Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner

142

A. N. Gendes, C. A. Desoer

Algebraic Theory of Linear Feedback Systems with Full and Decentralized Compensators

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Series Editors M. Thoma • A. Wyner

Advisory Board L. D. Davisson • A. G. J. MacFarlane • H. Kwakernaak J. L. Massey • Ya Z. Tsypkin • A. J. Viterbi

Authors A. Nazli Gende~ Dept. of Electrical Engineering and Computer Science University of California Davis, CA 95616 USA

Charles A. Desoer Dept. of Electrical Engineering and Computer Sciences University of Califomia Berkeley, CA 94720 USA

ISBN 3-540-52476-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52476-2 Spdnger-Verlag NewYork Berlin Heidelberg

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PREFACE

The algebraic theory of linear, time-invariant, multiinput-multioutput feedback systems

has developed rapidly during the past decade. The factorization approach is simple and

elegant; it is suitable for both continuous-time and discrete-time lumped-parameter system

models and many of its results apply directly to distributed-parameter systems. Major achieve-

ments of this algebraic theory are: 1) several equivalent formulations of necessary and

sufficient conditions for stability, where the concept of stability is defined with flexibility to

suit various applications; 2) the parametrization of all achievable stable input-output maps for

a given plant; 3) the parametrization of all compensators that stabilize a given plant; and 4) a

general method which is suitable to study different system configurations.

In this volume we aim to unify the algebraic theory of full feedback and decentralized

feedback control systems. Our main focus is the parametrization of all stabilizing compensa-

tors and achievable stable input-output maps for three particular feedback configurations: the

standard unity-feedback system; the general feedback system in which the plant and the com-

pensator each have two (vector-) inputs and two (vector-) outputs; and the decentralized con-

trol system in which the compensator is constrained to have a block-diagonal structure.

Several of the results we present are well-known to control theorists. We clarify and unify

the presentation of these results, remove unnecessary assumptions and streamline the proofs.

Among the new developments in this volume are a characterization of all plants that can be

stabilized by decentralized feedback and the parametrization of all decentralized stabilizing

compensators. The introduction to each chapter includes a list of the important results.

A good preparation for the material in this volume is a graduate-level course in linear sys-

tem theory and some familiarity with elementary ring theory.

We gratefully acknowledge the support of the Electrical Engineering and Computer Sci-

ence Departments of the University of California at Davis and Berkeley and the National Sci-

ence Foundation (Grant ECS 8500993). We thank Jackie Desoer and Giintekin Kabuli for their

patience.

Table of Contents

Chapter 1: INTRODUCTION .......................................................................................... 1

Chapter 2: ALGEBRAIC FRAMEWORK ..................................................................... 4

2.1 Introduction .......................................................................................................... 4

2.2 Proper stable rational functions ........................................................................... 5

2.3 Coprime factodzations ......................................................................................... 7

2.4 Relationships between coprime factorizations .................................................... 17

2.5 All solutions of the matrix equations XA = B , A X =/~ .............................. 26

2.6 Rank conditions for coprimeness ......................................................................... 32

Chapter 3: FULL-FEEDBACK CONTROL SYSTEMS ............................................. 36

3.1 Introduction .......................................................................................................... 36

3.2 The standard unity-feedback system .................................................................. 39

Assumptions on S ( P , C ) ....................................................................................... 39

Closed-loop input-output maps of S ( P , C ) .......................................................... 40

Analysis (Descriptions of S ( P , C ) using coprime factorizations) ........................ 43

Achievable input-output maps of S ( P , C ) ............................................................ 62

Decoupling in S ( P , C ) .......................................................................................... 64

3.3 The general feedback system ............................................................................... 67

Assumptions on Z(/~, C ) ........................................................................................ 67

Closed-loop input-output maps of E (/~, C ) ............................................................ 69

Analysis (Descriptions of Y(/~, C ) using coprime factorizations) ......................... 70

Achievable input-output maps of Z(/~, C ) ............................................................. 85

De, coupling in X(/~, C ) ........................................................................................... 86

v

Chapter 4: D E C E N T R A L I Z E D C O N T R O L SYSTEMS ............................................. 94

4.1 Introduction .......................................................................................................... 94

4.2 Two-channel decentralized control system ......................................................... 95

Assumptions on S ( P , C d ) ..................................................................................... 96

Closed-loop input-output maps of S ( P , C d ) ......................................................... 99

Analysis (Descriptions of S ( P , C d ) using coprime factorizafions) ...................... 100

4.3 Two-channel decentralized feedback compensators ........................................... I i0

4.4 Application to systems represented by proper rational transfer functions

.................................. ..... ............. ..... ............. ..... ... ......................................................... 129

Algorithm for two-channel decentralized Ru-s tab i l i z ing compensator design

° , . . . . . . . . . . * ° o . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 1

4.5 Multi-channel decentralized control systems ...................................................... 158

Assumptions on S ( P , C d ),,i .................................................................................. 159

Analysis (Descriptions of S ( P , C a )m using coprime factorizations) ................... 161

Achievable input-output maps of S ( P , C a )m ....................................................... 168

REFERENCES ................................................................................................................... 170

SYMBOLS .......................................................................................................................... 174

INDEX ................................................................................................................................. 175

Chapter I

INTRODUCTION

In this volume we present a unified algebraic approach to the study of linear, time-

invariant (lti), multiinput-multioutput (MIMO), full-feedback and decentralized feedback con-

trol systems. This approach applies to continuous-time as well as discrete-time, lumped-

parameter system models. Much of this theory applies directly to distributed-parameter sys-

tems (see e.g. the books [Blo.1, Cal.3, Fei.1, Vid.1] and the papers [CaLl, Cal.2]).

We use a factorization approach which is based on elementary ring theory. In order to

separate algebra from control theory, we collect all relevant purely al.~ebraic facts and

theorems in Chapter Two. We do not include basic definitions and properties of rings; elemen-

tary ring definitions (entire ring, principal ring, ideal of a ring, ring of fractions, etc.) can be

found in several texts in algebra [Bou.1, Coh.1, Jac.1, Lan.1, Mac.l]-- we recommend the brief

review in [Vid. 1, Appendix A, B]. The goal of Chapter Two is to present the conceptual tools

and key results needed for the systematic use of right- left- and bicoprime factorizations.

In Chapter Three we study two classes of feedback systems that have no restrictions on

the compensator structure. In Chapter Four we consider the restriction that the compensator

transfer-function is a block-diagonal matrix.

The main issues that we address in Chapters Three and Four are: closed-loop stability, the

parametrization of all stabilizing compensators and of all achievable stable closed-loop input-

output (I/O) maps. Three particular feedback configurations are considered: The first one is

the unity-feedback system, which we call S ( P , C ) ; this is the standard multiinput-

multioutput feedback system made up of a plant P and a compensator C , where there are no

restrictions on the compensator structure. The second configuration, which we call E(/~, C ) ,

2

is a more general interconnection where the plant and the compensator have inputs and outputs

that are not utilized in the feedback-loop; the unity-feedback system S ( P , C ) is a special case

ofZ(/~, C ). The third configuration we consider is the decentralized control structure; this is a

special case of the unity-feedback system S ( P , C ) , where only certain outputs are available

to be fed back to certain inputs; in this ease the compensator is constrained to have a block-

diagonal structure. The two-channel decentralized control system, which we study in detail, is

called S ( P, C a ) ; the results are extended to the multi-channel (m--channel) decentralized

control system S ( P, Ca ),,, •

Section 3.2 focuses on the unity-feedback system S ( P, C ) (Figure 3.1). It is well-known

that, for any MIMO plant P , there exists a dynamic feedback compensator C such that the

closed-loop system S ( P , C ) is internally stable. The parametrization of all stabilizing com-

pensators based on a right-eoprime factorization Np De -'1 , a left-coprime factorization

D7 t Np , or a bicoprime factorization Net D-1 NN + G of the plant P is a fundamental tool

in describing the achievable closed-loop performance of S ( P , C ) .

In Section 3.3 we consider the general feedback system E(/~, C ) (Figure 3.9). In this sys-

tem, the plant /~ and the compensator C both have two (vector-) inputs and two (vector-)

outputs. This configuration takes into account such cases where the regulated plant-output is

not necessarily the same as the measured output or where the plant is directly affected by exo-

genous disturbances.

In Sections 4.2 and 4.3 we study the two-channel decentralized control system

S ( P , C a ) (Figure 4.1). We characterize the class of all plants that can be stabilized by decen-

tralized output-feedback and parametrize all decentralized stabilizing compensators. In Sec-

tion 4.4 we consider systems that have rational transfer functions and we study the relationship

between decentralized stabilizability and decentralized fixed-eigenvalues. In Section 4.5 we

extend the results of Sections 4.2, 4.3 and 4.4 to the multi-channel decentralized control sys-

tems S ( P, C a )m (Figure 4.7).

3

We assume throughout that a transfer-function approach makes sense; in particular, the

plant and the compensator subsystems in the feedback-system have no hidden-modes associ-

ated with unstable eigenvalues so that their transfer functions describe the behavior of these

subsystems adequately for stability purposes.

The style and the results in Chapters Two and Three are inspired by [Vid.1, Net.l, Des.3];

some results are based on our original work (e.g. [Des.4, 5, 6, 7]). Vidyasagar's book [Vid.1]

has a list of references for previous related work. There is also a considerable amount of previ-

ous work on the existence issues for decentralized compensators and on various formulations

of decentralized fixed-modes (e.g. [And.l, 2, Day.l, 2, Fes.1, Tar.l, Vid.4, Wan.l, Xie.1]. The

characterization of all plants stabilizable by decentralized feedback and the parametrization of

all decentralized compensators that is presented in Chapter Four is a new development.

Chapter 2

ALGEBRAIC FRAMEWORK

2.1 INTRODUCTION

This chapter gives an integrated development of the algebraic facts that are used in the

following chapters. The reader is assumed to be familiar with basic properties of rings; the

material presented in [Vid. 1, Appendix A, B] provides sufficient background. More detailed

discussions on ring theoretic concepts can be found in [Bou. 1, Coh. 1, Jae. 1 (Sections 2.1-2.3,

3.7), Lan.1, Mac.l] .

Notation

H is a principal ideal domain.

J c H is the group of units of H .

I c H is a multiplicative subset of H , where 0 ~ l , 1 ~ I .

G := H / I = { n / d I n ~ H , d ~ I } is the ring of fractions of H with respect to I .

F := H / ( H \ 0 ) = { n [ d ] n , d ~ H , d * 0 } is the field of fractions of H .

Gs := { x ~ G [ ( l + x y ) -1~ G , for all y ¢ G } is the (Jacobson)radicalofG.

[]

Note that: (i) every element of I is a unit of G ; (ii) F ~ G ~ H ~ I ~ J ;

(iii) if x E I then x a G s .

Let a , b ~ H ; then a and b are associates (denoted by a ~ b ) iff there exists

u ~ J such that a = b u ; note that a ~ 1 iff a ~ J . ' . . . . is an equivalence rela-

tion on H .

5

The set of matrices with elements in H is denoted by m(H) ; this notation is used when

the actual order of the matrices is unimportant. Where it is important to display the order of a

matrix explicitly, a notation of the form A E H n° x ni, B E H n ix ni is used instead of A,

B ~ m(H). In the study of linear control systems, the set m(H) corresponds to stable sys-

tems; therefore we call a matrix H-stable iffA ~ m(H).

The identity matrix is denoted by I ; in some cases the order of the identity matrix is indi-

cated with a subscript as in I n .

Let A ~ m ( H ) ; then A is called H-unimodular (G-unimodular) iffA has an inverse

in m(H) (m(G), respectively); equivalently, A is H-unimodular (G-unimodular) iff

detA ~ J (detA ~ I ,respectively).

LetX e m ( G s ) , r ~ m ( G ) a n d Z ~ m ( G ) have appropriatedimensionssothat

X Y and Z X are defined; then X Y ~ m(Gs), z x ~ m(Gs),

( I + X Y ) - t ~ m(G) and (1 + Z X )-~ ~ m(G).

2.2 P R O P E R S T A B L E R A T I O N A L F U N C T I O N S

Let C+ := { s [ Re s > 0 } denote the closed right-half-plane and let C_ :=

{ s I Re s < 0 } denote the open left-half-plane of the field 112 of complex numbers. Let

U be a nonempty subset of C such that U is closed and symmetric about the real axis, and

C \ U =: D is nonempty; D is called a region of stability. Let

fi : = u t_, { ~ } .

In the study of continuous-time control systems, 1J D C+.

Let the ring of proper scalar rational functions of s (with real coefficients), which have no

poles in U be denoted by R u . The ring R u is a proper Euclidean domain, (which implies

that it is a principal ideal domain) with degree function ~ : Ru \ 0 ---> z+ defined by

8 ( f ) = number of l~-zerosof f .

6

Two functions f , g ~ Ru are coprime iff they have no common l~-zeros. The primes

in the ring R u are functions of the form s - a 1 (s - a )2 + c 2 s + b ' s + b ' ( s + b ) 2 and their associ-

ates, where a , b , c e R , a ~ U , - b ~ C \ I J , c > 0 .

Suppose that R u is the principal ideal domain n under consideration. By definition of

J , f e J implies that f is a proper rational function which has neither poles nor zeros in

1]. We choose I to be the multiplicative subset of R u such that f ~ I iff f (,,o) is a

nonzero constant in R ; equivalently, I c R u is the set of proper, but not strictly proper real

rational functions which have no poles in U. The ring of fractions R u [ I is denoted by

IRp (s) and consists of proper rational functions of s with real coefficients. The field of frac-

tions associated with the principal ideal domain R u corresponds to all rational functions of s

with real coefficients, denoted by l~(s ). The (Jacobson) radical of the ring lRp (s) is the set of

strictly proper rational functions, denoted by Rsp (s).

Suppose thatp ~ Rp(s ) ; thenp can be expressed as a fraction n / d , where n ~ R u

and d ~ I are coprime. The U-zeros of p are the same as the U-zeros of n ; hence,

p ~ l~sp (s ) if and only if n / 1 = n ~ R s p ( s ) ; t h e U - p o l e s o f p are the same as the

l l - ze ros of d .

Let A ~ m ( R u ) ; then A is Ru-unimodular iff detA has no U-zeros, i.e.,

8(detA ) = 0 . On the other hand, A is Rp (s)-unimodular iff detA has no zeros at infinity.

In the case that U is chosen to be C+, the ring R u corresponds to the set of scalar

transfer functions of bounded-input-bounded-output-stable (BIBO-stable), linear, time-

invariant, continuous-time, lumped systems; hence, R u is called the ring of proper stable

rational functions. Similarly, m ( R u ) is the set of matrix transfer functions of BIBO-stable

systems; hence, a matrix which has dements in R u is called an R u s t a b l e matrix. Note that

in most applications, it is desirable for the system to have poles in a region of stability C \ U

= K), which is more restricted than the open left-half-plane C_.

7

2 .3 C O P R I M E F A C T O R I Z A T I O N S

Suppose that p ~ G, the ring of fractions of H associated with I ; then p is equal to a

fraction x [ y , where x ~ H and y ~ I . In the fraction x ] y , x and y are not necessarily

coprime. Suppose that g e H is a greatest-common-divisor (g.c.d.) of x and y ; then there

are n , d ~ n such that x = n g and y = d g , where y ~ I implies that g ~ I and

d ~ I . T h e f r a e t i o n x / y is equivalent to the fractionn ]d , w h e r e n ~ H a n d d ~ I are

coprime; note that since g = u x + v y = u n g + v d g for some u , v ~ H, we have

u n + v d = 1 for some u , v ~ H.

Let P be a matrix whose entries are in the ring of fractions G of the principal ideal

domain H . In this section we define coprimeness in H and eoprime factorizations o f P over

m(H) and display some important properties o f coprime factorizations.

Definition 2.3.1. ( Coprime-fraction representations )

(i)

(ii)

The pair (Np , Dp ) , where Np , Dp ~ m(H), is called right-coprime (r.e.) iff there

exist Up , Vp ~ m(H)such that

v, o, + up N, = z , (2.3.1)

the pair ( Np , Dp ) is called a right-fraction representation (r.f.r.) of P ~ m(G) iff

Dp is square, detDp ~ I and P = NpDe- 1 ;

(iii) the pair (N t, ,Dr, ) is called a right-coprime-fraction representation (r.e.f.r.) o f

~' e mCG) iff (Np ,Op )isanr.f.r. of~' and (Np , D e ) i s r . c .

I f ( N p , Dp ) is an r.c.f.r, o f P then we call N e De- ' a right-coprimefactorization o f P .

8

(iv) The pair ( o~ . ~Tp ) . where ~p . rT, e mcH) . is called left-coprime (I.e.) iff there

exist Up , ~77 E m ( H ) such that

Np Up + D p Vp = I ; (2.3.2)

(v) the pair ( Dp ,/~p ) is called a left-fraction representation (Lf.r.) of P E n ' l (G) iff

/)p is square, det/)p ~ I and P = D f ' N p ;

(vl) the pair (/)p ,/Vp) is called a left-coprime-fraction representation (I.e.f.r.) of

P a m ( G ) iff ( /~. ,/qp )isanl.f.r . of P and (D-p ,Np )isl.c.

If (/)t , ' /Vt' ) is an 1.c.f.r. of P then we call D ; ' Np a left-coprimefactorization of P .

(vii) The triple ( Npr , D , Npt ) , where Npr , D , Npt ~ m ( H ) , is called a bicoprime

(b.c.) triple iff the pair ( Npr , D ) is r.e. and the pair ( D , Npt ) is l.c.

(viii) The quadruple (Npr , D , Npl , G ) is called a bicoprime-fraction representation (b.e.f.r.)

of e e m ( G ) iff the lriple (Npr ,D ,Npl ) is a bicoprime triple, D is square,

detD ~ I and P = NprD -1Npt+ G .

If ( Npr , D , Npt , G ) is a b.c.f.r, of P then we call Npr D-1 Np t + G a bicoprime fac-

torization of P .

[]

In the factorizations P = Np D f 1, P = DT1 N? and P = Npr D -1Npt + G, the matrices

Np, IV?, Npr, Npt are interpreted as "numerator" matrices and Dp, Dp, D are interpreted as

"denominator" matrices.

Equations (2.3.1) and (2.3.2) are called a right-Bezout identity and a left-Bezout identity,

respectively.

Note that every P e m ( G ) has an r.c.f.r. (Np ,D e ) , an 1.c.f.r. ( /gp, /¢p ) and a

b.c.f.r. ( Npr , D , Npt , G ) in H because H is a principal ring [Vid.1, Section 4.1].

9

L e m m a 2.3.2. ( Coprimeness af ter elementary operat ions )

Yt, Dr,

Let = E

Xp N~

are H-unimodular ; then

(|)

(ii)

Proof

an e,

the pair (Np , Dp ) is r.c. if and only if the pair (Xp , Yv ) is r.c.;

the pair ( D v , Np ) is l.c. if and only if the pair ( Yp, Xv ) is l.c.

[v,

equivalently, since E - l e m(H), there exist U x,

Vy Yv + Ux X t, = I ; equivalently, the pair ( Xp , Yp ) is r.c.

The proof of (ii) is entirely similar. 1"7

L e m m a 2.3.3. ( Products of units )

(i) Let a , b ~ H ; t h e n a b ~ J

(ii) Let c , d E H ; then c d ~ I

P roof

vy e m(H)

, where E , F ~ m(H)

(i) From Definition 2.3.1 (i), ( Np , Dp ) is r.e. iff there exist Up , Vp ~ m(H) such that

[] [] such that

if a n d o n l y i f a ~ J and b ~ J .

if a n d o n l y i f c ~ I and d ~ I .

(i) I f a , b ~ J , then a-1 , b-1 ~ H ; since H is commutative, 1 = a -1 a b b -1 =

( a b ) ( b -1 a-1 ) = ( b-1 a-1 )( a b ) ; hence, b- la -1 ~ H is the inverse of a b ~ H and

b -1 a -1 = a -1 b -1 , therefore a b ~ J . To show the converse, let a b =" u ; by assumption,

u -1 ~ H . Therefore, b ~ H has the inverse ( u - l a ) ~ H since ( u - l a ) b = 1 =

b ( u - l a ) , and hence, b E J . Similarly, a ( b u -1) = 1 = ( b u - 1 ) a implies that

( b u -1) ~ H is the inverse of a ~ H a n d h e n c e , a ~ J .

(ii) Since I is a multiplicative subset of H , c , d ~ I implies that c d E I . To

show the converse, let c d =: v ; then v -1 ~ G since v ~ I . Now

10

( v -1 c ) d = 1 = d ( v -t c ) , which implies that ( v -1 c ) • G is the inverse (in G ) of

d • H . Similarly, c ( d v -1) = 1 = ( d r -1 )c implies that ( d v - t ) • G is the

inverse (in G ) of c • H ; therefore, c • H and d • H are units in G and hence,

c e I a n d d e I .

Lemma 2.3.4. ( Uniqueness of eoprime factorizations )

Let ( Np, Dp ) be an r.c.f.r, and let ( /~p , Np ) be an 1.c.f.r. o f e • m(G) ; then

(i) ( Xp, Yt, ) is also an r.f.r. (r.c.f.r.) of P if and only if (Xp, Yp ) = ( Np R , Dp R ) for

some G-unimodular ( H-unimodular, respectively) R • m(H) ;

(ii) ( Yp, Xp ) is also an l.f.r. (1.c.f.r.) of P if and only if ( Yp, Xp ) = ( L / ~ p , L/Vp ) for

some G-unimodular ( H-unimodular, respectively) L • m(H).

Proof

(i) ( if ) If R • m ( H ) is G-unimodular, then detR • I . By Definition 2.3.1 (i),

detDp • I since (Np, Dr, ) is an r.c.f.r, of P ; if Yp = Dp R then by Lemma 2.3.3 (ii),

detYp = detDp detR • I . Since Xt, y;1 = We R R -1 0 ; 1 = Np D ; 1 = P , it follows

that (Xp , Yp ) is an r.f.r, of P .

So far we showed that if both Np and Dp are post-multiplied by a G-unimodular matrix

R, then the resulting pair (Xp , Yp ) = ( N p R , D p R ) is also an r.f.r, of P . Now if

n • m(H) is H-unimodular, i.e., if R -1 • m(H), then by the (fight-) Bezout identity

(2.3.1) we obtain

R-1Vp Dp R + R-I Up Np R = R-1Vp yp + R-1Ut, Xp = I ; (2.3.3)

therefore (Xp, Yp ) is also r.c. and hence, (X e , Yp ) is an r.c.f.r, o f P .

( only if) Let (Xp, Yp ) be an r.f.r, o f P ; then by Definition 2.3.1 (ii), detYp • I and

hence, yp-I e m ( G ) ; since (Np , Dp ) is an r.c.f.r, of P , we know that Np De -1 =

Xp Y71 . Now detDp e I implies that D71 • m(G); post-multiplying both sides of the

(fight-) Bezout identity (2.3.1) by Dp -1 , substituting Xp ypl for N e D71 = P and then

post-multiplying both sides by Yt, we obtain

11

VpYp + UpXp = D p l y p =: R , (2.3.4)

where R • m ( H ) since the left-hand side of equation (2.3.4) is a matrix with enlries in H

and R -1 = yT1Dp • G ; hence R • m ( H ) is G-unimodular. From equation (2.3.4),

Yp = Dp R and hence, Xp = Np1971Yp = Np R .

If the pair (Xp , Yp ) is also r.c., then there are matrices Vy , U x • m ( H ) , such that

vy rp + ux xp = I ; (2.3.5)

post-multiplying both sides of (2.3.5) by y p l , substituting Np DTl for Xp Yt~ 1 = P and

then post-multiplying both sides by De we obtain

vy Dp + ux Np = rFaD p = R -1 • m ( H ) . (2.3.6)

Since R • m ( H ) and R -1 • m ( H ) by equations (2.3.4) and (2.3.6), we conclude that

R • m ( H ) is an H-unimodular matrix when (Xp , Yp ) isr.c.

The proof of (ii) is entirely similar. I-1

We now consider coprime

• m ( G ) partitioned as

P l l P12

P21 P

factorizations of an (rio + n o ) x ( rii + ni ) matrix

• G(rlo+no) x (Tii+ni) where P e G n°xn i (2.3.7)

Lemma 2.3.5. ( Denominator matrices in triangular form )

Let /~ • re(G) be as in equation (2.3.7); then there exist Nix • H rl° xri~,

N12 ~ Hrio xn i , N21 ~ HnoXrli , Np • H n°xn i , Dll • H "qi xTli, D21 E H n ix~ i ,

Dp e H nixni , and /)II • Hat° Xqo, /~12 • Hri° Xno, ~p • Hnoxno ,

~711 • H I"1° x11i, ~712 e H r l ° x n i , N21 • H naxrli , l~lp • H n°xn i suchthat

12

(Np,oa) =: ( Nil

N21

N12

Np

Dll

I

D21

0 ) is an r.c.f.r, of /~ (2.3.8)

and

( a . , N ) ) =: ( Dll D12

o

NIl

I

N21

N12

;p ) is an 1.c.f.r. of /~ , (2.3.9)

wh~e

( N p , D p ) is an r.f.r, of P , and (Op , /Vp) is an l.f.r, of P .

Comment 2.3.6.

(i) In Lemma 2.3.5 it is only claimed that (Np, Dp ) is an r.f.r, and ( / ) p , Np ) is an 1.f.r. of

the 2-2 sub-block P of P ; these fraction representations of P are not necessarily coprime.

However, Np is right-coprime with Dp by equation (2.3.8) and L~p is left-eoprime with

(ii) Let P = N~ D f 1 = /~1A~p, where ( N~, D~ ) is an r.c. pair as in equation (2.3.8) and

( / )~ , N- b, ) is an l.c. pair as in equation (2.3.9); then

= Ng D f 1 =

Nil N12

N21 Np

pc? o

- o ; ' o ~ o C? o ; ' (2.3.10)

and

/~ = / ~ 1 / ~ =

0

N n N12

(2.3.11)

13

Proof of L e m m a 2.3.5

Since/~ e m(G), it has an r.c.f.r, over H (call it ( X , Y ) ) and an l.c.f.r, over H (call it

(f ,J?)).

(i) By the existence of the Hermite (column-) form [Vid.l, Appendix B], there exists an

H-unimodular matrix R ~ m ( H ) such that Dp := Y R ~ m ( H ) is in the lower-(block-)

triangular form given in equation (2.3.8), where we choose to denote the 2-2 entry of Df by

D e ; note that dot( Y R ) = detY detR = detDf E I . Note also that D 11 and Dp are

lower-triangular though this is not needed in the proof. Let Nf :=X R ~ m(H), where we

denote the sub-blocks in Nf as in equation (2.3.8), with Np ~ m(H) as the 2-2 sub-block.

Since R ~ m(H) is H-unimodular, by Lemma 2.3.4 (i), (N~, Df ) is also an r.e.f.r, of i ~ .

Now equation (2.3.8) implies that det( Y R ) = detDf = detD 11 detDp ~ I ; hence by

Lemma 2.3.3 (ii), detDll ~ I and detDp ~ I ; since P =Np D~ x by equations (2.3.7)-

(2.3.8), we conclude that (Np ,Dp ) is an r.f.r, o fP .

(ii) Equation (2.3.9) can be justified similarly: pre-multiplying Y by an H-unimodular

L ~ m(H), we obtain det/)f := L Y in the upper-(block-) triangular Hermite row-form of

equation (2.3.9); by Lemma 2.3.4 (ii), ( Dr, Nf ) is also an 1.c.f.r. of i~. Since detL detY =

de t /~ = det/~ u det/)p ~ I , by Lemma 2.3.3 (ii), det/~p E I ; hence we conclude that

)isanler of p []

Lemma 2.3.7. ( Generalized Bezout Identity )

Let ( N e , D e ) be an r.c. pair and let ( / ) e , Nt, ) be an 1.c. pair, and let /Vp Dr, = /~p Np,

whereNp ~ Hn°xni,Dp ~ Hnixni,Dp e Hn°xn°,f fp ~ Hn°xnl;thenthereare

matrices Vp ~ H nix ni, Up ~ H nix no, ~p ~. Hni x no, ~p E H n° x no such that

op -ap

(Equation (2.3.12) is called a generalized Bezout identity.)

l ni 0

0 ]no

(2.3.12)

14

Proof

Since ( N p , Dp ) i s r.c. and ( D p , b T p ) i s 1.c., where - N p D e + 19 e Np

matrices Up, V e , i f , V ~ r e ( H )

N e U + D e V = Ino;then

o e - f f

N e

such that V e D e + UpN e

In, Vp U - Up V

0 Ino

I ni

0

= O,thereare

:- I ni and

l no

Let Up :=De (Ve ~ - Up V ) - U and Ve := Np (Vp U - U e C¢ ) + ~ ; t h e n U e ,

e m ( H ) and hence, equation (2.3.12) follows. []

Corollary 2.3.8. ( Generalized Bezout identities associated with bicoprime triples )

Let (Net , D , Net ) be a b.c. triple, where Np r ~ HnoX n, D ~ H n x n ,

Npt ~ H n x ni ; then we have two generalized Bezout identifies:

(i) For the r.c. pair ( N e r , D ) there are matrices Ver ~ H n x n , Upr ~ H n×n°,

,~ ~ Hno x n, ~ e H n°x no, ~ ~ H n x no, ~ ~ Hno x no such that

V?r U?r D - U

Ne,

I n

0 In o

; (2.3.13)

(ii) For the l.c. pair (D ,Npl ) there are matrices Vet ~ H n x n , Upt ~ H nix n,

X ~ H nx ni, y E H nix ni, U ~- H n i x n , V ~ H n x n suchthat

D -Ne t

U V

Vpl X

- Upl Y

I n

0 I ni

(2.3.14)

15

Proof

(i) Since ( Np, , D ) is an r.c. pair, there exists an H-unimodular H-stable matrix, which

[o] [o] we call M r , such that Np r can be put in the Hermite form M r Np r = 0 . Par-

us, tition M r and label it as ; since M r is H-unimodular, ( Y, X ) is an l.e.

f

pair; note that X D = Y N p r . Following similar steps as in the proof of Lemma 2.3.7, there

exist i f , ~7

of the form

m(H) such that equation (2.3.13) is satisfied. Note that equation (2.3.13) is

Mr M7 "1 = I n +no (2.3.15)

(ii) Since (D , Npz ) is an 1.c. pair, there exists an H-unimodular H-stable matrix, which

[ I n 0 ] . Equation (2.3.14) follows along similar steps as in the proof of Lemma 2.3.7

after partitioning M t and labeling it as

pair, where D X

Vpt X

- Up1 Y

and noting that ( X , Y ) is an r.c.

-- Npt Y . Note that equation (2.3.14) is of the form

Ml -I M! = I n + ni • (2.3.16)

I"1

16

Remark 2.3.9. ( Doubly-eoprime-fraetion representations )

(i) Suppose that Q e m(H) is any arbitrary matrix whose entries are in H; then the general-

ized Bezout identity (2.3.12) implies that

= (2.3.17)

equation (2.3.17) is of the form

M M -1 = l ni +no , (2.3.18)

where M a re(H)isH-unimodular.

(|i) The pair ( (Np ,Dp ) , ( / )p ,~Tt, ) ) in the generalized Bezout identity (2.3.12) is called a

doubly-coprime pair. We do not need to assume that Dp and /)p are invertible matrices in

writing equation (2.3.12). Now if ( Nt, , Dp ) is an r.c.f.r, of P and ( L~p ,/Vp ) is an 1.c.f.r. of

P , then the pair ( ( N p , D e ) , (L~p, hTp ) ) in (2.3.12) is called a doubly-coprime-fraction

representation of P and Np Dp -t = L ~ 1 hTp is called a doubly-coprime factorization of P ;

note that in this case, by Definition 2.3.1 (ii) and (v), Dp and Dp are invertible matrices and

furthermore, Dp -1 e m(G) and /~;1 e m(G).

(iii) The pair ( ( Npr, D ) , ( Y , X ) ) in equation (2.3.13) and the pair ( (X, Y ) , ( D, Npl ) )

in equation (2.3.14) are also doubly-coprime pairs; note that we do not need to assume that D

is an invertible matrix in writing equations (2.3.13) and (2.3.14). However, if

(Npr , D , Npl , G ) is a b.c.f.r, of P ~ re(G), where G 6 m(H), then by Definition

2.3.1 (viii),D is aninvertible matrix and furthermore, D -I e m(G). []

17

2,4 R E L A T I O N S H I P S B E T W E E N C O P R I M E F A C T O R I Z A T I O N S

Let (Npr ,D ,Npt ,G ) be a b.e.f.r, o f P ~ m(G). In Theorem 2.4.1 we obtain an

r.c.f.r. (Np , Dp ) and an l.c~f.r. (D-p,/Vp ) for P from (Ner , D , Net , G ) . In Example

2.4.3 we apply Theorem 2.4.1 to the state-space representation of a matrix P that has rational

function entries.

Theorem 2.4.1. ( Doubly-coprime factorizations from bicoprime factorizations )

Let P E m(G) . Let (Np, ,D ,Np~ ,6 ) be a b.c.f.r, of P ; hence, equations (2.3.13)-

(2.3.14) hold for some Vet , Uer , X , Y , V , U , Vpt , Upt , X , Y , U , V ~ m(H).

Under these assumptions,

(NprX + G Y , Y ) =: (Np ,Dp ) isanr.c.f.r, o f P , (2.4.1)

( Y ,X Npt+ Y G ) =: (l~p ,Np ) isanl.c.f.r, o fP . (2.4.2)

Comment 2.4.2. ( Generalized Bezout identity for the doubly-coprime pair which is

obtained from a bicoprime triple )

Let ( Npr ,D ,Npt ) be a bicoprime triple; let G ~ mCH); let Vp, , U p r , X , Y , V , U ,

vpt , vpt , x , Y , u , v ~ m(H) be as in equations (2.3.13)-(2.3.14); then

( (Net X + G Y , Y ) , ( Y , X Npt + Y G ) ) is a doubly-coprime pair. A generalized Bezout

identity for this doubly-coprime pair can be obtained from equations (2.3.13)-(2.3.14) and can

be verified by direct calculation:

I v + rs v,~,. N,~t - rj r.rp,, a rs U,~r

- ' Y lVpt - ~ C ["

r -upj 6

up, x + o r 9 + Npr vpt 6 - a upt 6

In i

0 In,,

(2.4.3)

18

Note the similarity between equations (2.3.12) and (2.4.3). Equation (2.4.3) is of the form

lt~ A~ -1 = l ni +no (2.4.4)

Proof of Theorem 2.4.1

By assumption, P = N p r D - 1 N p t + G and equations (2.3.13)-(2.3.14) hold. Clearly

Npr X + G Y , Y , Y , X Nt, t + Y G ~ m(H) . We must show that ( Npr X + G Y , Y )

is an r.c. pair, where detY ~ I and that ( Y , X N p t + Y G ) is an 1.c. pair, where

detY ~ I :

Equation (2.4.3) implies that ( Npr X + G Y , Y ) is an r.c. pair and ( Y ,Y~ Npt + Y G )

is an 1.c. pair;, more specifically, if ( Nt, r X + G Y , Y ) =: ( N t, , D e ) and ( Y, X Npt + Y G )

=: (/~p ,/Vp ) as in equations (2.4.1)-(2.4.2), then

l lpDp + Up Alp = Ini , Np Up + Op Vp = Ino , (2.4.5)

where

Vp :~- V -I- U Vpr Npl - V Upr G , Up :~- U Upr ,

Up := Upl U , 17p :=17 + Npr Vpl U - G Upl U .

(2.4.6)

Now from equations (2.3.13)-(2.3.16), since M r M r 1 = M r 1M r = I n + no and M1-1 M r =

hit Mt -1 = In + nl , we obtain

[o 0] detD = det( 0 Ino

= det ( All -1 M t [ D

detD 0 /

M r M r 1 ) = det( _ ~ i n °

= detY detMr -a ,

I ni ) = det ( M1-1 0 g

,° 0] 0 Y Mr-1 )

(2.4.7)

1 I n - X U X |

- U Ini ] )

= detM/-1 detY (2.4.8)

19

Since M r , M t c m(H) are H-unimodular matrices, detM r ~ J and detMt -I E J ; furth-

ermore, since detD e I by assumption, equation (2.4.8) implies that

detY = detM/ detD ~ I (2.4.9)

and equation (2.4.7) implies that

detY = detM r detD ~ I . (2.4.10)

Now by equation (2.3.14), Npl Y = D X ; hence,

e Y = (Ne, D-1Npt+ G ) Y = N p r X -t- G Y . (2.4.11)

Similarly, by equation (2.3.13), Y N~r = f f D ; hence,

e = f (NprO-lNpl+ G ) = X Npt + Y O . (2.4.12)

From equations (2.4.9)-(2.4.10), we see that y-1 e m(G) and Y- t e m(G) ; therefore,

equations (2.4.11)-(2.4.12) imply that

P = (NprX + 6 Y ) y - 1 = 7 - I ( f , Np t + f 6 ) ; (2.4.13)

therefore (Np, X + G Y , r ) is an r.c.f.r, of P and ( Y ,X Npt + Y G ) is an l.c.f.r, o f e .

Example 2.4.3. ( Doubly-coprime factorizations from state-space representations )

Let H be R u as in Section 2.2. Let P ~ ~p(S) n°xni be represented by its state-space

representation (.4 , /~ , C , /~" ) , where /~ ~ ~ n x n , ~ ~ R n x n i , ~ E R n°xn

and ~. ~ ~no×nl. Let

e = ( s + a ) - l ~ [ ( s + a ) - l ( s I n - A ) ] - l / ~ +

( A , B ) be 1]-stabilizable and ( C , A ) be

( ( s + a ) - l C , ( s + a ) - l ( S l n - A ) ) is r.c.

( ( s + a ) - l ( s l n - X ) , B ) is I.e.

det [ ( s + a )-1 ( s I n - / ~ ) ] ~ I . Therefore,

- a ~ IR n C \ I ] ; then

= C ( s I n - A ) B +ft . . Let

l]-detectable; then the pair

over m(Ru) and the pair

over m ( R u) ; furthermore

(Npr ,D,Nt , t , G ) := ( ( s + a ) - l ~ , ( s + a ) - l ( s l n - A ) , B , E )

20

is a b.c.f.r, of P over m(Ru). Choose K ~ IR ni×n and F ~ ~ n x n o such that

(.4 - /~ K ) and ( .4 - F C ) have all of their eigenvalues in C\ 1] . Let

A k : = ( s i n - A + B K ) -1 and A / : = ( s i n - ,7 , + F C ) -1 ; (2.4.14)

note that A k , Af ~ m ( R u) n m ( ~ s p ( S ) ) ; since - a ~ C \ I ] , the matrices

( s +a )(s I n - A + B K )-I = ( s +a )A k ~ m ( R u) a n d ( s + a ) ( s I n - A + F ~ )-I

= (s + a ) / i f ~ m ( R u ) are Ru-unimodular. For this special b.c. triple

( ( s + a )-1 ~ , ( s + a )-1 ( s I - /~ ) , /~ ) , equations (2.3.13) and (2.3.14) become:

(s +a ) A f

- C A r I n o - C A y F ( s + a ) - l c Ino

= In +no f2.4.15)

( s + a ) - l ( s l n - A ) -B

(s + a ) - IK lnl

(s +a )At: ( s +a )A k

- K A k l nl - K A k

= In +hi .(2.4.16)

Matching the entries of equations (2.4.15) and (2.4.16) with those of (2.3.13) and (2.3.14),

respectively, we obtain a generalized Bezout identity for this special case from equation

(2.4.3):

Ini +KAfB-KAfFE

-CAfg - ( Ino- CAfF)E

K A / F

l n o - C a f F

I nl - K Ate

CAkB + E( I ni- KAkB )

-KAkF

I no+ CAkF- EKAgF

= I n i + n o (2.4.17)

Comparing the generalized Bezout identities (2.3.12)

( C A k B + E ( l n l - K A k B ) , ( I n i - K A k B ) ) is an r.c.

( ( l no - C A r F ) , C A r B + ( I no - C A r F ) E ) i s an l.c. pair, where

Up = K A y F , Vp =Ini + K A f B - K A f F E ,

and (2.4.17),

pair and

21

Up = K A k F , ~'p = 1no + C A k F - E K A k F .

Since A k , Af ~ mca~sp (s)), we have det ( ln i - K A k/~ ) ~ I and

det(Ino - C A r F ) ~ I . Furthermore, since (s + a )A k is Ru-unimodular,

det(ln~ - K A k B ) = d e t [ ( s + a ) - l ( s l n - A ) ] d e t [ ( s + a ) A k]

det[(s +a )-1 ( s I n - A ) ] ; similarly, since (s + a )At is Ru-unimodular,

det( lno - G A F F ) = d e t [ ( s + a ) - l ( s l n - A ) ] d e t [ ( s + a ) A f ]

d e t [ ( s + a ) - l ( s I n - ,4 ) ] ; therefore det ( I ni - K Ak B ) ~ det(Ino - C A L F ) .

We conclude that

(Np ,Dp ) := ( C A l e B + E( ln~ - KAI~B ) , ( Ini - K A k B ) ) (2.4.18)

is an r.e.f.r, of P over m(Ru) and

( /~p ' /Vt , ) := ( ( lno - C A L F ) , C A f B + (Ino - C A f F ) E ) (2.4.19)

is an 1.c.f.r. o f P over m(Ru). [ ]

Let ( N p , Dp ) , (Dp, Np ) , (Npr , D , Npl , G ) be any r.c.f.r., l.c.f.r, and b.c.f.r, of

e e m(G). By Lemma 2.3.4, any other r.c.f.r, is of the form (Np R ,Dp R ), where

R e re(H) is H-unimodular and any other 1.c.f.r. is of the form (L /~p , L/Vp ), where

L e m(H) is H-unimodular. By Theorem 2.4.1, any r.c.f.r. (Np ,Dp ) =

((Npr X + G Y )R , Y R ) for some H-unimodular R ~ re(H) and any l.e.f.r.

(Dp "Ne ) = ( L Y , L (XNp t + I" G )) forsomeH-unimodularL ~ r e ( H ) .

Suppose that ( (Np ,Dp ), (/~p ,/qp ) ) is a doubly-coprime pair as in equation (2.3.12)

and that (Np , Dp ) = ( (Npr X + G Y ) R , Y R ) for some H-unimodular R ~ m(H),

where X , Y ~ m(H) are as in equation (2.3.14). In Lemma 2.4.4 below, we show that

detDp, det/~p and detD are associates; thus, if any one of detDp , det/~p , detD is in I , then

the other two are also in [ . Consequently, the determinants of any r.c.f.r., any l.c.f.r, and any

h.c.f.r, of P ~ r e ( G ) are associates.

22

Lemma 2.4.4. ( Determinants of denominator matrices of coprime factorlzatlons )

Let ( (Np ,Dp ) , ( /~p , /Vp)) be a doubly-eoprime pair as in equation (2.3.12); let

o e m(H) and let ( N p , Dp ) = ( (Npr X + O Y ) R , Y R ) for some H-unimodular

R ~ r e ( H ) , where equations (2.3.13)-(2.3.14) hold. Under these assumptions,

detDp -- det/)p -- detD , (2.4.20)

and for all Q ~ re(H),

det (Vp - Np Q )

furthermore,

Proof

det ( Vp - Q/vp ) ; (2.4.21)

det[ (Vp - Q N p ) D p ] = det [/)p (Vp - Alp Q ) ] . (2.4.22)

Since (Np , Dp ) and ( L~p , ATp ) satisfy equation (2.3.12), equations (2.3.17)-(2.3.18) hold for

all Q e m ( H ) ; by equation (2.3.17),

M = . (2.4.23)

O t no - ' q . I no O Dp

Taking determinants of both sides of equation (2.4.23) we obtain

detDp detM = det/)p (2.4.24)

Since M ~ m ( H ) is H-unimodular, detM ~ J ; therefore equation (2.4.24) implies that

detDp ~ det/gp (2.4.25)

Now by Theorem 2.4.1, ( Npr X + G Y , Y ) is an r.c. pair and ( Y , X Npt + Y G ) is an 1.c.

pair since (Net ,D ,Nt, l ) is a b.c. triple by (2.3.13)-(2.3.14) and G ~ m(H) . By assump-

tion, (Np , Dp ) = ( (Npr X + G Y ) R , Y R ) for some H-unimodular R ~ m(H) ;

therefore, by equation (2.4.8), which is obtained from equations (2.3.14) and (2.3.16), we see

that

23

detDp = detY detR = detM l detD detR ;

since detM/ ~ J and detR ~ J , equation (2.4.26) implies that

detDp ~ detD

Finally, equation (2.4.20) follows from equations (2.4.25) and (2.4.27).

Now by equations (2.3.17)-(2.3.18),

I ni 0 I nl Up + O Dp

M = o lYp-upo. -Np 1.o-up(vp+eg p)

(2.4.26)

(2.4.27)

Vp-QASp 0

0 Ino

(2.4.28)

Taking determinants of both sides of equation (2.4.28) we obtain

det( lyp - Np a ) deft / = det( Vp - O ASp ) ; (2.4.29)

since M ~ m(H) is H--unimodular, equation (2.4.21) follows from (2.4.29). Now multiply-

ing both sides of equation (2.4.29) by detDp and using equation (2.4.24) we obtain

det( lY e - Np Q )detM detDp = det( lYp - Np Q ) det/gp = det( Vp - Q ASp ) detDp ;

(2.4.30)

hence equation (2.4.22) follows since det( Vp - a ASp ) detDp -- det [ ( Vp - a ASp )Dp ]

• .d ~e~p ~et(~p - Np e )--~et tap (Up - Np Q ) J []

Corollary 2.4.5. (Np ~ m ( G s ) i m p l i e s that detDp ~ I )

Let ( ( N p , Dp ) , ( / ~ p , ASp ) ) be a doubly-coprime pair satisfying the generalized Bezout

identity (2.3.12); let Np E m ( G s ) ; Under these assumptions,

detDp ~ I and detgp E I ; (2.4.31)

furthermore, for all Q ~ H,

det( Vp - Q ASp ) ~ I and det( 17p - Np Q ) ~ I . (2.4.32)

24

Proof

By assumption, (2.3.12) holds; therefore, (2.3.17) also holds for all Q ~ m(H). Now

~vp ~ mCGs)implies that (Up + Q/~p )Np ~ m(Gs)for all Q ~ m ( H ) ; therefore

( ln l - (Up + O Dr, )Ne )-a ~ m c G ) ; equivalently, det(Inl - (Up + Q De )Np )

I ; then by equation (2.3.17),

de t [ (Vp - O N p ) O e ] = d e t [ I n i - ( U p + Q D p ) N p ] E I . (2.4.33)

By Lemma 2.3.3 (ii), equation (2.4.33) holds if and only if detDp ~ I and

det( 17p - Q Ne) ~ I ; since detM ~ J , equations (2.4.31)-(2.4.32) follow from equations

(2.4.24) and (2.4.29). 17

Lemma 2.4.6. ( Denominator matrices of H-s tab le matrices )

Let (Ner , D , Net , G ) be a b.c.f.r, of P ~ m(G) ; then P ~ m(I--I) if and only if

D -1 ~ m(H) ;equiva len t ly , detD ~ J .

Proof

If D -1 ~ m ( H ) then P = Ner D -1 Net + G ~ m ( H ) since Npr, Net, G ~ m(H). To

show the converse, letNer D-1Net + G ~ m(H); thenNer O- l Net = P - G e m ( H ) .

By equation (2.3.14), Net D-I Net Uet = Net D -1 ( I n - D Vet ) =

Net 0 -1 - Ner Vet ~ m(H) and equivalently, Net 0 -1 • Furthermore, by equation

(2.3.13), Uer Ner D -1 = ( I n - Vpr D )D -1 = D -1 - Ver e m ( H ) and equivalently,

D -1 ~ m ( H ) . []

Comment 2.4.7.

(i) Let ( Npr, D , Npl ) be a b.c. triple over m(H) and let G ~ m(H); then following simi-

lar steps as in the proof of Lemma 2.4.6, we can easily show that Npr D -I Npt+ G ~ m ( G )

if and only if D -1 ~ m ( G ) ; but since D ~ m(H), D -1 e m(G) if and only if

detD ~ I .

(ii) Let (Alp , D p ) be an r.c.f.r, and (Dp , N p ) be an 1.c.f.r. of P E r e ( G ) ; then

( Np , D e , I , 0 ) and ( I , / 3 p , / ~ p , 0 ) are bicoprime-fraction representations of P and hence,

by Lemma 2.4.6,

o~ -1 • m(H) .

25

P ~ m(H) if and only if Dp -1 ~ re(H) and equivalently,

(iii) If P e m(H) then Lemma 2.4.6 implies that ( P , Ini ) is an r.c.f.r, and ( lno , P ) is an

l.c.f.r, of P ; by Lemma 2.3.4, any other r.e.f.r, of P is of the form ( P R , R ), where

,~ e m(H) is H-unimodular and any other 1.e.f.r. of P is of the form ( L , L P ), where

L e m(H)is H-unimodular.

(iv) Let (Npr ,D ,Npl ,G ) be a b.c.f.r, o f P ~ re(G), then detD ~ I is a characteristic

determinant o f P [Vid.1, Section 4.3]. It follows from Lemma 2.4.4 that detDp and det/)p are

also characteristic determinants of P . By Lemma 2.4.6, P • m(H) if and only if D is

n-unimodular and equivalently, the characteristic determinant of P is in the group of units J

of H .

(v) Let H be the ring R u as in Section 2.2. Let P ~ m(Rp(S)); letNp Dp -t =/)t~l/Vp =

Net D-l Np t + G be r.c., l.c., and b.c. factorizafions of P . Let the set of l J -zeros of detD be

denoted by

Z [ d e t D ] := { s o • 1~ [ detD (s o ) = 0 } ; (2.4.34)

note that detD (,,~) ~: 0 since detD ~ I . Lemma 2.4.4 implies that

Z [detDp] = Z [det/~p ] = Z [detD ] . (2.4.35)

An element d ~ 1] is a l~-pole o f P ~ m(IRp (s)) i f fd is an I ] -zero of a characteris-

tic determinant of P ; equivalently, d ~ 1] is an H-pole of P iff d E Z [ d e t D ] =

Z [ detDp ] = Z [ det/)p ]. Note that P has no poles at infinity since e ~ m ( R p (s)).

The McMillan degree of d e 1] as a pole of P is, by definition, equal to its multiplicity

as a zero of a characteristic determinant of P . []

2.5

26

ALL SOLUTIONS OF THE MATRIX EQUATIONS

X A = B , A - X = f f

In this section we consider all solutions for X and X over m(H) of the matrix equations

X A = B a n d A X = /~ ,where )~ = /~c /Vc , X = Dc , a = Np

[ - /Vp /)p ] (see (2.5.3) and (2.5.4) below).

Lemma 2.5.1. ( Parametrization of all solutions )

Let ( ( Np, Dp ), (Dp , Np ) ) be a doubly-coprime pair satisfying the generalized Bezout

identity (2.3.12). Consider the equations

~c Op + ~c N, -- B . (2.5.1) and

9 , Nc + g , o~ = g , (2.5.2)

where B e H ni × ni and/~ e H n° × no. Under these assumptions,

(i) ( / )c ,/Vc ) is a solution of equation (2.5.1) over m(H) if and only if

v~

for some O e m(H).

(ii) ( N c, D c ) is a solution of equation (2.5.2) over m(H) if and only if

Dc

o~

(2.5.3)

(2.5.4)

for some Q ~ m(H). []

27

Equation (2.5.3) is a parametrization of all solutions of the pair ( /~c,/Vc ) in (2.5.1) over

m ( H ) ; similarly equation (2.5.4) is a parametrization of all sohtions of the pair (N c, D c ) in

(2.5.2) over m ( H ) .

Proof

(i) ( i f ) Suppose that ( D c , N c ) is as in equation (2.5.3); then by equation (2.3.12),

['-] ( only i f ) By assumption, ( D c , N c ) satisfies equation (2.5.1). Let Q :=

-D~ V~ + N~ Vp ~ m ( H ) ; then

= [ B Q ] . (2.5.5)

Post-multiplying both sides of equation (2.5.5) by the H-unimodular matrix

and using equation (2.3.12), we obtain the solution given by equation (2.5.3).

The proof of part (ii) is entirely similar. []

v.

Remark 2.5.2.

(i) Suppose that B = I ni and B = I no in the matrix equations J~A = B and A X = B ;

then J~ is the left-inverse of A over m(H) and X is the right-inverse of .4 over m(H). In

Lemma 2.5.1, ff B = I n i , then (2.5.1) is a left-Bezout identity for the 1.c. pair (L~ c ,/Vc )

and i f B = I n o , then (2.5.2) is a right-Bezout identity for the r.e. pair ( N c, D c ); in this case,

if in addition L~ c N c = Nc De, then (2.5.1)-(2.5.2) imply that ( ( N c , O c ) , ( D- c ,N-c ) ) is a

doubly-coprime pair, where the associated generalized Bezout identity is:

28

Dp -N,

N, D c

In i

0 Ino

(2.5.6)

Comparing the generalized Bezout identifies (2.3.17) and (2.5.6), from Lemma 2.5.1, all solu-

tions of (2.5.6) over mfH) are given by

(5c .~L ) -- ((v,-Q :7, ).(,, +Q 5, )). (2.5.7)

(N~.~) = ((6, +0, Q ).(~7 -u, Q )). (2.5.8)

where Q ~ m(H).

The matrix Q ~ m(H) in equations (2.5.7)-(2.5.8) is called a (matr/x-) parameter in

the sense that all solutions of (2.5.6) for ( ( N c , D c ), ( Dc , lye ) ) are parametrized by the

matrix Q. Note that if 5 c N c = Nc Dc as in equation (2.5.6), then the (matrix-) parameter

Q ~ m(H) in (2.5.7) is the same as the (matrix-) parameter Q ~ m(H) in (2.5.8).

(ii) Suppose that P ~ m(Gs) and that (/Up, D e ) is an r.c.f.r., (t~ e , ~Tp ) is an l.c.f.r, of

P . Let the generalized Bezout identity (2.3.12) hold; then Np = P Dp ~ m(Gs) and

~Tp = 5 . e ~ m(Gs). By Corollary 2.4.5, ( V , - Q ~7, )-1 e m(G) and

(~7 e -Np Q )-1 e m ( G ) foral la e m ( H ) .

With P e m(Gs), suppose that the pair ( ( N c, D~ ), ( De , Nc ) ) satisfies the gen-

eralized Bezout identity (2.5.6); then the solutions in (2.5.7)-(2.5.8) have the property that

det( Vp - Q ATe, ) e I and deft 17 t, - N r Q) ~ I , (2.5.9)

fora l lQ ~ m(H). Let C :=DZ1Nc=NcD~-l; thenforal lQ ~ m ( H ) , C ~ m(G),

where by equations (2.5.7)-(2.5.8),

c = O : ' N ~ = (v,, - e ~ . ) - l (v , , + e 5 . )

c =uco;'=(6. +o.Q)(~_u.Q)-I (2.5.10)

are l.c. and r.c. factorizations of C ~ re(G).

29

(iii) Now suppose that P ~ m(G) but not in m(Gs); then and not in m(G )

either;, consequently, det( lie - Q Np ) and dot( ~Tp -Np Q ) are not necessarily in I for all

Q ~ m ( H ) . hthiscase, tic = <Vp-Qfip )and D c = (Vp -Np Q ) arevalid denomi-

nator mmrices for C E m ( G ) as in equation (2.5.10) only for those Q ~ m ( H ) such that

condition (2.5.9) is satisfied. One (conservative) way to ensure that condition (2.5.9) will be

satisfied is to choose Q ~ re(H) such that

ATe = ( up + a ffp ) s m ( G s ) ; (2.5.11)

in this case, detff c detOp = de t (1n , - ~7 c N p ) ~ I, which impl i e s that de t /~ c =

(let( Vp - Q / V p ) ~ I and hence, condition (2.5.9) is satisfied. Note that choosing

Q ~ m ( H ) such that (2.5.11) is satisfied guarantees/92 t e m ( G ) and hence, it follows

that N c = D-c-: Nc De is also in m ( G s ).

Note that those Q ~ m ( H ) that satisfy (2.5.11) actually parametrize all solutions of C

in equation (2.5.10) which are in m ( G s ) since C =/~clNc = N c D~ 1 E m ( G s ) ff and

only if/Vc e m ( G s ) and equivalently, N c ~ m ( G s ) .

Choosing a matrix Q e m ( H ) that satisfies (2.5.11) can be reduced to a simple

(scalar) sufficient condition as follows: Choose q ~ H such that

1 + q (det/~,) ~ G s , (2.5.12)

and take

QO := q ( det/~p ) Up/~-I ; (2.5.13)

note that (det/~t,)/~t~ 1 ~ H and hence, QO ~ m ( R u ) ; QO satisfies condition (2.5.11)

since ( I + q ( d e e p ) ) Up ~ m ( G s ) .

(iv) Suppose that H is the ring Rtl as in Section 2.2. I f P ~ m ( R s p (s)), then condition

(2.5.9) is satisfied for all Q ~ m(Ru) . If P ~ m(l~p (s)) but not in m(l% (s)), then

condition (2.5.11), which guarantees (2.5.9), is satisfied if and only if /Vc = ( Up + Q/~p )

30

• m ( H ) • mtRsp(s)) ; equivalently, Q • m ( R u) is such that

Q ( o o ) = - U p ( o o ) / ~ 7 1 (oo ) , (2.5.14)

where det/)p(*o) ~:0 sincedet/~p • I by Definition 2.3.1 (ii). Choosing Q • m(Ru)as

in equation (2.5.14) is a sujOicient condition for (V o - Q / V o )-1 E m0Rp(S)) and

equivalently, (170 - N o Q )-1 • m ( R p ( S ) ) . It is important to note that the matrices

C ~ m(Rp(S) ) that satisfy equation (2.5.10) are in m~sp(S)) if and only ff

Q e m ( R u ) ischosen as in (2.5.14).

(v) Suppose thatP • r e ( H ) ; then following the discussion in Comment 2.4.7 (Hi), an r.c.f.r.

can be chosen as ( P , ln i ) and an 1.c.f.r. can be chosen as (Ino ,P ) ; hence in the general-

ized Bezout identity (2.3.12), we can choose V o = In~ , 17o = In o , Up = Up = O. In

this case the solutions in (2.5.7)-(2.5.8) can be replaced by

(De ,Nc ) = ( ([ni - Q P ) , Q ) , (2.5.15)

(No,De) = (Q , (Ino - P Q ) ) , (2.5.16)

where Q • m(H).

If t ' e m ( H ) c~ m ( G s ) , t h e n

det(lni - Q P ) = det(lno - P Q ) e I (2.5.17)

for all Q a m(H); therefore,

C := ( In i - Q P ) - I Q = Q ( lno _ p Q)-I • m ( G ) , (2.5.18)

fora l lQ e m(H);furthermore, C • m ( G s ) if and only if Q e m(H) :~ m(Gs).

I fP • m ( H ) but not in m(Gs ) , then choosing Q • m ( H ) c > m ( G s ) is sufficient

to satisfy (2.5.17).

(vi) In Lemma 2.5.1, suppose that we started with an 1.c. pair ( D c , N c ) and an r.e. pair

(N c , D e ) satisfy the following generalized Bezout identity:

31

Dc - Uc In o

0 I ni

(2.5.19)

In this case, equation (2.5.1) is of the form/~X = B and equation (2.5.2) is of the form

["] I I I l f A = ~, whereX = D, , f = 5,, ~,, , ~ = - ~ 5~ , A =

Nc . Under these assumptions, ( N v , Dp ) is a solution of equation (2.5.1) over m(H)

if and only ff

-N~

G

D c -

N~

-4

B

(2.5.20)

for some Qp e mfH); s~arly, (Sp ,~p ) is a solution of equation (2.5.2) over mfH) if

and only if

V~

] (2.5.21)

for some Qv ~ m<H). t2

32

2.6 R A N K C O N D I T I O N S F O R C O P R I M E N E S S

In this section, the principal ideal domain H under consideration is the ring Ru of proper

stable rational functions as in Section 2.2.

Lemma 2.6.1. ( Rank test for right- or left-coprimeness )

(i) I~tN/ , E RunoX nl , Dp E R u nix nl ; then ( Np , D e ) is r.c. if and only if

ra [ o,s,] NI, ( s ) = ni ' for all s e IJ . (2.6.1)

(ii) I.,¢t/gp ~ R u n°xn° ,/qp ~ R u n°xni ; t hen (Dp ,Np )isl .c. if and only ff

r a n k [ D p ( s ) / V , ( s ) ] = n o , forall s ~ 1J . (2.6.2)

Q

Note that the rank tests for right-coprimeness and lefi-coprimeness in (2.6.1) and (2.6.2),

respectively, need to be performed only at the l l -zeros of detDp (equivalently, at the l~-zeros

of detL~p ), since these rank conditions hold automatically for all other s ~ L1 .

Proof

(i) (Np , D e ) is an r.c. pair if and only ff there is an Ru-unimodular m a r x E (labeled as

v~ vp ) such that

Vp (s ) Vp (s ) De(s)

Np(s)

[ ni

0

; (2.6.3)

since the matrix E has rank n i + n o for all s ~ U , equation (2.6.3) holds if and only if the

rank condition (2.6.1) holds.

33

(ii) Similar to part (i): the pair (D~, ATp ) is I.e. if and only if there is an Ru-unimodular

or -a. matrix F (labeled as ) such that

Dp(s) -Up( s )

Np(s) rTp (s ) = [ 0 Ino] ; (2.6.4)

since the matrix F has rank n i + n o for all s ~ (_1, equation (2,6.4) holds if and only if the

rank condition (2.6.2) holds.

Let

O

max rank M ( K ) (2.6.5) K E K

denote the maximum rank that the matrix M ( K ) achieves as K varies over the set K.

L e m m a 2.6.2

Let A ~ C TIxT, B ~ C px7 , ,4 ~ C f f x f , /~ ~ c f fXff

matrices.

(i) If for aU K E R p x q ,

rank[ B + K A ] <min { P , T } , then

rank A = max x ~ m e a )

be complex constant

(2.6.6)

rank[ B + K A ] . (2.6.7)

(ii) If for all /~

then

rank [ E

rank [ lff X

+ , 4 / ~ ] < ra in { ~ , ~ } , (2.6.8)

] = max rank[ /~ + / ~ / ¢ ] . (2.6.9)

34

Proof

We only prove part (i), the proof of (ii) is entirely similar.

(i) Let /~ be a p x r l real matrix that maximizes rank [ B + K A ] ; let r :--

rank [ B + K a ] ; by equation (2.6.6), this rank, r < rain { p , y } . Therefore, there is

a nonsingular complex matrix R • C 7xY, which corresponds to elementary column opera-

tions on complex matrices, and there is a nonsingular real matrix L e IRP x P, which

corresponds to row permutations, such that

[.0] (2.6.10)

where G is a ( p - r ) x r complex matrix and the zero in the bottom right is

( p - r ) x ( 7 - r ) ,with r <re.in { P , 7 } •

Let A R =: [ , 4 / ~ ] , where A • C 11xr and ,~ e cTIX(7-r): then

L ( B + / ~ A ) R ] A R =

I r 0

G 0 Now since R is a maximizer of

rank[ B + K A ] , f o r a l l K,2 e P , ( P - r ) x r l

o

r a n k ( L [B + ( K + L -1 ^ )A JR) = r a n k ( L ( B + K A ) R +

K2

AR) K2

[" ] ^0 ^ = rank G + K2 ~ K 2 A < r . (2.6.11)

By equation (2.6.11), rank K2 ~ = 0 for all /~2 and hence,/~ is the T l × ( y- r ) zero matrix.

Therefore,

35

rank A = r a n k ( 0 I A R ) = rank

I r 0

G 0

X o .= r

=ran [B+Ra] = max ra [B÷KA] r • m o R )

[]

Corollary 2.6.3

Let (Np , D p ) be an r.c.f.r., (Dp ,/Vp) be an 1.c.f.r., (Npr ,D ,Npt ) be a b.c.r.f, of

e ~ m(Rp(S)), where Np. Dp , /~ . , /Vp, Npr, D , Np, ~ m ( R u ) . Under these

assumptions, for each s o ~ t l ,

(i) there exists a real constant matrix K ~ R nix no such that

rank [ Dp (s o ) + K Np ( s o ) ] = ni ; (2.6.12)

(ii) there exists a real constant matrix /~

r a n k [ / ) , ( so )

E IR n ix no such that

+ Np ( So ) K ] = n o • (2.6.13)

Note that Dp ( s o )

(2.6.14)impliesthatrank [ Dp(s°)]Np(so)

that the pair ( Np , Dp ) is right-coprime.

Proof

We only prove part (i); the proof of (ii) is similar.

(i) Suppose, for a contradiction, that there is an So ~ t l such that, for all K ~ • n i × no ,

[ Op ( s o ) + K Np ( s o ) ] < n i (2.6.14) rank

C nlxni and Np ( s o ) ~ C noxni . By Lemma 2.6.2, equation

< n i ; but by Lemma 2.6.1, this contradicts the fact

[]

Chapter 3

FULL-FEEDBACK CONTROL SYSTEMS

3.1 I N T R O D U C T I O N

This chapter studies linear, fime-invariant (lti), multiinput-multioutput (MIMO) control

systems with full-feedback compensators: in particular, the classical unity-feedback system A

S( P , C ) (see Figure 3.1) and the more general system configuration Z(P, C ) (see Figure

3.9) are considered.

In the unity-feedback system S ( P , C ) , the plant has only one (vector-)input e and one

(vector)-output y ; this output is used in feedback to the compensator;, hence, the plant model

considers only additive inputs or disturbances (say u , u" ), which all affect the plant through

its actuators. More generally, however, there may be inputs (for example, disturbances, initial

conditions, noise, manual commands) which are applied directly to the plant without going

through the actuators; hence, the map from the directly applied inputs to the plant output may

be different from the map from the additive inputs to the plant output. Furthermore, the regu-

lated output variable of the plant (for example, tracking error, actuator states) may not be

accessible (e.g. some plant states), may not be directly measured (e.g. tracking error) or may

be different from the measured output (e.g. ideal sensor outputs); the measured output is util-

ized by the compensator. The configuration Z(/~, C ) represents a feedback connection which

takes such eases into account.

The unity-feedback configuration S ( P , C ) is studied in Section 3.2. Compensator

design using the configuration S ( P , C ) is called one-degree-of-freedom design (or one-

parameter design) due to the single free parameter matrix Q that parametrizes all

37

H-stabilizing compensators. The class of all achievable input-output (I/O) maps for

S ( P , C ) is obtained by using the class of all stabilizing compensators; this class is given L,

equation (3.2.58); all closed-loop I/O maps in the H-stabilized S ( P , C ) are affino maps in

the (matrix-) parameter Q . The problem of diagonalizing the FO map Hy u, in the

configuration S ( P , C ) is discussed for H-stable plants in Section 3.2.15.

The important theorems in Section 3.2 are Theorem 3.2.7 (H--stability conditions for

S (P , C ) in terms of coprime factorizations of the plant and the compensator), Theorem

3.2.11 (parametrization of all compensators that H-stabilize the plan0 and Theorem 3.2.16

(compensators that achieve decoupling for H-stable plants).

The system configuration Y.(P, C ) represents the most general intereonnection of two

systems, a plant/~ and a compensator C . This general system configuration is studied in Sec-

tion 3.3; the plant and the compensator each have two (vector-)inputs and two (vector-)outputs.

The measured output y of /~ is used in feedback, but the output z is the actual output of the

plant (the output in the performance specifications); the output signals z and y are not the

same. The input v is considered as a disturbance, noise or an external command applied

directly to the plant. The compensator output y ' , which is utilized by the plant in feedback, ,A,

can be considered as the ideal actuator inputs; the output z" of C can be used for perfor-

mance monitoring or fault diagnosis. The input v ' of C is considered as the independent

control input; for example, commands or initial conditions. The signals u and u", which

appear at the intereonnection of /~ and C model possible additive disturbances, noise,

interference and loading.

In the configuration E(P, C ) , intuitively only those plants which have "instabilities that

the feedback-loop can remove" can be considered for H-stabilization; these plants are called

Y_,-admissible. The restriction on the class of H-stabilizable i ~ is due to the feedback being

applied only through the second input e and the second output y . The 2-2 block of C is

essentially in a feedback configuration like S( P, C ) of Section 3.2; hence the set of all C that

H-stabilizes the feedback-loop is the same as for S ( P , C ).

38

The class of all C that H-stabilizes/~ is paramctrized by four H-stable (matrix-) param-

eters and hence, compensator design using the configuration 2;(/~, C ) is called four-degrees-

of-freedom design (or four-parameter design). The configuration ~ (/~, C ) can obviously be

reduced to two-parameter design by taking Cll = 0 and C12 = 0 ; but ~(/~, C ) is clearly

much more advantageous and general than two-degrees-of-freedom design with a two-input

one-output compensator [see, for example, Vid. 1]. The class of aU achievable maps for

Y-(/~, C ) , given by equation (3.3.70), involves the four compensator (matrix-) parameters

QI1, QI2, Q21, Q ; each closed-loop I/O map achieved by the H-stabilized ~ ( P , C )

depends on one and only one of these four (matrix-) parameters. Clearly, several independent

performance specifications may be imposed on the closed-loop performance of Y.(P, C ) . For

example, decoupling the I/O map Hzv, = Nl2 QR1 is independent of the I/O maps that are

affme functions in Q . On the other hand, in the unity-feedback configuration S ( P , C ),

diagonalizing the map Hyu, : u ' b-~y would depend on the choice for Q such that

N e ( tie + Q/~e ) is diagonal, and hence, diagonalizing the map Hy u, in S ( P , C ) may not

be possible for certain plants.

The problem of diagonalizing the closed-loop I/O map H~, from the external-input v '

to the actual-output z for the plant/~ is solved in Section 3.3.15; the !/O map Hzv, in the

configuration Y~(/~, C ) can always be diagonalized while preserving closed-loop stability.

The achievable I/O maps of a two (vector-) input two (vector-) output plant/~ (described

by its state-space representation) are calculated in Example 3.3.17.

The important theorems in Section 3.3 are Theorem 3.3.5 (H-stability conditions for

Y.(P, C ) in terms of coprime factorizations of /~ and C ), Theorem 3.3.9 (conditions for

Y-r-admissibility of/~ ) Theorem 3.3.10 (parametrization of all compensators that H-stabitize

/~ ) and Theorem 3.3.15 (class of all achievable diagonal Hzv, ); the compensators that diago-

nalize the I/O map H~,, from the independent control input v ' to the actual output z are

specified by equation (3.3.80).

39

3.2 T H E S T A N D A R D U N I T Y - F E E D B A C K S Y S T E M

In this section we consider the linear, time-invariant unity-feedback system S(P, C )

shown in Figure 3.1, where P :e boy represents the plant and C : e' boy' represents

the compensator. The externally applied inputs are denoted by g := u' , the inputs to the

plant and the ~mpensator are denoted by ~" := e ' , the plant and the compensator outputs

[] Y • the closed-loop input-output (I/O) map of S ( P , C ) is denoted are denoted by ~":= y , ,

by H~. : ~" ~ ~- .

U t e p

C

u

I+ P

Y

Figure 3.1. The unity-feedback system S ( P , C ).

3.2.1. Assumptions on S ( P , C )

(i) The plant P e GnoX ni .

(ii) The compensator C e G n ix no

(iii) The system S ( P , C ) is well-posed; equivalently, the closed-loop input-output map

H~-~ : ut ~ yp

Note that whenever P satisfies Assumption 3.2.1 (i), it has an r.e.fx., denoted by

( Np, O v ), an Lc.f.r., denoted by ( D v , N v ) and a b.e.f.r., denoted by ( N v r , D , Nv~, O ),

40

where Np ~ H n°×ni , Dp ~ H ni×ni , Dp ~ H n°×n° , Np ~ H n°×nl ,

Np r ~ Hno × n , D ~ H n × n , Np t ~ H n × ni. Hence, the generalized Bezout identities

(2.3.12), (2.3.13), (2.3.14) are satisfied for some Tip, Up, Vp , U p , Vt, r , Upr , X , Y , V ,

F ,v~,t ,up~ , x ,Y , u , v e m ( H ) .

Similarly, whenever C satisfies Assumption 3.2.1 (ii), it has an l.c.f.r., denoted by ( /)c ,/Vc )

and an r.c.f.r., denoted by ( N o , D e ) , where Dc ~ H m × n i , N,: ~ H m x n o ,

N c E H n ixn° ,D c E H n°xn° .

If S ( P , C ) is a lumped, continuous-time, linear, time-invariant system, then the princi-

pal ideal domain H under consideration is the ring of proper stable rational functions R u as in

Section 2.2; in that case we assume that P and C ~ m(l~p (s)).

3.2.2. Closed-loop input-output maps of S ( P, C )

Let Assumptions 3.2.1 hold; then the system S ( P , C ) in Figure 3.1 is described by:

I 0 Inl ]

F = ff + _In , , 0 Y ; (3.2.1)

[,o] ~-= o c ~

Substituting for b- from equation (3.2.1) into (3.2.2), we obtain

(3.2.2)

[,no ol o1[ o ,n ] 1 ( 0 Ini - 0 C -In, , 0 ) y" = 0 C u" (3.2.3)

o] o11o Writing ( 0 I ni - 0 C - I no 0 0 C as

-1 -1

" ° "1 [ ° " C Ini ln¢ 0 - I n i 0 "

41

it is easy to see that the !/O map H ~ : ff ~ y is in m(G) if and only if

C Ini ~ m ( G ) ; (3.2.4)

equivalently, Hy~ ~ m ( G ) ff and only if (In. + C P )-1 ~ m(G) if and only if

(Ino +P C )-1 E m(G). Note that (Ini +CP )-1 e m(G) if and only if

det(Ini +CP ) = det(lno +PC ) is a unit in G. In the case that S ( P , C ) is a lumped,

continuous-time, linear, time-invariant system where P , C ~ mORsp (s)) , condition (3.2.4)

is equivalent to

det( l ni + C ( ** ) P ( ** ) ) = det( I no + P ( ~, ) C ( ,,~ ) ) --k O . (3.2.5)

Now since Assumption 3.2.1 (iii) holds, condition (3.2.4) is satisfied. Therefore the

[ u ] [ Y ] i s i n m ( G ) where Hy~ is given in terms of closed-loop I/O map Hy- a : u ' ~ y '

(lni + C P )-1 in equation (3.2.6) and in terms of ( Ino + P C )-I in equation (3.2.7) below:

P ( I n i + C P ) -1 P ( I n i + C P ) - I C

H ~ = ; (3.2.6)

( I n i + C p ) - l - I n l ( I n i + C p ) - I c

( I n o + P C ) - I P ( I n o + P C ) ' q P C

- C ( I n o + P C ) - I P C ( I n o + P C ) - I

(3.2.7)

The equivalent expressions (3.2.6) and (3.2.7) for Hy~ are obtained from (3.2.3) and using the

following well-known matrix identities:

P ( I n i + C P ) -1 = ( l n o + P C ) - I P , ( l n i + C P ) - l C = C ( I n o + P C ) -1 , (3.2.8)

i n i _ ( i n i +C P )-lC p = ( ini +C P )-I = l n i _ C ( i n o + P C ) - lp . (3.2.9)

42

Definition 3.2.3. ( H-stability of S (P , C ) )

Thesystem S ( P , C ) i s sa id tobe H-stable iff Hy~ e m(H). []

If S ( P , c ) is a lumped, continuous-time, linear, time-invariant system, then the princi-

pal ideal domain under consideration is R u as in Section 2.2; therefore we say S ( P , C ) is

Ru-s table instead if H-stable.

Note that by Definition 3.2.3, the well-posedness of S( P , C ) is a necessary condition for

its H-stability. Each of the four transfer matrices in H ~ must be in m ( H ) for the closed-

loop system to be H-stable; in the case that P ~ m ( H ) as in Lemma 3.2.4 or in the case

that C e m ( H ) as in Lemma 3.2.5 below, checking the H-stability of S( P , C ) is reduced

to checking only one of the four matrices in Hy-ff :

Lemma 3.2.4. ( Closed-loop stability when the plant is H-s tab le )

Let Assumptions 3.2.1 hold and let P • m ( H ) ; under these assumptions, S ( P , C ) is

H-stable if and only if

Hy,~, := ( I nl + c J" y1 c ~ m ( H ) . (3.2.10)

Proof

( only if) By Definition 3.2.1, if S ( P , C ) is H-stable, then/-/y-~ e m ( n ) ; therefore

Hy,~, ~ m(H).

( if ) By assumption, Hy, u, • m ( H ) and P • m ( H ) ; using the matrix identities

(3.2.8)-(3.2.9) in the expression (3.2.6) for H ~ we obtain Hy u := P (Inl +C e ) - I ~.

P [ l n i - ( l n i + C p ) - l c ] = P [ l n i - H y , u , P ] ~ m ( H ) ; H r , , : = P ( I n I + C P ) - I C =

~' ny,~, ~ m ( H ) ; ny,~ := (Ini + c P )-l-lni =-(]ni + C e )-1 C P = -Hy,u,P

m(H). Therefore,

I and hence, S ( P , C ) is H-stable. []

m(H) (3.2.11)

43

Lemma 3.2.5. ( Closed-loop stability when the compensator is H-stable )

Let Assumptions 3.2.1 hold and let C • m(H) ; under these assumptions, S ( P , C ) is

H-stable if and only if

Hy u := P ( I n i + C P )-1 • m(n) . (3.2.12)

Proof

( only i f ) By Definition 3.2.1, if S ( P , C ) is H-stable, then H ~ ~ re(H) ; therefore

Hy. re(H).

( /f ) By assumption, Hy u • m(H) and C • m(H) ; using the matrix identities

(3.2.8)-(3.2.9) in the expression (3.2.7) for H ~ we obtain

Hy u H:y~,C Hy~ = - C Hy u ( In , - C Hy u ) C • m ( H ) (3.2.13)

and hence, S ( P, C ) is H-stable. []

If Assumptions 3.2.1 hold and if both the plant and the compensator arc H-stable, then

S(P,C) is H-stable if and only if (Ini+Cp)-l• m(H), cquivaicnfly,

( l n o + P C )-1 • m(H) ; i.e., when when P e m(S) and C • m(H), S ( P , C ) is

H-stable if and only if deft I nl + C P ) = det( I no + P C ) • J.

3.2.6, Analysis ( Descriptions of S ( P , C ) using coprime factorizations )

We now analyze the unity-feedback system S ( P , C ) using coprime factorizations over

m(H) of the plant and the compensator a'ansfer matrices; this analysis leads us to the charac-

terization of closed-loop H-stability in terms of coprime factorizations and the parametriza-

tion of all H--stabilizing compensators such that the closed-loop system is H-stable.

Assumptions 3.2.1 hold throughout this analysis.

44

(i) Analysis of S ( P , C ) with P = Np Dp -x and C = / ) c 1 b~ e

Let (Alp ,Dp ) be any r.c.f.r, of P e m(G) and let (D e ,N c ) be any l.c.f.r, of

C • m(G). The system S(P,C ) in Figure 3.1 can be redrawn as in Figure 3.2 below,

whereP = Np1971 andC = De -IN c;notethat Dp~p = e, y =Nt,~p ,where ~p

denotes the pseudo-state of P .

U t

+

r !

,-,e' i [ S-I V-L-q

U I - . . . . . . . . .

I e :1

I____ .a I

Figure 3.2. S ( P , C ) with P = Np Dp -1 and C : /~-1/Vc •

The system S ( P , C ) is then described by equations (3.2.14)-(3.2.15):

Np y 0 [o, [ 'n Equations (3.2.14)-(3.2.15) are of the form

Dttl {p = NHL 1 ff

N n R l ~p = Y - G n l ~ .

1[ u] b~c u ' ' (3.2.14)

olin] 0 u" (3.2.15)

DH I ] By Lemma 2.3.2, performing elementary row operations over m(H) on the matrix NHR 1

and elementary column operations over m(H) on the m a ~ [ N.,1 0.1 ] ,we conclude

that (NHR 1 ,DH1 ,NHL 1 ) is a b.c. triple. Since DHx, GH1 E m(H), it follows from Corn-

ment 2.4.7 (i) that

45

Hy~ = NHRI Dffl NHI.I + GH1 ~ m(G) (3.2.16)

(equivalently, the system S ( P , C ) is well-posed) if and only if detD H 1 ~ I . Since Assump-

tion 3.2.1 (iii) holds, condition (3.2.16) is satisfied and hence, detDH1 ~ I . Consequently,

(N//R 1, DH 1, NItL 1, Gtt 1 ) is a b.c.f.r, of Hy~ and hence, detD H 1 is a characteristic deter-

minant of H ~ .

(ii) Analysis of S ( P , C ) with P = /~-1 Nv and C = N c D c 1

Let (/gv '/Vv ) be any 1.c.f.r. of e ~ m(G) and let (N c, D c ) be any r.c.f.r, of

C e m(G). The system s ( P , C ) in Figure 3.1 can be redrawn as in Figure 3.3 below,

where P =/~1/Vv and C = Nc D ; 1 ; note that Dc ~c = e ' , y" = N c ~c, where ~ denotes

the pseudo-state of C .

U t

+

r" U

r

+ e i I

Figure 3.3. S ( P , C ) w i t h P = /~-INv and C = N cDc -1 .

The system S( P , C ) is then described by equations (3.2.17)-(3.2.18):

- D e Y 0

Equations (3.2.17)-(3.2.18) are of the form

1[ u] /~t' u" " (3.2.17)

0 u ' " (3.2.18)

46

DH2 ~c = NHL2 ~

NHR2 L ---- Y -- GH2 ~ •

As in Analysis 3.2.6 (i) above, it can be easily verified that (NnR 2, DH2, NHL2 ) is a b.c. triple

and that Hyg = NHR2DffI2 NHL 2 + GH2 E m ( G ) if and only if dctDH2 E I . Again by

Assumption 3.2.1 (iii), H~-~ ~ r e ( G ) and hence, detDH2 ¢ I . Consequently,

(NHR2, DH2, NHL 2 , GH2 ) is a b.c.f.r, of Hy~- and hence, detDn2 is a characteristic deter-

minant of/ /y-~.

(iii) Analysis of S ( P , C ) with P = Npr D-1 Np i + G and C = D e I N c

Let ( N ~ ,D ,Npt , G ) be any b.c.f.r, o f P ~ m ( G ) and let ( /)c ,No ) be any 1.c.f.r.

of C e m ( G ) . The system S( P , C ) in Figure 3.1 can be redrawn as in Figure 3.4 below,

note that D ~x = Npl e , y = Nt, r ~x + G e , whe,e P = N, , n -1 N,, + o and c -- & l ~7o

where ~ denotes the pseudo-state of P . "l

I

r , u I ~ - ~ I

u" e" : I - 'Z - " 7 [ - " 2 ~ : y" ~ + e ' ' ' ' ' + - Nc D c'

P II

Figure 3.4. S( P , C ) with P = N~r D-1 Npt + G and C = /~1 /~c -

The system S ( P , C ) is then described by equations (3.2.19)-(3.2.20):

D -Np!

y"

Npt 0

- co u I ' (3.2.19) u'

= - . (3.2.20)

0 Ini y" y" 0 0 u"

4 7

Equations (3.2.19)-(3.2.20) am of the form

DH3 ~3 = NnL3

NHR3[3 = Y - Gx3~ •

As in Analysis 3.2.6 (i) above, it can be easily verified that (NHR 3, DH3, NHL 3 ) is a b.c. triple

and that Hy~ = NHR3D~] NnL3 + GH3 ~ m ( G ) ff and only ff detDH3 ~ I . Again by

Assumption 3.2.1 (iii), Hy~ ~ m ( G ) and hence, detD#3 ~ I . Consequently,

(NltR3 ,DH3 ,Nm.a , GH3 ) is a b.c.f.r, of H ~ and hence, detDH3 is a characteristic deter-

minant of Hy~.

(iv) Analysis of S ( P , C ) with P = Nj,r D-1 Np I + G and C = N c Dc -1

Let ( N F , O ,Npl , a ) bc any b.c.f.r, o f P e m(G) and let ( N c , D c ) be any r.c.f.r.

ofC ~ m ( G ) . Thesys temS(P ,C) inFigurc3 .1canbcredrawnas inFigure3 .5be low,

whcr~ P = Npr D -! Npl + G and C = N c Dc "-1 ; note that D ~x = Npl e , y = N F ~x + G e ,

where ~ denotes the pseudo-state of P and D c ~ = e ' , y" = N c ~ , where ~ denotes the

pseudo-state of C.

. . . . . . . . . . . . . . . . . . . . . . . . "1

I

r . . . . . . . . . . . ~ U

r I I • u" e , I - ~ - - I ~ 1 - ~ - I , y ~ + e

I ; c I I J L

Figure 3.5. S ( P , C ) with P = Npr D-1 Ne t + G and C = N¢ D~ -1 .

The system S( P , C ) is then described by equations (3.2.21)-(3.2.22):

48

D

Npr

- Npl Nc

De+ G N c ~c

Npl

- G

0

I no

u] U p

(3.2.21)

N•r

0 I °llul Nc ~c y" 0 0 u ' (3.2.22)

Equations (3.2.21)-(3.2.22) are of the form

DH4~.~4 = NHL4ff

NxR4 ~ = Y - OH4 ~ •

As in Analysis 3.2.6 (i) above, it can be easily verified that ( NHR4, DH4 , NHL 4 ) is a b.c. triple

and that Hy~ = NHRnDff~ NHL 4 + GH4 ~. mfG) if and only if detDH4 ~ I . Again by

Assumption 3.2.1 (iii), Hy~ ~ m(G) and hence, detDH4 e I . Consequently,

(NHR4,DH4,NHL4, GH4) is a b.c.f.r, of Hy-ff and hence, detDH4 is a characteristic deter-

minant of H ~ . []

Theorem 3.2.7. ( H-stability of S ( P , C ) )

Let Assumptions 3.2.1 (i) and (ii) hold; let (Np , Dp ) be any r.c.f.r., ( /gp, /Vp ) be any l.e.f.r.,

(Npr, D , Net, G ) be any b.c.f.r, over re(H) of P ~ re(G) ; let ( /~c , Nc ) be any 1.c.f.r.,

(N¢, D c ) be any r.c.f.r, over re(H) of c ~ m(G). Under these assumptions, the follow-

ing five statements are equivalent:

(i) S ( P, C ) is H-stable ;

(ii) DH1 := [ /~c Dp + /Vc Np ] is H-unimodular ; (3.2.23)

49

(iv) Dn3 :=

D -Npt

;oN.r ao+;o is H-unimodular ; (3.2.25)

(V) Dtl 4 :=

O -N.,

Npr m c + O N c

is H-unimodular. (3.2.26)

[3

Note that each of statements (i) through (v) of Theorem 3.2.7 implies that the system

S ( P , C ) is well-posed; consequently, we do not need to state a well-posedness assumption in

the beginning of Theorem 3.2.7.

Proof

We prove the equivalence of statements (i) and (ii) of Theorem 3.2.7 using the system descrip-

tion in Analysis 3.2.6 (i): Suppose that Np D~ 1 is an r.c. factorization of P and/9c 1/qc is an

1.c. factorization of C ; then S ( P , C ) is described by equations (3.2.14)-(3.2.15). If

S( P , C ) is H-stable, then Hy-ff e r e ( H ) and hence, condition (3.2.16) holds; equivalently,

detOu, ~ I ; therefore (NHt¢I, DH1, NHL l , GH1 ) is a b.c.f.r, of Hy~. By Lemma 2.4.6,

Hy~ e m ( H ) implies that Dff~ a r e ( H ) . Conversely, if condition (3.2.23) holds, then

Dff] ~ m ( H ) andhence, ny~=NHR1Dff~ NHLI+GH1 ~ m(H).

The equivalence of statement (i) to any of Off), (iv) or (v) follows similarly from Analysis

3.2.6 (ii), (iii) and (iv), respectively. I-3

Definition 3.2.8. ( H-stabilizing compensator C )

(i) C is called an H-stabilizing compensator for P ( abbreviated as: C H-stabilizes P)

iff C e G ni x no and the system S ( P , C ) is H-stable.

(ii) The set

S ( P ) : = { C ] C H-stabilizes P }

is called the set of all H-stabilizing compensators for P in the system S( P , C ).

50

Coro l lary 3.2.9

Let Assumption 3.2.1 (i) hold; let (Np ,Dp ) be any r.c.f.r, and (/gp ,/Vt, ) be any 1.c.f.r. of

P e m ( G ) . Under these assumptions, the following four statements are equivalent:

(i) C H-stabilizes P ;

(ii) an 1.c.f.r. (/9c ,/Vc ) of C E m ( G ) satisfies

Gn. +#oN. ; (3.2.27)

(iii) an r.c.f.r. (N c, D c ) of C ~ m(G) satisfies

NpNc + DpOc : I n o ; (3.2.28)

(iv) a doubly-coprime-fraction representation ( (Nc ,Dcr ) , (Dc ,/Vc ) ) of C

m ( G ) satisfies

D. -N¢

Np D~

I ni

0

0

] no

(3.2.29)

P r o o f

Suppose that statement (i) of Corollary 3.2.9 holds; then by Definition 3.2.8 (i), C ~ m(G).

Let (/~* ,/V~ ) be any 1.c.f.r. and (De* , N c ) be any r.c.f.r, of C ; then by Theorem 3.2.7,

/9~ Op +/V~ Np =: L ~ m ( H ) is H-unimodular. By Lemma 2.3.4 (ii), (/~c ,/~c ) :=

( L - I / ) ~ , L -1/V~ ) is also an 1.c.f.r. of C ; but ( / ) c , Nc ) satisfies equation (3.2.27) since

( L -1D~ ) D e + (L -t N~ ) Nt, = In i , and hence, statement (ii) of Corollary 3.2.9 holds.

Now equation (3.2.27) implies that

d e t ( D c D t , + N c N t , ) = 1 = det/)c det(Inl + C P ) d e t D p . (3.2.30)

51

By Lemma 2.4.4, detDp ~ det/~ e and det/) c - - detD*; since det(Inl + C P ) =

det( I no + P C ) , equation (3.2.30) implies that

det/)p det( Ino + P C )de t / ) : = ae t ( / ) e D : + ; e N : ) - - 1 (3.2.31)

and therefore, (D- e D~ + Np N* ) =: R e m ( H ) is H-unimoaular. By Lemma 2.3.4 (i),

(N e, D e ) := (D e R - 1 N~ R-1 ) is also an r.c.f.r, of C ; but ( N e, D e ) satisfies equation

(3.2.28) and hence, statement (iii) of Corollary 3.2.9 holds.

Since Dp Np = Ne Dp and De Nc = Ne De ,statement (ii) also implies that the gen-

eralized Bezout identity (3.2.29) is satisfied and hence, statement (iv) holds.

Now suppose that C has an 1.c.f.r. ( / )e ,/Vc ) that satisfies equation (3.2.29), where

(Np , D e ) is any given r.e.f.r, of P ; then by Lemma 2.3.4, any other 1.e.f.r. of C is of the

form ( L / ~ c , L N¢ ) and any other r.c.f.r, of P is of the form (Np R , D e R ) , where L,

R e m ( H ) are H-unimodular matrices; therefore, condition (3.2.23) is satisfied for any

Le.f.r. of C and any r.e.f.r, of P since D H 1 = L R , and hence, by Theorem 3.2.7, S ( P , C ) is

H-stable. 0

Remark 3.2.10

Let (Npr , D , Nel , G ) be a b.e.f.r, of P ~ m ( G ) ; let Vpr , Upr , X , Y , V , U , Ve t ,

vet , x , r , v , v ~ m(H) be as in the generalized Bezout identities (2.3.13)-(2.3.14), and

let M r, M t e m ( H ) be the H-unimodular matrices defined in equations (2.3.15)-(2.3.16).

Then the denominator matrix/)Ha in equation (3.2.25) is H-unimodular if and only if

I n 0

DH3 M l =

~c(Np, v e , - ~ u p ~ ) - S c % , ~c (NerX + ~ r ) + 6e r

is H-unimodular, (3.2.32)

52

where ( / ) c , Nc ) is any 1.c.f.r. of C. From (3.2.32), DH3 is H-unimodular if and only if

Dc Y + Nc ( Nl,, X + G Y ) is H-unimodular. (3.2.33)

By Theorem 2.4.1, ( Npr X + G Y , Y ) is also an r.c.f.r, of P ; therefore condition (3.2.33) is

equivalent to condition (3.2.23) of the H-stability Theorem 3.2.7.

Similarly, the denominator matrix DH4 in equation (3.2.26) is H-unimodular if and

only if

MrDH4 =

In ( - Vpr Npt + Upr G ) Nc + Upr Dc

0 + f 6 + f o c

is H-unimodular, (3.2.34)

where ( N c, D e ) is any r.c.f.r, of C . From (3.2.34), DH4 is H-unimodular if and only if

( X Npl + f G ) N c + f D c is H-unimodular. (3.2.35)

By Theorem 2.4.1, ( Y ,)~ Npl + Y G ) is also an 1.c.f.r. o f P ; therefore condition (3.2.35) is

equivalent to condition (3.2.24) of the H-stability Theorem 3.2.7.

By Corollary 3.2.9, C H-stabilizes P if and only if an 1.c.f.r. ( / )c ,/~c ) of

c e m(G)sa t i s f ies

Dc Y +~lc (NprX + G Y ) = In, ; (3.2.36)

equivalently, an r.c.f.r. ( N c, Dc ) of C e m(G) satisfies

( X Npt + ~" G )N~ + f D~ = Ino

We now parametrize the set S ( P ) of all H-stabilizing compensators for P .

(3.2.37)

D

53

Theorem 3.2.11. ( Parametrization of all H--stabilizing compensators in S ( P , C ) )

Let Assumptions 3.2.1 (i) and (iii) hold; let (Np , Dr, ) be any r.c.f.r, and (D"p ,/Vp ) be any

1.c.f.r. of P ~ G n° x ni ; let Vp, Up, 17p, Up ~ m ( H ) be as in the generalized Bezout

identity (2.3.12). Under these assumptions,

S ( P )= { ( Vt, - Q Nj, )-l ( Up + Q Dr, ) I Q e H ni x n° ,det(Vt, -Q/V~, ) ~ I };

(3.2.38)

equivalently,

S ( P ) = { (LTt, +or, Q ) (17 , -Np Q )-1 I O E H nixn° ,det(17p - N p Q ) ~ I }.

(3.2.39)

Furthermore, corresponding to each compensator C ~ S ( P ) , there is a unique

Q e m ( H ) in the equivalent parametrizations (3.2.38) and (3.2.39). Equations (3.2.38) and

(3.2.39) are bijeefions from Q ~ m ( H ) toC ~ S ( / ~ ) .

Remark 3.2.12. ( H--stabilizing compensators based on a bicoprime faetorization ofP )

Let Assumptions 3.2.1 (i) and (iii) hold; let (Npr ,D ,Npl, G ) be any b.c.f.r, of

e ~ m(Gs) ; let Vpr, Upr, X , Y , V , U , Vpt, Upt, X , Y , U , V ~ m ( H ) be as in

the generalized Bezout identifies (2.3.13)-(2.3.14). By Theorem 2.4.1, (Npr X + G Y , Y ) is

an r.e.f.r, and (Y ,XNpt + Y G ) is an 1.c.f.r. of P ; therefore, by Theorem 3.2.11, the set

S ( P ) of all H-stabilizing compensators is given by

S I P ) = { ( V + U V p r N p I - U U p r G - Q ( X N p I + Y G ) ) - I ( U U p r + Q ~ ) I

Q ~ m ( H ) , det( V + U Vt, r Nt, t - U Upr G - Q ( £/Vpl + Y G ) ) E I } ; (3.2.40)

equivalently,

S ( P )= { (Upl U +Y Q )(17 + Npr Vpl U - G UpI U - ( N p r X +G Y )Q ) -1 ]

Q ~ r e ( H ) , det( 17 + Npr Vp~ ~7 - G Vp~ ~7 - (Npr X + G Y ) Q ) ~ I } . (3.2.41)

54

The equivalence of the representation (3.2.38) to (3.2.40) and of the representation (3.2.39) to

(3.2.41) is easy to see by comparing the generalized Bezout identities (2.3.12) and (2.4.3).

Proof of Theorem 3.2.11

By Corollary 3.2.9, C is an H-stabilizing compensator for P ff and only if an 1.c.Lr.

(/~c ,/Vc ) satisfies the Bezout identity (3.2.27); by Lemma 2.5.1, following Remark 2.5.2 (i),

all solutions of (3.2.27) over m(H) are given by equation (2.5.7). Similarly, C H-stabilizes

P if and only if an r.c.f.r. (N c, D c ) satisfies the Bezout identity (3.2.28); all solutions of

(3.2.28) over m(H) are given by equation (2.5.8).

Now ff C ~ m ( G ) is an H-stabilizing compensator, then the denominator matrix /)c

is (Vp - Q Np ) for some Q ~ m ( H ) ; by Definition 2.3.1 (vi), det(Vp - Q / V p ) ~ I.

Conversely, i fQ ~ m ( H ) ischosen sothatdet(Vp - Q AT~ ) ~ I , then (Vt, - Q / q t , ) isa

valid choice for the denominator matrix /)c. By Lemma 2.4.4, for all Q ~ m ( H ) ,

det( Vp - Q/Vp ) ~ det( 17p - Np Q ) . We conclude then that the set of all H-stabilizing

compensators is given by (3.2.38) and equivalently, by (3.2.39).

Now suppose that C is an H-stabilizing compensator; then from (3.2.38)-(3.2.39), an

1.c.f.r. ( /~c , /~c ) of C is given by

Q,] = : [ I n i Q 1 ] /¢1 , (3.2.42)

for some Ql ~ HniX no ; note that the matrix/~ ~ m(H) is H-unimodular by the gen-

eralized Bezout identity (2.3.12); similarly, an r.e.fx. (N c , D e ) of C is given by

Dc

-Q2

In o

-Q2

In o

(3.2.43)

55

for some Q2 e H h i × ' ° . But since ffZ 1 #c = Nc 0 ; 1 implies that gc Oc - t i c Nc = 0 ,

from (3.2.42)-(3.2.43) and the generalized Bezout identity (2.3.12) we obtain

-N~ -Q2

Dc In o

(3.2.44)

Therefore by (2.3.12) and by (3.2.44), we conclude that (lip -Ql~lt ,)-l(Ut, +Q1Dt, ) =

(ffp +Op Q2)( lTt , -Np Q2) -1 if and only if Q1 = Q2 -

Now suppose that C 1 e S ( P ) has an 1.e.f.r. ( / ) e l ,/Vcl ) and C 2 e S ( P ) has an

1.c.f.r. ( /~c2, A7c2 ) ; by (3.2.38) and (3.2.42),

for some Q 1 (~ Hn° x ni and

for some Q2 e Hn°x nl. From (3.2.45)-(3.2.46), C 1 = C 2 if and only if

(3.2.47)

since kl is H-unimodular, multiplying both sides of (3.2.47) by b~ -1 ~ re(H) , we obtain

/~-11 = /~-1 and /:~c 1 Q1 = /~c~ Q 2 ; i-c., Q1 = Q2. We conclude that C 1 = C 2

S ( P ) if and only if Q1 = Q2- Consequently, there is a unique (matrix-) parameter

Q e m(H) corresponding to each H-stabilizing compensator C ~ S ( P ) . []

56

Comment 3.2.13

(i) ( Parametrization of all H-stabilizing compensators when P c m(Gs) )

In addition to the assumptions of Theorem 3.2.11, suppose thatP ~ m ( G s ) ; then following

Remark 2.5.2 (ii), det( Vp - Q/Vp ) ~ I and equivalently, det( 17p - Np Q ) E I , for all

Q ~ m(H). Therefore, whenever e ~ m ( G s ) , Q ~ re(H)is a free (matrix-) parame-

ter in the equivalent parametrizations (3.2.38), (3.2.39), (3.2A0), (3.2.41). Hence

C = ( V p - Q ~ ' e ) - ' ( U e + Q D p ) = (Ue + D e Q ) ( V e - N p Q ) - I H-stabilizes

p ~ Gsno x ni for all choices of Q e H ni x no ; in particular,

C o := Vp 1 Up = 6/; V71 (3.2.48)

is an H-stabilizing compensator when P ~ m(G~ 1. with e = N, 0 ; 1 -- ~ ;1 ~7,, Fig-

ures 3.6 and 3.7 show the H-stable system S ( P , C ) , where

C = (vp-Q N, )-l(up -b e Dp ) and C = (Up -bOp Q )(Vp-Np O 1 -I , respcd2-

tively.

u' e" t o " %

+ :

C

U

r I

y ' [ + e : r - --~c----n,

I J p I .I

Y

Figure 3.6. The system S ( P , C ) ,

with P = Np D~ "1 = Dp-' Np , C = ( Vp - Q Np )-I ( U, + Q /)p ) .

57

If P ~ m(Gs), then det( Vp - Q/vp ) and det( Vp - Np Q ) are not necessarily in I

for all Q e m(H) ; in panacular, detVp and detlTp are not necessarily in I for any Vp and

17p that satisfy the generalized Bezout identity (2.3.12). In Figures 3.6 and 3.7, Q should then

be restricted to those H-stable matrices for which condition (2.5.9) holds; Vp and ~Tp should

be replaced by ( Vp - Q ° 1Vp ) and ( Vp -Np QO ), respectively, where a ° ~ mCH) is

chosen so that these are valid denominator matrices for the compensator. Note that following

Remark 2.5.2 (ii), if the (matrix-) parameter Q ~ m(H) in the representations (3.2.38)-

(3.2.39) is chosen as QO ~ m(H) given by (2.5.13), then det(Vp _QO~p ) ~ I and

equivalently, det( lYt, -Np Q° ) e I .

u . . . . . .

C

Figure 3.7. The system S ( P , C ) ,

with P = NpDp 1 = Dp-lNp, C -- (Up +Dp Q ) ( V p - N p Q )-1.

(ii) ( Observer-based compensator )

Suppose that S ( P , C ) is a lumped-parameter, continuous-time, linear, time-invariant system;

let P ~ m ( • p (s)) be represented by its state-space representation (A , B , C , E ) as in

Example 2.4.3. An r.c.f.r. (Np, Dp ) and an 1.c.f.r. (/~p,/~p ) of P are then given by (2.4.18)

and (2.4.19), respectively. Let

Vp :=In, + K A f ( B - F F . ) , Up . ' = K A / F ,

58

:= I . o , tY, : = K & r ;

by the generalized Bezout identity (2.4.17), Vp Dp + Up Np = l nl and

Dp Vp + NpUp = Ino . Now since the matrices A k := ( s l n - A + B K ) -1, A 1, :=

( s I n - A + F C )-t E m(Ru), defined in equation (2.4.14), are strictly proper, detVg =

det(lni + K Af ( / ~ - F / ~ ) ) ~ I and detlTp = det(lno + ( C - E K )A/~ F ) E I . Since

vp-1, lT~t E m ( R p ( s ) ) , Vp and ITp are already valid compensator denominator matrices

in this case for any P e m0R. <s)) ; therefore C O := V : 1 Up -- Up Vp I E m(Rsp (s))

is an H--stabilizing compensator, where

C o = ( ln i + K A : ( B - F F . ) ) - I K A f F = K A k F ( l n o + ( C - f f ~ K ) A k F ) - I

= K(sl n -A +BK +FC -FEK)-IF (3.2.49)

The compensator Co in the expression (3.2.49) is a full-order observer-based compensator,

note that this compensator is always strictly proper for all proper (and strictly proper) plants.

Now ( V p - Q b~p) and (17p-Np Q ) are also valid denominator matrices for all

Q e mCRu) such that det( Vp (.o) - Q (co) Np (oo)) -- det( 17t, (-0) - Np (oo) Q (oo)) , 0 ;

note that Vp(**) = I n l , Vp(oo) = Ino ' Np(oo) = Np(.o) = ff~ ; therefore in therepresenta-

tions (3.2.38)-(3.2.39) of the set S ( P ) of all H-stabilizing compensators, the (matrix-)

parameter Q ~ m ( R u ) should be chosen so that

det( lTp (**) - Np (~*) a (-0)) = det( I ni - a (**) ff~ )

(3.2.50) = det( I no - / ~ Q (**)) = det( Vp (**) - a (.o)/~p (co)) 4: o .

Note that condition (3.2.50) is automatically satisfied for all Q ~ m(Ru) n m( sp (s)).

Figure 3.8 shows the system S ( P , C ) , where the plant transfer function is obtained from its

state-space representation ( A , B , C , E ) as P = C ( s I n - A )-1/~ + /~. If the parame-

ter Q is chosen as a real constant matrix such that det(lni - Q E ) =~ 0, then £ is the

59

state of the compensator C in Figure 3.8; in this case, the state-space representation

( A c , B c , C c , E c ) is given by:

Xc = X - F 6 + ( ~ - r E)Cl,z,-Q ? . ) - I ( -K + Q 6 ) ,

~c = ( B - F F . ) ( I n i - Q E ) - I Q - F ,

Cc = ( I n i - Q ~ : ) - I ( - K + Q C ) , Ec = ( I n i - Q E ) - I Q , (3.2.51)

where Q ~ m(l~) is such that condition (3.2.50) holds, i.e., (l•i - Q/~ )-1 E m 0 R ) .

The compensator transfer function is obtained from Figure 3.8 as:

c = (i,,,-e ~.)-I(-K +Q 0)[ A:1-(~-r~.)(~,,~-e ~.)-I(_K +Q e)] -~

• [(/~-F E)(Ini-Q ~)-IQ -F ] + (Ini-Q ~)-IQ (3.2.52)

The compensator transfer function can he obtained also using the pseudo-state ~c : From Fig-

ur¢ 3.8,

:~ = A t(/~Q - F)~c ,

)Q] =,"

Therefore,

C = [ K A I ~ F + ( I n , - K A I ~ B ) Q ]

. [ i n ° + ( C _ E K ) A I ~ F _ ( 6 a l ~ ~ + ~ , ( i n i _ K A k ~ ) ) Q ] - I (3.2.53)

Note that this is in the right-coprime factorization form ( Up + Dp Q ) ( Vp - Np Q )-' as

in (3.2,39); to obtain the left-coprime faetorization form as in (3.2.38), we write

[ ,ni +K Af (B-F E)-Q (CA fB + (/no-CAsF)~")] ,'

Therefore,

C - [Ini

60

= [KA/F +Q(Ino-CA/F)] e'

+ K A/ (/~-F/~)-Q (CAf/~ + (In, , -6"A/F )/~)]-I

. [ K A / F + Q ( I n o - C A / F ) ] . (3.2.54)

In equations (3.2.52), (3.2.53) and (3.2.54) representing the compensator transfer function C ,

the (matrix-) parameter Q E m ( R u) is chosen so that condition (3.2.50) holds, i.e.,

det(Ini - Q (**) ff~ ) :/: 0 . Following Remark 2.5.2 (iv), the H-stabilizing compensator C

in (3.2.52) (and equivalently in (3.2.53) and (3.2.54)) is strictly proper if and only if Q E

m(H) c~ mORsp(S)) since equation (2.5.11) is satisfied if and only if Q(o,) =

-Up(**)/~-1 (**) = 0; this can also be seen from (3.2.53) and (3.2.54) since C(**) =

( in ~ _ Q (**)/~ )-1 Q (**). Note that by equation (3.2.5), condition (3.2.50) is equivalent to the

well-posedness of the system S ( P , C ) since det( I ni - Q (**)/~ ) = det( I ni + C (**) P (**)) =

det( Ino + P (**) C (**)), where C is given by the equivalent representations (3.2.52), (3.2.53)

and (3.2.54) and e (**) = E .

t . , , . . . . . . . . . . . . . . . . . . . . . . . .

u

Figure 3.8. The system S ( P , C ) , with an observer-based compensator.

61

(iii) ( Parametrization of all H-stabilizing compensators when the plant is H-stable )

In addition to the assumptions of Theorem 3.2.11, suppose that P E m(H); then following

Remark 2.5.2 (v), we can choose ( P , lni) as an r.c.f.r, and ( l n o , P) as an 1.c.f.r. of P . In

this case, from (3.2.38)-(3.2.39), the set S ( P ) of all H-stabilizing compensators is given by

S ( P ) = { ( [ n i - Q P )-IQ = Q ( I n o - P Q ) -1 ] Q ~ H nixn° ,

d e t ( l n i - Q P ) = d e t ( l n o - P Q ) ~ I } . (3.2.55)

If e ~ m(H) • m(Gs), then following Comment 3.2.13 (i) above, det( Ini - Q P ) =

d e t ( l n o - e Q ) ~ I , forall Q ~ m(H).

(iv) ( Parametrization of all strictly proper compensators )

Suppose that H is the ring Ru as in Section 2.2; let P ~ m(l~p (s)); then following Remark

2.5.2 (iv), the set of all strictly proper compensators C ~ m(•sp (s)) is given by the

equivalent representations (3.2.56) and (3.2.57) below:

{ (V,-Q;,)-I( U, +Q D,)IQ ~ mfRt0, Q(,*)=- u,f,~)~ff;1 (, ~) },

(3.2.56)

{ ( tTp + op Q ) ( ~Tp - Np Q )-' [ Q e m(Ru), Q (,~) = - o~-' (oo) tTp (~) } ;

(3.2.57)

note that D e Up = Up Dp by the generalized Bezout identity (2.3.12). Choosing Q

m(Ru) such that Q (~) -- - Up (o0)/~~ (oo) = - Dp --~ (oo) tTp (oo) as in equation (2.5.14) is a

sufficient condition for det( Vp - Q Nl, ) ~ I and equivalently, det( 17p - Np O ) ~ I .

(v) ( Parametrization of all plants such that S ( P , C ) is H-s tab le )

The conditions for H-stability of the system S( P , C ) arc symmetric in the plant P and the

compensator C. Suppose that Assumptions 3.2.1 (ii) and (iii) hold, where C ~ m(G) is

given and ( / ) c , Nc ) is an 1.c.f.r., ( N c, D c ) is an r.c.f.r of C ; let V c , Uc , 17 c , tTc

62

m(H) be as in the generalized Bezout identity (2.5.19). Under these assumptions, S ( P , C )

isH--stablewithP • m ( G ) ff and only if P is of the form

(6~ + o~Q~)<~- N~Q~)-~ = (vo- O,~7~)-~(c,o + ~,~o) (3.2.58)

for some Qp • m ( H ) such that det( Vc - Nc Qp ) - det( V c - Qp Nc ) (~ I . []

3.2.14. Achievable input-output maps of S( P , C )

An important consequence of the parametrization of all H-stabilizing compensators is

[u] [,] that the set of all achievable closed-loop I/O maps Hy-~ : u ' l---) Y' can now be

described explicitly using S ( P ) given in the equivalent representations (3.2.38) and (3.2.39).

The set

A ( P ) : = { //y-~ [ C H-stabilizes P }

is called the set o f all achievable IlO maps of the unity-feedback system S ( P , C ).

By Theorem 3.2.11, A ( P ) = { H ~ [ C • S ( P ) } , w h e r e S ( P ) i s t h e s e t o f

all H-stabilizing compensators given by (3.2.38) and equivalently, (3.2.39). We obtain the set

of all achievable I]O maps from the expression (3.2.6) for H ~ by substituting an r.e. factoriza-

tion Ng De-1 for P and an 1.c. factorization /)~-137c for C; from (3.2.38), D c l N c =

( V p - Q Np )-1 (Up +Q/~p ) for some Q • m(H) such that det(Vp-Q ff~, ) e I ,

where Vp , Up • m(H) as in the generalized Bezout identity (2.3.12):

Np (vp - e ̂ Tp ) Np(Vp +e~p) A(P)= { H~ = I

o , ( v , - o. ~,, ) - ~,,, op (up + a S p )

Q e m ( H ) , det(Vp - Q / ~ p ) e I } . (3.2.59)

The set A ( P ) of all achievable I/O maps can also be described as in (3.2.60) below, where

an 1.c. factorization/~1/qp for P and an r.e. factorization N c D~ -1 for C are substituted in the

63

expression (3.2.7) for H ~ ; from (3.2.39), N e O~ "1 = ( Up + Dp Q ) ( Vt, -Np Q )-t for

someQ e m(H) suchthatdet(Ifp -Np Q ) ~ I ,where tTp ,Vp E m(H) areasinthe

generalized Bezout identity (2.3.12):

(~,, -N,Q)~,, A ( P ) = { . ~ --

-(a, + o, Q )~,

i.o - ( ~ , - N, Q >a,

(a , + o , Q )o-,

Q ~ m ( H ) , det( ITt, - Np Q ) E I } (3.2.60)

The representation (3.2.59) and equivalently, (3.2.60), is a parametrization of all closed-loop

I/0 maps. Each of the four I/O maps of Hy~ in the equivalent representations (3.2.59)-(3.2.60)

is affine in the (matrix-) parameter Q ; the matrix Q ~ m(H) is a design parameter. Since

each achievable closed-loop map depends on the same (matrix-) parameter Q , the system

S (P , C ) is called a one-parameter design or a one-degree-of-freedom compensation scheme.

Now suppose that the plant P is H-stable; following Comment 3.2.13 (iii), the set

A ( P ) of all achievable I/O maps is given by

P ( In i - Q e )

A ( P ) = { H~ = - Q P

PO

I Q

Q ~ m ( H ) , det(lnl - Q P ) = det(Ino - P Q ) E I } . (3.2.61)

From the representation (3.2.59) we see that the plant output y =

E l[ Np ( Vp - Q/Vt, ) ( Up + O/~p ) u" ; hence, for all C ~ m ( H ) , the achiev-

able I/O maps Hy u and Hy u, both have the plant numerator matrix Np as a left factor. This

implies that the plant dynamics impose a constraint on the I/O maps Hy u and Hyu,.

64

3.2.15. Decouplingin S ( P , C )

Consider the unity-feedback system S ( P , C ); let Assumptions 3.2.1 hold. We now con-

sider the problem of decoupling in S ( P , C ).

Let ( N p , Dp ) be an r.c.f.r, of P and ( D p , Np ) be an 1.c.f.r. of P . Let Vp, Up, l~p,

Up be as in the generalized Bezout identity (2.3.12).

The system S ( P , C ) is said to be deeoupled fff S ( P , C ) is H-stable and the closed-

loop map Hy u, : u" b-~ y is diagonal and nonsingular.

A compensator C is said to decouple the system S ( P , C ) iff C ~ S ( P ) and the

map Hy u, : u" }---) y is diagonal and nonsingular.

By equation (3.2.59), Hyu, is an achievable map of S ( P , C ) if and only if

Ha,, = N e (Up + Q/~p ) (3.2.62)

for some Q e H ni × no such that det( Vp - Q / T p ) ~ I . There exists an H-stabilizing

compensator that deeouples S ( P , C ) if and only if there exists a Q ~ mfH) such that

det( V~ - Q/Vp ) ~ I for which the map Hy u, in (3.2.62) is diagonal and nonsingular.

Let rank P denote the normal rank of P ~ G n° x ni (i.e., the rank of P over the ring

G ). Since P = Np D~ q , rank P ~ rank Np ; since Np =- P D p , rank Np <_ rank P ; there-

fore, rank Np = rank P and similarly, rank Np = rank P . B y equation (3.2.62), if Hy u, is

an achievable map of S ( P , C ), then rank Hy u, < rank Np ; if the system S ( P , C ) is actually

de.coupled, then Hyu, is nonsingular and hence, rank Hyu, = no • But since N e ~ H n° × ni ,

n o = rank Hy u, < rank Np < n o implies that, if decoupling is achieved, then rank Np =

n o = rank P . Hence, a necessary condition for decoupling is that

rank P = no < ni (3.2.63)

In the case that P - is square, (i.e., Nt, and b~t, are also square), condition (3.2.63) means

that detP ¢ 0 (i.e., detNp ¢ 0 and det/V e ;~ 0 because detDt, "-" det/~p ~ I by

Definition 2.3.1).

65

In the rest of this section suppose that condition (3.2.63) holds and that the plant P is

H-stable; in this case, it is always possible to find a C ~ S ( P ) that decouples S( P , C ) :

From (3.2.61), Hy u, is achievable ff and only if

Hy u, = P Q (3.2.64)

for some O ~ m(H) such that det(Inl - O P ) e I . For k = 1, . . . , n o , let A/_, ~ H

be a greatest-common-divisor (g.c.d.) of the entries in the k-th row of P . Let

A L := diag [ ALl - . . AZ~ o ] ; (3.2.65)

the diagonal matrix A L E m(H) is nonsingular since ALk ~: 0 for k = I, • • • , n o ; furth-

ermore, the diagonal entries At, of A L are unique within factors in J. Let

p =: A L / 7 ; (3.2.66)

clearly, rank P < rankt ~ ; since detA L ~: 0 , r a n k / 7 = rank ( A L - l p ) ~ rank P ;

therefore, condition (3.2.63) implies that rank P = n o and hence, /7 E H n° x ni has a

right-inverse denoted b y / 7 1 . Note that /7 t ~ r e ( F ) . Write the i j - th entry o f / 7 t as

miJ , where dij ~ H dij :/: 0 and (mi j dij ) is a coprime pair over H . For di j mq , , ,

j = 1 , " " , n o , let ARj E H be a least-common-multiple (1.c.m.) of the denominators of

the entries in the j - th column o f / 7 t . Let

A R := diag [ AR1 . . . ARn o ] ; (3.2.67)

the diagonal malrix A R ~ m(H) is nonsingular since ARj :~ 0 f o r j = 1, . . . , n o ; furth-

ermore, the diagonal entries ARj of A R are unique within factors in J . (In the case that

F t e m(H), the denominators dij ~ J and hence, without loss of generality, A R is the

identity matrix In, , .) Now by definition, ARj = b O dq for some bq ~ H ; therefore the

ij-th entry o f / 7 1 AR is ~ ARj = mij bq ~ H ; hence,

/7~AR ~ m(H) . (3.2.68)

66

If we choose the (matrix-) parameter Q E m(H) as

where Qd

pensator

Q :__ jtTIAR Qd E m ( H ) , (3.2.69)

e m(H) is diagonal and nonsingular, then by (3.2.68), Q is H-stable. The com-

C = ( l n i - Q P ) - l Q = Q ( l n o _ P Q ) - I decouples S ( P , C ) , w h e r e Q is

chosen as in (3.2.69) and Qd ~ m(H) is a diagonal nonsingular matrix such that

det(Ini - Q e ) = det(lno - e Q ) = det(Ino - AL AR Qd ) ~ I . (3.2.70)

Condition (3.2.70) is automatically satisfied for all Qa ~ m(H) c~ m(Gs). If e

m(H) c~ m(G~), then AL ~ m(H)

satisfied for all diagonal, nonsingular Qd E

n y / l • -7.

n m ( G s ) ; in this case, condition (3.2.70) is

m ( H ) . ~ e diagonal map achieved is given by

AL AR Od (3.2.71)

If the map H~u, is required to be block-diagonal, then one way to achieve this is to choose

Qd e m(H) block-diagonal, where Q~ satisfies condition (3.2.70).

Theorem 3.2.16 below summarizes decoupling in the system S ( P , C ) when the plant P

is H-stable:

Theorem 3.2.16. ( Class of all achievable diagonal Hy u, )

Let P ~ H n°x ni and let rank P = n o ; then the set of all compensators that de, couple

the system S ( P , C ) is given by

{ f r IAR Qd ( I n o - AL AR Qd )-1 [ Qd ~ Hn°xn° isdiagonal, nonsingular

and det(#no - AL AR Qd ) e I } . (3.2.72)

Furthermore, the set of all achievable, diagonal, nonsingular maps Hy,,, : u ' b-~ y is given

by

{ AL AR Qd [ Qd ~ Hn°x no is diagonal, nonsingular

and d e t ( l n o - A L A R Qd) ~ I }. (3.2.73)

67

3.3 T H E G E N E R A L F E E D B A C K S Y S T E M

A A

In this section we consider the general linear, time-invariant feedback system E(P, C )

iv] [z] shown in Figure 3.9. where /~ : e I-~ y represents the ( Tlo + no ) x (Ili + nl ) plant

and C • e" ~ y , represents the( '%'+n i ) x ( ~ l i ' + n , )compensator. Theexter-

nally applied inputs are denoted by s2 := ~, , the plant and the compensator outputs are

U p

denoted by f := Y, ; the closed-Imp input-output map of E(/~, C ) is denoted by y '

:

~P

C

Z r V

y,

u

+

e

"1

z

¢-

Y

Figure 3.9. The feedback system E(/~, C ).

3.3.1. Assumptions on ~(/~, C )

(i) The plant P ~ G (rl°+n°) x (rli+ni) and is partitioned as in equation (3.3.1):

U r

+

P l l P12

P21 P

G (rl°+n°)×(rli+ni) , whereP ~ G n°xn i (3.3.1)

(ii)

68

The compensator C e G 01° "-I-hi ) x (Tli '-t-no ) and is partitioned as in equation (3.3.2):

C n C12 = ~ G ('q°'+ni)x('qi'+n°) , whereC ~ G nixn° (3.3.2)

C21 C

(iii) The system Y.(/~, C ) is well-posed; equivalently, the closed-loop input-output map

H ~ : ~ ~ ~" is in m ( G ) . t3

By Lemma 2.3.5, the plant P has an r.c.f.r. ( N f , D f ) and an 1.c.f.r. ( / ) f , Nf ) which

satisfy equations (3.3.3)-(3.3.4) below:

( N a , O ~ ) =: ( Nll N12

N21 Np

Dll

D21

0 ) ,

o~ (3.3.3)

where

Dll

0

D12 Nil

Op N'-21

N12 ) ,

( N p , D p ) is an r.f.r, of P and (D-p ,Ne) is an 1.f.r. of P .

(3.3.4)

By Lemma 2.3.5 applied to C, the compensator C has an 1.c.f.r. (/)~,, b~,,c ) and an r.c.f.r.

( N~, D~ ) which satisfy equations (3.3.5)-(3.3.6) below:

B{I g{2 = :

0 /~c

N 11 12

~Th ~7c ) , (3.3.5)

N^ ( c , D ~ ) = : ( NPll N'12

N'21 Nc

D tll

D t21 Dc

) , (3.3.6)

69

where

( / ) c , /Vc ) is an 1.f.r. of C and (N c ,D e ) is an r.f.r, of C .

By Lemma 2.3.4, any other r.c.f.r, of/~ is given by (Nf R , Df R ), where ( Np, Df ) is

the r.c.f.r in equation (3.3.3) and R ~ m ( H ) is H-unimodular. Similarly, any other 1.c.f.r.

of/~ is given by ( L / ~ f , L/~'f ), where (/)if,/VF ) is the 1.c.f.r. in equation (3.3.4) and

L ~ m ( H ) is H-unimodular. Note that the pair ( Nt, , Dp ) in equation (3.3.3) is not neces-

sarily r.c. and the pair (/~p ,/Vt, ) in equation (3.3.4) is not necessarily 1.c. Similar comments

apply to coprime-fraction representations of C .

3.3.2. Closed-loop input-output maps of E(/~, C )

Let Assumptions 3.3.1 hold; using (3.3.1)-(3.3.2), the system Y.(/~, C ) in Figure 3.9 is

described by:

111o 0 0 - P 12

0 Ino 0 - P

0 C12 ITI o, 0

0 C 0 Inl

z

Y Z t

y"

Pll

P21

= 0

0

Pt2 0 0

P 0 0

0 Cll C12

0 C21 C

V

U V t

U e

(3.3.7)

From equation (3.3.7), it is easy to see that Hya, : ~" ~ ~ ~ m(G) if and only if condition

(3.2.4) of Section 3.2 holds; (3.2.4) is equivalent to ( ln i +C P )-1 E m(G) and to

(Ino + P C )-1 ~ m(G); consequently, the system ~(/~, ~ ) is well-posed ff and only if the

unity-feedback system S ( P , C ) of Section 3.2 is weU-posed. By Assumption 3.3.1, since the

system E(P, C ) is well-posed, the matrix

T := ( l n l + C P )-1

is in m(G). The closed-loop I/O map H~,R is given in terms of T in equation (3.3.8) below;

note that it is also possible to wdte Hy£ in terms of (Ino + P C )-1 ~ m ( G ) , where

( I n o + P C) - I = i n o _ P ( in l + C P ) - I C = l n o - P T C :

7o

P l t - P t 2 T C P21 P12 T P12 T C21 Pt2 T C

(Ino - P T C)P2t P T P T C2t P T C

-C12(lno - P T C)P21 -C12P T Cl l -C12P T C21 C12(Ino - P T C)

- T C P21 T-In+ T C21 T C

Definition 3.3.3. ( H-stability of Y~(/~, C ) )

The system Y,(/~, C ) is said to be H-stable if( H:# ~ mfH).

(3.3.8)

3.3.4. Analysis ( Description ofT.(/~, C ) using coprime factorizations )

We analyze the system Y.(/~, C ) using coprime factorizations of the plant/~ and the com-

pensator C ; we only consider the case where a right-coprime factorization of/~ and a left-

coprime factorization of C is used.

Let Assumptions 3.3.1 hold. Let/~ = Np D# -1 , where ( N#,/9# ) is the r.e.f.r, of P given

in equation (3.3.3) and let C = / ~ 1 / ~ , , where ( D~,, N~ ) is the 1.c.f.r. of C given in equa-

tion (3.3.5); any other r.c.fx, of/~ is given by ( N# R ,/9# R ) and any other 1.c.fx. of C is

given by (/~ D-~ ,/~ N~ ), where e ~ r e ( H ) , E ~ re(H) are H-unimodnlar matrices.

^ + + + + [z] The system I;(P, C ) is redrawn in Figure 3.10; note that = e ' = y '

where L denotes the pseudo-state o f /~ .

Vt r . . . . . . . . I

I 'I-Tq u' e" ' IN~" i

I Z p r . . . . . . . . . . . . . . .

P t I Z

I , # I

ei~,,.~ 3.to. ~ e system Z(#, ~ ) ~m # = Np Df ~ and (~ = ~'~ ~ , .

71

The system E(/~, C ) is then described by equations (3.3.9)-(3.3.10) for any r.c.f.r, of/~ and

any 1.c.f.r. of C; note that the matrix R ~ mfH) appearing in these equations is

H-unimodular and that the H-unimodular matrix/~ e m(H) in (/~ D"# ,/7, N-cA ) cancels

out:

DFR

o O]N, R 0 Ino

[o o] 0 -Ini

. . °

z r

y"

I r i i + n i : 0

0

. . .

Y

U

V t

U t

0 I rio'+ ni

0

Z e

y"

g

Y

Z '

y"

(3.3.9)

(3.3.10)

Equations (3.3.9)-(3.3• 10) are of the form

By Lemma 2.3.2, performing elementary row operations over m(H) on the matrix NR

and elementary column operations over m ( H ) on the matrix [ N L /~H ] , we conclude that

(~R, BH, NL ) is a b.c. triple. Since Z~H e m(H) , it follows from Comment 2.4.7 (i) that

A A 1 Hy¢ = NR n ~ i~L e re(G) (3.3.11)

if and only if detOz e I (equivalently, the system y.(/2, ~ ) is well-posed). Since Assump-

tion 3.3.1 (iii) holds, condition (3.3.11) is satisfied and hence, det/~ H ¢ I . Consequently,

(NR,DH,NL) is a b.c.f.r, of H~¢ and hence, det/~ n is a characteristic determinant of H~,~,.

Using (3.3.3) and (3.3.5) in (3.3.9), we can rewrite/~H as:

72

[ Dll 0 ] .

D21 Dp

[ N ~2N21 /~ ~2Np

NcN21 NcNp ] i [

Theorem 3.3.5. ( H-stability of Z(/~, C ) )

[oo] 0 -lni

Jff ll D i2 ] 0 D e

R : 0

0 I ~o'4" ni

(3.3.12)

Let Assumptions 3.3.1 (i) and (ii) hold. Let ( N f , D f ) be the r.e.f.r, and ( D r , N f ) be the

1.e.f.r. of/~ given in equations (3.3.3)-(3.3.4); let (/9~,/V~ ) be the 1.c.f.r. and ( N~,, D~ ) be

the r.c.f.r, of C given in equations (3.3.5)-(3.3.6). Under these assumptions, the following

four statements are equivalent:

(1) g(/~, ~ ) is H-stable ;

(ii) /~tt is H-unimodular ; (3.3.13)

(iii) D 11 is H-unimodular , and (3.3.14)

~ t D 11 is H-unimodular , and (3.3.15)

[ Dc Dp + Nc Nv ] is H-unimodular; (3.3.16)

(iv) D n is H-unimodular , and (3.3.17)

D'11 is H-unimodular , and

[ ÷ ] is -un odo o

(3.3.18)

(3.3.19)

O

Note that each of statements (i) through (iv) imply that the system Z(/~, C ) is wall-posed; con-

sequenfly, we do not need a wcll-posedness assumption in Theorem 3.3.5.

73

Proof

Statement (i) is equivalent to statement (ii):

The system Z(/~, C ) is described by equations (3.3.9)-(3.3.10), where by Lemma 2.3.4,

(N~R, D~R ) is any arbitrary r.c.f.r, of/~ and (/~, D"~ ,/~ N"c' ) is any arbitrary 1.c.f.r. of

where R ~ m(H), tS e m(H) are H-unimodular matrices. If Z(/~, C ) is H-stable, then

e m(H) a~d hence, condition (3.3.11) holds; equivalently, detD~H ~ I; therefore

statements (i) and (ii) both imply that ( NR ,/~H, NL) is a b.c.f.r, of H~,#. By Lemma 2.4.6,

^ ^ 1 e m(H) implies thatD~ 1 ~ re(H). Conversely, if condition (3.3.13) holds, thenD~ ^ ^ I A

m(H) and hence, H~¢ = NR D~ NL ~ mCH).

Statement (ii) is equivalent to statement (iii):

From equation (3.3.12) we calculate detD~n using elementary operations over m ( H ) :

det/~ n = detD 1: det/~ 11 det(Dc De + Nc hip ) detR . (3.3.20)

By Lemma 2.3.3 (i), det/~// ~ J if and only if each of the factors in equations (3.3.20) is in

J ; equivalently, since R ~ m(H) is H-unimodular, condition (3.3.13) holds if and only if

all of (3.3.14), (3.3.15), (3.3.16) hold.

Statement (iii) is equivalent to statement (iv)."

By Lemma 2.4.4 applied to P and C, from equations (3.3.3)-(3.3.6) we obtain

detDt~ -- det/)t~ ( equivalently, detD 11 detDt, -- det/~H det/~p ) (3.3.21)

and

de t /~ -- detD~ (equivalently, det/~ I: detD c -- detD': l detD c ) .

Since det( Inl + C P ) = det(Ino + P C ), we can rewrite det/~//from (3.3.20) as

det/~ H = detDll det/2 Ix detDc det(Inl + C P ) detDp detR

= ( detD 11 detDp )( det/2 ~1 det/~c ) det( Ino + P C ) detR .

(3.3.22)

(3.3.23)

Since R ~ m(H) is H-unimodular, using equations (3.3.21)-(3.3.22) in (3.3.23) we obtain

74

det/~ n -- (det / ) l t det/) p ) (detD" u detDc )det ( lno +P C ) . (3.3.24)

Using P = DT1 Np and C w.. N c D~ -1 in (3.3.24) we obtain

det/~// -- detL~11 detD'l l det(Dp D e + Np N c ) . (3.3.25)

By Lemma 2.3.3 (i), detL~ H ~ J if and only if each of the three factors in (3.3.25) is in J; equivalently, condi-

tion (3.3.13) holds if and only if all of (3.3.17), (3.3.18), (3.3.19) hold. []

Comment 3.3.6

By Theorem 3.3.5, if E(/~, C ) is H-stable, then condition (3.3.14) implies that detD11 =

detDf(detDp )-1 ~ j ; equivalently, detDf N detDp. Similarly, condition (3.3.17)

implies that detDf N det/~t,. Condition (3.3.16) implies that (Np ,Dp ) is an r.c. pair and

condition (3.3.19) implies that ( / )p , /Vp ) is an 1.c. pair. Therefore, if Z(/~, C ) is H--stable,

o] o 1 then R := 0 Ini ~ m(H) is H-unimodular and L := 0 Ino

m(H) is H-unimodular;, hence,'there exists an r.c.f.r. ( Nff R , Dfl R ) of /~ given by

( N f R ,DpR ) = ( Nil N12

N21 Np

I'qi

D21

0

), n.

(3.3.26)

(and (Np , D e ) is a right-coprime-fracfion representation of P ) and there exists an 1.c.f.r.

( L O - f , L Np ) o f /~ given by

) = ( 11]o D I2

o

Nil

N21

N12

) , (3.3.27)

(and ( L~t, , ~Tp ) is a left-coprime-fraction representation of P ). Similar necessary conditions

apply to the compensator C : If Z(/~, C ) is H-stable, then/~

:= [ D Pll-1 m(H) is H-unimodular and/~

~ t --1 1 D ll 0 := 0 I ni

0 in ° e m(H) is H-unimodula~, hence,

75

there exists an 1.c.f.r. ( L D E , L N~, ) of C given by

111 o, D ~2

( L D $ , L N$ ) = ( o

~Th ~71: ) , (3.3.28)

(and (/9c ,/Vc ) is a left-eoprime-fracfion representation of C ) and there exists an r.c.f.r.

(N~,/~, O~/~ ) of C given by

( N ~ , O ~ ) = ( N' l l Ntl2

N 'el Arc

I rl i "

D P21

0

Dc ) , (3.3.29)

(and ( N c, D c ) is a right-coprime-fraction representation of C ). []

Definition 3.3.7. ( H-stabilizing compensator C )

(i) C is called an H-stabilizing compensator for P (abbreviated as: C H-stabilizes P )

iff C e G (Tl°'+ni) x (~i'+no) and the system Y(/~, C ) is H-stable.

(ii) The set

is called the

[]

S( e):= { c I ~ H-stabilizes fi }

set of all H-stabilizing compensators for f i in the system Z(/~, C ) .

Definition 3.3.8. ( Z--admissible plant/~ )

(i) P e m(G) is called Z--admissible iffthere exists a compensator ~ that H-stabilizes/~.

(ii) The class of all Y_,-admissible plants is the set of all plants/~ e G (rl°+n°) x ffli+ni ) for

which there exists a compensator C e G (rl°'+ni) x (Tli%no) such that the system

Z(/~, C ) is H-stable. []

76

Theorem 3.3.9. ( Necessary and sufficient conditions for Z~-admissibility )

Let P E G (rl°+n° ) x (l~i+ni) be as ill Assumption 3.3.1 (i); then the following three condi-

tions are equivalent:

(i) /~ is Z-admissible;

(ii) Qll

( Vp N21

N12

up

I~

- up N21

0

) is an r.c.Lr, of/~ (3.3.30)

and

I TIo (

0

-NI2 Up J~ll N12 Vp ) is an 1.c.f.r. of/~ , (3.3.31)

where (Np, Dp ) is an r.e.f.r, and (D-'p,/~Tp ) is an 1.c.f.r. of P ~ O n° x ni; the matrices

Up, Vp, Up, Vt, satisfy the generalized Bezout identity (2.3.12); b~ll ~ H ~° x~i, N12

HTIo x ni N21 ~ Hn° x rli are arbitrary H-stable matrices;

( i i i ) Pll - P12Dp Up P21 e m(H) (3.3.32)

and P 12 De ~ m(H) (3.3.33)

and L~p Pal ~ m(H), (3.3.34)

where (Np ,Dp ) is an r.c.f.r, and (/gp ,~7p ) is an l.c.f.r, of P ~ G n ° x n i ; the matrices

Up, Vp, Up, Vp satisfy the generalized Bezout identity (2.3.12).

Proof

Statement (i) implies statement (ii):

Suppose that/~ is Y_,--admissible; then by Definition 3.3.8, there exists a C ~ m(G) such that

~(/~, C ) is H-stable. Following Comment 3.3.6,/~ has an r.e.f.r. ( N~ R , Dp R ), given by

(3.3.26) and an l.c.f.r. ( L D f , L Np ), given by (3.3.27), where ( Np , Dp ) is an r.c.f.r, and

77

(D-p, ~7~, ) is an l.e.f.r, of P. Now since /~ = N¢ D f 1 = D"~ 1/~¢ implies tha t /~ Np =

/Vp D~ (and therefore, L D"~ Nff R = L/~1¢ D¢ R ), from (3.3.26)-(3.3.27) we obtain the

following equations:

~7~2t)p + (-lY~2)Np = N~2 ,

Nx2D21 - / )nN21 =Nl l - Nn •

(3.3.35)

(3.3.36)

(3.3.37)

(3.3.38)

Since ( Np, Dp ) is an r.c.f.r, and ( D-p,/Vp ) is an 1.c.f.r. of P , equation (3.3.35) implies that

( (NI,, Dr, ), ( L~,,/Vp ) ) is a doubly-coprime pair. Let V/,, Up, 17 , Ut, be as in the gen-

eralized Bezout identity (2.3.12). By Lemmz 2.5.1 (i), (/~12 ,/912 ) is a solution of equation

(3.3.36) over m(H) if and only if

v~ up (3.3.39)

for some Q e m ( H ) ; similarly, by Lemma 2.5.1 (ii), (D21 ,N21 ) is a solution of equation

(3.3.37) over m(H) if and only if

D21

N21

-e,

N21 (3.3.40)

for some Q e m(H) . Substituting for (/V12,/)12) and (D21,N21) from equations

(3.3.39)-(3.3.40) into (3.3.38) and using the generalized Bezout identity (2.3.12) we obtain

;12 -o12 ] D21

N21

e] N2~

= Nll - NIl • (3.3.41)

78

Using equations (3.3.39)-(3.3.40), the r.c.f.r. (Np,, R , Df R ) and the 1.e.f.r. ( L/)t~ , L N"p )

of/9 become:

(NpR ,OaR ) = ( Nll N12

Vp N21 - Np 6 Np - ~ N-21-D p o~ )~3.3.42)

111o -N12Up -QDp

o

Nil

N21

N ,2 v, - 6 ~ ,

X3.3.43)

o I [,o Let R := ~ I n~ and let/~ := 0 I no "

H-unimodular forall Q , Q ~ m ( H ) . Let Nn := Nu + NI2Q; byequation (3.3.41),

Nil + QN21 = Nll + N12Q =Nll (3.3.44)

Since R and /~ are H-unimodular, by Lemma 2.3.4, (Ng R R, Df R/~ ) is also an r.c.f.r.

and (E L D~,,, F, L ~ ) is also an Lc.f.r. of P ; but these are the r.c.f.r, given in (3.3.30) and

the 1.e.f.r. given in (3.3.31).

Statement (ii) implies statement (iii):

Suppose that statement (ii) of Theorem 3.3.9 holds. By (3.3.30) we have

A

Pll P12 Nil N12 [l"h 0 / 9 = =

P21 P Vp N21 Np O;1Up N21 Dp -1

/~11 + N12DTI Up N-21

v,, N2, + N. t,;' % N2~

N 12 D; 1

N. o;, ; (3.3.45)

by (3.3.31) we have

111o 8=

79

-N12 Up D-'p -1

0 D-p -I

Nu + N12 Up Dfl N21

Dp 1 N21

Nil N12 Vp

N'2t jVp

D;1Np

Using the expression for P 11 and P21 in (3.3.46) and for P 12 in (3.3.45), we obtain

P l l -- P12Dp Up P21 = /Qll E m(H)

PI2Dp = N12 (K m(H)

5pP21 = N"21 ~ m(H).

(3.3.46)

(3.3.47)

(3.3.48)

(3.3.49)

Statement (iii) follows then from (3.3.47)-(3.3.49).

Statement (iii) implies statement (i):

Suppose that statement (iii) of Theorem 3.3.9 holds. Let Q ~ m(H) be such that

det( Vp -Q/Vp ) e I ; note that (2.3.17) holds. Choose the compensator C as

-1 Cll C12 Irio' 0 0 0

C = =

c~1 c o vp-Q ~p 0 up +Q ~p

0 0

o (Vp-Q ~,,)-,(Vp +Q ap) e m ( G ) . (3.3.50)

With C = ( V e - Q Np )-1 ( Up + Q/ )p ) = /~-1 Nc as in (3.3.50), using the generalized

Bezoutidentity (2.3.17), T := ( Inl + C P )-I beomes

8O

r = op (vp - Q gp ) . (3.3.51)

Using (3.3.51), (2.3.17) and the compensator C e m(G) given by (3.3.50), we rewrite the

I/O map H~a" given by (3.3.8) in equation (3.3.52) below:

ell-el2Op(Up +Ol~p )e21 el2Dp(Vp-aNp) 0 P12Dp(Up +QDp) ( ~'p - Np o ) ap e ~l Np ( v,, - Q ~,, ) o Np ( up + e ~p )

0 0 0 0

(ap +op o. )n", ~'~x -(ap +rip o.)~p o np (up +o. ap )

(3.3.52)

Now conditions (3.3.32)-(3.3.34) imply that all of the maps in (3.3.52) are H-stable. Since

H~,¢ e m(H) with this choice of C ~ mfG), it follows from Definition 3.3.8 (i) that /~ is

~..--admissible. []

We now parametrize the set S( P ) of all H-stabilizing compensators for/~.

Theorem 3.3.10. ( Parametrization of all H-stabilizing compensators in 2;(/~, C ) )

Let Assumptions 3.3.1 (i) and (iii) hold. Let /~ E m(G) be F_,--admissible. Let (Np ,Dp )

be an r.c.f.r, and(/)p ,/Vp )beanl.c.f.r. of P E H n°×ni ;let Vp, Up, 17p, U/, beasin

the generalized Bezout identity (2.3.12). Under these assumptions, the set S ( /~ ) of all

H-stabilizing compensators C for P is given by equation (3.3.53) and equivalently, by

equation (3.3.54) below:

ITIo' -Q12/Vp

0 vp -Q~Tp

-I

Qli

Q21

QI~ z~p

u, + Qz~,

Qll e H rl°'×rl: , Q12 e H r i ° ' x n ° , Q21 e H nlxTli" , Q e Hni×no ,

det( Vp - Q b~t, ) ~ I } , (3.3.53)

{ d =

81

Qll QI2

Dp Q21 Up + D . Q

I131' 0

-Np Q21 l~p - Np Q

-1

I

Qll E H "q°'×'l]i" , Q12 e H ~ ° ' x n° , Q21 ~ Hnix 'q i " , Q ~ Hn/x no

det( 17p - Np Q ) E I } , (3.3.54)

Furthermore, corresponding to each compensator C ~ S( /~ ) , there is a unique Q 11, a

unique Q 12, a unique Q21 and a unique Q ~ m(H) in the equivalent parametrizations

(3.3.53) and (3.3.54). Equations (3.3.53) and (3.3.54) are bijections from Q l l , Q12, Q21,

a e m ( H ) to c e S(/~). Vl

Proof

Following Comment (3.3.6) and by Theorem 3.3.5, if C is is an H--stabilizing compensator for

/~, then C has an 1.e.f.r. (/~ D ~ , / r N~ ) given by (3.3.28) and an r.c.f.r. (Nz/~ , D , / ~ )

given by (3.3.29), where the 1.e.f.r. ( D c , N c ) and the r.e.f.r. ( N c , D c ) of C satisfy condi-

tions (3.3.16) and (3.3.19), respectively. By Lemma 2.5.1, the set of all solutions of (3.3.16)

for (/)c ,/Vc ) is given by (2.5.7) and the set of all solutions of (3.3.17) for (N c , D c ) is given

by (2.5.8). Note that (3.3.16)-(3.3.17) are equivalent to (3.2.29) by Corollary 3.2.9.

Now since C = D~ "l Nb' = N~ D~ -1 implies that /)~, N~, = /)~, N#, from (3.3.28)-

(3.3.29) we obtain the following equations:

/)c Ne = /Vc D c (3.3.55)

~ F ~2 Dc + ( - D 12) Nc = N'I2 , (3.3.56)

~ j /Vc ( -D '21 ) + /)c N'21 = N 21 , (3.3.57)

/V ~2D'21 - D {2 N'21 = N ' l l - N 11 (3.3.58)

82

Equations (3.3.55)-(3.3.58) for C are similar to equations (3.3.35)-(3,3.38) for /~. Following

similar steps as in the proof of Theorem 3.3.9 and using the generalized Bezout identity

(3.2.29), it is easy to show that ( A7 ~2,/~ 12 ) is a solution of (3.3.56) if and only if

(3.3.59)

and ( D "2t , N'21 ) is a solution of (3.3.57) if and only ff

D '21

N'2~

D c - lqp -6'

F" 21

(3.3.60)

for some {2' ~ m(H) and ~ ' ~ m(H). Substituting from (3.3.59)-(3.3.60) into (3.3.58)

and using the generalized Bezout identity (3.2.29) we obtain

[ / V i 2 /)12 ] D '21

N'2x

-6"

u l r

~ p = N'11 - N 11 (3.3.61)

[l i 0] Let L" := 0 in i and let R" := ~ , in,, ; the matrices L ' ,

R' ~ m(H) are H-unimodular for all Q ' , Q' e m ( H ) . Let Ql l := bT~l + Q'bT~l ;

by equation (3.3.61),

/VI1 + Q 'N21 = N l l + N'12 Q ' = Ql l (3.3.62)

Now let Q12 := N'12 and let Q21 := N ~x • Since L" a n d R ' are H-unimodular, by Lemma

2.3.4, ( L ' /~ D-'c,, , L"/~ N~ ) is also an 1.e.f.r. of C and ( N b,/2 R ' , D 6,/~ R ' ) is also an r.c.f.r.

of C , where

(L'L O~ ,L'L N e ) = (

Ir[o" - Q I 2 N p

o

QI1

Q21

Q12Dp ) , (3.3.63)

83

( N ~ R R ' , D # R R ' ) = ( Qll Q1,

D~, Q21 Nc

111i"

-Np Q21 Dc ) , (3.3.64)

and

( H o , ~ ) = < ( v , - o . ~ , ) , ( u , + Q H , ) ) , (3.3.65)

(,v~,z)o) -_ ('(O,, +~ , ,o . ) , (~ , -N , , o . )> , (3.3.66)

and Q ~ m ( H ) is such that d e t ( V p - Q / V p ) -- det(ITp-Ng Q ) E I . Thus we

showed that all H-stabilizing compensators C for /~ are as in the equivalent expressions

(3.3.53) and (3.3.54).

Now let C1 and C2 be two H-stabilizing compensators for P ; by (3.3.53)-(3.3.54),

^

C 1 =

Q,, + Q,,~'-~H~-;Q,1 Q, , (H, , + ~,, c ,>

and

~2 =

6.. + 6, ,~, D-:~6,, 6.,,(H, + ~, c , )

H~-~ 6.,, c,

(Vp-Q2fip)-'(up +Q2Bp ). From the proof of Th~orem 3.Z11, C, = C~ ifandon*y

if Q, = Q~ and #~, = ~ , . If c , = c , , ,hen , ~ ' = ( ~ , , + # , , C , ) =

(/)p +/Vp C , ) : D ~ i . NOW C, = C2 implies that: C12 = C12 = QlaDcl 1 =Qi2Dc-a 1

andhence, QI= = QI=; C=l = C21 = H~-llQ21 = 21 andhence, Q2I = Q21; CI1 - - ~,H:~ ^ ^ = ~ l l = Qll+Q12NpDc~Q2, = Q1,+612 Q21 andhence, Q,, = Q,1 since

Q1, = Q1, and Q21 = 6.21- We conclude that there is a unique set of(matrix-) parameters

Ql l , Q12, Q21, Q ¢ m ( H ) corresponding to each H-stabilizing compensator

~ g ( ~ ) .

84

Comment 3.3.11

(i) Suppose that /7 is X-admissible; then by Theorem 3.3.9, /7 is given by (3.3.45) and

equivalently, by (3.3.46). Now suppose that C is an H-stabilizing compensator for/7 ; then

by Theorem 3.3.10, ~ is given by

I~o" -Q12Np Qll Q12Dp = , (3.3.68)

0 ve - QNe Up + Qffp

where Q e m ( H ) is such that det(Vp - Q/Vp ) e I . Figure 3.11 shows X(/7, C)

where, from (3.3.30), /7 = Np D f 1 is given by

/~11 N12 ITli 0 /7 = Nff Df 1 = (3.3.69)

Vp N21 Alp D;1Up N21 D• 1

and C = D~ 1 N~" is given by (3.3.68). Note that the only maps in /7 and C that may not be

H-stable are Dp "I and/~-1 = ( Vp - Q b~p )-I, respectively.

Suppose that the ring H is the ring of proper stable rational functions R u as in Scction

2.2. In this case if /7 is Z,-admissiblc, then every H-pole of P11, P12 and of P21 is a

U-pole of P = Np Dp "I with at most the same McMillan degree. For C to be an

H-stabilizing compensator for /7, the U-poles of each of CIt, C12, C21 must be a subset

of the U-poles of C = Dc I N c , where C is chosen so that the feedback-loop is H-stable.

(ii) The class of all H-stabilizing compensators is parametrized by four matrices, Q 11, Q 12,

Q2~, Q e m(H) ; the matrix Q parametrizes the class of all C that H-stabilizes the

loop. We refer to design with the unity-feedback system S (P, C ) as one-degree-of-freedom

design because only one parameter matrix is available for design (see Section 3.2). In contrast,

for the more general system Z(/7, C ), there are four-degrees-of-freedom because C has four

completely free matrices in H, which can bc choscn to mcct performance spccifications. For

.example, the parameter Q21 can be used to diagonalizc the input-output map H~, : v' ~ z.

1,, s

I e p

85

i g t

I i U

I I

z

+

? Figure 3.11. The system Z(P, C ) with a T.-admissible plant P = Nf D f 1

and an H-stabilizing compensator C .

A A

3.3.12. Achievable input.output maps of~(P , C )

We can now describe the set of all achievable closed-loop YO maps

H~,~: u, ~ z y, of Y-(P, C ) based on the parametrization of all H-stabilizing U ' y '

compensators S(/~ ) given in the equivalent representations (3.3.53) and (3.3.54).

The set

is called the set of all achievable I10 maps of the system ~(/~, C ).

By Theorem 3.3.10, A(/~ ) = { H~'a" [ d e g (/~ ) }, where g (/~ ) is the set of all

H-stabilizing compensators given by (3.3.53) and equivalently, (3.3.54). We obtain the set of

all achievable I/O maps from the expression (3.3.8) for H~a' by using the expression for/~

given in (3.3.69) and the expression for C given in (3.3.68) and the generalized Bezout identity

86

(2.3.17). Note that C = ( Vt, - Q Ne )-1 (Ut' +Q Dr, ) - - (Up +De Q ) ( Vp -Nt, Q )-1

where Q G m ( H ) is chosen so that det( Vp - Q/Vt, ) - - det( I~p - Np Q ) ~ I .

Nll -NI2 Q N~ N~2 ( Vp - Q fi,, ) N~2 Q21

(~Tt, -N, Q ) fi2, Np(Vp-Qfip) NpQ2,

-- QI2/~21 -Ql2fip Qll

- ( u p -I-Dp Q )N21 - (Up -i-Dp Q )]Vp Dp Q21

N12 (Up +Q/~p )

Np (Up +Q D e )

Q12 Dp

~,, ( u,, +o. ~,, )

Q,,, Q,2, Q2,, Q e m(H) , de t (Vp -Q/vp ) - de t (Vp-Np Q ) E I } .

(3.3.70)

The representation (3.3.70) is a pararnctrization of all closed-loop WO maps. Each of the I/O

maps of H~,~ in (3.3.70) is affine in one of the (matrix-) parameters QI1, Q12, Q2~, Q ;

these matrices are the design parameters. The system E(P, C ) is called a four-parameter

design or a four-degrees-of-freedom compensation scheme.

If P l l = 0 and P21 = Ino, then the input v can be viewed as an additive disturbance

at the plant-output y . If P21 = In, , , then /~p = /~21 by (3.3.46). The disturbance-to-

output map H~, : v ~ y is given by ( l~p-Np Q )/V2, = (lTp - N t, Q ) /~p , which

depends on the parameter Q ~ m ( H ) . On the other hand, the external-input to output

maps H~,, -- N12Q21 and H~, = Np Q21 dependon a different parameter Q21.

3.3.13. Deeoupling in E(/~, C )

Consider the system Z(/~, ~ ) ; let Assumptions 3.3.1 hold; let/~ ~ m(G), partitioned

as in equation (3.3.1), be a Y.,--admissible plant. Assume that rli" = 1]o ; i.e., the number of

87

inputs v' and the number of outputs z are equal. We now consider the problem of

de.coupling in the system Z(/~, C ).

Let ( Np, D~ ) be an r.c.f.r, and (D-~, AT~ ) be an l.c.f.r, of ]~ ; since/~ is F.-admissible,

we can assume without loss of generality that (N~, D~ ) is given by (3.3.30) and (D~, N~ ) is

given by (3.3.31).

The system Z(/~, C ) is said to be decoupled iff Z(/~, C ) is H-stable and the closed-

loop map Hz,, : v" b--> z from the external-input v" to the actual-output z of the plant is

diagonal and no~ingular; a compensator C is said to decouple the system Z(/~, C ) iff

e S(/~ ) and the map Hzv, : v" b-~ z is diagonal and nonsingular.

By equation (3.3.70), Hzv, is an achievable map of Z(/~, C ) if and only if

Hzv, = N12Q2t (3.3.71)

for some Q21 e H nlxTl° (note that we assume Tli' = ~o ). fide.coupling is achieved, then

rank Hzv, = ~lo since Hzv, is nonsingular, hence, a neccssaxy condition for decoupling is that

rank NI2 = Tlo • But by (3.3.30), PI2 = N12 Dp -I (i.e., N12 = PI2Dp ) implies that

rank P12 = rank N12. Therefore, from now on we assume that

rank P12 = TIo < ni (3.3.72)

In the case that P 12 is square, O.e., 1]o = ni and hence, N12 is also square), condition

(3.3.72)mcansthatdetP12 g 0 (i.e.,detN12 # 0 becausedetP12 ~ 0anddetDp e I ).

In order to find compensators C that de, couple the system ~(/~, C ), we define two diago-

nal, nonsingular matrices A L and A R as follows:

Let A~ e H be a greatest-common-divisor (g.c.d.) of the elements of the k-th row of

N12. Let

AL :=diag [ At, 1 " " ALno] ; (3.3.73)

• • • , 7] o ; further- the diagonal matrix A L e m(H) is nonsingular since A/..t ~ 0 for k = I,

more, the diagonal entries A~ of A L are unique within factors in J. Let

88

N12 =: AL N12 ' (3.3.74)

where the matrix /~12 ~ Hri° x ni has full normal row-rank since rank N12 = rio and

. ^ 1 detA L ~: 0 ; therefore, N12 has a fight-inverse denoted b y / ~ 2 , note that N12 E m ( F ) .

^ I miJ Write the ij-th entry of NI2 as - ~ / j , where mij , dij • H , aij q: o and ( raij , dij ) is a

eoprime pair over H .

For j = 1, . ' . , r io, let AR/ ~ H be a least-common-multiple (1.e.m.) of the denomi-

nators of the entries in the j- th column ofNt2 . Let

AR := diag [ AR, " " ARgo] ; (3.3.75)

the diagonal matrix A R $ m ( H ) is nonsingular since ARj ~: 0 for j = 1, • • • , 11o ; further-

more, the diagonal entries AR/ of A R are unique within factors in J . (In the case that

^I N12 ~ m ( H ) , the denominators dij • J and hence, without loss of generality, A R is the

identity matrix 111o .)

Now by definition, ARj

^ I ralj N12 A R is ~ ARj = mij bij

= bij di/ for some bij • H ; therefore the ij-th entry of

e H ; hence,

^1 N12 A R ~ m(H) . (3.3.76)

Intuitively, if H is the ring Ru as in Section 2.2, then we can interpret the diagonal

matrices A L and A R as follows: Since rank N12 = 11o by assumption, z ~ 1] is a U-zero of

N12 if and only if rank N12( z ) < rio • Now ALk extracts the 1J-zeros that are common to all

elements in the k-th row of N12; A L "book-keeps" the l l -zeros of P12 = N12D71 that

appear in each entry of some row of N 12. Clearly, P 12 may have other 1J-zeros that cannot

be extracted by A L ; these 1J-zeros are the 1J-zeros of/~12 (equivalently, the 1J-poles of

/~1/2 ). Suppose for simplicity that P 12 is square: Multiplying N12 by the diagonal matrix A R

makes/~i-~ AR H-stable, i.e., cancels these tl-poles. Let s ~ 11 be a zero of A R (hence a

l~-zero of detN12 ); the multiplicity of s • 1J in detA R may exceed its multiplicity in

89

detN12. If detNl2 E n "q° × 1"1o has n zeros at s e 11, then detA R has at most n 11° zeros at

s e IJ ; so Ai R has at most as many l~-zeros as ( detN12 ) 111o .

Definition 3.3.14. ( Achievable diagonal input-output map Hzv, )

The set

~, ~,, ( /~) := { H~,, [ ~ H-stabit izes and the map Hzv, is diagonal and nonsingular

(3.3.77)

is called the set of all achievable diagonal nonsingular input-output maps Hzv, : v" ~ z .

EJ

Clearly, A z~' (/~ ) is a subset of the achievable v ' ~ z maps in A( /~ ) because

must be a H-stabilizing compensator, in other words, A ~,, ( i f ) is the set of all

N12 Q21 e m(H) that are diagonal and nonsingular. Thus we must choose the parameter

Q21 e m ( H ) so that N12Q21 is diagonal and nonsingular (see equation (3.3.71)). The

"minimal" restriction on Q21 to achieve diagonal Hzv, is given in Theorem 3.3.15 below:

Theorem 3.3.15. ( Class of all achievable diagonal Hzv, )

Let if e G (rl° +no ) x (rli +ni) be Y_c--admissible. Suppose that rli" = 11o • Without loss of gen-

erality, assume that an r.c.f.r, of/~ is given by (3.3.30) and an l.c.f.r, of/~ is given by (3.3.31).

Let P12 e G Tl° x ni have full normal row-rank; so rank N12 = Tlo • Under these assump-

tions, the set A zv' ( if ) of all achievable diagonal nonsingular input-output maps Hzv, is given

by

z~' (/~) = { AL AR Q:I [ Q21 e H Tl° × I]o is diagonal and nonsingular } ,

(3.3.78)

where A L and A R are the diagonal, nonsingular matrices defined by equations (3.3.73) and

(3.3.75), respectively.

90

Comment 3.3.16

(i) The square and diagonal input output map H~,, = A L A R Q21 ~ HTI° x 11o is achieved

by choosing the compensator parameter Q21 as

Q21 --'-- ~/2 AR Q21 , (3.3.79)

where Qzl ~ Hrl° x ~o is diagonal and nonsingular. By equation (3.3.76), if Q21 is chosen

as in (3.3.79), then Q21 ~ m(H). Therefore, to achieve diagonalization, from the set

S ( / ~ ) of all H-stabilizing compensators ~ , we must choose C21 = /~-1 a21 =

(vp - a ~7p )-1 Q2x as

C21 = ( V t , - Q Nt' )-1 /~1t2 AR 621 , (3.3.80)

where Q21 ~ HTI° x rio is a diagonal, nonsingular matrix and Q ~ H ni x no is chosen such

that d e t ( V p - Q / V p ) ~ I . Note that in equation (3.3.80), the (matrix-) parameter

Q ~ Hnix no is not used in diagonalizing the I/O map Hw, ; in the case that P e m ( G s ) ,

Q ~ m ( H ) is completely free since det( Vp - Q ATp ) ~ I for a11 Q ~ r n ( H ) . The /

other compensator parameters Q u and Q 12 am not used in diagonalizing the map Hw,

either.

Note that the parameter Q21 is restricted to be b~(2 AR Q2t for decoupling and hence,

can no longer be assigned arbitrarily in order to meet other design specifications; the only free-

dom left is the diagonal nonsingular matrix Q21 e r e ( H ) , in addition to the parameters

Qn,Q12 andQ.

0i) If H is the ring R u as in Section 2.2 and if P 12 is square for simplicity, then the "cosf'

of diagonalizing the map H~, is that the number of its U-zeros am increased. Since A L is a

factor of N12, Hzv, must have zeros at the 1J-zeros of A L ; the multiplicity of a 1J-zero of

H~v, may be larger than its multiplicity in detN12 due to the factor A R . If A L is the only

source of 1J-zeros of P 12 (equivalently, if N[~ ~ m(H) ) and if Q21 is chosen so that it

91

has no U-zeros, then the U-zeros of the diagonal Hr.,, have the same multiplicity as in

detN12 since in that case A R = 11.1o.

(iii) Although we chose to diagonalize the map H~,,, we could also diagonalize

H. , : v ' ~ y , the map from the same external-input v' to the measured output y of/~ (y is

the output used in feedback). In that case, assuming that Hi '= n o and that

rank P = n o < n i , we would define ARp, ALp and/~/, from N/, as we did above to obtain

AL, AR and /~12 from N12; the set of all achievable diagonal nonsingular maps Hy~,

would then be Pt ~,, ( f ) , where

yv" ( P ) = { ALp ARp 621 I Q21 E m(n) is diagonal and nonsingular } . (3.3.81)

The compensator pararnctcr Q21 must then be chosen as

N~ ARp Q.21 , (3.3.82)

where /~/ is the right-inverse of Np . []

Proof of Theorem 3.3.15

The map H~, is an achievable map of Z(/~, C ) if and only if H~,, = N12 Q21 for some

Q21 e m(H) . By equation (3.3.74),

H~, = NI2 Q21 = AL N12 Q21 (3.3.83)

for some Q21 e m f H ) . Now H~, ~ mfH) is diagonal and nonsingular if and only if

Q2~ ~ m(H) is such that A L/~12 Q21 is diagonal and nonsingular. Choose Q21 as in equa-

tion (3.3.79); then by equation (3.3.76), Q21 ~ mCH). Clearly, H~, = AL AR Q2~ is an

achievable diagonal nonsingular map.

A A

Now if H2v, is a given diagonal map achieved by Y.(P, C ) , then by equations (3.3.71)nd

(3.3.74), A L is clearly a factor of H~,. Now suppose, for a contradiction, that Q21 =

NI2A R Q21 , i.e., that

9 2

Hzv" = AL "~R Q21 , (3.3.84)

A where all (diagonal) entries of A R are the same as those of A R except the j - th entry, which

is a proper factor of ARj , i.e., for some 8j ~ J ,

ARj = l ~ j ~j . (3.3.85)

Since for i = 1, • • • , ni , ARj is a Lc.m. of dlj , some denominator, say the k-th row j-th

column denominator dk) has that factor 8 j , i.e., d M = 8j d t j - The kj-th entry of Q 21 is

then mtj ~Rj qj , where qj is the j-th (diagonal) entry of Q21- Since 5j is not a factor of a,j

~Rj and since ( mkj , dky ) is a coprime pair, the only way that the kj-th entry of Q21 will be

in H is if qj = ~jqj" for some q j ' E H; Q2I then becomes

, , oro,oro.,, . 1 ] " "" " • = "" " Q 2 1 =

A¢ AL AR Q 21 for some Q'21 e m(H). []

We conclude this chapter by applying some of the results of Sections 3.2 and 3.3 to a

Y_t--admissible plant/~ whose entries are proper rational functions.

Example 3.3.17

Let H be the ring Ru as in Section 2.2. Consider the two (vector-) input two (vector-) output

plant

P l l P12

P21 P

= (sln-A) -I B B +

o

where /~ ~ IR n x n , B e F, n×ni, C ~ ]R n°xn , ff~ ~ 1R n°xni, B ~ R n×rli and

~ R~o× n. Let (A, B ) be l~-stabilizable, ( C, X ) be U--detectable, let a ~ R and

thematricesK ~ F, ni × n , F ~ lRn × n° , Ak E Run × n , A/ E Ru n×n bedefinedasin

Example 2.4.3. Clearly,

93

(Np, ,D ,Np~ ,G ) := ( ( s + a ) - 1 ~ , ( s + a ) - l ( s t , ~ - X ) , B , ?. )

isab.c.f.r, o f P ¢ Rp(S) n°×nt

It is easy to check that/~ is ~--admissible by testing conditions (3.3.32)-(3.3.34) of

Theorem 3.3.9: From the r.c.f.r, and l.c.f.r, of P given by (2.4.18) and (2.4.19), Dp =

( I ni - K A k B ) andDp = ( l no - C Af F ) ; from the generalized Bezout identity (2.4.17),

Up = K Af F . Now condition (3.3.32) is satisfied since A k , Af • r n ( R u ) implies that

/~II = PII-PI2Dp Up P21 = CAk (In +BK Af )B

= C A k ( s l n - A + F C + B K ) A / B • m(Ru) ;

condition (3.3.33) is satisfied since

N12 = P12 Dp = ~ Ak B ~ mCRu) ;

condition (3.3.34) is satisfied since

N'21 = / )p P21 :-" ~ Af B E m(Ru) .

We can now write a right-coprime factorization o f /~ using (3.3.30) as: /~ = N¢ D [ 1

[ C A k ( I n + B K A : )gff C a k B 11.1 i 0

[ (Ino+CAkF-EKAkF)CAfB CAkB+E(In~-KAkB) -KAkFCAfB Ini-KAkB

The class of all Ru-stabilizing compensators for this P can be found from (3.3.53)-

(3.3.54) using (2.4.17), (2.4.18), (2.4.19). The set of all achievable maps Hzv, from the exter-

nal input v" to the actual-output z o f / ~ is given by:

{ Hzv" = FAt : /~ Q21 I Q21 6 R u nlxlli' }

Another I /0 map of interest is the map Hzv :v ~ z , which is often encountered in H. .

optimal design problems; the set of all achievable maps Hzv for the plant in this example is:

Q • Run/× no , det( Vp - Q/Vp ) ~ I } . []

-1

Chapter 4

DECENTRALIZED CONTROL SYSTEMS

4.1 INTRODUCTION

In Chapter Three we studied two system configurations: S ( P , C ) and ~,(P, ~ ) ; these

systems put no constraints on the structure of the feedback compensator, We now study the

consequences of restricting the compensator to be block-diagonal.

Restrictions on the feedback compensator structure are often encountered in large scale

systems. These systems have several local conUol stations; each local compensator observes

only the corresponding (local) outputs. Such decentralized control of systems results in a

block-diagonal compensator-matrix structure.

In this chapter, using the completely general algebraic framework of Chapter Two, we

obtain necessary and sufficient conditions on a plant P for stabilizability by a decentralized

dynamic compensator. Decentralized stabilizability conditions turn out to be that certain

canonical forms resembling the Smith-form must be satisfied by the coprime factorizations of

the plant P . When the compensator structure is required to be block-diagonal as in decentral-

ized output-feedback, finding the class of all stabilizing decentralized compensators is compli-

cated; the task is to find structured generalized Bezout identities associated with coprime fac-

torizafions plant P . For plants that satisfy decentralized stabilizability conditions, we

parametrize the class of all decentralized stabilizing compensators.

The two-channel decentralized control system S ( P , C a ) is studied in detail in sections

4.2 through 4.4; the results are extended to the m-channel decentr01i:,ed feedback control sys.

tern S ( P , Ca ),,, in Section 4.5. In Section 4.2, following the analysis using coprime factofi-

zations of P and C d , Theorem 4.2.5 gives four equivalent necessary and sufficient conditions

95

for the H-stability of S ( P , Ca ) . Section 4.3 contains two important results: Theorem 4.3.3

(Conditions on P = Np Dp -1 = D-p -1 N~, for decentralized H-stabilizability) and Theorem

4.3.5 (Class of all decentralized H-stabilizing compensators in S( P , C a ) ). In Section 4.4 it

is shown that the decentralized H-stabilizability conditions of Theorem 4.3.3 are equivalent to

the condition that the proper rational matrix P has no decentralized fixed-eigenvalues in the

undesirable region I] ; these conditions are shown to be equivalent to certain rank tests on

eoprime faetorizations of P in Theorem 4.4.4 (Rank tests on P = Np Dp -1 = D~ -1/~t, for

decentralized fixed-eigenvalues and H---stabilizability). Similar rank tests on state-space reali-

zations of P are given in Remark 4.4.6. It is interesting to note the relationship between decen-

tralized fixed-eigenvalues and transmission-zeros of the plant (Comment 4.4.8). Section 4.4.9

gives an algorithm for designing a two-channel decentralized H-stabilizing compensator in

the case that P has strictly-proper rational function entries; a simple example is also included

(Example 4.4.10). In Section 4.5, Theorems 4.5.4 and 4.5.5 are extensions of the main results

of Section 4.3 to the m-channel case; Comment 4.5.6 summarizes the rational functions case,

which was explained in detail for two-channels in Section 4.4. Finally in Section 4.5.7 we see

that the input-output maps achieved by the H-stabilized decentralized control system

S ( P, Ca )m (and the system S ( P , Cd ) in the ease that m = 2 ) are not afJine in the (matrix-)

parameters that describe all decentralized H-stabilizing compensators.

4.2 T W O - C H A N N E L D E C E N T R A L I Z E D C O N T R O L S Y S T E M

In this section we consider the linear, time-invariant, two-channel decentralized feedback

system S ( P , C a ) shown in Figure 4.1, where P : e2 I-~ Y 2 represents the plant

and C : ~ represents the compensator. The externally applied inputs are e 2' Y 2"

96

denoted by ~- :=

Ul u 2 U l t

u 2 t

, the plant and the compensator outputs are denoted by

Yl

Y l Y2

; the closed-loop input-output map of S ( P , C d ) is denoted by H ~ : ~ b-~ ~'.

Ul t e l t Yl" ~ Yl + :L.)-----------~

U 2" e 2 t

: C1

Cd

= c21

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J

U2

Y 2" ~ . 2 . _ ~ +

P

Y2

Figure 4.1. The two-channel decentralized control system S( P, C d ).

4.2.1. Assumptions on S ( P, Cd )

(i) The two-channel plant P e G n° x ni , where

Pl l P12 ]

P = P21 P22 '

PII e G n°lxnil , P12 ~ Gn°Ixni2 , P21 E G n°2xnil , P22 E G n°2x hi2

n o -----." 8ol + /'1o2 , tl i =: tlil + hi2

(ii) The decentralized compensator C d e G nix no, where

Cd = 0 C 2 ' C1 • G n i l x n ° l ' C2 E G ni2xn°2

97

(iii) The system S ( P , C d ) is well-posed; equivalently, the closed-loop input-output map

Hy~ : h" t---> y is in m(G). t3

Note that whenever P satisfies Assumption 4.2.1 (i), it has an r.c.f.r., denoted by

(N v , Dr, ) , an 1.e.f.r., denoted by (D-p,/Vp ) and a b.e.f.r., denoted by (Nvr , D , Nt, t , G ) ,

where the numerator and the denominator matrices can be partitioned as follows: In the r.c.f.r.

( Nt, ,Dp ) ,

Np ="

Npl • H n o x n i , D p = :

Dpl e H n i x n i , (4.2.1)

where Np 1 • Hn° 1 x nl , Np2 • H n ° 2 x m , Dp t • H n i l x ni ,Dp2 e H n~2xni .

I. the Lc f.r. ),

E H/*o X/' / i

(4.2.2)

where / )e l • H n ° × nol - - , - . , De2 • H n ° × n ° 2 , Npl • H n ° × m l Ne2 e H n°×ni2

In the b.c.f.r. ( Npr , D , Npt , G ) ,

Npr =-" Npr 1

Nw2

E [OllO1 ] • H n ° x n , Npl=: Npt 1 Npt 2 • H n × m , G-----: G21 G2 2 ,

(4.2.3)

where Nvr 1 e H n ° l x n , Np,2 • H n ° 2 x n N v , • H n × n i l Npt2 • H nxni2

Gll • H n o l x n i l , G12 • H n o l x n i 2 , G21 • H n o 2 x n i l , G22 • H n o 2 x n i 2 . I

D ~ H n x n

The generalized Bezout identity (2.3.12) for the doubly-eoprime pair

98

( (Np "De ), (Dr, ,1Vp ) ) is satisfied for some Vp, Up, Vp , Up ~ m(H). For the b.e.f.r.

(Npr, D , Npt, G ) of P we have the two generalized Bezout identities (2.3.13) and (2.3.14),

partitioned as follows: For the r.c. pair ( Npr , D ), there are matrices Vpr , Upr , X , Y , U ,

r7 e M ( H ) such that

Vpr Upr

-'2 f

O - 6

Np r

Vpr =:

-£

Upr l Upr 2 D - 6 Nprl vx

Ne,2 V2

I n 0

0 Ino

(4.2.4)

for the 1.c. pair ( D , Npl ), there are matrices Vpt, Upt, X , Y , U , V e m(H) such that

D -Npl

U V

Vpl

- Upz

X

=:

Y

D -Npl 1 -Npl 2

U V 1 V 2

Vpt X

-- Uptl Y1

-- Upl2 Y2

I n 0

0 Ini

(4.2.5)

If Ca satisfies Assumption 4.2.1 (ii), then C d has an l.c.Lr., denoted by (D e , N e ) and

an r.c.f.r., denoted by ( N c , Dc ) , where /~ ¢ H m x ni , Nc e H nix no, Nc ~ Hni x no,

Dc¢ H n° x no. Note that (/~c, Nc ) is an 1.c.f.r. of Ca if and only if (/~c 1, Nc I ) is an

1.c.f.r. o f C 1 and ( / ~ c 2 , N c 2 ) is an 1.c.f.r. o f C 2 , where / ~ c l E [_~nil×nil,

/~c2 e H nizx ni2,Ncl e H ml x no1 ,/Vc2 e H ni2xn°2 areasin (4.2.6) below:.

0 Dc2 , Ne = 0 N¢2 ' (4.2.6)

99

Similarly, (Nc, D c ) is an r.c.f.r, of C a if and only ff (N c 1, Dc 1 ) is an r.e.f.r, of C 1 and

(N~2,Dc2) is an r.c.f.r, of C 2, where Dcl ~ H n ° I x n ° t , De2 ~ H n ° z x n ° 2 ,

Net ~ H ni l x n° l ,No2 (~ H ni z x n° 2 areasin(4.2.7)below:

[ocl 0] 0] Dc = 0 Dc2 ' Nc = 0 N~2 ' (4.2.7)

4.2.2. Closed-loop input-output maps of S ( P , C a )

Let Assumptions 4.2.1 hold; the closed-loop I/O map H~-ff

given in terms of ( Inl + C a P )-1 in equation (4.2.8) and in terms of ( lno

n~

tion (4.2.9) below:

P ( I ni + Ca P )-1

( In i +Cd P ) - l - I n i

of the system S ( P , C a ) is

+ p Cd )-1 ill equa-

p ( in i +C d p )-1 Cd

( [ ni + Cd P )-I c d

; (4.2.8)

( in, , + p Cd )-l p

_Cd (ino +p Cd ) - l p

(ino + p Cd )-1 p Cd

Cd (lno +p Cd )-1

(4.2.9)

Note that equations (4.2.8)-(4.3.9) are the same as equations (3.2.6)-(3.2.7), where C d is

replaced by C .

Definition 4.2.3. ( H-stability of S ( P , C d ) )

Thesystem S ( P , C d ) i s sa id tobe H-stable iff Hfu ~ m ( H ) . []

100

4 . 2 . 4 . Analys is ( Descr ipt ions o f S ( P , C d ) using eopr ime factor izat ions )

We now analyze the decenlralized feedback system S ( P, C d ) using coprime factoriza-

dons over m(H) of the plant and the compensator transfer matrices.

Assumptions 4.2.1 hold throughout this analysis.

(i) Analys i s o f S ( P , C d ) with P = Np Dp 1 and C a = ~ 1 Nc

Let (Np ,Dr, ) be any r.c.f.r, o f P ~ m ( G ) , partitioned as in equation (4.2.1) and let

( / ) c ,/Vc ) be any 1.c.f.r. of C d ~ m ( G ) , partitioned as in equation (4.2.6). The system

S (P, C d ) in Figure 4.1 can be redrawn as in Figure 4.2 below, where P = Np D~ 1 ,

C 1 = DclNcl and C 2 = Dc~Nc2;notethat Dp ~p= e2 , Y2 = Np2 ~P'

where ~p denotes the pseudo-state of P .

Ul,- +

U2 •

e 1"

e 2'

u 1 ..... =~"-"="- ":"-"="-"="" q ....... i

r ~ • + , ~ , i y l ~ e 1 ~ i . ............................................... !

t_ . . . . . . . . . . . . t ! U 2 -

I . . . . . . . . . . . . 1 ~ •

i :Y2 2

I I - - I c 2 1 - - I , + ................................................. L.. _ _ l e d

Y l

Y2

Figure 4.2. S ( P , C d ) with P = NpDp -1 and C d = / ~ - l N c •

The system S ( P , C d ) is then described by equations (4.2.10)-(4.2.11) below:

101

D c l D p l + NclNt, 1

Dc2Dt,2 + Ne2Np2

Dc l

0

0 Ncl

0

0 Ul

U2

//1 t

1/2 t

, (4.2.10)

[,1] Np2

n,1 Dp2

1] Y2

YI '

Y2'

0 0

- l n i l

0

0 0 0 0 0 0

- l nl 2 0

0 Ul

0 u2

0 ul p

0 u2 p

(4.2.11)

Equations (4.2.10)-(4.2.11) are of the form

Dill ~p = NItL1 ff

DH 1 ] By Lemma 2.3.2, performing elementary row operations over m(H) on the matrix NttR 1

and elementary column operations over m(H)on the matrix [ NHL ~ DHI ] , w e conclude

that (NHR 1 ,Dn l ,NHL 1 ) is a b.c. triple. Since D n l , Gtt t ~ m(H), it follows from Corn-

ment 2.4.7 (i) that

H ~ =NnRIDf f l NHL l + GHI u m(G) (4.2.12)

if and only if detDnt u I. Since Assumption 4.2.1 (iii) holds, the system S( P, C d ) is well-

posed; therefore detD n I e I. Consequendy, ( NI.IR 1, DH 1, NHL 1 , GH 1 ) is a b.c.f.r, of H ~

and hence, detD n 1 is a characteristic determinant of H ~ .

(ii) Analysis of S ( P , C d ) with P = /~t~ l ATp and C d = N c D c 1

Let (/~p ,Np ) be any 1.c.f.r. o f P E m(G), partitioned as in equation (4.2.2) and let

(N e, D e ) be any r.c.f.r, of C d E m(O), partitioned as in equation (4.2.7). The system

S ( P , C d ) in Figure 4.1 can be redrawn as in Figure 4.3 below, where P = Np D f i ,

102

C~ = N ~ D ~ ~ and C2 = N~2D~ ~ ; note that D ~ I ~ I = e { , D ~ 2 ~ 2 = e2", y ( =

Nc 1 ~ 1, Y 2' = N~2 ~ 2, where, for i = 1, 2 , ~ i denotes the pse~lo-state of Q .

+

U 2 '

e 1 t

e 2 '

I . . . . . . . . . . . . 1

, " l .~c l • "['LVcl [ J I L I.-, I I ' I L.,, 1 l

I _ I

! !

[ - - -'Cd, ..................................................

+- L.!....i.....[ .......... ................. ]/ 1

Figure 4.3. S ( P , C d) with P = D ~ 1Np and C d = N c D c -1 .

The system S ( P , Ca ) is then described by equations (4.2.13)-(4.2.14) below:

g~2

I-}Vpl -}Vp2 D31 D p 2 ] U l

U2

U l ~' ,

U 2 '

(4.2.13)

-Dc~ 0

0 -D~2

Ncl o

0 Nc2

~c 1 Yl

= y2, Y l,

Y2

0

0

0

0

0 Ino 1 0

0 0 Ino 2

0 0 0

0 0 0

/g 1

t~ 2

/ /1 p

U 2 t

(4.2.14)

Equations (4.2.13)- (4.2.14) are of the form

103

DH2 ~ = NHL 2 ff

NHR2 ~ = Y - G x 2 f f •

As in Analysis 4.2.4 (i) above, it can bc easily verified that (NHR 2 , DH2, NHL 2 ) is a b.c. triple

and .that H~-~ = NttR2DtI-~ NHL 2 + GH2 ~ m(G) if and only if detDH2 ~ I . Again by

Assumption 4.2.1 (iii), H ~ ~ m(G) and hence, detDH2 ~ I . Consequently,

(NHR2 ,Dtt2 ,NItL2, Gtl2 ) is a b.c.f.r, of Hy£ and hence, dctDH2 is a characteristic deter-

minant of Hy£.

(iii) Analysis of S ( P, C d ) with e = Npr D-1 Np t + G and C d = D-~ 1 N c

Let ( Npr , D , Npl , G ) be any b.c.f.r, o fP ~ m(G), partitioned as in equation (4.2.3)

and let ( /~c , Nc ) be any 1.c.f.r. of Cd ~ re(G), partitioned as in equation (4.2.6). The sys-

tem S ( P , C d ) in Figure 4.1 can be redrawn as in Figure 4.4 below, where

e =N,,O-'N~+a and C~ =~:~g~ note that 0 ~ = U , , ~ e ~ + ~ ' , , , ~ e ~ , y~ =

Np, l ~x + G t l et + G t 2 e 2 , Y2 = Npr2~x +G21e] +G22e2 , where ~x denotes the

pseudo-state of P .

V .................................................................................. ] i r . . . . . . . . . . i ! " i

i~ C 1 ~ i " I I _1 ~1 I I i . . . . . . . . . . ~ i u2 i ( > - 4 o - 1 ~ 4 I I i

• i r . . . . . . . . . . ! i [ u 2 ~ " + e 2 ! e 2 ' ' J r-=---i r'=---71 ~. ~y " [+e~ i ~2 ++ U ~ i Y 2

Figure 4.4. S ( P , C d ) with P = Np, D-1Np z+ G and C d = D c 1N c .

104

The system S ( P, C d ) is then described by equations (4.2.15)-(4.2.16):

D -Npl 1 -Npl 2

NclNprl / ) c l +/Vc 1011 /Vcl O 12

Nc2Npr2 /Vc2 G21 /9c2 +/Ve2 G22

Yl"

Y 2"

Npl l Npl 2 0

0

0

0

Ul

//2

Ul p

U2 p

(4.2.15)

Nt~1 G11 G12

Npr2 G21 G22

0 Ini I 0

0 0 Inl 2

~x i yt i Y2

y l t =

Yl '

Y2' Y2'

Equations (4.2.15)-(4.2.16) are of the form

DH3 ~3 = NttL3 ff

NHR3 ~3 = Y - on3 ~ •

G l l

G21

0 0

G12 0 0

(/22 0 0

0 0 0 0 0 0

Ul] U2

Ul °

U 2 p

(4.2.16)

As in Analysis 4.2.4 (i) above, it can be easily verified that ( NHR 3, DH3, NHL3 ) is a b.c. triple

and that Hy u = NHR3Dff~ NHL 3 + GH3 • m(G) if and only if detDH3 • I . Again by

Assumption 4.2.1 (iii), Hy u • m(G) and hence, detDtt 3 • I. Consequently,

(NHR 3 ,DH3 ,NHL 3 , GH3 ) is a b.c.f.r, of Hy- a- and hence, detDH3 is a characteristic deter-

minant of H ~ .

105

(iv) Analysis of S( P , C d ) with P = Npr D -1Nvt + G and C d = N c De "1

Let ( N v r , D , N v t , G ) be any b.c.f.r, o f P ~ m(G) , partitioned as in equation (4.2.3)

and let ( /~c , b~c ) be any r.c.f.r, of C a ~ m(G) , panitioned as in equation (4.2.7). The

system S ( P , C a ) in Figure 4.1 can be redrawn as in Figure 4.5 below, where

e = N e r n - l N t , l + G and C d = D ; l f f l c : n o t e t h a t O ~ x = N t , l l e l + N p l 2 e 2 , r l =

Nprl ~ + G l l e l + G12e2, Y2 = Npr2~x +G21 el +G22e2, Dcl ~ 1 = e { , Dc2 ~c2 =

e2" , Yl" = Ncl ~cl , Y2" = Nc2~c2 , where ~x denotes the pseudo-state of P and for i

= 1 , 2 , ~ci denotes the pseudo-state of C i .

I ' r ....................................................................................

+- - ~ ~ I ' ' 1 i i + - - [ I ~"1+1 ['--"]~ax IYl

. . . . . . . . . . . . . . . . N . . . .

Figure 4.5. S ( P , Ca ) with P = NprD-1Np t + G and C a = N c D [ "1 .

The system S( P , C a ) is then described by equations (4.2.17)-(4.2.18):

106

D

Npr2

-Npt:N~:

Dot + G:z Arc:

G21Ncl

- Npl2 Nc2

G12 Nc2

De2 + G22 Nc2 ~2

N m N m 0

- G l l - G12 Inol

-G2~ -G22 0

0

0

Ino2

Ul

U2

Ul t

U 2"

(4.2.17)

Nprl GlINcl G12Nc2

Npr2 G21Ncl G22Nc2 0 No1 0

0 0 Nc2

I

~c2

Yl

Yl,

Y2

GI1

G2I

0 0

GI2 0 0

G22 0 0

0 0 0 0 0 0

Ul

U2

ttl t

U2 r

(4.2.18)

Equations (4.2.17)-(4.2.18) are of the form

DH4 ~4 = NHL4

NHR4 ~4 = Y -- GH4 ~ •

As in Analysis 4.2.4 (i) above, it can be easily verified that ( NHR4, DH4 , NHL 4 ) is a b.c. Iriple

and that H ~ = NHR4Dff~ NHL 4 + GH4 ~. m ( G ) if and only if detDH4 ~ I . Again by

Assumption 4.2.1 (iii), Hy~ ~ m(G) and hence, detDH4 ~ I . Consequently,

( NHR 4 , DH4, NHL 4 , GH4 ) is a b.c.f.r, of Hy-~ and hence, detDH4 is a characteristic deter-

minant of Hyff. []

Theorem 4.2.5. ( H-stability of S ( P , C d ) )

Let Assumptions 4.2.1 (i) and (ii) hold; let ( Np, Dp ) be any r.c.f.r., (/~p,/~p ) be any 1.c.f.r.,

(Npr , O , Npt, 6 ) be any b.c.f.r, over m(H) o f P ~ m ( G ) , partitioned as in equations

107

(4.2.1), (4.2.2) and (4.2.3), respectively; let ( / )c ,/Vc ) be any l.c.f.r., (Nc,D c ) be any r.c.f.r.

over m(H) of C ~ mtG), partitioned as in equations (4.2.6) and (4.2.7), respectively.

Under these assumptions, the following five statements are equivalent:

(i) S ( P , C d ) is H-stable ;

(ii) DH1 := [ lffcDt, +IVcN p ] = DL2Dp2 + Nc2Np2

is H-unimodular ;

(4.2.19)

= [ D'plDcl + NI, 1Ncl D-p2De2 + Np2Nc2] is H-unimodular ; (4.2.20)

(iv) D.3 := iVc Npr £ "b £ G

D -Npt 1 -Npl 2

£1Nprl £1+£1G11 NL2G21

NL2Npr2 /V¢I G21 £2+/Vc2 022

is H-unimodular ; (4.2.21)

(v) DH 4 :=

D -Npl N c

Npr Dc + G Nc

n -NpllNcl -GI2Nc2

Nprl Dcl+GllNcl G12Nc2

Npr2 G21Ncl Dc2+G22Nc2

is H-unimodular . [] (4.2.22)

108

Note that each of statements (i) through (v) of Theorem 4.2.5 implies that the system

S ( P, C a ) is well-posed; consequently, we do not need to state a well-posedness assumption

in the beginning of Theorem 4.2.5.

Proof

Follows from the system descriptions in Analysis 4.2.4 as in Theorem 3.2.7.

Remark 4.2.6

The denominator matrix D H 1 in equation (4.2.19) can also be written as follows:

DH1 =

Ocl 0 Ncl 0

o o

Dp 1

u.t N.2 0

~cl 0 0

o

Dpl Npl

t%

(4.2.23)

and detD H 1 can also be written as detD H 1 = det/)c det( I + C a P ) detDp . By normalization

and due to the block-diagonal compensator structure, DHX ~ m(H) is H-unimodular if and

only if there are block-diagonal matrices Vp : = / ) c , Up := /Vc • m ( H ) such that

Vp Dp + Up Np = Ini . (4.2.24)

Equation (4.2.24) is a Bezout identity where Vp , Up • m(H) are restricted to be block-

diagonal as shown in equation (4.2.23).

The denominator matrix DR2 in equation (4.2.20) can also be written as

- N ~ 0

0 -N~2

Dc I 0

0 De2

Dcl

0

0

0

0

-Nc2

De2

(4.2.25)

109

Following Remark 3.2.10, the denominator matrix DH3 in equation (3.2.21) is

H-tmimodular ff and only if Dc Y + Nc ( Npr X + G Y ) is H-unimodular, where

ffcr +gc(Np, X+OY)

o oo2

Y1

Y2

+

0

0

~2

Nprl X +Gll YI + G12 Y2

Net 2 X + G21 Y1 + G22 Y2

/~cl No1 0 0

0 0 ao2 ~2

Y1

NprlX +GII Y1 +GI2Y2

Y2

Npr2X + G21 YI + G22 Y2

(4.2.26)

Similarly, the denominator matrix DH4 in equation (3.2.22) is H-unimodular if and only if

( X Npt + Y G ) N c + Y D e is H-unimodular, where

(~?Np~ +re )N~ + ~D~

[ ~Np. + YI G11 + Y: G21 XNpl2+ Y1 G12+ Y2G22 ] Ncl

0

0

~ 2

+ [~, ~2] Dcl

0

0

D¢2

[~ XNpt 1 + Y1 Gll + Y2G21 ) Y1 -(XNp12+Yl~12+Y2~22) Y2 ]

N 0 i-- ¢1 Dcl 0

0 -No2 " 0 De2

(4.2.27)

4.3

110

TWO.CHANNEL DECENTRALIZED FEEDBACK COMPENSATORS

Throughout this section, we assume that the plant P satisfies Assumption 4.2.1 (i).

Definition 4.3.1. ( Decentralized H-s tab i l i z ing compensator C d )

(i) C d is called a decentralized H-stabilizing compensator for P

[ C l O ] G n i x n o H-stabilizes P ) i f f Ca = 0 C 2

H-stable.

(ii) The set

( abbreviated as: C d

and the system S (P , C a ) is

C 0 ] Sd(P) := { Ca = 0 C2 [ Ca H-stabilizes P }

set of all decentralized is called the

S( P , C d ).

H-stabilizing compensators for P in the system

where (Np, D e ) is any r.e.f.r, and ( D-p, A~p ) is any 1.c.f.r. of e e G n° × nl and Vp, Up,

~Tp, tTp ~ m ( H ) a~ as in the generalized Bezout identity (2.3.12).

equivalently,

S ( P ) : {(ffe+DeQ)(Vp-NeQ)-I I Q ~ Hnix no, det( I7 e - N e Q ) ~ I },

(4.3.2)

S ( P ) = t C ve - Q gp )-t ( vp + Q ) l Q an' × n° , det( Ve - O ) I ;

(4.3.1)

Comment 4.3.2

In Chapter 3 (Theorem 3.2.11) we showed that the set S ( P ) of all (full-feedback) compensa-

tors that H-stabilize P in the system S ( P , C ) is given by

111

The class of all decentralized H-stabilizing compensators S d (P) is more complicated.

(Note that S d (P) is a subset of S ( P ) ). For the existence of such decentralized compensa-

tors that H--stabilize P in the system S ( P , C a ) , the plant P must satisfy additional condi-

tions which are not required for the existence of full-feedback compensators that would

H-stabilize P in the system S ( P , C ) ; these conditions are due to the block-diagonal struc-

ture of the compensator. I-!

Theorem 4.3.3 below establishes the necessary and sufficient conditions on P for the

existence of decentralized H-stabilizing feedback compensators for P :

Theorem 4.3.3. (Conditions on P = Np Dp -1 = L~ "1/Vp for decentralized H--stabilizability)

Let P satisfy Assumption 4.2.1 (i); furthermore, let P ~ m ( G s ) ; then the following three

conditions are equivalent:

(i) There exists a decentralized H-stabilizing compensator C a for P ;

(ii) Any r.c.f.r. (Np ,Dp ) of P , partitioned as in equation (4.2.1), satisfies conditions

(4.3.3) and (4.3.4) below:

Dpl Ini 1 0

E1 R = , (4.3.3) Nt, 1 0 W~2

E2

Dp2

R =

0 Inl 2

W21 0

(4.3.4)

( i i i ) Any l.c.f.r. (D-p ,/Vp ) of P , partitioned as in equation (4.2.2), satisfies conditions

(4.3.5) and (4.3.6) below:

o- 1 0 [nol

- W21 0

(4.3.5)

112

- W12

0

0

I no2

(4.3.6)

where E 1 e H (ni2+n°l) x (nil+noD is H-unimodular, E 2 • H (ni:z+n°2) x (niz+no2)

is H-unimodular, R • H nix ni is H-unimodular and L • H n°x no is

H-unimodular; the matriees W12 ~ H n ° l x n i 2 and W21 ~ H n°2xn i l are

H-stable in equations (4.3.3) through (4.3.6).

Comment 4.3.4

(i) Statements (ii) and (iii) of Theorem 4.3.3 are equivalent for all P E m ( G ) . Statement (i)

implies statement (ii) for all P ~ m(G) ; the assumption that P ~ m ( G s ) is only needed

in proving that statement (ii) (equivalently, statement (iii)) implies statement (i).

(ii) Since 19 v N v = Np D e , the H-unimodular matrices E 1 ~ H (nil+n°D x (nil+nol)

E2 ~ H(ni2~-no2) x (ni2+no2) and the H-stable matrices W12 • Hnol x ni2,

W21 ~ Hno2 x nil in equations (4.3.3)-(4.3.4) are the same as the ones in equations (4.3.5)-

(4.3.6).

(iii) Let ( Np , Dr, ) bean r.c.f.r, of P ; then by Lemma 2.3.4 (i), ( X p , Yp ) := ( NpR, DpR ) is [x,] alsoanr.c.f.r, o f P ,wherethematr ixR ~ Hni x nl is H-unimodular. Let Xp := xp 2 ,

YP := ¥p2 . By Theorem 4.3.3, P can be H-stabilized by a decentralized compensator

C a if and only if some r.c.f.r. (Xp , Yp ) of P is of the form

Ypl x.1

. . °

Y.2 x.2

Dpl N.1

O.2 R =

Ei -~ 0

0 EE ~

Inn 0

0 W12 . . . . . .

0 lnl 2

W21 0

, (4.3.7)

113

where E 1 , E 2 ~ m(H) are H-unimodular and W12, W21 ~ m(H).

similarly, let (D-'p ,/V/~ )be an 1.c.f.r. o f P ; then (Yp ,J~p ) : = ( L / ) p , L Np) is also an

l.e.f.r, o fP,wherethemat f ixL e Hn°xn°isH-unimodular. LetYt, :=[ Ypl Y ~ 2 ] ,

J~t, := [ Xpl X~2 ] . By Theorem 4.3.3, P can be H-stabilized by a decentralized com-

pensator C a if and only if some l.e.f.r. ( Yp, Xt, ) of P is of the form

I 0 Ino t ! - W 1 2 0 ]

= - W21 0 0 lno 2 J E 1 0

0 E 2

(4.3.8)

where E 1 , E 2 E m(H) are H-unimodular and W12, W21 e m(H) are H-unimodular

matrices of appropriate dimensions.

(iv) Suppose that the plant P is H-stable; then following Comment 2.4.7 (iii), ( P , Ini ) is

an r.c.f.r, and (Ino ,P ) is an 1.c.f.r. o f P = P21 p~. . Clearly, conditions (4.3.3)-

(4.3.4) (equivalently, conditions (4.3.5)-(4.3.6)) hold whenever P E m(It). In this case,

Npl = P l l PI2 and Np 2 = P21 P22 ; hence, the H-unimodular

matrices in conditions (4.3.3)-(4.3.4) (equivalendy, conditions (4.3.5)-(4.3.6)) can be chosen

l nz ~ 0 [ l ni ~ 0 1 = , E 2 = , R = In~; then the matrices as E 1 - P l l Inol -P22 Ino2

W12 = P12, W21 = P21.

(v) Let P satisfy Assumption 4.2.1 (i). Now suppose that P12 = 0, i.e., P --

P21 Pz2 is lower block-triangular; then (Np ,Dp ) :--

114

IN 0]io. 0] ( N2t NZZ ' D21 D22 ) is an r.c.f.r, of P , where (N22, D99 ) is an r.c.f.r, of

P22- Since (N22, D22 ) is an r.c. pair, there is an H-unimodular matrix E 2 ~ m(H) such

that E 2 N22 = 0 . Clearly, condition (4.3.4) holds. Now condition (4.3.3) holds if

and only if ( N i l , D ll ) is also an r.c. pair, i.e., ( N i l , D 11 ) is an r.c.f.r, of P 11 ; by Theorem

4.3.3, in the case that P 12 = 0 , there exists a decentralized H-stabilizing compensator if and

only if ( N i l , D 11 ) is also an r.e. pair. If P is lower block-triangular, then Wl2 = 0 , W21

D21 ] [ Ini 1 0 = [ 0 lno2] E 2 N21 R =

E 1[ ] - lnl 2 0 E 2 N21 Inl 2

P l l P12 ] Similarly, ifP21 = 0, i .e . , P = 0 P22 is upper block-triangular, then(Np ,Dr, )

:= ( 0 N22 ' 0 D22 ) i s a n r ' c ' f ' r ' ° f P " w h e r e ( N l l ' D l l ) i s a n r ' c ' f ' r "

of P l l • This time condition (4.3.3) holds automatically. Now condition (4.3.4) holds if and

only if ( N 2 2 , D 2 2 ) is also an r.c. pair, i.e., (N22 ,D22) is an r.c.f.r, of P22; by Theorem

4.3.3, in the case that P2] = 0 , there exists a decentralized H-stabilizing compensator if and

only if ( N22, D22 ) is also an r.c. pair. If P is upper block-triangular, then W21 = 0 .

(vi) In Section 4.4 we show that, if H is the ring R u as in Section 2.2, then conditions (4.3.3)-

(4.3.4) (equivalently, conditions (4.3.5)-(4.3.6)) on P ~ I~p (s) n°x nl are equivalent to the

condition that the system S ( P , C d ) has no fixed-eigenvalues in I1 .

(vii) Suppose that P ~ m(Gs) is given by a b.c.f.r. (Npr ,D ,Npt , G ) and C d ~ m(G)

is given by an 1.c.f.r. ( D c , N c ) as in case (iii) of Analysis 4.2.4. Considering equation

(4.2.26), apply Theorem 4.3.3 to the r.c.f.r. (Np , Dr, ) := ( Npr X + G Y , Y ) of P ; then

equations (4.3.3)-(4.3.4) imply that P = Npr D-1 Np I + G ~ m(Gs) can be H-stabilized

115

by a decentralized compensator Ca if and only ff

Y1

E 1 R = NprlX + G l l Y I + G 1 2 Y 2

I ni 1 0

0 W~2

and (4.3.9)

E2

Y2

Np,2X + G21 Y1 + G22 Y2

R =

0

W21

I ni2

0

(4.3.10)

where E 1 e H (nil+n°O x (nil+nol), E2 e H (ni2+n°2) x (ni2+no2) and R e H nix ni

H-unimodular; W12 e H n°l xni2, W2 t e H n o 2 x n i l .

ar~

Similarly ff P is given by a b.c.f.r. ( Npr, D , Npl, G ) and C a is given by an r.c.f.r.

(N c, D c ) as in case (iv) of Analysis 4.2.4, then considering equation (4.2.27), we apply

Theorem 4.3.3 to the 1.c.f.r. ( D"p, Ne ) := ( f ' J~ Net + f G ) of P. Following equations

(4.3.5)-(4.3.6), P can be H-stabilized by a decentralized compensator C d ff and only if

0

- W2t

I noi

0

(4.3.11)

and

L [ - ( ~ N m + Y~G~2+Y2622) - - W12

0

0

Ino2

(0..3.12)

where E{ 1 e H (nil+n°l) x (nil+nol) , E~I E H (ni2+n°2) x (ni2+no2) and L ~_ H n°x no

areH-unimodular;, W12 e H n°lxni2 and W21 e H n°2xnil .

Equations (4.3.9)-(4.3.10) (equivalently, (4.3.11)-(4.3.12)) are useful in Section 4.4 where

we explain "rank-tests" for decentralized H-stabilizability in terms of the state-space

representation of P .

116

Proof of Theorem 4.3.3 Statement (i) is equivalent to statement (ii):

Suppose that statement (ii) holds, i.e., any r.c.f.r. (Np , Dp ) of P satisfies conditions

(4.3.3)-(4.3.4); then

E 1 0 . , . . , .

0 E2

Dpl

. . . R =

l ni 1 0

0 W12

0 Ini 2

W21 0

(4.3.13)

Refer to equation (4.2.23) and consider the decentralized compensator C a =

0

0] ~ c l fflc 2 , where l)c1,19c 2 , Nc1, IVc 2 are given by

C1 0 I = 0 C 2

(4.3.14)

Since e l ,e2 ~ m ( H ) , clearly/gcl ,Dc2 ,~Icl ,iVc2 ~ m ( H ) . Since for k = 1 ,2 ,E k is

H-unimodular, (/~ck, 'ffck ) is an 1.c. pair.

With (/)cl ,/Vcl) as in equation (4.3.14) and ( / )c2,Nc2) as in equation (4.3.15) and

(Np ,De ) as in equation (4.3.13), the denominator matrix DH1 in equation (4.2.23) is

H-unimodular since

DH1 =

/)cl 0 /V~I 0

0 & 2 0 N~2

Dp i

D: N,I N:

117

DPt Ini 0

= R -t ; (4.3.16) = 0 0 tic2 ~7c2 Dp2 0 I,,i2

N,2

note thatR ~ H nix ni is H-unimodular. Equation (4.3.16) implies that

/9c Dp + N¢ Np = R -I , (4.3.17)

where R e H nl × nl is H-unimodular. From equation (4.3.17),

det/~ c detDp = det(In, - Nc Np R )detR -I . (4.3.18)

Now ? ~ mfGs) implies that N v a m(G s) and therefore, Nc Nt, R ~ m(Gs) ;

hence, d e t ( l n i - N c N p R ) ~ I . But since detR e J , by equation (4.3.18),

det/~ c detDt, ~ I and hence, by Lemma 2.3.3 (ii), detDp E I and detffc ~ I ; but

det/) c = det 0 /~c2 e I if and only if det/~cl ~ I and det/)c2 e I , where,

from equations (4.3.14)-(4.3.15),

(4.3.19)

This proves that (I~cl ,/Vcl ) given by equation (4.3.14) is an l.c.f.r, of C 1 ~ re(G) a~d

(/)c2 ,/Vc2) given by equation (4.3.15) is a 1.c.f.r. of C~_ e m ( G ) . Now since equation

(4.3.16) implies that DH1 is H-unimodular, the system S ( P , C d ) is H-stable with this

01 choice of decenlralized compensator C a = 0 C2 = 0 Dc~ No2 "

Now suppose that statement (i) holds. Let C d = 0 C 2 be a decentralized

H-stabilizing compensator for P e m ( G s ) ; by Definition 4.3.1, C a satisfies Assumption

118

4.2.1 (ii) and the system S ( P , C a ) is H-stable. Let ( /~1 ,/Vel) be an 1.c.f.r. of C I and

(/~c2 ,/Vc2) be an 1.e.f.r. of C 2 . Let (Np ,Dp ) be any r.c.f.r, o fP , partitioned as in equa-

tion (4.2.1). By Theorem 4.2.5, since S ( P , C a ) is H-stable, the matrix DHI given by equa-

tion (4.2.19) is H-unimodular; hence, Dff] ~ m(H). LetDffi =: R and let

Dpl

Np~ O,,2 N:

R =:

D 11 D 12

Nil N12

D21 D22

N21 N22

By Lemma 2.3.4 (i), since R is H-unimodular, (Nt, R , Dp R

equations (4.2.19) and (4.3.20),

(4.3.20)

is also an r.c.f.r, of P . By

[ /~Cl ~1 0 0 ] o o a~2 t~¢2

[ /~cl /VCl 0 0 ] Dll D12

Nll N12

D21 D22

N21 N22

Dpl

N~

Np2

I nl 1 0

0 lnl 2

(4.3.21)

Let ( N c l , D c l ) be an r.c.f.r, of C1 and (Nc2,Dc2) be an r.e.f.r, of C2; then

-DclNcl+ZVclDcl = 0 and -Dc2Nc2+Nc2Dc2 = 0. By equation (4.3.21), it follows

from Lemma 2.3.7 that there are matrices Vcl, Ucl Vc2, Uc2 ~ m ( H ) , where

VclDcl + UclNcl = lno I and Vc2Dc2 + Uc2Nc2 = Ino2,suchthat

-Ucl ~cl

D11 -Nc l

Nl l Dcl

I nil

0

0

Ino 1

, (4.3.22)

/~2

-v~2

~ 2

v~2

D22

N22

-~2

Dc2

I ni2

0 I no2

(4.3.23)

119

Equations (4.3.22)-(4.3.23) are of the form E t Ei "l

E l :=

are defined as:

- U c l

Qcl

• E2 .'=

= I , E 2 E ~ 1 = l , w h e r e E 1 and E 2

fie2 7c2

-v 2 (4.3.24)

The matrices in equation (4.3.22)-(4.3.23) are H-unimodular; hence the matrices

E1 ~ H(ni l+nol)x (nil+nol) and E 2 ~ H (ni2+n°2) x (niz+no2) in equation (4.3.24) are

H-unimodular. Now let

WI2 :- - U c l D I 2 + VclN12 , W2I := -Uc2D21 + Vc2N21 . (4.3.25)

With the H-unimodular matrices E 1 and E 2 defined as in equation (4.3.24) and

WI2 E H n ° l x n i 2 and W21 ~ H n°2xn i l defined as in equation (4.3.25), by equations

(4.3.21) and (4.3.22) we get:

E 1 0

0 E 2

D n D12

NH N12

D21 D22

N21 N22

E l

0 E2

Opl

R =

I nl I 0

0 W12

0 Inl 2

W2~ 0

(4.3.26)

Equation (4.3.26) implies that any r.e.f.r. ( Np , Dp ) of P satisfies

Dp 1

Npl E? 1

0

l ni t 0

0 W12

0 lni 2

W21 0

R -1 , (4.3.27)

for some H-unimodular E 1 ~ H (nil+n°1) × (nil+nol) E2 E H (niz+n°2) × (ni2-i-no2),

R ~ H hi×hi and for some H--stable W12 ~ I"I ha l×hi2 , W21 ~ H no2×nil . Therefore

any r.c.f.r. ( Np , Dp ) of P satisfies conditions (4.3.3)-(4.3.4).

120

Statement (iO is equivalent to statement (1i0:

Suppose that statement (ii) holds. With E 1 , E2, R , WI2 and W21 as in conditions

(4.3.3)-(4.3.4), consider the following generalized Bezout identity, where L ~ H n° × no is

any arbitrary H-unimodular matrix:

o] [o o] R 0 0 E1 R inl2 0 E2

I 0 ln°l 1 L-1 -W21 0 E1

-W12 L-1

0 E2 [no2

[,,, o] R-1 Ei-I W12

[o E2l W21 0

[oo] E l l I no I 0 L

[00] E21 0 I no 2 L

Ini I 0 ] 0 lnl 2 0

0 Ino I 0 ]

0 Ino 2 ] (4.3.28)

In equation (4.3.28), let

I 0 Ino 1 ] :=L- 1 E l /~pl -- W21 0 '

1.[-2 0] 2 DP 2 0 lno 2 " (4.3.29)

(4.3.30)

Now equation (4.3.28) is of the form

v,l u~l vp2 v,2

-Np! Dpl -Np2 Dp2

121

Dp1 M~I

o,2 -~% N,2 v~

In i

0

0

D

ln~

(4.3.31)

where the H-stable matrices Vpl, Up1, Ve2, Up2, U,1, V,1, Up 2 , V,2 are defined in an

obvious manner by comparing equations (4.3.28) and (4.3.31).

Now we must show that the pair (D-p , A~p ), defined by equations (4.3.29)-(4.3.30), is an

l.c.f.r, of P : Since (4.3.28) is a generalized Bezout identity for the doubly-coprime pair

( (Alp, D, ) , ( 5 , , ~7, ) ), the pair ( D ~ , N/, ) is clearly I.e. Now since ( N p , Dp ) is an

r.c.f.r, of P , by definition, detDp • I ; but by Lemma 2.4.4, since

( (Np, Dp ), ( Dp, Np ) ) is a doubly-coprime pair,

det/)p "" detD, (4.3.32)

and hence, detD, • I . Now by equation ( 4 . 3 . 3 1 ) . - N , D , + D , N , =

- 1o 1 ÷ 1 1- 2o 2 ÷ - - - 0 - ; the fore, and

hence, (Np ,D e ) is an 1.c.f.r. o fP . SinceL • H n°x no is any arbitrary unimodular matrix,

by Lemma 2.3.4 (ii), any l.c.f.r, of P is given by equation (4.3.29) and hence, conditions

(4.3.5) and (4.3.6) are satisfied for any 1.c.f.r. of P .

It is entirely similar to show that statement (iii) implies statement (ii); the proof once

again follows from the generalized Bezout identity (4.3.28). []

Theorem 4.3.3 states that P • m ( G s ) can be H-stabilized by a decentralized compen-

sator C d if and only ff conditions (4.3.3)-(4.3.4) (equivalently, conditions (4.3.5)-(4.3.6)) are

satisfied. So in Theorem 4.3.5 below, in order to find the class of all decentralized

H-stabilizing compensators, we assume that any r.e.f.r. ( Np, Dp ) and any 1.e.f.r. ( 5 p , / q p )

of P • m ( G s ) satisfy these conditions in addition to Assumption 4.2.1 (i).

122

Theorem 4.3.5. ( Class of all decentralized H--stabilizing compensators in S ( P , C d ) )

Let P satisfy Assumption 4.2.1 (i); furthermore let P ~ m ( G s ) ; let any r.c.f.r. ( Np , Dp )

of P satisfy conditions (4.3.3) and (4.3.4) and equivalently, let any Lc.f.r. ( /~p, N~, ) of P

satisfy conditions (4.3.5) and (4.3.6) of Theorem 4.3.3. Under these assumptions, the set

S d (P) of all decentralized H-stabilizing compensators for P is given by

Sd(J') = { c , = o c2 = o 6 ; ~ c ~ I

Qll E H nilxnil , QI ~ Hnilxn°l , Q2 ~ nni2xn°2 , Q22 E H ni2xni2

Q 11 Q 1 W12 ] such that Q2 W21 Q22 is H-unimodular } , (4.3.33)

equivalently,

Sd(P) = C1 0 ] N,I O~ 1 0 ]

o c2 = o No2o~' I

- N ~ 0

De1 0 . . o o . ,

0 -N~2

0 Dc2

On . . .

0

0

. ° °

A

such that W21 Q1 Q22 is H-unimodular } . (4.3.34)

123

Comment 4.3.6

(i) Suppose that the plant P satisfies Assumptions 4.2.1 (i) and is block-triangular as in Com-

ment 4.3.4 (v). Suppose that conditions (4.3.3)-(4.3.4) (equivalently, (4.3.5)-(4.3.6)) of

Theorem 4.3.3 hold as explained in Comment 4.3.4 (v), where either the matrix W12 is zero

(P is lower block-triangular) or the matrix W21 is zero (P is upper block-triangular); then the

matrix

T := i011 o1 12] loll 0] [0 o1110 w211 Q2 W21 Q22 = 0 Q22 + Q2 0 w12 0

is H-unimodular (4.3.35)

Q H E H n i l × n i l and

i.l i 11 i 21 Dc 1 = E~ 1 Ino I ' Dc2 = E~ 1 Irto 2 '

block-triangular, then ( N c 1, De 1 ) and (No2, De2 ) are given by

(4.3.37)

where QI e m(H), Q2 e m(H) are any H-stable matrices of appropriate dimensions.

(ii) The easiest way to choose QI1, Q I , Q2, Q ~ e m(Ru) such that the matrix T in equa-

tion (4.3.35) is H-unimodular is to choose either one (or both) of Q1 and Q2 as the zero

matrix and then to choose both of Ql l and Q ~ as arbitrary H-unimodular matrices (or

without loss of generality, as the identity matrices of size nl 1 and hi2, respectively).

for all Q I , Q2 • m(H) and for all H-unimodular matrices

Q22 • Hn~2 x hi2, In this case, ( / )c I , ]Vcl ) and ( Dc2, ]Vc2 ) arc given by

Hence, ff P • m(Gs) is block-triangular as in Comment 4.3.4 (v) and if conditions (4.3.3)-

(4.3.4) of Theorem 4.3.3 hold, then the set S d (P) of all decentralized H-stabilizing compen-

sators in the expression (4.3.33) is parame~zed by two flee parameter-ma~ces Q 1 and

Q2 ~ m ( H ) . Similar comments hold for the expression in (4.3.34); if P ~ m(Gs) is

124

(iii) In Theorem 4.3.5, if P • m(G) instead of m(Gs), then the matrices Qll, Q1, Q2

and Qz~ e m(H) in the expression (4.3.33) should satisfy condition (4.3.35) and should be

chosen so that

lnil 1 detl~cl:=det([Qll Q1]E1 0 ) e I

Ini2 ] :-- det( L [ Q22 Q2j ] E2 0 ) e I . (4.3.38) and det/~c2

(iv) Suppose that the plant P • m(Gs) is H-stable as in Comment 4.3.4 (iv); then by

C1 Theorem 4.3.5, C a = 0

[.1.2] P 21 P 22 if and only if

0] C2 is a decentralized H-stablizing compensator for P =

(Qll-Q1Pll)-IQ1 0 1 Cd = 0 (Q22-Q2P~)-lQ2 '

for some Q 11, Q22, Q 1, Q2 e m(H) such that

(4.3.39)

Qll Q1P12 ] Q2P21 Q22 is H-unimodular . (4.3.40)

Proof of Theorem 4.3.5

We only prove equation (4.3.33); the proof of (4.3.34) is similar.

We fast show that, if Ca is given by the expression in equation (4.3.33), then Cd

H-stabilizes P : With (Np, Dp ) as in conditions (4.3.3)-(4.3.4) and ( / ) c , ATe ) given as in

the expression (4.3.33), we obtain

DH1 = [ : o

Dpl

~2 ~2

125

Qll QI] E1 : 0

0 " [Q22 ~ 2 ] E 2

Qll Q2W12

Q2W21 Q22

Ini I 0 ] EI"I 0 WI2

. . . [O,n] E21 W21 0

R-1

R -1 = T R -I (4.3.41)

The matrix on the right-hand side of equation (4.3.41) is H-unimodular since by assumption,

the matrix T in equation (4.3.35) and the matrix R in conditions (4.3.3)-(4.3.4) are

H-unimodular. Therefore,

Dc Dp T-1R + Nc Np T-1R = l ni . (4.3.42)

From equation (4.3.42),

det/) c detDp = det[ In, - IV c NI, T-1R ] detR -1 de tT . (4.3.43)

Now e s m(G~)implies that Np e m(G~) and th~fore, gc Np T-1R e m(Gs),

hence, det[ lni - Nc Np T-1R ] e I . By equation (4.3.43), since detR -l detT e J , w e

conclude that det/) c detDp ~ I ; hence, by Lemma 2.3.3 (ii), detDp e I and detDc e I ;

but det/) c = det 0 De2 e I if and only if det/) c 1 ~ I and det/9 c2 e I , where

Inll ]

lni2 ] 1~c2=[Q22 Q 2 ] E2 0

(4.3.44)

126

This proves that the 1.c.f.r. given by the expression (4.3.33), namely

(6~t , ~ t ) = [ a~,

is an 1.e.f.r. of C 1 and

N e l l = [ Q l l Qt ] E1 , (4.3.45)

P~¢2] = [Q22 Q2] E2 , (4.3.46)

is an 1.c.f.r. of C 2 . Now since equation (4.3.41) implies that Dtt 1 is H-unimodular, the sys-

tem S ( P , C a ) is H-stable with this choice of decentralized compensator

c a = It1 01 0]

o c~ = o 5:~c2 Now we show that any decentralized compensator C d that H-stabilizes P is given by the

expression in equation (4.3.33) for some (Qll ,Q1,Q2,Q22) e m(H) such that the

matrix T in equation (4.3.35) is H-unimodular:

[ 5;I ;,~1 o ] By assumption, C a = 0 D'c'~ No2

~7c2 ~ m ( H ) ; hence by Theorem 4.2.5, the

H-unimodular; equivalently,

H-stabilizes P , where/~, 1, De2, Nc l,

matrix DH1 in equation (4.2.19) is

/)c Dp Dff~ + /qc Np Dff] = Inl . (4.3.47)

By Lemma 2.3.4 (i), since O H l is H-unimodular, ( Np Dff~ , Dp Off~ ) is also an r.c.f.r, of

P ; since conditions (4.3.3)-(4.3.4) are satisfied by assumption, equation (4.3.47) implies that

there exists H-unimodular matrices E 1 , E 2 and R such that the r.c.f.r. ( Np D ff ~ , D p D ff ~ )

of P satisfies

[ 5cl ~c, o o ] o o 5c2 go2

Dpl

0~2 ~2

127

[ 5cl ~7ol o o ] = 0 0 /~c2 /Vc2

I . 0 Ei'~ 0 't WI2

. , .

0 lni 2 E2"I W21 0

R -1 = In l .

] Define

[o] E o] clearly, Q 1 and Q2 E m f H ) ; then by equations (4.3.48)-(4.3.49), we get

[~5~1 ~7ol] o 'o' o] R-1

E{1 W12

[o.]. E2"I W21 0

(4.3.48)

(4.3.49)

[oo] E~I Ino I 0 L

[oo] E21 0 I no 2 L

[nil

0

0

l ni2

Ol

0

0

L

Q2 (4.3.50)

Let the H-unimodular matrix R be partitioned as

Q n R12 R =: , QII E H n i l xn i l

R21 Q22

, R12 E H nil×hi2

R21 ~ Hni2xni l , Q22 ~ Hni2x ni2 (4.3.51)

Post-multiply both sides of equation (4.3.50) by the first H-unimodular matrix in (4.3.28);

then

0

128

0

[ lnll

0 I ni2

Qt 0

0 Q2 L

[,+ o] [o o] R 0 0 E1 R lni2 0

0 Ino I ] L-1 -W21 0 E 1

-W12 L-I

0 Ino2

E2

E2

[ [QI, +1]~1 [R,2-Q,+12 o]~2

~2, Q2w2, o]~1 [+22 +~]~2 Now by equation (4.3.52), since E 1 and E 2 are

[R12-Q1W12 O] E2 = Oand[R21-Q2W21 O] El=Oimplythat

R12 = Q1W12 and R21 = Q2W21 ;

therefore, by (4.3.51) and (4.3.53),

Oil Ol w12

Q2 w21 Q22 e m(H) is H-unimodular.

(4.3.52)

H-unimodular,

(4.3.53)

(4.3.54)

We have hence shown that, by (4.3.52), (/)el ,/Vcl ) and (/)c2 ,/Vc2) are of the form given

by the expression in (4.3.33) for some H-stable matrices Qtt ~ Hnitxnil ,

Q1 ~ Hn i l xn° t , Q2 ~ Hni2×n°2, Q22 ~ Hni2xni2 such that condition (4.3.35)

(equivalently, (4.3.54)) holds. []

129

4.4 APPLICATION TO SYSTEMS REPRESENTED BY

PROPER RATIONAL TRANSFER FUNCTIONS

In this section we consider the case where the principal ideal domain H is the ring R u as

in Section 2.2.

K 0 ] Consider the system S ( P , K d ) in Figure 4.6; let K d := 0 K 2 '

K l ~ R n i l x n ° l , K 2 ~. R n i2xn°2 . The system S ( P , K a ) in Figure 4.6 is the same as

S ( P , C d ) , where the dynamic decentralized compensator C d = 0 C2 ~ m(G)

of Figure 4.1 is replaced by the real constant decentralized compensator K a = 0 K 2 "

: ~ el" .................................... ~ ~ Y l U 1" +

/12" e2' P

L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J

Ul

+: U2

i + i

Y2

Figure 4.6. The constant-feedback decentralized control system S ( P , K d ).

Throughout this section we assume that Assumptions 4.2.1 of Section 4.2 hold, with H ,

G and G s replaced by R u , ~ p (s) and Rsp ( s ) , respectively. We also assume that the sys-

tems represented by the transfer matrices P , K and C d have no hidden-modes associated with

eigcnvalues in 11.

130

Let ( N t, ,D e ) be any r.c.f.r, of P ; equations (4.2.10)-(4.2.11) describing the system

S ( P , C d ) as in Analysis 4.2.4 (i) are now replaced by equations (4.4.1)-(4.4.2) describing the

system S ( P , K a ) with constant decentralized feedback:

Dpl .4- KI Npl

o : + K~N:

I nil

0

0 K~ 0

Ini 2 0 K 2

U l

/12

U 1"

tZ2 ~

(4.4.1)

Npl

D,,2

Yl

y2, _ Yl

y f

0 0 0 0

0 0 0 0

- l n i I 0 0 0

0 - I n i 2 0 0

/l 1

/12

U 1 '

U2 p

(4.4.2)

Descriptions of S ( P , K a ) analogous to equations (4.2.13)-(4.2.14), (4.2.15)-(4.2.16) and

(4.2.17)- (4.2.18) of Analysis 4.2.4 are also easy to obtain by replacing Dc I and Dc 2 with I nil

and l ni 2 , replacing N c l and/¢c2 wi thK 1 andK2,replacing De1 and D c 2 with I no 1 and

I no2 and replacing N c 1 and N c 2 with K 1 and K 2 , respectively, in each of these equations.

Corollary 4.4.1 below follows from the H-stability Theorem 4.2.5 for S ( P , C a ) :

Corollary 4.4.1. ( H--stability of S ( P, K a ) )

Let Assumptions 4.2.1 (i) and (ii) hold; let ( N p , Dp ) be any r.e.f.r., (D" e ,/Vp ) be any 1.c.f.r.

over mfRu) of P ~ m~p (s)) Under these assumptions, the following three statements

are equivalent:

(i) S ( P , K a ) is H-stable ;

,., o-., : [ o, + ] Dpt + K t N p l

is H-unimodular ; (4.4.3)

(iii) /~//2 := [ /)t,

131

[ D"pl + NplK1 D~2 + /VI, zK2 ] i s H-unimodular. (4.4.4)

[]

Note that each of statements (i) through (iii) of Corollary 4.4.1 implies that the system

S ( P , K a ) is well-posed.

Definition 4.4.2. ( Decentralized fixed-eigenvalue )

The plant P is said to have a decentralizedfixed-eigenvalue at s o ~ U with respect to K,/ i ff

s o ~ 11 is a pole of the closed-loop I/O map Hy~ of the system S ( P , K a ) f o r all

re e { 0 K2 I g l ~ Rn~×n°~, K2 ~ Nn~2xno2 } .

Remark 4.4.3

Since we assume that the plant represented by the transfer matrix P has no hidden-modes asso-

ciated with eigenvaiues in 1.1, it has no decentralized fixed-eigenvalues in 1.1 corresponding to

hidden-modes; it is clear that if P did have hidden-modes, these would remain as hidden-

modes in the closed-loop system for all real constant-feedback and hence, be associated with

decentralized fixed-eigenvalues.

Now s o ~ [1 is a l l -pole of the closed-loop I/O map H ~ of S ( P , K a ) if and only if

s o ~ ~1 is a zero of a characteristic determinant of S ( P , K a ) . By Corollary 4.4.1, since

/)t t l ~ /~H2, given in equations (4.4.3)-(4.4.4), are characteristic determinants of the

closed-loop I/O map H ~ of the system S ( P , Kct ) , so ~ LI is a decentralized fixed-

eigenvalue if and only if det/)H1 ( So ) = det/)tt2 ( so ) = 0 , for all K 1 , K 2 ~ m ( R ) ; i.e.,

for all

I + Np] (So )KI

K 1 ~ ~ n l I x no I

132

Dp~(So ) + r~lVp](so )

Op2(so ) + K 2 N p 2 ( $ o )

Dp2(So )+ Np2(So )K2 ]

, K 2 E ]R ni2xn°2

= 0

(4.4.5)

If s o ~ 1~ is a decentralized fixed-eigenvalue, then obviously s o ~ 1_1 is an eigenvalue

of the open-loop system P because with K 1 = 0 , K 2 = 0 , equation (4.4.5) becomes

det Dp2(So ) = det D p l ( s o ) D p 2 ( s o ) = 0 andhence , s o is a z e r o o f a

characteristic determinant of P ; this eigenvalue at s o ~ 1_1 remains a pole of the closed-loop

I /0 map H f i of the system S ( P , K a ) for all real constant decentralized feedback compensa-

tors. We prefer to call such s o ~ l_l afixed-eigenvalue rather than afixed-mode; although the

eigenvalue at so ~ [1 remains fixed irrespective of the constant decentralized compensator,

the eigenvector vo associated with the fixed-eigenvalue So ~ 1] depends on K 1 and K 2 .

Therefore the "mode" Vo es°t may change "direction" depending on the choice of constant

decentralized feedback; equivalently, the initial condition that sets up the mode Vo es°t varies

with K 1 , K 2 although the eigenvalue at s o ~ LI does not move. []

Theorem 4.3.3 gives the necessary and sufficient conditions on P e m ( G s ) for a

decentralized H-stabilizing compensator to exist. We now consider these necessary and

sufficient conditions in the framework of R u instead of the general principal ideal domain H ;

i.e., for the special ease that the plant transfer function P is a strictly proper rational function.

Theorem 4.4.4. ( Rank tests on P = Np Dp -1 =/~-1/~p for decentralized fixed-eigenvalues

and H--stabilizability )

Let P satisfy Assumption 4.2.1 (i); furthermore, let P ~ 1Tl(~sp (s)) ; then the following six

conditions are equivalent:

(i)

(ii)

133

There exists a decentralized H-stabilizing compensator C d for P ;

Any r.c.f.r. (Np ,Dp ) of P , partitioned as in equation (4.2.1), satisfies conditions

(4.4.6) and (4.4.7) below:

El ( s ) Dpx(s )

Npl(s ) R ( s )

l niÂ

0 W12(s ) (4.4.6)

E 2 ( S )

Dp2(s )

N : ( s ) R ( s ) =

0 ln i 2

W21 (s ) 0 (4.4.7)

(iii) Any 1.c.f.r. ( / )p ,/Vp ) of P , partitioned as in equation (4.2.2), satisfies conditions

(4.4.8) and (4.4.9) below:

D~1($ ) ] e1(~)-' = 0 Ino 1

- W 2 1 ( s ) 0 (4.4.8)

L ( , ) [ - ~ : ( ~ ) o - : ( s ) ] e2(s )-i = - W n ( s )

0

0

Ino2

(4.4.9)

where El ( s ) E R u (nil+n°l) x (nil+n°l) is Ru-unimodular , E2 ( s ) E

R u (niz+n°2)x(ni~n°D is Ru-unimodular, R ~ RU hi×hi is Ru-unimodular and

L E RU n°xn° is Ru-unimodular; the matrices W12 ~ R u n°lxn~2 and

W21 ~ RunO2 x nli are Ru-stable in equations (4.4.6) through (4.4.9).

(iv) Any r.c.f.r. (Np , Dp )

tions (4.4.10) and (4.4.11) below:

134

of P , partitioned as in equation (4.2.1), satisfies the rank condi-

Dpl(s ) rank

Nplfs ) > nil , forall s ~ 12 , (4.4.10)

(v)

rank Dp2(s )

Np2(s ) > hi2, forall s ~ I1 . (4.4.11)

Any 1.e.f.r. (/)p , Np ) of P , partitioned as in equation (4.2.2), satisfies the rank condi-

tions (4.4.12) and (4.4.13) below:

[ - _ ] rank - N p l ( S ) Dpl (S ) > nol , forall s ~ f i , (4.4.12)

rank [ - N p 2 ( s ) D-'p2(s)] > n o 2 , for all s ~ 1] . (4.4.13)

(vi) The plant P has no decentralized fixed-eigenvalues in 1].

Proof

We proved the equivalence of statements (i), (ii), (iii) in Theorem 4.3.3 for the general princi-

pal ideal domain H ; in Theorem 4.4.4 above, these three equivalent conditions are simply res-

tated for the special case of the ring R U of stable rational functions. Here we first prove the

equivalence of statement (ii) to statement (iv); the equivalence of statements (iii) and (v) can

be established similarly and we omit that proof. We then prove the equivalence of statements

(ii) and (vi).

135

Statement (ii) is equivalent to statement (iv):

(ii) => (iv) For any r.c.f.r. ( Nt, , Dp ) of P , since the matrices E 1 ( s ) , E 2 ( s ) , R ( s ) axe

Ru-unimodular, rank ( E 1 ( $ )

Dp1 (8)

Npl(s ) R ( s ) ) -- rank

%1(s)

Npl(s )

[ ni 1 0

rank

0 W12(s )

by the same reasoning, condition (4.4.7) implies condition (4.4.11).

(iv) => (ii) Condition (4.4.10) implies that there is an

Ru-unimodular matrix L 1 and an n i x n i Ru-unimodular matrix R 1 such that

> nil and hence, condition (4.4.6) implies condition (4.4.10);

L1 Dpl

N.1

I nil

R I =

0

0

( ni I + no 1 )x( tl i 1 + no I )

(4.4.15)

rank [ D2x D22 ] = hi2 , forall s e 1.] . (4.4.17)

By Lemma 2.6.1 (ii). equation (4.4.17) implies that the pair (/~22,/~21 ) is I.e.; hence, (recal-

ling the generalized Bezout identities in Corollary 2.3.8) there exist matrices V2t, U2t , X2,

r2. u2, v2 e m(Ru) such that

A

, N21 ~ Runo2x nil are some Ru-s tab le

A A

- D 2 1 D22

N21 0

Op2

L 2 ( R1 ) =

N,,2

where /~22 e R u ni2xni2, /~21 e R u ni2xnil

matrices and/~21, /~22 also satisfy

(4.4.16)

where /~12 e R u n°l x hi2 is some Ru-stable matrix.

Condition (4.4.11) on the other hand, implies that there is an (hi2 + no2 )×( ni2 + no2 )

Ru-unimodular matrix L 2 (corresponding to elementary row operations in R u ) such that

V2 U2 Y2

136

- U ~

X2 V2t

I nil

0

0

Ini2

(4.4.18)

Now since (Np ,Dr, ) is an r.c. pair, by Lemma 2.6.1 (i), rank Np

and L 2 are Ru-unimodular matrices,

= n i . But since L 1

r a n k (

L 1 : 0

0 ! L 2

Dpl(s ) Npl(s )

. , .

D p 2 ( S )

,v : ( s )

R1)

= r a n k

I nl 1

0

. ° .

D21 ( $ ) A

N2I ( S )

0

,G12 (s ) • . •

D22($ ) 0

= ni , for all s ~ 1] ; (4.4.19)

hence,

r a n k

/~12 (s)

B22(s ) = ni2 , f or all s • 1] . (4.4.20)

By Lemma 2.6.1 (i), equation (4.4.20) implies that (/~12,/~22 ) is an r.c. pair, and hence,

(recalling the generalized Bezout identities in Corollary 2.3.8) there exist matrices V2r , U z. ,

X2 , Y2 , U2, V2 ~ m(Ru) such that

V2r U2r 1hi2

0

0

I nol

(4.4.21)

Using the two generalized Bezout identifies (4.4.18) and (4.4.21), it can be easily verified that

equation (4.2.22) below holds:

V2 + U2 V2r/~21 U2 U2r

-f2 I;21

Now let

R 2 :=

Y2

/Q12X2

137

Y2 - U~

X2 V2z

-u2 E=

E RU nixnl ;

]nil

0

0

Inol

(4.4.22)

(4.4.23)

then R 2 is Ru-unimodular by equation (4.4.18). Let

Inil U2 V2,.

R := R 1 R 2

0 Ini 2

Run/X ni ; (4.4.24)

then R is Ru-unimodular by (4.4.15) and (4.4.23). Now let

E I "=

V2 + U2 V2r/~21 U2 U2r

-,~2/~2~ Y2

L 1 E R u (nil+n° I) x (hi l+no 1) ; (4.4.25)

then E 1 is Ru-unimodular by equations (4.4.22) and (4.4.15). Let

E 2 :=

~11i2 0 L 2 E R u (ni2+n°2) x (ni2+n°2) ; (4.4.26)

then E z is also Ru-unimodular by equation (4.4.16). Now let

W12 := X2 ~ RU n ' lxni2 and W21 := JQ21Y2 E R u n°2xnll (4.4.27)

Then from equations (4.4.23)-(4.4.27) we obtain

138

E 1 0

0 E 2

,,11 N~I • - . [R o,2

I Np:z J

[nil 0

o * ,

0

W21

0

W12

[hi2

0

(4.4.28)

Equation (4.4.28) implies that for any r.c.f.r. (Np , Dp ) of P , conditions (4.4.6) and (4.4.7)

are satisfied for some R u - u n i m o d u l a r matrices E t , E2 and R .

Statement (ii) is equivalent to statement (vi):

(vi) = > (ii) We will show that if statement (ii) does not hold, then statement (vi) does not hold

either. We proved above that statement (ii) is equivalent to statement (iv), therefore if state-

ment (ii) fails then statement (iv) also fails. Suppose, without loss of generality, that condition

(4.4.10) fails; i.e., there is an s o ~ t l such that

rank

Dpl(So)

Npl(So ) < nil • (4.4.29)

Equation (4.4.29) implies that for all K 1 E ~,.ni i x no 1,

r a n k [ D p l ( S o ) + K l N p l ( S o ) ] = r a n k ( [ I n i l K11

Opl (so)

Ne l (so)

< rank

Opl(So)

Npl(So) < nil . (4.4.30)

But equadon (4.4.30) implies that for all K 1 ~ Rnl I x no 1, K2 E F. hi2 x no2,

rank

Dpl ( S o ) + K1Npl ( S o )

h : ( s o ) + K2N:(so )

139

(4.4.31)

and hence, by equation (4.4.5) of Remark 4.4.3, this s o ~ 1.i is a decentralized fixed-

eigenvalue; therefore, statement (vi) fails.

(ii) => (v/) By assumption, conditions (4.4.6) and (4.4.7) hold for any r.c.f.r. ( Np , Dp ) ofP ;

suppose now, for a contradiction, that P has a decentralized fixed-eigenvalue at s o E 1_1.

Then by equation (4.4.5) of Remark 4.4.3, since the matrix R ~ RU nlx nl in conditions

(4.4.6)-(4.4.7)is Ru-unimodular, forall K 1 ~ IR nil x n°l , K 2 ~ ~ n i 2 x no2 ,

Ot, l ( So ) + KI Npx ( s o ) Dp1( So ) + Ki Npl ( s o )

rank R ( s o ) = rank < n i

Dp2(So ) + K2Np2(s o ) Dp2(So ) + K2Np2(s o )

Now let

Dpx (so)

Npl(so) R (S o ) =:

Dll (So) D12 (So)]

Nl1(So) Nl2(So

(4.4.32)

(4.4.33)

rank

DII (So)

NIl (So) = r a n k ( E l ( S o )

Dll ( S o )

Nil ( S o )

) = rank

0 = nil . (4.4.34)

Now by Corollary 2.6.3 (ii), equation (4.4.34) implies that there exists a real constant matrix

R 1 E I~ n~l Xnol such that

[ ^ ] rank D l l ( S O ) + K1Nl l ( s O ) = n i l ; (4.4.35)

i.e., the complex matrix ( D 11 ( So ) + K1 N I l ( So ) ) ~ cnilx nll is nonsingular. Let

Lemma 2.6.1 (i), the pair ( N i l , D 11 ) is r.c. because

• Since the matrix E 1 ~ RU (nil+n° t)x (nil+no 0 in condition (4.4.6) is Ru-unimodular, by

140

L1 := ( D l l ( S o ) + ~ l N l l ( S o ) )-1 and R 1 :=

note that L]

(4.4.36),

= LI[ Dll($o)

Now let

Inil - L I ( D I 2 + g l N 1 2 ) ( S o ) ]

0 lni 2 J ;

(4.4.36)

C nilxnil and R 1 ~ C nixnl are nonsingular. By equations (4.4.33) and

+ K 1 N l l ( S o ) D 1 2 ( s o ) + K 1 N l z ( S o ) ] R 1 =

z l ( o p l ( S o ) + RiNpl(So ) )

Dp2 ( So )

Ne2(So )

Inl I 0 ] .

(4.4.37)

R (s o )R 1 =:

Ini 1 0 D21( So ) D2:2 ( So )

N 2 1 ( s o ) N 2 2 ( s o )

(4.4.38)

Since L 1 and R 1 are nonsingular complex matrices, equations (4.4.32) and (4.4.38) imply

that for all K 2 ~ R ni2xn°2,

rank

A

Opl (So ) + X l N , l (So )

Op2(So ) + K2Np2(So ) R (So)

= rank

Ll(DplfSo ) + RxNi, x(So ))R (So)R~

(Dp2(s o ) + K2Np2(s o))R (s o )R 1

Ini I 0 0 ] = rank ( 0 Ini 2 K 2

I ni I 0

D21( so ) D22(So)

/V21 ( So ) N22 ( So )

= n i l + r a n k [ D 2 2 ( S o ) + K 2 N 2 2 ( S o ) ] <n i • (4.4.39)

141

But equation (4.4.39) implies that for all K 2 ~ ~ni2 x no2,

rank [ D22(So) + K z N 2 2 ( S o ) ] < h i 2 ,

and hence, by Lemma 2.6.2 (i),

l" -I max rank | D22(s o ) + K2N22(s o ) / = rank

t .

D 22 ( $o )

N22 ( $o )

Equation (4.4.41) implies that

rank

Ll(Dt~l(So ) + V. 1Npl(so )) Dp2 ( So ) Np2 ( So )

< hi2 •

I Inl t 0 R (So)R1 = rank D21(s o ) D22(s o )

N21 ( So ) N22 ( So )

(4.4.40)

(4.4.41)

< nil + ni2 . (4.4.42)

But the matrix R 1 ~ cni× ni is nonsingular and the matrix E 2 6 R u (ni2+n°2) x (ni2+no2) in

condition (4.4.7) is Ru-unimodular; therefore, condition (4.4.7) and equation (4.4.42) imply

that

rank (

[/lil 0

0 E 2

L~ (Op~ (so) + R~Np~ (so)) ° . .

op2 ( So ) %2(So)

R (s o )

= rank

lnil L1 (D12(So) + K1N12(So ) )

0 Ini 2

W21 ( So ) 0

< nil + hi2 (4.4.43)

This is clearly a contradiction since by elementary row operations on the second matrix in

equation (4.4.43), it is easy to see that the rank of the matrix in (4.4.43) should be exactly

equal to n i 1 + hi2 • We conclude that whenever conditions (4.4.6) and (4.4.7) hold, the plant

P can not have any decentralized fixed-eigenvalues in 1.1. []

142

Corollary 4.4.5. ( Rank test on P = Npr D-1 Nt, t + G for decentralized fixed-eigenvalues

and H--stabilizability )

Let P satisfy Assumption 4.2.1 (i); furthermore, let P e m(Rsp(S)) ; then the following

three conditions are equivalent:

(i) Any b.c.f.r. (Nt, r , D , Nt, 1 , G ) of P , partitioned as in equation (4.2.3), satisfies the

rank conditions (4.4.44) and (4.4.45) below:

rank

o (s) -Npl2 (s)

Nprl (s ) G12 -> n , for all s e 12 , (4.4.44)

rank

0 ( s ) -/vptl ( s )

Npr 2 ( s ) G21 > n , for all s e 1~ . (4.4.45)

(U) The plant P has no decentralized fixed-eigenvalues in 1~.

(iii) There exists a decentralized H-stabi l izing compensator C a for P .

P r o o f

We prove that conditions (4.4.44) and (4.4.45) are equivalent to conditions (4.4.6) and (4.4.7)

of Theorem 4.4.4; the equivalence of statements (i), (ii) and (iii) of Corollary 4.4.5 follow.

Note that the equivalent conditions (i) and (ii) of Corollary 4.4.5 are equivalent to (4.4.6) and

(4.4.7) for all plants P e m0Rp (s)) ; the assumption that P is strictly proper is only

needed to establish that these conditions are in turn equivalent to condition (iii).

By Theorem 4.3.3 applied to the r.c.f.r. (Alp, X + G Y , Y ) of P e m(lRsp (s)) as in

Comment 4.3.4 (vii), condition (4.4.6) is equivalent to

Y l ( S )

E l ( s ) R ( s )

Nprl(S )X (s ) + G I I ( S ) Y I ( S ) + G12(s ) Y 2 ( s )

143

ln~ t 0

0 Wt2(s )

and condition (4.4.7) is equivalent to

Y2(s )

E2( s )

Npr2(s )X ( $ ) + G 2 1 ( s ) Y I ( $ ) + G22(s ) Y 2 ( s )

(4.4.46)

R(s)

0 Inl 2

W21 ( s ) 0 (4.4.47)

where E 1 e R u (nil+n°Ox(nil+n°O , E 2 E R u (ni2+n°2)x(ni2+n°2) and R e R u nlxni

are Ru-unimodular; W12 e R u n°lxni2 , W21 • R u n°2xnil • Now by Theorem 4.4.4,

condition (4.4.6) is equivalent to condition (4.4.10); therefore, by (4.4.46), since E 1 and R are

Ru-unimodular matrices,

rank E 1 ( s )

r1(s )

Npr I ($)X ($)+GII ($)YI (S)+G12($ )Y2($ ) R(s)

= rank

r1(s )

Npr I(s )X (s )+GII(S )YI(s )+GI2(s )Y2(s )

for all s e

Now from the generalized Bezout idcntity (4.2.5),

D -Npl 1 -Npl 2

0 Ini I 0

Nprl GII (712

f i .

-Uvzl -Uvl2

X

Y1

Y2

nil ,

(4.4.48)

144

I n 0

= -- U p l l Y1 (4.4.49)

Nprl VpI -Gl l UpI1-G12Upl2 NprlX + Gll YI +G12Y2

Since the second matrix on the left-hand side of equation (4.4.49) is Ru-unimodular by

(4.2.5), for all s ~ 1] , the rank of the first matrix on the left-hand side is the same as the rank

of the matrix on the right-hand side; therefore, by equation (4.4.48),

z) (s ) - N m ( s ) - N m ( s ) rank 0 I nl 1 0

N p r l ( S ) G I l ( s ) a 1 2 ( s )

= n + rank

Y l ( s )

Nprl(S )X ($ ) + Gll($ ) Y I ( s ) + G 1 2 ( s ) r 2 ( s ) > n + n i l ,

for all s ~ 1] . (4.4.50)

By elementary row operations on the first matrix in equation (4.4.50), it is easy to see that

rank

D ( S ) - N p l l ( S ) -Npl2 ( s )

0 Ini I 0

N p r l ( S ) G I I ( $ ) G12(s )

= rank

D ( s ) - N m ( s )

Np,1 ( s ) G12 + nil >- n + nil , f o r a l l s ~ 11 . (4.4.51)

By equation (4.4.51), we conclude that condition (4.4.6) is equivalent to condition (4.4.44).

The equivalence of condition (4.4.7) to condition (4.4.45) can be established similarly using

equation (4.4.47).

145

Remark 4.4.6. ( Rank test on P = C ( s l n - A ) - I B + E for decentralized fixed.

eigenvalues and H-stabilizability )

Let P satisfy Assumption 4.2.1 (i); let ( A , B , C , E ) be the state-space description of

Pm(Rp(S)) given in Example 2.4.3. Following Example 2.4.3, (Nt, r ,D ,Npt ,G ) is a

b.c.Lr, of P , where

Npr -'~ I (s +a )-1 C1

(s +a )-I~2

D : = [ ( s

Npl :=[ B1

+a )- l (s l n - , 4 )] ,

[ -] E11 El2 B2 G := -

' E21 ff~22

where - a ~ R c~ e\ fi. X ~ m n x n , C1 E Px n° lxn ,

B1 E A n×nil , B2 E ]R nxni2, /~11 E [~nol×nil,

/~'21 E R no2×nil , ~'22 E ~?,no2×ni2.

Now by Corollary 4.4.5, the following two conditions are equivalent:

(i)

, (4.4.52)

C2 E ~nozX n ,

/~12 E •nol×ni2,

The state-space representation ( A , B , C , E ) o f P , partitioned as in equation (4.4.52),

satisfies the rank conditions (4.4.53) and (4.4.54) below:

rank s I . - X -B2

> n , for all s ~ 1J , (4.4.53)

rank c2 e21

> n , for all s ~ 12 . (4.4.54)

(i|) The plant P has no decentralized fixed-oigenvalues in 1J .

Furthermore, if P

also equivalent to:

146

E m(Rsp (s)), then the equivalent conditions (i) and (ii) above are

(iii) There exists a decentralized H-stabilizing compensator C a for P .

For the special b.c.f.r. ( Npr , D , Net • G ) given in equation (4.4.52), the rank conditions

(4.4.53)-(4.4.54) are the same as (4.4.44)-(4.4.45); for simplicity, we omitted the factor

1 in equations (4.4.53) and (4.4.54) since - a ~ 0 .

( s + a )

If P has any decentralized fixed-eigenvalues, then these are a subset of the eigenvalues;

hence, since they automatically bold for all other s e C, conditions (4.4.53)-(4.4.54) need to

be checked only at the l~--eigenvalues of ,4 , i.e., only for those s E 0 such that

det(s I n - , 4 ) = 0 . Similarly, conditions (4.4.44)-(4.4.45) need to be checked only at the

U-zeros of detD ( s ) .

Corollary 4.4.7. ( Decentralized fixed.eigenvalues of P remain in S ( P , C a ) )

Let P satisfy Assumption 4.2.1 (i). Let s o ~ O be a decentralized fixed-eigenvalue of P ;

then the closed-loop I/O map Hy~ of the system S ( P , C a ) also has a pole at s o ~ O for all

dynamic decentralized compensators C a .

Proof

Suppose that so ~ O is a decentralized fixed-eigenvalue of P ; then by Theorem 4.4.4, state-

0 . Suppose, without loss of generality, that condition (4.4.10) fails, ment (iv) falls at So E

i.e., that

rank

Opt (So)

Np l ( So ) < nil , (4.4.55)

where ( Nt, , Dp ) is any r.c.f.r, of P . Equation (4.4.55) implies that

rank ( [IDcl ( s o ) N~I (so)]

147

Dp1( so )

Ivpl (So) ) < rank

D~,I (So)

Npl ( So ) < nil ,(4.4.56)

for all / )e l ( So ) • c n i l x n l i , No1 ( S o ) • cn i l×n° t . But equation (4.4.56) implies that

the characteristic determinant D H 1, defined in equation (4.2.19), has a zero at s o E t l since

rank DH1 = rank

( Dc2Dp2 + ~Vc2Np2 )(so )

<rank [ ( / g c l D p l + N c l N p l ) ( S o ) ] + r a n k [ ( / ~ c 2 D p 2 + N c 2 N p 2 ) ( S o ) ] < n i l + ni2 ,

(4.4.57)

for all f f c l ( S o ), Ncl(So ), Oc2(So ), Nc2(So ) • m(c);consequently, s o • ~l is

a pole of the closed-loop I/O map Hy~ of the system S ( P , C a ) since it is a zero of the

characteristic determinant D H 1 of S ( P , C a ) for all C a = Dc -I Nc • []

Comment 4.4.8

(i) By Theorem 4.4.4, so ~ 1_1 is a decentralized fixed-eigenvalue of P if and only if either

rank

Dpl(So)

Npl(So) < nil (4.4.58)

or

rank

Dp2 ( so )

Np2 ( So ) < n~2. (4.4.59)

Now (4.4.58) and (4.4.59) cannot both hold at any s o • ~1 since ( N p , Dp ) is an r.c.f.r, of

P ; i.e., if conditions (4.4.10) and (4.4.11) both failed at some s o ¢ 11, then by (4.4.58)-

(4.4.59), this would imply that

148

rank

D e (so)

Np (so) < rank

D1,1 (so)

Npl(So) + rank

Dp2(So )

Ne2 ( So ) < nil + hi2 . (4.4.60)

But by ]_,emma 2.6.1 (i), equation (4.4.60) contradicts the fact that ( N e , D e ) is a r.c. pair.

The same comments apply to the rank conditions on a 1.c.f.r. ( / ) e , Ne ) ' on a b.c.f.r.

( N F , D , Npl , G ) and on the state-space representation ( A , B , C , E ) of P ; i.e, condi-

tions (4.4.12) and (4.4.13) cannot both fail, conditions (4.4.44) and (4.4.45) cannot both

fail, conditions (4.4.53) and (4.4.54) cannot both fail at any s o ~ ~1.

(ii) By Theorem 4.4.4, i f P has no decentralized fixed-eigenvalues in 1 ] , then conditions

Dpl Ini 1 0

(4.4.6)-(4.4.7) imply that the matrix can be put in the (Smith) form ^

Net 0 W 1

and at the same time, the matrix

Op2

can be put in the (Smith) form

0 Ini 2

v~2 o A A

where W 1 ^ 6 H n°l × ni2 and W 2^ ~ H n°2 × nil ; the only nonzero entries of W 1 and W 2

may be the ones on the diagonal; some o f these diagonal terms may actually be in J

(equivalently, be equal to 1 ). Therefore, conditions (4.4.6)-(4.4.7) imply that for j = 1 , 2 ,

Dpj

the first n o invariant factors of are equal to 1 ; if the n O-th invariant factor is zero

uej

at some So ~ [1 for either j = 1 or j = 2 , then s o ~ LI is a tixed-eigenvalue of P .

(iii) By Assumption 4.2.1 (i) on P , considering the b.c.f.r. (Npr , D ,Npl , G ), which is par-

titioned as in equation (4.2.3), we can write P as

P = P21 P22 = Npr2D-1Npl l + G21 IVpr2D-1Npt2 + G22

149

If ( Ner l , D , Npl 1, G It ) is a b.c.f.r, of P 11, then the plant P can have no decentralized

fixed-eigenvalues in l] ; similarly, if ( Net2 , D , Npt 2 , G22 ) is a b.c.f.r, of P22, then the plant

P can have no decentralized fixed-eigcnvalues in 1.] : To see this, note that if (Ner 1, D , Net 1 )

[ is a b.c. triple, then by Lemma 2.6.1 (i), rank Ner 1 ( s ) = n for all s ~ l~ since

( Npr 1, D ) is an r.c. pair (this shows that condition (4.4.44) holds) and by Lemma 2.6.1 (ii),

rank (this

shows that condition (4.4.45) holds).

In terms of the state-space representation of P as in Remark 4.4.6, this same sufficient

condition for P to have no decenlxalized fixed-eigenvalues in 11 is restated as follows: If

( C I , ( s I n - / ~ ) , B 1, El l ) is a l~-stabilizable and U-detectable state-space representation

of P 11, then then the plant P can have no decentralized fixed-eigenvalues in 1~ ; the same

conclusion follows if ( C 2 , ( s I n - A ) , B 2 ,E99 ) is a 1.'i-stabilizable and l~--detectable

state-space representation of P 22.

(iv) Suppose that s o ~ [1 is a decentralized fixed-eigenvalue of P ; then by Corollary 4.4.5,

either condition (4.4.44) fails or condition (4.4.45) fails. Now if (4.4.44) fails, then

( Npr I , D ,Npl 2 ) is not a b.c. triple and hence, P12 = Nprl D - t Nel2 + G12 has a hidden-

mode associated with the eigenvalue s o e (1 of P . If, on the other hand, (4.4.45) fails,

then ( N e , 2 , D , N e l l ) is not a b.c. triple and hence, Pal = N e , 2 D - 1 N e l l + G2I has a

hidden-mode associated with the eigenvalue s o ~ t l of P .

(v) Let not be equal to nll and let no2 be equal to ni2. Following Comment 4.4.8 (iv)

above, suppose that s o ~ [1 is a decentralized fixed-eigenvalue of P ; suppose, without loss

of generality, that this decentralized fixed-eigenvalue is due to the failure of condition (4.4.44)

at s o ~ ~1. Therefore,

and

But (4.4.62) implies that

150

D (so) ] rank Npr 1 ( so ) < n (4.4.62)

r a n k [ D ( s o ) -Npt2 ( So ) ] < n . (4.4.63)

D(so) -Nm(So) ] rank Nprl ( s o ) G11 ( s o ) < n + nil (4.4.64)

and (4.4.63) implies that

D (s o ) -Nt,12(s o ) ] rank Npr2 ( So ) G2 2 ( So ) < n + hi2 ; (4.4.65)

(note that here we assume ni2 = no2 ). Furthermore, the failure of (4.4.44) also implies that

[ D ( S o ) - N p t l ( S o ) - N p t 2 ( s o ) ] rank Npr l (S ° ) G l l ( S o ) G12(So ) < n + nil (4.4.66)

and therefore, by (4.4.66),

O ( s o )

rank Np, l ( s o )

Npr2 ( $o )

-Nm(so) -Nm(so) Gn(so) G12(So) G21 (so) G22 ( so )

< n + nil + ni2 . (4.4.67)

Assuming that P has normal rank equal to n i P l l has normal rank equal to nil = nol,

P22 has normal rank equal to t/j2 = n o 2 , e q u a t i o n (4.4.64) implies that P l l has a

transmission-zero, equation (4.4.65) implies that P22 has a transmission-zero, and equation

(4.4.67) means that P has a transmission-zero at the decentralized fixed-eigenvahe

s o ~ t l . It can be shown similarly, that if the decentralized iixed-eigenvalue at s o E L I

was due to the failure of condition (4.4.45) at s o ~ t l , then again, each of P11, P22 and

P have a transmission-zero at the decentralized fixed-eigenvalue s o ~ ~1. []

151

4.4.9. Algorithm for two-channel decentralized Ru-stabilizing compensator design

The proof of Theorem 4.4.4 ((iv) --> (ii)) suggests the following algorithm to find the

class of all decentralized Ru-stabilizing compensators for a strictly proper, two-channel plant

using Theorem 4.3.5. This algorithm is based on any r.e.f.r. (Nt, , D e ) of P ; similar algo-

rithms can be obtained based on any 1.e.f.r. or b.e.fx, of P as well.

Given: a plant P ~ Rsp (s)n°x ni which satisfies Assumption 4.2.1 (i).

Find: an r.c.fx. ( N p , Dp ) o fP ; partition Op andNp as in equation (4.2.1).

Check: that Dpl, Npl satisfy the rank condition (4.4.10) and that Dp2, Np2 satisfy the rank

condition (4.4.11). If either one of these conditions fails, then stop; there is no decentralized

Ru-stabilizing compensator for this plant.

Step I: Find Ru-unimodular matrices L 1 and R l ~ m(Ru) such that

Dpl [hi 1 0

L1 RI = , 0

where N12 E Ru n°~ x n~2 is some Ru-stable matrix.

(4.4.68)

Step 2: Find an Ru-unimodular matrix L 2 ~ m(Ru) such that

L2( Op2

RI) = -B21

A

N21 0

(4.4.69)

where /~22 ~ R u n~2xn~2 , /~2t ~ R u ni2xn'l , JQ21 ~ R u n°2xn~l

matrices and /~21, /~22 also satisfy

( D 2 2 , D 2 1 ) is anl.c, pa i r ;

are some Ru-stable

(4.4.70)

equivalently, rank [ D21 /~221 = hi2 ,f°rall s E U.

Step 3:

V2t, U92, X2, Y2,

152

Find a generalized Bezout identity for the 1.c.

V2

-/~21

U2, V 2 E

U2

B.

m(Ru) such that

Y2 - U2t

X2 V2t

pair ( D22 , O21 ) ; i.e., find matrices

I nil

0

0

I ni2

(4.4.71)

Step 4:

v2,, u2, ,,'72, f2 , v2, v2 ~ m(Ru) such that

u:.. -tT

-X2 Y2 /~12 V2

Find a generalized Bezout identity for the r.c. pair (NI2,/~22 ) ; i.e., find matrices

I hi2

0 l nol

(4.4.72)

Step 5: Define

e 2 ;-- Y2 - U2t

X2 V21

¢ R u nlxn~ ,-

Define

E 1 1=

R "=

V2 + U2V2r/~21

-~'2/~2~

RIR 2

I nil

0

U2 U2r

U2 Vz~

l ni2

RuniX ni

L 1 ~ R u (hi 1+no l) x (hi l+no 1) ;

(4.4.73)

(4.4.74)

E 2 :=

] ni2

/Q21U2/ U2 X'2

0

I na2

L 2 E RU (niz'l'n°2) x(ni2+no2) (4.4.75)

Finally define

153

Wa2 := X2 ~ RU n°xxni2 and W21 := /Q21Y2 e RU n°2×nil (4.4.76)

Step 6: Find matrices Ql l ~ Ru nilxnil, Q1 ~ Ru nitxn°l,

Q22 E RU ni2 x hi2 such that

Q2 E RU ni2×n°2,

Q 11 Q 1 W12

QzW21 Q22

is Ru-unimodular (4.4.77)

Step 7: Let

and let

(4.4.78) a n d / ) c 2 , /qc2 are given by (4.4.79). []

(4.4.79)

are given by

Example 4.4.10

We now follow the steps of Algorithm 4.4.9 to find a decengalized Ru-stabilizing compensa-

tor for the plant

p =

1 0

s - 2

1 - ( s + l ) s - 1 ( s - 1 ) ( s - 2 )

(4.4.80)

where nll= ni2 = 1. Let 1] be the set of all s ~ C whose real part is greater than or

equal to - 1. An r.c.f.r. ( Np , Dp ) of P is then given by

154

1 0 s + 4

1 0

s + 4

, o .

Now the rank conditions (4.4.10)-(4.4.11) hold since

rank

D, l ( s )

Np1(s ) = rank

s + l s + 4

s - 2 s + 4

s + l s - 1 s + 4 s + 4

1 0

s + 4

s - 1 s + 4

n i l = 1 , and

(4.4.81)

rank

Op2(s )

Np2(s) = rank

s - 2 0

s + 4

1 0

s + 4

n i 2 = 1 , for all s E 1] .

Step 1: One choice for the Ru-unimodular matrices L 1 and R 1 in equation (4.4.68) is:

Z 1 =

1 3

- 1 s + l s + 4 s + 4

, R1 =

s - 1 s + 4

0 - 1 (4.4.82)

and h e n c e , ]Q12 -- $ - 1 (S + 4 ) 2 "

Step 2: One choice for the Ru-unimodular matrix L 2 E m(Ru) in equation (4.4.69) is:

L 2 =

1 -(lls+14)s+4 l

1 ( s - 1 ) ( s - 2 ) s +4 (s + 4 ) 2

(4.4.83)

With L 2 as in equation (4.4.83), we get

A A

-D21 D22

A

N21 0

155

s - 2 1

$ + 4

S - 2 ($ + 4 ) 2

(4.4.84)

From (4.4.84), the pair (/~22 , /~21 ) -- ( 1 , -(s -2)

s + 4 ) is coprime.

A A

Step 3: A generalized Bezout identity as in (4.4.71) for the 1.e. pair ( D22 , D2t ) is given by

1 0

s - 2 1

s + 4

1

- ( s - 2 ) s + 4

0 [lo} °

0 1 (4.4.85)

Step 4: A generalized Bezout identity as in (4.4.72) for the r.e. pair (/Q12,/~22 ) is given by

1

- ( s - i )

0 1

S--1 (S + 4 ) 2 (S + 4 ) 2

:[lo 0 1

(4.4.86)

Step 5: By equation (4.4.73),

R

11s + 14 ( s + 4 ) 2

s - 2 s + 4

s - I s + 4

- 1

(4.4.87)

By equations (4.4.74)-(4.4.75),

E 1 =

1

- ( l l s + 14)

3

s3+ 12s2+ 15s +22

(4.4.88)

(S + 4 ) 3 (S + 4 ) 3

156

E 2 =

- ( l l s + 1 4 ) 1

s + 4

1 ( s - 1)(s - 2 ) s + 4 (s + 4 ) 2

= L 2 . (4.4.89)

By equation (4.4.76),

s - 1 s - 2 W12 - ( s + 4)2 ' W21 - ( s + 4)2 " (4.4.90)

Step 6: Choose Ql l = 1 , Q22 = 1 , Q1 = 0 and Q2 = 0 . Condition (4.4.77) is

satisfied with this choice of Qxt , Q1, Q2 and Q99 ~ Ru since

oll Qlw12] l Q2 W21 Q22 0 I

Step 7: By equation (4.4.78),

1

- ( l l s + 14)

( s + 4 ) 3

3

s 3 + 12s 2 + 15s + 22

( s + 4 ) 3

and by equation (4.4.79),

[ z~2 ~2 ]-- 1

:[1

1 3]

1

01 1

s + 4

- ( l l s + 1 4 )

s + 4

- ( l l s + 14) s + 4

( s - 1 ) ( s - 2 )

(s + 4 ) 2

,

(4.4.91)

(4.4.92)

157

Finally by equations (4.4.91) and (4.4.92),

ca =

;1,qo, o

0

0

- ( l l s + 1 4 ) s + 4

(4.4.93)

The decentralized Ru-stabilizing compensator Ca in equation (4.4.93) is itself Ru-stable;

Note that this is a not always possible, i.e., the plant need not be Ru-stabilizable by an

Ru-stable compensator in general. For example, if we chose Ql l = 15, Q22 = 1, Q1 =

- 1, Q2 = - 5 , then condition (4.4.77) is still satisfied since

QII Q1 w12

Q2 w21 Q22

15

- 5 ( s - 2 ) (s +4) 2

- ( s - l ) ( s + 4 ) 2

is Ru-unimodular. Repeating Step 7, we obtain another decentralized Ru-stabilizing com-

pensator for the plant in (4.4.80):

G =

44 s 3 + 528 s 2 + 2145 s + 2858

15 s 3 + 180 s 2 + 731 s + 974

0 - ( 1 6 s 2 + 4 3 s +66)

( s + 4 ) ( s - 1 )

(4.4.94)

The decentralized Ru-stabilizing compensator C a in (4.4.94) is not Rus t ab l e . []

158 4.5 M U L T I - C H A N N E L D E C E N T R A L I Z E D C O N T R O L S Y S T E M S

In this section we extend the results of Sections 4.3 and 4.4 to m-channel decentralized

feedback control systems, where m > 2 .

We consider the linear, time-invariant,

S ( P , C a )m shown in Figure 4.7, where P :

Cd:

e l t

e m"

yl t

Ym ~

m-channel decentralized feedback system

e l Yl

i }--* i represents the plant and

em Ym

represents the compensator.

f . . . . . . . . . . -.~

e l j ! ~ ' ~ Cl I !

Ul

+ Yl ~ f ~ ¢1

+ •

+

Yl Ul t

+

U m ~' e m P

ce

! !

!

• I I

I i

I I I 1

P

Ym

Figure 4.7. The m -channel decentralized control system S ( P, C a )m •

The externally applied inputs are denoted by ff :=

Ul

Um U 1 t

urn'

, the plant and the compensator

159

Y l

Ym " the closed-loop input-output map of S (P , Cd ) is outputs are denoted by y" := Y I' '

ym,

denoted by Hy~ : ff F-¢ ~' . We extend Assumption 4.2.1 to m-channels as follows:

4.5 .1 . Assumptions on S ( P , C d )re

(1) The m-channel plant P ¢ GnoX ni , where

n o = n o l + n o 2 + " " " + ?toot ) n i = n i l + hi2 + " '" + nim

(ii) The decentralized compensator C d ~ G nix no, where

C1 0 ... 0

0 C 2 - . . 0 C d = : : : : and for j = 1, . . . , m , Cy ~ G nij Xnoi

0 o - . . cm

(iii) The system S (P , C d )m is well-posed; equivalendy, the closed-loop input-output map

H ~ : ~ ~ ~- is in m(G). []

Note that whenever P satisfies Assumption 4.5.1 (i), it has an r.c.f.r., denoted by

( N e , D e ) , an 1.c.f.r., denoted by ( L~ e , b~ e ) and a b.c.f.r., denoted by (Npr , D , Npl, G ) ,

where the numerator and the denominator matrices can be partitioned as follows: In the r.c.f.r.

( Ne ,De ) ,

/~p = :

~ p l

E H n° X ni , D e = :

D p l

~2

o,m

H nixni , (4.5.1)

where, for j = 1 , . . . , m , Npj ~ H noy x ni , De j ~ Hno x ni

the 1.c.f.~. ( 5 e , Ne ) ,

160

. . . /~pm ] ~ H noxno ,

• . . Npm ] E n n o x n i , (4.5.2)

where, for j = 1 , . . . , m , D"pj E H n°x noj, Np i ~ HnoXnq

In the b.c.f.r. (Npr ,D ,Npl , G ) ,

Np,1

Np,2 Np, =: : ~ H n°xn

G = :

, N p l = : [ Npl , Npl2 . . . Npl m ] ~ H n x n i ,

G l l • . . Glm

G21 " '" G2m .

Gm 1 " " " Gmm

(4.5.3)

where, for j = 1, . . . , m , k = 1, . - - , m , Npr.i E H n°y x n

G~k ~ H n°jx nik ; D E H n x n .

, Npq ~ H nxniy

I f C d satisfies Assumption 4.5.1 (ii), then C d has an 1.c.f.r., denoted by ( / )c ,/Vc ) and

an r.c.f.r., denoted by ( N c , D c ) , where Dc ~ Hni x ni , Nc ~ Hni x n o Nc ~ Hni x no ,

D e E H n° x no. Let

0 P c =

0

" ' " 0

~ 2 " ' " 0

0 . . .

Ncl 0 . . . 0

0 ~7c2 . - - 0

0 0 " ' " No,,

; (4.5.4)

note that for j = 1, . . - , m , ( /~e , /Ve ) is an 1.c.f.r. of Cd if and only ff ( / ) ~ j , AT~j ) is

an 1.c.f.r. of C j , where /~cj ~ H nq x n i j , lqO ~ HniJ x noj. Let

D c =

D c l 0 • • • 0

0 De2 " " 0

0 0 " '" Dcr n

161

, ~ =

N~I 0 . . . 0

0 No2 . . . 0

0 0 " '" Nc~

; (4.5.5)

note that for j = 1 , • • • , m , ( N c, D¢ ) is an r.c.f.r, o f C a if and only if (Ncj , Dcj ) is an

r.c.f.r, of Cj , where Dcj • H n°j x noj , Nc j • Hn i j x noj E!

4.5.2. Analysis ( Descriptions of S ( P, C d )m using coprime factorizations )

We only analyze the m-channel decentralized system as in Analysis 4.2.4 (i); the other

cases of Section 4.2 are also easy to extend.

Assumptions 4.5.1 hold throughout this analysis.

Let (Np ,Dp ) be any r.c.f.r, o f P • m(G), partitioned as in equation (4.5.1) and let

( D c , N c ) be any l.c.f.r, of C a • m(G), partitioned as in equation (4.5.4). The m -

channel system S ( P , Ca )m is then described by equations (4.5.6)-(4.5.7) below:

De1 " " " 0 No1

0 . . . ~ 0

• "" 0

Ul

u m

U l t

:1 llm t J

, (4.5.6)

i 11 ylli0 000 N p m Ym 0 ' ' " 0 0 " ' " 0

Dpl ~t' = Yl ' - - l n i I " " 0 0 " " 0 : : •

Dpm Ym' 0 . . . . In i m 0 ' ' ' 0

Ul

Urn

Ul t

Um"

(4.5.7)

I"1

Theorem 4.5.3 and Theorem 4.5.4 below are obvious extensions to the m-channel ease

of Theorem 4.2.5 and Theorem 4.3.3, respectively; we state them without proof:

162

Theorem 4.5.3. ( H-stability of S ( P, C a )m )

Let Assumptions 4.5.1 (i) and (ii) hold; let (Np ,D e ) be any r.c.f.r, over m(H) of

P ~ m(G), partitioned as in equations (4.5.1); let ( / )c ,/vc ) be any 1.c.f.r. over m(H) of

c ~ r e ( G ) , partitioned as in equations (4.5.4). Under these assumptions, S ( P , C a )m is

H-stable if and only if the matrix D H l in equation (4.5.8) is H-unimodular.

DH 1

D c l D p l + Ncl Npt

DcmDpm + NcmNpm

0 0

• " " 0

" ° " O c m

0 ] Opl

(4.5.8)

Theorem 4.5.4. (Conditions on P = Np Dp -1 = D'-p -1/Vp for decentralized H-stabilizability)

Let P satisfy Assumption 4.5.1 (i); furthermore, let P ~ m ( G s ) ; then the following three

conditions are equivalent:

(i) There exists a decentralized H-stabilizing compensator Ca for P ;

(ii) Any r.c.f.r. (Np ,Dp ) o f P , partitioned as in (4.5.1), satisfies condition (4.5.9) below:

E, [ 0,i

e2 [ D,,2

Em[ Din'

R =

Ini i 0 0 "" 0

0 Wi2 W13 ... Wire

0 I n i 2 0 • • • 0

W21 0 W ~ . . . W ~

0 0 0 "'" In~

Wml Wm2 Wm3 "'" 0

(4.5.9)

163

(iii) Any 1.c.f.r. (/~p ,/Vp ) o fP , partitioned as in (4.5.2), satisfies condition (4.5.10) below:

, . [ [-,7,1 [_,7.. o-. . . .

0 Ino ~

- W2I 0

- W3I 0 :

--Wmx 0

- W12 0

0 Ino 2

-- 1~32 0 :

- W m 2 0

- W l m 0

- W2m 0

- W3m 0 :

0 I n n

, ( 4 . 5 . 1 0 )

where, for j = 1, . . . , m , Ej e H (nii+n°y) x (nq+noj) is H-unimodular,

R e H nix ni is H-unimodular and L ~ n n°x no is H-unimodular; the matrices

Wjk E H n°jx nit are H-stable for k = 1 , . . . , m ( note that Wjk = 0 when

tc = j ). []

We now extend the class of all decentralized H-stabilizing compensators in Theorem

4.3.5 to m-channels: The matrix T m in equations (4.5.13)-(4.5.14) below is defined as

Tm :-

Ql l QIWI2 Q1WI3 " '" QIWlm

Q2W21 Q22 Q2w23 " '" Q2W2m

Q3w31 Q3w32 Q33 "'" Q 3 W 3 m : : : :

QmW,,,1 O.,,,W,,,2 Q, .Wm3 " " Q,,~

QII 0 0 . . . 0

0 Q22 0 - . . 0

0 0 Q33 " '" 0 : : :

0 0 0 "'" Q , , ~

+

Q1 o o . . .

o Q2 o . . .

o o Q3 "'"

0 0 0 " "

0 0

0 W21

0 W31

Qm Wml

Wl2 w13 . . . Wlm

0 W23 " '" W2m

W32 0 "'" W3m

Wm2 Wm3 "'" 0

; (4.5.11)

the matrix ]~m is defined similarly by replacing each Q1 with Q1 and Qii with Q~i for

j = 1 , ' ' ' , m .

164

The following generalized Bezout identity, which is an extension of equation (4.3.28) to

m-channels, is useful to prove Theorem 4.5.5 below; the details of the proof can easily be

worked out following the same steps as in the proof of Theorem 4.3.5.

'I lil [ Ioio 1.2o o

R E 1 R E 2 •

Iv , ,

0 Ino 1

-W21 0

L-1 _W31 0

-w,,, i 0

E 1 L -1

- W I2 0

0 Ino:

-W32 0

-Win2 0

E 2 • . . L-1

" - W lm 0

-W2m 0

-W3m 0

0 In~

Era

E i - | R -1 E i -1 0 - . . 0 • -- I 10 . . . L

I0,.0 01 1 00 • L E~" W21 0 W23 "" W2..n R-I E~" 0 Ino 2

1| 0 0 0 "'" Inv, [-

F.£ [w,.1 w,.2 w,,,3 "" o _1[0 0 - . . 0 ] L

R-1 Era 0 0 , I no,, '

= Ini +no •

(4.5.12)

165

Theorem 4.5.5. ( Class of all decentralized H-stabil izing compensators in S ( P , C d )m )

Let P satisfy Assumption4.5.1 (i); furthermore let P ~ m ( G s ) ; let any r.c.f.r. (Np ,Dp )

of P satisfy condition (4.5.9) and equivalently, let any 1.e.f.r. ( / )p ,/Vt, ) of P satisfy condi-

tion (4.5.10) of Theorem 4.5.4. Under these assumptions, the set S d (P) of all decentralized

H-stabilizing compensators for P is given by

S d ( P ) = [ C 1 " " C m ] = diag [ D c 1Nc' "'" [

Qjj e H nij x nij , Qj e H nij x n,,j (4.5.13)

and equivalently, by

for j = 1, . . . , m ,

, such that T m is H-unimodular } ,

Dej

-62

~.jj ~ Hnoj xnoj , ~.j ~ Hnij xnoj

where the matrix T m is defined in equation (4.5.11) and Tm is defined similarly.

A

, such that T m is H-unimodular } , (4.5.14)

O

Comment 4.5.6. ( The rational functions case )

Let the principal ideal domain H be the ring Ru as in Section 4.4. The main results of Sec-

tion 4.4 can be extended to the m-channel decentralized control system S ( P , Cd )m as fol-

lows:

The plant P is said to have a decentralizedf~ed-eigenvalue at s o e tl with respect to

Kd = diag [ K1 "'" K,,, ] iff so e tl is a pole of the closed-loop I/O map of the system l

s(p. d) for a , { K. ] ]RnO × noj }.

Equivalently, s o

166

¢ 1J is a decentralized fixed-eigenvalue if and only if

OPl($°) + K1Npl($° ) det

Dr,. (so) + K., Np., (so)

- ,o,[ o ; , <,o >+ e , , <,o >+ <,o )K. ] =0

for all Kj ~ I~ no xn°j , j = 1 , . . . , m (4.5.15)

To extend Theorem 4.4.4 to the m--channel system S ( P , C a )m, let P satisfy

Assumption 4.5.1 (i); furthermore, let P E mfRsp (s)) ; then the following six conditions

are equivalent:

(i) There exists a decentralized H-stabilizing compensator Ca for P in the system

S ( P , Cd )m ;

(ii) Any r.e.f.r. ( Np , Dp ) of P , partitioned as in equation (4.5.1), satisfies condition (4.5.9)

of Theorem 4.5.4, where, for j = 1 , " " , m , Ej ~ R u (nO+n°j)×(nO+n°y) is

Ru-unimodular, R ~ R u hi×hi is Ru-unimodnlar and L ~ R u n°×n° is

Ru-unimodular; the matrices Wjk ~ R u n°j × nik are Ru-stable for k = 1, • • • , m ;

(iii) Any 1.c.f.r. (/~p,/Vj, ) of P , partitioned as in equation (4.5.2), satisfies condition

(4.5.10) of Theorem 4.5.4, where, for j = 1, . . . , m , Ej ~ R u (nO+n°j) x (nij+n°j) is

Ru-unimodular, R ~ RU ni×nl is Ru-unimodular and L ~ R u n°×n° is

Ru-unimodular;, the matrices Wit ~ RU n°J × na are Ru-stable for k = 1, - - • , m .

(iv) For k = 1 , - . . , m - 1, for all nonempty subsets A = { Ctl, . . . , s t } of

{ 1 , . - . ,m } , any r.e.f.r. (Nt, ,O e ) of P , partitioned as in equation (4.5.1),

satisfies the rank condition (4.5.16) below:

rank

Dpm ( s )

NI, aI ( $ )

D p ~ ( s )

N p ~ ( s )

167

> ~ n la j , fora l l s 6 I] ; (4.5.16) otj e A

(v) For k = 1, . - . , m - 1 , for all nonempty subsets A = { ~ 1 , " ' " ,Ctk } of

{ 1 , "-- ,m } , any l.c.f.r. ( / g p , / q p ) of P , partitioned as in equation (4.5.2),

satisfies the rank condition (4.5.17) below:

[ . . . . ] rank - N p a l ( s ) Dt, al ( s ) . . . . Nt, o~ ( s ) Dpot~ ( s ) > ~_~ niaj "

a j ~ A

for all s ~ I] ; (4.5.17)

(vi) The plant P has no decentralized fixed-eigenvalues in 11.

By Corollary 4.4.5, the six equivalent conditions above are equivalent to condition (vii)

below:

(vii) For k = l , . . . , m - l , for all partitions of the set { 1 , - - . ,m } into two (dis-

joint) subsets { o q , - . . , 0 c k } and { O~k+l , . . . , o ~ } , any b.e.f.r.

( Npr , D , Np t , G ) of P , partitioned as in equation (4.5.3), satisfies the rank condi t ion

(4.5.18) below:

D ( s )

Nproq ( s ) rank

NprOIk ( $ )

ring D = [ ( s

-Npz~+1 ( s ) . . . . Npz~ ( s )

Gm~+~ ( s ) . . . Galo~, ( s )

GcaOa+l ( $ ) . . . Ga~o~ ( s )

> n , for all s E I1 .

(4.5.18)

Condition (4.5.18) can also be written in the state-space setting as in Remark 4.4.6 by set-

-I-d )-1 ($1 n - A ) ] , Npraj = Caj , Npl~j = B~j •

168

Conditions (4.5.16) and (4.5.17) need not be checked for the entire set { 1, --- , m }

but only for (proper) subsets of it because since (N t , , Dp ) is r.c. and ( / ) p , Alp ) is 1.c.,

Lemma 2.6.1 implies that rank Np(s) = ni and rank Np(s) Dp(s) = n i , for

all s ~ 1J. Similarly, condition (4.5.18) needs to be checked for all disjoint pairs of subsets

neither one of which is all of { 1, -. • , m } ; condition (4.5.18) is automatically satisfied if

either one of the two subsets were all of { 1 , . . . , m } since ( N p r , D , N p t ) is a

bicoprime triple. []

4.5.7. Achievable input-output maps of S ( P , C d )m

For a simpler representation than we would obtain for H~-~, we look at the I/O maps of the

H-stabilized system S ( P , C d )m in a slightly different order.

Let Hy,yu~, :

Ul

U2 u.2"

Um

Um •

" Y l ' ] Y I | Y2 / Y2 | • Throughout this

y.,"

Ym I J

tion 4.5.1 (i); furthermore,

section, let P satisfy Assump-

let P ~ m ( G s ) ; let any r.c.f.r. (Np ,D e ) o f P satisfy condition

(4.5.9) of Theorem 4.5.4.

The set

Aa (P):= { ny,y~, [ Cd H-stabilizes P }

is called the set of all achievable I /0 maps of the unity-feedback system S ( P , C a )m •

By Theorem 4.5.5, Ad (e ) = { Hy,y~u, I Ca e Sd(e) }, where S d ( P ) is

the set of all decentralized H-stabilizing compensators given by the equivalent representations

(4.5.13) and (4.5.14).

Since the set S d (P) is a subset of the set S ( P ) of all H-stabilizing compensators for

169

the fuU-feedback system S ( P , C ) , the set A d ( P ) is also a subset of the set A ( P ) of all

achievable I/O maps of S( P , C ). The set A d ( P ) is obtained from equations (4.5.6)-(4.5.7)

by substituting for (Np ,Dp ) from equation (4.5.9) and for ( D c , N c ) from (4.5.13); the

matrix T m in equation (4.5.19) below is the H-unimodular matrix defined in equation

(4.5.11):

A a ( / , ) = { Hy,y.., =

[ I~,1 0 0 . ' . 0

Ei -1 W12 W13 . . . W1, n

0 Ini 2 0 " " 0 ]

E2"l WZl 0 WZ3 "'" W ~

[000 E";t win1 w~2 w,,,3 "'" 0

Qll QI ]E l 0 --. 0

0 [Q22 Q 2 ] E 2 "'" 0

o o

for j = 1 , . . . , m , O.jy ~ H nij x n o , Qj ~ Hni/ xno/

are such that T m is H-unimodular } . (4.5.19)

Since T m depends on the (matrix-) parameters Qjj and OJ ' the parametrization in (4.5.19) of

all achievable I/O maps in S ( P , C a )m is not affme in these (matrix-) parameters.

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S Y M B O L S

I/O

MIMO

a :--b

R

C

C+

Z

Z +

In dctA

H J I G

re(H) U

fi Ru

Rsp (s) r.c. 0.c.) r.f.r. (1.f.r.)

r.c.f.r. (1.c.f.r.)

b.c. (b.c.f.r.)

g.c.d.

l.c.m.

S(e,c)

S(P,Ca)

S(P,Kd) S(P,Cd ),,,

input-output

multiinput-multioutput

a is defined as b

real numbers

complex numbers

complex numbers with nonncgafive real part

integer numbers

nonncgativc integer numbers

n xn identity matrix

the determinant of matrix A

principal ring

group of units of H

a multiplicative subset of H

ring of fractions of H associated with I

Jacobson radical of G

the set of matriccs with cntrics in H.

a closed subset of C+

uu{~}

ring of proper scalar rational functions which arc analytic in l.l

ring of proper scalar rational functions with real coefficients

set of strictly proper scalar rational functions with real coefficients

right-coprime (left-coprime)

right-fraction (left-fraction) representation

right-coprime-fraction (lefl-coprimc-fraction) representation

bicoprime (bicoprime-fraction representation)

greatest-common-divisor

least-common-multiple

the unity-fccdback system

the general feedback system in which the plant and the compensator each have two (vector-) inputs and two (vector-) outputs

the two-channel decentralized feedback system

the two-channel decentralized constant feedback system

the m-channel decentralized fccdback system