Singular Control Systems (Lecture Notes in Control and Information Sciences)

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

1118 L. Dai

Singular Control Systems

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Series Editors M. Thoma • A. Wyner Advisory Board U D. Davisson • A. G. J. MacFarlane ° H. Kwakernaak J. L. Massey • Ya Z. Tsypkin • A. J. Viterbi Author Liyi Dai Division of Applied Sciences Harvard University Cambridge, MA 02138 USA

ISBN 3-540-50724-8 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-50724-8 Springer-Verlag NewYork Berlin Heidelberg Library of Congress Cataloging in Publication Data Dai, L. (Liyi) Singular control systems / L. Dai. (Lecture notes in control and information sciences ; 118) Bibliography: p. Includes index. ISBN 0-387-50724-8 : 1. Automatic control. 2. Control theory. 3. Linear systems. I. Title. I1. Series. TJ213.D25 1989 629.8-dc19 89-4091 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only pernlitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Bedin, Heidelberg 1989 Printed in Germany Offsetprinting: Mercedes-Druck, Berlin Binding: B.Helm, Berlin 2161/3020-543210

To

Yun

Wang

PREFACE

Singular network, tems,

systems

are found in engineering systems (such as

power systems, aerospace engineering,

economic systems,

electrical

and chemical processing),

circuit

social sys-

biological systems, network analysis, time-series analysis,

singular singularly perturbed systems with which the singular system has a great deal to do,

etc.

TheiF form also makes them useful in system modeling.

singular systems are called the descriptor variable systems,

In many articles

the differential/algeb-

raic systems, the generalized state space systems, etc. Singular systems are governed by

the so-called singular differential equations,

special

which endow the systems with many

features that are not found in classical systems.

terms and input derivatives in the state response, noncausality etc.,

between

making

Among these

are

impulse

nonproperness of transfer matrix,

input and state (or output),

consistent

initial

conditions,

the study of singular systems more sophisticated than of classical li-

near systems.

In my opinion,

this is the reason why singular systems have attracted

interest in recent years. Currently, although much effort has been made in the exploration of special properties for singular systems, the studies are mostly confined to the generalization of classical system theory;

of course,

this is also an important

task in control theory.

There are

mainly

two approaches in the studies:

latter approach is adopted in this book. summing up

the

development

geometric

and

algebraic.

The

The primary purpose of writing this book is

of singular control

system

theory and providing

the

control circle with a systematic theory of the systems. Some acquaintance with linear algebra and linear system theory is assumed.

Those not familiar with these prerequi-

sites can read the book admitting the results in appendices. near and time invariant.

of singular systems from the point of view of the system, an exclusive effort This makes

The systems are all li-

Technically, the book focuses on the mathematical treatment

to discuss

or, in other words, I make

the analysis and design of singular control systems.

possible the systematic design of singular control systems

applications.

in

practical

Some examples are used only to illustrate the results and design proce-

dures in this book.

The book is organized as follows. Chapter 1 discusses the state response structure for sion.

singular

systems and the state space equivalent forms

needed for later

Chapter 2 provides some fundamental concepts in the system analysis,

dlscusuch

as

Vl reachability, and deals back

so on.

controllability,

observability,

realization theory, transfer matrix,

These conc:epts are essential for system analysis and design.

with the control problem.

Chapter

The closed-loop behavior is given under state feed-

controls and detailed information on finite or infinite pole placement

obtained.

Chapters 4 and 5

3

may

investigate some aspects of control system design

be for

singular systems; the state observation problem is discussed in Chapter 4 and the dynamic

compensation problem is treated in Chapter 5.

show that under fairly general conditions,

Results in these

we may design

two

chapters

normal observers and dyna-

mic compensators which may be realized as in linear ~ystem theory. Chapter 6 is devoted to a special problem --- the structural stability, or, more precisely, tness of stability,

the robus-

which is not as crucial in classical linear system theory as

the singular system acse.

It is shown that to guarantee structural stability,

in

great

care must be excercised in singular systems. Chapter 7 presents results on the system analysis and synthesis through the transfer matrix method. Three problems are solved: the transfer matrix structure under state feedback control~

decoupling control,

and

input function observers.

Since the digital technique holds an important position in

practical

Chapter 8

system design.

basic design concepts are studied. trol

theory

systems.

Only

Chapter 9 contains basic results on optimal

explores discrete-time singular

con-

and Chapter IO contains a preliminary discussion on topics

for farther

research.

Liyi Dai Beijing, PRC August, 1988

CONTENTS

Preface Chapter 1

Solution of Linear Singular Systems

1

1-1. Introduction

1

1-2. Regularity of general singular systems

4

I-3. Equivalence of singular systems

I0

1-4. State response

16

I-5. Notes and references

22

Chapter 2

Time Domain Analysis

23

2-i. State teachability

23

2-2. Controllability, R-controllability and impulse controllability

29

2-3. Observability, R-observability,and impulse observability

38

2--4. Dual principle

46

2-5. System decomposition

5O

2-6. Transfer matrix and minimal realization

57

2-7. Notes and references

65

Chapter 3

Feedback C o n t r o l

67

3-1. State feedback

67

3-2. Infinite pole placement

78

3-3. Finite pole placement

87

3-4. Pore structure under state feedback control

95

3-5. Output feedback control 3-6. Notes and references Chapter 4

State Observation

98 I01 102

4-i. Singular state observer

102

4-2. Normal state observer

106

4-3. General structure o f state observers

118

4-4. Disturbance decoupling state observers

122

4-5. S t a t e o b s e r v e r with d i s t u r b a n c e estimation

128


131

Chapter 5

Dynamic Compensation for Singular Systems

132

5-I. Singular dynamic compensators

132

5-2. Full-order normal compensators

136

5-3. Reduced-order normal compensators

142

VIII 5-4. Singular compensators in output regulation systems

152

5-5. Normal compensators in output regulation systems

161


166

Chapter 6

Structurally Stable Compensation in Singular Systems

6-I. Structural

stability in homogenoous singular systems

6-2. Compensation function

on

structural stability via dynamic feedback

167 ]68 174

6-3. Structurally stable normal compensators

179

6-4. Structurally stable normal compensators in output regulation systems

184


196

Chapter 7

System Analysis via Transfer Matrix

197

7-I. Transfer matrix in singular systems

197

7-2. State feedback and transfer matrix: single-input single-output systems

201

7-3. State feedback and transfer matrix: multilnput multioutput systems

208

7-4. Singular input function observers

215

7-5. Normal input function observers

219

7-6. Decoupling control

224


230

Chapter 8

Introduction to Discrete-time Singular Systems

8-I. Finite discrete-time series

231 232

8-2. Solution, controllability and observability in discrete-time singular systems

239

8-3. State feedback and pole placement in discrete-time singular systems

245

8-4. State observation for discrete-time singular systems

250

8-5. Dynamic compensation for discrete-time singular systems

259


265

Chapter 9

Optimal Control

266

9-I. Introduction

266

9-2. Optimal regulation with quadratic cost functional

267

9-3. Time optimal control

272

9-4. Optimal control for discrete-time singular systems

275


284

Some Further Topics

285

IO-I. Large-Scale singular systems

285

10-2. Adaptive control

291

Chapter iO

10-3. Stochastic singular systems

294


301

Appendix A

Distribution and Dirsc Function

302

l× Appendix B

Some Results in Matrix Theory

306

Appendix C

Some Results in Control Theory

310

Appendix D

List of Symbols andAbbreviations

313

Appendix E

Subject Index

316

Appendix F

Bibliography

320

CHAPTER 1 SOLUTIONS OF LINEAR SINGULAR SYSTEMS

I-I.

Based on state space models, in modern control theory, ning of the 1960s. called

Introduction

System analysis and synthesis are the core features

which was developed at the end of the 1950s and the begin-

State space models of systems are obtained mainly using the so-

state space variable method,

whose advantage is that it not only provides us

with a completely new method for system analysis and synthesis, but it also offers us more understanding of systems. To get a state space model, weight,

temperature,

we need to choose some variables

and acceleration,

such

as speed,

which must be sufficient to characterize the

system of interest. Then several equations are established according to relationships among the variables. Naturally, it is usually differential or algebraic equations that form the mathematical model of the system, or the description equation. Its gen-

e r a l form i s a s f o l l o w s : f(~(t), x(t), u(t), t) = 0

(l-l.la)

g(x(t), u(t), y(t), t) = 0

(I-1.1b)

where x(t) is the state of the

system

composed of state

variables;

u(t)

is

the

y(t) is the measure output; and f and g are vector functions of ~(t),

control input;

x(t), u(t), y(t), and t,

of appropriate dimensions.

Generally (l-l.la) and (l-l.lb)

are called state and output equations, respectively. A special case of (l-l.l) used to describe the system of interest is

E(t)~(t)

= It(x(t), n(t),

t)

(1-1.2)

y(t) = J(x(t), u(t), t) where H,

J are appropriate dimensional vector functions in x(t),

u(t),

and t.

The

matrix E(t) may be singular. Systems described by (I-1.2) are the general form of the so-called singular systems proposed recently. If H,

J are linear functions of x(t) and n(t), another special form of (I-1.2) is

the linear singular system: Ex(t) = Ax(t) + Bu(t) y(t)

= Cx(t)

(1-1.3)

which is the main system studied in this

book.

Here,

x(t)E IRn, u(t)E~m, y(t)EIR r.

E, A E ~ nxn, B~IRnxm, and C~IRnxr are constant matrices. When E is nonsingular, system (I-1.3) becomes ~(t) = E-IAx(t) + E-iBu(t)

(]-] .4)

y(t) = Cx(t). It is the well-known classical linear system that will be termed normal

system here.

We shall not attempt to introduce the abundant results for normal systems here,

they

may be found in any book on control theory. For this reason, when we mention singular systems we always mean the singular E, i.e., rankE ~ q < n. In practical system analysis and control system design, established Thus,

in the form of (1-1.2),

many system models may be

while they could not be

described by

(1-1.4).

the studies of singular systems began at the end of 1970s, although they were

first mentioned in 1973 (Singh and Liu, 1973). In many articles, singular systems are called descriptor variable systems, tems,

generalized state space systems,

semistate sys-

differential-algebraic systems, singular singularly perturbed systems, degene-

rate systems, constrained systems, etc. Singular example, ses),

systems

appear in many systems,

power system,

such as engineering systems

(for

electrical networks, aerospace engineering, chemical proces-

social economic systems,

analysts, system modeling, and

network analysis,

so

biological systems,

time-series

on.

Example i-I.I. Consider a class of interconnected large-scale systems with subsystems of xi(t) = Aixi(t) + Biai(t)

(1-I.5)

bi(t ) = Cixi(t ) + Dial(t),

i = I, 2 . . . . .

where xi(t) , ai(t), and bi(t) are the substate, vely, of the ith subsystem. By denoting

N

control input, and output, respecti-

x1(t)

tat(t)

bl(t)

x2(t)

la2 (t)

b2(t)

x(t) =

a(t) =

[xi(t)

,

b(t)

La i ( t )

,

=

.

bN(t)

,

A = diag(A I, A 2 .... , AN),

B = diag(B I, B 2 . . . . .

B N)

C * diag(C I, C2, ... , CN),

D = diag(Dl, D 2 . . . . .

D N)

(I-1.5) may be rewritten as ~(t) = Ax(t) + Ba(t) b(t) = Cx(t) + Da(t)

(1-1.6)

3 Assume that the subsystem interconnection is the linear interconnection: a(t) = Lllb(t) + L12u(t) + rlla(t) + Rl2Y(t)

(I-1.7)

y(t) = L21b(t) + L22u(t) + R21a(t) + R22Y(t) where u(t) is the overall

input

measure output; Lij, Rij,

i, j = I, 2,

sions.

the

Equations (I-1.6) and (I-1.7)

form of (i-1.4). the

of

system

system.

In fact,

composed by

large-scale system;

y(t) is its

form a large-scale system

have proven that

equivalent

if we choose the state variable

dimen-

which is not in the

Singh and Liu (1973) and Petzold (1982) (I-1.6) and (I-1.7) could not be

On the other hand,

overall

are constant matrices of appropriate

[xT(t)

to

a

normal

aT(t)

bX(t)

y~(t)]~ (1-1.6) and (1-1.7) form the system

O0 0 [[~(t)

i° o

o

=

Ii

Oil,(t)

o o fix(t)

D

-I

R11- I

0 00Jl~(t)

r21

y ( t ) = [0 0 0 I ] [ x T ( t )

0

l[a(t)

L l l r12 lIb(t) L21 r22-1Jty(t) a~(t)

b~(t)

u(t) L12 [L22 J

yT(t)]~

which is in the form of (i-1.3). Example 1-1.2.

The fundamental dynamic Leontief model of economic systems

singular system. Its description model is (Luenberger,

is

1977):

x(k) = Ax(k) + B[x(k+l) - x(k)] + d(k)

(I-1.8)

where x(k) is the n dimensional production vector of n sectors;

output

(or production) matrix;

B[x(k+l)-x(k)]

A(~nxn

is an input-

Ax(k) stands for the fraction of production required

as input for the current production,

often appears

a

B 6 ~ nxn is the capital coefficient matrix,

is the amount for capacity expansion (Stengel in the form of capital,

d(k)

et

al,

1979),

is the vector that includes

and which

demand or

consumption. Equation (1-1.8) may be rewritten as

Bx(k+l) = (I-A+B)x(k) - d ( k ) . In multisector economic systems,

production augmention in one sector often

need the investment from all other sectors,

and moreover,

doesn't

in practical cases only a

few sectors can offer investment in capital to other sectors. Thus, most of the elements in B are zero except for a few. B is often singular. In this sense the system (1-1.8) is a typical discrete-time singular system. Example 1-1.3. extremely

When

administration is included,

complicated process.

the oil catalytic craking is an

Some companies reportedly found a description

model

for this process, whose simplication is 31 = AllXl + A12x2 + BlU + Flf O = A21Xl + A22x 2 + B2u + F2f

(I-1.9)

where

x I is a vector to be regulated, such as regenerate temperature, valve position,

blower capacity, etc;

x 2 is the vector reflecting business benefits, administration,

policy, etc; u is the regulation value; and f represents extra disturbances. Equation (I-1.9) is a standard singular system•

1-2.

Regularity of General Singular Equations

Let us consider the following general singular

equation:

E~(t) = Ax(t) + f(t) where E, A E ~ n x n

(1-2.1)

and f(t)£ R n

which is a special case of (I-1.3) when B = I . The n it has as many derivatives

function f(t) is assumed to be sufficiently differential; as needed.

For (I-2.1),

we are mainly concerned with the existence, uniqueness, and

structure o f its solution• For any

Lemma 1 - 2 . 1 ( G a n t m a c h e r , 1974).

two matrices E, AE R mxn, there always e-

xist two nonsingular matrices Q, P such that ~ QEP = diag(O, L1, t 2 ..... Lp, e~, e~ ..... L4, I, N) (1-2.2) ~ QAP = diag(O, Jl, J2 ..... Jp, J|, J~ ..... J~, AI, I)

oi]

where 0 E ~ m°xn°,

A1E ~hxh

10 •

Li =

0

o

..

-.

Ji =

1

~mix(mi+ 1 )

•*•°•o

1 0 ,

O1

i - i, 2, ..., p, O I L'= J

•

1

10

°

. ". 1 0

E ~(nj+l)xnj

d'= 3

0 1

,

j = I, 2 ....

, q,

[01 ]

N = diag(Nkl, Nkr . . . . .

0

Nki =

l "..'..

Nkr) E R gxg,

E Rkixki , 1

0

i -- I, 2, ... , r,

mo + ~m i + ~ (nj+1) + I k i + h = m i

j

i

no + ~. nj + ~ ( m i + l )

J

+Zki i

i

+ h ffin

~k i fig. i

Taking the coordinate transformation

x(t) = P~(t) and l e f t

multiplying E~(t)

(1-2.1)

= ~(t)

By e x p a n d i n g ( I - 2 . 3 )

by t h e n o n s i n g u l a r

+ ~(t),

~(t)

according

to (I-2.2),

m a t r i x Q, we h a v e (I-2.3)

= qf(t). i t may be r e w r i t t e n

as

(I-2.4a)

O~no(t ) = fmo(t) Li~mi(t) = JiXmi(t) + fmi(t),

i = I, 2 . . . . .

p

(I-2.4b)

L3~nj(t ) = J~xnj(t ) + fnj(t),

j = I, 2 . . . . .

q

(I-2.4c)

NkiXki(t) ~ Xki(t) + fki(t),

i = I, 2 ....

, r

(i-2.4a) (I-2.4e)

Xh(t) = AlXh(t) + fh(t) where Xk(t)E~k,

fk(t )ERk, and

~(t) = [X~o x~ ... X~p x~1 ... X~q Xtl ... xt~ x~f ~(t)

Thus,

flr f;):

= [ f ~ o f~l " ' '

f~p f~l " ' "

f~q f~1 " ' '

solving equation

(1-2.2) is

equivalent to solving

the equations

in (I-

2.4a~ e). Next we will examine the solvability of each equation. I. Equation (I-2.4a) can be solved only when fmo(t) = O. In this case, (I-2.4a) is an identical equation for any differentiable function Xno(t). Therefore, (I-2.4a) has either no so|u~ion or 2.

an

infinite number of solutioz~s.

{10]{01]

Equation (i-2.4b) is composed of a set of equations of the following form: I 0

.

z(t) =

"o ". I

0 .I .

z(t) + fz(t).

". "° 0

0

1

Expanding it, we obtain Zl(t) = z2(t)+fl(t) , ~ 2 ( t ) = z 3 ( t ) + f 2 ( t ) . . . . . Clearly,

Zk_l(t)

= zk(t)+fk_l(t

).

for any sufficiently differentiable function, Zl(t) , z2(t), ... , Zk(t ) may

be

determined

successively.

Therefore,

such equations have an infinite number

of

solutions. 3.

Every equation in (I-2.4c) is of the following form:

[i] 0 ".

0 I 1 ".

~(t) =

1

•

0

z(t) + fz(t) •

0 1

By expanding, we know that this equation is equivalent to Zl(t) = fl(t),

z2(t) = Zl(t) + f2(t) . . . .

~k(t) ~ Zk_l(t) + fk(t), Except the last one,

the equations may

~k(t) + fk+1(t) = O.

determine unique functions Zl(t) , z2(t), ...

• Zk(t ). But accounting for the last equation, zk(t) must satisfy Zk(~) + fk+l(t) =0. This shows that

the equation has no solution unless fk+l(t) satisfies the consistent

condition zk(t) + fk+l(t) = O. 4.

Equation (I-2.4d) consists of the following equations: 0

1

• .'..

z(t) = z(t) + fz(t) 1

0 which may he rewritten as z2(t) = Zl(t ) + fl(t),

z3(t) = z2(t) + f2(t) . . . .

Zk (t) = Zk_1(t) + fk_l(t), Beginning with the last one,

zk(t) + fk(t) = O•

Zl(t), ... , Zk(t ) may be success[rely determined for

sufficiently differentiable function f Z (t).

Hence (i-2.4d) has a unique solution for

any such fz(t). 5.

Equation

(I-2.4e)

is an ordinary differential equation, which

has a

unique

solution for any piece-wise continuous function f(t). To sum up, the necessary and sufficient condition for the existence and uniqueness of the solution to equation (I-2.1) is the disappearance of (I-2.4a)-(I-2.4c), or, in other words, QEP = diag(l, N),

QAP = diag(Al, I).

(I-2.5)

This condition is somewhat not so obvious and difficult to verify.

To obtain one

that is simpler to verify, we define the following. Definition I-2.1. For any given two matrices E, A E~nxn

the pencil (E,A) is call-

7 ed r e g u l a r i f t h e r e e x i s t s a c o n s t a n t s c a l a r , ~ C such t h a t ] aE+A I ~ O, or t h e p o l y nomial I sE-AI $ O. Lemma 1 - 2 . 2 .

(E,A) i s r e g u l a r i f and only i f two n o n s i n g u l a r m a t r i c e s Q, P may be

chosen such t h a t QEP = d i a g ( I n l ,

N),

QAP = d i a g ( A l , In2)

(1-2.6)

where nl+ n 2 = n, A 1 £ ~ n l x n l , N 6 ~ n2xn2 i s n i l p o t e n t . Proof.

Sufficiency:

Let ( 1 - 2 . 6 ) be t r u e .

We choose a E o(A 1) (such a s a r e nume-

r o u s ) . Then

Ia E + A I =

IQ-IIIp-IIIaQEP+QAPI = IQ-I Ilp-IllaI+AI 1¢ O.

Thus (E,A) is regular. Necessity: If (E,A) is regular, the definition shows the existence of a such that IaE+AI ~ O. Consider the pencil = (aE+A)-IE,

A = (aE+A)-IA.

It is easy to obtain A

A

^

=

I

-

a E.

On the other hand,

(1-2.7) it follows immediately from the Jordan canonical form decompo-

sition in matrix theory that there exists a nonsingular matrix T such that T~T -I = diag(El, E2 ) where El E ~ nlxnl is nonsingular, E2 E ~ n2xn2 is nilpotent. This means that I- aE 2 is ^ -I nonsingular. Let Q = diag(al I, (I- aE2)-I)T(aE+A) , and P = From (I-2.7) we ^-I ^ ^ -I ^ have QEP = diag(Inl, N), QAP = diag(A I, In2)~ where A 1 = E 1 (I- aE2), N = (I-aE2) E 2

T-1.

is nilpotent. Q.E.D. Equation solution square

for

(I-2.5)

shows that

if the differential equation (i-2.1)

any sufficiently differentiable function f(t),

and (1-2.5) holds.

has

the matrices

This is equivalent to the regularity of pencil

a unique must

be

(E,A)

by

Lemma I-2.2.

Hence, to guarantee the existence and uniqueness of solution for system

(I-1.3),

shall

we

hereafter assume that E and A are square and

regular

matrices,

unless otherwise specified. In this case, system (i-1.3) will be termed "regular". We often use (E,A,B,C) to denote the system (I-1.3) for convenience. The following theorem gives an easy criterion for regularity. Theorem l-2.1.(Yip and Sincovec, 1981).

The following statements are equivalent

(I).

System (I-1.3) is regular;

(2).

If Xo= KerA, X i = {x I AxEEXi_I}, it must be K e r E N X i = O, i= O, I, 2 .... ;

(3).

If Yo = Ker~, Yi = {x I A~x£E~Yi_I }, K e r E ~ ¥ i

(4).

Let

= O, i = O, I, 2, ...;

l

E

A "

E R kk+l)nxnk" " •

E

A

Then rankG(k) = nk, k = i, 2, ... ;

(5).

Let

I

F(k) =

E

A ".'. A E

]

~nkxn'k+l~,

E

A

)

Then r a n k F ( k ) = nk, k = 1, 2 . . . . .

Example I-2.1. E=

Consider the following matrix pencil: I 0

-

0

l

A= ,

-

2

0

0

1

.

Direct computation shows that lsE-AI = -4(s-I) 2 # O. Thus (E,A) is regular. This From

method verifying regularity is inconvenient when E and A are of high orders.

the viewpoint of computation,

Luenberger (1978)

proposed

another

criterion,

which is called the shuffle algorithm. C o n s i d e r t h e nx2n m a t r i x [E

If

E

(I-2.8)

A]

is nonsingular,

singular

(E,A) is regular and the algorithm

stops.

Otherwise,

E

is

block form

and by taking row transformation we may change (I-2.8) into the

of El [0

AI A2 ]

(i-2.9)

where EIE ~qxn is a full row rank with q = rankE.

Then we "shuffle" the second block

row in (1-2.9) to obtain [ El A2

A1 ] 0

(1-2.10)

If [El/A2] is nonsingular,

(E,A) is regular and the algorithm stops.

Otherwise,

we

repeat the algorithm. It has been proven that the algorithm would terminate in either of the f o l l o w i n g indicating

ways:

The m a t r i x

(E,A) i s r e g u l a r ;

or at

[ora,ed least

by the left ,~ columns

is

nonsingular,

one z e r o row a p p e a r s i n t h e m a t r i x ( 1 - 2 . 1 0 ) ,

which would indicate that (E,A) is not regular (Stengel et al, 1979). Example I-2.2.

Consider the matrix pencil (E,A) in Example 1-2.1. Then

0 1 1 1 1 0 1 ] [E

A] =

(1-2.11)

1 1 0 1 0 2 0 - I 0 1 1 - ] 0 1 ,

where E is singular. A row transformation in (1-2.11) yields

1

0

I

0

2

(|-2.12)

0

2 0 , [i 0 0I I - 2I°I} and shuffle of (I-2.12) gives:

0

1

-2

1 1

1 1 0 11

1

I

0

0

2

0

I 0 0 0

2 0

The left 3x3 matrix is nonsingular, indicating the regularity of pencil (E,A). we see

that

the row

A] is equivalent to the row transformation on

sE-A

with

Comments on shuffle al~orithm. transformation

on

[E

In

the shuffle

algorithm,

a

nonzero constant multiplying IsE-AI, i.e.,

(1-2.13)

I

tO

-A 2

Note the equation

[

-A 2

J

f[Ells [1]1

(E-2.14)

LA2J

From the second part in (i-2.14) we understand that the shuffle algorithm at one step is just multiplying the lower rows in by (-s).

sE-A ,

obtained from the row

trans[ormation,

This results in multiplying the polynomial IsE-AI by the same factor,

when

(E,A) is regular. The final step constitutes the following equation P(s)(sE-A) = sI - E ,

I P(s)l = ~s m ,

~ = constant ~ O.

Thus we may conclude that LsE-AI ~ O from above equation, or in other words, (E,A) is regular. The advantage of shuffle algorithm lies in its ease of computation.

10

1-3.

Equivalence of Singular Systems

As pointed out in the introduction,

the description equation for a real system or

mathematical model may be obtained by selecting appropriate state variables. lection is certainly not unique;

The se-

this in turn causes the nonuniqueness of the model.

We will now study the relationships between the state variables and models.

problem is clear,

If

this

an appropriate model may be established that is easy and convenient

for system analysis and/or design. Definition 1-3.1. tablished between any to represent a s

Let ~

two

denote a set, in which a certain relationship ~

elements~,

1 2 E ~ . The sign '~I~E2 '' will be used herein

relationship between[l and [2. If = satisfies

(1). (Reflexivity):

V E l E ~,

~I ~

(2). (Transitivity): v I1, I2, I3 E (3). (Invertibility): The relationship For

is es-

V I I, 12 E

El ' ~,

~,

ifll~12, if E I ~

[ 2 ~ [3, ~1~ 13 is

E 2, it must be E 2 e

true,

E1 .

m will be considered equivalence.

simplicity,

the time variable will be omitted in our discussion later unless

it is necessary to include it. The initial time is taken as to = O.

Consider the two singular systems: Ek = Ax + Bu

(1-3.J)

y = Cx

and E~ = ~ Y = ~

+ ~u .

(1-3.2)

Assume that there exist two nonsingular matrices Q and P such that x = P~

(I-3.3)

and QEP - E, QAP = ~, QB = B, CP = ~. Systems (1-3.1) and (i-3.2) will be called restricted system equivalent (r.s.e.). (]-3.3) is the coordinate transformation. From

the definition,

that possesses reflexivity, Example I-3.1.

restricted system equivalence is an equivalent relationship transitivity,

and invertihility.

Consider a simple circuit network as shown in Figure 1-3.1, where

voltage source Vs(t) is the driver (control input), R, L, and C stand for the resistor, inductor, and capacity, respectively,

as well as their quantities,

ltages are denoted by VR(t), VL(t), Vc(t), respectively.

and their vo-

Then from girchoff's laws,

11 we h a v e the following circuit equation (description equations):

1 o/j,i(t)l.

o

~

o

o

o o o,,~c(t~, 0

00J[VR(t)J

1-0

1

1

o

/v,.(o!

o

i

iVc(t)|+

0

1 [VR(t)J

R

l

-I

L

j

Vs(t)

(i-3.4)

Vs(t).

ecv~w

I(t) ~

Cl

Vc(t):

To Figure 1-3.1.

On the other hand, if we choose

f(t)

I

VR(t)+VL(t) VR(t)+VL(t)+Vc(t)

VR(t)

as state variable, it will be

1 -I

~=

0

-I/C

1

-I

0 0 I 0

I

2 0

~

li°°°l ° 111] [i] -I

0

1

0

0

0

•

[I/C-2R

+

Vs(t).

(1-3.5)

Both (1-3.4) and (1-3.5) are mathematical alodels that describe the circuit network in

Figure i-3.1.

Although (I-3.4) and (I-3.5) are different,

we can easily

verify

that they are r.s.e., with the transformation matrices

Q =

I'°°I I -I 0 1 l 0

0 2 I

I 0 0

p =

[0oo1 0 I 0 -I 0 0

Now we will introduce some common r.s.e, forms

0 -i I 0 0 1

.

in singular systems.

Consider

the

singular system

(I-3.6)

E~ ~ Ax + Bu y = Cx

where x ~ n ,

u~m,

y£~r

E, A t~nxn, B ~ n x m ,

and c ~ r x n

are constant

matrices.

12 1-3.1.

First Equivalent Form (EFI)

Lemma i-2.2

shows that:

For any singular system (I-3.6), there exist two nonsin-

gular matrices Q and P such that (1-3.6) is r.s.e, to

Xl = AlXl + BlU

(1-3.7a)

Yl = ClXl Nx 2 = x 2 + B2u

(1-3.7b)

Y2 = C2x2

(1-3.7c)

Y = ClXl + C2x2 = Yl + Y2 with the coordinate transformation Xl

=

Ix2 ] p-lx,

Xl£]Rnlxn2 ,

x2ER n2xn2

(]-3.s)

and QEP = diag(Inl , N),

QAP - diag(Al, In2),

QB = [BI/B2] , CP = [C 1

C2],

(1-3.9) where nl+n 2 = n, N E ~ n2xn2 is nilpotent. Equation form,

(1-3.7) is the EFI,

usually called the standard

decomposition. In this

subsystems (I-3.7a) and (i-3.7b) are called slow and fast subsystems respecti-

vely; Xl, x 2 are the slow and fast substates, respectively. Generally,

the matrices Q and P,

which transfer a singular system into its stan-

dard EFI, are not unique, resulting in the nonuniqueness of EFI, i.e., AI, B I, B 2, C I, C 2, N. Assume that Q and P are nonsingular and system (1-3.6) is transferred into its EFI, in other words, system (1-3.5) is r.s.e, to: Xl = AlXl + Bl u

Nx2 = ~2 + B2 u

(l-3.10a)

(l-3.10b)

Y2 -- ~2~2 Y = C1~1 + [:2~2 = Yl + Y2

(1-3.10c)

with the coordinate transformation p-lx = [Xl/X2]. The following theorem shows the relationship between the matrices Q, P and Q,

P,

and the system coefficient matrices. Theorem I-3.1.

Suppose that (1-3.7)-(1-3.9) and (I-3.10) are the EFIs for

system

13 _ _ Then n I = nl, n 2 = n2,

(i-3.6). T2E ~n2xn2

and there exist nonsingular matrices T I E ~

such that

Q = diag(Tl, T2)Q,

P = Pdiag(Tl, T 2)

A1

N = T2NT21

TIAITI 1

w

(1-3.11)

C i = CiTi I,

B i = TiB i, Proof.

nlxn 1

i = I, 2.

Since

IsE-AI = IQ-111p-IIIsI-AI

I = I Q-111p-IlIsI-~I

I

(1-3.12)

and n I + n 2 = n, and nl + n2 = n, we know that n I = nl' n2 = n2" Furthermore, if we denote

(I-3.13) tQ21

Q22

'

LP21

P22

the equations (1-3.8) and (1-3.10) show that Q-Idiag(l ' N)p-1 = ~-Idiag(i '

R)p-1, (I-3.14)

Q-Idiag(Al, l)p -I = ~-Idiag(Xl, I)P -I. Substituting (I-3.13) into (I-3.14), we know

Qll = P l l '

Q22 " P22

(I-3.15s)

QI2 = ALP12'

QzlA1 = P21'

(I-3.15b)

QI2 N = P12'

(I-3.15c)

Q22 ~ = NP22.

(i-3.15d)

Q21 =

NP21'

QIlA1 = A1Pll, The s u b s t i t u t i o n

of ( I - 3 . 1 5 c ) into (1-3.15b) y i e l d s

P21 = NP21AI'

(i-3.16)

P12 = A1P12N"

Note that matrices N and N are nilpotent. Thus P21 = NP21AI = ... = N n P21AI -n = O,

PI2 = 0

(1-3.17)

which, in conjunction with (I-3.15b) and (I-3.15c), yields Q21 = o,

QI2 = o.

(I-3.18)

Denoting T 1 = QII and T 2 = Q22' from above equation (1-3.18) and the nonsingularity of Q, P, Q and P, we may conclude that T I and T 2 are invertible. Equations (I-3.13), (1-3.15a), and (i-3.16)-(I-3.18) further show that Q = diag(T I, T2)~,

P = ~diag(T~ 1, r21).

14 Hence, equation (1-3.11) holds. Q.E.D. Although different EFIs may be obtained under different coordinate matrices,

transformation

this theorem assures the similarity property among these EFIs.

that EFI is unique in the sense of similar

Example I-3.2.

This means

equivalence.

Consider the circuit network in Example I-3.1, with L = C = R = I,

and a measure equation: y(t)

= V c ( t ) = [0 0 1 0 ] x ( t ) ,

x(t)

= [l(t)

VL(t )

Vc(t)

VR(t)] ~.

Then the system becomes

Ii °°°] [i 1°°] [il 0 0 0

1 0 0

0 0 0

i=

y = [0

0

I

0 0 I

-

0 0 1

0 1 1

x +

(1-3.19)

Vs(t)

O]x .

If we c h o o s e the following transformation p-I x = [Xl/X2],

X l E ~ 2,

[o11]

Q= 0

I 0 0-I 0 I

0 1 0

x 2 6 ~ 2,

[1ooo -I -I

P=

0 1

1

0

1 0 0 0

0 i

it will become

~1 = [-11-1]xlo +[I]Vs(t) o= x2+ [-~]Vs(t) y = [0

(1-3.20)

l]x 1 .

This is the EFI for system (1-3.19).

1-3.2.

Second Equivalent Form (EF2)

Let q = rankE. From matrix theory, we know that there exist nonsingular matrices Q1 and PI such that QIEPI = diag(lq, 0). By taking the coordinate transformation Pllx = [x I /x2] ' Xl E E q ' x2 E~n-q, system (1-3.6) is r.s.e, to ~I = AllXl + A12x2 + BlU 0 = A21x I + A22x 2 + B2u y = ClX 1 + C2x 2

(I-3.21)

15 where

QIAPI

=

[All [A21

A12 ] Q1B A22 ,

=

IBI] B2

CP 1 =

[C 1

C2]

.

Equation (I-3.21) is the second equivalent form (EF2) for system (I-3.6). transformation,

matrices Q1 and P1 are

not unique,

which results in the nonunique-

hess of EF2. Two EF2s may have a relationship too complicated to merit The EF2 the first second

clearly reflects the physical meaning of singular systems. equation is a

In this

differential one composed of dynamic

further study. In

subsystems,

is an algebraic equation that represents the connection between

(I-3.21), and

the

subsystems.

Thus, singular systems may be viewed as a composite system formed by several interconnected subsystems.

Furthermore, substates x I and x2 reflect a layer property in

some singular systems: one layer has a dynamic property (described by the differential equation); the other has interconnection, constraint, and administration properties (described by the algebraic equation). Singular value decomposition may be used in the decomposition of E instead of rank decomposition here. Example ]-3.3.

Consider the system in Example I-3.2. Under the coordinate trans-

formation p~Ix = [Xl/X2],

Xl E R 2

x2 E~2, with

P1

Q1 = 14'

i ooo]

0 0 1 0 = 0 1 0 0 0 0 0 1

system (I-3.19) is r.s.e, to

y = [0

l]x I + [00]x

2 .

I-3.3. Third Equivalent Form (EF3) Under the regularity assumption,

there always exists a scalars(such as are numer-

ous) such that IaE+AI~ O. Let the transformation matrices be Q2 = ( aE+A)-I'

P2 = In

Then Q2A = In - a( aE+A)-IE. Thus, system (I-3.6) is r.s.e, to

16

Ex

=

(I

(1-3.22)

aE)x + Bu

-

y = Cx where E = Q2 E and B = Q2 B, Obviously,

for

a fixed

which is the third equivalent form (EF3)

for

(I-3.6).

a , the EF3 is unique in the sense of algebraic equivalence

(similarity).

Example 1-3.4.

Consider system (1-3.19). Note that A -I exists. Then a may be

chosen as zero, and

Q2 = A-I

I 1

=

o

0

-01 -1 -1 1 1

0

1 0

P2 = I4 "

Thus, system (I-3.6) is r.s.e, to A

E~ = x + BVs(t)

(1-3.23)

y = Cx where t,.

E =

[ olo] [1 _

o o o

0 -I 0 1

0 0

;

=

,

1-4.

o

-I 0

co[o

o

1 o].

,

State Response

Consider the system E~

~ Ax

+ Bu

y = Cx where

x~IR n,

(I-4.1) u E ~ m,

y E ~ r.

As pointed out in Section I-3.1,

under the regularity

assumption on (i-4.1), there exist two nonsingular matrices Q and P, such that system (1-4.1) is r.s.e, to

Xl = AlXl + BlU'

Yl = ClXl

(1-4.2a)

N~2 = x2 + B2u'

Y2 = C2x2

(1-4.2b)

Y = ClXl + C2x2 = Yl + Y2 where X l E R n l ,

(1-4.2c)

x2 E R n2, n 1 + n 2 = n, and N i s n i l p o t e n t

whose n i l p o t e n t

index is de-

n o t e d by h, and QEP = d i a g ( I ,

N),

QAP = d i a g ( A 1 ,

I),

CP = [C 1

C2] ,

(1-4.3a)

17 QB = [BI/B2] .

p-Ix = [Xl/X2],

(1-4.3b)

Note that the slow subsystem (I-4.2) is an ordinary differential equation. a

It has

unique solution with any initial condition Xl(O ) (the initial time is supposed

to

be zero) for any piecewise continuous function u(t)" Xl(t) = eAltxl(O) + ~eAI(t-T)BIU(T)dT

.

(I-4.4)

Thus, Xl(t ) is completely determined by Xl(O), u(~)(0 ~T~< t). Suppose By

that the function u(t) is h times piecewise continuously

continuously taking derivatives wit~ respect to t on both sides of

differentiable. (I-4.2b), and

left multiplying both sides by matrix N, we obtain the following equations: N~ 2 = x 2 + B2u N2x~ 2) = N~ 2 + NB26 "'"

(i-4.5)

Nhx~ h) = m"h-lx2(h-l) + Nh-IB2u(h-I ) where x~ i) stands for the ith derivative of x2(t ). From the addition of these equations and the fact N h = O, we know readily that the solution x2(t ) is given by h-I x2(t) = - ~ N i B 2 i=o

u(i).

(I-4.6)

x2(t) is a linear combination of derivatives of u(t) at time t, which is certainly determined by the derivatives at time t. u(T), OCT ~t-e

For any scalar ~ > 0, the properties of

have no contribution to x2(t). This sho~.ts an interesting phenomenon

between the substates xl(t ) and x2(t): Xl(t ) represents a cummulative effect of u(T), 0x O. The substate of fast subsystem (I-4.2b) may also be obtained in another way. Taking the Laplace transformation on both sides of (I-4.2b), we have (sN-I)x2(s) = Nx2(O ) + B2u(s) where x2(s ) and u(s) stand for the Laplace transformation of x2(t) and u(t),

respec-

tively. From this equation we have h-I x2(s) = (sN-l)-l(Nx2(O)+B2u(s)) = - ~ Nisi(Nx2(0)+B2u(s)) • l=o Note that the Dirac function

~(t) has the Laplace transformation of

L[~(i)(tll = s i. Hence the inverse Laplace transformation of x2(s ) yields:

&(i)

h-t

x2(t) = - ~ i=o

I,-I

(t)Ni+Ix2(O) - ~--- NiB2u(i)(t) i=o

which is just the same as (I-4.9).

Example 1-4.1.

Consider the circuit network system in Example I-3.2:

llooo} o I

o ~=

0 0

0 0

0 0

y = [O

with

0

0

1

o) Ill

o o o -

0 I

0 1

1 1

Vs(t)

x+

-

(i-4.11)

O]x

the standard decomposition shown in (i-3.20).

From (1-4.6) and (I-4.4), the

slow and fast subsystems have the following state responses:

(i-4.12) respectively.

20

Furthermore, direct computation shows the state response of system (1-4.11) to be:

x(t) = [ xl(t)

]

where

-2si~t

xl(t) =

[ ~i

+

]x:~o>

2sin~t I

r-sin~3(t-T) _ ~/3

e-:(t-T)~

~

L

cosd~3(t-T)]

7:

Vs(~)dz

2sin :(t-*)

In this example, N = O; thus no impulse terms appear apparently in fast state response, nor do the derivative portions.

Example I-4.2. The following 2-order system (I-4.]3) Olt~2(t) I

tx2(t)

-I

has only fast subsystem, with coefficients 1 0 o]

-1

,

1 a

•

Therefore, its solution is = -

x2

Ni°-i-l-~t)

Xl(O)

x (0) -

i=1

NiBu-i-~t)( )" " i=o

= [- x2(O)~(t)u(t) + u(t) + u(t)]

(I-4.14)

Thus xl(t) = - x2(O)~(t) + u(t) + U(t) (I-4.15) x2(t) = u(t).

Two special cases are studied herein. I. Let the input function be the unit jump function u(t) Then

f(t-a)

[ 1

t >a> 0

10

t < a ,

a is a constant.

21 Xl(t) = - x2(O)~(t) + f(t) + ~(t-a) x2(t) = f(t). They are illustrated in Figure I-4.1.

X2

x1

k

r. U

t o

(a)

(b)

Figure I-4.1.

2. If the input function is 0

t

P

u(t) = g(t) = L t-~

t >

> 0 is a constant

then u(t) is continuous. But Xl(t) = - x2(O)~(t) + g(t) + f(t-~) x2(t) = g(t) whose locus is shown in Figure 1-4.2.

xlI 0

"

P

-

t

x2:ul

L t

0

(a)

P

(b) Figure 1-4.2.

Classical

system theory confirms that the state response is continuous,

that the input function is piecewise continuous (or

weaker).

However,

provided

this is

not

true in singular systems. Figures 1-4.1 and 1-4.2 show that when input derivatives are involved, either impulse terms may be created in the state response by any jump behavior in input due to the operation at the starting and closing switch actions

in

22 practice (Figure l-4.1(a)); or jump behavior appears in the state response, although the input function is continuous, havior appears at t = ~

as described in Figure i-4.2, which shows jump be-

in Xl(t) due to the discontinuous property in input deriva-

tives. It is worth pointing out that (i-4.10) is only a solution for (1-4.1) in the tribution sense, (I-4.10)

not a solution in the ordinary sense,

dis-

i.e., when applied~ equation

doesn't make the equality hold in the ordinary differential equation sense.

However, the equality holds in the distribution sense.

I-5.

Currently, argues

there

Notes and References

are mainly two opinions concerning the initial

condition. One

that the initial condition should be restricted to the consistent conditions;

while the other says that any possible initial condition should be acceptable. of its convenience in the system analysis, distribution may be found in Appendix A,

Because

this book adopts the latter. Knowledge of but one nmy read this

section accepting

only distribution solutions. Practical examples of singular systems may be found in Campbell (1982b),

and Rose

(1982), Itaggman and

BryanL (1984),

Kiruthi eL u[. (1980),

Campbell

Lewis (1986),

Luenberger (1977), Luenberger and Arbel (1977), Martens et e l . (1984), Newcomb (1981, 1982),

Petzold (1982), Singh and Liu (1973), Stengel et e l . (1979), and Wang and Dai

(1986a). Typical papers

concerning regularity are Yip and Sincovec

(1981),

and

Fletcher

(1986). Among papers oil r e s t r i c t e d system equivalence, Cullen (1984),

Dai (1987b),

t y p i c a l ones are Campbell (1982b),

Fuhymann (1977), Hayton et a l . (1986), Pugh and Shelton

(1978), Verghese et a l . (1981), Zhou e t e l .

(1987).

On the solution of distribution one may refer to Cobb (1983b, 1984), al. (1981), and CuIleu (1984).

Verghese et

CHAPTER 2 TIME DOMAIN ANALYSIS

The d e s c r i p t i o n tion the

e q u a t i o n of a p r a c t i c a l system may be e s t a b l i s h e d through

of the proper state variables.

system based on this description equation,

derstanding Using

through which we may gain a fair un-

of the system's structural features as well as its internal

time domain analys~s,

selec-

Time domain analysis is the method of analyzi.g

properties.

this chapter studies the fundamentals in system

theory

such as reachability," controllability, observability, system decomposition, transfer matrix, and minimal realization. The concepts explained in this chapter are essential for later chapters.

2-I.

State Reachability

Consider the regular singular system: E~ = Ax + Bu

(2-I.1)

y = Cx where x E ~ n ,

u6~m,

YEar

are its state,

control input, and measure output, respec-

tively; and E, A E Rnxn, B £ Enxm, C E ~rxn are constant matrices. Without loss of gene-

r u l i t y , q ~ raokE < n i s supposed. For system, a s p o i n t e d ouL in tile p r e v i o u s s e c t i ( m , two nonsingular matrices Q and P exist such that it is r.s.e, to (EFI):

Xl = AlXl + BlU N~ 2 = x 2 + B2u

(2-1.2)

y = ClX I + C2x 2 where x [ 6 ~

"1

, x26~

n2

, n I + n 2 = n, N 6 ~ n2xn2 is nilpotent,

the nilpotent index

is

denoted by h, and QEP = diag(I, N), QAP = diag(Al, I), p-Ix = [Xl/X2], For

CP = [C l

C2]

QB = [B1/B2].

the sake of simplicity,

in this chapter,

system (2-1.1) is assumed to be in

its standard decomposition form (2-1.2) unless specified,

llowever, the results

are

24 applicable to systems in the general form of (2-1.1).

We start

from the concept of reachable set to study Lhe state

structure.

Assume

that admissible control input is confined to u(t) E C ph-I ' which represents the (h-l)times piecewise continuously differentiable function set. Then from Section 1-4 we know that, when t > O, the state response for system (2-1.2) is Xl(t )

eAltx (O)

eAl (t-T)BlU(r)dT o

h-1

( 2 - i .3)

x2(t) = _ ~ NiB2u(i)(t). i=o Equation

(2-1.3)

indicates that when t > O,

u n i q u e l y d e t e r m i n e d by t h e i n i t i a l

state response x(t) = [xl/x2] is

c o n d i t i o n Xl(O), c o n t r o l

iul)Ut u(T), 0 ~ T ~ L, a.d

time point t. The reachable set may be defined as follows.

Definition reachable,

2-1.1.

Any v e c t o r w ( ~ n i n n - d i m e n s i o n a l v e c t o r s p a c e i s s a i d

to

be

if there exists an initial condition Xl(O), admissible control input u(t)

E Cph-I , and t I > O such that x(t I) = [Xl(tl)/X2(tl) ] = w. Let R(Xl(O) ) denote the reachable state from the initial condition Xl(O). Then R(xI(O)) = {w(Rn [ there e x i s t s an admissible control u(t) 6Ch-I and P t 1 > 0 such t h a t X ( t l ) = w} which is a subspace in n vector space. If we define <EIB>

= Im[B, EB . . . . .

En-IB]

where n is the dimension of E, ImB = {y

[ y = Bx, x E ~ m

},

KerE = ( x

[ Ex = O, x E E n}

we have the following lemma. Lemma 2-1.1.

For any nonidentically zero polynomial f(t), we define t

W(f,t) = I f(s)eAlSBIB~eA~Sf(s)ds" O

Then ImW(f,t) = for any t > O.

Proof.

To prove the result equals to the proof of nl-1 KerW(f,t) = f] KerBI(A~) i i=o

First, when xEKerW(f,t)

the definition indicates:

(2-1.4)

25 x~W(f,t)x = xZO = 0 i.e.,

t ~ t x~W(f,t)x = ~ x~ f(s)e A s1 m B~ Aes l f(s)xds= f [[Br Ae s 1 f(s)x[[2 ds= O o I 1 o 1 2

(2-1.5) 1

where I[" II 2 r e p r e s e n t s

the spectral

v e c t o r norm i.n t h e d e f i n i t i o n

of

II x 1~2 =

(x%)

~.

Since ":As

2

lisle i f(s)x[12

>0

is continuous, we know from (2-1.5) that

B eA Sf(s)x = 0, Since polynomial f(s)

o < s < t.

has a f i n i t e

number o f z e r o s (,. 0 .~ s ( t .

from

the

precu-

ding equation, we immediately have ~A~,s BI e 1 x

0 ~ s ~

= O,

t.

The arbitrariness of s yields nl-i x £ N Ker(Bi(A1)i). o

Thus KerW(f,t)

If x

~

nl-I ~ i ~ Ker(BI(AI) ).

nl-I ( K e r (nB l ( A ] ) i ) ,

the reverse

of this

process will

be e a s y t o p r o v e

x(

0

KerW(f,t),

i.e., n-| o

Ker(B¢(~) i) ~ ii

--

KerW(f,t)

"

Hence, the lemma is easy to obtain by combining these results.Q.E.D. Obviously, this process is similar to the one in the normal case. Lemma 2-].2.

For any h vectors x k ( ~n, i = O, ] l 2, ...

l

h-l, and t I > O, there

always exists a vector polynomial f ( t ) 6 ~ n whose order is h-l, such that f(i)(t I) = xi, Proof.

i = O, i, 2 . . . . .

The proof is straightforward

h-l.

by setting

f(L) = Xo + xl(t-tl) + "'" + Xh-l(t-tl )h-l" Theorem 2-1.1.

Q.E.D.

Let R(O) represent the state reachable set from the zero initial

condition (Xl(O) = O). Then R(O) = ~ 9 < N I

B2>

26 where " ( 9 " Proof.

is the direct sum in vector space. When Xl(O) = O, equation (2-1.3) shows

Xl(L) = IieAl(t_X)BlU(X)d~

h-1 NiB2u(i)(t), x2(t) = -~--~, i=o

,

t > 0.

Thus, ohvkous]y, x2(t) £ - Further use of the results in Appendix B shows that

eA1 t =

P o ( t ) l + ~l(t)A1 + . . . + ~ n l _ l ( t ) A T l - 1 .

Thus

t

Xl(t) = lleAl(t-')BlU(,)d,

=n~-~A~BIIo I~i(t-,)d, i=o

£

or in o t h e r words, x(L) ( (9 . T h e r e f o r e , R(O) G (~ .

(2-1.6)

On the o t h e r hand, for any x = [Xl/X2] E @ , with x 1 E , x 2 E , from x 2 E there exists x 2 i £ ~ n 2 , i = O, i, 2, ... , h-I such that h-l x2 = _ ~ NiB2x2i • i=o Akso from Lemma 2-1.2,

for any fixed t > O, there exists a polynomial f2(s) of order

h-I such that f~i~t) = x 2 9 Thus, if we impose the input control

u(t) = u l ( t ) + f2(t) it will be that t

t

Xl(t) =~oeAl(t-~)BlUl(~)dT + IoeAl(t-T)B2f2(T)dr

,

implying 7! £

where E l i s defined as ¢ ~I = Xl -

~eAl(t-t)B2f2(T)d*

For any fixed t > O, l e t f l ( s )

"

= s h ( s - t ) h. Then f l ( s )

i s not i n d e n t i c a I l y equal to

zero and Lemma 2-1.1 shows t h a t a vector z E$I nl may be chosen such t h a t W(fl't)z °

~1"

For ihe z (- I1~nl clioseii, we const.riivl

27 ~2...~ Al(t-s) z Ul(S) = r l t S ) m l e

Ox<s~t.

Direct computation would verify that

Xl(t)

= Xl - ~1 + ~ t e A l ( t - X ) B l U l ( x ) d r = Xl - ~1 + W ( f l ' t ) z ~o

= Xl

and h-I x2(t) = x 2 - ~ N i B 2 u ( i ) ( t ) . i=o Since

u}i)(t) = 0,

i = 0, I, ... , h-l,

it must be x2(t).. = x 2 • Thus R ( O ) ~ ~ < N

IB2>. The combination of it with (2-1.6) results in R(O)

= ~ ) < N I B2>. Q.E.D. Denote

l l ( x l ( O ) ) = { x = [ X l / X 2 ] [Xl= e A l t x l ( O ) ( ~ n l , which r e p r e s e n t s

the free state

If R is the reachable set

from a l l

R=

possible

reachable

s e t from s t a r t i n g

for system (2-1.2)

initial

condition

point Xl(O).

d e f i n e d a s tile u n i o n o f a l l

reachable

Xl(O) ( ~ nl , i . e . ,

U R(Xl(O)). Xl(O)

Earlier discussion R =

set

x2= O ( N n2 }

shows t h a t R t a k e s tile form o f

1,J

(R(O)+H(Xl(O)) = Rnl @ .

(2-1.7)

Xl(O) Obviously 0 (R. Equation (2-1.7)

p r o v i d e s us w i t h a p r e c i s e

x ( t ) a t any t i m e p o i n t t l i e s tI > 0

by t h e s t a t e

priate control

i n R,

response,

input u(t).

starting

Furthermore,

t o r s may be r e a c h e d i n any s h o r t Another direct up

the

any v e c t o r

form o f r e a c h a b l e s e t .

While t h e s t a t e

in R may be r e a c h e d a t a c e r t a i n

from a c e r t a i n

time

x1(O ) and d r i v e n by an a p p r o -

t h e p r o o f p r o c e s s a l s o shows t h a t such v e c -

period with a suitably

phenomenon may be s e e n i n r e p r e s e n t i o n

whole space as in the normal case,

chosen control (2-1.7):

u(t).

Instead of filling

the state response for singular

systems

lies on a fluid manifold in the space. This is a feature that differentiates the singular system from the normal one.

Example 2-1.1.

Consider the system described l~y (I-3.20):

28

,, [ I '

+[0

o= x2+ [-10 ]Vs(r) y = [0 where

(2-1.8)

llx I

Xl' x2 ~ ~2,

which is in the standard form for the circuit network in Example

[-3.2, with coefficient matrices

,,-[', 0'] According to (2-1.7), the reachable set from the zero initial condition is R(O) = ~ with

= ]m[B l

AIBI ] = ~ 2 ,

= ImB2 = ~ l ~ { o ] .

Thus R(O) = ~3 ~ {0}, and R = ~2 G) = R(O). In this example, R = R(O), or in other words, the reachable set for the whole system is identical to the reachable set from zero initial condition.

Example 2 - 1 . 2 .

Consider

t h e s e c o n d o r d e r s y s t e m in Example 1 - 4 . 2 :

For t h i s system, the reachable set f o r Lhe system and from tile zero i a i L i a l

cottdition

are, respectively, R = R(O) = ~ 2 ,

R(O) = = ~ 2 .

Both are whole vector space. Now, we will examine the state response of this system h - I x(L) = -~--~NlB2u(i)(t) i=o

u(t)

T h u s , f o r a n y w = [Wl/W2] E ~ 2 a n d u ( t 1) = w2,

[ u(t)+O(t) ] =

~(tl)

.

t 1 > O, t h e e q u a l i t y = w1 - w2

X(tl)

= w implies (2-1.10)

whose first equation fixes the value of u(t) at time point t I and the second assumes tile potential variation of u(t) at t I. Clearly, u(t) will increase or decrease greatly at t I if w I and w 2 defer remarkably. This control strategy is difficult to realize in the slow changing mechanical or chemical system, even though it may be realized.

and high energy may be required,

29 2-2.

R-Controllability,

Controllability,

and I m p u l s e

Controllability Characterizing the structural properties from various views,

controllability,

R-

controllability, and impulse controllability are three important concepts in singular systems. By using these concepts, we may have a fair understanding of controllability by control input.

The controllabilittes also reflect the dLfference of singular sys-

tems from normal ones. As before, only (2-1.2) is considered. Definition 2-2.1.

System (2-1.2) is called controllable if, for any t I > O, x(O)(

Rnand w E~n, there exists a control input u(t)(~h-lp such that X(tl) = w. This definition statesthaLunder condition starting

x(O), from

we may

the controllability assumption,

for any initial

always choose a control input such that the state

x(O) may arrive at any prescribed position in ~n

response

in any given

time

period, it is easy to see that the definition is a natural generalization of controllability concepts in the normal case. Rewriting system (2-1.2) in the slow-fast su0system form, we get:

Xl = AlXl + BlU Yl = ClXl

(2-2.1a)

Nx 2 = x 2 + B2u Y2 = C2x2

(2-2.1b)

Y = ClXl + C2x2 = Yl + Y2"

(2-2.1c)

Then we have the following theorem. Theorem 2-2.1. (i). Slow subsystem (2-2.1a) is controllable iff

V s6¢,

rank[sE-A, B] = n,

s finLte

(2-2.2)

where C represents the complex plane. (2). The following statements are equivalent. (a). The fast subsystem (2-2.11)) is controllable. (b). rank[B 2, NB 2 . . . . .

QIB

(c). rank[N

B2] = n 2.

(d).

rankle

B] = n.

(e).

For

= [BI/B2].

any n o n s i n g u l a r

Nh-IB2 ] = n 2.

matrices

Then -B2 i s of f u l l

( 3 ) . The f o l l o w i n g

statements

Q1 and P1 s a t i s f y i n g

row r a n k ,

E =Qldiag(l,

rankB 2 = a - r a n k E .

are equivalent.

O)P 1, L e t

3O (a). System (2-1.2) is controllable. (b). Both its slow and fast subsystems are controllable. nl-I

(c). rank[B 1, AIB 1 . . . .

, A1

(d). rank[sE-A

V s(C,

B] = n,

B1] = n 1 and rank[B2, s finite, and rank[E

NB2, . . .

, Nh-IB2 ] = n 2.

B] = n.

(e). The matrix

E D] =

-A E

B .. -A

B.

E

"B

n2x(n+m-i)n

O

n2x(n+m-1)n

or, equivalently,

D1 =

k

0

2

is of full row rank n . Proof.

We will prove the statements one by one.

( 1 ) . 'File slow subsystem ( 2 - 2 . 1 a ) es the c o n t r o l l a b i l i t y trollabte

definition

Ls a normal one, f o r which D e f i n i t i o n in the linear

2-2.1

system t h e o r y . Thus, ( 2 - 2 . 1 a )

becom-

is con-

iff rank[sl-A1,

BI] = h i ,

V s(C,

s fiuite.

(2-2.3)

Noticing that rank[sE-A, B] = rank[sQEP-QAP, QB] = rank[S~-Al

0 sN-I

and sN-| is invertible for any finite s ( ~ , rank[sE-A, B] = n 2 + r a n k [ s I - A 1 ,

B1 l B2 we have

B1]

which indicates that (2-2.3) holds if and only if rank[sE-A, B] = n, V s( C, and s is finite. This is (2-2.2). (2). According to the definition,

if the fast subsystem (2-2.1b) is

= ~ n2, or rank[B2, NB 2 . . . . . Furthermore,

o(N) = {s

Nh-IB2 ] = n 2. Thus (a) and (b) are equivalent.

(N, B2) is controllable if and only if

rank[sl-N, B2] = n2, where

controllable

I sE~,

V

s E o(N)

tsl-Nl = 0 } .

(2-2.4) is true if and only if rank[-N valence between (b) and (c).

Since N is nilpotent,

(2-2.4) a(N) = {0}. Equation

B2] = rank[N, B2] = n 2. Thus proves the equi-

31 To prove the equivalence between (c) and (d), we only need to note the fact rank[E

B] = rank[QEP

which means that rank[N

QB] = n I + rank[N

B2]

B2] = n 2 is equal to rank[E

B] = n.

The equivalence between (d) and (e) may be proven in a similar way. (3). Let (a) hold and Xl(O)=O, The controllability definition states that for any tl> 0 and

w E~n there exists an admissible control input u(t) 6 C h-I such that X(tl) P = w. This in turn assures that the reachable set from the zero initial condition R(O)

has t h e p r o p e r t y R(O) = ~ ) < N [B2> = ~ n , o r more p r e c i s e l y , A~I-IB1 ] = n 1 and r a n k [ B 2 , NB2 . . . . . if (c) holds,

On t h e o t h e r hand,

from t h e e q u a t i o n

R(Xl(O))

we know

Nh-IB2 ] = n2, which i s ( c ) .

rank[B l , A|B 1 . . . .

= R(O) + H ( X l ( O ) )

immediately that

R(Xl(O)) = ~n.

The system

is controllable. Preceding

discussion proves the equivalence of (a) and (c). The

combination

of these results with (I) and (2) proves the equivalence between

(b) and (c) as well between (a) and (c). Next we will prove the equivalence of (c) and (e). Let Q and P be the transformation matrices from (2-1.1) to (2-1.2), and = diag(q, Q . . . . .

Q) E m n2xn2

= diag(P, P . . . . .

P, Inm) E ~(n+m-l)nx(n+m-l)n

Then both Q and ~ a r e n o n s i n g u l a r . -A 1 0

Since

0

B1

-I

B2

1

0 -A 1

0

Bl

0

N 0

-I

B2

I

0

0

N

QDIP =

°i

t a k i n g e a c h m a t r i x b l o c k a s an e l e m e n t ,

-A 1

0

B1

0

- I

B2

I

0

B1

0

N

B2

Lhe f i r s t

row p l u s t h e t h i r d

by A1. and t h e n t h e f o u r t h row p l u s t h e s e c o . d row m u l t i p l i e d ^

A

rankD 1 = rankQD1P

row

by N, we have

multiplied

32 0

0

0 -I = rank

-A 21

0

B1

AIB l

I

0

B2

0

0

B1

NB 2

B2

I

0

-A l

0

0

0

0

-I

o°, l

0

B1

0

N

B2

row rauk iE alnl o . ly i f

T h u s , Dl i s o f l u l l

i I i s of f u l l

,°°

NB2

O

-A 1

O

N

0

I

B2 B1

.

B2

•

.

•

row raJlk. ConLinuing the process, we know t h a t t.he f u l l

row rank of m a t r i x

D I is equivalent to the full row rank of

°I

"' Nn-IB2

Nn-2B2

.

.

.

NB 2

B2

•

N o t i c i n g tile f a c t t h a t Ni = O, i ) h (h x< n2) and l l a m i l t o n theorem, which tlssures the e x i s t e u c e of

13o' ~ i '

"'"

, l~n1_l, such thaL

i n -1 A I = ~o I + ~IAI + ... + ~nl_iAl I ,

£ >j n I,

it mu~t be that rankD I = rank[Bl, AIB I, ... , AII-IB I] + rank[B 2, NB 2, .... Nh-IB2 ] x< n I + n 2 = n. Therefore, Dl (thus DI) is of full row rank if and only if both (AI, BI) and (N, B 2 ) are controllable, in other words, system (2-1.l) is controllable. This proves that (a) and (e) are equivalent. Thus we complete our proof. Q.E.D. Example 2-2.1.

Consider system (2-1.8). Direct computation shows that

rank[B I, AIBI] = rank [01 -11]= 2 rank[B2' NB2] = rank[-lo

O0 ] = 1 < 2.

From Theorem 2-2.1 we know system ( 2 - 1 . 8 )

tem is controllable.

is not c o n t r o l l a b l e ,

while

i t s slow suhsys-

Noticing that (2-I.8) is the standard decomposition of the sys-

33 tem

in Example 1-3.2,

trollable.

In fact,

we know that the circuit network in Example 1-3.2 is not conin the descriptor equation (I-3.4), VR(t) = RI(t) is an identi-

cal equation. Thus VR(t) - RI(t) = 0 (VR(t) - I(t) = 0 in Example I-3.2) is not controllable by control input.

Example 2 - 2 . 2 .

Consider the circuit

s y s t e n shown i n F i g u r e 2 - 2 . 1 .

R

L

C2

F i g u r e 2-2.1.

i n p u t . Choose Lhe s t a L e v a r i a b l e

in which the v o l L a g e sourse u e i s the c o n t r o l x =[uCl

Uc2

[2

II]

where Uc[, Uc2 , [ 1 ' 12 a r e t h e v o l t a g e s of CI, C2 aml t h e amperage of tile c u r r e n t s flowing over them.

According t o K i r c h o f f ' s

second law (Smith, 1966) we may e s t a b l i s h

[cOO]ooi ooo] [oj

the following state equation

01 0 0

0

2 -L

0

0 -I

0 I

1 0

0 0

0

0

I

0

0

R

0 0

x +

(2-2.5)

u

- l

with the measure output:

y = Uc2 = [0

I

0

(2-2.0)

O]x,

In this description equation, we have

I sC r a n k [ s E - A , B] = rank

0

0

sC 2

I

-i

• -I

0

0

-sL 0

-1

0

0

0

0

0

-R

-i

and

ranklE, B] = rank

Iio iio o o0 C2

0

0

0

-

= 4.

= 4, V s E ~ ,

s finite

34 Therefore, by Theorem 2-2.1, the circuit network (2-2.5)-(2-2.6) is controllable. Genrally speaking,

in the case of high-dimension n for system (2-1.2), criteria

in (11 and (2) in Theorem 2-2.1 are not suitable for computation since the system decomposition or its eigenvalues are needed, and are not easy to obtain. In such cases, the matrix criteria in (3) are of special importance in the verification of controllability. Definition 2-2.2.

Singular system (2-1.21 is called R-controllable, if it is con-

trollab]e in the reachable seL, or more precisely, for any prescribed t I > O, Xl(O)~R and w ER, there always exists an admissible control input u(tIEch-I such that X(tl) P =

W°

The R - c o u t r o l l a b i i [ t y ssible

initial

condition

guarantees Xl(0)

s h e d i n a n y g i v e n Lime p e r i o d Theorem 2 - 2 . 2 . (i).

Singular

our controllability

t o any r e a c h a b l e if

The f o l l o w i n g system (2-1.2/

the control statements

state

u(t/

f o r tile s y s t e m from any a d m i and this

is suitably

process

will

be f i n i -

chosen,

are equivalent.

is R-controllable.

( ~ ) . The s l o w s u b s y s t e m ( 2 - 2 . 1 a / i s c o n t r o l l a b l e . (iii),

rank[sE-A,

(iv).

rank[Bl,

(v).

Denote

B] = n, V s E C, and s i s

AIBI . . . . .

All-IBI ] = hi"

[4

-A

D2 =

B

E

B ". -A E

w h e r e k /> n 1 may be a n y p o s i t i v e is of full

finite.

B -A

B

kn x (n+m)k

n u m b e r ; f o r e x a m p l e , k = r a n k E o r n . The m a t r i x

Proof.

We f i r s t

prove the equivalence

By d e f i n i t i o n ,

f o r x(O) = O E R ,

between (i)

and ( i J ) .

the R-controllability

states

that

R(O) =

= ] R n l ~ . T h u s , = ~ n l . Slow s u b s y s t e m ( 2 - 2 . 1 a ) a b l e and v i c e v e r s a . Equivalence Criteria

Therefore,

among ( i i ) ,

(iv)

Theorem 2-2.1.

(fit),

(i)

and ( i i ) a r e

and ( i v )

it

is controll-

equivalent.

are direct

reuutts

o f Theorem 2 - 2 . 1 .

and ( v ) may be p r o v e n i n t h e same way a s i n t h e p r o o f o f ( 3 )

(e)

in

Q.F.D.

This theorem, combined with Theorem 2-2.1, lable i f

D2

row r a n k k n .

shows Lhat system (2-1.2)

is R-coaLroi-

is controllable, but its inverse i s not true.

Example 2-2.3.

According to Theorem 2-2.2, systems (2-1.8) and (2-2.5)-(2-2.6)

are R-controllable. But system (2-1.8) is not controllable.

35 Example 2-2.4.

From the definition of R-controllability,

the following singular

system with only fast subsystem: N~

=

x

Bu

+

(2-2.7)

y = Cx where N is n i [ p o t e n t ,

is always R - c o n t r o l l a b l e .

Definitions 2-2.1 and 2-2.2 become the results in linear system theory for

normal

systems.

Clearly,

the concepts of controllability and R-controllability are concerned with

only the terminal behavior for the system, hand,

on t h e o t h e r

exist

impulse

we have p r e v i o u s l y

terms

which is an element in vector space. But,

lloted [|l~it ill l h e s t a l e

response,

that are set out either by the initial condition

Lhere or

also

by

the

possible jump behavior in control input u £ C h-I and its derivatives. Therefore, it is P necessary to analyze the control effect on impulse terms in the stale response. Recalling the state

structure in

Section I-4,

there are no impulse terms in

substate Xl(t ) when u £ C h-lp and the impulse part in x2(t ) is determined by

h-I x2~(t) = -~--~,~(i-l)(t)Nix2(O) i=l and when u(t) m O, ~(u(i)(t)) x2~

h-I ->-~,NiB2~=(u(i)(t)) i=o

(2-2.8)

= O, we have

h-I = - ~. ~i-l~t)Nix2(O). i=l

(2-2.9)

Clearly, x2~ = 0 when x2(O ) = O. For ally element w in vector space, denote

I~(w,t) =

[0

]

[2z(w,t)

h-1 (i_ 1) ( [ - ~ ) N i w .

I2z(w,t) = ~, i=l

(2-2.10)

Then [ =o(X2(O),t) represents the impulse behavior in x(t) at the start point caused by only tile i n i t i a l x ( t ) aC

c o n d i t i o n x2(O),

l~(w,t)

i n c l u d e s a l l p o s s i b l e impulse terms in

~.

Since t h e impulse terms in s t a t e x ( t ) only appear in t h e s u b s t a t e x 2 ( t ) , x ~ ( t ) =

[O/x2~(t) 1. Definition 2-2.3.

System (2-1.2) is termed impulse controllable,

tial condition x(O), = 6 N a n d

w 6 ~ n2,

if for any ini-

there exists an admissible control input u

C h-[ such that P x~(t) = I~(w,t).

(2-2.11)

This d e f h H t [ o n i s s i m i l a r to t h e c o n t r o l l a h i ! i t y d e f i n i l ion. I t c h a r a c l e r i z e s the a b i l i t y to g e n e r a t e impulse terms by a d m i s s i b l e c o n t r o l . Under t h e impulse c o n t r o l l a -

36 bility, in

a suitable control in admissible set may be selected such that impulse terms

x(t) may "arrives" at any "element" in I~(l~,t) at any given time point

I~(]~n2,t)= {I~(w,t) t wE ~Rn2 }. Thus x~ Impulse portions

controllability in

Otherwise,

~=.

Here

may take any possible"value".

is important for the necessity to eliminate the

a system in which impulse terms are generally not expected

to

impulse appear.

strong impulse behavior may stop the system from working or even destroy

it. For example, in the singular system Ex = Ax + Bu + v, where v is white noise, any possible impulse terms may appear in x~(t) (Cobb, 1984). Thus, it requires that we must eliminate these impulse terms by imposing appropriate control inputs. This point will be seen more clearly later. Now we will explore the criteria for impulse controllability. Theorem 2-2.3.

The following statements are equivalent.

(a) System (2-1.2) is impulse controllable. (b)

its

fast

subsystem ( 2 - 2 . l b )

(c)

Kern + Im[B 2, NB2 . . . . .

(d)

ImN = Im[NB2, N2B2 . . . . .

is impulse controllable.

Nh-IB2 ] = • u2.

Nh-IB2 ].

(e) IroN + ImB2 + Kern = N n2. (f) Let the controllability decomposition of (N, B2) be

[ l I [] (

O

N22

,

O

)"

Then either N22 disappears or a matrix M exists such that [Ni2/N22 ] = [NII/O]M. (g) rank [AE 0E 0 ] B = n + rankE. Proof. The equivalence between (a) and (b) is obvious. The equivalence among (b)-(e) are give, in Cobb (1984). Now we prove the equivalence between (d) and (f). Assume that N11( • filx~1, N22 ( h-1 . .~ince ( N I l , B 2 1 ) i s c o n t r o l l a b l e , Ira[B21 ' N I I B 2 I ' " "" ' NI 1 B21 ] = Rnl . In t h -

~52xfi2

i s case, (d) holds i f f

[mN = Im

= Im[NB2, N2B2, . . . O

This

equation

, Nh-IB2 ] = Im

11

N22

is

"

true iff either

N22 d i s a p p e a r s or t h e r e e x i s t s an M such

[N12/N22 ] = [NI1/O]M, whJch i s the e q u i v a l e n c e between (d) and ( f ) . Moreover, note that

E

B

= rank

QAP

QEP

QB

= 2nl + rank

N

g2

that

37

=

2n I + rank

the c o n t r o l l a b i l i t y that r a n k [ N i l ,

Nil

NI2

0

O

0

0 I

N22 0

0 NIl

O NI2

0 B21

0

I

0

N22

O

of ( N i l , B21) and the n i l p o t e n t

property

of

N i l , N22,

B21] = ~1 a c c o r d i n g to Theorem 2-2.1 ( 2 ) . This r e s u l t s

rank A E

B

l

= n + n I + rank

we klloW

in"

2 -N22

Hence rank

0

A

E

B = n + rankE ~ n + n I + rank

iN N2] O

N22

holds i f and only i f rank

[Nll

N12 ]

0

[~II

N22

Since N22 is n i l p o t e n t , In t h i s c a s e ,

= rank

NI2N22] 2 N22 ]

t h i s e q u a t i o n holds it' and only if N22 d i s a p p e a r s or N22 = O.

t h e r e e x i s t s an M, N12 = NllM. This i s the e q u i v a l e n c e between ( f ) and

(g). Q.E.D. Clearly from Theorems 2-2.3 and 2-2.2,

a system is impulse controllable if it is

controllable. Its inverse is false. The

r e l a t i o n s h i p s among t h e s e t h r e e c o n t r o l l a l ) i l i t y

concepts may be des('rihed

the following diagram:

, ..R-controllability controllability J "x'~impulse controllability Here A

• B represents that B may be deduced from A.

Example 2-2.5. The singular system with only slow subsystem (or normal system): = Ax + Bu y = Cx is always impulse c o n t r o l l a b l e a c c o r d i n g to tile d e f i n i t i o n . Example 2 - 2 . 6 . rank

For the c i r c u i t

[E 0 A E

0 1 B

network ( 2 - 2 . 5 ) - ( 2 - 2 . 6 ) ,

we have

by

38

= rank

Thus,

it

is

CI

0

O

0

O

C2

0

0

0

0

-L

0

0

0

0

0

0

0

0

0

C1

0

0

0

0

= 7 = n + rankE.

o

0

I

0

0

C2

0

0

0

-I

I

0

0

0

O

-L

O

O

i

0

0

R

O

O

0

O

impulse controllable.

Moreover,

-1

its

controllability

h a s been p r o v e n in

Example 2 - 2 . 2 . Example 2-2.7.

Example 2 - 2 . 1

has proven that

system (2-1.8)

i s not c o n t r o l l a b l e .

But since rank

A

0r 0 ]a n= k6 =E n' + E

it is impulse controllable.

2-3.

Observability, R-Observability and Impulse Obser wfl~il i t y

Having discussed various controllabilities, we now introduce the dual concepts --observability, R-observability and impulse ohservahility,

which

characterize

the

ability of state reconstruction from measure outputs. Definition 2-3.1.

System (2-1.2) is observable if the initial condition x(O) may

be uuiquc, ly d e t e r m i u t : d by u ( L ) , The

observabiiity

observing the initial

states

y(t),

that

condition

0 x< t ,n2i = II2, altd i=l .

the f o l l o w i n g are true.

1. The subsystem (NII,B21,C2I) is both controllable and observable. 2. The subsystem

N21 N22

,

B221,

[C21 Ol

)

is controllable, but unobservable. 3. The subsystem

[C21 C23] ) 0

N33

,

is observable, but uncontrollable• 4. The subsystem (N44 , O, O) is neither control]able nor observable.

Denote QI = diag(Tl' T2)Q'

Pl = Pdiag(T~ 1' T21).

Obviously, under this transformation, system (2-5.1) is r.s.e, to

Xll x12

All =

x13 x14

o ,,, o l[Xl,1 ~22 ~2~ ~2~//x,~/ o ~ o //x~/

A21 0 0 0

A43

+

A44J~x14J

BI1 BI2 0 0 (2-5.5a)

,,,o

)"' ° 1['"I

fx"1

o

~, o

IF:.~/

/ x:~/

0

N43

N44Jl~24 j

Ix24 )

+

B2I B22 0 0

53

(2-5.5b)

y = [ell, O, C13, O, C21, O, C23, O][T;lxl/T21x2] where i=l,

Tilxi = [XillXi2/xi3/xi4], n.

2

.

xijE ~ 13,

4

i = i, 2, j = |, 2, 3, 4, ~j=l nij = hi' i = i, 2,

n| + n 2 = n, Nil is nilpotent, i = l, 2, 3, 4. Suitably rewrite

the coordinate vector

~i = [Xli/X2i]'

i = I, 2, 3, 4.

This means

tile combination

of corresponding s t a t e variables

substates.

Direct computation will verify that,

it* the slow

aml

fast

under such a coordinate transforma-

tion, system (2-5.5) is r.s.e, to

]E21E22 0

~23:24:2 E43

=

EaaJt x4

Xll °

X13 ° ][~1

X21

A22

~23

0

0

A43 A44JL ~'4

x2411~2 (2-5.6)

y

=

[El, O, C3' 0][~1/ff2/~3/~4]

where Eii = diag(l, Nii);

Aii = diag(Aii' I),

i = I, 2, 3, 4

Eij = diag(O, Ni.j);

~ij -- diag(Atj, 0),

i ~ j, i, j = I, 2, 3, 4

Bi = [Bli/D2i]'

i = I, 2;

Cj = [CIj, C2j],

-

i ~ I, 3, 4

are constant matrices, ~iEH~ st, ~i = nli + n2i' i = I, 2, 3, 4, ~-~'~i = n. i=l

From tile c o n t r o l l a b i l i t y

and o b s e r v a b i l i t y

and the decomposition of (2-5.3)-(2-5.4),

criteria

(Theorems 2-2.1 and

it is easy to know that the system (2-5.6)

has the following properties. a. The subsystem (EII,AII,BI,CI)

is both controllable and observable.

b. The subsystem

LE21 is c o n t r o l l a b l e ,

g22

'

A21 ~22

but unobservable.

c. The subsystem

'

2-3.1),

B2

' [C 1 0 ]

)

54

11 El3

(

All

13]

[B1], ) 0 [~1 ~2]

I

E33

, 0

A33 j ,

is observable, but uncontrollable. d. The subsystem (E44, ~44' O, O) is neither controllable nor observable. Both

(2-5.2) and (2-5.6) are canonical forms for system (2-5.1) under

restricted

system equivalence. The

significance

o[ canonical form (2-5.6) lies in its abilily

controllability (observability)

substates from uncontrollable

to separale

(unobservable)

the ones.

Thus it characterizes the inner structure of a system. Decomposition (2-5.6) not only provides us with a canonical form, but its constructive proof also gives us a computation al)proach to obLain it: We first take staudard decomposition on the system to obtain the slow-fast subsystem decomposition. will

take controllability and observability decomposition on each subsystem,

ting in r.s.e,

system (2-5.5).

Then we resul-

Finally, (2-5.6) is ready by rearranging the coordi-

nate vector in (2-5.5).

Particularly, let Xl = [xl/~2 ] and x2 = [~3/~41" Then (2-5.6) becomes

0

A22JL72 J

(2-5.7)

Y = [~i' C2][~1/~2] where ~ij' Aij' Bi' Ci' i, j = I, 2, are certain constant matrices, and (EI[,A|],BI) is controllable. (2-5.7) is called the controllability canonical form for system (2-5.]). If we set ~i = [~1/~3 ] and ~2 = [F2/F4 ]' system (2-5.6) may

o ^

E21

^

A

=

E22JLx21

^

A21

be

rewritten us

I ~

A22J[x2

U

+

B2

(2-5.8)

y = [C I, O][xl/x2] where Eij' Aij' Bi' el' i, j = i, 2, are certain constant matrices, and (Ell,All,e I ) is observable. This form is called the observabi]ity canonical form of system (2-5.1). If we set ~I = [×i/x21/x22 ] and ~2 = [x23/x24]' noticing (2-5.4), system (2-5.2) (thus system (2-5.1)) is r.s.e, to

55 A12 ][~1

0

E22J ~2 .l

~1

0 (2-5.9)

y = [C I, c 2 | [ x l l x 2] where rank[Ell, BI] = ~I' [Eli' B1 ] i s of f u l l composition is called normalizability

row rank and ~22 is nllpotent. This de-

(the definition is given in the next chapter)

decomposition. It will be needed later.

Example 2 - 5 . 1 .

Consider system (I-3.19), vhose standard decomposition is given by

(1-3.20): = [-I o

=

1

×2

-1 1 0 ]Xl + [ 0 ]Vs ( t ) (2-5.I0)

+ [-t/°lVs(t)

y = lO

llx I

Since

o)[o'] i s both c o n t r o l l a b l e

and o b s e r v a b l e ,

s.bsyslom.

i f we d e . a t e

Note tt.lt

not o b s e r v a b l e ,

we need to make d e c o m p o s i t i o n on only t h e

x 2 = ]x21/x22 ] in ( 2 - 5 . 1 0 ) ,

and x22 i s n e i t h e r .

Hence, system ( 2 - 5 . 1 0 )

fast

x2[ is c o n t r o l l a b l e

(thus (1-3.19))has

hut

tile f o -

llowing canonical form:

0!

01 ,, O 0 !,_ 0 Xl . TM ~ , 0f]x21

o- ~ 0

-'

Example 2 - 5 . 2 .

t

l°

1

0 ' 0 ' O|[Xl

- - -~--'-

"x

o.,}.,0[121

[ 0

0 ' 0 ' lJLX22

(2-5.tl)

1 ', 0 ~ O ] [ X l / X 2 1 / x 2 2 ] .

As g i v e n in Examples

is c o n t r o l l a b l e ,

To o b t a i n

[ 1

=

0 , Ot-oJlX22 y = [0

(2-2.6)

1 -i00

2 - 2 . 2 and 2 - 3 . 2 ,

tile c i r c u i t

network ( 2 - 2 . 5 ) -

but ,lot o b s e r v a b l e .

its c a n o n i c a l

we begin by determining its s t a n d a r d

decomposition,

decom-

position,

via the method given by Lemma ]-2.2. In this system, matrix A is nonsingu-

far. Thus

a = 0 certainly satisfies laE+AI ~ O. Let QI = A-I' P1 = 14 . Then

El = QIEPI = A-IE =

-RC I

0

0

0

-RC l u

0 -L C2 0

0 0

0

0

I

[ C1 A

similarity

0

LransformaLlon of E l and s e p a r a t i o n

Q1API = 14"

of i t s

(2-5.12)

.Jordan b l o c k s b e l o n g i n g

to zero eigenvalue from those belonging to nonzero eigenvalues, we know that

56

{i0oo]

T=

1

1 0

o o 1 0

0

0

R

will transfer E l into

-RC

0

0

0

-RC 1

0

-L

0

TE1T-1 =

Denoting

C2

0

0

0

0

(2-5.73)

0

Q = TQ 1 and P = Pl T-I , from (2-5.12) and (2-5.13), it is easy to obtain

QEP = TEIT-1,

QAP = 14.

Thus, the coordinate

(2-5.14)

transformation

[Xl/X2 ] = p - l × = Tx = [ u c l / U c 2 / I 2 / ( R I I + U c l ) ] will

transfer

(2-2.5)-(2-2.6) _-RC1

0

into its

Decomposition

o,

l

0 ~ OJ

y = Uc2 = [0 (2-5.15)

1

o:o/rx,1

+

u

0-0

( 2 - 5 . t 5)

0 [ O][Xl/X2].

~s not only t h e c a n o n i c a l

form as well. In this decomposition, vable; but substate x 2

form

,oool

0 t 0

o

l~---O-

canonical

the

form but t h e o b s e r v a b l e c a n o n i c a l

substate x I is both controllable and obser-

is only controllable.

Corresponding

this fact refers to that x 2 = RI 1 + Uc] is not observable, 2-2.1,

in which RI l + ucl = u e could not be

By using EF3, the coordinate

which is obvious in Figure

from measure Uc2.

we may obtain the canonical decomposition

transformation

lity decomposition

determined

to its original system,

more directly. First take

to obtain EF3; then take controllability

on (E,B,C),

and finally

detailed process is omitted here. Interested

the canonical

and observabJ-

form is obtainable.

readers may easily do this.

The

57 Transfer Matrix anti Minimal Realization

2-6.

Based on the state variable concept,

the state space description method for system

ana|ysis and synthesis obtains its state space

model (or description

the physical sense of variables and their relationships. to understand the inner structure of systems. reflects from

the outer structure:

input to

output.

input-output

To compare,the

or transfer

relationship

state space method may characterize

properties but is sometimes too complicated

from

On the other hand, the transfer matrix relationship,

for practical interests;

sfer matrix is u s u a l l y unable to characterize

c i n c t dependence r e l a t i o n s h i p

equation)

Thus, this method allows us

inner p r o p e r t i e s ,

between i n p u t and o u t p u t .

the inner

while the tran-

it provides a

suc-

We begi.n our d i s c u s s i o n from

the Laplace t r a n s f o r m a t i o n . The I,aplace t r a n s f o r m a t i o n L [ f ] of a f u n c t i o n f ( t ) L[f] =

i s d e f i n e d as

foT-Stf (t)dt

which is assumed to exist. It has the basic property:

L[ nf I +l~f2]

=

aL[fl]

+I~LIf 21

LI~I = s U I f l - f ( O ) .

Taking Laplace t r a n s f o r m a t i o n on both s i d e s of t h e system e q u a t i o n

E~ = Ax + Bu

(2-6. I )

y = Cx we have sEx(s) - Ex(O) = Ax(s) + Buts) y(s) = CX(S). Under the assumption of regularity

(sE-A) -I exists. Thus

y ( s ) = C(sE-A)-I(Ex(O) + S u ( s ) ) . When x(O) = O, t h e p r e c e d i n g e q u a t i o n y i e l d s the i n p u t - o u t p u t r e l a t i o n s h i p : y ( s ) = G(s)u(s)

(2-6.2)

G(s) = C(sE-A)-IB

(2-6.3)

in which

is called tile transfer matrix f o r system (2-6.1). Example 2-6.1. G(s) =

Consider system (i-3.19).

C(sE-A)-IB

Its transfer function is

58

= [o

o

[ o el[el

I]

-1

o

s

l

O

O

0

-1

-1

o

-1

O

-1

1

=

-1

s2+s+l

which is a strictly proper function as in the normal case.

Example 2 - 6 . 2 .

singular

The

s y s t e m w i t h only a f a s t s y s t e m

N~ = x + Bu

(2-6.4)

y = Cx where N is nitpotent, has a transfer maLrix G(s)

= C(sN-I)-IB

where h i s t h e n i l p o t e n t

=

-CB

-

sCNB

-

...

sh-IcNh-IB

-

index o f N. T h e r e f o r e ,

the transfer

matrix of system (2-6.4)

is a polynomial matrix.

Let Q and P be n o n s i n g u l a r

and E = QEP, A = QAP, B = QB, C = PC, t h e n tile t r a n s f e r

m a t r i x G(s) h a s t h e form G ( s ) = C ( s E - A ) - I B = CP(sQEP-QAp)-IQB = C ( s E - A ) - I B which

states that r.s.e, systems

(2-6.5)

have the same transfer matrix.

great advantage in system theory. For systems with structures analyze

This property is a

that are too complex to

and~or design, we ,lay need to study an r.s.e, form convenient to our discus-

sion and find its properties in which we are interested. As previously

proven,

for a given

system (2-6.1),

matrices Q and P such that (2-6.1) is r.s.e,

there exist two

to tile canonical

form

nonsingular

(2-5.6).

Direct

but tedious computation shows that

(;(s) = C(sE-A)-]B = C1(sgll-Xll)-igl. Therefore, and

the transfer matrix for a system is determined by the controllable

observable

subsystem,

tttlobservable [)arts. important control

part

(2-6.61

and

has no relationship to other

Accounting for this,

of a system:

the transfer

the controllable

and o b s e r v a b l e

effect of input on the measure outputs,

uncontrollable

matrix reflects part,

or

o n l y l h e most which

is

the

without any knowledge of control on

the state. Further calculation

tells us

G(s) = ~ I ( S E I I - ~ I I ) - I B I where Gl(S) = Cll(SI-All)-IBll Another distiuguishing o f t h e normal s y s t e m ,

= Gl(S ) + G2(s )

(2-6.7)

, G2(s ) = C 2 1 ( S N l l - I ) - I B 2 1 .

feature

may be s e e n from ( 2 - 6 . 7 ) :

whose t r a n s f e r

matrix is strictly

Different

proper,

from t h e c a s e

the transfer

matrix

59 of singular systems generally have two parts. the slow subsystem and is strictly proper; determined by the fast subsystem.

One is Gl(S), which is determined

by

the other is a polynomial G2(s), which is

In the general case, G2(s ) ~ O. Different from the

strict properness in the normal case, G(s) for a singular system is rational,

but not

necessarilly proper. This is a special feature of s~ngular systems. Example 2-6.3.

Consider again the system (I-3.19) with standard decomposition (I-

3.20). Its transfer matrix is G(s) = CI(SI-AI)-IB 1 + C2(sN-I)-IB 2 = Cl(sI-Al)-IBl

(C 2 = O)

= 1/(s2+s+l) which c o i ~ l c i d e s w i t h t h e r e s u l t s r.~.e,

in Example 2 - 6 . 1 ,

wit.I) much l e s s c o m p u t a t i o n s i n c e

form i s u s e d .

For a single input single o u t p u t system, its transfer matrix

c(s)

hn_isn-l+ ... + b l S + bo =

snl+ anl_lSnl-l+

...

+ als + a o

is a rational transfer function. In general, n I < n. G(s) is not proper if bn_ 1 ~ O. Thus, viewing from the point of trans[er matrix, the ol)ly difference between the normal and singular systems is the strict properness of transfer function.

If it ks strictly proper, tile input-

output relationship may be described by a normal system, otherwise, the singular system is used. As for multi-input multi-outpuL systems,

the difference lie8 in the pro-

perness of the transfer matrix. It is worth pointing out that a singuJar system may have a strictly proper funct. inu, a s t h e c a s e of s y s t e m ( l - 3 . t9)

in Example 2-6.1,

i f tile c o n t r o l l a b l e

and o b s e r -

vable s u b s y s t e m i s normal.. As p o i n t e d out e a r l i e r ,

two r . s . e ,

we can prove tile f o l l o w i n g

s y s t e m s have t h e same t r a n s f e r

matrix.

Further,

Lheorem.

Theorem 2-6.1. Two controllabie and observable systems(E,A,B,C) and (E,A,B,~) have the same t r a n s f e r

matrix

if

and o n l y

if

they a r e

r.s.e.

Proof.. By noticing (2-6.7) the proof is easy. Q.E.D. This theorem assures that the transfer matrix is not only determined by controllable and observable subsystem but is determined uniquely. If controllability and oilservability assumptions are not guaranteed, this theorem may be not true. For example, as shown in ExampJes 2-6.1 and 2-6.3, system (i-3.19) and the system

6O

= [ - 1u - 10 Ilx y = [o

+ [ 10 ] (2-6.8)

llx

have the same transfer function I/(s2+s+l).

to each o t h e r

But they are not r.s.e.

since they are of different orders.

As mentioned earlier, for a given singular system, we may obtain its transfer matrix. C o n v e r s e l y , we w i l l certain

be i n L e r e s t e d

in t h a t

for a given transfer matrix G(s),

s i n g u ] a r system may be found whose t r a a s f e r

called realization Definition

maLrix i s C ( s ) . T h i s i s the

a so-

t h e o r y . To be p r e c i s e , we d e f i n e the f o l l o w i n g .

2-6.|.

Assume t h a t G(s) E~rxm is a r a L i o n a l m a t r i c e s .

If

there exist

matrices E, A, B, C such that G(s) = C(sE-A)-IB where E, A E ~ nxn,

B E ~ nxm,

CE~

xn

are constant matrices,

the system

Ex = hx + Bu

y = Cx

(2-6.9)

will be called a singular system realization of G(s), simply, a realization of C(s). Furthermore,

~f any other realization has an order greater than n, system (2-6.9)

will be called the minimal order realization or, simp]y, minimal realization. Example 2-6.4.

Both systems (I-3.19) and (2-6.8)

are

the realization of transfer

matrix G(s) = l/(s2+s+l). However, system (2-6.8) is of lower order than system

(1-3.19). Lemma 2 - 6 . 1 .

Let g(s) be a r a t i o n a l function g ( s ) = b ( s ) / a ( s ) .

e x i s t s a s t r i c t ly proper raL iona]

function gl(s)

Then t h e r e always

and a polynomial g 2 ( s ) such that

g(s) = gl(S) + g2(s). Proof.

Let d = deg(b(s)),

polynomial.

~= deg(a(s)), llere deg(.) represenLs the degree of

a

If d < ~, g(s) is sLrictly proper. In this case, the lemma is Lrue for

gl(s) = g(s), g2(s) = O; otherwise, d >/~, from the polynomial division theorem,

we

know that there exisL polynomials q(s) and r(s) such that b(s) = q(s)a(s) + r(s)

and

deg(r(s)) < deg(a(s)). Therefore,

g(s) =

b(s) a(s)

Let g l ( s ) = r ( s ) / a ( s )

= r(s) + q(s). a(s) and g2(s) = q ( s ) , then g l ( s ) i s s t r i c t l y

proper, g2(s) i s a pol-

ynomial, and g ( s ) = g l ( s ) + g 2 ( s ) . Q.E.D. Theorem 2 - 6 . 2 . Any rxm r a t i o n a l matrix C(s) may be r e p r e s e n t e d as

61 G(s) = Gl(S) + G2(s) where Gl(S) is a s t r i c t l y

(2-6.10)

proper r a t i o n a l m a t r i x and C2(s) i s a polynomial m a t r i x .

Proof. Let G(s) = (g'lj(S))rxm' where gij(s) is the element at the ith row and jth

column, which i s r a t i o n a l . From Lemma 2-6.1 t h e r e e x i s t s t r i c t l y 2 g l j ( s ) and polynomial g i j ( s ) such t h a t 1

2

gij(s) = gij(s) + gij(S),

i = l, 2 . . . . .

proper f u n c t i o n

r, j = 1, 2 . . . . .

m.

Thus, l h e theorem is t r u e by setting

Gl(s) = (g]j(s)),

Q.E.D.

G2(s ) = (g~j(s)).

Consider t h e t r a n s f e r m a t r i ×

Example 2 - 6 . 5 .

(;(s) ~ [

1 -=---•

s2+s+l

/

4 3 s +s -s s2+s+l

~--

(2-6.1 I)

l-

Since s 4 +s 3 -s

_ s2 . -I

l

+

s2+s+l

s2+s+l

G(s) may be rewriLten in the form of C(s) = Gl(s ) + G2(s), with

Gl(S) = s2+s+--'~---

,

.

Lemma 2 - 6 . 2 . For any polynomial matrix P ( s ) ,

t h e r e always e x i s t m a t r i c e s N, B2, C2,

N is nilpotent, such that P(s) = C2(sN-[)-IB 2. Proof. Consider the strictly proper matrix G(s) = - sP(~), 1 I which may be viewed as

the t r a n s f e r matrix of a normal system. Thus, a c c o r d i n g to l i n e a r system t h e o r y , there e x i s t m a t r i c e s N, B2, C2, such t h a t ~(s)

=

-

1 p ( ! ) = C2(sI_N)-IB2"

s

Noticing that G(s) has only zero poles, N has only zero elgenvalues. Thus, N ~s nilpotent. Direct computation will results in P(s) = C2(sN-I)-IB2 . Q.E.D. llence, systenl

N~ = x + B2u ,

y = C2x

may be viewed as a singular system realization for P(s). Speaking from this point,

62

Lemma 2-6.2 shows that any polynomial matrix P(s)

has a singular system

realization

with only a fast subsystem. Theorem 2-6.3.

Any rational matrix may have a realization of (2-6.9), which sati-

sfies G(s) = C(sE-A)-]B; furthermore, the realization is minimal if and only if the system is both controllable and observable. Proof. According to Lemma 2-6.2, any rational matrix G(s) may have a decomposition C(s) = Gl(s) + G2(s), in the strict

which Gl(S) is strictly proper and G2(s) is polynomial. For

properness of Gl(S),

t i o n AI, B i , C1, s a t i s f y i n g

from J i n e a r

system theory,

it always has a realiza-

Gl(S) = CI(sI-A1)-IB1 .

Also, Lemma 2-6.2 shows the existence of ni]potent N, B2, C2, such that

G2(s) =

C 2 ( s N - [ ) - I B 2. Let E = diag(I,N), It is easy to verify

A = diag(Al,

I),

B = [B1/B21,

C = [C 1, C2].

(2-6.12)

that

G(s) = Gl(S) + G2(s) = C(sE-A)-IB. The the system determined by (2-6.12) is a realization of C(s). As f o r t h e s e c o n d c o n c l u s i o n , posed

i n t o the sum o f a s t r i c t l y

of its

realization

we n o t e thaL any t r a n s f e r

m a t r i x G ( s ) may be decom-

p r o p e r G l ( S ) and a p o l y n o m i a l G 2 ( s ) ,

i s the sum o f t h o s e o f G i l a )

anti G 2 ( s ) .

Thus, system ( 2 - 6 . 9 )

minimal realization iff (AI,BI,C I) and (N,B2,C2) are minimal, (At,BI,CI) ans

(N,B2,C2) are minimal realizations

and tire o r d e r

or

is a

equivalently,

iff

of strictly proper matrices Cl(S )

and C2(s), respectively, indicating both (AI,BI,C l) and (N,B2,C2) are controllable and observable. This in turn shows the minimal realization (2-6.121 is controllable and observable. Q.E.D. The f i r s t : and s e c o n d p o r t i o n s of realizations

o f Theorem 2 - 6 . 3 ,

and minimal r e a l i z a t i o n s .

a metltt+tl t o f i n d

realizatitms.

in which G[ i s s L r i c L l y

First

decompose m a t r i x C(s)

p r o p e r and G2 i s p o l y n o m i a l ;

t i o n s o f G l ( S ) and C 2 ( s ) .

Combining t h e m a t r i c e s

be really t o g e t t h e minimal r e a l i z a t i o n A direct

result

respectively,

show t h e e x i s t e n c e

M e a n w h i l e , t h e p r o o f a l s o p r o v i d e s us w i t h illtt) two p+lr+ts G = {;1+(;2,

t h e n f i n d t h e minimal

i n t h e way o f ( 2 - 6 . 1 ) ,

realiza-

and we w i l l

(2-6.9).

o f t h e c o m b i n a t i o n o f Theorems 2 - 6 . 2 and 2 - 6 ° 3 i s t h e f o l l o w i n g

corollary. Corollary

2-6.1.

Ex~,mple 2 - 5 . 5 . l/(s2+s+l).

Any two minimal r e a l i z a t i o n s

Both systems

Since system ( 2 - 6 . 8 )

(l-3.19)

anti

is controllable

for a rational

(2-5.8)

are

matrix are r.s.e.

realizations

of

G(s)

and o b s e r v a b l e ~ i L i s m i n i m a l . But

=

63 ( I - 3 . 1 9 ) is not. For a g i v e n p o l y n o m i a l m a t r i x , P ( s ) = Po + P1 s + " ' " + Pk-1 sk-1 E I~ rxm Lemma 2-6.2 a s s u r e s

(2-6.13)

Lhe e x i s t . e n c e of B2 E IRa2 xm, ('2 E i~ rxn2 and tile ni l p o t e n t

mat r i x

NER n2xn2such t h a t P(s) = C2(sN-I)-IB2 .

(2-6.14)

We now introduce the Silverman-Ho algorithm to search the mintmal realization. Define

M O

-Po

-P1

"'"

-Pk-2

-PI

-P2

"'"

-Pk-I

-Pk-1 0

E R kr x km

=

-Pk-2 -Pk-I . . . . . .

0

-Pk-I

......

0

"'" ,.,

0 0

-P1 -P2

MI =

-P2 -P3

I

, . .

-Pk-1

L

0

0

-Pk-I 0

. o .

E ~kr x km

. . ,

0

. . . . . .

0

0

......

0

and s e t ~2 = raakMo" Then t a k e f u l l

rank decompo~Ltion

M o = LI.L 2

where L1

(2-6.15)

IRkrxi~2 , L 2 E l~~2xkm are of full column and row rank, respectively.

Let B2 and C2' respectively, be the first m columns of L 2 and the first r rows

of

L 1 , and

= (LIL I)

-1

I:

~

~

EIMIL2(L2L 2)

-I

Then m a t r i c e s N, B2 and ~2 form a minimal r e a l i z a t i o n Proof. 6.3.

(2-6.16)

• for P(s).

Existence of minimal realization is proven in Lemma 2-6.2 and Theorem

2-

There exists controllable and observable triple (N, B2, C2), N is nilpotent,

such that (2-6.14) holds, i.e., Po + Pi s + ... + Pk_l sk-I = - C2B 2 - C2NB2s - ... - c2Nk-IB2sk-I Equalling the coefficients of the same order on both sides of this equation, we have

64 i = O, I, 2 . . . .

- Pi = c2NiB2' (If k is less than

, k-l.

the nilpotent index h of N,

the coefficients of the higher order

terms will be considered zero). Thus M ° = HI.H2,

(2-6.17)

M 1 = HINH 2

where

1[ 1 : [ C 2 / C 2 N / . . , / c 2 N k - 1 ] ,

H 2 = [B 2, NB 2 . . . . .

our assumption (N,B2,C2) is minimal.

Recalling

Therefore,

Nk-IB2|. H I and II2 are of full co-

lumn and row rank, respectively. Noticing n 2 : r a n k l l l

= rankll2, we have ~2 = rankM ° = rankll I = n 2.

Furthermore,

from (2-6.15) and (2-6.17) it is easy to see LIL 2

(2-6.18)

= HIH 2 •

Left multiptying

L

both s i d e s of ( 2 - 6 . 1 8 )

•

I

by and paying attention to the inverLibi,= -I lity of I/iLl, we obtain L 2 = ( L I L I) LIHIH 2, By selection, n 2 = rankL 2 = rankll2, and if T-~- (L1LI)

Lltt 1 (~ lln2xn2

we know thaL T i.s n o n s i n g u l a r

and

L2 = TH2 = [TB2, TNB2, . . . A c c o r d i n g Lo t h e s e l e c t i o n B2

(2-6.19)

, TNk-JB2].

of ~ 2 ' from ( 2 - 6 . 1 9 )

we have

= TB2"

]'he s u b s t i t u t i o n ing t h £ s e q u a t i o n

(2-6.20)

of (2-6.20) +-c

into

by II 2 and n o t i c i n g

(2-6.18) that

results

I[2H 2

in I,ITII 2 = IIlH 2. R i g h t m u l t i p l y -

is nonsingular,

we know t h a t

[C2T-]/C2NT-I/.../c2Nk-IT-I ].

L I = IIIT-I Thus

~'2 = C2T - l " Furthermore, and ( 2 - 6 . 1 7 ) ,

(2-6.2])

Lhe s e l e c t i o n

method of N in ( 2 - 6 . 1 6 ) ,

together

with L 2 = TIt2, L 1 =lll T-l,

yields -c

= (LiLl)-

1'¢

*:

•

LIIIINH2L2(L2L2 )

-I

= TNT-1

.

(2-6.22)

Equations (2-6.20) - (2-6.22) show that the real izatio~ has the same order with the

minilm,l

oue ( N , B 2 , C 2 ) ,

mal reatization

aln] t h e two r e a l i z a t i o n s

for P(s). Q.E.D.

~ire s i m i l a r .

Thus ( N , B 2 , C 2 )

is a mini-

65 The method for finding the minimal realization given here is relatively simple.

Example 2 - 6 . 7 . C o n s h t e r t h e polynomial m a t r i x G2(s) in Example 2 - 6 . 5 : G2(s) = [0 / s2-1l.

Clearly,

iLs minimal r e a l i z a t i o n

P(s) = s2-1 is obtained.

i s ready when t h e minimal r e a l i z a t i o n

of polynomial

Next, we find its minimal realization using the method men-

tioned e a r l i e r . For P(s), we have

Mo =

[10 ] 0 -I

-I 0

MI = ,

[o 0] -i 0

Note that M o is nons]ngular. Thus we may choose [| 0

0],

~2 = [1

0

= (LILI)

0 0

0 0

.

L I = M ° and L2 = 13. T h e r e f o r e , "B2 =

-I], and according to (2-6.16),

LIMIL2(L2L2)

[ cil

=

0

1

.

And t h e s i n g u l a r

}|ence, (N, B2' 52) is a minima] realization oF P(s) = s 2 - I.

sys-

tem

o o]

0

0

I

0

yo[o

ill

~=x+

0

u

0

o_1 °°Ix

is a minimal r e a l i z a t i o n

of G2(s ) .

2-7.


Section 2-I is due to Yip and Sincovec (1981) and the theory of various coatrollabilities, observabilities and duality is mainly from Cobb (1984). The matrix criteria under

three

(1987).

There

equivalent

for singular systems, 1983;

definitions

and Zhou e t a ] .

or terms of controllability and observability

such as structural controllability (Aoki et al,

1983; Murota,

Yamada and Luenberger, 1985a, b), Di-controliability (Pandolfi, 1980); strong

controllability (Yip

forms are mainly from Dai (1987b, 1988d)

are other

and

( C o u t i a n and T a r n ,

S[ncovec, 1981).

1982;

Verghese eL a l ,

h partial list of other papers

observability includes Bender and Lauh (1985),

1981), on

Campbell (1982b),

C-controllability controllability

and

Christodoulou

and

Paraskevopoulos (1985), Cu|len (1986a), Kalyuthnaya (1978), Lewis and Ozcaldiran (1984), Lewis (1985a), Zhou et al. (1987).

66 It needs (Verghese

to be pointed out et

al.

(Verghese et 51.

1981),

that under the strong restricted

the impulse controllabtlity,

system

controllability

equivalence at

1981) (which is the same as impulse controllability here),

infinity antl the

controllability of the fast subsystem become the same concept. Sections 2-5 and 2-6

(1986b).

are from Verghese et al. (1981),

Dai (1988c)

and

Cullen

CHAPTER 3 FEEDBACK CONTROL

We u s e t h e t e r m f e e d b a c k c o n t r o l feedback

control,

dynamic o r s t a t i c of a s y s t e m ' s

static

In practical

behavior,

and r e s p o n s e

engineering

3-1.

Consider the following

singular

that

the

and d e s i g n ,

characterize

s p e e d . But g e n e r a l l y , This inspires

is the content

and o f c o n L r o l

static

used methods to change

a t t h e same t i m e .

The i m p r o v e m e n t o f t h e p r o p e r t i e s

f e e d b a c k and

system analysis

i s b a s e d on some f e a t u r e s

are not fulfilled

t o p i c s of c o n t r o l

to state

t h e m o s t conmlonly

properties.

properties

as s t a b i l i t y , teristics

one of

to refer

output sysLems's

the evaluation the system,

ul[

such

of the charac-

the study of adjustment.

o f s y s t e m d e s i g n and f o r m s many

theory.

StaLe Feedback

system:

E~ = Ax + Bu

(3-1.1)

y=Cx

where xE ~ n , u 6 ~m, and y E ~r a r e i t s pectively;

E, A ( ~ n x n ,

B ( ~ nxm,

staLe,

and C ( ~ r x n

used t h a t q ~= r a n k E < n and s y s t e m ( 3 - 1 . 1 ) l,et the s t a t e

feedback c o n L r o l

control

input,

are constant

and m e a s u r e o u t p u t , matrices.

It

is regular.

have Lhe form o f

u = KIX - K2x + v

wbere the fir',~t

term is praport ional

(3-1.2)

feedbuck and t h e st, t'olld i s d e r i v n t i v e

which may be viewed as the "speed" feedback,

K2E~mxn

res-

is also ass-

v(t)

i s the new i n p u t c o n t r o l ;

foetlback, and K 1,

are constant matrices. Control (3-1.2) is the proportional and derivative

(P-D) s t a t e

feedback.

[n singular systems, control input in the form of (3-1.2) is often used. The closed-loop system formed by (3-I.I) and (3-1.2) is (E + BK2)x = (A + BK1)x + By

(3-] .3) y = Cx Particularly,

when K2 : O, the feedback c o n t r o l

(3-1.2)

becomes

68 = KlX + v

u

(3-1.4)

which is a pure proportional corresponding

closed-loop

(P-) state feedback,

as in the normal system case.

system is

Ex = (A + BK l)x + Bu y = Cx.

To

hereafter

assume t h a t

(3-1.3)

and ( 3 - 1 . 5 )

3-1.1.

Stability

It

( 3 - I .5)

guarantee the existence and uniqueness of

will

are

feedback control

solution for any

Thus,

Definition 3-1.1.

Singular system (3-l.l) ~ 0 for

ll 2 x< a e - ~ t l l x ( O )

By definition,

slabilily

i s l h e most

otherwise, importaut

we s t r e s s

ll2 ,

x(t)

satisfies

t > O.

(3-1.6)

stability under the Lyapunov sense, or asymptotic

I sE-A [ ~ a n l s n i + . . .

characteristic

polynomial

+ als

singular,

tents.

We

stabi-

= O.

+ ao

for sysLem (3-I.1).

IsE-AI = 0 are called finite poles for the system, is

a

the polynomial

f(s) the

iu

is culled stable Lf there exist scalars

L > 0 iLs s t a t e

lity. When system (3-1.1) is stable and u(t) m O, t > O, lJmt~ooX(t) We c a l l

it could not

properly

we are most concerned with Lhe stability of closed-loop systems.

a , ~ > 0 such t h a t when u ( t ) IIx(t)

both

regular.

o r be d e s L r o y e d .

system. Therefore,

control input, we

i s c o n f i n e d t o t h e s e t such t h a t

is wellknown that a practical system should be stable,

work p r o p e r l y

The

t h e number o f f i n i t e

will

pole set for

use

o(E,A) ={s

the system, and

will

s finite,

E =

In c i . r c u i t

0

0

1

0

0

0

0

0

0

0

0

0

IsE-AI = O} as

o(A).

[i00]

network (1-3.19),

A =

_

,

The characteristic polynomial of this system is f(s) = I sE-AI

or poles for simplicity.

be s p e c i f i e d

{i000]

Example 3 - 1 . 1 .

sWs satisfying

p o l e s i s a l w a y s s m a l l e r t.han n f o r

I sEC,

o(I,A)

The f i M t e

0

0

0

0

0

1

1

1

!

.

to

f(s)

=

Since E

singular

sys-

d e n o t e Lhe [ i n i L e

69

-

i

-10 0

0s 0

-I

-1

=

00 -1

-I

Thus the finite pole set is

Theorem 3-1.1.

= s 2 + s + 1 = (s+½+ ~ . ,i s t s + ~1- ~ i ) .

a(m,A)

~1

= {-

-d~i,

- ~1

+~i}

.

System (3-1.1) is stable ~f and only if

o(m,A) C £-, where C- =

( s I s6C,

Re(s) < O}

represents the open left half complex plane.

Proof. When u(t) ~ O, t > O, the state response x(t) of system (3-1.1) satisfies E~ = Ax,

x(O) = xo.

(3-1.7)

Note that system (3-1.1) (thus (3-1.7)) is regular. There exist two nonsingular matrices Q and P such that (3-1.7) is r.s.e, to EF]: ~l(t) = AlXl(t),

Xl(O) = Xlo

(3-1.8a)

Nx2(t ) = x2(t),

x2(O) = X2o

(3-1.8b)

where QEP = diag(l, N), QAP = diag(Al, I), x(t) = P[Xl(t)/x2(t)] , and x ° = P[Xlo/X2o]. Thus, system (3-1.8) has the solution

XL(L)

=

eAltxlo t>O.

Xz(t) = 0

Since x2(t) = O, t > O, inequality (3-1.6) holds in and only if Xl(t) s a t i s f i e s

IlXl(t)ll 2 < II P 1121 me-I~tllx(O) tl2,

t > O,

which is equivalent to o(AI)CC-. On the other hand,

o(E,A) =

e(QEP,QAP) =

o(diag(I, N), diag(Al, I)) = o(AI) ,

which means that (3-1.6) is true ~f and only if

Example 3-1.2.

o(E,A) = a(Al)C'{:-. Q.E.D.

From Example 3-1.1, system (1-3.19) has the pole set 1

o(E,A) =

(- ~-

~3 ., , - ~1 + ~ ,3 .} .

Thus, it is stable by Theorem 3-I.i.

[,oo

Example 3-1.3.

E=

[ooo

Consider system (2-2.5). Ilere

0

C2

0 0

0 0

0 -L 0

A= ,

0

0

I

0

-I I

I 0

0 0

0 R

70

and direct computation shows its characteristic polynomial is IsE-AI = (RCIs+ I)(s2C2 L + I). Since R, C I, C2, L > 0 for a real system, we have

1

li,

-I i}.

-(E,A) = (- R-~I , C2L

C2L

From Theorem 3-1.1 it is unstable. In fact, it is an oscillation system whose output uc2 (obtained from direct computation from (2-2.5)-(2-2.6)) satisfies (3)r~ CIC2RLuc2 x-) + C2Lu~)(t ) + CIRUc2 (t) + Uc2(t) = Ue(t).

(3-1.9)

Assume that the system is s t a t i c at beghming, Uc2(O) = Uc2(O ) = u(2)ro c 2 x J~ = O,

aud

Ue(t) = ~(t) is the unit impulse term at the starting point. To solve Uc2(t), we take the Laplace transformation on both sides of (3-1.9). Then

Uc2(S) =

I (s2C2L+I)(CIRs+I)

(CIR)2

=

(CIR)2+C2L +_L(

1

C2L

sCIR+I

(CIR)2+C2 L

l

s +i

J

s -i

I

C R

( - -

1

2 CJ~2L s C~2L+i

+ - -

1

)

s C~2L-i

)1.

The inverse Laplace transformation yields the represention of Uc2(t): Uc2(t )

C1R (CIR)2+C2 L

t e- C-~ _

C2L

(

(C R)2+C L 7 2

uc2(t)

0

=- t

Figure 3-1.1.

C1R cos t 2C]~ drc,.2L 2

-

sin t

.

71 F i g u r e 3 - 1 . 1 shows t h e l o c u s o f t h i s

3-1.2.

oscillation

system.

Stabilizability and detectabil~ty

Definition

3-1.2.

feedback ( 3 - 1 . 4 )

System

(3-1.1)

is called

such t h a t t h e c l o s e d - l o o p

From D e f i u i t i o n

skabilizal)le

system (3-1.5)

3 - 1 . 2 and Theorem 3 - | . 1 ,

if there exists

a state

is stable.

we can p r o v e t h e f o l | o w i n ~

Ihoorom.

Theorem 3-1.2. System (3-1.1) is stabilizable if and only if r a n k [ s E - A , B] = n,

V s EC+,

s finite

(3-1.10)

where ~+ represents the closed right half complex plane;

~+ = ~ s

~ s~£,

Re(s) ~0},

which, in turn, is equivalent to the stabilizability of the slow subsystem. Proof. Necessity:

According to Definition 3-1.2 and Theorem 3-I.1, system (3-].1)

is stabilizable iff there exists a K IE ~mxn rank[sE-(A+BKl)]

= n, V s E C + ,

such that a(E,A+BKI)CC-,

i.e.,

s finite.

Noticing rank[sE-(A+BKl) ] = rank[sE-A, B][I/KI] , we have rank[sE-A, B] ) n. O[l the other hand [sE-A, B] is of n rows. Thus, equality holds. It is just (3-1.10). SuEficiency: Assume that (3-1.10) holds. From Lemma i-2.2 there must exist nonsingular matrices Q and P, such that system (3-1.1) is r.s.e, to

~1 = AlXl + Blu

(3-].lla)

Nx2 = x 2 + B2u

(3-1.lib)

y = ClX 1 + C2x 2 where QEP = d i a g ( I , x = P[xl/x2] ,

N);

QAP = dlag(Al, I);

XlE~nl,

x2ENn2;

QB = [BI/B2] ;

CP = [Cl, C2];

n I + ,i2 = n.

Paying attention to the nilpotent property of N, from (3-1.10) and (3-1.11) we have rank[sE-A, B] = rank[sQEP-QAP,

QB] = n 2 + rank[sI-Al, BI] = n,

V sE~+, Thus r u l J k [ s l - A p zability

131] = tl - ~l2 = h i , V s ( ~ +, und s i s f i . i t e ,

o f t h e slow s u b s y s t e m . T h e r e f o r e ,

a(AI÷BIKI)C~-.

s finite.

Let g I = [ K I ' O]P-I E ~mxn.

a(F,A÷BKI) =

indicating

tile s t a b i l i -

a m a t r i x K1E ~mXnlmay he c h o s e n such t h a t Direct

computation results

a(QEP,Q(A+BK1)P ) = a ( A I + B I K I ) C C - .

in

72 The system is stabilizable by definition and Theorem 3-1.1. Q.E.D. (3-1.10)

gives

the stabilization criterion,

which shows that the

property is determined uniquely by its slow subsystem.

stabilization

The system is stabilizable if

its slow subsystem is stabilizable, otherwise it is not. Combining it with Theorem 31.2, we have the following corollary. Corollary 3-1.1.

System (3-1.1) is stabilizable if it is R-controllable,

but the

inverse is not true.

Example 3-1.4.

System (1-3.19) is stable.

Thus it is stabilizable;

The unstable

system (2-2.5) is also stabilizable because it is R-controllable by Example 2-2.3. In fact, for this system we need only to set RC I K 1 = [I-3RC I, sRCI-LRCIC 2, ~----3LRCI, 0].

(3-1.12)

~2 The f e e d b a c k c o n t r o t

would be

U e ( t ) = K1x + v = (l-3RC1)Ucl where v ( t )

Thus, As

+ (3RC1-LRCIC2)uc2 +

t s the new voltage s o u r c e .

Direct c o m p u t a t i o n shows

~(E,A+BK 1) =

c

the closed-loop

{-1,

-1,

canonicol

(3-i .14)

C-.

system is stable.

h a s been p o i n t e d o u t i n

nonsingular

-1}

(3-1.13)

C2 -3LRCI)I 2 + v

matrices

Section

2-5,

Q[ and P1 s u c h t h a t

for regular

(3-].1)

system (3-1.1)

is r.s.e,

to

tile

there

exist

controllability

form ( 2 - 5 . 8 )

(3-].15) y = [C 1, C 2 ] [ X l / X 2] whe re

QIEP 1 &

,21

1 0

E22J ,

CI' 1 = [C 1, C2], and ( E l i , A l l , B 1 )

QIAPI =

All

AI2 ]

0

A22

,

1

x = PlJXl/X2l ,

is controllable.

It is easy to verify

from ( 3 - 1 . 1 0 )

that

the following

coroilary

holds.

73 Corollary 3-1.2.

System (3-1.1) is stabilizable if and only if in its

This corollary

canonical

a(A22 ) C C-.

form (3-1.15) A22 is a stable matrix,

shows that the stabilizable system has only stable

uncontrollable

poles. It also shows that there exists a feedback control u(t) = f(x(t), t)

(3-1.16)

where f(x(t), t) is a vector function of x(t) and t, such that the closed-loop system formed by (3-|.|) and (3-].]6) is stable Jf and only if it is stabLlizable. Thus

any

feedback in the form of (3-1.16) cannot stabilize system (3-[.}) if it is not stabilizable. This indicates the equivalence of the existence of (3-1.2) stabilizing (31.3) and that of (3-1.4) stabilizing (3-1.5). The stabi]izability characterizes

the controllability of the system's

stability.

The dual concept of stabilizability is detectability which is defined as follows.

Definition

3-1.3.

System ( 3 - 1 . 1 )

is

detectable

il

i t s dual system (E~,A~,I~,

C ~) is stabilizahle, This is a mathematical definition. In the next chapter, we wlll give another defLnition.

A direct result of Definition 3-1.3 and Theorem 3-1.2 is the following theo-

rem.

Theorem 3 - l . 3 .

System ( 3 - 1 . 1 )

rank[sE-A/C] = n,

i s d e t e c t a b t e Lf and only ][

V s6~+,

s finite,

which i s e q u i v a l e n t to t h e d e t e c t a b i l i t y Thusstabilizabilityand alone.

of i t s slow s u b s y s t e m .

detectability

a r e d e t e r m i n e d by t h e slow subsystem and

by

This indicates the fact that they really only characterize properties of slow

subsysLem although Lhey are defined for the whole system. The

R-observabiliLy,

along with Theorem 3-1.3,

deduced from R-observability,

Example 3 - 1 . 5 .

shows that detectability may be

but its inverse is not true.

Consider the system

[,oo] I

-I

0

0

0

0

[i -2°] Ii]

~ =

y = [1

-

0

1

0

2

1

x

+.

~]x.

We have

rank

= rank c

s+l

0

1

-s-I

0

-2

-1

o

•

= 3=11,

P

sE~ +,

s fLnite

74 It is detectable by Theorem 3-1.3, but when s = -l,

rank _[ t Thus i t

i

= rank C

0 0 0 -2 1 o

s=-I

is not R-observable.

0 -1 ¼

=

2 < n.

From Corollary 3-1.2 we could also obtain a further corollary. C o r o l l a r y 3-I .3.

E21 be

an

E22

l,eL

,

A21

A22

,

[C 1, Ol )

observability canonical form for system (3-1.7)

where (EII,A|I,BI,CI) is observable. detectability

0

r.s.e,

equivalence,

is that the system has only stab]e unobservable poles, i.e., o(EII,A]I)

Example 3 - l . 6 .

[,

under

Then, the necessary and sufficient condition of

IL i s easy to v e r i f y from C o r o l l a r y 3-1.3 t h a t

0

0

o] I

[0 ] o

x+

y

I0

is not detectable.

3-1.3.

Normal i z a b i ] i t y

Definition 3-1.4.

System (3-1.1) is called normalizable if a feedback control (3-

1.2) may be chosen such that its closed-loop (3-1.3) is normal, i.e., i E+BK2[ # O.

While

for

singular.

any matrix K 16 ~ m x n

the closed-loop system under (3-1.4) is

Under the assumpti.on of n o r m a ] i z l d ~ i l i t y ] o r system ( 3 - 1 , ] ) ,

appropriate enabling

(3-I .17)

feedback

control

the application

we may choose

(3-1.2) such that its closed-loop system

of results in normal, system theory to

always

singular

is

normal, ones.

As

long as condition (3-1.17) is satisfied, the closed-loop system (3-1.3) would become = (E+BK2)-I(A+BK1)x + (E+BK2)-IBv

(3-1.]8) y = Cx and its plain feature is its n finite poles, normalizabil~ty assumption,

without any infinite poles.

Under

a singular system's infinite poles may be set to

the

finite

via feedback (3-1.2). Thus, no impulse terms appear in the closed-loop state response.

75 Theorem 3-1.4.

The following statements are equivalent.

(a). SysLem (3-1.1) is normalizable. (b). rankle

B] = n.

(c). The fast subsystem (1-3.11b) is controllable. (d). In (3-1.11), rank[N, B2] = n 2. (e). For any nonsingular matrices QI and PI satisfying QIEPI = diag(l, 0), Q1B

=

[BI/B2 ]' B2 is of full row rank n-rankE. Proof. Equivalence among (h)-(e) has been shown in Theorem 2-2.1.

We need only to

prove the equivalence of (a) and (b). Assume that

(a) holds.

By definition there

exists a matrix K2E ~mxn

such

that

IE+BK21 ~ O, i.e., rank[E+BK2] = rankle Thus, ranklE

B| = . ,

B][I/K2] = n.

which is ( b ) .

C o n v e r s e l y , i f (b) h o l d s , from where QI and P1 a r e n o n s i n g u t a r ,

t h e maLrix rank d e c o m p o s i t i o n QIEPI = d i a g ( I , O), and d e n o t i n g QIB = [BI/B2], we know from ( e ) t h a t

B2 must be of f u l l row rank n - r a n k E ,

i n d i c a L t n g B2B$. > O i s symmetric: p o s i t i r e d e f i -

fi21Pl I E ~mxn. Then

n i t e . Let K2 = [0

I E+BKI = I Q; 1. pylll ~2B~ m~

o.

System (3-1.I) is normalizabie, i.e., (a) is true. Q.E.D. As previously pointed out, for a given singular system, st abilLzability and detectal}ility

characterize

opposite,

reflecting

normalizabL1ity

its

s l o w subsystem p r o p e r L i e s ,

while .ormalizability

only iLs fast subsystem properties. system may he changed i . t o

a normal one v i a s u i t a b l e

of feedback (3-1.2). Example 3 - 1 . 7 .

[ oooo]

C o n s i d e r system ( 1 - 3 . 1 9 ) . S i n c e

B] = rank

rank[E

0 0 1 0 0 0 0 0 0 0000-I

0

= 3 < n = 4

t h i s system i s not n o r m a l i z a b l e a c c o r d i n g to Theorem 3 - 1 . 4 . E×ample 3 - 1 . 8 . C1 =

i s the

system,

clarifies the difference between singular and normal systems ....

nnr.ual i z a h l e s i . g u l a r

E

As for the whole

In system ( 2 - 2 . 5 ) - ( 2 - 2 . 6 ) ,

0

0

0

0

C2

0

0

0

0

-I,

0

0

0

0

0

0 B

=

I!

a

s e l e ( ' l i()a

76 and

rank[E

B] = 4.

Thus it is normaltzable. K2 = [0

0

Let

-11.

0

We have

0

0 O

O

C2 0 O -L

O

O

O

1

C10 IE+BK 2 I

=

0

O

By d e f i n i t i o n ,

Example 3 - 1 . 9 .

= - I.CIC 2 ~ O.

the singular

system with only slow subsystem

(a

normal system) = Ax + Bu,

y = Cx

is normalizable. From Theorem 3 - 1 . 4 and c o n t r o l l a b i l i t y zabie

if

it

is controllable.

under r.s.e.,

Thus, if

tile n e c e s s a r y

tem ( 3 - 1 . 1 )

is

that

(3-1.15)

and s u f f i c i e n t

E22 i s n o n s i n g u l a r

Decomposition (2-5.9)

criterion,

A system is impulse controllable it

(1-3.19))

[s impulse controllable.

a system is normali-

the controllability

condktion

canonical

for the normalizability

form

of sys-

in (3-1.15).

is the canonica[

example,

is

we s e e t h a t

normalizability

decomposition.

if it is normalizable;

the inverse

is false. For

has been pointed out in Example 2-2.7 that system (2-1.8)

(thus

system

But it is not normalizable,

as p r o v e n

is dual normalizable

dual system is norma-

in

Example

3-1.7.

3-1.4.

Dual n o r m a l t z a b i l i t y

Definition

3-1.5.

System (3-1.1)

if

its

lizab|e. We have the following theorem.

Theorem 3 - 1 . 5 .

SysLem ( 3 - 1 , 1 )

is dual normalizable

i f and o n l y i f

ranklE/C]

= n.

It has been shown in Section I-3 that any regular system (3-I.I) is r.s.e, to EF3:

Fx = ([ - ~ E ) x y= where

o~ satisfies

theorem.

+ Bu

(3-1.9)

Cx IaE+A| # O, Under this equivalent

form we can prove the following

77 Theorem 3-1.7.

Consider system (3-I.19).

(I). It is stabilizable if and only if rank[s[-E, B] = n,

V s ~ 0, s E ~ +, ^

s finite. ^

Or equivalently, the uncontrollable poles of (E,B) must be stable or zero. (2). It is normalizable iff rankle

B] = n.

(3). I t i s detectable i f f r a n k [ s I - E / C ] = n, V s ~ O,

sE~. +,

s finite.

( 4 ) . I t i s dual n o c m a l i z a b l e i f and o n l y i f r a n k l e / C ] = n. Proof. We need only to prove (1) and (2) s i n c e (3) and (4) may be o b t a i n e d by tilt: dual p r i n c i p l e . (1).

According to Theorem 3 - 1 . 2 ,

lizability

the n e c e s s a r y and s u f f i c i e u t

c o n d i t i o n of s t a b i -

f o r sysLem ( 3 - 1 . 1 9 ) i s

rank[s~,-(l-~),

BI = rank[(s+o¢)~-I,

BI = n,

V s E ~ +,

This h o l d s i f s+ot = O; o t h e r w i s e , by s e t t i n g Z = I / ( s + o t ) ,

s finite.

i t may be r e w r i t t e n a s

^

rank[ kI-E, B] = n,

V /k ~ O,

k E ~+,

A

finite,

which is (I). (2). Statement is obvious. Q.E.D.

So f a r ,

we have l n t r o d u c e d v a r i o u s c o n c e p t s in s i n g u l a r system t h e o r y ,

controllability, R-controllability, vability, lily,

impulse controllability, observability, R-obser-

impulse observability, normalizability, dual normalizability, staIHlizabiAmong t h e s e c u n c e p t s t h e r e e x i s t both s i m i l a r i t i e s

and d e t e c t a b i l i t y .

[erences.

Their

relationships are complicated,

sysLem from different points of view. trollabilily

and o b s e r v a b i l i t y

controllability, properties

including

R-observability, and

singular

while con-

c h a r a c t e r i z e tile p r o p e r t i e s f o r the whole systom,

for slow subsystem;

normalizability,

but they c h a r a c t e r i z e the

As for their intrinsic properties,

and d i l ' -

stabilizability,

and d e t e c t a b i l i t y

and impulse c o n t r o l l a b i l i L y ,

dual n o r m a l i z a b i l i t y for the fasl

R-

characterize

impulse o b s e r v a b i . l l t y ,

subsystem.

These

roncepls

cannot be s u b s t i t u L e d f o r each o t h e r , To sum up,

the r e l a t i o n s h i p s

among t h e s e c o n c e p t s may be d e s c r i b e d by t h e f o l i o -

wing diagram:

controllability

/

\

R - c o n t r o l lahi 1iLy ~. n o r m a l i z a b i l i t y

• stabi lizabi lily ~, impuIse c o n t r o l l a b i l i t y

R - c o n t r o l labi I i t y + uorlmkl izabi I it y

78 R-observability observability

J

%

~

stabi]izability

dual normalizability

,

impulse observability

R-observability + dual normalizability

where A

~B

represents that B may be deduced from A, and A ~

B indicates the equi-

valence of A and B.

3-2.

Practically, that

Infinite

it is expected to change

Pole Placement

a system's properties by control inputs so

the closed-loop system possesses the expected properties.

Many facts show that

pole placement is important in affecting tile dynamic und static properties of a ear time-invariant system. which studies system

the

liu-

Thus, pole placement is a basic probelm in system design,

selection of feedback control

so that poles

lie at the expected places in the complex plane.

of the

For singular

closed-loop systems,

not

only finite poles, but infinite ones also affect the response properties of a system. In this and the next sections, we will study the pole placement problem for

infinite

and finite poles, respectively.

3-2.1.

Infinite pole structure in singular systems

First, we will briefly outline the concept

of infinite pole structure in singular

systems. Consider singular system (3-I.I): E~ = Ax + Bu

(322.1)

y = Cx

For

system (3-2.1) the finite pole structure is determined by the zero structure

sE-A

at

finite s and

its infinite pole structure is defined as isomorphic

to

zero structure of sE-A at infinity, which in turn is isomorphic to that of IE - A

of the at

S

s=O. For any regular system (3-2.1) there always exist two nonsingular matrices Q and P such that QEP = diag(Inl, N),

QAP = diag(Al,

where n I + n 2 = II, N nill)utent. Note that

1,12)

79 Q(sE-A)P = diag(sl-Al, sN-l).

Then, the finite while the i n f i n i t e

pole structure of

system (3-2.1) is completely determined hy AI,

pole s t r u c t u r e is isomorphic to the zero s t r u c t u r e of

-IN- I

(3-2.2)

S

at s = O . Therefore, i f N = O, from (3-2.2) we know that poles, and v i c e versa. When N ~- O, let

system (3-2.1) has

no

N = GF be

the

infinite (3-2.3)

full rank decomposition of N,

in which C and F

rank, respectively. Thus, an irreducible realization of

are of full column

iN -

and row

i is (following the no-

S

tation of Yerghese et al., 1979):

..........

R(~) =

-4-....

c I-I

V(s)

(3-2.4)

I W(s)

,

i.e.,

-IN S

I = V(s)T-I(s)U(s)

+ W(s).

According to the Smith-McMillan d e f i n i t i o n of the pole-zero s t r u c t u r e of a r a t i o n a l matrix (Rosenbrock, 1970; Verghese eL a l . , 1979), (3-2.4) has the same f i n i t e zero structure as (3-2.2). Since the zero structure of (3-2.4) is determined by [sI+FG

O]

0

-I

,

or sl + FG, we have the following proposition.

Proposition 3-2.1.

If N = O, system (3-2.1) has no infinite poles, otherwise,

infinite pole structure is siomorphic to the zero structure o[ FG. Note that N is nilpotent. Assume that

I 0 0

~ u2-11

lln2xn2

Then i.t has the f u l l rank decomposition N = GF,

G =

[Ion2-1 ]

F = [0

In2_l].

Thus FG=

0 0

in2_2]

~

:R(n2_ 1 ),(n2_ 1 )

,

its

8O This

shows

that

when N is a Jordan block matrix,

system (3-2.1) is

determined

the infinite pole

structure

of

for the nonexistence

of

by 1-order lower Jordan block.

Note that N = 0 is the necessary and sufficient condition

both infinite p o l e s and impulse terms in the state r e s p o n s e of system (3-2.1). Proposition

3-2.2.

No i m p u l s e terms e x i s t

i f and only i f t h i s system has no i n f i n i t e From the that

preceding discussion,

i.e.,

poles, of infinite poles is

lower than tile o r d e r o f ,Jordan b l o c k s

by one number.

se structure of singular systems, determined

r e s p o n s e o f system ( 3 - 2 . ] )

the (geometric) multiplicity

o f z e r o in N but minus o n e ,

longing to zero eigenvalue)

in t h e s t a t e

(be-

In such senses, and from the state respon-

the impulse terms in state response of (3-2.1) are

by the pole multiplicity

at infinity.

there are no impulse terms in state response;

If the system has no infinite poles,

conversely,

if there exist

infinite

poles,

there exist impulse terms in state response. The higher the multiplicity,

higher

the order of derivatives

highest multiplicity In

a

expected

parctical

in impulse terms.

the

Its highest order is equal to the

of infinite poles. system,

the impulse term and its derivatives

in the state response,

otherwise,

are

usually

not

strong impulse behavior may saturate the

system so that it could not work or may even destroy it. Thus, except for a few cases (such

as on impulse generating mechanism),

state response.

This probelm is equivalent

the closed-loop

system has no

impulse terms are always avoided to selecting

infinite poles,

in the

feedback controls such

or N = 0.

this is the main

that

problem

studied in this section.

Example 3 - 2 . 1 .

In the s t a n d a r d d e c o m p o s i t i o n ( 1 - 3 . 2 0 ) , N = O.

Thus,

system ( I -

3.19) has no infinite poles or impulse terms in its state response. ExamDle 3 - 2 . 2 .

C o n s i d e r t h e system

I0 0 00] I

0

0

l

x=x+

[10]

0

u

0

y=Cx. In this system,

N is a third-order Jordan block belonging to zero eigenvalue.

it has an infinite pole of multiplicity

3-2.2.

Infinite pole placement

Consider the pure proportional

(P-) state feedback control

(3-2.~)

u = Kx + By where

Thus,

2. Impulse terms exist in its state response.

v(t) is the new control input.

Then (3-2.1) and (3-2.5) form

the closed-loop

84 system E~ = (A+BK)x + By y = Cx Note system.

(3-2.6)

that the infinite pole structure Thus,

feedback

is determined by the fast subsystem

of

a

control achieves its control goal by changing the slow-fast

subsystem structure. To begin, we have the following lemma. Lemma 3-2.1. The closed-loop system (3-2.6) has no infinite poles if and only if deg(IsE - (A+BK)I) = rankE. Proof.

For

singular system (3-2.6),

there exist nonsingular matrices Q

and

P

such that it is r.s.e, to EFI

Xl = AlXl + BIV Nx2 = x 2 + B2v where N i s n i l p u t e n t .

From t h e p r e c e d i n g

discussion,

system (2-3.6)

h a s no i n f i n i t e

p o l e s i f and o n l y i f N = O. T h u s , deg([sE - (A+BK)[) = deg(Isl - All) = rankE.

Q.E.D.

Theorem 3-2.1. For system (3-2.1), there exists u feedback control (3-2.5) so that the closed-loop system has no infinite poles if and only if it is impulse controllable. Proof.

Lemma 3-2.1 shows the existence of feedback (3-2.5) such that

the closed-

loop system (3-2.6) has no infinite poles iff there exists a matrix K satisfying d e g ( I s E - (A+BK)I) = r a n k E . Necessity:

(3-2.7)

Let (3-2.7) hold. Then there exist nonsingular matrices T 1 and T 2 such

that T1ET 2 = d i a g ( I q ,

q = rankE.

0),

Denoting

TIAT 2 =

[A A12] [l] TIB =

A21

A22

•

B2

KT

= [KI, K2],

,

from (3-1.5) we have rankE = deg(lsE - (A+BK)I) = deg(IsTiET 2 - TI(A+BK)T21 ) = deg( SIq - All- BIK 1 - A21- B2K 1

-AI2- BIK 2 ] ) ~ q. -A22- B2K 2

I

82 The equality holds if and only if I -A22- B2K2~ ~ O, i.e., lA22+ B2K21 @ O, indicating the normalizability of (A22,B2), rank[A22, B2] = n-rankE. Thus, system (3-2.1) impulse controllable.

is

Sufficiency: Assume that system (3-2.1) is impulse controllable. For the preceding decomposition, by Theorem 2-4.2 we know rank[A22, B2] = n - rankE (A22,B2) is normalizable. Therefore, there exists a matrix K 2 such that IA22+ B2K21 @ O. Setting K = [0

K2]T21, direct computation yields

deg(lsE-(A+BK) l) = deg( I sI-AII -A21

-AI2- BIK2 1 ) = rankE, -A22- B2K 2

which assures the authenticity of (3-2.7). Q.E.D. This theorem

further

infinite poles, eliminate

the

relationship

among

impulse

controllability,

impulse term in the state response in the closed-loop system

state feedback ones,

reveals the

and impulse terms. Impulse controllability guarantees the ability to control.

via

P-

In this process the infinite poles are placed to the finite

indicating the closed-loop system has at most rankE finite poles

and no inFi-

nite poles. Thus, the other infinite poles are driven to Finite positions. Example 3-2.3. Consider the following singular system

[o0o] I

0

0

1 0

0

y = [I

~=x+

u

(3-2.8) O

-I]x.

Itere we have

Io00] [i00]

E = I O O O 1 0 ,

A=

This system has only infinite poles,

IO O1,

B= O.

with pole set {~,

~.

The multiplicity of

the infinite pole is two. Thus, impulse terms are certain to appear in the state response. It may

be verified that system (3-2.8) is impulse controllable.

There exists

for example,

K:[O

1

I]

that satisfies rankE = deg(IsE-(A+BK)J) = 2. For such Ks the closed-loop system

K,

83

[ooo] 111 ] Ill 1

0

0

0

1

0

~=

y = [1 has no i n f i n i t e

0

poles.

0

1

0

0

x +

0

v

0

(3-2,9)

-llx And t h e c l o s e d - l o o p

(3-2.9)

h a s tile c h a r a c t e r i s t i c

polynomial

of

I sE - (A+BK) I = - (s 2 + s + I), which i J n p l i e s

Thus

feedback

o(E,A÷BK)

=

{-

~ -

i,

- 2 ÷

i}.

N(~ i111"iHilc p o l e s ~Jl~l~e~Jr in ( 3 - 2 . 9 )

control drives the two infinite poles in (3-2.8) to the finite

posi-

tions.

Moreover,

according to the proof,

under the impulse controllability

assumpLion,

there exists a matrix K satisfying deg(IsE-(A+BK)l) = rankE if and only if lA22 + B2K 2 I~ O. Inspired by this point,

(3-2.10) we have the following algorithm for finding matrix K.

We

first take rank decomposition: T3A22T 4 = diag(Ifi, O),

/B= rankA22

where T 3 and T 4 are nonsingular. And then donate T3B 2 =[ B211 B22J • Under Lhe a s s u m p l _ i o n o f i m p u l s e c o n t r o l l a b i l i t y , f u l l row r a n k .

r a n k [ A 2 2 , B2] = n - r a n k E .

B22

is

of

Let

We have IA22+ B2K21 =I T3111T;IIIB22B221 ~ O. Thus, matrix K = [O

K2]T21 satisfies

(3-2.7).

Example 3-2.4.

T1 =

Consider system (3-2.8). The nonsingular

0

0

1

0

matrices

T 2 = 13 ,

transform

TIET 2 = diag(I 2, 0), in which A22

TIB = [O

0

I]~:, TIAT 2 = T 1

O, B 2 ~ I. The scalar K 2 ~ 1 satisfies [A22+ B2K2[ ~ I ~ O. Thus matrix

84

K = [0

K2]T21 = [0

0

1]

would satisfy deg(IsE-(A+BK)l) = deg(s2+ l) = 2 = rankE.

In standard decomposition, there exist two nonsingular matrices Q and P, such that regular system (3-2.1) As r.s.e, to EFI: ~I = AlXl+ BlU Nx 2 = x 2 + B2u

(3-2.1])

y = ClX 1 + C2x 2 where x|E ~nl,

×2 E ~n2,

n I + n 2 = n, and

QEP = diag(I, N),

QAP = diag(Al, I),

CP = [Cl, C2]

x1

Under decomposition (3-2.11), the P-state feedback u = Kx + v becomes u = KlXl + K2x2 + v

w h e r e KP = [ K I ' K2 ] '

(3-2.12)

KIE ~ m x n l '

K2 E ~mxn2"

A. When K2 = O, f e e d b a c k c o n t r o l

(3-2.12)

becomes

u = KlXl + v. In t h i s it

feedback form,

t h e slow s u b s t a t e

(3-2.13) only substate

feedback control,

B. I f KI = O, f e e d b a c k c o n t r o l

x 1 i s used f o r f e e d b a c k p u r p o s e s .

We t h u s c a l l

slow feedback hereafter.

(3-2.12)

becomes

u = K2x2 + v in which o n l y t h e f a s t back c o n t r o l , C.

In

substate

is derived.

We w i l l c a l l

it

the fast

substate

l eed-

fasL feedback hereafter.

general,

We w i l l c a l l

(3-2.14)

feedback control

i t Lhe combined s l o w - f a s t

(3-2.12) substate

includes

b o t h slow

feedback, state

and f a s t

substates.

feedback hereafter.

While the standard decomposition characterizes the system's inner

structure,

the

slow and fast feedbacks show the effect of partial state feedback on the whole system structure. Theorem 3-2.2. 2.12)

such

Given system (3-2.1),

there exists

a state feedback control

that the closed-loop system has no infinite poles if and only

if

(3there

exists a fast feedback (3-2.14) such that the closed-loop system (3-2.6) has no infi-

85 nite p o l e s . Proof. and

By Theorem 3 - 2 . 1 ,

t h e impulse c o n t r o l l a b i l i t y

for (3-2.1) is the necessary

sufficient condition for existing state feedback (3-2.6) such that

poles

exist

in its closed-loop system.

no

infinite

The impulse controllability for (3-2.1)

equivalent to the impulse controllability of the fast subsystem.

Thus,

is

from Theorem

3-2.1 there exists a matrix K2 such that deg(IsN - (I+B2~2)I) = rankN and vice versa. For such a K2' we have deg(IsE - (A+BK)I) = deg(IsQEP - Q(A+BK)PI) = n I + deg(IsN - (I+B2K2)I) = rankN + n I = rankE indicating the conclusion. Q.E.D. This theorem shows that the pole structure at infinity is determined only by structure

of

the

fast subsystem,

revealing that the fast. subsystem

the

reflects

the

system's pole structure at infinity.

Previously,

we studied the distribution of infinite poles under P-state feedback.

For the more general P-D state feedback control u = KlX - K2~ + v where K I, K2E I~mxn,

(3-2.15)

the closed-loop system formed by (3-2.1) and (3-2.15) is

(E+BK2)~ = (A+BKI)x + Bv

(3-2.16)

y = Cx. Under

the assumption of normalizability,

we know that matrix K 2 E ~mxn

may

be

chosen such t h a t I I"+BK21 ~ O. T h e r e f o r e Theorem 3 - 2 . 3 . feedback

Assume t h a t system

(3-2.1) is

normalizable.

Then

c o n t r o l ( 3 - 2 . 1 5 ) such t h a t the c l o s e d - l o o p system ( 3 - 2 . 1 6 )

there

exists

i s normal,

and

thus has no infinite poles. We noticed in the proof process of Theorem 3-2.1 that for any matrix K,

we always

have tile i n e q u a l i L y : deg( IsE - (A+BK)I) x< rankE. Therefore,

under feedback control (3-2.5) the closed-loop system (3-2.6) is always a

singular s y s t e m , hand,

we

closed-loop

see

no m a t t e r what c o n d i t i o n s a r e imposed on t h e s y s t e m .

On tile

from Theorem 3 - 2 . 3 t h a t under t h e a s s u m p t i o n o f n o r m a l i z a b i l i t y

system is normal ( w i t h n f i u i t e

other the

p o l e s ) by s u i t a b l e s e l e c t i o n of feedback

(3-2.15), which i s i m p o s s i b l e f o r P - s t a t e feedback ( 3 - 2 . 5 ) .

86 Since feedback control (3-2.5) is a special form of (3-2.15), trol

if a feedback

(3-2.5) may be chosen such that the closed-loop system (3-2.6) has

poles,

con-

no infinite

we may always be able to choose a suitable (3-2.15) such that (3-2.16) has no

infinite poles (thus impulse terms in state response are eliminated).

But the inverse

is false. For instance:

Example 3-2.5. Consider the system

ioo I [oo] [o] 0 0

0 0

l 0

~=

0 0

1 0

0 1

x+

(3-2.17)

u

0

in which

[0L01

E=

0 0

0 0

1 0

A=

[100]

,

0 0

1 0

0 1

B=

i0] 1 0

,

,

n = 3, rankE = 2, and rank

Thus,

system

E

0

O]

A

E

B

(3-2.17)

= 4 < 5 = n + rankE.

is not

impulse

conLrollable.

For any P-state

v, there always exist impulse terms in its closed-loop

feedback

u = Kx +

system.

But, if we choose u = [-l the closed-loop

0

0]~ + v = - K2x + v,

system would be

I°°l [°°1 Iil 1 0

0 0

1 0

~ =

0 0

Since the characteristic

1 0

0 1

x +

polynomial

v.

(3-2.18)

of closed-loop

system (3-2.18) is

deg(Is(E+BK 2) - AI) = deg(s2+ I) = 2 = rank(E+BK2), closed-loop its

state

ponse not

system (3-2.18) has no infinite poles; thus there are no intpulse terms in

response.

This

may be e l i m i n a t e d

impulse

shows that by d e r i v a t i v e

impulse

lerm~ of a singular

feedback

even in the

case

system

in s t a t e

when t h e

system

resis

controllable.

Theorem 3-2.4

the closed-loop

rank

(Dai,

1988j).

There

exists

a P-D s t a t e

feedback

(3-2.15)

system (3-2.16) has no infinite poles if and only if

A

E

B

0

= n + rankE.

such

that

87 3-3.

Distribution

of

dynamic and s t a t i c

poles,

Finite

Pole PlacemenL

especially

properties

finite

poles,

has a profound effect

on tile

of a system.

Consider the system E~ = Ax + Bu y = Cx where x E ~ n, ctively,

u E ~ m,

(3-3.1) yE~ r

are its

state,

control

input,

and m e a s u r e o u t p u t ,

feedback coutrol

aud tile P - s t a t e u = Kx + v

where v E ~ m

(3-3.2)

The c l o s e d - l o o p

system is

E~ = (A+BK)x + By y = Cx. Lemma 3 - 3 . 1 .

(3-3.3)

System (3-3.1)

tric set

tric set

K satisfying

If system (3-3.1)

A with n I elements

1( such t h a t

the closed-Loop

on t h e c o m p l e x p l a n e , finite

pole set

singular matrices

1-2.2,

Q and P s u c h that: i t

is R-c~mlrol-

f o r any synmm-

there always exist~

occur

i.e.,

for any regular

(stabilizable),

~r(E,A+BI() = A ( ~ - ) .

i s t h e seL i n w h i c h c o m p l e x s c a l a r s

By Lemma

(3-3.3)

I sE-(A+BK) I i/ O.

is l~-controllahle

order o f Lhe slow s u b s y s t e m i n ( 3 - 3 . 1 ) , Proof.

i f and o n l y i f

is R-controllable

labIe f o r a n y f e e d b a c k g a i n m a t r i x Theorem 3 - 3 . 1 .

respe-

in c o n j u g a t e

a gaL. matrix

tlere the pairs,

symme-

and n I i s

the

u I = deg(I sE-A I).

system (3-3.1)

is r.s.e,

there

always exist

two n o n -

t o EVI:

Xl = AlXl + BlU

(3-3.4a)

HA2 = x 2 + B2u

(3-3.4h)

y = ClX 1 + where XlE ~ n l ,

C2x2

x2E • n 2 ,

(3-3.4c) n 1 + n2 = n, N is nilpotent,

QEP = diag(l, N),

QAP = diag(Al,

I),

and

CP ~ [C1, C2]

(3-3.5) QB = [ B1

Under

,

the assumption

Xl

,

of R-controilability

3 . 4 a ) h a s t h e same p r o p e g t y .

O(Al+ B1K I) = A ( c E - ) .

(stabiiizability),

Thus, for any A there

slow subsystem

exi.sts a matrix

~ that

(3-

satisfies

88 Let the feedback control be u = Klx 1 + v = [ K I , O l p - l x + v. The closed-loop

system is

~1 = (AI+BIKI)Xl + BlV

(3-3.6)

N~2 = x 2 + B2KlX 1 + B2v y = ClX 1 + C2x 2. Lo t h e nilpotent property of N,

Paying atLention

the closed-loop

pole set of (3-3.6)

(thus (3-3.3)) is o(E,A+BK)

=

o(QEP,Q(A+BK)P)

o(AI+BIKI) :

A

( C ~-)

where K : [KI, O]P -I. Q.E.D. In t h e p r o o f p r . c e s s if our

we s e e t h a t

feedback control

is a slow

feedback.

Therefore,

purpose in imposing feedback control is to stabilize a system or pole

ment, it is sufficient However, will exist

to finish this task by applying slow feedback.

the main disadvantage

of such feedbacks

impulse terms in the state response

O, which is often unexpected

0 0

0 0

is that since n I ~ rankE,

in the closed-loop

there

system (3-3.3) if N

in practice.

[oo] [!oo] 1]

Example 3 - 3 . 1 .

place-

In the system

l 0

~=

1 0

x+

0 1

1

(3-3.7)

u

n 1 = 1, n 2 = 2, and

[o 0,l

N =

)

,

AI =

)

This system is controllable. The

BI =

,

B2

)

ro ]

L

=

0

ClearLy,

~=

0 -2

1 0

x+

impulse terms appear in the state

Therefore, infinite

t 0

o u r aim

poles.

0 1

A =

{-i }.

that

(3-3.8)

v .

response.

to impose the feedback c o n t r o l

This requires

Assume that

case

system is

[°° 1 [°iJ [] 0 0

In t h i s

O]x + v

alnl t h e c o r r e s p o n d i n g c l o s e d - l o o p

0 0

•

Thus it is R-controllable.

o ( A I + B I K I ) = { - 1 } h o l d s by c h o o s i n B K[ = - 2 . u = KlX 1 + v = [ - 2

l

not only

should

are the finite

place both finite poles

placed

on

and the

89

arbitrarily assigned positions, but the impulse terms are also eliminated in the state response in closed-loop systems. Thus, both finite and infinite pole placement should be considered at the same time, in other words, the c]osed-loop system should have rankE finite poles (thus no infinite poles) that lie at arbitrarily assigned positions. Theorem 3-3.2. Let system (3-3.1) be controllable. Then for any symmetric set A with rankE elements on the complex plane, there always exists a gain matrix K such that the closed-loop system (3-3.3) has the finite pole set o(E,A+BK) = A . Proof. The proof is constructive. Let QI' PI he nonsingular matrices satisfying QIEPI = diag(Iq, 0),

q = rankE.

By denoting QIAPI

[ All AI2 I

=

A21 A22

QI B =

[ BI ] 132

x = pTlx

,

,

system (3-3.1) is r.s.e, to EF2 (on]y state equation is needed here): lq

x

[°l 0

[ All

L

0

AI2

A21 A22

1[

Accounting for the c o n t r o l l a b i l i t y

u.

(3-3.9)

132

assumption,

system (3-3.1) is impulse c o n t r o l -

lable. By Theorem 2-4.2, rank[A22, B2] ~ a-q, (A22, B2) is normalizable. Thus, matrix K2 E ~mx(n-q) may be chosen such that B2K21 ~ O.

I A22 +

Setting ~ = [0

a

(3-3.10)

K21P~1 and

u = Kx + u 1 = [0

K2]~ + u I

(3-3.11)

in which Ul(t) is the new input for the system, 3.11) and (3-3,9) is

[ Ioq ~ ]x

[ AIIA21 AI2+BIK2]A22+B2K2 =

~

+

the closed-loop system formed by ('3-

(3-3.12)

[ BIB2]u I L

"

From (3-3.10) (A22+B2K2)-1 exists, and the two matrices Q2' P2 defined as [q

_ (AI2+B 1K2) (A22+B2K2)-1 ]

Q2 =

0

P2 =

q - (A22+B2K2)-IA2I

I

n-q o

(A22+B2K2)-I

90 are nonsingular.

Direct computation shows

Q2diag(Iq, 0)P 2 = d i a g ( I q , 0)

f

Q2

all

AI2+BIK 2 ]

A21

A22+B2K 2

P2 = diag(Al' i)

(3-3.13)

where ~I = All - (A12+ B1K2)(A22+ B2K2)-IA21

BI = BI - (AI2+ BIK2)(A22+ B2K2)-lB2'

B2 = B2"

By defining = p~l~ = l,~lp~Ix from (3-3.13),

system (3-3.12) is r . s . e ,

d i a g ( I q , 0)~ = d t a g ( ~ l ,

to

I)~ + [BI/BelUl,

(3-3.14)

which is R - c o n t r o l l a b l e according to Lemma 3-3.1. The pair (Al' ~l ) is controllable. Since A1E ~ q x q for any symmetric set A with q = rankE elements, a matrix 7 2 may bc chosen to s a t i s f y a(A I + BIK I) = A.

Define

ul = r:t~1, o ] ~ + v . The overall

feedback control is

u =

in

which

K

Kx

+ u! = K x

= K + [71

+ v

Olp~lv~ 1. For such a feedback gain matrix K, it is easy t:o ve-

rify that the closed-loop system (3-3.3) has the finite pole set of o(E,A+BK) =

o(A I + BIKI) = A

or system (3-3.3) has A as its finite pole set. Q.E.D. The constructive proof also provides us with a design process to find gain matrix K.

Example 3-3.2, formation matrices

System (3-3.7) in

B×ample 3-3.1 is controllable,

Under the trans-

[,oo]

91

QI =

I

PI :

o

,

o

o

I

o

1

o

,

[o o°I,o]

it is r.s.e, to

o B--~-l-o J

(3-3. ] 5)

~--VI-;

Thus, we have

1. Then I A22 + B2K21 = 1 t~ 0 and

Let K 2 =

A1 : All

_

-

(AI2+ B1K2)(A22+ B2K2) IA21

=

1

[0

-1] -1

BI = B I - (AI2+ B1K2)(A22+ B2K2)-IB2 = [ -(~ ]

P2

=

- (A22+ B2K2) A21 (A22+ B2K2)-I

=

0

1

0 -1

0

1 .

Note that

rank[Bl, therefore rankE

=

(AI'

AIBI]

: rank [ _ 0

1I ]

=2,

BI ) i s indeed c o n t r o l l a b l e .

2 stable finite poles -I, -l, or

The c l o s e d - l o o p A

:

system i s assumed t o

have

{-l,-I } . By setting

"El : [-4' 2l we have

I sl

(AI+BIKI) I = (s+l) 2.

-

Thus, o(AI+B1K1) = A = [-1, -1 }. in this case, the feedback gain matrix K is K = K + [KI'

O]p]IP11

=

[-4

1

2]

and tile corresponding closed-loop system (3-3.3) for this special system becomes

[1oo] 0 0

0 0

1 0

~ =

0 -4

1 ]

0 3

x +

[,] 0 l

It is easy to verify that the closed-loop

v

(3-3.16)

pole set is the expected (-I, -l} and no

92 infinite poles (thus no impulse portions) exist in the closed-loop system.

Summing up the results in this and last sections,we have proves the following main result of pole placement for singular systems. Theorem 3-3.3. a. control b.

Consider system (3-3.1).

Closed-loop

There exists

a gate matrix

the most rankE finite c.

(3-3.3) has at most r a n k E finite poles

system

for any

feedback

(3-3.2).

poles)

If system (3-3.1)

chosen

so

that

K such that

i f and o n l y

is R-controllable

closed-loop

(3-3.3)

h a s no i n f i n i t e

if system (3-3.1)

is

has

ha~

impulse controllable.

and i m p u l s e c o n t r o l l a b l e ,

system (3-3.3)

pole~ (thus

a m a t r i x K may be

rankE arbitrarily

assigned

finite

poles. Under rankE

the controllability assumption,

arbitrarily

Theorem 3-3.2 states that (3-3.3) may have

assigned p o l e s by s u i L a b l y chosen feedback,

shows t h a t the number o f most p o s s i b l e f i n i t e Similarly

we have the f o l l o w i n g

Corollgry 3-3.1.

There exists

deg(IsE-(A+BK) l) = rankE if and

T h i s theorem

further

p o l e s i s rankE,

corollary, a matrix K

only if

satisfying

both

o(E,A+BK) C ¢;-

system (3-3.1) is stabilizable

and

and impulse

controllable. In

other words,

stabilizability and

impulse controllability

are necessary

and

sufficient conditions for the existence of K such that the closed-loop system (3-3.3) not only

is stable

Lhe slate

response).

Corollary 3-3.2.

b u t a l s o h a s no i n f i n i t e

There

exists a matrix

d e g ( l s E - (A+CC) I) = rankl", a r e s a t i s f i e d

poles

(thus eJiminates

C such that.

i f ;rod o n l y i f

bot:h

impulse terms

o(E,A+GC) C ¢ -

system (3-3.1)

in

and

is detect.hie

and impulse o b s e r v a b l e , This conclusion

We next

will

be n e e d e d l a t e r .

consider the

combined proportional and derivative (P-D)

state

feedback

control u(t)

= KlX(t ) - K2~(t ) + v(t)

under which the closed-loop

(3-3,17)

system is

(E+BK2)~ = (A+BK1)x + By

(3-3.1s) y = Cx.

Lemma 3-3.2.

Assume that

system (3-3.1) is R-controllable

(stabilizab]e).

Then

93 (E+BK2,A,B,C) is R-controllable (stabilizable) for any K 2 6JRmxn that

satisfies

Is(E+BK2)-A l~ O. Remark. As shown in Example 3-2.5, derivative state feedback may change the impulse controllability of a system; this

lemma clarifies that:

Neither proportional nor

derivative s t a t e Feedback corttro] can change the R-control [ a b i l i t y ( s t a b i ] i zabi]Jty) of a s y s t e m . Proof.

By d e f i n i t i o n o f R - c o n t r o l l a b i l i t y

3.l) is R-contro[lahle (stabitizable)

rank[sE-A, B] = n,

V

( rank[sE-A, B] = n,

attd Theorems 2 - 2 . 2 - '3-1.2,

system ( 3 -

iff

s(C,

s finite

V sE ~.+, s finite).

On the other hand, rank[s(E+BK2)-A, B] = rank[sE-A, B] []K2s =

rank[sE-A, B].

V s~C,

O[ ]

s finite.

Thus, system (E+BK2,A,B,C) is also R-controllable (stabilizahle). Q.E.D. Theorem 3-3.4. Let A

be an arbitrary symmetric set with n elements on the complex

plane. Then controllability is the necessary and sufficient condition for the existence of (3-3.17) such that the closed-loop system (3-3.18) has the finite pole set: o(E+BK2,A+BK I) = A. Proof. Sufficiency:

Under the assumption of contro]l~fl~ility, system (3-3.1) is R-

controllable and normalLzable.

Matrix K 2 exists so that I E+BK21 ~ O. System (E+BK 2,

A,B,C) Ls R-controllable by Lemma 3-3.2. Noticing that it is a normal system ((E+BK2)-IA, (E+BK2)-IB) i s c o n t r o l l a b l e ,

and

f o r ~my A, a m a t r i x g l may be s e l e c t e d

that satisfies a((E+B/2)-IA + (E+BK2)-IBKI) = Hence,

the closed-loop system (3-3.18)

o((E+BK2)-I(A+BKI) ) = A. determined by K I and K 2 has the finite

pole

set : =(E+BK2, A+BKI)= Necessity:

The

=((E+BK2)-|(A+BKI) ) =

A.

existence of n finite poles in (3-3.18) shows IE+BK21 ~ O.

system (3-3.1) is n o r m a l i z a b l e ; R - c o n t r o l l a b i l i t y

Thus

i~ a d i r e c t r e s u l t o f a r b i t r a r i n e s s

in p o l e p l a c e m e n t . Q.E.D. Therefore,

under

the controllability

assumption,

feedback c o n t r o l ( 3 - 3 . 1 7 ) may

not only drive a l l infinite poles to finite positions, but further make the closed-

94 loop system normaL. This feature could not be achieved via proportional feedback (3-3.2).

Example 3-3.3. System (3-3.7) is controllable. Let

A = {-I,-I,-I} be its expected

closed-loop pole set, which could be achieved by derivative feedback (3-3.17) only. Let K 2 = [0

L

O]. Then l E+BK21 ¢ O. The feedback

u = - K2~ + u l and system

(3-3.19)

form t h e c l o s e d - l o o p

(3-3.7)

[ o I [oo] 0 0

0 l

1

~ :

0 0

0

I 0

0 1

sysLem

[il

x +

u I.

(3-3.20)

Noticing that

[ [ ol] [o]

E+BK 2 =

0 0

0 1

1 0

is nonsingular, system (3-3.20) may be written as

=

0 l

For any matrix and

(3-3.21)

has

1 0

x +

K1 = [ a l ,

the

u 1.

(3-3.21)

a3] , the

a2,

characteristic

s 3 _ (l+a2)s2 Assume t h a t

1 0

closed-loop

system

formed

by u I = KlX + v

polynomial

+ (a2-1-a3)s

+ (l+al+a3).

the ex pected c l o s e d - l o o p pole set

is ( - l , - l , - I }

. [f

we J e t a l

=

8,

a2 =

-4, and a 3 = -8, direct computation yields the P-D state feedback: u = [0

-I

01~ + [8

-4

-8]x

+ v

(']-3.22)

and the closed-loop system

]

0

[ 0

x =

0

i

0

8

-4

-7

which has the finite pole set

Theorem 3-3.5. exists

a

feedback

x+

0

v

l

{-I,-I,-I }.

Let system (3-3.1)

be stabilizable and normalizable.

control (3-3.17) such that the closed-loop system

normal with only stable finite poles.

Then

there

(3-3.18)

is

95 3-4.

3-4.1.

Pole Structure

Pole structure

under S t a t e Feedback C o n t r o l

under pure proportional

(P-)

slate

feedback

Consider system (3-3.1) E~ ~ Ax +

Bu

y = Cx xERn,

and i t s

(3-4.])

P-state

feedback (3-3.2)

u = Kx + v .

As p o i n t e d o u t

(3-4.2)

ill tile l a s t

sect£on,

to p l a c e t h e p o l e s o f t h e c l o s e d - l o o p

if our aim to apply

feedback (3-4.2)

Js merely

system

E~ = (A+BK)x + By y = Cx to e x p e c t e d

positions,

(3-4.3)

the feedback gain matrix

exists in its selection.

We naturally think of

K is

not unique.

Much

feasibility

using this feasibility

to improve

the closed-loop properties. Many

facts have shown that not only pole positions but also pole structure or the

Jordan forms in the coefficient matrix influence the properties of system (3-4.3) especially

the dynamic p r o p e r t i e s .

using t h e f e a s i b i l i t y

The s o - c a l l e d

in the selection

pole structure

placement problem

of feedback gain matrix

is

K to place closed-loop

system's Jordan structure. Without trollable, As

loss of generalLty,

stated previously,

avoided.

'rh[s

structure. finite

i n LhLs s e c t L o n we assume t h a t system ( 3 - 4 . 1 )

is con-

and rankB = m, B E ~ n x m . t h e i m p u l s e terms a r e h a r m f u l

requires

The l a t e r

elimination

problem is equivalent

p o l e s o r no i n f i n i t e

in a system

and

o f t h e i m p u l s e Lerms w h i l e we a r e to requiring

system (3-4.3)

should

placing

be

pole

to have rankE

poles.

To begin, take the rank decomposition on matrix E. Let q = rankE. Then two nonsingular matrices Ql and P1 exist such that QIEPI = dLag(lq, 0). I f we d e n o t e

=

QIAP1 Xl E N q

X

A21

A22

x2 E ~ n - q

,

QI B = KP

it 2

,

= [ ~ 1 . ~2] ,

=

P

!

x2

,

96 the description equation in closed-loop

system (3-4.3) is r.s.e, to EF2:

~I = (AII+B1K1)Xl + (AI2+B1K2)x2 + BlV

(3-4.4)

0 = (A2I+B2KI)x I + (A22+B2K2)x2 + B2v. Noticing the results in the last section, closed-loop system (3-4.3) has no infinite poles [f and o. ly i[ p A22+ B2K2 [ ~ O.

(3-4.5)

Under t h i s assumption, the inatrices

Q2 = [I0q - (AI2+BIK2)(A22+B2K2)-I ] In-q lq P2

0

[_ =

(A22+B292)-IA21

(A22+B2K2)-1 ]

would transfer system (3-4.4) into Xl = (~1 + BIKI)X1 + B1v

(3-4.6)

O = £2 + B2KI~I + B2v

where A1 = All - (AI2+BIK2)(A22+B2K2)-IA21 B1 = B! - (AI2+BIK2)(A22+B2K2)-IB2

~1

(3-4.7)

Xl

If (3-4.5) is satisfi.ed,

from ( 3 - 4 . 6 ) we know t h a t the c l o s e d - l o o p pole s t r u c t u r e

is determined u n i q u e l y by the matrix ~'1 ÷ B1K1 for a given K,,.~ ltowever, the matrix K2 (thus matrix K) t h a t s a t i s f i e s ( 3 - 4 . 5 ) is not unique. For d i f f e r e n t K we may obt a i n d i f f e r e n t system ( 3 - 4 . 6 ) . But we can prove t h a t the c o n t r o l l a b i l i t y

indices

of

(AI, gl) are independent of the s e l e c t i o n of K, provided ( 3 - 4 . 5 ) i s s a t i s f i e d . Theorem 3-4.1 (Da~, 1988e).

Assume t h a t system ( 3 - 4 . 1 ) ~s c o n t r o l l a b l e and rankB

C C = m. Let 9~ £ 1.72 x< . . . 4 If m be the c o n t r o l l a b i l i t y i n d i c e s of (A 1, B1). Then they a r e uniquely determined by system ( 3 - 4 . 1 ) and are independent ef the s e l e c t i o n

of K, provided ( 3 - 4 . 5 ) i s f u l f i l J e d . We d e f i n e

lt~ ~< b ~ x< . . . x< Lfm c as the c o n t r o l l a b i l i t y

i n d i c e s for system ( 3 - 4 . 1 )

under P - s t a t e [eedback. I t s computation a l g o r i t h m i s as t o / l o w s .

First,

find Lhe ma-

97 trix K E ~ mxn

such that

deg(IsE - (A+BK)I) = rankE. Then for such a K, controllability to

calculate

determine the standard decomposition;

and,

Einally, compute the

indices for its slow subsystem, which is what we want. Another way is

the controllability

indices

f o r (A 1, B 1) d e t e r m i n e d

by ( 3 - 4 . 7 )

ill

the

decomposition of EF2 for system (3-4.1). The following are our main results on pole s t r u c t u r e placement. Theorem 3 - 4 . 2 ( D a i , 1 9 8 8 e ) . Assume t h a t s y s t e m ( 3 - 4 . 1 ) " ' " ~ ~mc a r e i t s c o n t r o l l a b i l i t y indices under P-state

2, .... that

is controllable. feedback.

V~ (

Let Pi(S),

i = 1,

m, be polynomials satisfying Pi_l l Pi(S). Then there exists a matrix K

{pi(s)}

x'~

such

is the set of invariant polynomials of the slow subsystem coefficient

matrix for the closed-loop system (3-4.3) if and only if m

m

i =j

i=j

c

j = I, 2, ... , m,

m

where ~ i=l

3-4.2.

D( i = n,

o([ = d e g ( p i ( s ) )

Pole Structure

, i = 1, 2 . . . . .

under Proportional

m.

Imd D e r i v ~ l t i v o (P-D) S t l l t e I"cedback

Now we will consider the P-D state feedback control u = K I x - K2~ + v ,

where KI, K 2 ( ~ m x n .

(3-4.8)

System (3-4.1) and (3-4.8) form the closed-loop s y s t e m

(B+BK2)~ = (A+BKI)X + By (3-4.9) y=Cx

.

Sitice we h a v e a s s u m e d t h a t trix

system (3-4.1)

is controllable,

it

is luormalizablo.

Ma-

k 2 exists to satisfy

i E+BK21 ~ O.

(3-4.10)

in this case, system (3-4.9) becomes

(~ + ~K1)x + y = Cx

(3-4.11)

where

= (E+BK2)-IA,

B = (E+BK2)-IB.

The controllability assumption of system (3-4.1) shows that (A, 6) is controllable

98 Let

I1~ ~< I ~

~< . . .

~< I1 me be t h e c o n t r o l l a b i l i t y

d e p e n d e n t o f K2 p r o v i d e d liability

indices

By n o t i c i n g

(3-4.10)

holds

for system (3-4.1)

the closed-loop

eorem according to the linear

(I)ai,

indices

1988e).

for

( ~ , B ) . They a r e i n -

Such i n d i c e s

u n d e r P-D s t a t e

feedback.

system form (3-4.11),

we e a s i l y

are called

to.It.-

have the following

th-

system t h e o r y .

Theorem 3-4.3. Assume that system (3-4.1) B2c ~ ... ~< lame are controllabiJity

is controllable

and rankB = m;

p~ x
/ n-r by the fact M E N ncxn.

a normal observer

In Theorem 4-2.1

we

~'I[ of order n-r for system (4-3.1). Thus min nc = n-r.

Q.E.D. Therefore,

observers designed following the process in Theorem 4-2.1 are of mini-

mal order among observers of the form ~]I" Theorem 4-3.6. Assume that system (4-3.1) is controllable and ~'M is observable. Y]'M is a normal observer for system (4-3.1) and ME]~ ncxn, HEI~ ncxn, J E ~ nxn such that l.

if and only if there exist matrices

L

Bc = AcH + MB.

2. MA - AcL = GC. 3. ME - L - ]]B2(YR~Z)-IP 2 = O, 4.

J]B2(RY'C)-IB2 =

i.

I = FcL + FC + J .

5. H = FcH + JPQB, 6.

H B 2 ( R R t ) - I N = O,

JPQE = O.

o (Ac) c ~-.

Here every

item is defined

4-4.

by T h e o r e m 4 - 3 . 1 .

Disturbance Decoupling State Observers

We note that the systems we have studied so far have no extra disturbance

inputs,

which often exist in practice. When such disturbances exist as input in a system, the asymptotical

stale estimatioa is expecLed not Lo be affected by unknown disturbances

so that we may have a precise estimation of state. This is the thought of disturbance decoupling state observers (DDSO). Consider the system with extra disturbance inputs: E~ = Ax + Bu + Mf

y = Cx + DE where x E R n ,

u E~m,

measure output; fit) E ~ d

(4-4.1) year

are its state, control input (or reference signals), and

E, A, B, C, M, D

are constant matrices of appropriate dimensions;

is extra disturbance input satisfying the following differential equation fit) = mf(t)

(4-4.2)

where L E ~d×d. Since stable disturbance has no effect on state estimation, an asymptotical behavior,

we will hereafter assume that system (4-4.2)

which is

is unstable,

123 i.e.,

~ ( L ) E C +.

De[initLon 4-4.1. Consider system (4-4.1) and the Following dyna,lic system EeOc(t) = Acxc(t)

+ Bcu(t) + Gy(t)

w(t) = FcXc(t ) + Vy(t) + llu(t)

where x c E ~ nc,

w ~n

(4-4.3)

Ec, Ac, Be, G, Fc, F, and I{ are constant matrices of approp-

riate dimensions, such that lim t~

(x(t) - w(t)) = 0

for any initial conditions x(O), xc(O) and f(t) satisfying (4-4.2). I.

When rankE c < nc, system (4-4.3) will be termed a singular DDSO for system (4-

4.1). 2.

Otherwise, rankE c = nc, and E c = Inc is assumed, system (4-4.3) is called

a

normal DDSO. Theorem 4-4.1.

Assume that system (4-4.1) is controllable when f(t) m O. Then the

system Ec~ c = Acx c + B e u

+ Gy (4-4.4)

W

=

XC

is a singular DDSO

for system (4-4.1)

if and only if there exists a matrix T E ~nxn

such that (|). Ec = TE, (2).

A c = TA - GC,

B c = TB.

o(Ec,Ac)CC-.

(3). CD = TM. Proof.

Necessity:

Let (4-4.4) be a DDSO [or system (4-4.1). Then for any initial

condition x(O), xc(O ) we have lim t~

(x(t) - w(t)) = lim (x(t) - xc(t)) = O. t--~

Particularly, when f(O) = 0 (in which case f(t) m 0), system (4-4.4) would be a state observer for system E~ = Ax + Bu,

y = Cx

(4-4.5)

Therefore, as shown in Corollary 4-3.1,

there exists a matrix T E ~nxn

satisfying

(1) and (2). Let e(t) = x(t) - w(t) = x(t) - Xc(t). By left multiplying (4-4.1) by T

and then substracting (4-4.4),

paying attention to

124 assumption (I) and (2), we obtain Ec~(t ) = Ace(t ) + (TM-GD)f(t). Since o(Ec,Ac)CC-,

and l i m t _ ~ e ( t )

= 0 holds for any initial conditions x(O), xc(O)

and any f(t) satisfying (4-4.2), we know limt~oo~(t) = O, lim

(TM-CD)f(t) = 0,

V x(O), xc(O), implying

V f(0).

Hence TM-GD = O, which is (3). Sufficiency is easy to obtain by the inverse process. Q.E.D. This theorem shows the general structure of DDSOs. From

the condition E c = TE,

a

singular system doesn't have full-order DDSOs of the form ~c = AcXe + Bcu + Cy

Now we

Xc •

=

W

will discuss the existence condition

and design methods for one

kind

of

singular (or normal) DDSOs. Theorem 4-4.2. System (4-4.1) has a singular DDSO of the following form E~ c = Axc + Bu + G(y-Cxc) (4-4.6) W

XC

=

if and only if there exists a matrix G E ~ n x r I.

a(E,A-GC) C C-.

such that

2. M = GD.

Proof. This is a special case of Theorem 4-4.1 when T = I . Q.E.D. n Theorem 4-4.2

shows that the existence condition of singular DDSOs is independent

of the equation that f(t) satisfies.

Therefore,

if (4-4.6) [s a DDSO for system (4-

4.1) with respect to disturbance satisfying (4-4.2), disturbance f(t),

and vise versa.

it must be a DDSO for arbitrary

With this property,

(4-4.6) is called a singular

DDSO for any disturbance f(t), which needs not satisfy equation (4-4.2). I n Lhe c a s e o f r a a k D = d ~ r , Then we c a n p r o v e t h e f o l l o w i n g Theorem 4 - 4 . 3 .

System (4-4.1)

w i t h no J o s s o f g e a e r a l i t y ,

h a s a DDSO o f t h e form ( 4 - 4 . 6 )

bance f(t) if and only if

rank

C

= d + n,

V s E ~ +,

s finite.

Prgof. Denote

C=

C2

,

CIE

we a s s u m e D = [ I d / O ].

theorem.

~dxn

C2E ~(r-d)xn

for arbitrary

distur-

125 From D = [Id/O], we know

rank

-- rank D

C2

0

0

Id

V s E ii. ,

Thus

rank

I EA M 1 c

m n + d ~

rank

D

r AMCl1 L

= n,

V s E C +, s finite

c2 (4-4.7)

where ~

denotes if and only if.

Necessity:

Assume that

system (4-4.1) has a singular DDSO (4-4.6) for

arbitrary

disturbance f(t). By Theorem 4-4.2, there exist a matrix G satisfying o(E,A-GC) C C

, M = GD.

Denoting

G = [GI, G2] ~

G1E ~nxd,

G2EN n x ( r - d ) ,

and noticing D = [fd/O], C = [CI/C2] , from M = GD we have M = GI, and

o(E,A-G1C1-G2C2) =

o(E,A-MCI-G2C2) c ~-.

Thus rank(sE-A+MCl+G2C 2) = n,

V s E C+, s finite.

Therefore we have

I

sE-A+MCI+C2C2

= rank

0 = n,

c2

j

c2

j

V s E ~ +, s [iniLe.

Combining it with (4-4.7) we immediately obtain

rank [sE-A -M] C D ' = n + d,

V sE~+,

s finite.

Sufficiency: From (4-4.7) we have sE-A+MCI] rank

= n, L

V sE~+,

s finite,

C2

i.e., (E, A-MCI,C 2) is detectable. Therefore, there exists C 2 E ~ n x ( r - d ) o(E,A-MCI-G2C2) C C-. The following relationship holds a(E,A-GC) C ¢-,

M = GD,

such that

126 G = [M, G 2 ] .

by simply setting

According to

Theorem 4-4.2 we know

system

(4-4.6)

determined by such a C is a DDSO for system (4-4.1). Q.E.D. Let us define the disturbance transition zeros of system (4-4.1) to be the complex scalars satisfying rank

C

D

< n + min(r, d).

Since d(r, the necessary and sufficient condition for the existence of singular DDSOs for

arl)itri~ry

disturbance

i8

that

transition

disturbance

all

zeros

are

in the

open

left complex plane s no unstable transition zeros exist. This is the physical sense of what

is stated

(such

as

in Theorem 4-4.3.

D = 0 and M ~ 0),

Theorem 4-4.3 a r e

It is worth noticing that if rankD = d

the DDSO (4-4.6) muy not exist although

is

false

conditions

in

sutisfied.

Example 4-4.1. Consider the system with disturbance inputs:

{°°I [°°I [°I [il 0 0

0 0

1 0

~ =

y = {l

o

0 0

1 0

0 1

x +

1 0

0 1

u +

f(t)

(4-4.8)

Olx + r(t)

in which

[oo]

E =

0

0

I

0

0

0

C = [I S£nce

0

fool {oo] iol

A =

,

0],

0

I

0

0

0

1

B =

I

2

,

1

(4-4.9)

D = I

= rank

C

M =

0

sEAMJ 1100 I os2

rank

I

,

D

0 1

0-I 0 0

= 4 = n+d

1 1

V sE~+,

s finite,

' (4-4. I0)

by

Theorem 4-4.3 we know that system (4-4.8) has a singular DDSO of the form (4-4.6)

for arbitrary disturbance f(t). Note

that

D = I.

It wiii be

G = M

provided (4-4.6) exists.

Thus

a DDSO

for

arbitrary disturbance is readily given by Theorem 4-4.2:

[°l°I Ii°°l [°°1 [°1 0 0

W

0 0

1 0

=

X C •

~c =

I 0

0 1

xe +

i 0

0 1

Verified from (4-4.10) the transition zero set is

For the same reason as pointed out before, realize. 4.1).

u +

2 1

y

(4-4.11)

(-I, -I} ~ C-, which is stable.

singular DDSOs are often difficult

to

To overcome this difficulty we now consider the normal DDSOs for system (4-

127

Theorem 4 - 4 . 4 .

Consider system (4-4.1).

Assume Lhat Lhere e x i s L s u m a t r i x (;EH~rlxr

satisfying 1.

o(E,A-CC) C C - .

2. d e g ( I s E - ( A - G C ) l) = rankE.

3. M = CO. Then this system always has a reduced-order normai DDSO of the following form: ~c = Acxc + Bcu + GeY (4-4.12) w = F c x c + Fy + l[u where xc E ~q. Under Lhe a s s u m p t i o n ,

Proor.

from Theorem 4 - 4 . 2 we know t h a t

fc~r t h i s

maLrix G E

~nxr, the system

E~ c = A~c + Bu + C(y-C~c) (4-4.13) = xc

w

is a DDSO for system (4-4.1). appear

in

condition 2 shows

Moreover,

the state response of (4-4.13).

that no

impulse

two nonsingular matrices

Thus,

terms Q and P

exist, such that system (4-4.13) is r.s.e, to ~I = AclXl + Blu + GIY (4-4.14)

0 = x 2 + B2u + G2Y p[ x I w

Iq

=

+

] x2 in_q

=

where

QEP = d t a g ( I q , QB =

0),

Q(A-GC)P = d i a g ( A c l '

QG = B2

,

Xc = P G2

In-q )'

x 1 E R q,

,

x2

x2 E ~ n - q

,

Apparently, if we choose xc = x I, A c = Acl, B c = B I, G c = G I, and

F

=P

0

,

I

n-q

,

I

n-q

G2'

system (4-4.14) is (4-4.12). Q.E.D.

Corollary

4-4.I.

for the existence

D e L e c t a b L i i t y and i m p u l a e o b s e r v a b i l i L y

are necessary condiCions

o f normal DDSOs f o r s y s t e m ( 4 - 4 . 1 ) .

Therefore, conditions in Tbeorem 4-4.4 are somewhat strong. Example 4-4.2. It has been proven in Example 4-4.1 that the system

128

[oi!] [~o0],[oo] [!I 0 0

0 0

~c =

-2 -1

1 0

0 1

+

i 0

u +

1 (4-4.15)

w = x

e

is a DDSO for system (4-4.8).

Note that there is no impulse portion in state respon-

se of this DDSO. We can choose the nonslngular matrices

Q=

0 0

1 0

P=

1 0

,

0 I

0 0

such that

W.e

[io 0] 0

=

II

o

~--~-,~-8 Let the coordinate

10101 1-2 _ ....

o

transformation

is r.s.e,

=

,

e-tXc = [×2x 1 J ' System ( 4 - 4 . 1 5 )

q(^-gc)P

o It_ -

0,

1

be

x 2 E ~ 2,

x 2 E ~ 1.

to

(4-4.16)

0 = x 2 + [0 - 1 ] u - y w = P[Xl/X2] which may be further rewritten as

~=[o 2],I ~ + [ o IIo+[~] ' w=

I°iI [i-il I-'l 1 0

Xl+

u+

0 0

y

which is just a normal DDSO in the form of (4-4.12).

4-5.

State Observer with Disturbmlce Estimation

Previously given DDSOs disturbance. may

As a r e s u l t ,

be r e q u i r e d

give an asymptotical

state estimation independent

no i n f o r m a t i o n o f d i s t u r b a n c e

in e n g i n e e r i n g

design.

For t h i s

of extra

i s g i v e n in s u c h DDSOs, which

reason,

we now c o n s i d e r

observer that estimates both state and disturbance at the same time.

Such

the slate observers

are called stale observers w~th disturbance estimation. They are one kind of DDSOs. Consider system (4-4.1). Let

129

F" =

0

Then z E ~n+d.

~

Id

['"1

X =

0

L

,

g =

[°] 0

,

~ =

[C

D],

z =

f

By combining ( 4 - 4 . 1 ) w i t h ( 4 - 4 . 2 ) , we o b t a i n Lhe f o l l o w i n g system

+

Xz

=

~u

(4-5.1)

y = ~z.

According t o

the r e s u l t s

of Theorems 4 - 1 . 1 and 4 - 2 . 2 ,

when the augmented

system

(4-5.1) is detectable, it has a singular state observer:

E~'c = (A-GC)zc + ~u + ~y (4-5.2) W

where

:

ZC

Zc E ~n+d.

Wheu system ( 4 - 5 . l )

i s d e L e c t u b l e and impulse o b s e r v a b l e ,

it

has

a normal state observer of the following form Zc : Aczc + Bcu + Gy

(4-5.3) w = l,'cZc + Fy + flu where z c E R nc,

and n c ~ rankE + d,

Both (4-5.2) and (4-5.3) are slate observers with disturhance esLimation for

sys-

tem (4-4.1). The estimation of disturbance is given by :

lO IdlW A

such that limt_oo(f

- f) = O, V

f(O), x(O), zc(O).

Meanwhile, state estimation is given by ~ = [I n

O]w such that lim

t ~

(~

-

x) : o,

v x(O), zc(O), f(o). Theorem 4 - 5 . 1 . C o n s i d e r system ( 4 - 4 . 1 ) . 1. I£ (E,A,B,C) i s d e t e c t a b l e , system ( 4 - 4 . 1 ) has a s i . g u l a r 2. I f (E,A,B,C) i s d e t e c t a b l e and impulse o b s e r v a b l e ,

DDSO ( 4 - 5 . 2 ) .

system ( 4 - 4 . 1 ) has a normal

DDSO of t h e form ( 4 - 5 . 3 ) . It

must be p o i n t e d out t h a t t h e DDSOs g i v e n h e r e esL[mate d i s t u r b a n c e a t Lhe c o s t

of h i g h e r o r d e r . It is interesting

to view t h e c l o s e r e l a t i o n s h i p between DDSOs, g i v e n in the l a s t

section and the state observer with disturbance estimation here. Assume that rankD = d and the condition in Theorem 4-4.3 rank[ sE-A C is satisfied.

-MD ] = n + d,

V sE~+,

s finite,

Then system (4-4.1) has a singular DDSO (4-4.6) for arbitrary

bance [(t). Since

distur-

130

rank

_

= rank

islAM1 [sEAM] O C

sl-L D

) rank

~ n + d,

C

D

Y s ( ¢+, s finite. System (4-5.1) is detectable. a DDSO

As a direct result of Theorem 4-5.1,

with state estimation,

tained. Sometimes, (Wang aim Dai,

from

whose output, disturbance

system (4-4oi) has

estimation may he

DDSO (4-5.2) may exist even though rankD = d is

not

ob-

satisfied

1987b), hut (4-4.6) may not.

Similarly,

when rankD = d and conditions

in Theorem 4-4.4

are

satisfied,

Theorem 4-4.4 we know system (4-4.1) has a DDSO (4-4.12) for arbitrary

from

disturbance.

Also note that

rank

I

~

= rank

[sEAM] 0 C

sl-L D

V s(~+, System (4-5.1) is detectable.

i A+c o]

= rank

0 C

4.1

we

Hence if we choose G = [G/O] it will be = rankE,

system (4-5.1) is impulse observable.

can conclude

= n + d,

s finite.

deg(IsE-(A-CC) i) = deg(Isl-Lll sE-(A-GC)I) or in other words,

sI-L D

As a result,

from Theorem 4-

that system (4-4.1) has a normal DDSO with disturbance

estima-

tion. These arguments show an important normal)DDSO

principle.

for arbitrary disturbance,

the disturbance

If system (4-4.1) has a singular

(or

we may choose an appropriate DDSO from which

is estimated.

Example 4-5.1. Consider the system E~ : Ax + Mf (4-5.4)

= Lf

y = Cx i n which

E =

[i 0] All l] M=[° '] 0

0

'

O

1

,

1

2

,

L =

1

0

(4-5.5)

c=[10l. For

this

s y s t e m D = O, M ~ O, Theorem 4 - 4 . 4

[E E : : (E,A,B,C) rver

is

(4-5.3)

0 [C

D] :

detecLable with

[]oL a p p l i c a b l e .

[iooo] I

o] 1

is

= [1

o o o

o

0 0

0 0

I 0

0 I

0

0

0].

as w e l l

disturbance

~=

,

as i m p u l s e

estimation.

L

observable.

since

[i_,o,]

^ .

0

However,

=

o

, I 2

0 0

0 0

0 -1 i 0

We may c o n s L r u c t

,

a s/aLe

ohse-

131 Therefore, the conditions given by Theorem 4-5.1 are much looser than before. But, on the other hand,

the DDSO given

by this theorem

is not applicable to arbitrary

disturbance since the model of disturbance is needed. Finally, we would like to further explain. System (4-5.1) is impulse observable i f and only if s y s t e m (4-4.1)

Theorem 4-5.2.

is impulse observable when f(t) ---O. Proof. The conclusion is easy to obtain by noticing

rank

= rank

0

~ LO

0

E

0

0 0

0 C

1 D

= 2d + r a n k

0

c

.

Q.E.D. T h i s shows t h a t

impulse observability

~ a L i s f y a normal d i f f e r e n t i a l

[ s £ndependenL o f d i s t u r b a n c e

inputs,

which

equation.

4-6.


This chapter is based on Wang and Dai (1986b,

1987a, 1987b). The observer problem

has been considered by several researchers. In EL-Tohami et al. (1983) another method based on observers

pseudo-inverse was used to studied this problem. for

continuous-time

Carroll (1987), and Wang (1984).

The other works

singular systems are Saidahmed (1985),

on state

Shafai

and

CHAPTER 5 DYNAMIC OOMPENSATION FOR SINGULAR SYSTF~S

As

pointed out in Chapter 3,

working system.

stability is a basic property possessed by

a well-

Therefore, it should be the first thing considered in control system

design. To achieve this, dynamic compensation is the common means usually used, which constructs a dynamic compensator using the measure output of original system as

its

input and its output acting on the original system as a feedback control input,

such

that

the whole closed-loop system is stable.

to realize the state feedback control.

This method also provides an

approach

Dynamic compensation has many other

purposes

such as t r a c k i n g and model m a t c h i n g . Here only t h e compensation f o r s t a b i l i t y

purpose

is studied, as space is lim£ted. Dynamic compensation is one form of dynamic feedback control of outputs.

5-I.

Singular Dynamic Compensators

Consider the linear singular system E~(t) = Ax(t) + Bu(t) y(L) = Cx(t)

(~-I.1)

where x(t)E ~n, u(t) E ~ m , y ( t ) E ~ r a r e E, A E ~ n x n

B E ~ nxm,

its state, control input, and measure output;

C E ~ rxn are constant maLr[ces. Without loss of generaJity, we

assume thaL system (5-1.1) ks regular and rankE = q < n. The

so-called

singular

dynamic compensator (compensator for

simplicity)

is

a

dynamic system in tlle following form Ec~c(t ) = Acxc(t ) + Bey(t) u(t) = Fcxc(t ) + Fy(t) where Xc(t )E R nc,

Ec, Ac, Be, Fc, F

(5-1.2) are constant matrices of appropriate dintensions

and (5-1.2) is regular, such that the closed-loop system Ex(t) = (A+BFC)x(t) + BFcxc(t ) Ec~c(t ) = Acxc(t) + BeY(t) is stable, i.e.,

(5-1.3)

133

(5-1.4) 0

Ec

, L BeC

Ac J

Figure 5-1.1 is the diagram of compensators.

+

v(L)

~i ÷

reference

E~=Ax+Bu

J

U(L)

I

I

y(L)

y=CX system (5-1.1)

E(:~c=Arx(:+Bcy U=FcXc+Fy compensator ( 5 - 1 . 2 )

Figure 5-1.1

Theorem 5-1.1.

Assume that system (5-1.1) is stabilizable and detectable. Then it

has a singular compensator of the form E~ c = (A-GC)x c + Bu + Gy u

:

(5-{.5) Kx c

where G ( ~ nxr and K E ~ mxn Proof. sfies

satisfy

o ( E , A - G C ) c C - and

o(E,A+BK) C~-, respectively.

By assuming the stabilizability, there exists a matrix K El~ mxn

o(E,A+BK)CC-. u

=

Tile state feedback determined

by

that sati-

K:

Kx

(5-1.6)

stabilizes the closed-loop system Ex = (A+BK)x. Note that slate feedl)ack centre[ cannot

I)e

realized (l[rectLy in general case.

We

now consider its realization via state observers. It

is assumed that system (5-1.1) is detectable,

such that

a matrix G E ~ nxr can be chosen

o(E,A-{;C) CC-. Thus

Ex:c = Axc + Bu + G(y-Cxc) W = XC iS a singular state

observer

for

system

(5-1.1) such that

limt_oo

=Kxc=Kw in lieu of (5-1.6), we obtain the closed-loop system E~ = Ax + BKx C Ec~ c = (A-GC)x c + BKx c + Gy

(w-x) = O. Using

u

134

whose pole set may be obtained by direct computation: BK A_OC+BK] ) =

a( foe ~] A [, GC Therefore,

a(E,A+BK) U o(E,A-GC) C C-.

the system determined by (5-1.5) is a singular compensator for system (5-

I.I). Q.E.D. As mentioned in Chapter 4, one aim of designing observers is to realize state feedback. From this theorem we see further that the dynamic compensator is merely a combination of state feedback and observer. The separation principle holds in the closed-loop pole set,

i.e., the pole set of

the closed-loop system is the union of the pole sets under state feedback and observer that can be designed separately,

which makes this method convenient for application.

Combining this discussion with the results on pole placement, we easily obtain tlm following corollary. Corollary 5-1.1. (observable), poles

If system (5-1.1) is R-controllable (controllable),

R-observable

a dynamic compensator (5-1.5) may be designed such that the 2n I finite

of its closed-loop system can be arbitrarily assigned to any symmetric set

the complex plane,

especially in the open left hall plane.

in

Here n I is the order

of

its slow subsystem. It has

been proven in Theorem 5-1.1

that detectability and sLabilizabttity

are

sufficient conditions for the existence of dynamic compensators. We can prove that it they are also necessary. Theorem 5-1.2.

System (5-1.1)

has a dynamic compensator

if and only if

it

[s

stabilizable and detectable. t)roof.

The

sufficiency

i s given l)y Theorem 5 - 1 . I .

We o n l y ueed

to

prove

its

necessity. Let stable,

(5-1.2) i.e.,

be a compensator f o r system ( 5 - I . 1 ) .

(5-1.4)

hohls,

[sE-(A+BFC)

c l o s e d - l o o p system

= n + n c,

V sE~+,

s finite.

(5-1.7)

sEc-A c

llence

rank[sE-(A+BFC)

-BFc] ) n + nc,

NoLicing t h a t the matrix [sE-(A+BFC) rank[sE-(A+BFC) Thus) r a n k [ s E - A ,

B] = n,

is

-BFc ]

rank -BcC

Then i t s

or i n a n o t h e r form

-+

g s 6 C , s finite.

-BFc] is of n rows, we know

-BFcl = rank[sE-h Y s E C +, s f l n [ t e ,

[i

B] -FC

o] = n,

-F c

so system ( 5 - 1 . t )

V s E ~+, s f i n i t e . [s s t u l ) i l i z a b l e .

135 In a similar way we can prove that it is also detectable. Q.E.D. Theorem 5-1.2 is the result for the normal system when E = I. It is worth pointing out that the dynamic compensator itself is not necessarily stable although the

over-

all closed-loop system (5-1.3) is stable. Example 5 - I . I .

Consider the circuit network (2-2.5)-(2-2.6):

E~ = Ax + Bu (5-1.8)

y = Cx with coefficient matrices 0

0

C2 0

0

Cl 0

0

E=

0 0

0

O-L

0

0

c=[o

]

0

-I

1

0

0

o

0].

ooiI [o 0

l

0

1

0

0

0

0

R

0

B=

A=

0

,

-I

It has been proven in Chapter 2 that it is stabilizable and detectable. Therefore, it has a compensator (5-1.5). We ]lave verified in Example 3-].4 that the matrix K K = [I-3RCI,

3RCI-LRCIC2'

RC 1 -~2 - 3LRCI'

O]

satisfies a(E,A+BK) =

{-i, -I, -lJ C C-.

For the sake of simplicity in discussion, Then K = [-2

2

-2

0]. Let G = [-9

0

-9

it is assumed that CI = C2s=3 R = L = I. O]~. We have [sE-(A-CC)[

+ s 2 + los

+ I, which may be verified to be a stable polynomial by Eouth criterion (Fortmann and Hitz, ]977). Thus

a(E,A-CC)CC-.

For such matrices K and G, by Theorem 5-I.], system (5-1.8) has a compeasator:

[oo!} [iol [o} 0

I

0

0 -I

0

0

u = [-2

0

~c

=

0 2

-2

O]x c.

-

0

I

0

0

0

0

-2

2

1

xc +

-9

0

Y

(5-1.9)

136

5-2.

Full-Order Normal Compensators

Because of the difficulty in realizing singular systems, although singular compensators exist theoretically, they are often difficult to physically realize. To cancel this

disadvantage,

we use naturally normal compensators,

which imply

compensators

described by the following normal system: ½c = Acxc + BcY u = Fcx c + Fy where XcE ~nc,

(5-2.1)

Ac, Bc, Fc, and F are constant matrices of appropriate dimensions.

Compensation function of normal compensators may be described by Figure 5-2.1.

+,.

v(t)

'+

,[ E~=Ax+Bu y=Cx

u(t)

re[ereuce

]

y(t)

system ' ----]

I

1 I I .._1

..........................................

compensator

Figure 5-2.1.

Normat Dynamic Compensator

D e f i n i t i o n 5 - 2 . 1 . Let ( 5 - 2 . 1 ) be a dynamic compensator f o r system ( 5 - 1 . 1 ) . C o r r e s ponding to the two c a s e s ,

n c = n,

n c < n,

system ( 5 - 2 . 1 ) i s c a l l e d t h e

full-order

or r e d u c e d - o r d e r , r e s p e c t i v e l y , normal c o m p e n s a t o r s f o r system ( 5 - 1 . 1 ) . Only will

furl-order

c o m p e n s a t o r s a r e d i s c u s s e d in t h i s s e c t i o n ,

be s t u d i e d in t h e next s e c t i o n .

1 . 1 ) , we c o n s i d e r t h e P-D s t a t e u = KlX - K2~

reduced-order

To d e s i g n a normal compensator f o r system

ones (5-

feedback (5-2.2)

137 where K 1, K 2 E ~ m X n a r e

constant matrices.

Theorem 5-2,1. Assume that system (5-1.1) is normalizable. If it is detectable and stabilizable, a full-order compensator exists. Proof.

The proof is constructive. Under the assumption ~mxn may he chosen t o s a t i s f y

matrix K2E

of

normalizability,

I E+BK2 I ¢ O. By Lemma 3 - 3 . 2 ,

(5-2.3)

tile system (E+BK2,A,B,C) i s a l s o s t a b i l i z a b [ e ,

implying

Lhat a

Kl E

exists, such that

~mxn

a(E+BK2,A+BKI) ~ C-. The

a

matrices K l and K 2

(5-2.4)

satisfying

(5-2.3)-(5-2.4) determine

feedback (5-2.2)

and

thus determine the closed-loop system (E+BK2)~ = (A+BKI)X, which is stable by (5-2.4).

(5-2.5)

Since x and ~ are generally inaccessible,

feedback

(5-

2.2) cannot be realized directly. Now we consider its realization via observers. Note

the a s s u m p t i o n of d e t e c t a b i l i t y ,

a matrix G 6Nnxr

may be chosen t h a t s a t i s f i e s

a(E,A-GC) < ¢-.

(5-2.6)

Therefore, according to Theorem 4-1.1, E~ c = (A-GC)x c + Bu + Gy

(5-2.7)

is a singular observer for system (5-1.1). Let the control be u = KIX c - K2~ c instead of (5-2.2).

(5-2.8)

Substituting it into (5-1.1) and (5-2.7),

we obtain the closed-

loop system

0

E+BK 2

Xc

GC A-GC+BK 1

xc

with finite pole set

E+BK2 which

,

CC

A-CC+BKlJ

shows that the closed-loop system (5-2.9) is stable with a pole set of the

union of these of feedback and observer. Combining (5-2.7) and (5-2.8), we see that if A c = (E+BK2)-I(A-GC+BKI) B c = (E+BK2)-Ic

138 F c = K 1 - K2(E+BK2)-I(A-GC+BKI) F

= - K2(E+BK2)-IG

it will be ~c = Acxc + B c Y

(5-2.m) u = Fox c + Fy.

Q.E.D. Apparently,

whenever

usual compensator Theorem

5-2.1

Corollary

for

is the following 5-2.1.

in a d d i t i o n ,

E = In,

Assume

System (5-2.10)

K 2 may be taken as K 2 = O.

the normal

systems.

A direct result

of Corollary

is

the

3-3.2

and

and detectable,

if,

corollary.

that system (5-1.1)

is stabilizable

i t is normalizable and impulse o b s e r v a b l e , a compensator can be designed

such t h a t the c l o s e d - l o o p system has n+rankE s t a h l e l illiLe p o l e s . Thus no impulse

terms exist in the state response

Example 5 - 2 . 1 .

Consider the c i r c u i t

t a b l e , and aormalizable. Choosing

K2

= [0 =

neLwork ( 5 - 1 . 8 )

C 1 = C 2 = R = L = I is assumed 0

A

a(E+BK2,A+BKI)

0 -i],

we have [E+BK2[

-I,

is stable -I}

~

Similarly, o(E,A-GC)

delec-

lor simplicity. And for any matrix

KI,

A

0 0 1-1

1 0

0 0

I

0

I

0

=

=

,

-

.

shows

- (A+BKI) { = lsl - (A+BKI)I

= s 4 + (a4_l)s 3 + als2 + (a3+a4-1)s

-L,

which i s s t a b [ l i z a h l e ,

= -i ~ O.

For any K I = [al, a2, a3, a4] , direct computation

which

system.

a(A+BKI) , [n which

= (E+BK2)-I A =

I s(E+BK2)

of the closed-loop

+ a]+a2-1 ,

if we let a I = 6, a 2 = -4, a 3 = O, and a 4 = 5.

o(E+BK2,A+BKL)

={-l,

~;-. when G is taken to be G = [-9

0

-9

O] ~.

C C-.

Let KI, K2, and G be defined

as previously,

0 0

Ac = (E+BK2)-I(A_GC+BK) =

B c = (E+BK2)-Io

= [-9

F c = K 1 - K2A c = [I

0 0

9 0

9 0

1 -10 -5 4 O] ~ I]

we have

0 1

0 0

1 1 0

0 -4

from Example

5-1.1 we know

139

F = - K2(E+BK2)-IG : - K2B c = 0.

By Theorem 5-2.1, we obtain the full-order compensator for system (5-1.8):

[0 0 I

~c

0

0 =

1 -I0

-5 u = [l

0

I

0

0

0

4

0 -4

O

l]x c-

Xc +

9

Y

0

(5-2.11)

The closed-loop system formed hy (5-2.11) and ( 5 - l . 8 ) has seven f i n i t e poles, thus no impulse terms exist in its state response.

Generally speaking, Singular systems are often not normaliznhte, so the design method in Theorem 5-2.1 is not a p p l i c a h l e . Now we consider designing compensators for general singular systems. To begin, we consider the normalizability decomposition (25.9) for system (5-I.I).

First,

nonsingular maLrices Q and

l' such that

for any regular system (5-I.I),

QEP = diag(Inl, N),

there

exist

two

(Lemma 1-2.2):

QAP = diag(Al, In2)

QB = [BI/B2],

CP = [C 1 C21,

where n 1 + n 2 = n, N is n i l p o t e n t , Taking c o n t r o l l a b i l i t y decomposition of (N, B2), from linear system theory there e x i s t s a nonsingular matrix T such that TNT -I = [ Nil O

where (NIl , B21 )

NI2 ] N22

,

is controllable,

Nil E • n3xn3,

N22( ~ n4xn4 are n i t p o t e . t ,

n 3 + n4

= n2. Let PI = F'diag(lnl, T- l )

QI = d[ag(In I' T)Q,

p;1 x = [ ~1~2 ], ]

~1 E ~nl+n3'

x2 6 ~ n4

Then system (5-1.1) is r.s.e, to

Ell~l + EI2~I = AII~I + B1 u (5-2.12)

N22~2 = ~2 y = CI~I + C2~2 where 211 = diag(In I' NIl)'

A1 = diag(Al' In3)'

El2 = [O/NI2]' (5-2.13)

BI = [BI/B21]'

CI = [CI' C21]'

C2T-1 = [C21' C2 ]"

140 According

to our decomposition,

(N22 , B21 ) is controllable.

Thus (EII,AI,BI,CI) is

normalizable. System (5-2.12) is a normalizability decomposition. Furthermore, from rank[sE-A, B] = rank[ sEII-AI O

and the i n v e r t l b l l i t y

BI sEl2 O

] = n,

V s E ~+, s finite

sN22-In4

of sN22-In4 we know t h a t ( E l l , A 1 , B I , C 1 ) i s s t a b i I i z a b l e

if sys-

tem ( 5 - 1 . 1 ) i s s t a b i l i z a b l e .

(EII,AI,BI,C1)

Similarly

i s d e t e c t a b l e i f system ( 5 - 1 . 1 ) i s d e t e c t a b l e .

T h e r e f o r e , i f system ( 5 - 1 . 1 )

Ell ~ A1 z =

y :

+

is stabilizable

and d e t e c t a b l e ,

the subsystem

B1u

(5-2.14)

~1 ~

is normalizable, stabilizable, and detectable.

Theorem 5-2.1 shows that there exists

a normal compensator ~c = AcXc + BcY u = F c x c + F~ where xcE ~n-n4,

0

(5-2.15)

such that the closed-loop system

In_n4

is stable. Using the m a t r i c e s defined by ( 5 - 2 . 1 5 ) ,

we c o n s t r u c t the f o l l o w i n g system

Xc = AcXc +BcY u = F c X c + Fy. By

direct computation,

I.I)

and

(5-2.17) we may verify that (5-2.17) is a compensator for system

the closed-loop systems (5-1.1)-(5-2.17) and (5-2.16) have the

same

(5slow

subsystem structure (Dai and Wang, 1987b). Summing up this dLscussion, we thus prove the following theorem. Theorem 5-2.2. designing

Designing a normal compensator for system (5-1.i) is equivalent to

its normalizable subsystem (5-2.14) Jn the sense that their closed-loop

systems have the same slow subsystem. A direct result of Theorems 5-2.1 and 5-2.2 is the following theorem. Theorem 5-2.3.

If system (5-1.1) is stabilizable and detectable,

it has a normal

compensator of order n-n4: Xc = Ac×c + BcY u = F c x c + Fy

(5-2.1s)

141 In

this theorem the conditions are the same as in the normal system case.

Combi-

ning it with the necessity in Theorem 5-1.2 yields the following theorem. Theorem 5-2.4.

System (5-1.1) has a normal compensator (5-2.1) if and only if

it

is both stabilizable and detectable. This

t h e o r e m r e v e a l s an i n t e r e s t i n g

phenomenon.

The c o n d i t i o n s f o r tile e x i s t e n c e

of a singular compensator are the same as those for a normal one; a p p r o p r i a t e in system design for i t s

c o n v e n i e n c e in r e a l i z a t i o n .

the latter is more This is a f o r t u n a t e

thing for system d e s i g n .

The design algorithm of normal compensators may be summarized as follows. I. Take normalizability decomposition

for a given system to obtain its normalizab-

ie subsystem (5-2.14) if the system is not normalizable. 2. Under the stabilizability and detectability assumption, choose matrices KI, K2, and G that satisfy I EII+BIK 2 1 ~ O,

a(EII,AI-GCI) ~ ¢-,

o(EII+BIK2,AI+BIKI) c C . 3. Compute the matrices a c = (EII+BIK2)-I(AI-GCI+BIK I) B c = (EII+glK2I-IG F c = K 1 - K2A c F

=

-

K2(EII+BIK2)-IG.

Then dynamic system (5-2.18) is a normal compensator for system (5-1.1).

From Corollary 5-2.1, and Theorems 5-2.2 and 5-2.3,

we can further prove (Do[ and

Wang, 1987b) the following corollary. Corollary 5-2.2.

Assume that system (5-1.1) is stabilizable and detectable.

is impulse controllable and impu]se observable,

If it

Jt has a normal compensator (5-2.[7)

such that its closed-loop system has nc+rankE stable finite poles. Tilerefore, the closed-loop system has no infinite poles and no impulse terms exist in its state response.

142 5-3.

R e d u c e d - O r d e r Normal Compensators

In control system design,

the higher the order of a compensator,

the more compo-

nents needed for its realization, increasing investment and raising the cost, and the lower

the reliability of the control system,

decreasing its efficiency.

Therefore,

the principle in system design should be lowering the order of compensators as far as possible when allowed. Now

we will consider the existence and design methods of reduced-order

compensa-

tors. Theorem 5-3.|. Assume that the singu]ar system E~ = Ax + Bu y = Cx is stabilizabie

(5-3.1)

and detectable.

1£ i t

is dual normai[zable and raakC = r,

it has a

normal compensator of order n-r of the following form Xc = Acxc + BeY (5-3.2) u = F c x c + Fy where x c6 • nc,

n c = n-r.

Proof. According to tile stabllizability assuml, Lion, there exists a matrix K £ ~mxn satisfying

a(E,A+BK) C C-. Subject know t h a t

(5-3.3)

t o t h e a s s u m p t i o n rankC = r , there exist

nonsingular

11° ]

E21

In_ r

QAP = ,

from t h e p r o o f p r o c e s s o f Theorem 4 - 2 . 1

matrices

[

P and Q

All

A12

A21

A22

we

such that

1

QB =

,

[ 11 B2

,

(5-3.4) CP = [ T r ' O] and that system (5-3.1) has a reduced-order normal state observer of order n-r:

~c = Acxc + BcU + GY w = Pcxc + ~y such that l l m t ~ ~ (w-x) = O, Ae = A22 - G2A12,

V x(O), xc(O ) in which Bc = B2 - G2BI

= A21 - G2AII - (A22-G2A12)(E21-G2Ell)

Fc = P [ O / I n - r ] '

F = P[]r/-E21+C2Ell]

(5-3.5)

143

and G2E ~(n-r)xr

is any matrix satisfying (5-3.6)

o(A22-G2A12) ~ ¢-. Consider the control law u = Kw = KFcx c + KFy where K is defined in (5-3.3). We can prove that the following system

~c = Acxc + Bcu + GY

(5-3.7)

u = Kw = KFcx c + KFy is just a normal compensator for system (5-3.1). [n fact, substltutlng (5-3.7) into (5-3.f), we obtain the closed-loop system

0

In_ r

Xc

BcKFC+~C

Ac+BcKFcJ[ Xc

"

By denoting KP = [KI, K2] , paying attention to (5-3.3)-(5-3.6), and making elementary transformation, we know that the closed-loop pole set is

.[E° ][ABK. 0

In_ r

, BcKFC+~C

QEP 0 0

]

In_ r

Ac+BcKF c

[QAP+QBKFCP QBKFc ] ) ,

BcKFCP+GCP

Ac+BcKF c

[E0 0] IAa1iKl 12E2G2 AI2B12] -

a(

o(

E21

I

0

0

0

I

,

A21+B2KI-B2K2(E21-G2EII)

A22

G+BcKI-BcK2(E21-G2EII )

0

Ell 0

0] [

E21

0

J 0

I E21-G2EII

AII+BIK I

I

A21+B2K 1 ,

_

A21-G2AII+BcK 1

B2K 2

AI2

BIK 2

]

A22

B2K 2

)

0

)

Ac+BcK 2

Ac+BcK 2

= o(E,A+BK)U o(A22-G2AI2 ) ~ ¢ . Thus, the closed-loop system is stable, with the separation principle of pole sets. Setting Ac = Ac + BcKFc,

Bc = G + BcKF,

F c = KFc,

F

= KF,

system (5-3.7) is a normal compensator of order n-r. Q.E.D.

The process also proves tile following corollary. Corollary 5-3.1. If system (5-3.1) is R-controllable and observable, it always has

144 a normal compensator (5-3.2) of order n-rankC whose pole set may be arbitrarily assigned.

Example 5-3.1. From Example 4-2.1, the singular system

[oo I [oo] [oo1 0

0

1

0 0 0

~ =

y = [I

0

0

1

0

0 0 1

x +

1

0

u

0 1

(5-3.9)

O]x

has a normal observer of order n-r = 2 of the form

[il I°°l

w=

y +

1 0

We have

2

0

xc

1

such that l[mt~oo(x-w ) = O,

1

(5-3.10)

V x(O), xc(O). Let the feedback gain matrix be

.

a(E,A+BK) = ( - 1 , -1 | C C - .

By Theorem 5 - 3 . 1

we may c o n s t r u c t

its

normal corn-

pensator: ~c = [ - 2 -I

l

-1 -I .Ixc

u°[O oo ] X c

+

+[_o]y [o]y

whose closed-loop system is

1o000][1

00 01 0I 00 O0 ][J 0 0 0

0 0 0

0 0 0

0 1 0

0 0 1

with the finite pole set

~c

0

1

0

0

0

1

0

1

2

0

0 -1

0 0

0 -2 -1 0 -1 -1

x

xc

q {-I, -I, - + 3. ~l, -~1 - ~ i } , which is the

union

of poles o f

state feedback and normal observer. Theorem 5-3.2. Assume that system (5-3.1) is stabilizable and detectable. If it is impulse observable and impulse controllable,

a normal compensator of order q = rankE

may be designed: Xc = AcXc + BeY u = F c x c + Fy

where x c E ~nc,

(5-3.11)

nc = rankE, such that its closed-loop system has nc+ rankE stable fi-

nite poles. Thus the closed-loop system has no infinite poles.

145 Proof. By assumption, system (5-3.1) is impulse controllable and impulse observable, From Theorem 3-5.1, we know that a matrix KIE ~mxr may be chosen that satisfies deg(IsE - (A+BKIC)I) = rankE.

(5-3.12)

Consider the feedback control u = K1Y + v

(5-3.13)

where v is the new control input,

and, when applied to (5-3.1), the closed-loop sys-

tem is E~ = (A+BKIC)X + Bv y = Cx.

(5-3.14)

Accounting for equation (5-3.12), via suitubJe selection of transformation matrh~es Q and P system (5-3.14) is r.s.e, to Xl = AlXl + Blv 0 = x2 + B2v

(5-3.]5)

y = CIX 1 + C2x 2 where XlE ~q, x2E ~ n - q

and

QEP = diag(lq, 0),

Q(A+BKIC)P = diag(Al, In_q),

QB =

CP ~ [C1, C2] , B2

,

B1E ~qxm Furthermore,

B2 E ~(n-q)xm

C1E ~rxq,

C2 E ~ r x ( n - q )

the stabilizability and detectability

(AI,BI,C|) is stabilizable and detectable. be chosen such that

a(AI+BIK2) C~-, and

Thus,

of system (5-3.1)

shows that

two matrices K2E~mXq, G2(~ qxr may

a(AI-G2CI) CC-.

Now we will construct

the

observer for substate x I in (5-3.15):

~c = (Al-G2Cl)Xc + (BI+G2C2B2)v + G2Y

(5-3.16)

w=x c where XcE ~q, and limt~ ~(w-xl) = O,

V x(O), xc(O).

Let the control be v = K2w = K2x c From (5-3.13) and (5-3.16) we know the overall input satisfies: Xc = (AI-G2CI)Xc + (BI+G2C2B2)K2Xc + G2Y u = KlY + K2x c

(5-3.17)

146 Applying it to the original system (5-3.1), the closed-loop system is

0

G2C

AI-G2CI+B] K2+G2C2B2K2 J[ Xc J •

Its finite pole set is

0 =

I

, [ G2C

AI-G2CI+B1K2+G2C2B2K2

o(AI+BIK 2) U o(AI-G2C 1) ~ C-.

T h e r e f o r e , ( 5 - 3 , 1 7 ) i s a compensator

f o r system ( 5 - 3 . 1 ) .

I t s o r d e r i s nc=rankE

and

t h e c l o s e d - l o o p p o l e number i s 2rankE = nc+ra.kE, which i s our c o n c l u s i o n . Q.E.D. Compared w i t h the s i m i l a r r e s u l t of C o r o l l a r y 5 - 2 , 2 ,

compensator ( 5 - 3 . 1 7 ) i s of a

lower order while they both finish the same design purpose; their closed-loop systems have no infinite poles. Theorem 5-3.3. Singular system (5-3.1) has a normal compensator ~c = Acxc + BcY (5-3.18) u = Fcx c + Fy

where X c E ~ nc, such that its closed-loop system has no infinite poles, i.e., E

0

deg(

]

[A+BFC s -

0

Inc

L BcC

if and only if it is stabilizable,

BFc ]I) = nc + rankE Ac

(5-3.19)

detectable, impulse controllable,

and impulse ob-

S u f f i c i e n c y i s g i v e n by Theorem 5 - 3 . 2 . Tile s t a h i l i z a b i l i L y

and d e t e c t a b i l i -

servable. ProoF.

t y p a r t s o f n e c e s s i t y have been proven by Theorem 5 - 2 . 4 .

For the remaining part of necessity, we note that for matrices Ac, Bc, Fc,

and

F

we have deg(

s -

0 and

the

Inc

) ~ n c + rankE

Bee

Ae

equality holds if and only if

deg(lsE-(A+BFC)[) = rankE,

indicating

the

impulse controllability and impulse observability for system (5-3.1). Q.E.D. SLngular systems with no infinite poles, or, equivalently, with no impulse portion in their state response, are an important class of systems that possess the property ef mtructural stability, or robustness of stability, which will be studied thoroughly in the next chapter.

147 Example 5-3.2.

Consider the compensation problem for the simplified model of che-

mical process given in the introduction of Chapter I: Xl = AllXl + A12x2 + BIU 0

= A21x I + A22x 2 + B2u

y

= Clx I + C2x 2.

(5-3.20)

Here x I E R nl, x 2E R n2, n I + n 2 = n. System (5-3.20) is in the EF2 for s~ngular system (5-3.1). Assume that system (5-3.20) is stabilizable, detectable, impulse controllable, and impulse observable. Then rank[A22/C2] = rank[A22 , B2] = n 2. that a matrix K 2E ~mxr exists satisfying

Lemma 3-5.1

IA22 + B2K2C21 ~ O.

has proven

(5-3.21)

For such a matrix K2, we define tile control u = K2Y + v I

(5-3.22)

where v[ is tile new input. When applied to (5-3.20), it forms the closed-loop system ~I = (AII+BIK2CI)Xl + (AI2+BIK2C2)x2 6 BlV 1 0

= (A21+B2K2CI)Xl + (A22+B2K2C2)x 2 + B2v I

y

= Clx I + C2x 2.

(5-3.23)

Noticing (5-3.21), A22+B2K2C 2 is invertible. Defining the transformation matrices In I Q =

0

(AI2+BIK2C2)(A22+B2K2C2)-I -

In2

Inl P =

]J

0

_ (A22+B2K2C2)-l(A21+B2K2CI)

]

(A22+B2K2C2)-I

then Q and P are nonsEngular. Under the coordin~ite trnnsformatEon

=

]

system (5-3.23) is r . s . e , to 41 = Aog1 + BoY1 0 = 52 + B2v1 y = Cog I + C2g 2 where

glE Nnl,

~2E • n2

148 -1 Ao = A l l

+

B1K2C1 - (A12+B1K2C2)(A22+B2K2C 2)

(A21+B2K2C1)

Bo = B1 - (AI2+B1K2C2)(A22+B2K2C2)-lB2 c o

=

c 1

-

(5-3.24)

C2(A22+B2K2C2)-l(A21+B2K2C1).

By assumption, system (5-3.20) is stabilizable and detectable. Then, (Ao,Bo,C o) is stabilizable and detectable. Thus, suitable matrices KI,G 2 may be selected to satisfy

O(Ao+BoKI) C ~-, Now,

O(Ao-G2Co)CC-

following the proof process of Theorem 5-3.2)

(5-3.25) we can construct the normal com-

pensator for system (5-3.20): Xc = Acxc + B c Y

(5-3.26)

u = F c x c + Fy where xc( N nl, A c = A o - G2Co + BoK1 + G2C2B2K1 Bc = G 2 (5-3.27)

Fc = K 1 F

= K2

and their closed-loop system has no infinite poles, or no impulse terms exist in their closed-loop state response. In summary,

the design algorithm of normal compensators for system (5-3.20) is as

follows. Under the sufficient conditions for the existence of normal compensators as in Theorem 5-3.3. I, Choosing matrix K 2 so that (5-3.21) is fulfilled. 2. For such K2, c a l c u l a t i n g 3. Choosing

BoK 1 ) and

matrices

Ao, Bo, CO according to (5-3.24).

KI, G 2

satisfying (5-3.25) with expected pole sets

a(A o +

a(Ao-G2Co).

4. Calculating coefficient matrices Ac, Bc, Fe, and F in accordance with (5-3.27). 5. Then (5-3.26) is the designed normal compensator with desired properties.

Compensators

discussed so far have a common feature:

Their output,

or

feedback

control u(t), is a linear combination of measure output and the state of compensator. Now we will give one result for another type of compensators.

149 Theorem 5-3.4 (Dai and Wang, 1987b). able,

and impulse observable.

Let system (5-3.1) be stabilizable,

detect-

Then it always has a compensator of order no greater

than rankE of the following form: ~c = Acxc + Bcu + Gy w = F c x c + Fy + ilu

(5-3.28)

u=Kw where

x c EI~ n c ,

wE~R n,

n c ~< rankE,

and a l l

matrices

are constant,

such t h a t

tile

closed-loop system has no infinite poles. The compensator is a combination of P-state feedback control and the normal

state

observer ill the form of (4-2.18). The structure of compensator (5-3.28) is shown in Figure 5-3.1.

u(t)

"t E~=Ax+Bu,y--Cx [

v(t) reference

y(t)

system (5-3.1)

~.

w(t) I xc=Acxc+Bcu+Gy [

w=FcXc+Fy+Hu

I.

IS

j-

compensator ( 5 - 3 . 2 8 )

Figure 5-3.1.

Example 5-3.3,

Dymmlic Compeusator ( 5 - 3 . 2 8 )

In the proof of Theorem 5-3.2, if the Luenberger reduced-order ob-

server for system (5-3.15) is used instead of the same algorithm is processed,

the full-order observer (5-3.16),

and

the final compensator is in the form of (5-3.28) in-

stead of (5-3.11).

So far, we have discussed They

existence and desLgn methods for various

have two common f e a t u r e s .

separation

They a r e o b t a i n e d b a s e d on s t a t e

princLple holds in their

observer separately.

Thus, i t

When E = I n i s a l l o w e d , systems in linear

design,

compensators.

observers,

and tile

e n a b l i n g us t o d e s i g n t h e f e e d b a c k

and

is convenient for use.

all

system theory.

the results

become

the well-known results

f o r normal

150 While

all other compensators are in the form of dynamic output feedback

in Figure 5-2.1, Its

output (which acts as the input for original system) visibly

u(t),

as shown

the compensator in the form of (5-3.28) adopts a different version. depends

on

input

thus not possessing the form of direct dynamic feedback. Viewed from the point

of transfer relationship,

besides the dynamic output feedback compensation,

includes a series compensator,

it also

which acts as a feedforward controller. This property

may be seen clearly from Figure 5-3.2.

u(t)

+ +=

v(t)

"I Ex=Ax+BUy=cx " y(t)

ref. system (5-3.1)

KWl(t) ~ic=AcXlc+Bcu

xlc(O)=O wl=FcXlc+l[u feedforward

Kw2(t)

IK--~°

w2

X2c=Acx2c+Gy I x2c(O)=xc(O) w2=FcX2c+FY feedback

Figure 5-3.2.

For

a given system,

However,

the

more

Compensator (5-3.28)

the compensators that stabilize it are by no

the requirements,

such as lower order or

means

normality,

unique. the

less

freedom in their design. As

proven in Section 2-5,

for any regular system (5-3.1) there always exist

nonsingular matrices Q and P, such that under tile coordinate transformation

p-ix = [~I/X2/X3/X4 ]'

ViE ~ni'

system (5-3.1) is r.s.e, to its canonical form:

~-~'~i = n, i=l

two

151 Ell 0

El3 0

]

E21

E22

E23 E24

0

0

E33 0

0

0

E43

i,3 ~2

E44J

=

liAll ° A13° ll ll [iB21 21 A22

x4

A23 A24

~2 +

0

A33 0

x3

0

A43

54

A44

(5-3.29)

y

=

[c 1, o,

c 3,

o][~1/~2/~3/~]

where the subsystem El|X 1 = AllX 1 + FlU (5-3.30) =

cl~ 1

is both controllable and observable. If Ec~ c = Acxc +BcY (5-3.31) u =Fcx c + F~ is its compensator.

Here, Ec may be singular. Then the system determined by

Ec~c = Acxc + Bcy (5-3.32) u = FcXc + Fy is a compensator for system (5-3.1) since when applied to (5-3.1) the closed-loop system

[~ 011~1 o

E

~,+0~c ~ii]

~c

L BcC

~,

Ac j t x c

has the finite pole set

0

o([ Eli

Ec

,

BcC

Ac

0 } [AII+BIFC 1 BIFcl ) V o(E22,A22) U a(E33,A33 ) U ~(E44,A44),

0

Ec

,

BcC 1

Ac

which is obviously stable when the original system (5-3.1) is stabilizable and detectable. Sparked by these thoughts, the general desigu algorithm of compensators (singular or normal) may be summarized as follows. Under the sufficient conditions of stabilizability and detectability for system (5-3.1). i. Taking canonical decomposition to obtain (5-3.29) when it is not.

152 2. Choosing suitable feedback u = KI~ 1 - K2~ l

(5-3.34)

(or u = KI~ I)

to stabilize subsystem (5-3.30). 3. Constructing for subsystem (5-3.30) the state observer (H may be zero matrix): Ec~ c = Acx c + Bcu + G~ w = F c x c + F~ + Hu such t h a t l[mt~oo(w-x)

= O,

V x(O), xc(O).

4. Tile system Ec~ c = AcXc + gcu + Gy w = F c x c + Fy + Hu

u = Klw - K2~ is a compensator for system (5-3.1).

5-4.

Singular Compensators in Output Regulation Systems

Now we will consider the compensation problem for singular systems with extra disturbance. whose

In

this section we will study the deterministic disturbance

initial condition is unknown.

twofold properties:

In this problem,

with

models

our compensators should

have

resistance of the disturbance and stability when disturbance dis-

appears, the latter property is termed internal stability. We would like to explain the disturbance.

Assume that the dlsturhance f(t) satis-

fies the singular equation Elf(t) = Aff(t)

(5-4.1)

where Ef and Af are constant, and system (5-4.1) is regular. From the state represention, its fast substate is always zero once started, contributing nothing to the system.

Thus,

we

will hereafter only consider the disturbance f(t) that satisfies

an

ordinary differential equation (or the slow subsystem form): f(t) = Lf(t). Consider the singular system with external disturbance f(t):

(5-4.2)

153 E~(t) = Ax(t) + Bu(t) + Mf(t) f(t) = Lf(t) (5-4.3)

y(t) = ClX(t) + C2f(t) z(t) = DlX(t) + D2f(t) where x ( t ) 6 ~ n i s

state,

u ( t ) E ~m i s t h e c o n t r o l i n p u t , f ( t ) £

y(t)6 ~r

~1~ i s t h e d i s t u r b a n c e

satisfying

(5-4.2),

i s t h e measure o u t p u t and z ( t ) 6 ~ d i s t h e v e c t o r t o be

regulated.

E, A, B, M, L, C1, C2, DI, and D2 a r e consLanL m a t r i c e s of al~l[ropriutc d i -

mensions. S i n c e ( a s y m p t o t i c a l l y ) tically

s t a b l e d t s L u r h a n c e i n p u t has no e f E e c t s on asympto-

regulated output, we hereafter assume that

Let u(t) ~ O, t >I O.

o(L) 6¢ ÷.

If system (5-5.3) is stable when f(t) m O, t ~ O,

i.e.,

limt~oox(t) = O, V x(O), system (5-4.3) will be termed internally stable; if lim (z(t) - Zr(t)) -- O, V x(O) t~oo holds for reference signal Zr(t), system (5-4.3) will be called output regulation. If Zr(t ) satisfies a singular differential equation, by using the state augmention method, the problem when Zr(t ) ~ 0 may be changed into the problem when Zr(t ) = O. Thus we will assume that Zr(t ) ---O, t >/ O. Example 5-4.1.

When u(t) m O,

t >i O,

singular system (4-5.8) with

disturbance

input becomes

IOlO1 joel [o] 0 0

I ~(t) =

0

I 0 x(t)+

2 f(t)

0

0

0

0

1

0

z(t) = [i

O

1

(5-4.4)

0]x(t)

whose solution is

x(t)

=

[-z~(t) - ~(t)] | 2f(t) ~(t)| -

[

J

-f(t)

(5-4.5)

~(t) = - 2~(t) - ~(t). Let the disturbance f(t) satisfies f(t) = O, [.e.,

a jump signal. From (5-4.5) we

know that system (5-4.4) is not only [uternal]y stable but also output regulation. If the disturbance f(t) satisfies f(t) = f(t), is no longer output regulation. Example 5-4.2. When f(t) = O, the system

[°°I [! !] [!I 0 0

1 0

0 0

z = [0

~= 1

0 0

l]x + f

x+

also from (5-4.5),

system (5-4.4)

154 is not stable. Thus it is not internally stable, or output regulation.

Consider the dynamic compensator Ec~c(t) = Acxc(t) + BcY(t) u(t) = mcxc(t) + Vy(t) where xc(t) E Rnc

(5-4.6)

is its state; Ec, Ac, Bc, Fc, and F are constant matrices of appro-

priate dimensions. Systems (5-4.6) and (5-4.3) form the closed-loop system

E

O ]{~c

BcC

Ac J[xc ]

L BcC2

If

= Lf

(5-4.7)

y = CI× + C2f z = DlX + D2f.

Definition 5-4.1. Let (5-4.6) be a dynamic system. If I. Closed-loop system (5-4.7) is internally stable, i.e., o (

(5-4.8)

O

Ec

,

BcC

Ac J

2. System (5-4.7) is output regulation, i.e., lim t~

z(t) = O,

V x(O), xc(O), f(O),

(5-4.9)

we will call (5-4.6) an output regulator for system (5-4.3), or regulator without confusion. (a)

If rankE c < n c, system (5-4.6) is termed a singular output regulator (SOR).

(b)

Otherwise, rankE c = nc, E c = Inc is assumed without loss of generality, sys-

tem (5-4.6) is termed a normal output regulator (NOR). Only SORs are considered in this section. Lemma 5-4.1. Assume that (E, A) is a regular pencil. Then the matrix equation AV - EVL = M

(5-4.10)

has a unique solution for any matrix M if and only if o(E,A)~ o ( L ) = ~. Proof.

(5-4.11)

Under the regularity assumption for (E,A), by Lemma I-2.2 there exist non-

singular matrices Q and P such that QEP = diag(Inl, N),

QAP = diag(Al, In2)

155 where n I + n 2 = n, N E ~n2xn2

is nilpotent.

Left multiplying Q on both sides of

(5-

4.10), we obtain diag(Ai, I ) p - I v - d i a g ( I ,

N)p-lvL = QM.

(5-4.12)

By denoting p-Iv = [VI/V2],

QM = [MI/M2],

equation (5-4.12) becomes AIV I - VIL = M 1

(5-4.13a)

V 2 - NV2L = M 2.

(5-4.13b)

It may be verified that equation (5-4.13b) has a unique solution

h-I V2 = ~ , NJM2 Li i=O for

any

L and M2,

Sylvester

(5-4.14)

where h is the nilpotent index of N.

equation,

which

Equation (5-4.13a)

has a unique solution for any matrix M 1 if and

is only

a if

(Appendix B) o(Al)na(L) Thus,

from

o(E,A) =

= O. o(Al) ,

we know that (5-4.10) has a unique solution

for

any

matrix M if and only if (5-4.11) holds. Q.E.D. Le,una 5-4.2 (Elementary Lemma). Consider the singular system E~ = Ax + Mf

(5-4.t5)

= Lf z = DlX + D2f with

o(E,A) C ~-.

The property

limt~ooz(t) = O,

V x(O), f(O) is true if and only

if there exists a matrix V satisfying AV - EVL = M

(5-4.i6)

DIV = D 2. Proof. Su[ficiency: For any matrix V ( ~nxp satisfying (5-4.16), we define Vf, which

possesses the dynamic equation E& = Aw, z = Dlw.

w = X +

o(E,A)CC-

we

a(L) c ~+, it will

be

Thus, by

have tim z(t) = lira Dlw(t ) = O, V t~oo t~oo Necessity:

Since we

a(E,A)tl~(L) = @.

have assumed

Lemma 5-4.2 shows

(5-4.10). Let w = x+Vf. We have

x(O), f(O).

that

o(E,A) ~ ~-

and

that a unique matrix V

exists

that s a L i s r i e s

156 E~

=

Aw

z = Dlw + (D2-DIV)f. Using

o(E,A)C C- once more, l i m t _ ~ w ( t )

= O, V w(O). Therefore, l i m t ~ ~ z(t) = 0

is true for any f(O) (thus f(t)) if and only if iim t~

(D2-DLV)f(t) = O,

together with

o(L)C~+,

V

f(O),

implying D 2 = D]V. Q.E.D.

This lemma is needed in later discussions. Example 5-4.3. In system (5-4.4),

io,o]

E=

0 0

0 0

1 0

Matrix V = (I-E)-IM = [3 AV

-

EVL

=

A=

, 3

ioo] 0 0

1 0

0 1

M=

,

[o1 2 l

.

1]~ is the unique solution of the Sylvester equat[~n

M.

llowever, DLV = 3 ~ D 2 = O. Thus, this system is not output regulation according to Len~a 5-4.2 in which L = I. This coincides with tile result in Example 5-4.1.

A dlrect result of Lemma 5-4.2 is the following corollary. Corollary 5-4.1. closed-loop

system

Assume (5-4.6)

that

(5-4.6) is a

dynamic compensator

is internally stable.

such

Then the closed-loop

output regulation if and only if two matrices VIE ~nxp,

that

the

system

is

V2E • ncxp exist such that

(A+BFCI)V 1 + BFcV 2 - EVIL = M + BFC 2 BcCIV I + AcV 2 - EcV2L = BcC 2

(5-4.~7)

DIV I = D 2. Theorem 5-4.1. Consider the following singular system with disturbance input: E~ = Ax + Bu + Mf = mf y = ClX

z = D2x If I. (E,A,B,CI) is stabilizable. 2. rank[B

M] = rankB.

(5-4.18)

157 3.

sE-A 0 C1

rank

-M ] sI-L = n + p, 0

V s~+,

s finite,

system (5-4.18) has an SOR of the form (5-4.6). Proof.

By assumption

l

a matrix KIE ~mxn

may be chosen to satisfy

a(E,A+BKL) C

¢-; assumption 2 indicates the existence of a matrix K 2 satisfying (5-4.19)

M = BK 2. Apparently,

the feedback control

(5-4.2o)

u = KlX - K2f when applied to (5-4.18), stabilizes its closed-loop system

(5-4.21)

El = (A+BKI)X. Thus l[m t ~ z

= D|limt~ ~ x

= O,

V x(O).

Note that feedback control (5-4.20) cannot be realized directly.

To realize it we

consider the following composite system

(5-4.22) y = [c I

It is detectable

Ol[x/f].

according to assumption 3,

assuring the existence of its singular

state observer:

([oA ~]-G[C10])xc W

=

+[~]u + GY (5-4.23)

XcP

where G satisfies

a([ E O] [ A M]_ G[C1 0])6- ¢0

I

,

0

(5-4.24)

L

such that limt~oo (w - I x / f ] ) Using the control u = [K 1

= o,

v x(O),

xc(O),

f(o).

-K2]x e is lieu of (5-4.20),

its substitution into

(5-

4.23) yields Ec~ c = Acx c + B c Y (5-4.25) u where

= F c x c + Fy

XcE ~n+d, and coefficient matrices defined as

158 [ E 0 ]

Ec

0

Ac = [ A

I

,

M]_G[C1

0

B c = G, F c = [K 1

O] + { B ]

L

-K2] ,

0

[KI -K2]

(5-4.26)

F = O,

Hence (5-4.25) is an SOR for system (5-4.18). In fact, Lhelr closed-loop stsrem has finite pole set

o(

0

I

,

BcC

Ac

= o(E,A+BK 1) U o( implying the closed-loop

)

0

I

,

0

L

- G[CI

01) c c

system is internally stable.

From (5-4.19) and (5-4.26), equation (5-4.17) is fulfilled with the matrices Vl = 0 E ~n+d

V 2 = [O/-Id] E ~ ( n + d ) x p

Therefore, Corollary 5-4.1 shows (5-4.25) is an SOR for system (5-4.18). Q.E.D. This

theorem shows a sufficient condition for the existence and design method

of

SORs for systems wiLh disturbance not directly involved in its measure and regulated outputs.

If

Dl = I , z = x. The

r e g u l a t e d v e c t o r i s the sl:ale whnse r e g u l a t o r i s c a l l e d

s t a t e r e g u l a t o r . The f o l l o w i n g Theorem 5-4.2 i s a r e s u l t concerning the s t a t e r e g u l a tor. Theorem 5-4.2. Consider the system E~ = Ax + Bu + Mf = Lr Y = Cl x

(5-4.27)

If rnnkM = p, system (5-4.27) has a singular stale regulator (5-4.6) if and only if 1. (E,A,B,C) is stabilizable.

2. rank[B

MJ = rankB.

3. (

0

I

,

0

L

, [C I

0])

is detectable. Its proof is omitted here (see Dai and Wang, 1987d). The

theorem

gives the sufficient and necessary condition

of the singular

stale

159 regulators only for systems with rankM = p. systems,

The case is much complicated

for general

or even for system (5-4.18).

As seen in Theorems

5-4.1 and 5-4.2,

4.22) is given as a sufficient be (asymptotically)observed (regulator).

of the composite

which indicates

in order to be rejected

This is in accordance

may we mention

the detectability

condition,

by applying

with our common knowledge,

the problem of disturbance

rejection

system

that the disturbance a dynamic

otherwise,

without knowledge

(5-

should

compensator by no means

of the disturb-

ance itself. The

output

particular

regulator

is a special

goal, resisting

dynamic

[~xample 5-4.4. Consider

the singular

feedback mechanism

that

achieves

of disturbance on the regulated

the influence

a

vector.

system

[1o01 [ 0] [00] [01 0 0

I 0

I 0

~ =

y = [l Z

=

0 I -i 1 -2 0

x +

0 I

u +

0

I -I

f

(5-4.28)

00]x

X

with the jump disturbance

input f(t) satisfying

= O. Now consider

(5-4.29) the design of its state regulator.

[°°I

E=

0 0

I 0

I 0

C 1 = [I

0

0],

Since rank[sE-A, Furthermore,

rank

A=

,

[i

I -I -2 0

B= ,

C 2 = O,

D 1 = 13,

B] = 3 = n, V s E C + ,

s finite,

rank[B

0 Cl

In system (5-4.28)

[ii] [i] M=

,

D 2 = O, system

-

,

L = O.

(E,A,B,C)

is stabilizable.

M] = 2 = rankB and

sl-L 0

= 4 = n + p,

V s ( C+, s finite,

i.e.)

( 0

,

0

m

)

[CI

0])

is detectable. From Theorem 5 - 4 . 2 ,

we a r e s u r e o f t h e e x i s t e n c e

of a singular

state

regulator.

Le

Let

g l --

[,oo} 0

0

0

.

We have o(E,A+BKI)

=

|-I,

-0.5}C

C-.

(5-4.30)

160 If we choose the feedback control

u =

0

0

0

x +

(5-4.31)

-I

when applied to system (5-4.28),

its closed-loop

E~ = (A+BKI)X , is stable according

system

z = x

to (5-4.30). That is, l i m t ~ o o z(t) = O for any x(O).

To realize the feedback control

(5-4.31), we consider

the composite

[i°°;Ill[ °fill[!il 1 0 0

1 0 0

0

y--[1 which

corresponds

~ f

0 1-1 1 1 -2 O 0 0 0

=

o

o

0 -5

+

u

(5-4.32)

o][x/f],

to system ( 5 - 4 . 2 3 ) ,

fact, matrix C = [0

x f

system

and i s d e t e c t a b l e as t e s t i f i e d

earlier.

In

2] ~ fills the equation

o ( [ 0 E 0 [ , [ A M [+ G[C1

(-1,

0])=

-1)C £-.

[ooo] [ oo] [OO]ofl]

T h e r e f o r e , system 0 0 0

I 0 0

i 0 0

0 0 I

Xc

0 I -I I 6 -2 0 -I -2 0 0 0

=

Xc +

0 I

I 0 0

u +

_

y

W = XC is a singular observer

for system (5-4.32).

Let the control input

u

=

[~oo~] Xc 0

0

0 -1

[1ooo] [!_1oo]

be used to s u b s t i t u t e 0 0 0

u = according

i 0 0

l 0 0

[~

o 0

( 5 - 4 . 3 1 ) . Then the s i n g u l a r s t a t e r e g u l a t o r i s

0 0 1

xc

= -

0 0 0

+

Xc

Y 2 (5-4.33)

~]

o 0 -i

to Theorem 5-4.1.

its closed-h>op

I -I -2 0 0 0

Xc For any constant disturbance

system internally

f(t), system (5-4.33) makes

stable and output regolat~on.

161 5-5.

Normal Compensators in Output Regulation Systems

Compared with singular compensators, normal ones are more convenient to physically

r e a l i z e anti more i n s e n s i t i v e

to e x t e r n a l n o i s i n p u t s .

For a given singular system with disturbance input E~ = Ax + Bu + M[ = Lf

(5-5.1)

y = ClX + C2f z = Dix + D2f where

the items are determined by (5-4.3),

its normal output regulator (NOR) is

an

output regulator in the form of

~c = Acxc +BcY

(5-5.z) a = Fcx c +

where x cE ~ nc,

Fy

Ac, Be, Fc, and F a r e c o n s t a n t m a t r i c e s of a p p r o p r i a t e d i m e n s i o n s .

System ( 5 - 5 . 1 ) and compensator ( 5 - 5 . 2 ) form t h e c l o s e d - l o o p system

= Lf

(5-5.3)

y = CI× + C2f z = D1x + D2f. As

a

special case of E c = I whenever it is allowed in Corollary 5-4.1,

we

have

Corollary 5-5.1. Corollary 5-5.1. Consider system (5-5.1). Let closed-loop system (5-5.3) be internally stable. Then closed-loop system (5-5.3) is output regulation if and only if two matrices VI6 ~ n x p

V26 • ncxp exist such that

(A+BFCI)V 1 + BFcV 2 - EVIL = M + BFC 2

BcCIV I + AcV2 - V2L = BcC 2 I~ll' I ,-I~

The

key

regulators

.

difference is

(5-5.4)

in

the design of normal regulators from

that

that either a normal observer is used to observe the

of

state

singu]ar and

the

162 or derivative

disturbance,

Theorem

state feedback

have a more complicated

normal regulators

5-5.1. Consider

tile singular

is

used in the design process.

form than singular

In generalp

regulators.

system with disturbance

E~ = Ax + Bu + Mf

{ = Lr

(5-5.5)

y = ClX z = DlX in whi.ch disturbance I. (E,A,B,CI) 2. ra,k[B 3.

E

Proof.

in m e a s u r e and regulated

outputs.

If

is stabilizable.

0

,

L ], [C]

system (5-5.5)

O])

a l w a y s has an NOR i n t h e form o f ( 5 - 5 . 2 ) .

The p r o o f i s c o n s t r u c t i v e .

there exist

two n o n s i n g u l a r

f

Ell

QEP = [

L O where E l i ,

visibly

M] = rankB.

is detectable,

2.12),

is not involved

N22

A1 6 ~ n l x n l ,

is normalizable.

E12

]

As s e e n i n n o r m a l i z a b i l i t y

decomposition

(5-

m a t r i c e s Q and P such t h a t QAP =

,

[A, O

N22( • n2xn2,

]

QB = ,

[]

(5-5.6)

0

n I + n 2 = n, N22 is nilpotent,

and (EII,AI,BI)

Let

QM =

M1 E ~ M2

,

M 2 ( Rn2xP.

,

Then from assumption

2 we have M 2 = O.

Therefore,

under Lhe c o o r d i n a t e

transforma-

tion,

x =

system

(5-5.5)

0

P

[x 1 x2

is r.s.e,

I'122 jt~ 2 j

XlE N n l ,

x 2 E R n2,

to

0

[ = Lf y = CllX 1 + C12x 2 z = DllXl + D12x 2

°If ll I JLx2J

u

] (5-5.7)

163 where

CIP = [Cll, C12],

DIP =[DII, DI2].

(5-5.8)

It amy be verified that the subsystem Ell~ ! = Aix I + Blu + Mlf = Lf

(s-s.9)

= CllXl = Dllxl

possesses the following properties. I. (EI],A],BI,CII) is stabilizable and norn~l~zable. 2. rank[Bl, MI] = rankB I. 3. Q

I

, 0

L

, [CII, 0])

is detectable. From assumption i, there exist matrices K21

E mxnl and RIIE ~ mxnl satisfying

~(EII+BIK21,AI+BIKII ) C ~-.

rank(Ell+BiK21 ) = n I,

(5-5.10)

Assumption 2 indicates a matrix K E ~ nlxp may be chosen to satisfy (~-5.11)

M l = BIK. Clearly, the control law

(5-5.12)

u = KllX ] - K21~ ] - Kf such that limt~ooz(t ) = Dllimt~ ~ x l ( t ) = O,

V Xl(O ).

To realize the control (5-5.12), consider the system

(5-5.13) y = [ell,

Ol[xl/f]

By assumption 3, this system is detectable. Thus there exists a matrix G E ~ rx(nl+p) such that

o(

-

0

Ip

Therefore, the system

,

L

C[C11, 0]) c C-.

(5-5.14)

164

Ell 0 ]

J

0

[ Al

~c

Ip W

=

=

(

u+Cy

0

(5-5.15)

X C

i s a s i n g u l a r o b s e r v e r f o r system ( 5 - 5 . 1 3 ) ,

where x c 6 ~ n l + P .

Using x c in lieu of [x/f] in (5-5.12), we obtain the control

u = [KII,

-K]x c - [K21, 0]~ c ,

which, when applied to (5-5.15), results in the system Xc = Acxc + B c Y u = Fox c + F~

where

- G2CII

L

Ell+BIg21)-IC1 ] Bc =

G =

(5-5.16)

G2 F c = [KiI-K2[(EII+BIK21)-I(AI-GIC|I+BIKll), F

-K!

= - K21(EII+BIK21 )-I.

It can be proven that the compensator Xc = AcXc + B c Y (5-5.]7)

u = Fox c + Fy

i s an NOR f o r s y s t e m ( 5 - 5 . 5 )

(Da[ and Wang, 1987¢1). Q.E.D.

Example 5-4.4. Consider system (5-4.28)-(5-4.29).

This system is normalizable,

Lhe m a t r i x

satisfies

rank(E+BK21 ) = 3 = n° I f we l e t

9-3 it

will

be

0

o(E+BK21,A+BKll) =

Theorem 5 - 5 o l , u =

[-1,

we c h o o s e t h e c o n t r o l 9 -3

0 -i

Xc -

-1,

-I}C

as 0

0

~-.

Following the proof process

of

and

165 where x c is determined

by

[0001 [ 00] [001 0

1

1

0

0 0

0 0

0 0

0 1

~c

0

=

1 -!

6 -2 -2 0

l

0-I 0 O

Xc +

o

l

I 0

0 0

(5-5.]9)

u +

which is a singular observer for system (5-4.28)-(5-4.29). According to

the results of Theorem 5-5.1,

from (5-5.18)

and (5-5.19) we obtain

the NOR

which

xe

=

u

=

15 -4 -i 6 -2 0 -2 0 0

makes its closed-loop

Xc +

Y

system both internally stable and output regulation

for

any c o n s t a n t d i s t u r b a n c e f ( t ) . Comparing same.

The

the normal

conditions of Theorems 5-4.1 and 5-5.1 we see that regulator may be designed

they

tf tile system has a singular

are

the

one,

the

former is certainly of application significance. As indicated

from the proof process of Theorem 5-4.1,

form of (it exists under certain conditions

if a normal observer

in tile

l)y Chapter 4):

~c = AcXc + Bcu + Gy u = FoX c + Fy is used in lieu of (5-4.23) to observe the state o£ (5-4.22), 2.1 the proof process will yield the following

Theorem 5 - 5 , 2 (Oai and Wang, 1987d). L e t ( E , A , B , C 1 ) 1-3 i n Theorem 5 - 5 . 1 be s a t i s f i e d ,

instead of Theorem 4-

theorem.

be n o r m u l i z a b l e ,

and rankC 1 = r . Then s y s t e m ( 5 - 5 . 5 )

assumptions h a s an NOR o f

o r d e r n + p - r o f t h e form Xc = AcXc + BeY

u = FcXc + Fy. Its proof is omitted here. As for the normal state regulator, Theorem 5-5.3.

we have the following theorem.

Assume that D 1 = I n, rankM = p.

input f(t) has a normal state regulator Xc = Acxc + BeY u

= F c x c + Fy

System ( 5 - 5 . 5 )

with disturbance

166 if and only if assumptions i-3 in Theorem 5-5.1 are satisfied. Proof. its proof is given by combining Theorem 5-5.1 with Theorem 5-4.2. Q.E.D. It is worth pointing out that the output regulators (singular or normal) in either the

last section or this section consist of a mode] of external

a(Ec,Ac) =

o(L)Uo(A),

lves are not stahle.

or

o(Ac) =

disturbance,

i.e.,

a(L) U o(A). Therefore, the compensators themse-

This is a property that must exist in order to

achieve

output

regulatioH.

5-6.


Singular compensators were first given in Wang (1984). In this chapter,

compensators are discussed parallel to linear system theory. For

details of some results whose proofs are omitted here, be useful.

Dai and Wang (19g7b,d)

would

CHAPTER 6 STRUCTURALLY STABLE COMPENSATION

Mathematical

models

are

IN SINGULAR SYSTEMS

often used to describe systems of interests

~n

system

ana|ys[s and synthesis, llowever, choosing the appropriate models is a somewhat dif[icult

task even though it is important in control theory.

main approaches les;

and the identification

approximation

of

or

method.

the system,

accurately described. ronments

Currently,

in obtaining the model: from the physical relationships

Furthermore,

The model itself, on the other hand, is only an

nominal

points.

In

to

be

influenced by many external factors such as envithe coefficients

for a system are

fixed; deviations exist in them. In this sense, coefficients Here,

two

among variab-

since any practical system is too complicated

executive components,

uncertainties.

there are

we will call them perturbations, some cases where perturbation

by

no

means

in a system always have

defined as the deviation from

is serious,

crucial problem

may

occur, as shown in the following example.

u(s) input

I

s+l

I

~

y(s) output

L

y(s) output

s3+s2+5s+5"l

Figure 6.1

I

u(s) ~nput

s+l I s3+s2+5s+4"9 [

Figure 6.2

The system described by the input-output according to tile Routh crLterion

relation shown in Figure 6.1 is unstable

(Fortmann and llitz, 1077). llowever, if errors exist

in modeling and the system is m£smodeled as shown in Figure 6.2. The system ~s stahLe.

168 Note that only a minor error 0.2 occurs between these two models to make a completely opposite conclusion on the stability of the system.

In this chapter, perturbations

we will study the structural stability for singular systems with

(or uncertainties,

deviations,

errors in other terms) in coefficient

matrices, as well as their compensation strategies.

6-1.

Structural Stability in Homogenous Singular Systems

It is wellkuown t h a t if the homogeneous normal system x(t) = Ax(t) where x ( t ) 6 ~ n, A E ~ nxn

is stable, i.e.,

o(A)KC-,

a positive constant 6 > 0 exists such that

a(A+ ~A) ~ Cfor arbitrary deviation Here

IIAII

~A in coefficient matrix A provided

~A is within

lJ~A]J < 6.

is any consistent matrix norm. This property is termed structural stabili-

ty here. Thus a stable homogeneous system is structurally stable. But this statement is unfortunately not applicable to the singular system E~(t) = Ax(t) where x(t) 6 ~ n , E, A E ~ n x n

(6-1.1) are constant matrices. It is assumed that system (6-I.i)

is regular and q ~ rankE < n. The stability of system (6-I.I) is sensitive to parameter perturbations,

and thus makes the structural stability more complicated than the

case in normal systems. Example 6 - 1 . 1 .

C o n s i d e r tile s i n g u l a r

[oo] [ 0 1 0 0 0 0

~ =

o]

0 0 1 0 1 0

It is stable with finite pole set

hA =

[0o o1 0 0

0 0 0-¢

system

x.

(6-1.2)

o(E,A) = i-l} ~ ¢-. When a perturbation

e> 0 is a scalar, ,

is present, we have a(E,A+ ~A) = ~-I, Thus,

1 E } .

the perturbed system E~(t) = (A+ ~A)x(t) has an unstable finite pole

it is unstable) no matter how small ¢ (thus

~A) is.

~

(thus

169 In

system (6-I.I),

The deviation

E and A are called the structural parameters for the

system.

~A caused by any reason is termed perturbation in the parameter matrix

A. Definition 6-1.1. formed by l i s t i n g

Data point

P~E,A}

for system (6-1.1)

t h e e l e m e n t s o f E, A i n a r b i t r a r y

is defined

as a vector

order.

For example, listing the elements in coefficient matrices in system (6-1.2) in the row order PI{E,AI = [1 0 0 0 l 0 0 0 0 -1 -1 0 0 0 1 0 1 O] ~ o r i n t h e column o r d e r

P2tE,A} = [I both PIiE,A}

Data

0 0 0 1 0 0 0 0 ~1 0 0 -1 0 1 0 1 01~

and P2iE,A)

point

are data points for system (6-1.2).

may be either a columa or a row vector,

fleet J.t iS restricted

to

a

column vector. For any vector a 6 ~ n , Us(a) = { x [

b > O, the set II x - a l l

< 6 , x6~ a

is termed a neighborhood at a with radius 6 , here

uxn

is the consistent vector

norm.

Definition 6-1.2.

Assume that

system (6-1.1) is stable,

o(E,A) d ~-.

If

there

exists a certain neighborhood Ug(PiE,A}) at data point ]'{IS,A} such that tile stability of system (6-1.1) is preserved in the presence of arbitrary perturbations of structural

parameters in this neighborhood,

we will term the

system (6-1.1)

structurally

stable at data point P{E,A) . By definition,

a structurally stable system is stable.

But the inverse statement

is not true, as shown in Example 6-1.1. Theorem 6-I.i. point P(E~

Let system (6-1.1) be stable.

It is structurally stable

at data

if and only if rankE = n.

Proo f . If rankE = n, system (6-1.1) becomes x = E-iAx. Noticing that E is nonsingular in a certain neighborhood at P{E) , the previous system is structurally stable. This gives the sufficiency. Next we will prove tile necessity. Assume that system (6-1.1) is structurally stable at data point P(E}. If rankE # n is false,

rankE < n.

Note that

o(E,A)CC

(implying regularity).

There must exist

two nonsingular matrices Q and P such that system (6-1.1) is r.s.e, to EFI:

170 AlXl

~I =

N~ 2 = x 2 where x I 6 ~ nl, x26 • n2, n I + n 2 = n, and QEP = diag(Inl, N),

QAP = dtag(Al, I).

j01 ]

Without loss of generality, we assume that N is in the form

0

N=

1

]Rn2xn2.

".'. • " 1

0 Let the perturbation he

~E = Q-I

Onlxn 1 0

0 O(n2_l)xl

0 0

0

e

Olx(n2_l)

where e > 0 i s a s u f f i c l e n t l y teristic

small scalar.

polynomiaI of perturbed

indicating

that

the perturbed

e > 0 is.

Therefore,

flicts

w i t h our a s s u m p t i o n .

T h i s t h e o r e m shows

~A = O,

Then d i r e c L c a l c u l a t i o n

(-1)n2-1(es

s y s t e m h a s an u n s t a b l e system (6-l.1)

Thus,

g i v e s tile c h a r a c -

n2 - 1) pole

Re(s) > 0

is not structurally

no m a t t e r

stable,

that a singular

s y s t e m c o u l d n o t be s t r u c t u r a l l y

how con-

stable

unless

c a r e must I)e done in tile d e t e r m i n a t i o n

E i n sysLcm m o d e l i n g so a s t o a v o i d l a l s o con(:lust(ul on s L u b i l i L y

Theorem 6-1.2.

which

rankE = n. Q.E.D.

i t becomes u aormal o n e . Ilence, p a i n s t a k i n g matrix

i{nxn,

p-I

system:

I s ( E + bE) - A I = I Q-III p - 1 1 1 s I - h l l

small

]

of

to hupl)en.

Assume that system (6-1.1) is stable. It is structurally stable at

d a t a p o i n t P{A} i f anti o n l y i f deg(IsE-AI)

= rankE,

(6-1.3)

or in other words, the system has no infinite poles. Proof.

Sufficiency: If

o(E,A)~ ~- and (6-1.3) holds, there must exist two non-

singular matrices Q and P such that QEP = d i a g ( I q ,

0),

QAI~ = d i a g ( A l ,

Denote

8A = ~-J[ ~AII 8A12 SA21

~A22 ] ~-l.

1),

q = rankE.

171

From the inequality

tt ¢~A22H < I1[ ~A21,

~A22]11 < II Q~APII ,

we know that II~A2211 is sufficiently small when II~AII is. (I+ ~A22) is nonsingular. Thus we have lsE-A- ~AI = IQ-III p-ll[Slq-Al- hall k

- ~AI2

- $A21

]

- (I+ ~A22)J

I~-IIIp-III-(I+ ~A22)llS[q-A 1- ~AII- ~A12(I+ ~A22)-1 ~A21I .

=

Under the assumption o(A I) = o(E,A)C£-, ciently small the perturbation ~AII +

it is obvious that when ~A is suffi-

~AI2(I+ ~A22 )-I ~A21

may be so small that o(A! +8A11+ 8AI2(I+~A22 )-! gA21)c C-, i.e., o(E,A+ ~A)C C-, indicating that system (6-I .I) is structurally stable at data point P{ A}. Necessity:

Let

system (6-1.1) be structurally stahle at P{A},

Since system (6-t.1) is regular,

with o(E,A)C C.

there exist nousiugular matrices Q and P such List

it is r.~.e, to EF[: x1

=

where x 1 6 ~nl,

AlXl'

Nx2

=

x2

x 2 ( N n2, n I + n 2 = n, N ( N n2xn2 n[Ipotent. Without loss of generali-

ty, we assume 1

0 N

1

~n2xn2

=

1

0 and the perturbation term

~a = Q-I

[

CA

Onlxn 1 0 o

0

O(n2_i)xn2 [-,, o . . . . .

1

p-1

o]

tlere v < 0 is sufficiently small. In this case, we consider the polynomial -I IQ IIP IIsE-(A+ ~A)I = lsl-AiI

s

" v 0

"" "

"-

172 =

I

s I - A l l ( - l ) n 2 ( v sn2-1+1)-

(6-1.4)

Under the assumption of structural stability for system (6-I.i) at P~A},

o(E,A) ~ C-

when Ivl > 0 is sufficiently small. (6-1.4) is a stable polynomial. Thus it must be n2 = I, or equivalently, N = O, which is equivalent to deg(IsE-Al) = rankE. Q.E.D. A

simple

structurally

but inaccurate explanation of this theorem is that

system (6-1.1)

is

stable at data point P~A} if and only if no impulse terms exist in its

state response.

This sparking result implies that when impulse terms are involved in

the state response, the singular system is more sensitive to parameter variations.

Example 6-l.2. For system (6-1.2), we have deg(IsE-Al) = I < 2 = rankE < n = 3. By Theorem 6-1.2, this system is not structurally stable at either P{A] or P(E,A~, which coincides with the results of Example 6-I.I. Example 6-1.3. Consider the system Ex = Ax

(6-1,5)

in which x E ~R3 and

E:

[ioo] 0 0

0 0

0 0

A=

[ioo] 0 0

,

I 1

i 0

.

For this system we have deg(IsE-Al) = deg(s+l) = 1 = rankE.

Thus,

system ( 6 - 1 . 5 ) is

structurally stable at P(A~ according to Theorem 6-1.2.

In

a real system,

the parameters often don't hold an equivalent

position.

Some

parameters (often nonzero ones) may be uncertain, the perturbation occurs, while some are completely accurate, especially the zero elements. Furthermore, some elements may

have d e v i a t i o n s oa only one d i r e c t i o n , time of a mechanism,

for example,

w i t h the increase of

operating

the temperature of its components always tend to increase

ins-

tead of decrease. In this sense, perturbation in a system often subjects to constraints t h a t may be g e n e r a l l y described as

g( BE, £A) = O.

(6-I.6)

We d e f i n e U1 = | P ~ E + ~ E , A + ~ A }

[

g( ~E, ~A) = O }

(6-1.7)

as a constrained neighborhood at data point P[E,A), on which the structural stability has the fotlowing theorem. Theorem 6-1.3 (Dai and Wang,

1987c).

Let

system (6-1.1)

be

stable

deg(IsE-A[). If U 1 = { P(E+ g m , A + g A ~

[ deg(Is(E+~E) - (A+~A)]) - ql = O }

and ql

173

for any sufficiently small [ ~E, ~A] and P{E+~E,A+~A} E UI, we have o(E+ ~ E,A+ ~A) c C-. In this

case (6-1.1) is called constrained structurally stable

at

data

poJnt

P{E,A) with constraints U I. The condition given in this theorem is only sufficient. The perturbed system (6I.I) may also be stable for perturbation revoking this condition as shown in the following Example 6-1.5. Example 6-1.4.

For system (6-1.2),

deg(l sE-A]) = I < rankE = 2.

Thus it is

not

structurally stable at P(A} by Theorem 6-1.2. floweret, let the perturbation be in the form of

SE = O,

~A =

When the perturbation quation

[ ~all ~a21 ~a31

~a12 ~a22 ~a32

5a13 ] (6-1.8) ~0231 .

~A is sufficiently small, I$a321 < I, I~a231 < 1 and the e-

deg(Is(E+ ~E) - (A+~A)I) = deg(

t[

- ga21

s- ga22

- ~a31 -1- ~u32

Jl

-1-

a23

)

.

= 1 = deg(Isg-AI) holds. Therefore, by Theorem 6-1.3 the perturbed sysLem (E+ ~E)~ = (A+~A)x [s stable for such perturbation provided i t is s u f f i c i e n t l y small. Although the perturhation is constrained in this example, its wide form enables it to r e p r e s e n t most o f tile r e a l c a s e s .

Example 6-1.5. Consider the system (6-1.2) with perturbation

~a = o,

~A =

[

~al 1 ~a12 ~al3] 0

0

~a22

~a23

~a32

~a33

~a33 > O.

(6-1.9)

,

Its characteristic polynomial

is(E+ SE)-(A+ gall = (s+l- ~all)(-1)( ~a33s+l+ ~a23+ ~a32+ ~a23~a32 -

~a22~a33 ).

together with ~a33 > O, clearly shows o(E+ ~E),A+ ~ A) c C- provided ~A is sufficiently small to satisfy I ~all I < I,

I ~a23+ ~a32+ ~a23~a32- ~a22Sa33 I < I.

174 This

is a single-direction perturbation problem since

~a33 > 0

is

constrained.

This example shows that a perturbed system may be stable even though Theorem 6-1.3 is not applicable here. Theorem 6-1.3 is inconvenient for use for its complicated form. Of interests often is

its special form such as these in Theorems 6-1.1 and 6-1.2.

A further result

is

the following corollary. Cprollary 6-1.1. Let system (6-1.1) be stable, rankE = deg(IsE-Al), and

U2 = {P{E+~E,A+~A}

[

rank(E+~E) = rankl~

Then system (6-1.1) is structurally stable on U 2. The

combination of this corollary with Theorem 6-1.2 shows that system (6-1.1) is

structurally stable if the slow-fast subsystem structure is preserved in the presence of perturbation. This property is in accordance with common sense.

The p h y s i c a l s e n s e o f Theorem 6 - 1 . 3 l i e s lar

in t h a t t h e s t a b i l i t y

of a s t a b l e

sing.-

system is preserved if the perturbation doesn't create a new pole (or the number

of poles remains the same), which is often unstable.

6-2.

Compensation Function on Structural Stability

via Dynamic Feedback

As discussed Jn the last chapter, compensators

for singular systems.

under certain conditions we may design

But we must be aware of tile fact that the design

procedure is applicable only to the case of accurate coefficient matrices. certainties are unavoidable to exist in any real system,

only

[ [ n i s h tile s t a b i l i z a t i o n

t o be s t r u c t u r a l l y

stable,

are

Since un-

the compensators should not

achievement bug a l s o g u a r a n t e e tile c l o s e d - l o o p systems

making them a p p l i c a b l e i n r e a l c o n t r o l system d e s i g n .

For s i m p l i c i t y and p r a c t i c a l tors

dynamic

significance,

in t h i s c h a p t e r only normal

compensa-

d[st-ussed.

C o n s i d e r the s i n g u l a r system E~c(t) = Ax(t) + Bu(t) y(t) = Cx(t) where x ( t ) (

~n,

y(t)(

~r,

(6-2.1)

u(t)(~m

puL, r e s p e c t i v e l y ; 1';, A( ~nxn.

are its stale,

. t e a s . r e o u t p u t , and c o n t r o l ill-

BE 1~nxm, C E ~ rxn a r e coustanL m a t r i c e s . The

system

is assumed to be regular with singular E, and the normal dynamic compensator:

X c ( t ) = AcXc(t) + BeY(t) u(t) = Vcxc(t ) + Fy(t)

(6-2.2)

175

where Xc(t)E • nc, Ac, Bc, Fc, and F are constant matrices of appropriate dimensions. System (6-2.1) and compensator (6-2.2) form the closed-loop system =

0

I

(6-2.3)

Xc

BcC

Ac J L x c

•

D e f i n i t i o n 6 - 2 . 1 . C o n s i d e r system ( 6 - 2 . 1 ) and i t s compensator ( 6 - 2 . 2 ) . The c l o s e d loop system ( 6 - 2 . 3 ) i s assumed t o be s t a b l e ,

o(

0

I

The compensator

(SSNC)

,

BcC

Ac

(6-2.2) will

i.e.,

) C ¢-.

be termed a

(6-2.4) structurally

stable

normal

compensator

a t d a t a p o i n t P{E,A,B,C} i f t h e c l o s e d - l o o p system i s s t r u c t u r a l l y

stable

at

P[E,A.B,C}; I t w i l l be termed a c o n s t r a i n e d SSNC a t PIE,At with c o n s t r a i n t s U1 i f ( 6 2.3) i s c o n s t r a i n e d s t r u c t u r a l l y

Theorem 6-2.[.

s t a b l e a t PtE,A} with c o n s t r a i n t s U1.

Compensator (6-2.2) is an SSNC

at data point P{A,B,C~

for sysLem

(6-2.1) if and only if the closed-loop system (6-2.3) is stable and deg(

s 0

) = ne + rankE.

Bee

I

Ac

Proo..f. S u f f i c i e n c y i s shown i n Theorem 6 - 1 . 2 . Now we w i l l prove n e c e s s i t y . For any r e g u l a r system ( 6 - 2 . 1 ) ,

t h e r e e x i s t two nonsLngular m a t r i c e s Q and P such t h a t

QEP = d i a g ( I n l , where n I + n 2 = n,

N),

AlE • n l x n l ,

QAP = diag(A1, In2) NE~n2xn2

in w r i t i n g , and w i t h o u t l o s s of g e n e r a l i t y ,

N =

'.

.

is nilpotent.

Far t h e sake of s L m p l i c i t y

we assume

6 • n2xn2.

Let the perturbation be

~A

=

Q-Idlag(O,

~A22)P-I E R nxn,

Then i n t h e p r e s e n c e o f such p e r t u r b a t i o n ,

~A22 = [e, O ....

, O]

( Rlxn2.

t h e p e r t u r b e d c l o s e d - l o o p s s y s t e m has t h e

characteristic polynomial

sLrA ABcOcCcFc]

176

=]Q-ip-ll

sI-(AI+B1FC 1 ) -BIFC 2 sN-(l+ aA22+B2FC 2) -B2FC 1 -BcC 2 -BcC I

-BIF c -B2F c sl-A c

(6-2.s)

Let bll

b12

b21 B2FC 2 =

•

"'"

bln2 l

b22

...

b2n 2

. .

. , .

bn21 bn22 .--

bn2n 2

•

Next we consider the term of the highest order (HOT), which is

HOT(&(s))

= [ Q-1p-1 [[ HOT(I sI - (AI+B1FC 1 )[ ) "HOT( IsN-(I+~ A22 +B2FC2) [)HOT( IsI-A c I) = I Q-I p-1 II HOT(I sN-(I+ ~A22 +B2FC2) I )st' l+nc.

(6-2.6)

Since HOT(IsN - (I+~A22+B2FC2)I)

-I -ls

bll "'"b12..

s

1 " ". Sl

= IIOT( I

-c 0 0

-

-

bn21

bn22

.b In2

] )

bn2n2J

= (-e-bn21)sn2-1 from (6-2.6) we know the term of the highest order in ~(s) is -b (-~

)sn+nc-I

(6-2.7)

n21

On the other hand,

it is assumed that the closed-loop system is stable, i.e., (6-

2.4) holds. Therefore [A+~A+BFC ]Bc C

BFc] 1 Ac J

is either a positive or negative scalar for all sufficiently small perturbations e (thus, ~A). Without loss of generality we assume a > O. Combining it with (6-2.7) we have a(s) = (-¢-bn21)sn+nc-l+ ... + a .

(6-2.8)

Under the assumption of structural stabi]ity for tile c]osed-]oop system at P(A,B,C}, A(s) is a stable polynomial for any sufficiently small e, either positive

177 or negative. Thus, by noticing -

a > O, we know

bn21 > O.

In this case we have s

-

)

BcC Ac

=

nc

+

rankE,

which is the condition in the theorem. Q.E.D.

Example 6-2.1.

According to Theorem 6-2.1, it may be testified that the compensa-

tors given in Examples 5-2.1, 5-3.1, and 5-3.2 are all SSNCs at P~A,B,C~

for systems

(5-1.8), (5-3.9), and (5-3.20), respectively.

Combining Theorem 6-2.1 with Theorem 5-3.2, we have the following theorem. Theorem 6-2.2.

System (6-2.1) has an SSNC (6-2.2)

only if it is stabilizable, detectable, Therefore,

at data point P{A,B,C} if and

impulse controllable, and impulse observable.

the concepts of impulse controllability and impulse observability have

a close relationship with structural stability. This theorem states only the existence of SSNCs in the form of (6-2.2). The design method may follow that in Chapter 5. The combination of Theorems 6-1.2 and 6-2.1 yields the following corollary. Corollary 6-2,1. the

closed-loop

If (6-2.2) is an SSNC at data point P{A,B,C} for system (6-2.1),

system

(6-2.3) is definitely

structurally

stable

at data

polnt

PiA,B,C,Ac,Bc,Fc,F }. In

engineering,

electric deviations

the compensators may be realized either using hardwares such

components, occur due

in manufacturing,

control meters, to

factors,

etc.,

or on a computer;

such as aging of components,

and environmental influence;

as

In the former case, admissible error

in the latter case, uncertainties exi-

st because of the restriction on word length. Therefore, perturbation always exist in the

realization

of

compensators.

What is stated in Corollary 6-2.1 is

sufficiently small perturbations are allowed without revoking stability.

that

some

This endows

the SSNC with the practical significance. Similar to the proof process of Theorem 6-2.2,

it is easy to prove the

following

theorem. Theorem 6-2.3.

The normal compensator (6-2.2)

could not be an SSNC at P~E}

for

system (6-2.1). This fact is unfavorable for system design. This nevertheless weakens the value of usage of compensator (6-2.2).

178 To have a further understanding of this point,

we examine the

singular system in

the EF2:

In

(6-2.9a)

0

= A21x 1 + A22× 2 + B2u

(6-2.gh)

y

= ClX 1 + C2x 2.

this EF2,

system rizes

Xl ~ AllXl + A12x2 + BlU

dynamic equation (6-2.9a) may be sometimes viewed

formed by s u b s y s t e m s ;

the intercoanection

meanwhile,

the

as

a

composite

a l g e b r a i c e q u a t i o n ( 6 - 2 . 9 b ) characte-

among t h e s e s u b s y s t e m s . In the system m o d e l i n g , provided we

have a fair examination of system structure so as not to mistake the dynamic equation for the algebraic one, the matrix will suffers

no perturbation; in case if happe,s,

the perturbation should be the constrained perturbation witll the constraints of fixed rank

rank(E+~E)

= rankE.

Just as the order of a normal systems, separate analysis.

system should be determined in advance in the modeling

in the analysis of singular systems,

the dynamic part from the algebraic part. In this sense,

we should have the ability This is a basic task

the closed-loop system (6-2.3)

in

is structurally stable

Concerning the constrained perturbation, we have the following theorem. Theorem 6-2.4. Let closed-loop system (6-2.3) be stable and

0

s -

I

BcC

Ac J 1

).

Then (6-2.2) is a constrained SSNC at data point P{E,A,B,C} U1 =

{P{E+~E,A+~A,B+gB,C+~C}

[

with constraints Ul:

nc + ql =

I} )

o

I

L

Bc(C+~C)

Ac

J.

to

system

Corollary 6-2.1.

nc + ql --~ deg(

of

o

by

179 6-3.

As

Structurally Stable Normal Compensators

shown in Theorem 6-2.3,

structurally s t a b l e for a s i n g u l a r pensators

normal compensators of the form of (6-2.2) cannot

for system (6-2.1)

system at all

structural

of other forms.

a t d a t a p o i n t P { E , A , B , C } , To c o l m t r u c t parameters

P{E,A,B,C},

we m u s t c o n s i d e r

If the derivative of y(t) is available, we

may

be

an SSNC com-

consider

compensators of the following form ~c(t) = Acxc(t) +

BcY(t) + Hc~(t) (6-3.1)

u(t) where x c E • n c ,

= Fcxc(t)

+ Fy(t)

+ fig(t)

Ac, Be, He, F c , F, and H a r e c o n s t a n t

matrices

of appropriate

dimen-

sions.

First, if we denote it(t) = Xc(t) - IlcY(t), compensator (6-3.1) may he written as ~c(t)

=

Ac~c(t ) + (Bc+AcHc)Y(t) (6-3.2)

u(t) = FcEc(t ) + (F+FcAcHc)Y(t) + H~(t). The dynamic equation doesn't involve the differential term y(t). Therefore,

without

loss el generality, we may assume that IIc = 0 in (6-3.1), so as to become ~c(t) = Acxc(t) + BeY(t) u(t)

= Fcxc(t)

Theorem 6-3.1. P(E,A,B,C}

+ Fy(t)

+ H~(t).

Compensator (6-3.2)

(6-3.3)

is an SSNC

for system (6-3.1) at data

(i).

SysLem (6-2.1) is stabilizuble and detectable.

(2).

The s y s t e m i s n o r m a l i z a b l e

Proof.

point

if and only if the following two conditions are fulfilled:

Necessity:

and d u a l n o r m a l i z a b l e .

The c o m p e n s a t o r ( 6 - 3 . 2 )

and s y s t e m ( 6 - 2 . 1 )

form the closed-loop

sysLem

[,

only a

204 Let y~(s) = Ge(s)u(s) be the perturbed output response, the response deviation caused by perturbation. I we have ey(S) = y(s) - yg(s) = U(S) = ~, ey(t) = - ee t, with the matter

small

be

llence

e/(l-s)

t ~ O,

response locus shown in Figure 7-2.3, how

and ey(t) = y(t) - y¢(t)

Then when u(t) is a unit jump signal,

e ~ 0 is.

showing that ]imt~ooley(t)l = ¢~

This implies that unstable zero-pole

cancellation

no is

impossible. This point should be paid special attention in the design of PPS.

ey(t)

0

Figure 7-2.3

The loop

preceding discussion shows that by using a dynamic compensator, system (7-2.2) ~ y

lized via compensators.

he a PPS.

Note that state feedhack control

the

closed-

i~ usually

Thus we discuss this problem for state feedback control.

reaWe

separately study this problem for P-, and P-D state feedbacks.

7-2.1. Pure proportional (P-) state feedback Consider Lhe P-sLate feedback: K = |kl, k 2 . . . . .

= Kx + v,

u

where v E R m is t h e 1~ew input,

kn]

Feedback control (7-2.9)

(7-2.9) and system (7-2.]) form the

closed-loop system E~ = (A+bK)x + by y

=

(7-2.10)

CX.

Under the assumption of regularity, such that system (7-2.1)

there exist two nonsingular matrices Q and P

is r.s.e, to

diag([nl, N)x = diag(A I, In2)~ + [bl/b2]u

(7-2.11~J)

205 (7-2.11b)

y = [c 1, c2]~ where ~ = p - l x , n 1 + n 2 = n, A1 ( R n l x n l , QEP = d i a g ( I n l , N),

QAP = diag(A1, I n 2 ) ,

Qb = [ b l / b 2 ] , Subject

are

cP = [ c l , c21.

to the controllability

controllable pairs.

controllability

assumption

( A l , b l ) and (N,b2) they are assunled to be iu the

for system (7-2.1),

Without loss of g e n e r a l i t y , form

canonical 0 I

N 6 ~ n2xn2 i s u i l p o t e n t , and

0

1

AI =

".

"

E

]{n l x n l ,

b1 =

~ ~ nl

(

Rn2xn2,

b2 =

E • n2

0 1

N

0 1

=

. I °

0 1 where "~" represents

0

the possibly

nonzero elements.

n1 K = ~p-1

[0,

R

Let

a O, l ,

l , O,

(7-2,12)

O]

Then the c h a r a c t e r [ s t [ c polynomial of c l o s e d - l o o p system (7-2.10) i s IsE - (A+bK)I

= IsQEP - Q(A+bK)P I

0 bI ] ~ = [ [ s I ; A1 s N _ l ] - [ b 2 J I = constant

Thus,

there exist

matrices Q and P such that (7-2.10)

two nonslngular

is r.s.e,

~x = ~ + boy

to

(7-2.13)

y = Co~

where x" = p - I x , and

ol

0

]

1

= QEP =

Qb = h O = °°

O(A+bK)~ = I ,

l

0

0

1 co

_--

o

c v = [c'~, c 2 . . . . .

Note that only a f a s t subsystem e x i s t s in (7-2.13).

c°l.

For any [eedl,ack v = KoE + ~J,

206 K 0 = [k~, k ; . . . . .

k~],

e

iS the new input (or reference

system formed by v and (7-2.13)

signal),

the

closed-loop

is

N~ = (I+boKo)~ + bo¢

y

=

Co~

with input output relationship Denoting

I

y ( s ) = Co(SN - ( I + b o K o ) - l b o ¢ ( S ) .

~? = k.x,i o = I, 2, ... , n, we have z 1

x = ~,

y(s) = - Co*[XI - (N-boKo T)]-lboq(s)

c |o + c ~

+ ...

+ cOT n - I n -o ~;, ~o n - I + xn kl + + "'" + n c ? s n - ] + c ~ s n-2 + . . . =

-

kO n-1 Is

From (7-2.14) trans[er

unless zero-pole

of order not greater

Particularly,

+ c° "

+ ... +

we see that the P-state

function

any polynomial

,o n-2

+ K2s

¢(s)

(l+k~)

~(s).

(7-2.14)

feedback cannot change the numerator

cancellation

occurs.

But its denomLnator

of the

may take

than n-l.

i f we c h o o s e Ko = [ - c ~ , - c ~ ,

""

, -c°n-l, -c,~-lI,

transfer

functio.

(7-2.14) becomes y(s) =

~'(s),

(7-2.15)

which is a PPS. This corresponds

Example 7-2.3. Consider

y = [t ~r

with the result of dynamic

the two-order

compensation.

system

0]x.

any state feedback

system has the following

in the form of input-output

u = Kx + v = [kl, k2}x + v, its closed-loop

transfer

relationship:

y<s = [l 01(s[ 0 = [f we select the closed-loop ctice.

_(l+k2)s

I + (l_kl_k2)

v(s).

k I = -l, k 2 = -I, it will be y(s) = v(s). system has only infinite

poles,

this design

Noticing

that in this case,

is not applicable

in pra-

207 Example 7-2.4. The following system

[o 01 = 1]x

y = [o

and the state feedback u = [k I

k2]x + v for.) tile closed-loop system

(7-2.16) y = [o

1]x,

which has the input-output relationship s+l _(l+k2)s + l+kl_k 2 v(s).

y(s) =

Choosing k I = k2 = -2,

we have y ( s ) = ( s + l ) / ( s + l ) v ( s )

T h i s s y s t e m h a s no i n f i n i t e

poles.

And s t a b l e

= v(s),

zero-pole

which i s a l s o a I'|'S.

cancellation

occurs here,

7-2.2. P-D state feedback Now we consider the P-D state feedback:

(7-2.17)

u = KlX + K2x + r(t).

According to selected

the

previous discussion,

such that (7-2.10) is r.s.e, to

a P-state feedback u = Kx + v(t) may (7-2.13).

For any feedback

be

(7-2.17),

we

denote

^

K1 = K1 - K = [ k ] ,

1 k2,

...

, k~]p - I '

2

K 2 = [k~, k 2 ....

, k~lP -1

A

v = KIX + K2R + r ( t ) . Then t h e c t o s e d - l o o p

s y s t e m formed ( 7 - 2 . 1 )

and ( 7 - 2 . 1 7 )

is

(E-bK2)~ = (A+bKI)X + b r ( t ) y

=

(7-2.18)

CX,

which has the transfer relathmship o n-I o n-2 ClS +c2s +. • .+c n y(s) = c(s(E-bK2)-(A+bKl))-ibr(s) =

2 n 2 1 n-I .+(_l_kln) r(s) klS +(k2-kl )s +..

(7-2.19) showing

that as in the case of P-state feedback,

under any P-D state feedback

in

the form of ( 7 - 2 . 1 7 ) the c l o s e d - l o o p system ( 7 - 2 . 1 8 ) has a f i x e d numerator unless zero-pole cancellation occurs. However, its denominator may take any polynomial

of

208

order not greater than n.

7-3.

State Feedback and Transfer Matrix: Multiinput Multioutput Systems

The results

on transfer matrix structure u n d e r state feedback

for

single-input

s i n g l e - o u t p u t systems, a l t h o u g h s i m p l e , ore apl)lical, l e to m u l t i input m u l t i o u t p u t systems:

E~ = Ax + Bu y = Cx

(7-3.1)

where x E ~ n , u E]R m, y6 ~r

are its state, control input, and output,

and E, A E ~nxn,

C 6 ~ rxn are constant m a t r i c e s . System ( 7 - 3 . 1 )

B6 ~nxm,

respectively; is assumed

t o be r e g u l a r and rankE < n. C o n s i d e r t h e P-D s t a t e

feedback control

u = KlX + E2~ + Fv

where KI, K26 ~mxn,

F ~mxm

(7-3.2)

are constant matrices, F is nonsingular, and v(t)

is

the new input. Under tile control (7-3.2), system (7-3.1) has the closed-loop system (E-BK2)~ = (A+BKI)X + BFv (7-3.3) y = Cx

with the transfer matrix GF(S) -~ C(s(E-BK2) - (A+BKI))-IBF, whose

structure

Subject

is dependent of the selection o f KI, K2, F.

to the controllability and o b s e r v a h i l i t y ^

for ally

(7-3.4)

a satisfying IaE+AI ~ O,

assumptions for system

(7-3.]),

is controllable and observable

(Section

^

(E,B,C)

2-4), where = (aE+A)-IE,

fi = (aE+A)-IB.

(7-3.5)

System (7-3.1) is r.s.e, to its EF3: (7-3.6)

y = Cx

with the closed-loop transfer matrix ^

^

GF(S) = C((s+a)E - sBK 2

_1 A -

(I+BKI))

A

^

^

^

BF = C((s+a)(E-BK2)-(I+B(KI-aK2)))-IBF. (7-3.7)

209 We will start from (7-3.7)

to study the transfer matrix structure under P-D state

feedback control. Theorem 7-3.1 (Dai,

1986b).

Let system (7-3.1) bc

controllable and ^

rankB = m. Then for any two a's, aI ~ a2, satisfying I aE+AI ~ O,

observable,

a

^

A

(EI,BI) and (E2,B2)

have the same controllability indices. Here, ~i = (aiF'+A)-IE' Bi = ( a t E+A)-|B' J = 1,

2. A

This

theorem

A

shows that the controllability indices o[ (E,B) are independent

the selection of a provided

a satisfies I aE+AI ~ O.

of

We call these numbers the cun-

trollability indices for system (7-3.1) and denote them by

~ ~ ~ ~ ... < ~.

(7-3.8)

Particularly, when A is nonsingular,

a may be chosen as zero. The controllability

indices of (E,B) are those of (A-1E,A-IB). A

Similarly, also

if rankC = r, we may prove that the observability indices of (E,C) are

independent of a

satisfying IaE+AI ~ O.

We define them as

the

observability

indices for system (7-3.1) and denote them by o

Example 7-3.1. Consider the system

-I

i

0

0

~

-

-I

1

y = [0

In t h i s system,

=

0

0

x

+

0

0

u

0

(7-3.9)

1]x.

matrix A i s n o n s i n g u l a r .

Thus,

the c o n t r o l l a b i l i t y

i n d i c e s for

t h i s system are t h a t of (A-1E,A-1B). Since E = A-IE =

O 0.5

0 0.5

i]

B ffiA-IB ffi

,

which has the controllability indices I and 2,

[o

1

O I

0.5

system (7-3.9)

, has

controllability

indices 1 and 2. ^

It

may be testified that this system is observable.

with observaI)illty

index 3, which is that

Hence,

(E,C)

is observable

for system (7-3.9).

Let system (7-3.1) he controllable, rankB = m, and

~

~

ff~^~^... ~ o ~

controllability indices for system (7-3.1), which are also of (E,B).

be

the

210 Denote

B = [bl, b2, ... , bm].

According

to the definition

of controllability

^

indices,

and by rearranging

the order of columns

assume that B is in its original ^

^~-1

M = [bl,gbl,...,E

in B,

without

loss of generality

we

order such that the matrix

,, ~2c - I

^

bl, b2,Eb2,...,E

•

^ b2,

,~o.Cm_l._

... , bm,Ebm,.,

bm]

is uonsingular. i Denote di = ~ , j=L

~jc, . i. =. 1. .2 .

.

m, and T k is the cl. Lhl row in M -l, k = l, 2,

, m. We construct

. . .

^ (y~-I

•

T = [TI/TIF,/.../TIE By t h e c o n s t r u c t i n g scalar,

,ff~-]

^

C-I

/T2/T2E/.../T2E

method,

/"'/Tm/TmE/'"/Tm

we know t h a t T s a t i s f i e s

showing t h e n o n s i n g u l a r i t y

~m

].

ITMI = ( - 1 ) d, d i s

of T. For t h e n o n s i n g u l a r

(7-3.10) a

positive

m a t r i x T c h o s e n i n such

a way, we d e f i n e = TFT -I = (Eij),

B = TB,

C = CT -I.

(7-3.11)

Then O 0 L.

:

1 0

0 l

,.,

... ...

0 0

c

E R

...

~i x

~c

j

II

0 ,.T.-

0 .~

0 ..~

,,,

..,

0 0

0 0

0 0

... ...

1

(7-3.i2)

t**

zj =

E..

0

0

0

0

...

~

~

,,,

0

0

=

0

0 0 I

where "." represents On the other hand,

0

i~j

0

O"

0 0 6

0 O

Q

g

i

•

0

I

@ @ @

0

~nxm ,

0

9@@

.

I

@ m

a

U

,,,

0

C

0 @ 0

°°,

0

c

5 i x aj

°,°

,,°

0

O 0

0

the possibly equation

I

1 nonzero

(7-3.7)

elements.

shows

GF(S) = C(gE - I - BE1 - K2u ) - I ~ F '

(7-3.13)

211 where ~ = s+u, K] = K 1 - K2a . Let E 6 ~mxn

be the matrix formed by the d. rows of E, B

m

1

m

be an mxm nonsingular

matrix formed by the d i rows of B, =A

K1

--1

KIT

=

'

K2

K2T-I

.

(7-3.]4)

S(.) = diag(

, i = l, 2 . . . . .

m).

I

It is easy to verify that

Thus we have (E.-

(I+BKI+BK2.))-IB = S(.)((Em-B32).S(.)

- BmKI{(.) - l)-IBm.

Rearranging the co|umns in (Em-BmK2)~S(N) - BmKIS(N ) - I so as to he written as (Em-BmK2).S(.) - BmKIS(.) - I = KS(.), where S(~) = diag(

c [.~i .....

~ , I]~, i = I, 2 . . . . .

m).

Paying attention to the form of S(g) and the nons[ngulariky of Bm' assure that

K is determined uniquely by K I,

K 2,

and,

conversely,

from (7-3.14) KI,

K 2 may

we be

se]ected to satisfy (7-3.17) once K is fixed.

Thus CF(S) Denoting

R(s)

=

=

Cg(~)(KS(,))-IBmF.

CS(~). N(s)

=

~S(,). f

= (BmF)-|K, we have

GF(S ) = R(s)N-l(s). Therefore, itself

and

greater than

(7-3.15)

from this process we see that R(s) is determined completely by the system N(s)

may take any nonsingular matrix whose order of ith column

is

not

~ic by suitable selection of KI, K 2, F.

In summary we have proven the following theorem. Theorem 7-3.2. Assume that system (7-3.1) is controllable, P-D state feedback (7-3.2),

rankB = m.

the closed-loop transfer matrix GF(S) has the

Then under following

212 structure: CF(S) = R ( s ) N - l ( s ) where

R(s)

is

(7-3.16)

determined by the system itself and N(s) may

take

polynomial matrix whose order of ith column is not greater than For

any

nonsingular

c G[.

multivariable system (7-3.1), it is called a pure prediction system (PPS)

if

there exists a matrix K such that y(s) = Ku(s). A direct

result

o f Theorem 7 - 3 . 2 i s t h e f o l l o w i n g

Theorem 7-3.3.

Assume that system (7-3.1) is controllable, rankB = m. There exist

feedback gain matrices KI, K2, PPS

if

theorem.

and F

such that the closed-loop system (7-3.3) is

and only if a matrix K E ~ r x m

exists satisfying R(s) = CS(s) = KT(s),

a

where

T(s) is nonsingular. Example 7-3,2.

For system (7-3.9),

a = 0 satisfies I aE+AI ~ O, and

2. Example 7-3.1 gives

1

=

2 0

2 0

1

O]

l

1

0

0

~ =

M = [b I, b2, Eb2] =

0

1

3

I

0

I

1

0

-~

0

1

1

1

= [bl,

b2]

and

M -I

=

I

1

0 4

1 2 -5

-~

0 o

.

Thus we have T I = [I, -3' I], T2 = [_4, -3' O], and

T =

T11 T2^

T2F.

=

1

3

1

3

_4 _2

o

3 2 -~

0

Direct computation shows that

3 2 5

1

1

2

0

2 1 -~

i

o

½

O

I] 7 -1

,

=

. . . .

i

a~ = i, O5 =

213 1

E=TET -1--"

0

0

1

0

0

~

--cr -I - [i

Theorem 7-3.2,

we

,

O 0

1 ~1.

0

i

Therefore, we have R(s) = [ 1 By

B=

1

~].

know that

for this system the general

of

structure

its

closed-loop system under feedback u = KlX + K2x + Fv is GF(S) = [ i

~lN-l(s),

where N(s) is nonsinguiar and 2

S(s)

all+al2s

a21+a22s+a23s ]

a3]÷a32s

a41+a42s+a43s

2

=

Choosing all = a41 = l, and the others are zero, we simply have G(s) = [ l,

l ~]-

In viewing of the preceding results,

prob-

we can now consider the model-matching

lem. The

so-called

model-matching problem may be stated in such a way:

transfer matrix Gm(S) (the model),

For

a given

we find the feedback matrices KI, K2, F such that

GF(S ) = Cm(S), or the closed-loop system takes Gm(S) as its transfer matrix. Let Gm(S)

be a given model.

We make decomposition on the matrix Gm(S)

Gm(s ) = Rm(S)Nml(s)

(7-3.17)

where Rm(s), Nm(s ) a r e rxm and mxm p o l y n o m i a l m a t r i c e s , Nm(S) is n o n s i n g u l a r ,

and

Rm(s) and Nm(s) are r i g h t eoprime, i . e . , rank

Nm(s)

= m,

Y s £ C, s f i n i t e .

Theorem 7 - 3 . 4 . Let sysLem ( 7 - 3 . 1 ) be c o n t r o l l a b l e , rankB = m, Gm(S) = Rm(s)Nml(s) be an rxm transfer matrix. Then there exist feedback gain matrices KI, K 2, F such that GF(S) = Gm(S) if and only if there exists a nonsingular polynomial matrix H(s)~ ~{mxm such that 1. R ( s ) = lCm(s)H(s).

2. The order of the ith column of Bm(S)H(s) is not greater than ~ c m.

i = 1, 2, . . . ,

214 Proof.

Sufficiency:

For any Nm(s)H(s) satisfying condition 2,

constant matrix M such that Nm(s)H(s ) = MS(s).

Let K = M.

there exists

Then K1, K2, F

may

u be

chosen from K. And we have CF(S) = R(s)N-I(s) = Rm(S)ll(s)(Nm(s)H(s)) -I = Gm(s). Necessity: Let KI, K2, F be gain matrices such that GF(S) = R(s)N-l(s) = Rm(s)Nm1(S). Then we have R(s) = Rm(s)Nml(s)N(s) = Rm(s)adj(Nm(s))N(s)/INm(S)l

(7-3.18)

Furthermore, under the decomposition. Rm(S) and Nm(s) are right coprime. There must exist two matrices Dl(S) and D2(s) such that Dl(S)Nm(S) + D2(s)Nm(s) = ImRight multiplying both sides of this equation by adj(Nm(s))N(s ) yields Dl(S)Rm(s)adj(Nm(s))N(s) + D2(s)Nm(s)adj(Nm(s))N(s) = adj(Nm(s))N(s ). Noticing the fact Nm(s)adj(Nm(S)) = INm(S)II and (7-3.18), we know adj(Nm(S))N(s ) = H(s)INm(S)l , where H(s) = D1(s)R(s ) + D2(s)N(s), indicating that Nml(s)N(s) = adj(Nm(s))N(s)/INm(s)t = ll(s) is nonsingular. Hence

N(s) = Nm(s)H(s ) and R(s) = Rm(s)Nml(s)N(s) = Rm(s)ll(s). This is condition I. Condition 2 is easy to obtain according to Theorem 7-3.2. Q.E.D.

215 7-4. Singular Input Function Observers

Consider the singular system (7-3.1): E~(t) = Ax(t) + Bu(t)

(7-4.1) y(t)

where x ( t ) ( E n ,

Cx(t)

=

u(t)~m,

y(t)E~r

ra.kE < n, and sysLem (7-4.|) is regular.

De[initiun 7-4.1. Consider system (7-4.1) with initial condition x(O);

y(t) is

its measure output when t ~ O. If there exist matrices Ee, Ac QsEe-Ac[~ 0), Bc, and i n i t i a l

C e,

c o m l i t i ( m xc(O) such t h a t Ec~c(t ) = Acxc(t ) + Bcy(t) u(t) = Ccxc(t)

where xcE • nc,

(7-4.2)

and equation (7-4.2) holds for any control input,

we will call sys-

tem (7-4.1) input function observable, and (7-4.2) is an input function observer (IFO). The IFO,

or inverse system,

is a system

to construct precisely (observe)

input

funct[on u ( t ) using the measurement of o r i g i n a l system as i t s input. The [FOs may be described by Figure 7-4.1.

J Ecxc=Acxc+BcY

Ex=Ax+Bu I y=Cx

u(t) (unknown)

y(t)

-]

fi = Ccxc

~(t)

= u(t)

(known)

Figure 7-4.1.

Input Function Observer

Let C(s) = C(sE-A)-IB

(7-4.3)

Gc(s) = Cc(SEc-Ac)-IBc

(7-4.4)

and

be the transfer matrices of (7-4.1) and (7-4.2), respectively. Then we have tlle foil-

216 owing theorem. Syste~l (7-4.1) is input f u n c t i o n observable if and only if there

Theorem 7-4.1.

exist constant matrices Ec, Ac, Be, and Cc, such that Cc(s)G(s ) , Im . Proof. Necessity:

(7-4.5)

Assume that system (7-4.1) is input function observable.

There

exist constant matrices Ec, Ac, Be, Cc, and initial condition xc(O) such that (7-4.2) h o h l s f o r any u ( t ) .

Taking L a p l a c e t r a n s f o r m a t i o n

on b o t h s i d e s

of (7-4.2)

we o h t a i n

u(s) = Cc(sEc-Ac)-l(Ecxe(O) + BeY(S)).

(7-4.6)

Moreover, from (7-4.1) we have y(s) = C(sE-A)-l(Ex(O)+Bu(s)). Substituting it into (7-4.6) yields u(s)

= Cc(sEc-Ac)-I[Ecxc(O) + B c C ( S E - A ) - I ( E x ( O ) + B u ( s ) ) ] .

Since this hohls for any u(s), it must be that C c ( s E c - A c ) - I B c C ( s E - A ) - I B = Im,

which is (7-4.5). Sufficiency:

Sufficiency is proven

constructively.

Let Ec, Ac, Bc, and C c exist

satisfying (7-4.5). We construct the system Ec~ I = Acz I - BcCz 2 + BeY , E~ 2 = Az2,

Zl(O) = 0

z2(O ) = x(O)

= CcZ 1 . By taking Laplace transformation on both sides of these equations we know that ~(s) = CcZl(S) = Cc(sEc-Ac)-l(BcY(S)-BcCz2(s)) Cc(sEc-Ac)-l[BcC(sE-A)-l(Ex(O)+Bu(s))- BcC(SE-A)-IEx(O)]

=

= Cc(S)C(s)u(s)

= u(s).

^

u and u a r e

identical,

Thus t h e

[Ec 0

E

[Ac

z2

0

A

z l ( O ) = o, u = [C c

system

z2 (7-4.7)

z2(O) = x(O)

O ] [ Z l / Z 2]

i s an IFO f o r s y s t e m ( 7 - 4 . 1 ) .

Hence, t h i s

system is input function

observable.

Q.E.D.

217 the substate z 2 in (7-4.7) is unobservable and the transfer matrix

Obviously,

of

(7-4.7) is Ge(S) . As shown in Chapter 2, any rational matrix has a singular system realization. Combining the two preceding theorem, we have the following result. Theorem 7-4.2. System (7-4.1) is input function observahle iff its transfer matrix G(s) is left invertible. Further from Theorem 7-4.2, we have the following curoltary. Corollary 7-4.1. other words,

If system (7-4.1) is input function observable,

to determine inputs from measurements,

r ) m,

or

in

there must not he fewer measures

than inputs.

[looo] [,ooo] [ o]

Example 7-4.1. Consider the following four-order system

0 0 0

1 0 0

0 0 0

0 0 0

:}=

0 0 0

2 0 0

0 1 0

0 0 1

01 1 -1 0 O-I

× +

u

(7-4.8) y=

O 0 0 0 x 2-110

with transfer matrix -2(s+l)

s+l

0

0

G(s) = C(sE-A)-IB =

s

-1

s-2

s-2

which is left invertible. By Theorem 7-4.2, its IFO exists. It has been shown in Example 7-1.2 that

Gc(S)

..1. t s+l

s+2

1]

s s+l

s+--i s+2

2

=

[loooj [ooo]

is a left inverse of G(s). Gc(S)G(s) = Im; m = 2. Choosing

Ec=

Cc =

0 0 0

1 0 0

0 0 0

1

0

-1

-1 -1

0 0 0

Ac

= , 0 ]

0 -1

0 -2 0 0 0 0

0 1 0

0 0 1

BC =

,

1

0

0

0

1

0

0 1

0 1

1 2

)

218

it will be Gc(S) : C c ( s E c - A c ) - l B c .

oo632,1, [oo1

Thus, from Theorem 7-4.1 we may

f o l l o w i n g IFO f o r s y s t e m ( 7 - 4 . 8 ) :

i°°° 1

0

0

0 0

0 0

0 0

0 •

-

0 !

0

0

0 0

1 0

0 0

0

0

0

0 0 0 0

~2

i1

=

,1

-I -I

0

0

0

1

0

2-1

0

1 -2

1

0

2 0

0 2

(} o[LZ2J + 0

0

0

0

0

Zl(O) = O, n

0

0

oo

0

0

1

construct

tile

1

1

Zl]

o]

0

?

1

2

Y

o

z2(O) = x(O)

(7-4.9)

o o o o op, ] 0 -I

0 0

0 OJLz2J

which i s o f o r d e r 8.

IFOs o b t a l n e d i n such a way a r e g e n e r a l l y o f h i g h o r d e r . BuL i t has been shown (EI-Tohami et al., 1985) that a process may be imposed on it to obtain IFOs of lower order. Furthermore, from rank(s[~

=

n

+

~1-[~

~] ) = rank[ : ~ - A - ~ ] [ ~ - ( s E l A)-IB]

rankC(sE-A)-IB

and Theorem 7-4.2, we have the following corol]ary. Corollary 7-4.2. System (7-4.1) is input function observable if and only if

rank(s

+ 0

0

) :

n + m

C

is false for no more than n+m

s.

This condition may be easily verified by a slight modification algorithm in Section i-2.

of the "shuffle"

219 7-5.

Normal Input Function Ohservers

For the difficulty in the realization of singular systems, singular IFOs are difficult to realize physically. and design methods for normal IFOs,

as pointed out before,

Now we will consider the existence

which are easier to realize than singular ones.

The normal IFO is an IFO in the form of Xc = Acxc + BeY (7-5.1) u = F c x c + Fy. Generally,

exist

a singular system does not necessarily

[FOs for sonic s i n g u l a r systems.

have normal

may have normal ]FOs. There definitely exist us normal Theorem 7-5.1.

IFOs. llowever,

there

IL has beeu proveu t h a t only s i i l g u l a r sysI('.IS

If there exists a norms]

[FO (7-5.1)

IFOs for normal systems. for system (7-4.1),

E must l,e

singular. Proof. Let (7-5.1) be a normal IFO for system (7-4.1). According to Theorem 7-4.1, we have [F + Fc(sI-Ac)-IBc][C(sE-A)-IB] If E is nonstngular,

= Im.

without loss of generality,

we assume E = I n, and tile preceding

equation becomes [F + Fc(SI-Ac)-IBc]C(sI-A)-IB

= I m.

which holds for any s. But this is obviously false if we set s ~

in the preceding

equation. This conflict shows E is singular. Q.E.D. Thus o n l y s i n g u l a r transfer lar

s y s t e m s have tile p o s s i b i l i t y

m a t r i x i s d e t e r m i n e d by t h e c o n t r o l l a b l e

s y s t e m may have a n o r m a l IFO o n l y i f

its

to p o s s e s s a n o r m a l [1'~. S i n c e and o b s e r v a b l e s u b s y s t e m ,

controllable

the

a singu-

and o b s e r v a b l e s u b s y s t e m i s

singular.

For any r e g u l a r

ssytem (7-4.1),

there exist two nonsingular matrices Q and P such

that [t is r.s.e, to EFI:

il = AlXl + BlU (7-5.2)

N~ 2 = x 2 + B2u y = ClX I + C2x 2

where p-Ix = [Xl/X2], x I E R nl,

x2

• n2'

n 1 + n 2 = n, and

220 QEP -- diag(Inl , N),

QAP = diag(A I, In2),

QB = [B1/B2],

CP = [C 1 C21,

N E ~ n2xn2 is nilpotent with a nilpotent index h. Subject to EFI (7-5.2), £he trans[er matrix C(s) is (7-5.3)

G(s) = C(sE-A)-IB = CI(SI-AI)-IBI + C2(sN-I)-IB2 . Let us take observability decomposition on (N,C2) to obtain

TNT-1 =

I,°l 1 N21 N22

e2T-1 = [C21

O]

,

where T i s nonsingular and (Nll,C21) i s o b s e r v a b l e . By denoting Tx 2 = [ x21 ]

x2i

EI~ n2i

,

i = 1 2 ,

)

L x22 ] , B2i EI~ n2txm ,

TB2 = [ B21 ] B22 J ,

i = 1,2,

n21 + "22 = n2' system ( 7 - 5 . 2 ) (thus system ( 7 - 4 . 1 ) )

is r . s . e .

LO

~1 = AlXl + BlU NlI~21 = x21 + B21u

(7-5.4)

N21121 + N22~22 = x22 + B22u y = ClX 1 + C21x21. Theorem 7-5.2. mxn2| such that |.

System (7-4.1) has a normal IFO if and only if there exists an M E

MB21 = Im,

2.

MNII = O.

P r o o f . N e c e s s i t y : [.et ( 7 - 5 . 1 )

be a normal [FO. Then by Theorem 7-4.1

[F + Fc(SI-Ac)-IBc]C(sE-A)-IB = Im. Expanding it we obtain Im = [F + Fc(sI-Ac)-IBc][CI(SI-A1)-IB I + C21(SNlI-I)-IB21 ] = [F + Fc(SI-Ac)-IBc]CI(sI-AI)-IBI

it will

be

221

h-i

i i

Fc(Si_Ac)_iBcC2l h-i

- FC21 i__~O - . s NllB21 =

i i

(7-5.5)

i__~O " . s =NIIB21

Since

(7-5.6)

si(sI-Ac )-I = si-ll + sS-2Ac + ... + A~(sI-Ac )-I,

equation (7-5.5) becomes h-l

lm = [F+Fc(s[_Ac)-lgc]Cl(si_hl)-lgl

i _ Fc(sT_Ac)-lffcC2lB - l,,C21 i=~9 si NIIB2I

h-I _

~Fc[si-Ii+si-2Ac

+ .,. + Ai(sI_A )-IBcC21NIIB2 li

i=l - Fc(SI-Ac)-IBcC2]B21. Equaling

the

coefficients of correspondiug terms at both sides of this

equa t/on

ytetds

-

h-i FC21B21 - ~ F c A i=l

i-I i c BcC21NI]B21 = Im.

(7-5.7)

On the other hand, by combining (7-4.1) and (7-5.1), we know that u(s) = Fcxc(S) + Fy(s) =

Fc(sI-Ac)-l×c(O) + [F+Fc(SI-Ac)-IBc]C(sE-A)-IEx(O) + [F+Fc(SI-Ac)-IBc]C(sE-A)-IBu(s)

holds for any x(O), xc(O), and u(s). P a r t i c u l a r l y ,

s e t t i n g u(s) = 0 r e s u l t s in

Fc(sI-Ac)-lxc(O) + [F+Fc(SI-Ac)-IBc]C(sE-A)-IEx(O) = O. Keeping in mind that the preceding equation haids for any x(O),

by using

(7-5.6)

to expand this equation, and equaling the constant terms on both sides of it, we have FC2N + FcBcC2N2 + ... + FcA c11-2BcC2N h

= O,

or equivalently, h = O. FC21NII + FcBcC21N~I + ... + FcA h-2 c BcC21NII

(7-5.8)

Let h-2 h-| M = - (FC21 + FcBcC21NII + ... + FcA c BcC21NII ). Eq~ons

(7-5.7) and (7-5.8) are conditions 1 and 2.

Sufficiency:

Assume that matrix M satisfies conditions I and 2.

both sides of the second equation in (7-5.4) with M we obtain

Left multiplying

222 (7-5.9)

u = - Mx21. Oil the other hand, system

satisfies

the dynamic

(7-5.4)

NIl

that tlle substate

equation:

i ,o ] IAl 0

~mplies

}" =

C 1 C21

0

I-B21M

0

0

~ +

I°] 0

~.

(7-5.10)

I

By d e c o a q ) o s i t i o n , (NII,C21) i s o b s e r v a b l e . T h e r e f o r e tlm m a t r i c e s

[io]

Nll

and

C2lJ are of full

0

Nil

CI

C21

co]umn r a n k s . Thus, a m a t r i x |l may be chosen so t h a t

|I

0

Nil

C[

C21J

Left multiplying

= I.

both sides

of ( 7 - 4 . 1 0 )

by |I we have (7-5.1])

= Ac~ + Bc~ where

[,

Ac = 11 0

I-B21M

O

Denoting

O

x c = ~ - BeY,

]

Be = tl ,

(7-5.11)

Xc = AcXc + B c Y '

I,°] 0

I

and (7-5.9)

.

respectively

become

xc(O) = ~(O)-BcY(O) (7-5.12)

u

= FcX c + Fy

where B c = AcBc, system

(7-4.1).

F c = - M[O

I], F = - H[O

TIB c. Thus

(7-5.|2)

is a normal

IFO

for

Q.E.D,

Some more convenLcnt conditions may be deduced from tile conditions in t h i s rem.

Let T I and T 2 be any two nonsingular TIN|IT 2 = diag(I~,

0),

matrices

~ = rankNll

satisfying

theo-

223 and denote TIB21 = [BI/B2]. Theorem 7-5.2 implies the following.

C o r o l l a r y 7-5.1. o n l y if rankB 2 = Or,

System ( 7 - 5 . 1 )

has a normal IFO

in the form of ( 7 - 5 . 1 )

i l and

m,

in other

words,

the dimension of the ohservable

suhsystem

i s no l e s s

than the

number o f i n p u t s . Corollary

7-5.1

further

Corollary. 7-5.2. only if

rank[E

shows

Dual n o r m a l i z a b l e

system (7-4.1)

B} = r a n k E + m, w h i c h i s c o n v e n i e n t

Ires a mJrmal IFO ( 7 - 5 . 1 )

The d e s i g n o f n o r m a l IFOs may f o l l o w t h e s u f f i c i e n t

Example 7-5.1.

i f and

for testtfieation. proof

in Theorem 7 - 5 . 2 .

Example 7-4.1 has shown that system (7-4.8) has a singular

IFO (7-

4.9). For system (7-4.8), dual normalizable,

m = r = 2,

and rank[E

n = 4,

rankE = 2, aml rank[E/C]

B] = 4 = rankE + m.

Hence,

= 4.

by Corollary

Thus

it is

7-5.2,

this

system has a normal IFO in the form of (7-5.1). In fact,

if we let x = [xl/x2] , Xl, x 2 ( ~2, system (7-4.8) may be written as =

2

O]u

1

0

= x2 - u

y

=

(7-5.13)

0 2-

xI +

0

0 0

1

x2 .

Clearly,

(7-5.14)

u = x2 • Moreover,

system (7-4.8) shows that x satisfies

oOlO 00 00 2-2 00

0 0 0 0

[i°° -

-1

1

Left multiplying

tl=

and r e w r i t i n g ,

2 0 0 0

~=

0 2 0 0

1 0 1 0

0 1 0 1

1

0 0

0 0

both sides of (7-5.15)

, 0 0 0

o 1 0 0

~.

x +

1

o 0 0 0

we o b t a i n

o 0 0 0

o 0 0 1

o 0 0 0

o 0 1 2

by

o i o

o o I

(7-5.15)

224

[2°,1][oo 1 o 0 0

Xc =

2 0 0

o 0 0

xc(O) = ×(o)

[o o u=

0

102 000 000

xC +

y

0 O O] 000

-

0 1

0 0

1 2

(7-5.16)

y(O)

o] +[0 0

0

0

I

Xc

1

0

2

Y

ooo]

where

XC

:

X

0 0 0 0 0 1 1 0 2

--

y'

which is apparently a normal IFO for system (7-4.8). Furthermore,

to left

multiply

(7-5.13) and (7-5.14) show that if we use the matrix

both sides of the third

u = -

-2

1

Xc

1

where x c = x I. The substitution

=

2

1

×c ÷

equation in (7-5.13)

0

it

will

be

Y

of it into the first equation in (7-5.13) yields

1

0

2

Y

(7-5.18)

xc(O) Xl(O) =

which,

together with (7-5.17),

forms another normal IFO for system (7-4.8) of order

2, which is lower than the preceding one.

7-6.

In

a multivariable

several control

control system.

inputs,

control

Decoupling

system,

which r e s u l t s

Decoupling control

every scalar

determined

ly,

as

by o n l y one c o n t r o l

input-output

i s a kind of c o n t r o l relationship.

stratesy

a[fected

by

relationship

in

that

can make

the

One o u t p u t may be c o m p l e -

i n p u t . Such s y s t e m s a r e c a l l e d

long as the decoupling i s achieved,

several independent single input,

output is generally

in a tempi i c a t e d

c l o s e d - L o o p s y s t e m have a s i m p l e i n p u t - o u t p u t tely

control

decoupled. Obvious-

the mulLivariable system is sp]iL

into

single output systems. The advantage of decoupling

225 control

lies in its simplicity and reliability,

especially when the control is par-

tially executed by a human factor.

7-6.1. Dynamic decoupling control Definition

7-6.1.

A singular

fer matrix is diagonally

system is called

(dynamically) deconpled

if

its

trans-

nonsingular.

Consider the multivariable singular system E~ = Ax + Bu (7-6,1)

y = Cx

where x 6~n, u 6 ~ m ,

y E~r

tively; E, A E ~ n x n

B 6~nxm,

are its state, control input, and measure output, respecC E~rxn

are constant matrices; rankE < n. System

(7-

6.1) is regular. Let the feedback control be the P-D state feedback u = Klx + K2~ + Fv where K I, K 2 6 ]Rmxn,

F E ~mxm

(7-6.2) are constant matrices.

System (7-6.1) and control (7-6.2) form the closed-loop system: (E-BK2)~ = (A+BKI)X + BFv (7-6.3) y=Cx with the transfer matrix (7-6.4)

GF(S) = C[s(E-BK2) - (A+BKI)]-IBF. S i n c e CF(S ) i s an rxm m a t r i x ,

t o g u a r a n t e e t h a t Gl,(S ) i s d i a g o n a l we s t i p u l a t e

r =

m ~ n and F i s n o n s i n g u l a r . I n this section, system (7-6.1) is assumed both c o n t r o l l a b l e

By definition,

and observable.

the decoupling problem is to find feedback matrices KI, K2, F such

that GF(S ) is diagonally nonsingular, i.e.,

GF(S) = diag(gl(s) , g2(s), .... gm(S)),

gt(s) is nonzero polynomial, i = I, 2, ... , m. We have seen

in Theorem 7-3.2 that if system (7-6.1) is controllable and rankB

=

m, the closed-loop transfer matrix under P-D state feedback (7-6.2) has the form: GF(S ) = R(s)N-l(s) where

R(s) is independent of the selection of KI,

(7-6.5) K2, and F. N(s) may take any non-

c Let N(s) singular polynomial matrix whose order of ith column is not greater than ffi"

226 be denoted by No(S) when K 1 = K 2 = 0 and i" = I. Then

(7-6.6)

G(s) = C(sE-A)-IB = R(s)N~l(s).

Theorem 7-6.1. For system (7-6.1), there exists feedback control (7-6.2) such that

is decoupled if[" i t s t r u n s f e r m a t r i x (;(s) is nOllsingu-

the c l o s e d - l o o p system ( 7 - 6 . 3 ) lar. Proof. Necessity:

Let KI, g 2, F be matrices such that (7-6.3) is decoupled.

(iF(S) i s d i a g o n a l l y Sufficiency:

nonsingular,

If

implying the IIonsingularity

G(s) i s n o n s i n g u l a r ,

from ( 7 - 6 . 6 )

we know R ( s )

Note that the order of ith column of R(s) is not greater than

g2, 1' such Lhat N(s) = R ( s ) , and GF(S ) = I i s d i a g o n a l l y Thus a n e c e s s a r y c o n d i t i o n If

system (7-6.1)

theorem

shows t h a t

f o r any a

~i" c

is nonsingular. There exist

nons[ngular.

Kl

Q.E.D.

f o r d e c o u p l i n g i s t h a t rankB = rankC = m = r .

may he d e c o u p l e d v i a f e e d b a c k c o n t r o l the transfer

satisfying

Then

o f G(s) by ( 7 - 6 . 6 ) .

(7-6.2),

the

preceding

m a t r i x G(s) = C ( s E - A ) - I B i s n o n s i n g u l a r .

l a E - A J ¢ O, and IG(a)l

Note

that

= I C ( u E - A ) - I B I ~ O,

GF(S ) = C(s(E-BK2) - (A+BKI))-tB = C ( s E - A ) - I B [ I - ( K I + S K 2 ) ( s E - A ) - I B I - I F (7-6.7) and C(sE-A)-IB = C(aE-A)-IB + aC(aE-A)-I(sE-A)-IB

- sC(aE-A)-I(sE-A)-IB.

By setting K] = -aFC(aE-A) -I, K 2 = FC(aE-A) -I, F = [C(,E-A)-IB] -I, we have GF(S) = I,

i m p l y i n g tile c h , s e d - l o o p By n o t i c i n g Corollary

m = r,

7-6.1.

s y s t e m i s a I'I'S.

from C o r o l l a r y

There exists

stale

7 - 4 . 2 and Theorem 7 - 6 . 1 , feedback control

we have t h e f o l l o w i n g .

(7-6.2)

such t h a t

the ch,sed-

loop system (7-6.3) is decoupled if and only if the following matrix pencil

([0 is reguiar,

which i s e a s y t o t e s t i f y

Particularly,

via the "shuffle"

algorithm.

when E = I n , Theorem 7-6.1 is applicable

to the normal system

= Ax + Bu

y = Cx. This means that 6.2),

the

(7-6.9) if the state feedback u = Kx + v is extended to tile general form (7-

decoup]ing problem may have a wider feasibility

control such that the closed-loop system is a PPS.

so as to find

a feedback

227 7-6.2. Static decoupling Now we will consider the static decoupling problem. Definition 7-6.2. System (7-6.1) is called static decoupling if it is stable and lim s~O

C(s) = Jim

C(sE-A)-IB

(7-6.10)

s~O

is a diagonally nonsiugular matrix; lims_~O C(s) is ass.reed Lo exist finitely. Under

the control (7-6.2),

the static decoupling problem is to choose

gain

ma-

trices KI, K2, F s u c h that I i m GF(S) = l i m C(s(E-BK 2) - (A+BK1))-IBF = d i a g ( g l , s~O

g2 . . . . .

gm )

s~ 0

where gi ~ 0 is scolar, i = I, 2, ... , m. The static decoupling properly states an asymptotical decoupling behavior, withouL mention the decoupling property for any finite time period. Theorem 7-6.2.

For system (7-6.1),

there exists a feedback control (7-6.2) such

that the closed-loop system (7-6.3) is static decoupl~ng if anti only

if it is stabi-

] izable and rank Proof. is static le,

C

0

Assume t h a t decoupling.

and s y s t e m ( 7 - 6 . 1 )

= n + m. there

(7-6.11)

exist

Therefore,

feedback matrices o(E-BK2,A-BKI)CI:

is stabilizable.

KI, K2, and F s u c h t h a t

(7-6.3)

. The c J o s e d - l o o l ) s y s t e m i s s t u h -

A-BK 1 i s n o n s i n g u l a r .

In t h i s

case,

[ira GF(S ) = lira C [ s ( E - B K 2 ) - (A+BKI)]-IBF = - C(A+BKI)-IBF s o0 s--O

is nonslngular,

i.e., (7-6).12)

rank C(A+BK])-IB = m. On the other hand, for such a matrix gl,

rank

C

= rank

= ra,k C

Combining it with (7-6.12) Conversely, a matrix

if (7-6.11)

0

we c o n c l u d e t h a t

(7-6.13) C

(7-6.11)

i s t r u e and s y s t e m ( 7 - 6 . 1 )

-C(A+BK1)-I B

.

holds. is stal, tiizab[e,

we may

choose

K1 s u c h t h a t

o(E,A+BKI) C C and thus both A+BK I and C(A+BKI)-IB are nonsingular according to (7-6.13).

(7-6.14) Setting

228 F = - [C(A+BKI) -IB]-I

K 2 = O, we have

lim GF(S ) = lim C[s(E-BK2) s-~O s~O

- (A+BKI)]-IBF ~ I m.

therefore,

for such matrices KI, K2, and F

decoupling.

Q.E.D.

Remark 1. A n e c e s s a r y

condition

the cIosed-loop system (7-6.3) is staLic

for static

decoupling

Remark 2. K1 and K2 a r e u s e d o n l y t o s t a b i l i z e

i s r a n k B = r a n k C = m.

the closed-loop

system.

F is chosen

to assure the static d e c o u p l i n g . Remark 3. As seen in the proof process, closed-loop

there exists a feedback (7-6.2) such that

system (7-6.3) is static decoupling if and only if a P-state feedback

n = KlX + Fv

(7-6.14)

may be c h o s e n s u c h t h a t

the closed-loop

system

E~ = (A+BKI)x + BFv (7-6.15) y = Cx is static Thus,

decoup]ing. for

static

decoupliag

p r o b i e m we p r e f e r

using the P-sLate

feedback

(7-

6.14) for its simplicity.

Example 7-6.1. Consider the singular system

0 1 0 0 0 0

~=

0 0 0 0 1

[1 0 Y = O 0 1 which is skabilizable.

0 0

u

(7-6.16)

0

x,

The f e e d b a c k g a i n m a t r i x

0 0 0

=

saLisfies

O]

×+

l

e(E,A+BKI) =

{-1,

-1 } C ¢ - .

In this system n = 3, m = r = 2, and

rank Thus,

by

C

0

= 5 = n + m.

Theorem 7 - 6 . 2 ,

(corresponding

to (7-6.15))

In fact, if we select

there

exists

is static

a (7-6,14) decoupling.

such that

the closed-loop

system

229

F = [C(A+BKI)-IB]-I 0

-1

and define

u

=

[0

0 o0 I x

-I

-2

[o_,]

+

_

2

v

the closed-loop system is

I1°il li '°] li] 0 0

i 0

~ =

-

-2 I

0 I

x +

-

2 0

v

I 0 O] Y=

0

0

i

x,

This may be verified by selecting v to be constant input,

which is static decoupling. in which case lim y(t) = v. t~o~

Concerning the relaLionship

between s L a t i c

d e c o u p l i n g and dynamic d e c a u p l i n g ,

Lhe

following has been proven (Dai, 1988f): Theorem 7 - 6 . 3 . it

may be s t a t i c

Thus, decoupting

System ( 7 - 6 . l )

may be d e c o u p l e d v i a P-D s l a t e

decoupled via

when ( 7 - 6 . 1 1 )

state

holds,

feedback ( 7 - 6 . 2 )

feedback (7-6.2).

t h e r e a r e two d e s i g n ways f o r o u r s e l e c t i o n :

or static decoupling.

if

The former decouples the system instantly

dynamic without

guaranteeing the stability for the closed-loop system, while the latter, assuring tile closed-loop stability, cannot achieve the instant decoupling property. IL is worLh pointing out that the inverse ol Theorem 7-6.4 is false.

Examnle 7 - 6 . 5 .

In t h e s y s t e m

{00] [01i] 10] 0 0

1 l

0 0

~=

0 0

0 0

x+

0 0

1 0

u (7-6.]7)

[1 Y =

0

0 0

O] I x

m = r = 2, n = 3. S i n c e

rank [ AC

0B ]

=4 may be chosen such that (8-3.4) is r.s.e, to N ~ ( k + i ) = 7~(k) + B y ( k )

where x(k) = F~(k),

(8-3.8)

and QEP = N, Q(A+BK)P -- In, QB = B,

N is.nilpotent and its nil-

potent index is denoted by h. According to (8-2.5), the state response of (8-3.8)

is

h-I x(k) = Px(k) = - >"-~ pNiBv(k+i) i=O showing that x(k) is independent of former inputs;

x(k) indicates the control func-

tion of present and future inputs v(k), v(k+l) . . . . .

v(k+h-l), revealing some prope-

rties of [uture inputs. Especially, when {v(k)) is not available in advance, its properties may be conjectured from that of output {y(k)}. This is one special feature of discrete-time singular systems. The f r e e

response

The f o l l o w i n g Corollary

of (8-3.8)

two r e s u l t s

8-3.1.

i s ~ ( k ) m O, k ~ O.

are useful.

If system (8-3.1)

is controllable

and n 2 > O, t h e r e

always exisLs

a matrix that satisfies (8-3.7). Corollary 8-3.2.

If system (8-3.1) is observable and n 2 > 0 (rankE < n),

always exists a matrix G such that deg(IzE-(A-CC)l)

there

= O.

i ooo] [11oo] [o]

Example 8-3.5. Consider system (8-2.7):

0 0 0

1 0 0

0 0 0

0 1 0

x(k+l) =

y(k) = [0

1

0 0 0

10]x(k)

l 0 0

0 1 0

0 0 1

x(k) +

1 ! -I

u(k).

(8-3.9)

250 It

i s easy t o t e s t i f y For any m a t r i x

IzE-(A+BK)I Thus,

if

that

K = [kl,

the system is k2, k3,

= - k3 z3 + ( l + 3 k 3 - k 4 ) z 2 -

we c h o o s e k 1 = - l ,

8-4.

controllable

and n 2 = 2 > O.

k 4 ] , we have (2+3k3-2k4-k2)z

k 2 = O, k 3 = O, k 4 = 1, i t

SLate Observation

for

will

l)iscrete-Time

+ l÷k3-kl-k2-k4.

be I z E - ( A + B K ) I

,gingular

= 1.

.qystems

C o n s i d e r t h e singular system Ex(k+l)

= Ax(k)

y(k)

= Cx(k)

+ Bu(k)

(8-4.1) where x(k) ( ~ n

u(k) ( ~ m

y(k) ( Rr,

E, A ( E n x n

B ( ~nxm,

C ( ~rxn

are constant

matrices, rankE < n is assumed. The

state

observation

problem for system (8-4.1)

i s to c o n s t r u c t

a dynamic system

o f t h e form E c x c ( k + l ) = Acxc(k ) + Bcu(k ) + C y ( k )

(8-4.2) w(k) = F c x c ( k ) + F u ( k ) + lly(k)

where ×c(k) 6~nc,

w(k) E ~ n ,

Ec, A c( ~ncXnc,

Bc, G, Fc, F, H are constant matrices,

and [zEc-Ac[ # O, such that the asymptoticai state reconstruction is achieved: lim (w(k) - x(k)) = O, k--~

V

x(O), xc(O).

In general, the state observers for discrete-time singular systems may be constructed in ways analogous to those of the continuous-time case. For system (8-4.1), we consider the following dynamic system e~(k+l) = A~(k) + Bu(k) + G(y(k)-C~(k)) In (8-4.3),

(8-4.3)

the first two terms on the right-hand side model the dynamic equation

of system (8-4.1), while tile last term is employed to amend the mismatch in the modeling process.

If ~(0) = x(O), the states of (8-4.3) and (8-4.1) are identical, i.e.,

~(k) = x(k), k ~ O. Let

e(k)

= x(k) - ~(k) be tile estimatLon error between x(k) and

its

estimatioll

~(k). Then e(k) satisfies the dynamics Ee(k+l) = (A-CC)e(k),

e(O) = x(O) - ~(0)

If system (8-4.1) is detectable,

a matrix G may be selected so that

(8-4.4) o(E,A-GC) C

2G1 U+.

Thus system (8-4.4)

ymptotically

is stable,

reconstructs

limk~ooe(k)

= O,

V e(O),

implying

that x(k) as-

the state x(k).

In summary we have proven the following. Theorem 8-4.1. ~nxr

If system (8-4.1)

such that (8-4.3)

Furthermore,

there must exist a matrix

satisfying

is observable,

IzE - (A-GC)I

= ~(k),

from Corollary

= constant.

matrices Q1 and PI may be chosen so that (8-4.4)

N~(k+l)

that

x(k) = ~(k), k = O, I . . . . .

If system

Therefore,

k ~ O, i, 2 . . . .

perty is termed exact reconstruction. Theorem 8-4.2.

to

k ~ O, I, 2 . . . .

e(k) = Pl~(k) = O,

such

8-3.2

In this case two

is r.s.e,

where QIEP 1 = N, QI(A-GC)P 1 = I n, e ( k ) = PlY(k) ~md N i s n i l p o t e n t .

or in other words,

G

for system (8-4.1).

if rankE < n and system (8-4.1)

there exists a matrix G E ~ n x r nonsingular

is detectable,

is an observer

(8-4.1)

x(k) and ~(k) are identical.

Hence we have proven is observable,

the state of system (8-4.1)

This pro-

the following.

rankE < n, there exists a matrix G

is reconstructed

exactly

by

an observer

(8-

4.3).

What

is

conditions, beginning

by

emphasized

in

the

of system (8-4.1) may be reconstructed

state

an observer

t h i s theorem is t h a t w i t h o u t any

of tile form (8-4.3).

s i n g u l a r s y s t e m s . Normal s y s t e m s d o n ' t

Example 8-4.1.

Consider

This reflects

knowledge exactly

of

at

a special

iniLial the

1

0

0

0 0

0 I

0 0

the system

x(k+l) =

y(k) = [0

1

for which the matrix C = [I eorem 8-4.2,

of

have such a p r o p e r t y .

I°°°l [°°°1 [°I 0

0 0

very

feature

1

1

0

0

0 0

0 0

1 0

0 [

x(k) +

1

u(k)

0 2

(I~-4.5)

I -1]x(k) 0

0 -i]: satisfies

IzE - (A-CC) I = 1 = constant.

By Th-

the observer

[o00] 11 i io] i] 0 0 0

reconstructs

Obviously,

1 0 0

0 0 1

exactly

0 0 0

x(k+l) --

1 0 0

1 0 1

0 l 1

0 0 0

x(k) +

I 0 2

u(k) +

0 0 -I

y(k)

the state of system (8-4.5) at every step.

if (8-2.2)

is the EFI for system

(8-4.1),

Theorem 8-4.2 holds,

provided

252 system (8-4.1) is R-observable, rankE < n and C 2 ~ O. As seen in Section 8-2, a realization of discrete-time system meeds future inputs. Thus the singular observer, though it exists theoretically, is difficult, or even impossible,

to physically realize, especially the state reconstructor. We use the term

state reconstructor

to refer an observer that reconstructs exactly the state

for

a

singular system. Thus, we now consider the design of normal observers. The

normal observer for system (8-4.1) is a kind of observer (8-4.2) when E c = I.

Analogous to that in the continuous-time case, we may prove tile following two Lheorems (Dai, 1988a). Theorem 8-4.3. If system (8-4.1) is detectable, dual normalizable, rankC = r,

it

has a reduced-order normal observer of order n-r of tile form: xc(k+l) = Acxc(k ) + Bcu(k ) + Gy(k) w(k) = Fcxc(k ) + Fy(k) where xc(k) ~ n - r

and l i m k ~

(w(k)-x(k)) = O,

V x(O), xc(O).

Such o b s e r v e r s a r e c h a r a c t e r i z e d by t h e i r measure outpnLs w(k) which d o n ' t inclmle input signal u(k) visibly.

As pointed out in Section 4-3,

if (kc,Fc) is observable,

such observers are of the lowest order of n-r. Theorem 8-4.4.

Assume that system (8-4.1) is delectable and Y-observable. Then it

has a state observer of order not greater than rankE of the following form:

x c ( k + l ) = Acxc(k) + Bcu(k) + gy(k) w(k) = Fcxc(k) + Fy(k) + Hu(k) in which xc(k ){ Rd, lim k~ As

d ~ rankE, w(k) E ~ n

(w(k) - x(k)) = O,

mentioned

earlier,

such that

V x(O), xc(O ).

the causal relationship generally doesn't

exist

between

state and input in discrete-time singular systems. Any state x(k) at time k cannot be determined by initial condition and inputs u(O), u(1), ... , u(k), normal systems. 4.4.

as in the case of

However, an interesting phenomenon is shown in Theorems 8-4.3 and 8-

Under some conditions,

the state x(k) of discrete-time singular system (8-4.1)

may be asymptotically estimated by former outputs y(O), y(1),..., y(k), together with former inputs u(O), u(]), ..., u(k). And the estimation error may be made arbitrarily small when time step k is sufficiently large. These two kinds of observers may be designed in the same way as in the continuoustime case.

253 Now we will consider the design of normal observers for system (8-

Example 8-4.2. 4.5).

0

1

o]

0 0

0 I

1 1

In this system, rankC = 1 and the nonsingular matrices

Q=

I

0

0

0

0 0

I 0 0 0

0 !

p:

-

,

0

satisfy QEI:' = d i a g ( O ,

I3),

CP = [ | [ 0

=

oB

Oil -11

o

Ol

=

* o 0

1

1

,

By denoting p-Ix(k) = [ xl(k) ] x2(k)

Xl(k) 6 I~,

x2(k) 6 ~3,

,

system (8-4.5) is r.s.e, to x1(k) = y(k) x2(k+l) =

g -~ [0

[00] 1 0

0

1 1

0 1

0 y(k)

x2(k)

(8-4.6)

l]x2(k) = O.

Let

Then G 2 satisfies

~(

[ °oI 1 0

t

o

[

I

-

[o

o

11) = { o ,

~, 5

which i s within the unit circle on the complex plane, server f o r s u b s t a t e

)'

1 l/6J

hence we may construct the ob-

x2(k):

~2(k+l) = (

I 0 1 1

- G2[O

0

l])~2(k) +

u(k) +

y(k) + G27 (8-4.7)

such that limk~oo(x2(k)-~2(k)) = O, V x(O), ~2(0). Combining it with (8-4.6) we thus obtain the normal observer for system (8-4.5) of the form:

254

xc(k+l ) =

1 -1

1

xc(k) +

1 -~

w(k) = A

where x c ( k ) = x 2 ( k ) , reduced-order

0 0 0

1 0 1

0 1 1

such that

observer

o

[1° 1

u(k) +

2

Xcq/ k o,

changed i n t o f i n d i n g a m a t r i x G6 ]¢nxr s a t i s f y i n g e l ( k ) = O, And

e2(k) = O,

the minimum-time s t a t e

r i n g system ( 8 - 4 . [ 5 )

k >j k o.

reconstrucLion

such t h a t

is

d e g ( I z E - ( A - C C ) l ) = rankE such t h a t

its

state

problem i s c o n s e q u e n t l y changed i n t o

seL-

r e a c h e s and keeps a t z e r o i n minimum s t e p s . A

Without

loss

of generality,

Section 7-3 and by d u a l i t y , 0 1

we assume

there exists .

.

. *

0

•

0

a n o a s i n g u l a r m a L r i x T such L h a t

. *

.

0 I

..........

rankC I = r . A c c o r d i n g t o t h e r e s u l t s

°

~_2 ............... • 0 * 1 0 I

=~I

T-1

:

•

* . . . . . . . . . . . . . . .

...

•

_~_~_ ~I *I

•

~l

. .

•

I

•

i

* * * •

1 ~I .

__

I -I- . . . . . . . . . . . . .

*

0

:

~

0

]

:! .I I I

*

I0 I 1

0

l

l

t I I i

~,, "

: "" |

, , to

dr

(Yr

in

256

^

C1. = C1T

-1

--

0

0

o

o

0

0

1

0 ...

.i

0~_ 0

~.". . .

...

O 0 O

.,.

O 0 0

"'"

O

...

1

Oi

...

11

0

~i

~.ii 0"-'~-,.-. . .

.*li

d

o fir

r

r

where "~" represents the possibly nonzero elements, Let Ao be the qxr matrix formed by the gular matrix formed by the _

_--I

Co = A°C° '

q = rankE = ~-~,di, 11o= max{tit}. [= 1

q°tht columns of AI and Co be the rxr no.sin-

g~th columns of CI" If we set

^

^

(;12 = O,

Cll = TGo,

direct calculation shows 0 1

0 0

0

1 "• 0 1

0 0 1

0 0

0

1 ~l-Go~l

•

""

=

"0

:I

I

I

1 0 I I

0 I

O O

O 1 "•

O

I O Thus, ~1-Go~ 1 is nilpotent, (AI-CoCI)~° = O. For such a G O we have ^

^

h

h

A

A

A

_

A

A

(AI_(,I ICI_GI IC2([_G12C2) IGI2C I )~o Therefore, k o =

-I ~, observer

(A~-C~C~)

'" = T ( X : % e t ) V ° T

-~

=

o.

~/o ' and ^

C = G 1 + Q1 I G I I / G 1 2 I

Thus

=

(8-4.12)

= G1 +

Q~tI'rGo/°1"

reconstructs

exact]y the state of system (8-4.1) in

steps. Moreover, by c o n s t r u c t i o n

the m a t r i x G s a t i s f i e s

deg(IzE-(A-GC) l) = rankE.

there exist nonsingular matrices Q2 and P2 such that Q2EP2 = diag(lq, 0),

Q2(A-GC)P2 = diag(A o, I).

Hence,

257 As a result, under the coordinate transformation

[~l(k) ] ~(k) = P2 ~2(k)j

Q2B

J

=

[0] B2

'

q2G

[0]

~

G2

'

Ct'2 ~

[¢1' ¢21

.observer (8-4.12) is r.s.e, to ^

^

Xl(k+l) = AoXl(k) + BlU(k) + GlY(k) ^

0 = x2(k) + B2u(k ) + C2Y(k) , which may be written in another form: Xc(k+l ) = AoXc(k ) + BlU(k) + GlY(k)

w(k) = P 2 [ ~ ] x c ( k ) + P2 where

-

u(k) + P2 _G2

2

xc(k) = ~l(k), such that w(k) - x(k) = O,

k ~ k O,

V

x(O), xc(O).

Therefore, this observer is what is required. This establishes the theorem. Q.E.D. Clearly, this observer is a minimum-time state reconstruction observer under a design method such as Theorem 7-4.5.

Example 8-4.3. For the singular system

[iooo] [looi] [°12 1

0

0

0 0 0 0 1 0

x(k+l)

=

1

1

0

0 0 1 0 0 0

x(k) +

1

0

u(k)

j (8-4.16)

y(k) = [0

1 1 -]]x(k)

under the coordinate transformation

Q =

I o00} 0 1 0 0 00-II 0 0 1 0 ,

P =

[,ooo] 0 1 0 0 0 0 1 0 0111

and p-Ix(k) = [Xl(k) l x2(k) it is r.s.e, to

Xl(k+l) = [

1 0 0 1 1 0

010

xl(k)

(8-4.17a)

258 x2(k) = - y(k) (8-4.17b) 0 -

[0

The s u b s t a t e for substate

llXl(k).

x 2 ( k ) i s g i v e n by - y ( k ) .

Now we consider constructing the observer

Xl(k).

Let G =

[1

(

0

3

2]':. Then

[oo I

Therefore,

1

I

o

0

1

0

-c[o

o

11) 3 =

we may c o n s t r u c t

xc(k+l)

=

[ o} 1 0

1 -3 I -2

toil 1

1 -3

0

1 -2

the following

xc(k ) +

[i]

state

u(k) +

= o.

observer

[i]

for substate

Xl(k):

y(k)

s u c h that X l ( k ) - x c ( k ) = O, C o m b i n i n g it with (8-4 . 1 7 ) ,

k ) 3,

and x ( k ) = P [ x l ( k ) / x 2 ( k ) ]

Xc(k+l) =

1 1 0

0 -1 1 -3 1 -2

w(k) =

1 0 0 0

0 1 0 1

is a normal state ted exactly

observer

by ( 8 - 4 . 1 8 )

Moreover, I E+GCI ~ O.

V x(O), xc(O).

0 0 1 1

]

Xc(k ) +

xc(k) +

I°) l 2

0

0

is obvious that

the system

y(k)

u(k) +

(8-4,m) y(k)

-1

for system (8-4.16).

in three

, it

And t h e s t a l e

x ( k ) may be r e c o n s t r u c -

steps.

if system (8-4.1) is observable, Let A = (E+GC)-IA and

there exists a alatrix G E]~nxr so that

~o be the observability index of (A,C), which has

been proven to be a fixed value independent of the selection of G(~nxr satisfying I E+GCI ~ O.

Then we can construct state observers for system (8-4.1) of another form. Theoreln 8 - 4 . 6 .

If system (8-4.1)

is observable,

it has a normal state

observer

Xc(k+l ) = AcXc(k) + BcU(k) + GcY(k) (8-4.19) w(k) = FcXc(k) + Fy(k) such that its state x ( k ) may be reconstructed exactly in

~o steps.

Such observers are characterized by their output in which the inputs are not directly, which are different from observers of the form (8-4.11).

used

259 8-5. Dynamic Compensation for Discrete-Time Singular Systems

For discrete-time singular system (8-4.1): Ex(k+l) = Ax(k) + Bu(k)

(8-5.1) y(k) = Cx(k) its state compensation problem os to construct tile dynamic compensators of the form Ecxc(k+l ) = AcXc(k) + BeY(k) u(k) = FcXc(k ) + Fy(k) where xc(k ) E]R nc,

(8-5.2)

Ac, Be, Ec, Fc, and F are constant matrices and system (8-5.2)

is

such LJlat the closed~loop sysLem

regular,

y ( k ) = [C is stable,

O][x(k)/xc(k)]

i.e.,

o(

) c U+. 0

Ec

,

BcC

Ac

Thus closed-loop system (8-5.3) is asymptotically stable. Systems (8-5.2) and (8-5.3) ar e g u a r a n t e e d

t o be r e g u l a r .

I f Ec i s n o n s i n g u l a r , wise, i t

is a singular

we w i l l

way t o t h a t

a noralat (dynamic) compensator;

other-

compensator.

For t h e d i s c r e t e - L i m e a simitar

term (8-5.2)

for

singular

system (8-5.1),

its

the contLnutms-time case,

in

c o m l m a m J t o r s may be d e s i g n o d i n a form o f o b s e r v e r - b a s e d

con-

trollers. Another control

problem

inputs

u(O),

tant k is transferred sense,

is

studied u(1) ....

in this

section

such that

to zero in finite

is the deadbeat

the state steps.

Lhe d u a l p r o b l e m of t h e e x a c t s l a t e

control

problem: Find

x(k) of system (8-5.1)

The d o a d h e a t c o n t r o l

a t any i n s -

problem,

in

reconsCructLou problem in the last

a sec-

tion.

8-5.1. Singular compensators Following are two results on singular compensators for system (8-5.1). Theorem 8-5.1. System (8-5.1) has a singular compensator of the form of (8-5.2) if

260 and only if it is stabilizahle and detectable. Theorem 8-5.2. n.

Assume that system (8-5.|) is controllable and observable, rankE
0 steps,

or at least one

However,

this theorem shows that,

troller,

which is in the form of a singular compensator,

singular

system (8-5.1) from arbitrary initial condition to zero in zero steps,

state step.

different from the normal case, the deadbeat con-

from the initial instant. Example 8-5.1. Consider the singular system

can transfer the stole

of or

261

i oo] 1

1

0

0

0

0

x(k+[)

[2,0] [i]

~

0

3

0

0

0

!

x(k)

u(k)

+

-

(8-5.a) y ( k ) = [I -I which i s boLh c o n t r o l l a b l e

2]x(k) and o b s e r v a b l e .

It is easy to testify that the two matrices K = [1

0

C = [1

i],

2

0.51 z

satisfy IzE-(A+BK)I

= I,

IzE-(A-GC)I

= -3.

From Theorem 8-5.1 we can construct the following deadbeat controller

l 0

1 0

0 l

xc(k+l) =

u(k)

=

[1

0

-I -1.5

5 -3 0.5 -1

Xc(k) +

2 0.5

y(k)

l]Xc(k)

the state of system (8-5.8) is driven to and kept at zero from the initial

such that instant.

8-5.2. Normal compensators Now we wilt cnnsider the normal compensators

for system (8-5.1):

Xc(k+l ) = AcXc(k) + Bay(k) (8-5.9)

u(k) = Fcxc(k) + Fy(k) where x c ( k ) £ • n c , Theorem 8 - 5 . 3 .

Ac, Bc, Fc, and F a r e c o n s t a n t m a t r i c e s System ( 8 - 5 . 1 )

of appropriate

dimensions.

h a s a normal c o m p e n s a t o r i n t h e form o f ( 8 - 5 . 9 )

if

and only if it is stabilizable and detectable. Thus,

the existence conditions for normal and singular systems are the same. But,

for the noncausality

in singular systems,

the causal normal compensators have an ad-

vantage for practical system design. Therefore,

normal compensators are often adopted

for control system design. It has been proven that although, have

a

deadbeat controller

s y s t e m c o u l d n o t be i d e n t i c a l l y (8-5.9)

to

transfer

under certain conditions,

in the form of (8-5.9), zero. Generally,

the state

of system (8-5.[)

the state

system (8-5.1) of

the

may

closed-loop

we can d e s i g n a normal c o m p e n s a t o r t o and keep i t

at zero

in

finite

steps. To consider

this problem,

we note that the compensators are often designed

on state feedback realized via a state observer.

based

Thus, let as consider the observer-

262 based normal compensator: Xc(k+l)

: AcXc(k)

w(k) = Fcxc(k)

+ Bcu(k) + Gy(k) + Fy(k)

(8-5.10)

u(k) = Kw(k) in which

w(k)

is

the output of

observer

such that limk~oo(w(k)-x(k))

= O, V x(O),

xc(O). Let s y s t e m ( 8 - 5 . 1 )

be c o n t r o l l a b l e

4-3.4 we know that tile necessary

and ( A c , F c ) be o b s e r v a b l e .

and suffictettt conditEous

Then,

from Theorem

for w(k) to be an observer

for state x(k) are Bc

2.

MA - AcME = GC.

3.

I = FcME + FC.

4.

=

MB.

1,

o (Ac) < U+.

M is a constant Denoting

matrix.

e(k) = xc(k) - MEx(k),

from (8-5.1) and (8-5.10)

we have

i.e.,

E x ( k + l ) = (k+BK)x(k) + BKFce(k) e(k+l) = Ace(k),

(8-5.12a) (8-5.121))

e(O) = Xc(O) - MEx(O).

Let K be chosen such that

I zE - (A+BK)I = c o n s t a n t Then t h e r e e x i s t

(8-5.13)

~ O.

n o n s i n g u t a r m a t r i c e s Q2 and P2 s u c h t h a t

(8-5.12a)

is r.s.e,

N~(k+l) = ~ ( k ) + Be(k)

to (8-5.14)

where

= Q2EP2 , Let ~ be

Q2(A+BK)P 2 = I n ,

the ni]potent

index of N.

QzBKFc = B ,

From ( 8 - 5 . 1 3 )

x(k)

= P~(k).

and ( 8 - 5 . 1 4 )

we know t h a t x ( k ) and

e(k) are given by

x(k) = - P2 7 ~ , NIBe(k+i) i=O

e(k) = Aie(O). Thus,

to transfer

the state x(k) to zero in minimum time is to find the matrix A c

263 so that e(k) reaches and keeps at zero in minimum time, or to determine matrix A c that the q satisfying A~ = 0 is minimum. lem

so

As a result, the deadbeat controller prob-

via compensator (8-5..10) is equivalent to finding matrices K and A c

subject

to

w(k) being an observer for x(k). Particularly,

((E4GIC)-IA,C) G1 ~ ~nxr

if

is

system

(8-5.1) is observable,

the observability index

a f i x e d v a l u e i n d e p e n d e n t of t h e s e l e c t i o n

nf m a t r i x G1

~o

of

provided

satisfies IE+GICI ~ O. Thus we can prove the following.

Theorem 8-5.4.

Let system (8-5.1) be controllab]e and observable, rankE < n. Then

it has a normal compensator in the form of (8-5.10) such that the state of system (85.1) is driven to and kept at zero from an arbitrary initial condition in duo steps. Proof.

Under the assumption of ohservability,

a matrix g E ~mxn

may be chosen so

that I zE - (A+BK)I = constant ~ O. Hence,

as long as the feedback control a(k) = Kx(k) is applied to (8-5.1), the state

of the closed-loop system is identically zero.

Since x(k) is inaccessible,

we

next

realize it via state observer. Note

the

assumption that system (8-5.1) is observable.

satisfying I E+GICI ~ O. matrix

Go may be c h o s e n

There exists a matrix G 1

Via a similar procedure as that used in the last section,

a

so t h a t

((E+GIC)-IA - GoC) p° = O. For such C I, G o , we choose G 2 = (E+GIC)G °ancl construct the state observer

(E+G2C)~(k+I) = (A-G2C)~(k) + Bu(k) + G2Y(k+l) + GlY(k). Direct c o m p u t a t i o n x(k)

(8-5.15)

shows that

= ~(k),

k ~

70"

By defining Xc(k) = ~(k) - (E+GIC)-IG2Y(k), system (8-5.15) may be written as

x c ( k + l ) = Acxc(k) + Bcu(k) + OcY(k) wOO = ~(k) = FcXc(k) + Fy(k) where

Ac = (E+G1C)-I(A-G2C), Bc = ( E + c 1 c ) - I B , Gc = (E+GIC)-IG 2 + Ac(E+G1C)-IG1 F

= (E+GIC)-IG1 ,

Fc = I.

(8-5.16)

264 According to the previous discussion, we know that the compensator xc(k+l ) = (Ac+BcK)xc(k) + (Gc+BcKF)y(k) u(k) = Kxc(k) + KVy(k) formed

by feedback u(k) = Kw(k) and observer (8-5.16) is a deadbeat

system (8-5.1)

controller

for

such that its state from arbitrary initial condition ls driven to and

kept at zero in ~ o steps. Q.E.D.

As shown in Example 8-5.1,

Example 8-5.2. observable,

rankE = 2 < n = 3.

the matrix G I = [0

0

[21 j

observability

o°

0.5

0.5

index of ((E+G1C)-IA,C)

i s 3.

Let (;o = [-4

((E+GIC)-IA - GoC) 3 = O. For these matrices chosen as earlier, we have

G2 = (E+GIC)G o = [-4

12

4.5]

Ac = (E+GIC)-I(A-G2C) = (E+G1C)-IA - GoC 6 -18 14.25

-3 18 -12.75

Bc = (E+G1C)-IB = [0

8 ] -32 -24

1

Off

G c = (E+GIC)-IG 2 + Ac(E+GIC)-IG 1 = [0

= (E+GIC)-IG 1 = [0 6 -17 14.25

Ac+BcK =

Gc+BcKF = [0

0.5

0

-3 18 -12.75

0

0.25] ~

0.5] ~

8]

-31 -24

0.25] ~

KF = 0.5.

By Theorem 8-5.4, t h e compensator Xc(k+l) = | u(k) = [1

controllable

and

IzE-(A+BK)I = 1 and

]

2

will be

F

is

1] ~ satisfies IE+GIC; ~ O. Direct computation shows

E+oI )-IA= and t h e

system (8-5.8)

The matrix K = [I 0 i] satisfies

- 7 14.25 0

18 -12.75

1

-81 Xc(k) + -24 J

1]xc(k) + 0.5y(k)

I° ] 0.5 O. 25

y(k)

16

12.25] ~.

IL

265 i s a d e a b e a t c o n t r o l l e r f o r s i n g u l a r system ( 8 - 5 . 8 ) such t h a t x(k) = O,

xc(k) = O,

V x(O), xc(O),

8-6.

k ) 3.


Discrete-time singular systems were first studied by Luenberger (1977). 8-1

and 8-2 are based on Dai (1988d).

and

Dai

and Wang (1987a),

respectively,

Campbell and Rodriguez (1985),

Sections

Sections 8-4 and 8-5 are based on Dai (1988a) and other p a p e r s such as

Ei-Tohami et al. (1987),

Bender

(1987),

Lewis (1983a, 1984, 1985a),

and Luenberger (1977, 1979, 1987). The as

relationship between continuous-time and discrete-time singular systems

discretization of continuous-time systems

nuous-time systems deserves further work.

such

and control of discrete data on conti-

CHAPTER 9 OPTIMAL CONTROL

9-1.

So

far,

assign a

Introduction

in the control system design we placed our design on one

tile closed-loop

poles to arbitrarily

prescribed

positions.

principle:

to

On one hand, such

design principle is difficult to handle since evaluting the performance ofasystem

from pole distribution is not an easy task; on the other hand, in the pole assignment the feedback control is not unique. closed-loop

properties,

Thus,

tile freedom should be used to improve the

such as fast response, minimum-energy

consumption.

These aims

require choosing a best control law from all feasible strategies. This is the optimal control

problem,

loosely speaking.

It is worth poinLing o u t that deviations always exist in a rear system model, thus cause the optimal control to lose its expected property.

In some cases, a nonoptimal

control law may have better practical effect than a ~heoretically these

reasons,

lJowever, the

control system design.

For s i m p l i c i t y ,

optimal one due

to

optimal method provides a more effective approach

in

In practice,

this approach often proves reliable and useful.

we s t u d y o n l y the s i m p l e

linear

singular

system:

E~(t) = Ax(t) + Bu(t)

where x(t)(

I~n,

u ( t ) ( ][{m a r e

and B ( n~nxm a r e c o n s t a n t

(9-1.1)

its

maLrices.

slate It

and c o u t r o l

inpuL, respecLively;

is assumed Lhat system (9-1.1)

E, A( ~ n x l |

is regular

and

rankE < n. In r e a l

systems,

the control

input

u(L) o f t e n

subjects

to a constrainL

use U to denote the collection set of u(t) subject to constraint the constraint set, trel.

For s i t l g u l a r

ficiently

or simply, the constraint. syslems,

continuously

s l m u l d I)e f i r s t

the p i e c e w i s e s u f -

function since derivatives of control input are

included in the state of singular systems. The constraint tion of state, control, and time t, or

which is called

Any u(t)~ U is called admissible con-

the a d m i s s i b l e cont. r . I

differentiable

n,

a . We MmlJ

m(x(t),

m

often is a vector func-

u(t), t). We also suppose that n tar-

get $ is given, which is a constraint on terminal state. For system (9-I.i), let J be a functional J(x(t), u(t), t) of state,

control,

and time, which is called a cost functional.

(9-1.2) The optimal control

267

problem for system (9-I.I) with respect to the target S, constraint U, initial time t = O, and initial state x(O) is to determine the admissible control u~(t)~ U such that the t e r m i n a l s t a t e reaches S and the cost f u n c t i o n a l J i s minimized (or maximized): min

J(x(t),

u(t),

t) = J(x~(t),u~(t),t)

(9-1.3)

u( U

(or

max

J(x(t),u(t),t)

= J(x*(t),u*(t),t)

).

u( U

The cost J i s chosen w i t h respect to the prohlem in we are Js

represent

different

optimal control,

meaning of our optimal problem.

interested.

UsuaIJy,

it

is

Differen! the

time-

the cheap control(minimum energy consumption), etc.

For normal systems, the optimal solution for general problems may be obtained from the

wellknown maximum principle.

general

singular

systems.

However,

there doesn't exist such a principle for

Much effort has been made toward

solving

this

problem

(Lovass-Nagy el at., 1986; Lewis, 1985b; Bender and Lauh, 1987b; Dai, ]988g). At present, the problems concerning optimal control are treated by changing them into optimal problems for normal systems, then solving via normal system theory.

9-2. Optimal Regulation with Quadratic Cost Functional

Consider the singular system E~(L) = Ax(t) + Bu(t)

(9-2.I)

with the quadratic cost functional !

J = J [ x ~ ( t ) F x ( t ) + u ~ ( t ) ( : u ( t ) ]dr

(9-2.2)

o

where F and G are symmetric positive definite.

This is a problem

with an

infinite-

interval cost functional. It is Supposed that the admissible control is tile piecewise sufficierltly this

continuously differentiable

cost i s often adopted i n p r a c t i c e .

energy,

function.

For i t s simple form for a n a l y s i s ,

Since imposing conLrol means consuming

of

the optimal c o n t r o l w i t h q u a d r a t i c cost f u n c t i o n a l i s also c a l l e d cheap con-

tro]. Cobb (1983a) has proven the f o l l o w i n g Lheorem. Theorem 9 - 2 , 1 . Consider ( 9 - 2 . 1 ) w i t h q u a d r a t i c cost f u n c t i o n a l ( 9 - 2 . 2 ) . Then Lhere e x i s t s an o p t i m a l c o n t r o l u ~ ( t ) such t h a t cosL f u n c t i o n a l ( 9 - 2 . 2 ) i s minimized i f only i f

and

system ( 9 - 2 . 1 ) i s s t a b i l i z a b l e and impulse c o n t r o l l a b l e .

We now

consider s o l v i n g the optimal c o n t r o l under the assumption of e x i s t e n c e and

268 uniqueness

of, the solution.

Assume that

system (9-2.1)

matrix K E ~ m x n

is stabilJzable

and detectable.

Then there

exists

deg(IsE - (A+BK)I) = rankE. Let tile preliminary

[eedback control

(9-2.3)

he

u = Kx + v, where v E ~ m

a

satisfying

(9-2.4)

is the new input. When applied

to system

(9-2.1),

the closed-loop

system

is

(9-2.5)

E~ = (A+BK)x + B y . .

According to tile s e l e c t i o n of matrix K,

equaLion (9-2.3) hohls. Thus, there exisL

nonsingular m a t r i c e s QI and P1 such t h a t QIEPI = d i a g ( I q , 0 ) ,

q = rankE.

QI(A+BK)P 1 = d i a g ( A l , I ) ,

By denoting

P1Ix =

x2 , Xl(

]Rq

x2E ]{n-q

(9-2.6)

system (9-2.5) is r.s.e, to Xl = AlXl + BlV 0

(9-2.7)

= x 2 + B2v.

From (9-2.7),

x 2 -- - B2v. Combining

it with (9-2.6) we have

fxj. i, oli:l. . iP1:1[ x,] tP1 o1[,o-!ti:11 u

K

Thus cost functional

I

Kp I

(9-2.2)

I

~2

KP I

I

0

"

becomes

F 0

F

II

(9-2.8)

where ]

E °}i :;I II 0

H

G

P1

F

0

KPI

0 g

P1

0

0

-

0:2

Note t h a t the matrix d i a g ( F , G) i s

KP1 i.JL0

~2 "

symmetric p o s i t i v e d e f i n i t e and matrix

(9-2.9)

269

i P, :1[ I

0 0

KPI

-B 2 l

is of full column rank. and so are

matrices

We know that matrix (9-2.9) is symmetric positive

F and C. Therefore,

cost

functional

(9-2.9)

definite,

may b e f u r t h e r

writt-

en as oo

J =

(9-2.10)

J (x~FXl + w~Gw)dt

where _ - 1 It,~ = F - HG Paying

attention

to

the

w = v + -1HZXl

.

relationship:

i ~ ~l ~ I ~-~-'~ ~I ~ "li~ ~l"l 0

.

and nonsingularity

of

0

I

the

matrix

[~_rx,,] 0

I

(9-2.11)

H~ ~

f

~9~,~

,

,

we know that matrix ~ is symmetric positive definite. Substituting

v = w - ~-lll~Xl

into

(9-2.7)

we o l ) L a i n

Like f o l l o w i n g

system

~I = AlXl + BlW

(9-2.13)

AI = A1 - BIG-IHx"

(9-2.14)

where

Thus, the optimal control problem for singular system (9-2.1) is changed into problem

the

of finding the new control w for normal system (9-2.13) such that cost func-

tional (9-2.10) is minimized. To solve this problem using the result in normal system theory, we need to prove the following lemma. Lemma 9-2.3. (AI' BI) is stabilizable. Proof. This is a direct result of the assumption of stabilizability for system (92.1), also noticing equation (9-2.14) and the stabilizability of (AI, BI). Q.E.D. By applying the quadratic regulation problem for normal systems in Appendix C, the optimal

control for system (9-2.13) minimizing the cost functional (9-2.10) is given

by w ~ = - G-IB~Mx~ where x~

is the optimal trajectory,

(9-2. ]5) the symmetric positive definite matrix M is the

270 unique solution of the following Riccati equation MA I + AIM - MBIG-IBI M + F = O,

(9-2.16)

and the minimum value of g is min J = x~(O)Mx|(O). Therefore,

by summing up this discussion, we obtain the optimal control u~(t) for

singular system (9-2.1): -i

u~(t) = Kx~(t) + v~(t) = Kx~(t) + w~(t) - G = (K - G

(H +BIM)[I

.=

II x~(L)

O]Pll)x~(t )

(9-2.17)

and the optimal trajectory x~(t): E~(t) Moreover,

= [A+B(K-~

(H +BIM)[I

O]mll)]x~(t).

(9-2.18)

direct computation shows that system (9-2.18) is stable and no

impulse

terms exist in optimal trajectory x~(t), the minimum value of J is sin J = x~(O)(PT)-I [ ~ ] M [ I Hence,

O]PllX(O).

in summary of this discussion,

(9-2.19)

if the solution of optimal control

for s i n g u l a r system (9-2.1) with q u a d r a t i c cost functLonal (9-2.2) e x i s t s

problem

and i s u n i -

q u e , the problem may be solved according to the following procedure.

I. Verify the stabilizabllity and inpulse controllability 2. Choose matrix K so that deg(IsE-(A+BK)l)

for system (9-2.1).

= raakE.

3. Calculate the matrices AI, Bl, B2, QI, and P1 according to (9-2.7).

4. D e t e r m i n e tile maLrices ~', II, ~,) F) and ~1 acc'or(lin]g to (9-2.9),

(9-2.11),

and

(9-2.14). 5. Find the unique solution of Riccoti equation (9-2.16).

6. [ ' i n a l l y obtain and (9-2.18). In

the optimal conLrnl uC'(t) and t r a j e c t ~ r y x~:(t)

using

(9-2.17)

the optimal regulation problem with quadratic cost functional for normal

sys-

tems, to obtain the solution we need to solve a Riccati equation of order n, the same as the order of the system. order

rankE < n

However,

as seen from (9-2.16),

is needed for this problem.

a Riccati equation

of

This will be convenient when rankE

Ls

much smaller than n. Optimal control (9-2.17) is in the form of state feedback.

Example 9-2.[. Consider the singular system

271

0 0

0 0

1 0

~ =

x +

u

(9-2.20)

with cost functional J~x'Zx

J = This

+ 2u~u)dt.

system is controllable,

(9-2.21)

thus it is stabilizable amd impulse controllable.

The

procedure is applicable in this case. In this system, rankE = 2, n = 3, F = I3, and G = 2. The matrix K = [O

1

O] satisfies deg(IsE-(A+BK)I)

= rankE. Direct verification

shows that under the nonsingular matrices

Q1 =

0 0

1 -1 0 1

PI =

-1 1

,

we have QIEPI = diag(12, 0),

A1

=

0

-i

,

QI(A+BK)PI = diag(Al, I),

=

-

,

=

For this system, equations (9-2.9), (9-2.11), and (9-2.14) give F = diag(l, 4),

G = I,

[o'-o'] To s o l v e

Riccati

M[O

0

M=

mI [m 2

F = diag(l, 3)

[o]

equation

(9-2.16),

10]M

+ [0~

which here 0]

_

is

M[00]M

= 0'

(9-2.22)

we denote

Performing

the

m2 m3].

i.d icated

matrix multiplication,

from (9-2.22) we obtain the

equat ions: 2 2m I - m2 + 1 = 0 m2 - m1 - m2m3 = 0 2 - 2m2 - m3 + 3 = O,

which has the unique solution m[ = 8+6f2, Thus

m 2 = -3-2f2,

m 3 ~ I+2f2.

three

272 [8+6vr2 M = According

-3-2J2]

-3-2/2

to

i+2~

(9-2.17)

problem, respectively,

u.(t)

=

IK

.

and (9-2.18),

the optimal control and trajectory

this

for

is G-I(H~+B~M)[I

O]PTl]x*(t) = [-3-272, I, 2J~]x*(t) I

and

~

~*(t) =

~ 0

1

L-3-2~

which has finite pole set ~ - ~ ,

-4~}.

x*(t),

1 l+2~J Hence the closed-loop system is stable

and

no impulse terms exist in its optimal trajectory.

9-3.

Time-Optimal Control

In the time-optimal problem, or fast control problem, the cost functional is taken as (the initial time is t = O) J = t,

which

(9-3.1)

represents the time period taken to drive the state from one position to

ano-

ther. Consider system (9-2.1): Ex = Ax + Bu

(9-3.2)

with two fixed states: the initial state x(O) and the terminaJ stale x(tl). Its timeoptimal control is to find the control u*(t) such that the state from x(O) is

driven

to x(t i) Ln a minimum period, i.e., t I = min J . As

shown in thc concept of c o n t r o l l a b i l i t yin Chapter 2,

strained

the

i f the control i s

s t a t e of c o n t r o l l a b l e system (9-3.2) may be driven from

initial state to any terminal state in an arbitrarily short period. problem will lose interest. Therefore,

an

un-

arbitrary

In this case our

the time-optimal problem is often studied with

a constrained controL. Let the constraints be bounded control: luil ~ ai,

u = [Ul, u2, ... , Um], i = 1, 2 . . . . .

and u(t) is p[ecewise sufficientlycontinuously

m

(9-3.3)

differentiahle.

Furthermore, for the sake of simplicity, the terminal state is assumed to be zero,

273 i.e.,

x(tl)

= O.

For any r e g u l a r

system (9-3.2),

there

exist

two n o n s i n g u l a r

matrices

Q and P s u c h

that it i s r.s.e, t o

~1 = AlXl + BlU

(9-3.4)

N~2 = x 2 + B2u where AlE ~ a l x n l ,

N E~ n2xn2, n I + n2 = n, N is ntlpotent.

Now we will examine the time-optimal problem of two extreme cases.

9-3.1. The slew subsystem Consider the slow subsystem xL = AlXl + BlU' It

is

a normal system.

xl(O) = wl"

(9-3.5)

From linear system theory we know that the optimal

control

under constraints (9-3.3) that drives any initial state of system (9-3.5) to zero t h e minimum t i m e i s t h e s w i t c h i n g

9-3.2.

The f a s t

subsystem

For t h e f a s t

subsystem

N~2 = x 2 + B2u ,

control,

in

which L a k e s t im e x t r e m e v a l u e s .

x2(O ) = w2,

(9-3.6)

when t > O, its state solution is h-I x2(t) =-~-~,NiB2u(i)(t), i=O

t > O,

(9-3.7)

where h is the nilpotent index of N. Therefore if we choose u(t) m O, t > O, be

x2(t ) m O,

t > O.

the initial instant. any

short period.

it will

Note that the initial condition exists in the impulse terms at Thus,

But,

the state may be driven to zero by admissible control in

the time period generally cannot be zero.

This shows

that

generally no optimal solution exists for time-optimal problems for fast subsystems. Definition 9-3.1. Consider system (9-3.2) with control constraints (9-3.3). Let > 0 be any prescribed scalar.

If there exist admissible control u~(t) and time point

t I such that the state x(t) is driven to zero at tl, i.e., x(t I) = O, but there exists no admissible control such that the state x(t) is driven to zero at any period ter

than t - c , the control uS(t) is called

called the

shor-

E-suboptimal control and the time t I is

c-suboptimal time.

By definition, the e-suboptimal control always exists for a fast subsystem. Clear-

274 ly, the e-optimal control is practical only for small

e, and it is acceptable if

is sufficiently small. In general we have the following theorem. Theorem 9-3.1. Let t I be the optimal time for subsystem (9-3.5) and

fi~(t) be

the

optimal control, x~(t I) = O. Then for any ¢ > O, there exists an a-suboptimal control u~(t) such that the state of system (9-3.2) is driven to zero at time tl+ ¢, x(tl+¢) = O. Proof.

Let t I be the optimal time for subsystem (9-3.5) and 5~(t) be the

optimal

control, 0 K t < tl, such that x~(tl) = O. Noting the slow substate solution we know

x~(t l) = oA]tixl(O ) + J~otleAl(tl-t)l~l[V~(t)dt

= O.

Let the new control be

u~(t) = { ~(t)

O~ O, b > O, and c > O.

Heyer, Kalman, and Y a c u b o v i t c h have proven tile [ o l l o w i n g well-kuown p o s i t i v e Lemma 1 0 - 2 . ! .

Let G ( s ) be a s t r i c t l y

proper r a t i o n a l

f u n c t i o n with a

lemma.

realization

(A,b,c): G(s) = c(sl-A)-Ib. Then G(s) is positive (strictly positive) iff there exist a vector positive definite

m a t r i c e s P and Q s u c h t h a t

'] and

symmetric

292 1. A~P + PA = - Q.

~>

2. Pb - c~ = 0 (Pb - c~ =JP~ ,

0).

Consider the system described by the input-output relationship:

y(s) = C(s)u(s)

(10-2.2)

in which G(s) may be not proper. Thus G(s) may be the transfer function of a singular system. In

the

model reference control problem,

our aim is to find the control u(s)

so

that the output y(s) is the prescribed signal: y(s) = Yr(S),

Yr(S) = Gm(S)r(s )

in which Yr(S) is the reference output, or the expected signal; r(s) is the reference input; and Gm(S ) is a given transfer function, or the model.

Yr ( t ) +

r(t)

y(t)

+

e(t)

u(t)

Figure

The g e n e r a l which

8

is

form o f model r e f e r e n c e the adjustable

Fo[lowing

Model R e f e r e n c e C o n t r o l

adaptive

parameter.

is to choose the parameter ty.

10-2.1.

8

the principle

control

i s shown i n F i g u r e

The t a s k o f model r e f e r e n c e

such that

e(t)

Thus

from

Figure

e(s)

I

-

in normal systems,

C(s)o~Y.(s)

]0-2,1

= y(s)

we s e e

it

.

that

the

error

- Yr(S)

= G(s)(r(s)

+8Xv(s)) - Cm(s)r(s)

10-2.1,

in

control

t e n d s t o z e r o when t i m e t e n d s t o i n f i n i [s supposed that

vector transfer function F(s) and a constant vector 8 ~ s u c h that

Cm(s) =

adaptive

e(t):

there

exists

a

293 Cm(S) = 1 + Gm(S)e~F(s)

=

[r(s) +e~(s)]

- Cm(s)r(s)

Gm(S)e~v(s ) - Cm(S)e*~F(s)C(s)[r(s)+e~v(s)]

= Gm(S)~v(s)

(10-2.3)

where # = O - 0 ~ is the adjustable error vector. If there exists a rational function L(s) such that the transfer matrix Gm(S) = Gm(s)g(s ) is strictly positive real, we choose = L(s)) , where

~

(IO-2.4)

is a vector to be determined.

According to (10-2.3) we have

e ( s ) = Cm(S)L(s)*~v(s) = ~ m ( S ) ~ v ( s ) . On the other hand, let (A,b,c) be the minimal realization of Gm(S), c = [I, 0,..., 0]. Then = Ax + b ( ~ v )

(1o-2.s) e

Therefore,

~

Cx,

from Lemma 10-2.1, there exist symmetric positive definite matrices P and

Q satisfying I. A~P + PA = - Q. 2. Pb = c ~. Let A > O, and T be any symmetric positive definite matrix, and = -

I

X Tve.

(10-2.6)

Next we verify that in this case there will be l i m t ~ ~ e ( t )

= O. Consider

the Lya-

pun()v Function:

V = x~Px + A~T-I~ Since = x~(PA + A~P)x + 2x~Pb(~v) + 2 ~ T - I ~

,

we know V = - xXQx. Thus system (10-2.5) [s stable and l i m t ~

In summary, we know from this discussitm

e ( t ) = O.

that if we choose the control (10-2.7)

u=r+~ where

~

is determined

in (10-2.4) and (10-2.6), such a control u

loop system stable, i.e., limt~ ~ e(t) = O.

makes

the closed-

294 Figure 10-2.2 shows the control strategy.

• Gm(s) J

Yr(t)

r(t)

e(t)

F i g u r e 1 0 - 2 . 2 . A d a p t i v e C o n t r o l Scheme ( 1 0 - 2 . 6 )

As may be seen from Figure 10-2.2,

in this adaptive process,

used to compare with t h e r e f e r e n c e s i g n a l Yr(S),

the

output y(s)

tn evaluuLe t h e e r r o r e ( t )

is

from whi-

ch we determine the control, and to reduce the error e(t) to zero.

10-3.

So

far,

Stochastic Singular Systems

singular systems studied are the deterministic

items are deterministic;

systems in which all

even in the presence of external disturbance,

ce i~ supposed t o be d e t e r m i n i s t i c .

the

the disturban-

In r e a l problems, c o n t r o l system d e s i g n based on

such system models can meet with most of our demands. However, on the other hand, any real

systems are unavoidably influenced

influence.

Therefore,

Meanwhile,

trol

be studied

in

inputs, but also

stochastic error always exists in measure output. Although these sto-

such factors must be considered

system.

environmental

subject to the limited precision in measu-

chastic factors may be neglected in most cases, required,

factors such as

such systems have not only the deterministic

unavoidably stochastic inputs. rement methods,

by stochastic

in some cases where high precision

to obtain a good performance for the

is

con-

Thus tlre system model naturally becomes the stochastic model that will this section.

control for discrete-time

We briefly discuss the problem of filtering

singular systems.

and

LQG

295 Consider the following discrete-time stochastic singular system: Ex(k+l) = Ax(k) + Bu(k) + Dw(k) (10-3.1)

y(k) = Cx(k) + Fw(k) k = O, I, 2, ... where the state x(k) E ~n,

the control input u(k)E ~ m

the external stochastic disturbance w(k) E ~ P ; ~rxn

F~rxp

the measure output y(k) ~ D~r,

E, AE ~ n x n

BE~nxm,

DE~nxp,

C 6

are constant matrices. This system is assumed to be regular and rankE

=

~ f'(x)~(x)dx =

= lira

((~(x)f(x)lT T

r~oo

lim

jT~(X)df(x)

W~ o o

_

-

T f 9'(x)f(x)dx) -T

°°f(x) 9' (x)dx = - .

Example A-3. Consider the unit jump function

I(x)

I

1

when x > 0

I

O

when x ~ O.

We have >

=

-

t

J

-

=

-

J

'(x)dx

= ~(o) = < ~ , ~ > indicating

~(x) = d % ( x ) .

Thus the derivative of the unit jump function is the Dirac

function.

By Definition A-4, the ith order derivative f(i~x)

= (_l)i,

of f(x) is uniquely defined by

i = I, 2 . . . .

(A.3)

The arbitrary order derivative of a normal function may not exist. (A.3) we see that a distribution has any order derivative. fferentiation

of discontinuous

points r. 6 ~ ,

from

function that we intrduce the nation of distribution.

Assume that f is a distribution. g(x),

However,

In fact, it is for the di-

If there exists a piecewise continuous

i = O, ±l, ±2, ...,

and the number of

1

r.

is

function

finite

on any

1

bounded set, such that f = g, the distribution

x E (Yi, is

~i+l )'

i = O, ±1, ±2 . . . . .

called piecewise

continuous.

We use D'

to

denote the set of

distributions on ~ and ~' to denote the piecewise continuous distribution on ~. P Let f(x) be a real function on ~, which has derivative except the point

f(~o+O) =

lira f(ro+$) , £ ~ O+

f(~o-0) =

-co and

lira f(~o+E) t-*O-

exist. Thus ~,f

= f(ro+O) - f(~o-O)

represents the jump at ~ O of function f. Furthermore, its

derivative

Then we have

let f'(x) be the distribution derivative of function f(x) and ~'(x) be in the normal sense,

which is assumed to exist at the point x E ~.

305

=

lim { -

~

]

f(x)~'(x)dx -

f(x)~'(x)dx

},

E>

O.

Integrating by parts and taking limitations, we obtain

lira { - f ~ l ~

O. a

6

¢.

It is the solution

of

309 3.

JlA+Bfi ~

Norm

flAil +

MBfl , for any A, B E c n x m

II A If is called consistent

mensions, The norm

[lABI[

~

if for any two matrices

A and B of appropriate

lIA llliBI[ .

llxaJ of vector x is the special

case of matrix normal when x E cnxl.

di-

APPENDIX C

SOME RESULTS IN CONTROL THEORY

The following system ~(t)

= Ax(t)

y(t)

= Cx(t)

where x(t)6 ~n, tem theory

x ( O ) = xo

+ Bu(t),

(c.1)

u(t) 6 ~ m

and y(t)E ~r,

which is termed

is the classic linear system in linear sys-

the normal system here for the sake of distinction with

singular systems; x o is the initial condition.

C-I. State response and stability The solution of the differential equation in (C.1) is called the state response of system (C.1). It is given by x(t) = eAtxo +

[teA(t-~)Bu(~)d~ -o

System (C.I) is called (asymptotically) (C.I) is stable if and only if

.

stable if l i m t . ~ x ( t )

= O,

Y x o.

System

o(A)CC-.

C-2. Fole and pole structure The

poles of system (C.I) are eigenvalues of the coefficient matrix A.

The

structure for system (C.1) indicates the eigenstructure in the Jordan form of A.

C-3. Controllability and observability System (C.I) is controllable iff rank[sI-A,

B] = n,

V s 6 C,

or equivalently, rank[B, AB . . . . .

An-IB] = n.

System (C.I) is observable iff rank[sI-A/C] = n,

Y s ~ C,

pole matrix

311 o r equivalently, rank[C/CA/.../CA n-l]

= n,

V s (~¢.

C-4. Pole placement Let the state feedback be u ( t ) = Kx(t) + v ( t ) .

(C.2)

When applied to system ( C . I ) , i t s closed-loop system is

~(t) = (A+BK)x(t) + By(t).

(C.3)

If system (C.I) is controllable, the closed-loop poles of (C.3) may be arbitrarily assigned via feedback (C.2). C-5. Controllability indices and observability indices Let

system (C.I) be controllable.

numbers

defined in the following way.

Its controllability indices are a set of Assume that B is of full column rank,

wise, delete the corelated columns in B. Let

Vl $

~2 ~ "'" ~

Pm

real other-

be the controlla-

bility indices of (A,B). Then P 1 i s tile smallesL number of i in the following series B,

AB,

A2B,

... , AiB,

when corelated column appears in it. that is

Delete this column and the corresponding columns

create this column in B and all corresponding columns in this series. Then the

second

I)2

smallest number in this series when corelated columns occur. Delete

all tile corresponding columns as before, and so on, unLil the controllability indices of (A,B) are defined.

Vm is called tile index of (A,B).

The observability indices of (A,C) are the controllability indices of (A~,C~). C-6. Controllability and observability decomposition For any system (C.I) there exists a nonsingulor matrix T such that

TAT -I =

All

O

AI3

O

A21 O

A22

A23

0

A33

A24 0

0

0

A43

A44

CT-1 = [C l

0

C3

B] TB=

,

O],

where (All, BI,C i) is controllable and observable;

(

[c 1 o1) LA21

A22

,

B2

,

B2 O O

312 is controllable but unobservable; and

A33

,

,

is observable but uncontrollable.

C-7. Optimal regulation with quadratic critierion For system (C.l) and quadratic cost functional

J =

~

(xxSx + u~Ru)dt,

where S and R are symmetric posStive definite, system (C.I) is stabilizable. Then the optimal control that minimizes the cost functional J is u = - R-IB~Mx, where the symmetric matrix M is the unique positive definite solution of the ing Riccati equation ~

+ A~M - MBR-IB~M + S = O.

The minimal vaiue of J is min J = x~(O)Mx(O).

C-8. Transfer matrix The transfer matrix of system (C.I) is G(s) = C(sI-A)-IB. It is a strictly proper rational matrix, i.e., lim s~ooG(s)

= O.

follow-

APPENDIX D

LIST OF SYMBOLS AND ABBREVIATIONS

D-I. Symbois The following IAI

A -!

symbols are adopted

=

the determination

=

the transpose

in this book~

of matrix A;

of matrix A;

__ the inverse of matrix A when it exists,

A>(~)O

= Matrix A is symmetric

positive

A-IA = AA -I = I;

(nonnegative)

definite;

[A/B]

=

is used to denote the matrix

=

Im[B, AB . . . . .

adj(A)

=

the adjoint matrix of a matrix A, adj(A)A = IAII;

£

= the complex plane

Ck

= set of k times continuously

ck P

= set of k times piecewise

An-!B],

differentiable

continuously

= set of mxn complex matrices;

Cn

= set of complex

{+

= the closed right half complex

deg(. ) (i)

u[ A;

functions;

differentiable

functions;

vectors;

= the open left half complex

plane, ~+ = {s [ s ~ ¢ ,

Re(s)

) 0};

plane, ~- = C - C+;

= the degree of a polynomial; = a superscript

of a function,

is the ith order derivative

Im(s)

= the complex

Im(A)

= the range of a matrix A, Im(A) =

In

= the nxn identity

Ker(A)

= the kernal

L[.]

= the Laplace

part of a complex

scalar s; {Y

I Y = Ax >~

(or unit) matrix;

(null) of matrix A, Ker(A) = transformation

= set of real numbers

~mxn

;

n = the dimension

~mxn

C-

[~]

of a function;

(the real axis);

= set of mxn real matrices;

{ x l Ax = O };

of a function;

314 ~n

= set of real vectors;

rankA

= rank of matrix A;

Re(s)

= the real part of a complex scalar s;

U+

= the unit circle on the complex plane, U + = { z i z 6 ~, I z[ < 1 } ;

xx

= the impulse part of a function x at time -~ ; = the jump part of the function x at time

~(t-~)

z ;

= the Dirac (or Delta) function; = coverdot = the derivative of a function = the expectation

of a random variable;

o(A)

= the spectrum of matrix A, i.e., the eigenvalue set of A;

a(E,A)

= the set of finite eigenvalues of regular matrix pencil (E,A); = belonging -- b e i n c l u d e d

to; in;

U

= the

union

fl

= the

intersection

= the

direct

of sets; of

sum o f

two vectors

I

= be d i v i d e d

V

= for any, for arbitrary; = identical,

with

sets;

~.~

linear

divided

spaces; exactly;

equal everywhere;

= not identically __a

no r e m a i n d e r ,

or

= by definition,

equal to; defined as;

= notation of a set.

D-2. Some abbreviations DDSO

= disturbance decoupling state observer;

EFI (2,3) = the first (second,

third) equivalent

gcd

= greatest common divisor;

HOT

= highest order term;

IFO

= input function observer;

NOR

= normal output regulator;

p-

= pure proportional;

form;

315 P-D

= proportional and derivative;

PPS

= pure prediction system;

r)s,e.

= restricted system equivalent;

SOR

= singuiar output regulator;

SSNC

= structurally

stable normal compensator:

SSNOR

= structurally

stable normal output regulator.

APPENDIX E SUBJECT

Algorithm shuffle, 8, 218, 226

INDEX

impulse, see impluse observability index, 311

Silverman-}|o, 63

indices, 96, 98, 209, 311

of general compensator design, 151

R-, see R-controllability

of normal compensator design, 141

Y-, see Y-controllability

148

Cost functional, 266 quadratic, 267, 277, 312

Causality, 234, 242 Compensator, 132, 175, 285

Data point, 169, 186

full-order normal, 136

Deadbeat control, 259

normal, 136, 161, 261

Decentralized

reduced-order normal, 136, 142 singular, 133, 152, 259 Condition

control, 285 dynamic compensation, 285 Decomposition, 12, 50-56, 311

admissible initial, 18, 22, 243

normalizability, 55

complete, 232

standard, 12, 23

initial, 17, 232 terminal, 232 Constraint, 172, 266, 272 Control

system, 12, 50 Decoupliag disturbance, 123 dynamic, 224-225

adaptive, 291

control, 224

admissible, 24, 266, 295, 297

static, 227

deadbeat, see deadbeat control

Detectability,

decentralized, 285

Dirac function, see function

decoupling, see decoupling

Distribution, 18, 302

hierarchical, 285 LQG, 297

71, 105, 246

as solution, 18 D~sturbance,

122, 152, 184

optimal, see optimal

DDSO, 122

stochastic, 294

estimation,

time optimal, see optimal

transition zero, 126

Controllability

128

Dual

about, 185

normalizability, see normalizability

canonical form, 54

principle, see principle

decomposition, 54

system, see system

317 Dynamic

terms, 18, 35, 80

compensation, see compensator

Input function observer, see observer

decoupling, see decoupling

Internal

feedback, see feedback control

model, 185

programming, 277

stability, see stability

Equivalence, 10

Kirchoff's law, I0, 33

algebraic, 16 restricted system, 10 similar, 14 Equivalent form first (EFI), 12, 51 second (EF2), 14, 47 third,(EF3), 15, 48

Laplace transformation, 19, 57, 305 Lemma elementary, 155 positive, 291 Leontief model, 3 Local state feedback stabilization, 289 Lyapunov

Feedback control dynamic Output, 174, also see

f u n c t i o n , see f u n c t i o n stability, see stability

dynamic compensation output, 98, 150 P-D state, 67, 207, 225 P-state, 68, 204 state, see state feedback

Model internal, see internal matching, 213 modeling, 285

Feedforward, 150 Finite discrete-time series, 232-238 Fixed mode finite, 287 infinite, 287

Neighborhood, 169 constrained, 172 Normal

Fixed polynomial, 287

compensator, see compensator

Function

DDSO, 123, 127

Dirac, 17, 302-304

IFO, 219

jump, 20

NOR, 154

Lyapunov, 293

observer, see state observer

test, 302 unit jump, 20

system, see system Normalizability, 74 dual, 76

Hamilton theorem, 32, 306 Observability, 38, 238, 244 from, 185 Impulse controllability, 29, 35 observability, 38, 43

impulse, see impulse observability index, 209, 258, 311 indices, 209, 311

318 R-, see R-observability

internal model, 190

Y-, see Y-observability

maximum, 267

Observer input function (IFO), 215 Luenberger, ii7 normal IFO, 219 singular .IFO, 215 State, 102, 250 Optimal

R-controllability, 29, 34, 237, 244 Readability, 185 Realization, 60 minimal, 57, 60 Recurrence backward, 233

control, 266, 272 regulation, 267, 312 time, 272

f o r w a r d , 233 Regularity matrix pencil, 6

trojectory, 269 Output

system, 4, 7, 101, 232 Riccati equation, 270

feedback control, see feedback control regulation, 153, 184

R-observability, 38, 41, 238, 244 Routh criterion, 167

response, 17, 241 Output regulator, 154, 185 normal, 154, 185 singular, 154

Singular DDSO, 123 IFO, 215

SSNOR, 185

observer, see state observer SOR, 154 Perturbation, 169 elementary assumption of, 186 Pole

constrained structural, 172

finite, 68, 87 infinite, 78

Stabilizability, 71, 246

Pole placement, 78-90, 245, 311 finite, 87, 92 infinite, 78, 80, 84 Pole structure, 95, 98, 310 infinite, 78, 95 Polynomial matrix, 60-61 m i n i m a l , 307

Principle d u a l , 46

State feedback control, 67-101, 201, 208, 245 filtering,

295

observation, teachability,

characteristic, 68

strictly,

internol, 154 structural, see structural

set, 68

Positive real,

system, see system Stability, 68-69, I01, 245

reachable set, response,

291

23, 236 25-27, 236

16, 241, 310

State estimatiou, 291

102-131

295-296

State observer, also see o b s e r v e r disturbance decoupling (DDSO), 122 full-order normal, 106 general structure of, 118

319 general structure of normal, 120

Target, 266

minimum-time, 254

Transfer matrix, 57, 197, 201, 208, 312

normal, 103, 106

Transformation

reduced-order normal, 106, 113

coordinate, 6-7, I0

singular, 102

time-scale,

198

with disturbance estimation, 128 State reconstruction,

see state observer

Structural

Y-controllability, 237, 244 Y-observability, 238, 244

constrained SSNC, 175, 178 parameter, 169 SSNC, 175, 179, 184 stable compensation, 167 stability, 168-170, 174 Sylvester equation, 155, 308 System continuous-time, I, 231, 243 decomposition, see decomposition discrete-t~me, 231 dual, 46, 246 large-scale, 2, 285 multiinput multioutput, 208 normal, 2, 310 output regulation, 152, 161, 184 PPS, 202, 212 single-input single-output, 201 singular, I, 2, 167 stochastic, 294 E-Suboptimal control, 273 time, 273 Substate fast, 12 slow, 12 Subsystem, 12, 59 backward, 240 fast, 12 forward, 240 slow, 12, 140

Zero-pole cancellation, 204

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