the toolbox to understand the

theory or to apply on a design. Non-robust control researchers can study the code pro-

vided by the toolbox together with the papers referenced therein to become familiar

with the robust control theory. Students can use the toolbox either for robust control

research or for realistic design studies. It seems that the Robust-Control Toolboz can

serve people from a variety of backgrounds.

Robust _ Robust

Control / _ Control

Toolbox _ NN Toolbox

Robust-Control Toolbox

Robust

Control

Toolbox

Toolbox

@

Robust

Control

Toolbox

Fig. 5.1 The importance of the Robust-Control Toolboz.

310

6 Summary

The Robust-Control Toolboz

• Provides a bridge between modern robust control theory and the real-world

applications.

• Has the most up-to-date Robust-Control theories and algorithms.

• Is in readable M-files, so it's educational.

• Contains the tools one needs to do robust control system design, analysis and

model reduction.

• Is direct, powerful and easy-to-use.

References

[1] R. Y. Chiang and M. G. Safonov, Robust-Control Toolboz. So. Natick, MA: Math-

Works, 1988.

[2] R. Y. Chiang, Modern Robust Control Theory. Ph.D. dissertation, USC, 1988.

[3] M. G. Safonov and R. Y. Chiang, "CACSD using the State-Space L °_ Theory

- A Design Example", IEEE Trans. on Automat. Contr, vol. AC-33, no. 5, pp.

477-479, 1988.

[4] M. G. Safonov and D. J. N. Limebeer, "Simplifying the H _ Theory via Loop-

Shifting", Proc. IEEE Conf. on Decision and Control, Austin, TX, Dec. 7-9,

1988.

[5] M. G. Safonov, D. J. N. Limebeer and R. Y. Chiang, "Simplifying the H °°

Theory via Loop-Shifting, Matrix Pencil and Descriptor Concepts", to appear

Int. J. Control 1989.

[6] M. G. Safonov, R. Y. Chiang and H. Flashner, "H _ Control Synthesis for a

Large Space Structure," Proc. of American Contr. Conf., Atlanta, GA. June

15-17, 1988. Submitted to J. of Guidance and Control, June, 1989.

[7] M. G. Safonov, Robustness and Stability Aspects of Stochastic Multivariable Feed-

back System Design, Ph.D. dissertation, MIT, 1977. Also, M. G. Safonov, Sta-

bility and Robustness of Multivariable Feedback Systems. Cambridge, MA: MIT

Press, 1980.

[8] J. Tekawy, M. G. Safonov and R. Y. Chiang, "Computer Algorithms for Multi-

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