Multivariable Control: A Graph-theoretic Approach [Reprint 2021 ed.] 9783112480588, 9783112480571

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Mathematica/ Research Multivariable Control Kurt Reinschke

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K. J. Reinschke Multivariable Control

Mathematical Research

Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Karl-Weierstraß-Institut für Mathematik

Band 41 Multivariable Control by K. J. Reinschke

Mathematische Forschung

Multivariable Control A Graph-theoretic Approach

by Kurt J. Reinschke

Akademie-Verlag Berlin 1988

Autor: Prof. Dr. Ing. Dr. rer. nat. Kurt 0 . Reinschke Ingenieurhochschule Cottbus Wissenschaftsbsreich Informatik und P r o z e & e t e u e m n g Cottbus 7500

Die Titel dieser Schriftenreihe werden vom Originalmanuskript der Autoren

reproduziert

ISBN 3-05-500452-3 ISSN 0138-3019 Erschienen im Akademie-Verlag Berlin,DDR-1086 Berlin,Leipziger Str.3-4 ©

Akademie-Verlag Berlin 1988

Lizenznummer: 202 • 100/411/88 Printed in the German Democratic Republic Gesamtherstellung: VEB Kongreß- und Werbedruck, 9273 Oberlungwitz Lektor: Dr. Reinhard Höppner LSV 1085 Bestellnummer: 763 791 4 (2182/41) 03800

Preface This monograph is addressed to engineers engaged in control systems research and development, to graduate students specializing in control theory and to applied mathematicians interested in control problems. The author's objective is to present a graphtheoretic approach to the analysis and synthesis of linear time-invariant multivariable control systems. After 1950, the demands of control practice led to the development of control methods for plants having more than one input ahd/or output. These multivariable control systems have attracted much attention within the so-called "modern control theory", which is primarily based on state-space methods. The state-space theory has provided the control engineers' community with profund new concepts such as controllability and observability and has clarified essential system theoretic questions which were poorly understood in the "classical control theory". Nevertheless, seen from the point of view of practising engineers, the state-space theory has also serious disadvantages: - The plant is modelled by ordinary matrix differential equations where the entries of the occurring matrices are regarded as numerical values which are exactly known. Experience shows that the results of the controller design may be largely sensitive to small variations of the chosen numerical values for the matrix entries in the description of the plant. In practice, however, the engineer has to cope with more or less uncertain and varying plant parameters. - The procedures for the plant analysis and controller synthesis are based on cumbersome matrix manipulations. Control engineers lose desirable "feeling" and visual insight. - In the case of large-scale systems the order of matrices to be investigated increases rapidly.. Sparsity, as a typical feature of large-scale systems, must be taken into consideration. This requires specialized advanced matrix techniques. After 1970, the "geometric approach" to linear multivariable control arose as an attack "against the orgy of matrix manipulation" (see Wonham, 1974). This geometric approach is based on an abstract coordinate-free representation of linear vector spaces. The comparatively simple language of matrix arithmetic is translated into the more abstract language of high-dimensional vector spaces. Unfortunately, this level of abstraction does not correspond to the traditional kind of reasoning of control engineers . The "graph-theoretic approach" is another attempt to overcome the disadvantages of the state-space theory: The given control system is modelled by a suitably chosen graph representation. The system properties under investigation may be expressed by properties of the graph. That is why the investigator obtains a better insight into the structural nature of interesting properties. Provided he succeeds in showing that a desired property holds generically (i.e. independently of numerical parameter values), the indeterminate parameters may be considered to be degrees of freedom during further steps of design or optimization. If the graph-theoretic characterization shows that a desirable property does not hold, the investigator is able to suggest system modifi-

5

cations with the aid of which the desired property could be fulfilled. As for largescale systems, it should be realized that the graph-theoretic system representation reflects exactly the non-vanishing couplings. Zero matrix entries do not at all appear in the graph representation. Consequently, the graph-theoretic approach has proved to be especially suitable for sparse large-scale systems. The reader of this book is expected to have some previous acquaintance with control theory. He should have taken, or be taking concurrently, introductory courses in frequency-responce methods and in state-space methods. Some working knowledge in graphtheory would be an advantage, although it is not necessary. The Appendices 1 and 2 provide the reader with all the graph-theoretic tools needed for our purposes. In Appendix 1, the basic concepts are explained and the notations used in this book are introduced. In Appendix 2 several possibilities for graph-theoretic interpretations of determinants are dealt with. A crucial role plays an interpretation of determinants by cycle families published by A.L. Cauchy as early as in 1815 and re-invented by C.L. Coates in 1959. The main part of this book consists of four chapters. Chapter 1 starts with an appropriate graph-theoretic representation of large-scale dynamical systems. Based on this representation, important system properties such as decomposability, structural controllability and observability can be checked easily. Besides, the concept of "generic validity of a system property" is discussed in some detail. Chapter 2 - b cover material on controller synthesis. In Chapter 2, static state feedback is assumed. The

problem of pole placement is re-

considered, seen from the graph-theoretic point of view. The graph-theoretic approach to two classical problems - disturbance rejection and decoupling by static state feedback - supplies nice new results. In Chapter 3, static output feedback is assumed. Based on the graph-theoretic interpretation of the open-loop and the closed-loop characteristic polynomial coefficients, poles and zeros of multivariable systems - including zeros at infinity and their multiplicities - are characterized both algebraically and graph-theoretically. Then, the well-known problem of arbitrary pole placement by static output feedback is attacked. A necessary and sufficient criterion for local pole assignability and a new sufficient condition for global pole assignability are derived. In Chapter 4, an outline is given for further exploitation of the graph-theoretic approach to controller synthesis. Apart from static output feedback under structural constraints, the remaining problems - dynamic controllers, implicit system descriptions, non-linear plants, mixed system consisting of logical and dynamical components - are just briefly mentioned. This reflects the author's deliberate intent to leave the book "open" at the far end. That is, only the direction is shown in which further results by means of graph-theoretic tools may be found. There seems to be much Jeyond the confines of ths monograph, that should be tackled graph-theoretically. A few words about references Eire in order. Each chapter is divided into sections. Thus, Section 23 refers to the third section within the second chapter. For purposes of reference, formulas, theorems, lemmas, corollaries, examples and figures are numbered

6

consecutively within each section. A reference such as Fig. 23.11 refers to the eleventh figure of Section 23. In referring to the bibliography the name of the author and the year of publication of the source contained in the bibliography are written down. To indicate the end of a proof a full triangle, A , is used. Finally, I would like to acknowledge the help and support I have had from other people. I am grateful to the Director of the Zentralinstitut für Kybernetik und Informationsprozesse der Akademie der Wissenschaften der DDR, Prof. V. Kempe, for the possibility to work in this field over a period of some years. A debt of gratitude is owed to Prof. M. Thoma, President of the IFAC, and to Prof. H. Töpfer (Dresden). They encouraged me to prepare such a monograph. I have profited by (mainly epistolary) discussions with many experts in other countries, in particular with Prof. 0.1. Franksen and Prof. P.M. Larsen of the Danish University of Technology, Prof. F.J. Everns (London), Prof. D.D. Siljak (Santa Clara, Calif.), Dr. N. Andrei (Bukarest), Prof.D. Hinrichsen and Dr. A. Linnemann (Bremen), Prof. H. Schwarz and his co-workers (Duisburg), Dr. K. Tchon (Wroclaw), Dr. A.J.J, van der Weiden (Delft), Dr. L. Bakule (Prague). In this country, we have had stimulating debates concerning the topics of this book in the Control Theory Group headed by Prof. K. Reinisch (Ilmenau). A careful study of parts of the manuscript was undertaken by my colleagues Dr. J. Lunze and Dr. P. Schwarz. They helped to improve the presentation by their criticism. This support is most gratefully acknowledged. Another debt is acknowledged to Mrs. Schöpke who turned the author's rough pencil drawings into neat and workmanlike figures. Thanks are dure to the staff of the Akademie-Verlag, especially Dr. R. HSppner, for friendly and effective cooperation. Last but not least, I greatly appreciate all the various forms of help of my wife while the manuscript was in preparation.

7

Contents C h a p t e r 1.

Digraph modelling of large-scale dynamic

systems

11.

Mapping of state-space m o d e l s into digraphs

11

12.

Structure matrices and their associated digraphs

17

13.

Appropriate state enumeration

20

13.1

Decomposition based on connectability properties

20

13.2

A n algorithm for reordering the states

22

13.3

Some properties of irreducible structure matrices

27

14.

Structural controllability, structural observability, and structural

15.

completeness

14.1

Input-connectability and structural controllability

14.2

Criteria of structural controllability

34

14.3

Structural observability

41

a n d structural completeness

Do structural properties hold generically ?

C h a p t e r 2.

Digraph approach to controller based on static state

21.

30 31

44

synthesis

feedback

Pole placement by static state feedback

49

21.1

Problem formulation

49

21.2

Graph-theoretic characterization and algebraic reinterpretation of the closed-loop characteristic-polynomial

coefficients

51

21.3

Pole placement for single-input systems

21.4

Pole placement

62

21.5

Determination of all the feedback matrices

for multi-input systems

^9 which

provide the desired pole placement 22.

74

Disturbance rejection

79

22.1

Problem

22.2

A necessary and a sufficient condition

formulation and preliminary results

22.3

Compensation of the full variety of

22.4

A n algorithm

disturbance rejection

82 rejectable

disturbances

8

79

for

88 for disturbance rejection

95

23.

Digraph approach to noninteraction controls by means of state feedback 23.1 23.2 23.3 23.4

Chapter 3. 31.

31.2 31.3 32.

109 111

characteristic

Transfer functions and their graphical interpretation Feedback dependencies of the closed-loop characteristic-polynomial coefficients Graph-theoretic interpretation of the closed-loop characteristic-polynomial coefficients

119 120 128 137

Poles and zeros of multivariable systems

142

32.1

142

32.2 32.3 32.4 33.

103 108

Digraph approach to controller synthesis based on static output feedback

Transfer function matrices and closed-loop polynomials in graph-theoretic terms 31.1

102

An introductory example Problem formulation A necessary condition for decoupling by static sate feedback A sufficient condition for decoupling by static state feedback

Poles and zeros of single-input single-output systems Poles and finite zeros of multivariable systems Graph-theoretic characterization of structural properties of finite zeros and poles Infinite zeros of multivariable systems and their graph-theoretic characterization

145 156 169

Pole

placement by static output feedback

175

33.1 33.2

Problem formulation and preliminary considerations Necessary and sufficient conditions for local pole assignability

175

33.3 33.4

Conditions for global pole assignability Numerical computation of the smooth submanifold

193

M e IR m * r

208

belonging to a given regular p-value

180

9

Chapter 4. A n outline for further exploitation of the graphtheoretic approach to controller 41.

synthesis

Static output feedback under structural constraints

214

41.1

215

Controllability

and observability of systems

under structurally constrained output 41.2

feedback

Fixed modes and structurally fixed m o d e s

218

42.

Dynamic c o n t r o l l e r s

224

43.

Semi-state system description

229

44.

Digraph approach to nonlinear

systems

and automated complex systems

235

Appendix Al.

A2.

Introduction to Graph theory

237

Al.l

Graphs and digraphs

237

Al.2

Paths, cycles, and trees

240

A1.3

Graph-theoretic characterizations of square matrices

244

D i g r a p h s a n d determinants A2.1

Computation of determinants with the aid of weighted

A2.2

247

Computation of characteristic polynomials in several v a r i a b l e s with the aid of weighted

A2.3

247

digraphs digraphs

Graph-theoretic characterization of characteristicpolynomial coefficients and their

253 258

sensitivities

References

263

Subject index

271

10

C h a p t e r 1. D i g r a p h modelling of large-scale systems 11

Mapping of state-apace models Into digraphs

There are several possibilities to describe dynamic feedback mathematically. In this monograph, we base on the state-space

systems descrip-

tion. This means,we assume the plant under Investigation to be modelled by equations of the form xt(t) = f1(x1(t)Ix2(t),..,xn(t);u1(t),..,um(t);t)

(11.1)

for i = 1,. . ., n Yj(t) - g j ( x 1 ( t ) , x 2 ( t ) , . . , x n ( t ) ; u 1 ( t ) , . . , u n i ( t ) ; t )

(11.2)

for J » 1,... ,r The plant can be controlled by different control strategies. In the case of static state feedback the control law is given by u k ( t ) » h ® ( X l ( t ) ,x 2 (t) ,.. ,x n (t) ;t)

for l< = 1

,m

(11.3)

for k = l....,m

(11.4)

In the case of static output feedback we have uk(t) = h°(yi(t)

y r (t);t)

Usually, the column vectors

and

x(t) a ( x 1 ( t ) , x 2 ( t ) , . . . , x n ( t ) ) '

(11.5)

u(t) = ( u 1 ( t ) , u 2 ( t ) , . . . , u m ( t ) ) '

(11.6)

y(t) - ( Y l ( t ) , y 2 ( t ) , . . . , y p ( t ) ) '

(11.7)

are called state vector, input vector, and output vector, The symbol

means

respectively.

transposition.

With a given dynamic feedback system (11.1), (11.2), ciate a directed graph (digraph) G

s

(11.3) we asso-

defined by a vertex-set and an

edge-set as follows: The vertex-set is given by m input v e r t i c e s denoted by II, 12,..., Im, by n state v e r t i c e s denoted by 1, 2,..., n, and by r output v e r t i c e s denoted by 01, 0 2 , . . . , Or. The edge-set results from the following

rules:

If the state-variable Xj really occurs in f 1 ( x , u , t ) , i.e. Sfj/Oxj $ 0 , then there exists an edge from vertex J to vertex i.

11

If thje input-variable u^ really o c c u r s in fj^x.u.t), i.e. dfj/au^ * 0 , then there exists an edge from input vertex Ik to state vertex i. If the state-variable x ± really o c c u r s in gj(x,u,t), i.e. Qg^/axj^ 4 0, then there exists an edge from v e r t e x i to vertex O j . Finally, if the state-variable x t occurs in h®(x,t), what is generally assumed in the case of state feedback, then there exists an edge vertex i to vertex

from

Ik.

For illustration, a characteristic part of a digraph G 8 has been sketched in Fig. 11.1.

Fig. 11.1 Similarly, with a given dynamic system

(11.1), (11.2), (11.4) we asso-

ciate a digraph G .

g It has the same vertex-set as G .

The state edges indicating a ^ / a x ^

* 0 , the input edges indicating

3f i /òu | < * 0 , and the output edges indicating 3 g j / 3 x i * 0 result

from

the same rules as in case of G 8 . Instead of feedback edges leading

from state v e r t i c e s to input v e r t i c e s

we have now feedback edges from output vertex Oj to input vertex Ik if and only if y^ occurs in h£(y

,t).

Fig. 11.2 shows a characteristic part of a digraph G°. Obviously, the digraphs G s and G° contain less information than the equations (11.1, 11.2, 11.3) and (11.1, 11.2, 11.4),

respectively.

We shall say that the digraphs G s and G° reflect the structure of a closed-loop system with state feedback and with output feedback,

12

re-

spectively. A s far as small-scale systems are concerned,

for example, systems w i t h

n = 2 or n = 5 state variables, it seems to be unnecessary to investigate the system structure separately. In case of large-scale

systems,

however, we should start with a structural investigation in any case. The author hopes to convince the reader of this book that the digraph approach is extremely useful, in particular for large-scale

systems.

Fig. 11.2 A striking feature of large-scale systems is their sparsity. This structural property becomes evident in the digraphs G s or G°. These digraphs reflect a priori only the non-vanishing couplings of the 2 system. So, instead of n state edges we have really to take into account only a small percentage of this number in most

applications.

Moreover, the digraphs G s and G° give us an immediate impression of the information flow within the closed-loop systems. The of the following c h a p t e r s will demonstrate

considerations

that the investigator

re-

ceives more insight into the structural nature of properties under investigation than with the aid of the conventional numerical

treatment.

He should exploit the recognized structural properties for design purposes before he performs numerical

computations.

In this monograph, we deal with the digraph approach for linear multivariable control systems. This m e a n s the plant is modelled by matrix

13

e q u a t i o n s of the

where

form

A x > Bu

(11.8)

Cx

(11.9)

x(t) e IR n . u(t) e

R m , a n d y(t) e IR r . The m a t r i c e s A , B, C

real e l e m e n t s a n d are of d i m e n s i o n nx.n, n X m , In c a s e of static state feedback w e

rxn,

have

respectively.

have

u • Fx

(11.10)

a n d in case of static o u t p u t

feedback

u = Fy

(11.11)

A s a rule in c o n t r o l p r a c t i c e ,

the feedback m a t r i c e s F are

subjected

to structural c o n s t r a i n t s because some of the feedback g a i n s may not be chosen

freely. A s s u m e

the a d m i s s i b l e

feedback p a t t e r n to be c h a r a c -

t e r i z e d by the freely c h a n g e a b l e e n t r i e s of F. If all the e n t r i e s of F a r e p r e s u m e d to be freely a s s i g n a b l e ,

then we shall s o m e t i m e s use

the

symbol E i n s t e a d of F. In o r d e r to i n v e s t i g a t e m u l t i v a r i a b l e c o n t r o l s y s t e m s we shall s u i t a b l y c h o s e n square c o m p o u n d

consider

matrices.

It w i l l be seen to be useful that the total i n f o r m a t i o n a b o u t the o p e n l o o p system is s u m m a r i z e d in a c o m p o u n d square m a t r i x

Co 0 0 \

0

A

B

0

0

(11.12)

In c o n t e x t of c o n t r o l l a b i l i t y

of o r d e r

in context of o b s e r v a b i l i t y

of o r d e r

i n v e s t i g a t i o n s the following

square

appropriate.

(11.13)

n+m

investigations.

(11.14)

r+n

in c o n t e x t of both c o n t r o l l a b i l i t y a n d

14

r+n+m,

\

c

matrix w i l l prove to be most

«1 "

of o r d e r

observability.

0

C

0

0

A

B

of o r d e r

r+n+m

(11.15)

E O O in c o n t e x t of o u t p u t 0

C

0

0

A

B

F

0

0

feedback, p o s s i b l y s t r u c t u r a l l y

of o r d e r

constrained.

r+n+m

(11.16)

In A p p e n d i x A l . 3 , it h a s been d i s c u s s e d that there a r e several

possibi-

l i t i e s of c o n s t r u c t i n g an a s s o c i a t e d d i g r a p h having a o n e - t o - o n e

corre-

s p o n d e n c e w i t h a g i v e n square m a t r i x . In the s e q u e l , w e shall base on the s e c o n d g r a p h - t h e o r e t i c matrix c h a r a c t e r i z a t i o n i n t r o d u c e d in A l . 3 . Definition

11.1

Let Q be a g i v e n square matrix of o r d e r q. Q m a y be r e p r e s e n t e d by a d i g r a p h G(Q) w i t h q d i f f e r e n t v

l'

v

v

2'•••'

q"

T h e r e e x i s t s an edge

v e r t e x v^ if and

(v^, v.)

o n l y if the e n t r y q ^

T h e e d g e w e i g h t is e q u a l

vertices

from v e r t e x v^ to

of Q d o e s not

vanish.

to the n u m e r i c a l v a l u e of 2, and the p l a n t

3, m

equations

.8, 11. 1

'. \ X 1 X

2

a

B a

B

a

0 k

Yl

\

ll

C

31

12 0

a

32

>CM

11

0

0

0

For this e x a m p l e

x

r

X

2

X

3

b

+

ll

0

0

b22

0

0

(11.17)

(11.18)

"23

system,

the d i g r a p h G ( Q 3 ) h a s bei d r a w n in F i g .

11.3.

12

Fig.

11.3

15

Generalized symbolic represent tions of the digraphs G ( Q q ) , G f Q ^ , G(Q 2 ), , and G(Q 4 ) are shown in the Figures 11.4.1, b, c, d, and e, respectively. The hyper-edge symbols ^ correspond to matrices which may be regarded as generalized edge weights. The hyper-vertices u, x, and y of the generalized digraphs of Fig. 11.4 are associated with the input vector, the state vector, and the output vector, respectively.

Fig. 11.4

b) A

16

d)

c) A

e)

12

Structure matrices and their associated digraphs

In the framework of the traditional control theory, the entries of the matrices A, B, C, F are regarded as numerical data given with 100 percent precision. For physical reasons, however, the parameters involved in the entries of A, B,... are only approximately known. Consequently, it seems to be more adequate to regard the noet entries of A, B, ... a* indeterminate. Only some entries which are often precisely zero have exact numerical values. In context of "structural controllability" introduced by C.T. Lin in 1974 and of related "structural investigations" one has been used to take into account only the "structure" of the matrices A, B,.... This means, instead of numerically given matrices A, B,... the corresponding structure matrices [A], [B], ... of the same dimensions are considered. Definition 12.1 The elements of a structure matrix [Q] are either fixed at zero or indeterminate values which are assumed to be independent of one another. A numerically given matrix Q is called an admissible numerical realization (with respect to [Q]) if it can be obtained by fixing all indeterminate entries of [Q] at some particular values. Two matrices Q* and Q" are said to be structurally equivalent if both Q' and Q" are admissible numerical realizations of the same structure matrix [Q], We shall denote the indeterminate entries of a structure matrix by "L" and the entries fixed at zero by "0" or, often more conveniently, by an empty place. Example 12.1; Consider a plant with m = 2

inputs and

n = 6

state-variables,

outputs that is described mathematically by

r = 3

the following matrices introduced by Eq. (11.8) and Eq. (11.9):

A

C

\

0 0 0 0 0 a 24 a25 21 0 0 0 0 a36 0 0 a33 0 . B a 0 0 0 0 0 0 0 42 0 b 0 0 0 0 a a 51 54 55 0 0 0 0 0 b62 0, there is an

s e S

if, for each

such that the Euclidean

9(3,r) * 6.)

To give an example of a structural property let us introduce the notion of "structural rank of a rectangular structure matrix" Definition

[Q].

12.3

A set of Independent entries of

[Q] Is defined as a set of indeter-

minate entries, no two of which lie on the same line (row or column) . The structural

rank (for short, s-rank) of [Q] is defined as the

maximal number of elements contained in at least one set of Independent

entries.

It should be noted that the s—rank of

[Q] is equal to the maximal

rank

(in the usual numerical sense) of all admissible numerical matrices Q , s-rank

[Q] =

max rank Q Q e [Q]

(12.3)

In the literature, the notations "generic rank" and "term rank" are used with the same meaning as "structural rank" (see, for example, Oohnston et al. 1984 or Andrei 1985 and the numerous references cited there).

19

13 13.1

Appropriate state

enumeration

Decomposition based on connectabilitv

properties

For large systems there exists a high degree of sparsity in the system matrix A . The associated digraph G([A]) reflects this sparsity in a most evident manner. Moreover, the digraph representation of [A] has an important invariance property: The enumeration of the vertices does not play any role. In other words, the digraph G([AJ) is invariant with respect to permutation transformations of [A], A n appropriate

re-

ordering of the vertices, however, has proved to have many useful implications, especially in case of large systems. For this purpose, we try to decompose the digraph G([A]) into subgraphs based on connectability properties between its vertices. Such a decomposition of the digraph G([A]) should always be made as tion of large-scale

basis for the investiga-

systems.

In Appendix A 1 . 2 , n o t i o n s and notations suited for the

decomposition

of digraphs are explained. We have to look for subgraphs of G( [A]) whose vertices are strongly connected. Here let us remind the

reader

of the fact that two v e r t i c e s j and i are said to be strongly connected if a path exists from vertex j to vertex i as well as a path from vertex i to vertex j. The subset of v e r t i c e s strongly connected to a given vertex i forms an equivalence class K(i) within the set of all the n v e r t i c e s of G([A]). Each equivalence class of strongly connected vertices,

together with

all the edges incident only with these vertices, constitutes a subgraph G([Q]) belonging to a square submatrix

[Q] of

[A]. In terms of

matrix theory, the property of strong connectability of G([Q]) called irreducibilitv of [Q] (see, for example, Gantmacher

is

1966).

If G([A]) does not contain a cycle that touches a given vertex i then we shall say that the vertex i constitutes an "acyclic" class. The corresponding the corresponding main diagonal of

equivalence

subgraph G([Q]) is the isolated vertex i, and

square submatrix

[Q] is a zero element placed on the

[A],

The set of equivalence classes can now be enumerated in such a way that transitions from equivalence classes of lower indices to equivalence classes

of higher indices are

The reordered structure matrix

impossible.

[A] results from [A] by a permutation

transformation, [A] = P 1 [ A ] P

20

(13.1)

1 2 3 1» 5 6 7 8 9 10 11 12 1 3 1 5 16 1 2 3 V 5 6 7 [A]

L

L L L

L

L L L L

L L L

L L

L L L L L L L L

e 9 10 11 12 13 11» 15 16

which cannot be used any longer, we look for

the next non-zero column which lies to the right within the jth hyperrow. This column is denoted by Rj. In the same way as in Lemma 13.1 it can be proved that both (AJJ - M , matrix A

Rj) and the modified square sub-

have full rank.

The column where Rj stems from must not be used in the sequel.... The next modified main diagonal submatrix will be A ^ . . . . This process is continued until the newly added column belongs to B, i.e. the last modified main diagonal submatrix Ap^ arises from (A v

PP

32

- XI, R ) where R stems from B. ' P' P

The desired regular

nxn

submatrices ^¿¿f ^ j i ' submatrices (A

submatrix of (A - X I , B) is yielded by the ^pp

t0

9 e t h e r w i t h all the other main diagonal

- Xl) which were not changed during the choice p r o c e s s

described above. T h i s completes the proof. Example 14.2: We shall illustrate the simple basic idea of the foregoing proof by an example (see Fig. 14.2). 1 2 3

1 * i * n, of the closed-loop

characteristic

polynomial may indeed be obtained with the aid of the cycle

families

of width i in G(Q) as stated in Theorem 21.1.

A

Let P - (Pj. P 2

Pn)'

(21.13)

be the column v e c t o r encompassing the coefficients of the closed-loop characteristic polynomial

(21.5). Each coefficient can be

represented

a s a sum Pi "

p

i

+

h

i(F)

(21.14)

where p°, p 2 . . . . . p°

are the coefficients of the characteristic

poly-

nomial of the open-loop system (21.2), i.e. det( s I - A) - s n + p ° s n - 1 + ... + p ^ j S + p° and

h^CF)

is a multilinear form of feedback

(21.15)

gains.

From Theorem 21.1 it can now be e a s i l y seen how the coefficients pend on the feedback

Pl

de-

gains:

Theorem 21.2 The v e c t o r (21.13) depends on the feedback gains as follows. .o

m y-

n y-

- e"(bi,Ab1,A2bi)

-p^

P°2 (21.25)

P?

P°n-1 M

ej (b^^.Abj

.An-2b1.An-1b1)

P°n-2 o Pi

For

j = 1, 2,..., n, all these equations can be compressed into the

matrix equation (21.22) that w a s to be proved.

Finally, it should be noticed that the matrix equation

(21.22) can also

be understood using purely graph-theoretic arguments. For this purpose we multiply both sides of (21.22) by e^ from the left and by e^ from the

right, -l.e'Ak~1bi+

p ° . e j A k ~ 2 biJ++. . . + P°_ 2 ejAb.+ p ° _ 1 . e ' b i = - 1 - p J 3

A l l the terms occurring in (21.26) admit of a graph-theoretic

(21.26) interpre-

tation. They are subgraphs within the digraph 'A

b/

(21.27)

e- 0 The coefficients p ^

( ^ = 1, 2,...,I4

n = 17

state vertices. The edge sequence

forms a simple path from which all state verti-

ces can be reached. At this point we recall the historically first characterization of s-controllability published by C.T. Lin in 1974. Using a more flowery language, saying "stem" and "bud" insted of "simple path" and "cycle", respectively, he denoted a spanning subgraph of (21.29) having the

62

structure Illustrated In Fig.21.4 as a "cactus". (More exactly: A 1 cactus' associated w i t h [A.b] is a spanning input-connected graph consisting of a simple path with the initial vertex I, p 0) v e r t e x disjoint c y c l e s and p distinguished edges each of which connects exactly one cycle with the path o r with another cycle.) A s a consequence, Lin formulated his main result as follows: Lemma 21.2 The pair [A,b] is s-controllable

if and only if its associated

digraph is spanned by a cactus. We shall here prove another graph-theoretic characterization of s - c o n trollability for single-input

systems.

Lemma 21.3 The pair [A,b] is s-controllable if and only if there exist w i t h i n the digraph (21.29) n cycle families such that each of these cycle families contains another feedback edge and has a different w i d t h . Proof: (a) Sufficiency: Assume there exist n different cycle families as specified in Lemma 21.3. Consider such a set of cycle families. Each of the n feedback edges involved starts from another state vertex j and leads to the input vertex, where

J = 1, 2

n. Each feedback edge belongs to a

cycle consisting of a simple path from the input vertex to the state vertex j and the feedback edge under consideration. This implies the input-connectedness of every state vertex

J = 1, 2,..., n.

By assumption, there is a cycle family of width n in the digraph

(21.29),

Thus both the conditions for s-controllability stated in Theorem

14.2

are

fulfilled.

(b) Necessity: If the pair [A,b] is s-controllable then such a set of n cycle as specified in Lemma 21.3 can be constructed as

families

follows:

Choose in the given digraph (21.29) a cycle family of width n, c o n taining a feedback edge, and, additionally, a minimal set of input and state e d g e s such that the input-connectedness of all state v e r t i c e s rem a i n s valid. In the sequel, we consider the subgraph G chosen e d g e s and all feedback

made-up by the

edges.

The digraph G is nothing else than a c a c t u s in the sense of Lin,

comp.

Lemma 21.2, completed by n feedback edges. It should be realized

that

the reduced digraph G is still associated with an s-controllable pair [A.b] C

[A.b].

The set of final v e r t i c e s of all input edges in G is denoted by Sj and its cardinality by s^. If

Sj> 1

then

of the S^-vertices

belong

to cycles in G. Denote one of these Sj-1 state v e r t i c e s by 1. If

Sj » 1

then Sj consists of one state v e r t e x - w h i c h is denoted by 1. 63

In both cases, the input edge (1,1) together with the feedback

edge

(1,1) constitute a cycle family of w i d t h 1. Now we try to continue the path from I to 1. The set of final v e r t i c e s (* 1) of all state edges starting in 1 is denoted by S 2 , its cardinality by s 2 . If

S2" 1

s 2 > 1 then

t le

'

S

2"

v e r t i c e 8

belong to cycles in G

not yet touched previously. Denote one of these s 2 - l state v e r t i c e s by 2. If

s 2 = 1 then S 2 consists of one state vertex which is denoted by

2. In both cases, the edge sequence {(1,1), (1,2), (2,1)}

constitutes

a cycle family of w i d t h 2. A g a i n we try to continue the path ces ( $

I

1

2. The set of final verti-

{ l , 2}) of all state edges starting in 2 is denoted by S 3
1 then s 3 - l of the S^-vertlces belong to cycles in G not yet touched previously. Denote one of these S3- 1

state

v e r t i c e s by 3. If s 3 = 1 then S^ consists of one state vertex w h i c h is denoted by 3. The edge sequence {(1,1), (1,2), (2,3), (3,1)} constitutes a cycle family of width 3. ... A f t e r e finite number of steps, say continue the path

I

1

2

steps, it will be impossible to

... -*• Jj.

Now we look for a state set S^ w i t h the maximal index J,

1 * J < J^,

in w h i c h at least one state vertex h a s not been denoted yet. Let the maximal index be J 2 . The state v e r t i c e s contained in

K-

v* v*}

are spanned by a cycle family that constitutes a permanent part of all further cycle families to be constructed in the following. If there are "s. > 1 v e r t i c e s in S not denoted a s yet then "s.. - 1 Jg J2 J2 v e r t i c e s of S. belong to cycles not touched previously. Denote one of J2 _ these s H - 1 v e r t i c e s by J 11 + l . If s H = 1 then denote the corresponding J2 J2vertex of Sj

by j^+1.

The edge sequence { ( 1 , 1 ) ,

(j2-l.J2). (J2'Ji+1)-

(1,2)

forms a cycle. This cycle together with the cycle 1 state sets -{s. Ji ,..., S 32*1 | of w i d t h Jj^+1.

(Ji*1-1)}

family spanned by the

used previously constitute a cycle

family

Next we try to continue the process of path prolongation starting the path I

1

...

v e r t i c e s ($ {l, 2,..., J

J2 2
(A-sCI)"1Bw2,..,(A-sCI)"1Bwn]

(21.70)

Multiplying both sides by F from the left and taking into account (21.69) we get -(w1#w2

wn) = F[(A-s°I)-1Bw1,

From this follows

... , ( A - s ^ I ) - 1 B w n ]

(21.67).

Obviously, the m-vectors

may be interpreted as parameter vectors.

Any set of parameter vectors (o^Wj < oí 2 w 2 scalars o ^

(21.71)

,« n

° ¿ n w n^

non-vanishing

gives the same results as the set (w^ f Wg , .. . ,w n ) .

Therefore, we have n(m-l) free parameters. A In an explicit manner Eq. (21.67) was first derived and recognized as "a procedure for computing F" by B.C. Moore in 1976. G. Roppenecker has used this relation in 1981 and later on. There the reader can find many interesting applications, see also 0 . Follinger 1986.

78

22

Disturbance

22.1

rejection

Problem formulation and preliminary

results

Consider an externally disturbed system mathematically described by the equations (22.1)

X(t) - A x(t) + B u(t) + D v(t) y(t) where

c

(22.2)

x(t)

x ( t ) e R n , u ( t ) e IRm, and

y(t)e

Rr

symbolize the state vector,

the input vector, and the output vector as above. The newly introduced vector

v(t) € IR^

represents the disturbances. The

nxq

matrix D is

assumed to be real and time-invariant. The problem of complete disturbance rejection is to find a state feedback matrix F such that (22.3)

u(t) « F x(t) so that the disturbances v(t) have no influence on the outputs

y(t).

Fig. 22.1 shows the generalized digraph associated with the square matrix 0

C

0

0

0

A

B

D

0

E

0

0

0

0

0

0

of order

r+n+m+q

(22.4)

A Fig. 22.1 All entries of the

mxn

matrix E may be freely chosen.

In Fig. 22.2 a characteristic part of the digraph G ( Q 5 ) has been sketched.

79

Fig. 22.2 Now we are able to formulate the problem of complete disturbance jection in graph-theoretic

re-

terms.

Map the disturbed plant (22.1), (22.2) into the digraph G ( Q & )

asso-

ciated with 0

C

0

0

0

A

B

D

0

0

0

0

0

0

0

0

Complete disturbance

(22.5)

rejection means that we are able to compensate

simultaneously each of the paths leading

from a disturbance vertex

to an output vertex by a path leading from the same disturbance vertex to the same output vertex via a state feedback

edge.

More definitely, each of the simple paths in the digraph G([Qg]) leading from a disturbance vertex to an output vertex must contain a state edge (j,i) that can be compensated by edge (Ik,i)j- with the aid of an admissible

pairs(J,Ik),

feedback matrix

F e [£].

Compensation takes place if and only if

'U For the example system shown in Fig. 22.2 complete disturbance

(22.6) rejec-

tion is impossible because the path -{(v^,!), (i.yj)} cannot be a f f e c t e d

80

by any state feedback (22.3). This observation can be generalized at once. Lemma 22.1 Complete disturbance rejection is impossible if there is in G(Qg) a path of length 2 from a disturbance vertex to an output vertex, or, said in algebraic terms, complete disturbance rejection is impossible if there are admissible realizations C e [ C ] and D e [ D ] such that C D - 0

(22.7)

Looking for admissible feedback matrix candidates F G [E] by which disturbance rejection might be achieved two important observations must be taken into account. First, feedback edges with non-vanishing weights can produce new paths from disturbance vertices to output vertices. Second, only those state edges can be compensated whose final vertices are incident with an input edge.

81

22.2

A necessary condition and a sufficient condition for disturbance rejection

The maximal subset of the output-connected state vertices whose output-connectedness cannot be compensated by state feedback

is

denoted by V . ' max The state subset V

max

may be determined as follows: '

First, reverse the edge orientations in the digraph G([Q q ])

associated

with Q

o

0

C

0

A

o' B

0

0

0

Introduced in (11.12)

Second, pursue each of the paths starting in any output vertices until a state vertex adjacent to an input vertex has been reached. The set of all state vertices touched during this process constitutes the subset V . max A formal algorithm for the determination of

v

m a x

has been written down

below (Fig. 22.11). By V Q we denote the subset of state vertices in G([Qg]). see (22.5), adjacent to disturbance vertices, i.e. V Q = { i : i e { l , 2 , . . , n j , (Dj.i) exists for some J e { l , . . , q } | Lemma 22.1 may now be extended to a more comprehensive structural condition for complete disturbance

(22.8)

necessary

rejection.

Theorem 22.1 For complete disturbance rejection it is necessary that the state subsets V m a x and V Q are disjoint, V

max n

V

D

* 0 •

(22.9)

must not be adjacent to disturbance Proof: Each state vertex of V — — max vertices. Else disturbance rejection by state feedback would be out of the question.

A

Assume condition (22.9) to be met. In other words, the disturbances structurally characterized by [D], see (22.1), act directly only on states of 7

82

max " i 1 '

2

"}XVn,ax

(22

'10)

In o r d e r to decouple

from the system outputs the effect of

disturbances

ire have to choose the feedback gain matrix F in such a w a y that the w e i g h t s of all p a t h s from disturbance v e r t i c e s to output vertices v a nish in the clo9ed-loop system. For this purpose, we try to compensate all e d g e s of an appropriately chosen cut set of e d g e s in G([Qg]). Definition 22.2 A disturbance-output cut set, for short, D-0 cut set, in the digraph G([Qg]), see (22.5), is defined as a minimal set of state e d g e s whose removal w o u l d disconnect all paths leading from one of the disturbance v e r t i c e s 01

Dl,..., D q

to one of the output v e r t i c e s

Or.

The attribute "minimal" indicates that no proper subset of those state edges is capable of cutting each of the

disturbance-output

p a t h s simultaneously. The set of initial v e r t i c e s (final vertices) of all edges of a D - 0 cut set is denoted by T q

(TJ).

E v e r y D - 0 cut set induces a natural partitioning

of the state v e r t i c e s

a s well as of the input vertices. For illustration, see Fig. 22.3.

Fig. 22.3 A detailed description of all the subsets of v e r t i c e s occurring in Fig. 2 2 . 3 gives Definition 22.3 In the digraph G ( [ Q 6 ] ) . see (22.5), consider a D - 0 cut set with the initial vertex set T and final vertex set T,. o l The subset of state v e r t i c e s involved in paths from any disturbance v e r t i c e s to T

o

is denoted by V ' o

83

The subset of state v e r t i c e s involved in paths from Tj to any output v e r t i c e s is denoted by

V^

The subset of the remaining state v e r t i c e s neither contained in V Q in V 1

nor

is denoted by V 2 >

The subset of inputs from which all paths to any

output v e r t i c e s pass

through T q is denoted by 1(0). The subset of Inputs, not contained in 1(0), from which all p a t h s to output v e r t i c e s pass through Tj^ is denoted by 1(1). The remaining inputs neither contained in 1(0) nor in 1(1) form the input subset 1(2). These partitions -of the state set and of the input set are associated with a corresponding partitioning of the structure matrices [A] and [B], see Fig. 22.4.

Fig. 22.4 Now we can formulate a sufficient condition for disturbance by state

rejection

feedback.

Theorem 22.2 A given system (22.1), (22.2) admits of complete disturbance

re-

jection by state feedback (22.3) if there e x i s t s a D-0 cut set in the digraph G([Qg]) such that the rank of the associated matrix

Bt

sub-

formed by the common entries of the T h r o w s

and

the i(l)-columns of B is equal to the cardinality of T^, i.e. rank B T

84

J.J. = cardiTj).

(22.11)

In Fig. 22.4, the submatrix

BT

has been marked by double

1

hatching.

Proof of Theorem 22.2: The trivial case in which there are no paths between disturbance vertices and output vertices has been tacitly excluded in the following consideration. Select a 0-0 cut set and determine the associated partitioning of the state-variables and of the inputs as indicated by Definition 22.3. Accordingly, the structure matrix [E] can be subdivided into sections as sketched in Fig. 22.5. V —i-Vo 1 — -•j T 0 * Ti I 1 I HO) 1 I | 1 I 1 I 1 I 1(1) 1 I | •

—

t

c

J tI

1 1 l

1(2) ± _

1 i Fig. 22.5

We have to choose a state feedback matrix F e [E ] in such a way that the closed-loop system matrix A + BF has the shape shown in Fig. 22.6. —

v0

-r -

vi —

~~*1 To

Ì ° 1

V

..

i Ti

l

i

° o o o o o o

T

o o o o o o

,

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

Vi i _

f 1

2 —

o o o o o o o o o o o o

12

Fig. 22.6 1

85

This

means

B

T

B

(V1\T1)WV2,I(2)

A T

l'

F

I(l)Ul(2)

l f

T

+

o

B

I(1)V/I(2),

F

V

I(2),Vo F

T1.I(1)V>I(2)

"

\T

q

0

°

o

(22.12)

0

(22.13)

I(1)UI(2),T

0

"

0

(22.14)

Putting

F

I(l)Ul(2).V

F

o

\T

0

Eq. (22.25) is solvable if and only if (22.26) The choice of the feedback

90

matrices

holds.

F

"1(2).V.

I (v 2 ) , V ' F I (v1 ) , V ' ' max '' max

[comp. Fig. 22.9)

does not Influence the rejection of the disturbances under investigation. Example 22.2: Consider the example system sketched in Fig. 22.10. Since V

max

=

i4'

5

'

6

7

'

'

8

'

9

10,

'

17

VD = {

a n d

J"

13, 15, 1 6 }

are disjoint, the necessary condition (22.9) is met. Here the matrix equation (22.25) reads a s follows; a

4,ll 0

0

"8,12

0

3

0

4,2

°

f

b 8,3 3

9,12

f

9,3

2,ll

f

2,12

3,ll

f

3,12

Obviously, b

rank

rank

%,!(!)

0

42 0

b

83

0

b

93.

cardfSj)

2 < 3

(22.28)

Complete disturbance rejection with the aid of state feedback for all the disturbances meeting condition (22.9) is impossible because c o n d i tion (22.26) is not

fulfilled.

T h i s does not necessarily imply that rejection of the actual disturbances exhibited in Fig. 22.10 is impossible. Let us try to apply Theorem 22.2. Choose the edges (3,11), (14,12) as disturbance-output cut set. T h i s implies the following

subdivision of state v e r t i c e s and inputs (comp.

Fig. 22.3 and Fig. 22.8): T0 ={3. V

14}

= -{ 2, 3, 13, 14, 15, 16}

T. = •( 11, 12} V^^ = { 4, 5, 6, 7, 8, 9, 10, 11, 12,

17}

1(0) = 0 1(1) = { I I .

14}

1(2) = { 1 2 ,

13}

The sufficient condition (22.11) is met because of rank B T

1(1)

=

rank

b„ „ „ 11,1 0

0

2 = card(T^)

12,4;

91

Therefore, feedback matrices F supplying disturbance rejection do exist. Suited feedback matrices may be obtained with the aid of (22.15), (22.16) and (22.17) as follows: ij

0

for

0

for

1. 2, 3, 4 i - 2, 3

0

'11,3 0

b

a12,14

J - 2, 3, 13, 14, 15, 16

and

ll,l 0

J • 2, 13, 15, 16

and

'1,3 f

, 4.3

12,4

'1,14 '4,14

Hence, '1.3

'1.14

11,3

4,3

0,

4,14

- _

12

"12,4

while the remaining feedback gains may be freely chosen. Whence 1 2

in

3 4 k

1 L L L L

2 0 0 0 0

3 L 0 0 0

4 L L L L

6 L L L L

5 L L L L

7 L L L L

8 L L L L

9 10 ii 12 13 14 15 16 17 L L L L 0 0 0 0 L L L L L 0 0 0 0 L L L L L 0 0 0 0 L L L L L 0 L 0 0 L

= The entries underlined constitute the submatrix I(1) T numerically fixed by condition (22.17). The other L-entries are indeterminate. Thus, there are 44 feedback gains that need not be fixed in order to ensure complete disturbance rejection. These degrees of freedom should be used for the simultaneous solution of other controller synthesis requirements.

If condition (22.26) is not fulfilled then complete rejection of all disturbances meeting condition (22.9) can still be achieved after augmenting the system by appropriately chosen additional inputs. Suppose the investigator to be able to introduce cardfSj^) - rank Bg additional inputs acting directly on states of S^. Denote the set of additional inputs by 1(a). Let the additional Inputs act on S^ in such a way that ran,

< ^ . K U U K a )

"

card

(22.29)

Then the augmented system of linear equations V 92

S

o

+ ^,1(1)1/1(8)

F

I(l)\JI(a).S o

- 0

(22.30)

o r , e x p l i c i t l y written, ajt +

H

bjkfki

"

0

for

JgSi-

l e s 0

( 22.30')

with Ik e l ( l ) U l ( a ) becomes solvable. In the augmented system, therefore, complete rejection of any disturbances meeting (22.9) i s possible by means of state feedback. Example 22.3: We continue to Investigate the example system discussed before. Because of (22.28) we heve to introduce c a r d ( I ( a ) ) - c a r d f S ^ - s-rank[Bg

jjj)]

» 3 - 2 - 1

additional input. Thus, 1(a) - { i s } . Let 15 act on the ninth stats-variable, see Fig. 22.11.

F i g . 22.11 The augmented structure matrix Bg V ^,1(1)^1(8)

0

j ( i ) L / I ( a ) becomes

0

0

b8,3

0

b9,3 b9,5

0

(22.31)

Obviously, rank B ^ > I ( 1 J U I ( a )

• 3.

93

This ensures the solvability of (22.30). For the example under investigation Eq.(22.30) can explicitly be w r i t t e n down: a

0

4,11 a

b

4.2

°

8,12

8.3 0

9,12

b

0

f

0

f

2,ll

f

2,12

3,ll

f

3,12

lf5,ll

f

5,12

9,3

b

9,5

3,ll=

f

5,ll=

One gets f

'JL11

2,ll

>4.2 1

'5,12

. 9,5

(_a

9,12

a f

+

f

'3,12

8,12 8,3

b9-3 a . 8,12) 8, 3

The edges associated with bg

5

, f 2 11' ^3 12

marked by open arrows in Fig. 22.11.

94

2,12=

anc

'

12

have been

22.4

A n algorithm

for disturbance re.1ecti.on

A feedback design procedure that ensures complete rejection of any disturbances meeting condition (22.9) can now be briefly

stated.

Step 1: Consider the governing system equations (22.1), (22.2) and form the structure matrices [A], [B], [C],

[D].

Step 2: Inspect the structure matrices [B], [C], [D] and write down the [c 1 ],

structure row v e c t o r s [b'] (

[d*]

where [b']

1

. [c1^

1

[d'L

1

[L mJ [0

"{

U

(22.32)

else if the ith column of TC] contains at least one L

^0

IL =•{ (0

for

if the ith row of [B] contains at least one L

else if the ith row of [0] contains at least one L

(22.33)

(22.34)

else

i = 1, 2,..., n.

Form the structure row v e c t o r [r1]

=

[c^Afd'J

(22.35)

by elementwise conjunction, . ÎL [r ]. =< 1 0 We have

if both

[c 1 ]14

= L

i.e. and

else [r 1 ] * (0, 0,..., 0)

[ d '1L

= L

for

i = 1

n.

if and only if the relation

(22.7) holds, comp. Lemma 22.1. In this case, complete

distur-

bance rejection is impossible. Otherwise, go on to the next step. Stepr 3: Determine the subsets V

and S. of state variables. These max 1 subsets may be found with the aid of the following algorithm (see Fig. 22.12):

95

Fig. 22.12 The non-vanishing elements of the structure row vectors [ v m a x ] and [s 1 ] obtained by the outlined algorithm give the desired sets v m a x and Sj, respectively, V

96

max " { l !

1 6

i1'

2

"}•

"

(22

'36)

Sj^ - { i s

1 e { 1, 2

n},

S t e p 4 : Form t h e s t r u c t u r e [r2] -

[s1]1 - L}

(22.37)

row v e c t o r

[v"ax]A[d-]

(22.38)

o If

[r

] * 0

then complete d i s t u r b a n c e

rejection i s

comp. Theorem 2 2 . 2 . O t h e r w i s e go on t o t h e n e x t S t e pr 5 :

U s i n ga V „ end S„ d e t e r m i n e t h e s t a t e max x the p a r t i t i o n 1 ( 1 ) , 1(2) of the i n p u t 7

max " {

SQ - { i :

l !

1 6

41'

i £ { l ,

2

2

n},

«{iki

ke{l

«},

[bik]

1(2)

» {ik:

ke{l

in},

Ik $

Step 6 : Consider the p a r t i t i o n i n g o f Possibly a f t e r

for

- L for j e s J

(22.40)

1 € S 1 }-

(22.41)

1(1)}

(22.42)

t h e m a t r i c e s A , B, F I n d u c e d by

subsets determined i n the foregoing

reordering of

t h e s t a t e and i n p u t into

sets,

steps. t h e ma-

s u b m a t r i c e s as shown

22.9.

in

2

Complete r e j e c t i o n o f a l l

disturbances with

( 2 2 . 2 8 ) , makes demands on t h e Step 7 :

card(S^)

solving

(23.37)

for

there are many solu-

tions. Each of them may be used in the sequel. The other submatrices of G are given by (23.29). 7. For each

cr = l,2,...,s

compute

F y r ^jcr ' max

solving

(23.30).

The remaining entries of F may be used for other purposes, in particular, for pole assignment. The desired decoupling has been achieved now. 8. The decoupled system has B = B-G

and

A = A + B-F

and as system matrix, respectively. For put w0'

as input matrix

cr = 1,2,...,s, the new in-

subvector "

F

I*,{l,2,...n}x

+

^Sf.l^)"1

controls the given output subvector

B

sf,I(l) y°~

ul

2

Fig. 23.15 Nevertheless, decoupling is easily possible, see Fig. 23.15. Introducing a new input vector s W

1

,W2

\

U

a

+

,u2

a

-f15x5l

1

U

1

«

0

.U2

+

15 0

one obtains a decoupled system shown in F i g . 23.15.b. The allowable (i.e. noninteraction controls preserving) feedback matrices F have a structure as 1 2 3 4 5 6 7 L L 0 0 f15 L 0 where f 1 5 - ^ 0 0 L L L 0 L 117

The seven F - e n t r i e s marked by "L" m a y be arbitrarily chosen. A s a conclusion from the foregoing considerations, we can

strengthen

Theorem 23.2 a s follows: Theorem 23.3 For decoupling by static state feedback of each of the

systems

meeting (23.23) it is necessary and sufficient that the rank dition (23.34)

con-

holds.

The analogy between Theorem 22.5 and Theorem 23.3 should be noticed. The meaning of the phrase "each of the systems meeting

(23.23)" in

Theorem 23.3 can be made quite clear with the aid of Fig. 23.8. The rank condition (23.34) reflects only the adjacency relations bet1 2 s 1 and S 1 , S^ S^. If (23.34) is met

ween the "hyper-vertices" u

then decoupling is possible independently of the actual realization of all the "hyper-edges" V

max ^

S

1

for

=

u1

u2 = * V m a x .

V ^

1,2,...is. If .(23.34) is not met then decoupling

is impossible for at least one realization of these

118

hyper-edges.

C h a p t e r 3 . D i g r a p h a p p r o a c h to controller synthesis b a s e d on static output feedback 31

Transfer function matrices and closed-loop polynomials In graph-theoretic

characteristic

terms

In Chapter 3, we consider closed-loop systems obtained by applying static output

feedback

u(t) = F y(t)

(31.1)

to the plant ¿(t) = A x(t) + B u(t)

(31.2)

y(t) = C x(t)

(31.3)

x ( t ) e lRn,

where

u ( t ) e IRm,

y(t)e

R r , rank B « m, rank C - r.

From (31.1), (31.2), (31.3) we get, by the Laplace

transform, (31.1-)

U(s) = F Y(s) s X(s) - x(O) = A X(s) + B U(s)

(31.2') (31.3')

Y(s) = C X(s) This set of e q u a t i o n s can equally be written as I 0

0

y

- A

-B

X

0

I m

u

-C

r si

n

-F where 1 , 1 , r n spectively.

and I

m

0 B

x(0)

(31.4)

0

are unit matrices of dimension r, n, and m, re-

119

31.1

Transfer functions and their graphical

Multiplying

interpretation

the second hyper-row of (31.4) by (sl n - A)

from the left

and substituting the result into the first hyper-row, we obtain Y(s) = C ( s l n - A ) - 1 B U(s) + C ( s l n - A ) - 1 x ( 0 )

(31.5)

O n the right-hand side of (31.5), the so-called transfer matrix

function

T(s) occurs,

T(s) = C ( s l n - A ) _ 1 B

(31.6)

The (j.i)-entry of T(s) is the transfer function of the plant from input u^^ to output yj (1 < i < m,

l < j < r)

tdi(s) - c'(sln - A ) - 1 b 1 - [det(sl n - A ) ] - 1 c ' ( s l n - A ) a d J Z

bl

11(9)

d e t ( s l n - A) The denominator is the open-loop characteristic polynomial

introduced

in (21.15), d e t ( s l n - A) » s n + p ° s n _ 1 + ... + p°_jS + p° The calculation of

det(sl

- A)

(31.7)

has been discussed in Section 21 and

in Section A 2 . 3 . In Theorem A 2 . 5 , the rules how to get the polynomial coefficients

p°

(1 * i * n) from the digraph G(A) have been derived.

Here, we merely add some comments concerning large-scale

systems.

If the number of state v a r i a b l e s is not too small (say, n > 10)

then

the determinant

(>1)

(31.7) of order n splits up into a product of t

subdeterminants of order

n^, n 2 , ... , n t in most cases:

t det( si - A) - T T det(sl - A , .), n jL'j n± ^iij Each submatrix

is

t ZZ

iml

n. - n i

(31.7')

associated with an equivalence class of

strong-

ly connected v e r t i c e s within the digraph G(A), comp. Section 13. Corollary 31.1 If the square matrix A is not irreducible, i.e., the digraph G(A) is not strongly connected, then there exist

t > 1

equivalence

classes of states which correspond to square submatrices associated strongly connected subgraphs

120

G(A,.

.),

anc

'

Before applying Theorem A2.5, to the t d i s j o i n t s u b g r a p h s a b l y u s e d to d e t e r m i n e E x a m p l e 31»Is C o n s i d e r the The c o r r e s p o n d i n g

G

the d i g r a p h G ( A )

(A(n))*

s h o u l d be

Then E q . (31.7)

be

reduced profit-

the d e t e r m i n a n t d e t ( s l n - A ) . 16x16

m a t r i x A e x p l a i n e d in F i g .

r e d u c e d g r a p h h a s b e e n d r a w n in F i g .

13.1.

31.1.

® ® ©

© © ®

8 Fig.

31.1

The s u b g r a p h a s s o c i a t e d w i t h the e q u i v a l e n c e c l a s s K 5 in Fig. 13.1 supplies a s

5

subdeterminant

(5) 4 (5) 3 (5) 2 (5) (5) + p^ 's + p£ 's + P 3 s + p^ 's + p£ ' (5) for

T h e d e t e r m i n a t i o n of the c o e f f i c i e n t s

p

the aid o f T h e o r e m A 2 . 5 is left to the

reader.

The

det(sl5 -

- 1,2,3,4,5

s u b g r a p h a s s o c i a t e d w i t h the e q u i v a l e n c e c l a s s Kg in Fig.

supplies a s

4

+

p