Finite Zeros in Discrete Time Control Systems (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari 338 Jerzy Tokarzewski Finite Zeros ...

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Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

338

Jerzy Tokarzewski

Finite Zeros in Discrete Time Control Systems With 11 Figures

Series Advisory Board

F. Allgöwer · P. Fleming · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · A. Rantzer · J.N. Tsitsiklis

Author Jerzy Tokarzewski Military University of Technology Faculty of Mechanical Engineering Institute of Mechanical Vehicles and Transport ul. Kaliskiego 2 00-908 Warszawa Poland [email protected]

ISSN 0170-8643 ISBN-10 ISBN-13

3-540-33464-5 Springer Berlin Heidelberg New York 978-3-540-33464-4 Springer Berlin Heidelberg New York

Library of Congress Control Number: 2006924582 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by editors. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany (www.ptp-berlin.com) Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 89/3141/Yu - 5 4 3 2 1 0

Preface

During the past three decades the notion of zeros of multi-input multioutput control systems (known also as multivariable zeros) has received considerable interest from researchers. A large number of definitions (not necessarily equivalent) of multivariable zeros can be found in the relevant literature. In this light, the term “zero” has become ambiguous. On the other hand, this indicates that the question of extending to multi-input multi-output systems of the well understood concept of zeros of singleinput single-output systems does not seem to be an evident task. The purpose of this monograph is to present a self-contained description of the geometric approach to the analysis and interpretation of multivariable zeros of discrete-time control systems in the context of the problem of zeroing the system output. The dynamical interpretation of zeros is fundamental to the understanding of several basic problems occurring in linear system theory. The point of departure for our discussion, which is held in the state space framework, is a rather simple observation that the first nonzero Markov parameter of a linear discrete-time system comprises a large amount of information concerning zeros, output-zeroing inputs, maximal output-nulling invariant subspaces and the zero dynamics. To the best of my knowledge, this is the only book which starting from definition of invariant zeros goes as far as a general characterization of output-zeroing inputs and the corresponding solutions, explicit formulas for maximal output-nulling invariant subspaces and for the zero dynamics. The invariant zeros are treated as the triples: complex number, nonzero state-zero direction, input-zero direction. Such treatment is strictly connected with the output-zeroing problem and in that spirit the zeros can be easily interpreted even in the degenerate case (i.e., when each complex number constitutes an invariant zero). Simply, in the degenerate case, to each complex number we can assign an appropriate initial state and an appropriate real input sequence which produce a nontrivial solution of the state equation and the identically zero system response. Clearly, when zeros are treated merely as complex numbers, such interpretation is impossible.

VI

Preface

The basic mathematical tools used in the analysis are the Moore-Penrose pseudoinverse and singular value decomposition (SVD) of a matrix. The objective of this monograph is to render the reader familiar with a certain method of analysis of multivariable zeros and related problems. The bibliography, which by no means is complete, includes those items which were actually used. The minimal mathematical background that is required from the reader is a working knowledge of linear algebra and difference equations. The reader should be familiar with the basic concepts of linear system theory. In order to make this volume as self-contained as possible some basic concepts and results used throughout the book are described in the Appendix. This monograph is directed primarily to an audience consisting of researchers in various fields, PhD candidates and mathematically advanced undergraduate or beginning graduate students. It can also be used by control engineers interested in a proof-oriented state space approach to the relevant topic that goes beyond the classical approach. I wish to express sincerest gratitude to Professor Tadeusz Kaczorek for his valuable comments and suggestions which enabled me to improve essentially this book. I also would like to thank Professor Tadeusz Kaczorek for offering me the opportunity to present main results of this book in a series of seminars devoted to geometric methods in control theory, held in Warsaw University of Technology, Institute of Control and Industrial Electronics. My special thanks go to my wife Małgorzata and to our children Anna and Jan for their support during the long hours of writing. I am very grateful to Jerzy Osiecki for his technical support. Warsaw, December 2005

Jerzy Tokarzewski Military University of Technology, Warsaw and Automotive Industry Institute, Warsaw

Contents

1 Introduction.......................................................................................... 1.1 Smith Zeros and Decoupling Zeros............................................. 1.2 Scope of the Book........................................................................ 1.3 Glossary of Symbols....................................................................

1 2 5 7

2 Zeros and the Output-Zeroing Problem............................................... 2.1 Definitions of Zeros..................................................................... 2.2 Decoupling Zeros......................................................................... 2.3 Invariant and Transmission Zeros................................................ 2.4 The Output-Zeroing Problem....................................................... 2.5 Invariant Zeros and Output-Zeroing Inputs................................. 2.6 Relationships Between Smith and Invariant Zeros...................... 2.7 Algebraic Criteria of Degeneracy................................................ 2.8 Exercises......................................................................................

9 9 13 18 21 25 36 45 53

3 A General Solution to the Output-Zeroing Problem............................ 3.1 Preliminary Characterization of Invariant Zeros.......................... 3.1.1 Invariant Zeros in Proper Systems........................................ 3.1.2 Invariant Zeros in Strictly Proper Systems........................... 3.2 Output-Zeroing Inputs.................................................................. 3.2.1 Output-Zeroing Inputs for Proper Systems........................... 3.2.2 Output-Zeroing Inputs for Strictly Proper Systems.............. 3.3 Exercises.......................................................................................

65 65 66 67 71 73 80 91

4 The Moore-Penrose Inverse of the First Markov Parameter................ 4.1 System Zeros in Strictly Proper Systems..................................... 4.1.1 Decoupling and Transmission Zeros..................................... 4.1.2 First Markov Parameter of Full Rank.................................... 4.2 System Zeros in Proper Systems.................................................. 4.2.1 Decoupling and Transmission Zeros..................................... 4.2.2 Proper Systems with Matrix D of Full Rank......................... 4.3 Systems with Vector Relative Degree.......................................... 4.4 Exercises.......................................................................................

107 107 109 112 120 121 122 128 137

VIII

Contents

5 Singular Value Decomposition of the First Markov Parameter.......... 5.1 Invariant and Smith Zeros in Strictly Proper Systems................ 5.2 A Procedure for Computing Zeros of Strictly Proper Systems.. 5.3 Invariant and Smith Zeros in Proper Systems............................ 5.4 A Procedure for Computing Zeros of Proper Systems............... 5.5 Exercises.....................................................................................

149 151 165 173 186 192

6 Output-Nulling Subspaces in Strictly Proper Systems....................... 6.1 Systems with the Identically Zero Transfer-Function Matrix.... 6.2 First Markov Parameter of Full Column Rank........................... 6.3 Invariant Subspaces.................................................................... 6.4 SVD and the Output-Zeroing Problem....................................... 6.5 Zeros, Output-Nulling Subspaces and Zero Dynamics.............. 6.6 Exercises.....................................................................................

197 198 204 208 222 231 237

7 Output-Nulling Subspaces in Proper Systems.................................... 7.1 Matrix D of Full Column Rank.................................................. 7.2 Invariant Subspaces.................................................................... 7.3 SVD and the Output-Zeroing Problem....................................... 7.4 Zeros, Output-Nulling Subspaces and Zero Dynamics.............. 7.5 Exercises.....................................................................................

245 245 249 260 267 269

8 Singular Systems................................................................................. 8.1 Definitions of Zeros.................................................................... 8.2 Relationships Between Invariant and Smith Zeros.................... 8.3 Sufficient and Necessary Condition for Degeneracy................. 8.4 Zeros and SVD of Matrix E....................................................... 8.4.1 Invariant Zeros.................................................................... 8.4.2 Output Decoupling Zeros.................................................... 8.4.3 Input Decoupling Zeros....................................................... 8.5 Markov Parameters and the Weierstrass Canonical Form.......... 8.6 Invariant Zeros and the First Markov Parameter........................ 8.6.1 First Markov Parameter of Full Column Rank.................... 8.6.2 SVD of the First Markov Parameter.................................... 8.7 Exercises.....................................................................................

271 271 274 274 275 276 279 280 281 283 287 290 295

Epilogue................................................................................................. 297 Appendix................................................................................................ A Appendix A................................................................................... A.1 Reachability and Observability.............................................. A.2 Canonical Decomposition of the State Space........................

299 299 299 302

Contents

B Appendix B.................................................................................... B.1 The Moore-Penrose Inverse of a Matrix................................. B.2 Singular Value Decomposition of a Matrix............................ B.3 Endomorphisms of a Linear Space over C............................. C Appendix C.................................................................................... C.1 Polynomial and Rational Matrices.........................................

IX

305 305 308 309 312 312

References.............................................................................................. 317 Index....................................................................................................... 323

1 Introduction

The determination of zeros has received considerable attention in recent years. The zeros are defined in many (not necessarily equivalent) ways (for a survey of these definitions see [16, 40, 58]) so that the term “zero” has become ambiguous. There are four main groups of definitions: (a) those originating from Rosenbrock’s approach [16, 22, 27, 40, 42, 53–55, 56, 90] and related to the Smith or Smith-McMillan form, (b) those connected with the concept of state-zero and input-zero directions introduced in [40] (see also [15, 48, 56]), (c) those employing the notions of inverse systems [37], and (d) those based on the module-theoretic setting [7, 58]. Although many authors before 1970 alluded to the concept of zeros of multi-input multi-output (MIMO) systems, Rosenbrock [53, 54] is credited with the first definition of zeros of MIMO systems (multivariable zeros). Rosenbrock’s first multivariable zeros, termed the decoupling zeros, were related to the notions of reachability (controllability) and observability. The most common definition of multivariable zeros [53, 40, 90] employs the Smith canonical form of the system (Rosenbrock) matrix and determines the zeros as the roots of the so-called zero polynomial. In the sequel, these zeros will be called the Smith zeros. They may be defined as the points of the complex plane where the rank of the system matrix falls below its normal rank (recall that the term “normal rank” means “rank” in the field of rational functions). The zeros of a proper transfer-function matrix are defined [90, 16, 42] from its arbitrary minimal (i.e., reachable and observable) standard state space realization as the points where the system matrix (formed from matrices of the minimal realization) loses its normal rank. These zeros will be called the Smith zeros of a transfer-function matrix. When the transfer-function matrix has full normal rank, the above definition is equivalent to the Desoer-Schulman one [14, 10]. Smith zeros, decoupling zeros and Smith zeros of a transfer-function matrix are involved in several problems of control theory, such as zeroing the output [2, 4, 6, 20, 31], tracking the reference output [20], disturbance decoupling [4, 41, 61, 91], noninteracting control [20], output regulation [4, 20, 41, 91] or robust servomechanism design [17].

J. Tokarzewski: Fin. Zeros in Diskr. Time Contr. Syst., LNCIS 338, pp. 1–8, 2006. © Springer-Verlag Berlin Heidelberg 2006

2

1 Introduction

A different definition of zeros, also based on the system matrix rank test, was given by Davison and Wang [13, 19, 40, 58]. All the above definitions of multivariable zeros, although deceivingly simple, consider zeros merely as complex numbers and for this reason create some difficulties in their dynamical interpretation in the context of the output-zeroing problem. For instance, the dynamical interpretation of zeros of a transfer-function matrix introduced by Desoer and Schulman [14] (for continuous-time systems) includes the Dirac distribution and its derivatives in control vector, i.e., signals rather difficult for implementation. In all likelihood in order to overcome these difficulties, MacFarlane and Karcanias [40] introduced the notions of state-zero and input-zero directions and formulated the so-called output-zeroing problem. Unfortunately, in [40] the Smith zeros were not directly related to this problem. The Smith zeros and zero directions were, under various simplifying assumptions concerning the systems considered, extensively discussed in the relevant literature [15, 34, 35, 39, 40, 48, 51, 52, 56, 57, 59, 60, 92] (other concepts of zeros were discussed in [13, 19, 37, 49, 50]). Another definition of multivariable zeros, employing the system matrix and state-zero and input-zero directions, was introduced (for continuoustime systems) in [63] and applied to their algebraic and geometric characterization in [80]. In this book we extend the above approach to discretetime systems.

1.1 Smith Zeros and Decoupling Zeros Consider the discrete-time system S(A,B,C,D) with m inputs and r outputs x(k + 1) = Ax(k ) + Bu (k )

(i)

y (k ) = Cx(k ) + Du(k )

,

k = 0,1,2,... ,

where x, u and y are respectively the n-dimensional state vector, mdimensional control vector and r-dimensional output vector, and A, B, C, D are real matrices of appropriate dimensions. The most commonly used definitions of multivariable zeros employ the Smith canonical form of the system matrix. Recall that for S(A,B,C,D) in (i) the system matrix is the (n+r)x(n+m) pencil (ii)

zI − A − B  , P (z) =  D   C

whereas the transfer-function matrix is the rxm rational matrix

1.1 Smith Zeros and Decoupling Zeros

(iii)

3

G ( z) = C(zI − A ) −1 B + D .

Let us briefly recall the Smith canonical form of the polynomial matrix P(z) [9, 18, 23, 57]. It is well known that for P(z) there exist unimodular matrices (n+r)x(n+r) U(z) and (n+m)x(n+m) V(z) (recall that a square polynomial matrix is said to be unimodular if it is invertible over the ring of polynomial matrices, i.e., its determinant is different from zero and does not depend upon z) and a (n+r)x(n+m) polynomial matrix Ψ ( z ) of the form

(iv)

ψ1 ( z )   Ψ (z) =    

.

    ψ q (z)  0  0

such that P ( z ) = U ( z )Ψ ( z )V ( z ) . Here Ψ ( z ) is called the Smith canonical form of P(z) when polynomials ψ i (z) are monic (recall that ψ i (z) is said to be monic if the coefficient associated with highest power of z is equal to 1) with the property that ψ i (z) divides ψ i +1 (z) for i = 1,..., q − 1 , and q is the normal rank of P(z) ( Ψ ( z ) can be obtained from P(z) by elementary operations (Appendix C)). The polynomials ψ i (z) are called the invariant factors of P(z) [45]. Their product ψ(z) = ψ1 (z)...ψ q (z) is called the zero polynomial of P(z) (or of S(A,B,C,D)). The roots of ψ ( z ) are called the Smith zeros of (i). The transmission zeros of S(A,B,C,D) are defined as the Smith zeros of its minimal (reachable and observable) subsystem (see Appendix A.2). These zeros will be called in the sequel the Smith transmission zeros of S(A,B,C,D). Rosenbrock's first multivariable zeros of S(A,B,C,D) were termed the decoupling zeros [53, 54]. The Smith zeros of the pencil

(v)

[zI − A

− B]

are the input decoupling (i.d.) zeros (they are unreachable ( r ) modes of A), whereas the Smith zeros of the pencil (vi)

zI − A   C   

4

1 Introduction

are the output decoupling (o.d.) zeros of the system (they are unobservable ( o ) modes of A). Rosenbrock defined also the input-output decoupling (i.o.d.) zeros. The i.o.d. zeros of the system are those o.d. zeros which disappear when the i.d. zeros are eliminated. The zeros of the transfer-function matrix G(z) are defined as the roots of the numerator polynomials of the Smith-McMillan form of G(z) [9, 23, 40]. These zeros can be equivalently defined (cf., [16]) as the Smith zeros of the system matrix obtained from any given minimal state space realization of G(z) and for this reason they will be called in the sequel the Smith zeros of G(z). Since all the mentioned above definitions of zeros are based on Smith zeros of appropriate matrix pencils, they do not relate the defined objects in a clear way with the state space approach to the analysis of discrete-time linear systems. This in turn creates certain difficulties in geometric interpretation of these objects. These difficulties become easily seen when we try to give dynamical interpretation of multivariable zeros (in the context of the problem of zeroing the system output) in the state space framework restricting simultaneously admissible inputs to the class of all real-valued sequences of natural numbers. Moreover, a more detailed analysis indicates that for characterizing the output-zeroing problem the notion of Smith zeros is too narrow. Such observation can be motivated by simple numerical examples of minimal and asymptotically stable systems in which there are no Smith zeros, while the maximal output-nulling subspace is nontrivial (i.e., there are nontrivial output-zeroing inputs producing simultaneously nontrivial solutions of the state equation). In order to overcome the above difficulties and to give simple dynamical interpretation of multivariable zeros in the framework of elementary theory of difference equations, we extend in a natural way (with the aid of zero directions) the concept of Smith zeros. x o  Recall [40] that for a given complex number λ a vector   satisfying  g  (vii)

 x o  0 P (λ )   =   ,  g  0

where P(z) denotes the system matrix for (i), is called the zero direction corresponding to λ (where x o is the state-zero direction and g is the input-zero direction).

1.2 Scope of the Book

5

Now, we can consider the set of all those complex numbers such that for each its element there exists a zero direction with a nonzero state-zero direction (i.e., x o ≠ 0 ). The elements of this set are called in the sequel the invariant zeros (cf., [63–84]) of the system S(A,B,C,D) in (i). In other words, a complex number λ is an invariant zero of this system if it is possible to assign to λ a zero direction with a nonzero state-zero direction (note that in this definition the condition x o ≠ 0 is essential and it can not be omitted). As we shall see in the sequel, the invariant zeros defined above have the following basic properties: • to each invariant zero there corresponds a real initial condition and a real-valued input sequence which generate the identically zero system response and a nontrivial solution of the state equation (whose trajectory is entirely contained in the maximal output-nulling subspace); • the set of invariant zeros is invariant with respect to the same set of transformations of the system (i) as the set of Smith zeros; • the set of Smith zeros is contained in (or equal to) the set of invariant zeros. Of course, the set of Smith zeros is at most finite. The set of invariant zeros may be infinite (then the system (i) is called degenerate). However, if the set of invariant zeros is infinite, it is equal to the whole complex plane. On the other hand, if the set of invariant zeros is finite (or empty), it is equal to the set of Smith zeros (in this case the system (i) is called nondegenerate). In a nondegenerate system (i) the Smith and invariant zeros have the same algebraic multiplicities (i.e., they are exactly the same objects); • the set of invariant zeros is empty if and only if the maximal outputnulling subspace is trivial. This equivalence does not hold for the set of Smith zeros (i.e., there exist systems in which the set of Smith zeros is empty, while the maximal output-nulling subspace is nontrivial).

1.2 Scope of the Book This book is organized as follows. The question how the classical concept of the Smith zeros of a multi-input multi-output linear discrete-time system S(A,B,C,D) can be generalized and related to the state space methods is discussed in Chapter 2. Nothing is assumed about the relationship of the number of inputs to the number of outputs nor about the normal rank of the underlying system matrix. We introduce first the necessary definitions and discuss the question of interpreting the invariant zeros in the context of the

6

1 Introduction

output-zeroing problem. To this end, we employ throughout the book the formulation of the output-zeroing problem given by Isidori [20]. The invariant zeros are treated as the triples: complex number, nonzero state-zero direction, input-zero direction. Such treatment is strictly connected with the output zeroing problem and in that spirit these zeros can be easily interpreted even in the degenerate case. Relationships between the invariant and Smith zeros are also examined. Finally, a simple sufficient and necessary condition for nondegeneracy is provided. Chapter 3 is devoted to the state space characterization of the outputzeroing problem. A general solution to the problem in the form of necessary conditions for output-zeroing inputs is derived. The presented approach is based on the first nonzero Markov parameter. Recall [10] that the transfer-function matrix (iii) can be expanded, in a neighbourhood of the point z = ∞ of the complex plane, in the power series (viii)

G ( z) = D + C( zI − A ) −1 B = D +

CB CAB CA l B + + . .. + + ... , z z2 z l +1

where matrices D, CB, CAB,..., are called the Markov parameters of (i). As we shall see, the first nonzero Markov parameter carries a large amount of information that can be used in the algebraic and geometric characterization of invariant zeros. In Chapter 4 we present some algebraic characterization of invariant zeros using the Moore-Penrose pseudoinverse (see Appendix B) of the first nonzero Markov parameter. The basic tool that is employed for the zeros characterization and computation is the Kalman four-fold canonical decomposition theorem (see Appendix A). In Chapter 5 a more detailed characterization of invariant zeros is presented. It is based on singular value decomposition (SVD) (see Appendix B) of the first nonzero Markov parameter and comprises a number of criterions which enable us to decide the question of degeneracy/nondegeneracy of an arbitrary system (i) in at most a finite number of steps and in case of nondegeneracy to compute all its invariant zeros. In particular, it is shown that the proposed recursive procedure for the computation of invariant zeros preserves the set of invariant zeros, the set of Smith zeros as well as the zero polynomial. Chapters 6 and 7 are devoted to the geometric characterization of invariant zeros. It is shown, in particular, that although in a degenerate system any complex number is an invariant zero, however the output-zeroing inputs generated by these zeros keep the trajectories of the corresponding solutions of the state equation inside the maximal output-nulling subspace. The singular value decomposition of the first nonzero Markov parameter

1.3 Glossary of Symbols

7

enables us also to determine explicitly the maximal output-nulling subspace as well as to show that invariant zeros generate this subspace. Only in one case a recursive procedure (analogous to that mentioned above) is needed. Of course, it is shown that besides the sets of invariant and Smith zeros and the zero polynomial this procedure preserves the maximal output-nulling subspace as well. Finally, we show that the class of all systems of the form (i) can be decomposed into two disjoint subclasses: of nondegenerate and degenerate systems. In nondegenerate systems the Smith and invariant zeros are exactly the same objects which are determined by the zero polynomial. The degree of this polynomial equals the dimension of the maximal outputnulling subspace, while the zero dynamics are independent upon control vector. In degenerate systems the zero polynomial determines merely the Smith zeros, while the set of invariant zeros equals the whole complex plane. The dimension of the maximal output-nulling subspace is strictly larger than the degree of the zero polynomial, whereas the zero dynamics essentially depend upon control vector. By using simple numerical examples we show that in the general case the Smith zeros do not generate the maximal output-nulling subspace and for this reason they do not characterize completely the output-zeroing problem. In Chapter 8 we try to extend the approach presented for standard linear discrete-time systems to singular discrete-time systems (see, e.g., [12, 23, 25] for basic results concerning such systems). The sets of exercises form an integral part of the book. They are designed to help the reader to understand and utylize the concepts and results covered. In order to retain the continuity of the main text, some important results are stated in the sets of exercises.

1.3 Glossary of Symbols ∃

∈ ⊆

a∈A A⊆B

⊂ ∪ ∩ ⇒ ⇐

A⊂B A∪B A∩B p⇒q p⇐q

there exists a is an element of A (a belongs to A) set A is contained in (or equal to) set B (A is a subset of B) set A is contained in (and not equal to) set B union of set A with set B intersection of set A and set B p implies q (equivalently, “not q” implies “not p”) q implies p

8

1 Introduction

⇔

p⇔q

:= ∅ R C

A := B

R n (C n ) R p xq ( C p xq )

AT Ker A Im A det A σ( A )

A+ V⊕W diag( A, B, C)

f :X → Y

p if and only if q (equivalently, p implies q and q implies p) the set A is by definition the set B the empty set field of real numbers field of complex numbers (Note: for z ∈ C , Re z and Im z denote the real and imaginary part of z; z denotes the complex conjugate of z) the set of n-tuples of elements belonging to R , respectively C the set of pxq matrices with entries in the field R, respectively C the transpose of matrix A the null space or kernel of matrix A (or linear map A) the range or image of matrix A (or linear map A) the determinant of matrix A the spectrum of matrix A (equivalently, the set of eigenvalues of A, i.e., σ( A ) = {λ ∈ C: det(λI − A ) = 0} ) the Moore-Penrose inverse of matrix A the direct sum of two vector spaces a diagonal matrix with matrices A, B and C as block diagonal elements f is a function or map, mapping a domain X into A⊆X by Y; for codomain the f (A) = {y ∈ Y :y = f ( x ), x ∈ A} we denote the image of the set A under f; for B ⊆ Y by f −1 ( B) = {x ∈ X :f ( x ) ∈ B} we denote the inverse image of B under f.

2 Zeros and the Output-Zeroing Problem

In this chapter we introduce the definition of invariant zeros and we give simple geometric interpretation of these zeros in the context of the outputzeroing problem. We indicate also that, according to the adopted definition, the question of finding out all invariant zeros constitutes a nonlinear problem. Relationships between invariant and Smith zeros are considered and the concepts of degenerate and nondegenerate systems are introduced. Moreover, simple algebraic criterions of degeneracy/nondegeneracy are formulated. In order to characterize decoupling zeros of individual kinds (o.d., i.d. and i.o.d.), the four-fold Kalman decomposition theorem has been employed (see Appendix A).

2.1 Definitions of Zeros Consider a discrete-time system S(A,B,C,D) (Fig. 2.1) with m inputs and r outputs of the form x(k + 1) = Ax(k ) + Bu(k ) , y (k ) = Cx(k ) + Du(k )

k ∈ N = {0,1,2,...} ,

(2.1)

where x(k ) ∈ R n , u(k ) ∈ R m , y (k ) ∈ R r , and A, B, C, D are real matrices of appropriate dimensions. We assume B ≠ 0 and C ≠ 0 . By U we denote the set of admissible inputs which consists of all sequences u(.): N → R m . If D = 0 , then the first nonzero Markov parameter of (2.1) is denoted by CA ν B , where 0 ≤ ν ≤ n − 1 , i.e., CB = ... = CA ν −1B = 0 and CA ν B ≠ 0 . The system (2.1) is called proper if D ≠ 0 ; otherwise the system is said to be strictly proper. In the sequel we make frequently use of the four-fold Kalman canonical decomposition of the system (2.1) (see Appendix A.2) in which matrices A, B, C and the state vector x are decomposed as


10


Fig. 2.1.

 A ro  0 A=  0   0

A12

A13

A ro 0 0

0 A ro 0

A14  A 24  A 34   A ro 

B r o  B  B =  ro   0     0 

C = [0 C ro

0 C ro ]

(2.2) x ro  x  ro  x= x ro     x ro 

and D remains unchanged. Although the form (2.2) of (2.1) is not unique, however in any such form the orders n ro , nro , n ro and n ro of the corresponding submatrices on the diagonal of the A-matrix as well as the characteristic polynomials of these submatrices (up to a constant) remain unchanged. One form can be obtained from any other by a similarity transformation of the state space (i.e., by a change of coordinates in the state space). In this way the following objects are well defined. The elements of σ( A ro ) (i.e., eigenvalues of A r o ) are called the reachable and unobservable ( ro ) modes of (2.1). Analogously, we define ro-modes as eigenvalues of A ro (these are poles of the system transfer-function matrix G(z)), ro -modes as eigenvalues of A r o and ro -modes as eigenvalues of A ro . Moreover, n ro , n ro , nro are

2.1 Definitions of Zeros

11

uniquely determined by the order of the state matrix (n), the degree of the transfer-function matrix ( nro ) and rank defects of the reachability and observability matrices ( nr and no ) as follows (see Appendix A.2) n ro = n − n ro − n r , n r o = nro + n r + n o − n , n ro = n - n ro − n o .

(2.3)

By S( A r o , B r o , C ro , D) we denote the minimal (i.e., reachable and observable) subsystem of (2.2). For Markov parameters of (2.2) and of S( A r o , B r o , C ro , D) the following relations hold CA l B = C ro A lr o B r o

for

l = 0, 1, 2, . . .

and, consequently, for transfer-function matrices we have G (z) = D + C(zI − A ) −1 B = D + C r o (zI r o − A ro ) −1 B r o = G r o (z) ,

where I r o denotes the identity matrix of the order n ro and G ro (z) is the transfer-function matrix for S( A r o , B r o , C ro , D) . We adopt the following definition of zeros of the system (2.1). Definition 2.1. [80] (i)

A number λ ∈ C is an invariant zero of the (strictly proper or

proper) system (2.1) if and only if there exist vectors 0 ≠ x o ∈ C n (statezero direction) and g ∈ C m (input-zero direction) such that the triple λ, x o , g satisfies

 λI − A − B   x o   0  .  =  C D   g  0 

(2.4)

The system is called degenerate if it has an infinite number of invariant zeros; otherwise the system is said to be nondegenerate. (ii) The transmission zeros of (2.1) are defined as the invariant zeros of its minimal (reachable and observable) subsystem. The unobservable ( o ) modes of (2.1) are called the output (iii) decoupling (o.d.) zeros. (iv) The unreachable ( r ) modes of (2.1) are called the input decoupling (i.d.) zeros.

12


(v) The input-output decoupling (i.o.d.) zeros are defined as the unreachable and unobservable ( ro ) modes of the system (2.1). ◊ Remark 2.1. Note that the notion of degeneracy used in Definition 2.1 is not equivalent to that commonly used in the literature [10, 16] which means that the system matrix P (z) is not of full normal row nor full normal column rank (see Sect. 2.8, Exercises 2.8.5 and 2.8.6). ◊ By  x o  0 ZI: = {λ ∈ C: ∃ 0 ≠ x o ∈ C n ∧ ∃ g ∈ C m (P (λ)   =   )}  g  0

(2.5)

we denote the set of all invariant zeros of (2.1). Consider a system of nonlinear algebraic equations zx − Ax − Bu = 0 Cx + Du = 0

(2.4a)

consisted of n+r equations with n+m+1 unknowns in z, x, u . According to Definition 2.1 (i), a number λ is an invariant zero of (2.1) if and only if there exist vectors x o ≠ 0 and g such that the triple λ, x o ≠ 0, g satisfies (2.4a), i.e., this triple is a solution of the system (2.4a). Observe that the condition x o ≠ 0 is essential and it can not be omitted (otherwise, Definition 2.1 (i) becomes senseless, i.e., for any system (2.1) each complex number may be an invariant zero). In this way the question of seeking all invariant zeros of the system (2.1) constitutes a nonlinear problem. As we shall see in the sequel, there are only the following three cases concerning the number of invariant zeros of the system (2.1): the set ZI may be empty, i.e., ZI = ∅ , it may contain a finite number of zeros, or any

complex number is an element of ZI , i.e., ZI = C .

Definition 2.2. A number λ ∈ C is a transmission zero of a proper transfer-function matrix G (z) if and only if λ is an invariant zero of a given minimal (reachable and observable) state space realization of G (z) . G (z) is called degenerate if and only if it has an infinite number of the transmission zeros; otherwise G (z) is called nondegenerate. ◊ Since all minimal (irreducible) state space realizations of a given proper rational transfer-function matrix G (z) are equivalent, i.e., one from the other can be obtained by a linear nonsingular transformation of the state

2.2 Decoupling Zeros

13

space ( R n ), as well as the invariant zeros are invariant under such transformations, the transmission zeros of G (z) defined above do not depend upon particular choice of the minimal realization of G (z) . Example 2.1. In a strictly proper single-input single-output (SISO) system S(A,b,c) (2.1) (in SISO systems matrices B, C, D will be denoted in the sequel respectively by b, c, d, and the transfer-function will be denoted by g(z)) let 1 1 0  0   A = 0 0 0 , b = 0 , c = [0 1 0] . 1 0 0 1 x o   1o  o Denoting x =  x 2  , we can write (2.4) as  o  x 3  (λ − 1) x 1o − x o2 = 0 λ x o2 = 0 − x 1o + λ x 3o − g = 0 x o2 = 0 .

 0  Since for any λ ≠ 0 and g ≠ 0 the triple λ, x =  0 , g satisfies (2.4), g / λ  0 o   the system is degenerate. The triple λ = 0, x = 0 , g = 0 also satisfies 1 o

(2.4). ◊

2.2 Decoupling Zeros As is shown in this section, the adopted above definitions of decoupling zeros of different kinds (o.d., i.o.d. and i.d.) are equivalent to the classical definitions of these zeros [53, 54] (cf., Chapter 1).

14


It is clear that i.d., o.d. and i.o.d. zeros are invariant under similarity transformations of the state space (see Exercise 2.8.1). Lemma 2.1. A complex number λ is an o.d. zero of the (strictly proper or proper) system (2.1) if and only if λ is a ro -mode or a ro - mode of (2.1). Proof. By the Rosenbrock definition (see (vi) in Section 1.1) λ is an o.d. zI − A  zero if and only if the pencil   loses at λ its normal column rank n  C  or, equivalently, if and only if (λI − A)x o = 0 and Cx o = 0

(i)

for some 0 ≠ x o ∈ C n . Of course, (i) implies that Re x o and Im x o belong to the unobservable subspace n −1

X o := f Ker CA l ⊆ R n .

(ii)

l =0

If (2.1) is expressed in the form (2.2), then this subspace takes the form

{

}

X o = x ∈ R n : x ro = 0, x ro = 0 and (i) can be written as

(iii)

(λI − A )x o − A x o = 0 ro r o ro 13 ro .  o λ I − A x = 0 ( )  r o r o ro 

The relations in (iii) imply in turn (by reductio ad absurdum) that λ ∈ σ( A ro ) or λ ∈ σ( A ro ) . Thus we have shown the implication: λ is an o.d. zero of (2.1) ⇒ λ is a ro -mode or a ro -mode of (2.1). The converse implication follows easily from (2.2). In fact, if λ ∈ σ( A ro ) , let us take into account an eigenvector x or o associated with λ . Then the pair

λ,

x o   ro  o  0 ro  x = 0   ro   0 r o 

satisfies (i), i.e., if λ ∈ σ( A ro ) , then λ is an o.d. zero of the system (2.1).


15

In order to show that each element of σ( A r o ) is an o.d. zero of (2.1), we consider two cases. If λ ∈ σ( A ro ) and λ ∈ σ( A ro ) , then λ is an o.d. zero. In the second case, when λ ∈ σ( A ro ) and λ ∉ σ( A ro ) , let x or o denote an eigenvector of A r o associated with λ . Then the pair  x or o    o  0 ro  λ , x =  o  , where x or o = (λI r o − A ro ) −1 A13 x or o , x r o  0 ro   

satisfies (i), i.e., λ is an o.d. zero. ◊ Lemma 2.2. A complex number λ is an i.d. zero of the (strictly proper or proper) system (2.1) if and only if λ is a ro -mode or a ro -mode of (2.1). Proof. Recall that, by the Rosenbrock definition (see (v) in Section 1.1), λ is an i.d. zero if and only if [zI − A − B ] loses at λ its full normal row rank n. Now, from (2.2), viewed as the two-fold decomposition into a reachable and an unreachable part, it follows that the considered pencil loses its normal rank if and only if λ ∈ σ( A ro ) or λ ∈ σ( A ro ) . ◊ As is well known [46], any square matrix is unitarily similar to an upper triangular matrix (in which the diagonal consists of all eigenvalues of the original matrix). Denote by U r o , U r o , U r o , U r o unitary matrices which transform appropriate diagonal submatrices of A in (2.2) into their upper triangular forms Tr o , Tro , Tro , Tr o . Then the nxn matrix T = diag(U r o , U ro , U ro , U r o )

transforms A in (2.2) into its upper triangular form A T without disturbing the Kalman block structure of A. Now, removing from A T the diagonals of Tr o and Tro , we eliminate all input decoupling zeros from the system. The output decoupling zeros which disappear in this process are all the elements of the diagonal of Tr o – and these are all the input-output decoupling zeros of the system. From Lemmas 2.1 and 2.2 and from the above discussion concerning input-output decoupling zeros we obtain the following.

16


Corollary 2.1. The Kalman form (2.2) of the (strictly proper or proper) system (2.1) displays individual kinds (o.d., i.d. and i.o.d.) of decoupling zeros (including multiplicities). a)

The o.d. zeros of (2.1) are the roots of the polynomial χ o.d (z) = det(zI r o − A ro ) det(zI r o − A r o )

(i)

zI − A  which is equal to the zero polynomial of  .  C 

b)

The i.o.d. zeros of (2.1) are the roots of the polynomial χ i.o.d ( z) = det(zI r o − A ro ) .

(ii)

The algebraic multiplicity of a given i.o.d. zero is, by definition, equal to its multiplicity as a root of the polynomial χ i.o.d (z) . c)

The i.d. zeros of (2.1) are the roots of the polynomial χ i.d (z) = det (zI r o − A ro ) det (zI r o − A r o )

(iii)

which is equal to the zero polynomial of [zI − A − B ] . Proof. b) The proof follows from the discussion preceding the corollary as well as from the adopted definition of multiplicities of input-output decoupling zeros. a) For the proof we only need to show that (i) is equal to the zero polyzI − A  nomial of the pencil   . However, by virtue of (2.2), we can write  C  this pencil as

(iv)

zI ro      

− A ro 0 0 0 0

− A12 zI ro − A ro

− A13 0

0 0 C ro

zI ro − A r o 0 0

− A14 − A 24

   − A 34  . zI ro − A ro   C ro 

Interchanging suitably rows and columns in (iv), we obtain


− A13 zI r o − A ro

− A12 0

0 0 0

0 0 0

zI ro − A ro 0 C ro

A Since the pair (  ro  0

A 24  , [C ro A ro 

(v)

zI ro      

− A ro 0

(vi)

17

− A14 − A 34

   − A 24  . zI ro − A ro   C ro 

C ro ] ) is observable, the submatrix

zI ro − A ro  0   C ro

− A 24  zI ro − A ro   C ro

of (v) has full column rank ( n ro + n ro = no ) at any λ ∈ C . Consequently, it can be transformed by elementary row operations into the Hermite row ~  A o (z) ~ form (see Appendix C)   , where A o (z) is square (of the order  0  ~ no ), upper triangular and unimodular, i.e., det A o (z) = const ≠ 0 . In this way, by elementary row and column operations, we can transform (iv) into the form

(vii)

zI ro     

− A ro

− A13

0 0

zI r o − A ro 0

0

0

zI ro − A ro  0  0 

− A13 zI r o − A r o

 x  , ~ A o (z)  0  x

where the submatrix (viii)

0

   ~ A o (z) x x

is square (of the order n ro + n ro + no = n ). The determinant of (viii) equals (const ) χ o.d (z) . This means that the zero polynomial of (iv) and, zI − A  consequently, of   (cf., Exercise 2.8.1) equals χ o.d (z) , as claimed.  C 

18


c) For the proof it is sufficient to show that (iii) equals the zero polynomial of the pencil [zI − A − B ] . Using (2.2) and interchanging appropriate columns, we can write this pencil as

(ix)

zI ro     

− A ro 0 0 0

 A ro Since the pair (   0

− A12

− B ro

zI ro − A ro 0 0

− B ro 0 0

A12  , A ro 

− A14  0 − A 24  . zI ro − A ro − A 34   0 zI ro − A ro 

B r o   B  ) is reachable, the submatrix  ro 

zI ro − A ro  0 

(x)

− A13

− A12 zI ro − A ro

− B ro  − B ro 

of (ix) has full row rank ( n ro + n ro = n r ) at any λ ∈ C . Consequently, it can be transformed by elementary column operations into the Hermite col~ ~ umn form 0 A r (z) , where A r (z) is square (of the order nr ), upper tri~ angular and unimodular, i.e., det A r (z) = const ≠ 0 . The remaining part of the proof follows the same lines as the proof of a). ◊

[

]

2.3 Invariant and Transmission Zeros As it easily follows from Definition 2.1 (i), the set of invariant zeros in (2.5) is invariant under similarity transformations of the state space (i.e., nonsingular transformations of the form x' = Hx ). Moreover, ZI is invariant under constant state feedback and is preserved also under constant output feedback. Furthermore, ZI remains unchanged after introducing to the system (2.1) a nonsingular pre- or postcompensator (Fig. 2.2). Lemma 2.3. The set ZI of invariant zeros of the (strictly proper or proper) system (2.1) is invariant under the following set of transformations: (i) (ii) (iii) (iv) (v)

nonsingular coordinate transformations in the state space, nonsingular transformations of the inputs, nonsingular transformations of the outputs, state feedback to the inputs, output feedback to the inputs.

2.3 Invariant and Transmission Zeros

19

Fig. 2.2.

Proof. (i) Let x' = Hx , where det H ≠ 0 , be a similarity transformation of R n . As is well known, it transforms S( A, B, C, D) in (2.1) into the system S( A ' , B' , C' , D' ) , where A ' = HAH −1 , B' = HB , C' = CH −1 and D' = D . Using Definition 2.1 (i), it is immediately seen that a triple λ, x o ≠ 0, g satisfies (2.4) for S( A, B, C, D) if and only if λ, x' o = Hx o , g satisfy (2.4) for S( A ' , B' , C' , D' ) . Thus ZIS( A , B, C, D) = ZIS( A ', B ', C', D') . (ii) Consider a mxm nonsingular matrix V and the transformation of the inputs in (2.1) of the form u = Vu' . Then S( A, B, C, D) (2.1) passes into the system S( A, B' , C, D' ) , where B' = BV and D' = DV . It is straightforward to verify that a triple λ, x o ≠ 0, g satisfies (2.4) for S( A, B, C, D) if and only if the triple λ, x o , V −1g satisfies (2.4) for S( A, B' , C, D' ) , i.e., ZIS( A , B, C, D) = ZIS( A, B ', C, D') .

(iii) Consider a rxr nonsingular matrix U and the transformation of the outputs in (2.1) of the form y ' = Uy . It transforms S( A, B, C, D) into the system S( A, B, C' , D' ) , where C' = UC and D' = UD . Using (2.4), we obtain the following equivalence: a triple λ, x o ≠ 0, g satisfies (2.4) for

20


S( A, B, C, D) if and only if this triple satisfies (2.4) for S( A, B, C' , D' ) ,

i.e., ZIS( A , B, C, D) = ZIS( A, B, C', D') . (iv) Into S( A, B, C, D) we introduce the control law u = u'+Fx , where u' is a new input and F is a mxn real matrix. In this way we obtain a new (closed-loop) system S( A ' , B, C' , D) with the input u' , where A' = A + BF and C' = C + DF . Now, it is not difficult to observe that a triple λ, x o ≠ 0, g satisfies (2.4) for S( A, B, C, D) if and only if the triple λ, x o , g' = g − Fx o satisfies the condition (2.4) for S( A ' , B, C' , D) , i.e., ZIS( A , B, C, D) = ZIS( A ', B, C', D) .

(v) Consider a mxr real matrix T and introduce into S(A,B,C,D) the control law u = u '+Ty assuming simultaneously that I r − DT is nonsingular. The system S( A, B, C, D) becomes now a new (closed-loop) system S( A ' , B' , C' , D' ) with the matrices A' = A + BT(I r − DT) −1 C , B' = B + BT(I r − DT) −1 D , C' = (I r − DT) −1 C and D' = (I r − DT) −1 D . It is straightforward to verify that the following equivalence holds: a triple

λ, x o ≠ 0, g satisfies (2.4) for S( A, B, C, D) if and only if this triple satisfies the condition (2.4) for the system S( A ' , B' , C' , D' ) , i.e., ZIS( A , B, C, D) = ZIS( A ', B ', C', D') . For strictly proper systems the proof follows the same lines. ◊

Lemma 2.4. Any transmission zero of a strictly proper or proper system (2.1) is its invariant zero. Proof. In the sense of Definition 2.1 (i) (cf., also (i) in the proof of Lemma 2.1) each output decoupling zero is also an invariant zero (which is not the case when the Smith zeros are considered – see Exercise 2.8.4). This implies in turn that any transmission zero of the system (2.1) (see Definition 2.1 (ii)) is also its invariant zero. In order to show this implication, we consider the form (2.2) of (2.1). Let λ be a transmission zero of (2.1), i.e., by Definition 2.1 (ii), let a triple λ, x oro ≠ 0, g satisfy (λI ro − A ro )x oro = B ro g C ro x oro + Dg = 0.

We discuss separately two cases. If λ ∉ σ( A ro ) , then the triple

2.4 The Output-Zeroing Problem

21

 x oro   o  x λ, x o =  ro , g , where x oro = (λI ro − A ro ) −1 (B ro g + A12 x oro ) ,  0     0 

satisfies Definition 2.1 (i), i.e., λ is an invariant zero of (2.1). In the second case, when λ ∈ σ( A ro ) , then λ ∈ ZI as an o.d. zero of (2.1). ◊

2.4 The Output-Zeroing Problem The point of departure for the geometric interpretation of invariant zeros is the following formulation of the output-zeroing problem for the (strictly proper or proper) system (2.1) [20]. Find all pairs (x o , u o (k )) consisting of an initial state x o ∈ R n and an admissible input u o (k ) such that the corresponding output y (k ) is identically zero (i.e., y (k ) = 0 for all k ∈ N or, what means the same, y (k ) ≡ 0 ). Any nontrivial pair of this kind (i.e., such that x o ≠ 0 or u o (k ) is not identically zero) will be called the output-zeroing input.

In each output-zeroing input (x o , u o (k )) , u o (k ) should be understood simply as an open-loop real-valued control signal which, when applied to (2.1) exactly at the initial state x(0) = x o , yields y (k ) = 0 for all k ∈ N . The internal dynamics of (2.1) consistent with the constraint y (k ) = 0 for all k ∈ N will be called [8, 20] the zero dynamics of the system. Remark 2.2. The same symbol ( x o ) is used to denote the state-zero direction in Definition 2.1 (i) and to denote the initial state in the definition of output-zeroing inputs. According to Definition 2.1 (i), a state-zero direction x o must be a nonzero vector (real or complex). Otherwise, this definition becomes senseless (for every system (2.1) each complex number may serve as an invariant zero). According to the formulation of the output-zeroing problem, the initial state x o must be a real vector (but not necessarily nonzero). If a state-zero direction x o is a complex vector, then it gives two initial states Re x o

22


and Im x o (see Remark 2.4 and Lemma 2.8) and, of course, at least one of these initial states must be a nonzero vector. ◊ If in an output-zeroing input (x o , u o (k )) is u o (k ) ≠ 0 for some k ∈ N , then we say that the transmission of the signal u o (k ) , applied to (2.1) at the initial state x o , has been blocked by the system. Thus the transmission blocking property of (2.1) is a particular case of the output-zeroing property. Remark 2.3. Recall that a solution of the state equation in (2.1) passing at k = 0 through a point x o ∈ R n and corresponding to an admissible input u(.) ∈ U ( u(k ) ∈ R m ) is unique and has the form k

(i)

x(k + 1) = A k +1x o + ∑ A k −l Bu (l ), k = 0,1,2,... , x(0) = x o l =0

or equivalently,

(ii)

for k = 0 xo   k −1 . ◊ x(k ) =  k A x o + ∑ A k −1− l Bu(l ) for k = 1,2,...  l =0 

Lemma 2.5. The set of all output-zeroing inputs for a strictly proper or proper system S(A,B,C,D) (2.1) complemented with the trivial pair (x o = 0, u o ( k ) ≡ 0) forms a linear space over R.

Proof. In fact, if (x1o , u1o (k )) and (x o2 , u o2 ( k )) are output-zeroing inputs and give respectively solutions of the state equation x1o (k ) and x o2 (k ) , then from linearity of (2.1) and from the uniqueness of solutions as well as from the fact that the set U of admissible inputs forms a linear space over R it follows that each pair of the form (αx1o + βx o2 , αu1o (k ) + βu o2 ( k )) , with arbitrarily fixed α,β ∈ R , is output-zeroing and yields the solution αx1o (k ) + βx o2 (k ) . ◊

In the linear space consisting of all output-zeroing inputs and the trivial pair we can distinguish a subspace consisting of all pairs of the form (x o = 0, u oh (k )) , where u oh (.) ∈ U and u oh (k ) ∈ KerB ∩ KerD for all

2.4 The Output-Zeroing Problem

23

k ∈ N (note that if D = 0 , then Ker D = R n and KerB ∩ KerD = KerB ). Each pair of this kind affects the system equations in the same way as the trivial pair, i.e., it gives the identically zero solution of the state equation

and y (k ) = 0 for all k ∈ N . Furthermore, if (x o , u o (k )) is output-zeroing and gives a solution x o (k ) , then the pair (x o , u o (k ) + u oh (k )) is also output-zeroing and gives the same solution as (x o , u o (k )) . Lemma 2.6. If (x o , u o (k )) is an output-zeroing input for a strictly proper or proper system (2.1) and x o (k ) denotes the corresponding solution, then the input u o (k ) applied to (2.1) at an arbitrary initial condition x o ∈ R n yields the solution (i)

x(k ) = A k (x o − x o ) + x o (k ) , k ∈ N ,

where x(0) = x o , x o (0) = x o , and the system response (ii)

y (k ) = CA k (x o − x o ) .

Proof. Simple proof by verification that (i) and (ii) satisfy equations in (2.1) is omitted here. Now, the assertion follows by the uniqueness of solutions. ◊ The above discussion and Lemma 2.6 show in particular that if (x1o , u1o (k )) and (x o2 , u o2 ( k )) are output-zeroing inputs and give respect-

ively solutions x1o (k ) and x o2 (k ) and (2.1) is asymptotically stable, then the input signal αu1o (k ) + βu o2 (k ) applied to the system at an arbitrary initial state produces asymptotically vanishing system response, i.e., y (k ) → 0 as k → ∞ . In this way in an asymptotically stable system (2.1) the set of all output-zeroing inputs enables us to generate a class of input sequences which are asymptotically attenuated by the system. Finally, we do not associate output-zeroing inputs of the form (x o = 0, u oh (k )) , where u oh (.) ∈ U and u oh (k ) ∈ KerB ∩ KerD for all k ∈ N , with invariant zeros, since (as Examples 2.2 and 2.3 show) they can exist independently upon these zeros. We associate these pairs simply with the trivial pair (x o = 0, u o ( k ) ≡ 0) .

24


Example 2.2. Consider a minimal (reachable and observable) and asymptotically stable system (2.1) with the matrices 0  − 1 / 3 , A= − 2 / 3  0

 − 3 1 0 1 B= , 0 0 1 0

1 0 C=  . 0 1 

This system has no invariant (transmission) zeros, although output-zeroing inputs of the form (x o = 0, u oh (k )) exist. In fact, since C is nonsingular, there is no triple λ, x o ≠ 0, g which could satisfy the relations λx o − Ax o = Bg, Cx o = 0 (cf., Definition 2.1 (i)). On the other hand,

each admissible input u oh (k ) satisfying Bu oh (k ) = 0 for all k ∈ N and the initial state x o = 0 yield the solution of the state equation x o (k ) = 0 for all k ∈ N and, consequently, y (k ) ≡ 0 . ◊ Example 2.3. Consider a minimal and asymptotically stable system (2.1) with the matrices 0  − 1 / 3 , A= − 2 / 3  0 0 0 C = 1 0 , 0 1

 − 3 1 0 1 B= , 0 0 1 0

− 3 1 0 1 D =  0 0 0 0  .  0 0 0 0 

This system has no invariant (transmission) zeros, however output-zeroing inputs of the form (x o = 0, u oh (k )) exist. In fact, the relation (2.4) may be written in the form

(i)

1 (λ + ) x 1o = −3g1 + g 2 + g 4 3 2  g1  (λ + ) x o2 = g 3 3 g  o x  , where x o =  1o  , g =  2  . − 3g1 + g 2 + g 4 = 0 g3   x 2    x1o = 0 g 4  o x2 = 0

2.5 Invariant Zeros and Output-Zeroing Inputs

25

Now, it is easily seen that there is no triple λ, x o ≠ 0, g satisfying (i). On the other hand, each admissible input u oh (k ) satisfying simultaneously Bu oh (k ) = 0 and Du oh (k ) = 0 for all k ∈ N together with the initial state x o = 0 produces the solution x o (k ) = 0 for all k ∈ N and y (k ). ≡◊ 0

The above examples indicate also that there are systems which possess the transmission blocking property although they do not possess invariant zeros (note also that systems in Examples 2.2 and 2.3 have no Smith zeros – see Sect. 2.6, Proposition 2.1 (i)).

2.5 Invariant Zeros and Output-Zeroing Inputs In this section we shall show that to each invariant zero we can assign an output-zeroing input. Remark 2.4. In order to show that each invariant zero generates an outputzeroing input, it is convenient to treat a strictly proper or proper system (2.1) as a complex one, i.e., admitting complex valued inputs, solutions ~, ~ y. and outputs which are denoted respectively by u x and ~ ~ Of course, it is well known fact that if x (k ) is a solution of the state equation in (2.1) (treated as a complex system) corresponding to an input ~ (k ) and to an initial condition ~ u x o , then Re ~ x (k ) is a solution which o ~ passes at k = 0 through the point Re x and corresponds to the input ~ (k ) . Analogously, the pair Im ~ ~( k ) satisfies the state equaRe u x ( k ) , Im u xo . tion in (2.1) and the initial condition Im ~ ~ (k ) ) is such that it gives also ~ y (k ) = 0 for Furthermore, if a pair ( ~x o , u o o ~ ~ (k )) are ~ ~ all k ∈ N , then the pairs (Re x , Re u (k )) and (Im x , Im u output-zeroing inputs for (2.1) and give respectively solutions Re ~ x (k ) ~ and Im x ( k ) . ◊

As we show below (see Lemma 2.8), Definition 2.1 (i) clearly relates invariant zeros (even in the degenerate case) to the output-zeroing problem. To this end, we need first the following result. Lemma 2.7. [74] If λ ∈ C is an invariant zero of a strictly proper or proper system (2.1) and a triple λ, x o ≠ 0, g satisfies (2.4), then the input

26

(i)


~ (k ) =  g for k = 0 u  k λ g for k = 1,2,...

applied to (2.1) (treated as a complex system) at the initial condition x o yields the solution (ii)

 x o for k = 0 ~ x (k ) =  k o  for k = 1,2,... λ x

y (k ) = 0 for all k ∈ N . and the system response ~

Proof. Simple proof by inspection is left to the reader as an exercise. ◊ Lemma 2.8. [74] Each invariant zero of the (strictly proper or proper) system (2.1) determines an output-zeroing input. More precisely, if a triple λ = λ e jϕ , x o ≠ 0, g satisfies (2.4), then the pairs for k = 0  Re g ) (Re x o , u o (k ) =  k  λ (Re g cos kϕ − Im g sin kϕ) for k = 1,2,...

and for k = 0  Im g ) (Im x o , u o (k ) =  k  λ (Re g sin kϕ + Im g cos kϕ) for k = 1,2,...

are output-zeroing inputs for (2.1). Proof. Since matrices A, B, C, D in (2.1) are real, from (2.4) it follows that if λ ∈ C is an invariant zero of (2.1), i.e., a triple (i)

λ = λ e jϕ , x o = Re x o + jIm x o , g = Re g + jIm g

satisfies (2.4), then (2.4) is fulfilled also for the triple (ii)

λ = λ e − jϕ , x o = Re x o − jIm x o ,

g = Re g − jIm g ,

i.e., λ is also an invariant zero of (2.1). By virtue of Remark 2.4 and Lemma 2.7, the triples (i) and (ii) generate two real initial conditions and two real-valued inputs for (2.1) which produce the identically zero system response. More precisely, the pair


27

(Re x o , u o (k )) ,

(iii) where

Re g for k = 0 ~ (k ) =  u o (k ) = Re u  k Re (λ g ) for k = 1,2,... Re g for k = 0   1 k = 1 k λ g + λ g for k = 1,2,...  2 2

(iv)

Re g for k = 0  = k ,  λ (Re g cos kϕ − Im g sin kϕ) for k = 1,2,...

is an output-zeroing input and yields the solution of the state equation  Re x o for k = 0 x(k ) = Re ~ x (k ) =  k o  Re(λ x ) for k = 1,2,...  Re x o  = 1 k o 1 k o λ x + λ x  2 2

(v)

for k = 0 for k = 1,2,...

 Re x o for k = 0 . = k o o (Re cos k Im sin k ) for k 1 , 2 ,... x x λ ϕ − ϕ =  

Analogously, the pair (vi)

(Im x o , u o (k )) ,

where Im g for k = 0 ~ (k ) =  u o (k ) = Im u  k Im(λ g ) for k = 1,2,...

(vii)

Im g for k = 0   = 1 k 1 k − j λ g + j λ g for k = 1,2,...  2  2

28


for k = 0  Im g = k ,  λ (Re g sin kϕ + Im g cos kϕ) for k = 1,2,...

is an output-zeroing input and gives the solution  Im x o for k = 0 x(k ) = Im ~ x (k ) =  k o  Im(λ x ) for k = 1,2,...

(viii)

  = −  

Im x o 1 1 j λk x o + j λk x o 2 2

for k = 0 for k = 1,2,...

 Im x o for k = 0 . ◊ = k o o   λ (Re x sin kϕ + Im x cos kϕ) for k = 1,2,...

Example 2.4. In (2.1) let 1/ 3 0   0  A= 0 0 1 / 3  , − 1 / 3 − 2 / 3 − 1 / 3

0 0 B =  0 1 , 1 0 

 − 2 − 1 0 . C= 1 0 0

This system is minimal (i.e., reachable and observable), asymptotically stable and degenerate (see Sect. 2.7, Proposition 2.2). On the other hand, it has no Smith zeros. In particular, for any given ϕ ∈ R the triple  1 0  3 + cos ϕ + j sin ϕ o   λ = cos ϕ + j sin ϕ , x = 0 , g =   1   − 1 3  

satisfies (2.4), i.e., λ = cos ϕ + j sin ϕ is an invariant zero. According to Lemma 2.8, to λ = cos ϕ + j sin ϕ we can assign the following outputzeroing inputs  1 0 cos kϕ + cos(k + 1)ϕ  ( x o = 0 , u o (k ) =  3  ), 1   − ϕ cos k 1 3  

k∈N ,


29

and  1 0  3 sin kϕ + sin(k + 1)ϕ   ( x = 0 , u o ( k ) =   ), 1   − sin kϕ 0 3   o

k∈N .

The corresponding solutions of the state equation are equal respectively  0  x (k ) =  0  coskϕ o

and

 0  x ( k ) =  0  , sin kϕ o

k∈N .

Note that due to the asymptotic stability, each of the above input signals u o (k ) applied to the system at any initial state yields asymptotically vanishing system response (cf., Lemma 2.6). The triple λ = 0,

0  x = 0, 1 o

 1/ 3  g=  − 1 / 3

also satisfies (2.4), i.e., λ = 0 is an invariant zero of the system. The corresponding output-zeroing input and solution is of the form  1/ 3  0   for k = 0  − 1 / 3 o   ( x = 0 , u o ( k ) =  ) 0  1   for k = 1,2,...   0

and  0   for k = 0  0  1  x o (k ) =    0  0 for k = 1,2,...    0

. ◊

30


Remark 2.5. In this remark we show that o.d. zeros are responsible simultaneously for nontrivial free evolutions and zeroing the system output. Recall that free evolutions in (2.1) are those solutions of the state equation which correspond to the condition Bu(k ) = 0 for all k ∈ N , i.e., when the state equation is not affected by inputs. Free evolutions are said to be nontrivial if the aformentioned solutions are not identically zero. 1. For strictly proper systems ( D = 0 ) we prove the equivalence: a strictly proper system (2.1) is observable ⇔ nontrivial free evolutions and zeroing the system output do not appear simultaneously. ⇒ ) Suppose that the system is observable and nontrivial free evolutions together with zeroing the system output take place. Then there exists an input sequence u(.) ∈ U and an initial state x(0) ≠ 0 such that for all k ∈ N we have Bu(k ) = 0 and y (k ) = Cx(k ) = 0 , while the solution of the state equation x(k ) = A k x(0) is not identically zero. However, since y (0) = Cx(0) = 0 , y (1) = CAx(0) = 0 ,…, y (n − 1) = CA n −1x(0) = 0 , the observability assumption implies x(0) = 0 and, consequently, x( k ) = 0 for all k ∈ N which contradicts the assumption of nontrivial free evolutions. ⇐ ) For the proof it is sufficient to show that if the system is unobservable, then nontrivial free evolutions together with zeroing the system output take place. This implication, however, is obvious, since (see

Lemma 2.1) we can find a number λ ∈ C and a vector 0 ≠ x o ∈ C n such that (λI − A )x o = 0 and Cx o = 0 and then (cf., Lemma 2.7) ~ (k ) =  g for k = 0 , where Bg = 0 , u  k λ g for k = 1,2,...

gives  x o for k = 0 ~ and ~ y (k ) = 0 for all k ∈ N . x (k ) =  k o  for k = 1,2,... λ x

Thus in a strictly proper system (2.1) nontrivial free evolutions together with zeroing the system output exist if and only if the system is unobservable (i.e., if and only if o.d. zeros exist). 2. For proper systems ( D ≠ 0 ) the implication – S(A,B,C,D) is unobservable ⇒ nontrivial free evolutions together with zeroing the system output take place – is valid. However, as the following example shows, the converse implication does not hold in general, i.e., we can meet nontrivial


31

free evolutions and zeroing the system output also in observable systems. Example 2.5. Consider a minimal system (2.1) with the matrices − 3 1 0 1   − 1 0 0 0  , D= , C= , B= A= .     1 0 0 0  − 2 0 1 0 0 0 The triple λ = 0 , x o =   , g =   represents invariant zero at zero.  − 1 1 The pair  g for k = 0 ) (x o , u o (k ) =  0 for k = 1,2,...

is output-zeroing and, since Bu o (k ) = 0 for all k ∈ N , it produces the nontrivial free evolution  x o for k = 0 . x(k ) = A k x o =    0 for k = 1,2,... 3. As it follows from the discussion in points 1 and 2 above, in both cases D = 0 and D ≠ 0 with o.d. zeros we can always associate the n −1

output-zeroing inputs (x o ≠ 0, u o (k ) ≡ 0) with x o ∈ X o = h Ker CA l l =0

o

(i.e., x lies in the unobservable subspace for (2.1)). ◊ For observable systems we can formulate the following characterization of invariant zeros. Lemma 2.9. [74] Consider an observable strictly proper or proper system (2.1). Then a triple λ, x o ≠ 0, g satisfies (2.4) if and only if the input (i)

~ (k ) =  g for k = 0 u  k λ g for k = 1,2,...

and the initial condition x o ≠ 0 yield ~ y (k ) = 0 for all k ∈ N . Moreover, in the triple under considerations is g ≠ 0 , and the solution ~ x (k ) correo ~ sponding to x and u (k ) takes the form (ii)

 x o for k = 0 ~ . x (k ) =  k o  for k = 1,2,... λ x

32


Proof. If λ ∈ C is an invariant zero, i.e., a triple λ, x o ≠ 0, g satisfies (2.4), then, as we know from Lemma 2.7, the input sequence of the form (i) applied to the system (treated as a complex one) at the initial condition ~ x (0) = x o yields a solution of the state equation of the form (ii) and the identically zero output sequence. It remains to show that g ≠ 0 . However, supposing g = 0 in the triple under considerations, we would obtain in (2.4) (λI − A )x o = 0 and Cx o = 0 which would contradict the observability assumption. Therefore we must have g ≠ 0 . In order to prove the converse implication, we should show that if a triple λ, x o ≠ 0, g is such that the input (i) applied to the system at the inx (0) = x o yields ~ y (k ) = 0 for all k ∈ N , then this triple itial condition ~ sat-isfies (2.4) and g ≠ 0 . The idea of the proof is as follows. We shall show first that for each k ∈ N the identity CA k ((λI − A)x o − Bg ) = 0 holds. Then, by virtue of the observability assumption, we will have (λI − A )x o − Bg = 0 . The second identity in (2.4) (i.e., Cx o + Dg = 0 ) will follow immediately from the condition ~ y (0) = 0 . Finally, the relation g ≠ 0 will follow as a simple consequence of the observability assumption. Now, in view of the assumptions, we can write for k = 1,2,... (iii)

~x (k + 1) = A~ x (k ) + Bgλk ~ y (k ) = C~ x (k ) + Dgλk

and for k = 0

(iv)

~ x (1) = Ax o + Bg ~ y (0) = Cx o + Dg

as well as (v)

~ y (k ) = 0 for all k ∈ N .

From (iii), (iv) and (v) it follows that


33

~ x (1) = Ax o + Bg 0 = Cx o + Dg ~ y (1) = C~ x (1) + Dgλ = 0.

(vi)

Now, using (vi), we can verify that (vii)

C((λI − A) x o − Bg ) = Cλx o − CAx o − CBg = λCx o − C~ x (1) = λCx o + λDg = 0.

Then we show that for each k = 1,2,... is (viii)

CA k ((λI − A)x o − Bg ) = λC~ x (k ) − C~ x (k + 1) .

In fact, the left-hand side of (viii) can be written as λCA k x o − CA k +1x o − CA k Bg .

(ix)

On the other hand, we know that the solution of the state equation in (2.1) corresponding to the initial condition ~ x (0) = x o and to the input of the form (i) may be written as k ~ x (k + 1) = A k +1x o + ∑ A k − l Bgλl .

(x)

l =0

Premultiplying both sides of the above relation by matrix C, we obtain (xi)

k CA k +1x o = C~ x (k + 1) − ∑ CA k − l Bgλl . l =0

Introducing the right-hand side of (xi) into (ix) and making use of (x), we obtain the relations CA k ((λI − A )x o − Bg ) k x (k + 1) + ∑ CA k − l Bgλl − CA k Bg = λCA k x o − C~

l =0 0

= λC [ A k x o + A k −1Bgλ + ... + ABgλk −1 ] − C~ x (k + 1) k −1 = λC [ A k x o + ∑ A ( k −1) − l Bgλl ] − C~ x (k + 1) = λC~ x (k ) − C~ x (k + 1)

l =0

which prove (viii). Furthermore, we can write

34


~ y (k ) = C~ x (k ) + Dgλk = 0 ~ y (k + 1) = C~ x (k + 1) + Dgλk +1 = 0.

(xii)

Multiplying the first identity in (xii) by λ and comparing the result with the second identity, we observe that λC~ x ( k ) + Dgλk +1 = C~ x (k + 1) + Dgλk +1 = 0 x (k ) − C~ x (k + 1) = 0 . This means that the rightand, consequently, λC~ hand side in (viii) is equal to the zero vector. This fact together with (vii) proves that CA k ((λI − A)x o − Bg ) = 0 for all k ∈ N . Now, the observ-

ability assumption implies (λI − A )x o − Bg = 0 . Moreover, from (vi) we have Cx o + Dg = 0 . Thus the triple under considerations satisfies (2.4), and g ≠ 0 follows from the observability condition. In the case D = 0 the proof follows the same lines. ◊ Note that in Lemma 2.9 the observability assumption is essential. The ~ (k ) following example shows that the implication – if an input sequence u y (k ) = 0 for all k ∈ N , of the form (i) applied to (2.1) at x o ≠ 0 gives ~ then the triple λ, x o , g satisfies (2.4) – does not hold if this assumption is neglected. Example 2.6. Consider an unobservable system (2.1) with the matrices 1 0 B = 0 1 , 1 1

1 0 0  A = 0 1 0 , 0 0 1 

C = [1 0 1]

− 1  1  as well as the triple λ = 1, x =  0 , g =   .  − 2  1  Naturally, it is easy to verify (see Remark 2.3) that the constant input 1  u(k ) =   , k ∈ N , applied to the system at the initial state x(0) = x o  − 2 gives y( k ) = 0 for all k ∈ N , but the triple under considerations does not satisfy (2.4). ◊ o


35

Remark 2.6. [75] Throughout this remark a strictly proper or proper system (2.1) is assumed to be minimal (i.e., reachable and observable). It is possible to observe that the adopted definition of invariant zeros (Definition 2.1 (i)) does not create ambiguity in case of coincident zeros and poles, i.e., it enables us to distinguish between them. To this end, however, we will need the following state space characterization of poles. The following statements are equivalent: (i)

λ ∈ σ(A ) ;

(ii)

there exist vectors 0 ≠ x op ∈ C n and 0 ≠ h ∈ C r such that (λI − A )x op = 0 and Cx op = h ;

(iii)

~ (k ) ≡ 0 applied to (2.1) at the initial condition x o ≠ 0 the input u p  h ~ y (k ) =  k hλ identically zero).

yields

for k = 0 for k = 1,2,...

(note that

~ y (k )

is not

In this way a complex number λ representing a pole of the system may be treated as a triple λ, x op ≠ 0, h ≠ 0 which satisfies (ii) and possesses dynamical interpretation given by (iii) (in order to pass in (iii) to real values use Remark 2.4). Moreover, we use the state space characterization of invariant zeros that follows immediately from Definition 2.1 (i) and Lemma 2.9. The following statements are equivalent: (iv)

λ ∈ C is an invariant zero;

(v)

there exist 0 ≠ x o ∈ C n and 0 ≠ g ∈ C m such that (λI − A )x o = Bg and Cx o + Dg = 0 ;

(vi)

~ (k ) =  g the input u  k gλ

for k = 0 for k = 1,2,...

~ (k ) is not (note that u

identically zero) applied to (2.1) at the initial condition x o ≠ 0 y (k ) = 0 for all k ∈ N . yields ~ We can discuss now two special cases of coincident poles and zeros.

36


At first, consider a strictly proper ( D = 0 ) system (2.1) and let λ be its common zero and pole, i.e., let a triple λ, x o ≠ 0, g ≠ 0 satisfy (v) and let a triple λ, x op ≠ 0, h ≠ 0 satisfy (ii). It is clear that we must have Bg ≠ 0 (otherwise, the system would be unobservable) as well as the vectors x o and x op have to be linearly independent. In the second case, let λ be a common zero and pole of a proper ( D ≠ 0 ) system (2.1). Note that we can meet here the situation Bg = 0 , but then we must have Dg ≠ 0 (otherwise, the system would be unobservable). Moreover, in such a situation, we can even have x o = x op . As an appropriate example we can take the system of Example 2.7. ◊ Example 2.7. Consider the system as in Example 2.5. The triple 0 0 λ = 0, x o =  , g =    − 1 1

represents invariant zero at zero, whereas the triple 0 1 λ = 0, x op =   , h =   1 0

represents system pole at zero. Thus λ = 0 , treated as a pole, can be distinguished from λ = 0 , treated as an invariant zero, merely by different dynamical interpretations (iii) and (vi). ◊

2.6 Relationships Between Smith and Invariant Zeros The following result establishes relationships between the set

Z S := {λ ∈ C : rank P (λ) < normal rank P (z)}

(2.6)

of Smith zeros and the set ZI (2.5) of invariant zeros in the system (2.1) (strictly proper or proper). Note that in (2.6) the Smith zeros of (2.1) are determined (see (iv) in Sect. 1.1) as all those distinct complex numbers at zI − A − B  drops below its which rank of the system matrix P (z) =  D   C normal rank.

2.6 Relationships between Smith and Invariant Zeros

37

Proposition 2.1. [82] For a strictly proper or proper system (2.1) the following claims hold: (i) If λ ∈ C is a Smith zero of (2.1), then λ is an invariant zero of (2.1), i.e., Z S ⊆ Z I ; (ii) (iii)

System (2.1) is nondegenerate if and only if ZI = Z S ;

System (2.1) is degenerate if and only if ZI = C .

Proof. (i) The assertion follows by showing that if P(z) loses its normal rank at λ (i.e., λ is a Smith zero of (2.1)), then there exists a solution x o    of the system  g   x  0  P (λ )  =   u  0

(iv)

of n+r linear equations with n+m unknowns such that x o ≠ 0 . Since λ ∈ Z S , we have

rank P (λ) < normal rank P (z) ≤ n + min{m, r} ≤ n + m ,

i.e., columns of P (λ ) are linearly dependent over C . This means that the set of nonzero solutions of (iv) is not empty. We shall discuss separately two cases. − B  Suppose first that the submatrix   of P(z) has full column rank and D 0  let a vector   , where g ≠ 0 , be a nonzero solution of (iv). This yields g  − B  0  D  g = 0 and, consequently, g = 0 . This contradiction proves that in     any nonzero solution of (iv) we must have x ≠ 0 , i.e., λ satisfies Definition 2.1 (i). − B  In the second case, when   has rank m' < m , we can assume, D without loss of generality, that the first m' columns of that matrix are linearly independent.

38


− B' Denote by   the matrix composed of these columns, i.e.,  D' 

(v)

− B  − B' − B' '  D  =  D' D' '    

− B   − B' and rank   = rank   = m' . D  D' 

zI − A − B' It is clear that P ' (z) =  has the same normal rank as P (z) . D'   C Moreover, rank P (z) = rank P ' (z) at any fixed z ∈ C (where rank is taken over the field of complex numbers). This enables us to write rank P ' (λ ) = rank P (λ ) < normal rank P (z) = normal rank P ' (z) ≤ n + min{m' , r} ≤ n + m' ,

i.e., columns of P ' (λ ) are linearly dependent over C . Now, by virtue of the first part of the proof (referred to the system S( A, B' , C, D' ) ), we conx o  clude that there exists a vector   , where x o ≠ 0 and g '∈ C m' , such  g '   x o  0 g ' that P ' (λ)   =   . Finally, taking g =   , where 0 ∈ C m − m' , we  g'  0 0  x o  0  obtain P (λ)  =   , i.e., λ satisfies Definition 2.1 (i) for S(A,B,C,D).  g  0

(ii) We shall show (by reductio ad absurdum) that nondegeneracy of (2.1) implies ZI = Z S (since Z S is at most finite, the converse implication is obvious). To this end, suppose that (2.1) is nondegenerate and Z S ⊂ ZI , i.e., there exists a λ o ∈ C such that λ o ∈ Z I and λ o ∉ Z S . This means that for λ o there exists a solution of (iv) with a nonzero state-zero direction and, consequently, normal rank P (z) = rank P (λ o ) < n + m . We shall show the contradiction: any λ ∈ C such that λ ∉ Z S satisfies

λ ∈ ZI (i.e., (2.1) is degenerate). We discuss separately two cases.


Suppose

first

that

− B  D  

has

full

column

rank

and

39

let

rank P(λ) = normal rank P(z) , i.e., λ ∉ Z S . This means that rank P(λ) = normal rank P( z) = rank P(λ o ) < n + m and, consequently, for λ there exists a solution of (iv) with a nonzero

state-zero direction, i.e., λ ∈ ZI . − B  In the second case, when   has rank m' < m , we assume that the D − B  first m' columns of   are linearly independent (and the notation as in D x o  (v) is used). Denote by   , where x o ≠ 0 , a nonzero solution of (iv) g o 

corresponding to λ o , i.e., λ o x o − Ax o = Bg o , Cx o + Dg o = 0 . Since we − B' − B  have Im   = Im   (where Im M denotes a subspace spanned by  D'  D columns of matrix M), we can find a vector g '∈ C m ' such that − B' − B  o o o  D  g o =  D'  g ' , i.e., λ o x − Ax = B ' g ' , Cx + D' g ' = 0 and, con    sequently, rank P ' (λ o ) < n + m' . Thus we have rank P ' (λ o ) = rank P (λ o ) = normal rank P (z) = normal rank P ' (z) < n + m'.

Now, if λ is not a Smith zero of (2.1), then we must have rank P' (λ) = rank P(λ) = normal rank P(z) < n + m' .

However, from the relation rank P' (λ) < n + m' it follows that the equation  x  0  P ' (λ )  =   has a solution with a nonzero state-zero direction. u' 0 Consequently, λ is an invariant zero of (2.1) (compare with the last part of the proof of (i)).

40


(iii) It suffices to show that if (2.1) is degenerate, then ZI = C (the con-

verse implication is obvious). If (2.1) is degenerate, then Z S ⊂ ZI , i.e., there exists an invariant zero λ o which is not a Smith zero. By virtue of

(i), each λ ∈ Z S is also an invariant zero. Hence we only need to show that each λ ∈ C such that λ ∉ Z S is an invariant zero. To this end, we proceed with the proof analogously as in the proof of (ii). ◊

Proposition 2.1 tells us that each Smith zero of a strictly proper or proper system (2.1) is also its invariant zero. Furthermore, the set ZI of invariant zeros may be empty, finite or equal to the whole complex plane, and when (2.1) is nondegenerate, the sets of Smith zeros and of invariant zeros coincide. The following claim follows immediately from Definition 2.1 (i) and Proposition 2.1 (i). Corollary 2.2. If λ ∈ C is a Smith zero of the strictly proper or proper system (2.1), then there exist vectors 0 ≠ x o ∈ C n and g ∈ C m such that the triple λ, x o , g satisfies (2.4). ◊ Thus, although the Smith zeros are defined merely as certain complex numbers (i.e., the roots of the zero polynomial), however to each Smith zero we can always assign a nonzero state-zero direction ( x o ) and an input-zero direction (g) and, consequently, by virtue of Lemma 2.8, a clear dynamical interpretation in the context of the output-zeroing problem. In nondegenerate systems we can identify the notion of Smith zeros with the notion of invariant zeros. Corollary 2.3. If the strictly proper or proper system (2.1) is nondegenerate, then for a given complex number λ the following statements are equivalent: (i) λ is an invariant zero of the system; λ is a Smith zero of the system; (ii) (iii) λ is a root of the zero polynomial. Proof. Since the equality ZI = Z S means that λ is an invariant zero if and only if λ is a Smith zero, the equivalence of (i) and (ii) follows from Proposition 2.1 (ii). The statements (ii) and (iii) are equivalent by virtue of the definition of Smith zeros (see Chap. 1, Sect. 1.1). ◊


41

Definition 2.3. If the strictly proper or proper system (2.1) is nondegenerate and λ ∈ C is its invariant zero, then the algebraic multiplicity of the invariant zero λ is defined as the multiplicity of λ in the zero polynomial. ◊ Remark 2.7. In Chapters 4 and 5 we shall show that Definition 2.3 is well posed. This will be done by proving that in any nondegenerate system (2.1) its invariant zeros are characterized by a certain polynomial that is equal to the zero polynomial. For a nondegenerate transfer-function matrix we define the algebraic multiplicities of its transmission zeros using Definitions 2.2 and 2.3. For a degenerate system (2.1) we do not define multiplicity of its invariant zeros, although the notion of the algebraic multiplicity of Smith zeros still makes sense. ◊ Corollary 2.4. Let the system matrix P (z) corresponding to a strictly proper or proper system (2.1) have full normal column rank. Then (2.1) is nondegenerate, i.e., ZI = Z S . Proof. In view of Proposition 2.1 (i) and (ii), it is sufficient to show that any invariant zero is also a Smith zero. From Definition 2.1 (i) it follows, however, that if λ is an invariant zero of (2.1), then columns of P (λ ) are linearly dependent over C. Thus we can write rank P (λ ) < n + m = normal rank P (z)

which means that λ is a Smith zero of (2.1). ◊ Corollary 2.5. In a strictly proper or proper system (2.1) let the first nonzero Markov parameter have full column rank. Then (2.1) is nondegenerate, i.e., ZI = Z S . Proof. In view of Corollary 2.4, it suffices to show that P(z) has full normal column rank. In the proof we use singular value decomposition (SVD) (see Appendix B) of the first nonzero Markov parameter. Moreover, the cases D = 0 and D ≠ 0 are treated separately. If in (2.1) is D = 0 and the first nonzero Markov parameter CA ν B has

full column rank, then SVD of CA ν B we can write as CA ν B = U Λ V T ,

42


where m ≤ r and U ∈ R rxr and V ∈ R mxm are orthogonal and M  Λ =  m  with mxm diagonal and nonsingular M m . Denote  0   C  B = BV = B m and C = U T C =  m  ,  Cr − m 

where Cm consists of the first m rows of C , and observe that M m = Cm A ν B m . Now, we can write

 zI − A − B  P (z) =   0   C

(i)  I = 0 rxn

0 nxr  zI − A − B   I  U T   C 0  0 mxn

0 nxm  . V 

On the other hand, P (z) may be written as

(ii)

zI − A − B m    P (z) =  Cm 0 ,  Cr − m 0  

zI − A − B m  where P ' (z) =   is square. In order to show that P(z) has 0   Cm

full normal column rank, it is sufficient to note that det P ' (z) is a nonzero polynomial. However, forming an nxn matrix K ν := I − B m ( Cm A ν B m ) −1 Cm A ν

and using well known identities for determinants (cf., Exercise 2.8.25), we obtain the desired result (iii)

det P ' (z) = z − m(ν +1) det M m det (zI − K ν A) ≠ 0 .

Suppose now that matrix D has full column rank. Writing SVD of D D  as D = U ΛV T , where Λ =  m  and mxm D m is diagonal and nonsin 0  gular, and taking into account the same notation for matrices B and C as in the previous case, we obtain


(iv)

 zI − A − B  P (z) =   Λ   C 0 nxr   zI − A − B   I  I =  T  D  0 mxn 0 rxn U   C

43

0 nxm  . V 

On the other hand, P (z) may be written as

(v)

zI − A − B m    P (z) =  Cm Dm  ,  Cr − m 0  

zI − A − B m  where P ' (z) =   is square. Moreover, we have the followDm   Cm ing relation

(vi)

−1 det P ' (z) = det D m det (zI − ( A − B m D m C m ))

which means that P ' (z) and, consequently, P(z) is of full normal column rank. ◊ In the remaining part of this section we consider a r x m transferfunction matrix G (z) and assume that a strictly proper or proper system (2.1) constitutes its n-dimensional minimal (irreducible) state space realization and P(z) stands for the system matrix of (2.1). Making use of Proposition 2.1 and Corollary 2.4 as well as of the well known relation [10] normal rank P (z) = n + normal rank G (z)

(2.7)

which is valid without any conditions concerning the normal rank of G(z) (for the proof of (2.7) see Appendix C), we can relate Definition 2.2 to other commonly known definitions of zeros of a transfer-function matrix. To this end, we take into account the Desoer-Schulman definition and the definition of Smith zeros of G(z) [16]. Recall that the latter is based on a minimal state space realization of G(z) and defines the Smith zeros of G(z) as the Smith zeros of P(z), i.e., as the points of the complex plane where the rank of the underlying system matrix drops below its normal rank. Note that in this definition we assume nothing about normal rank of G(z). On the other hand, the definition of zeros of G(z) attributed to Desoer and

44


Schulman [14, 10] concerns merely transfer-function matrices with full normal rank and exploits the matrix coprime fraction description G (z) = D l−1 (z)N l (z) . Then a number λ ∈ C is said to be the Desoer-

Schulman zero of G(z) if and only if rank N l (z) at z = λ falls below its normal rank, i.e., rank N l (λ) < min{m, r} . In other words, these zeros are the Smith zeros of N l (z) . The Desoer-Schulman zero of G(z) can also be defined by using dynamic equations [10]. Namely, λ ∈ C is a DesoerSchulman zero of G(z) if and only if  λI − A − B  rank P (λ) = rank  < normal rank P (z) = n + min{m, r} , D   C

i.e., if and only if λ is a Smith zero of (2.1). Now, we can formulate the desired result. Corollary 2.6. [74] For a rxm transfer-function matrix G(z) let a strictly proper or proper system S(A,B,C,D) (2.1) denote its n-dimensional minimal state space realization. Then: a) If λ ∈ C is a Smith zero of G (z) (as in [16]), then λ is also a zero of G (z) in the sense of Definition 2.2 (i.e., there exist vectors 0 ≠ x o ∈ C n and 0 ≠ g ∈ C m such that the triple λ, x o , g satisfies (2.4), where P(z) is determined by the matrices of S(A,B,C,D)).

b) If G (z) has full normal rank and λ ∈ C is its Desoer-Schulman zero, then λ is also a zero of G (z) in the sense of Definition 2.2. c) If G (z) has full normal column rank, then λ ∈ C is a zero of G (z) in the sense of Definition 2.2 if and only if λ is a Desoer-Schulman zero of G (z) (this result fails when G (z) is not of full normal column rank). d) If G (z) is square and of full normal rank, then the following statements are equivalent: (i) λ ∈ C is a zero of G (z) in the sense of Definition 2.2; (ii) λ ∈ C is a Desoer-Schulman zero of G (z) ; (iii) det P (λ) = 0 . Proof. a) The proof follows immediately from appropriate definitions of zeros and from Proposition 2.1 (i) when applied to the minimal realization of G(z).

2.7 Algebraic Criteria of Degeneracy

45

b) In this case we have the relation rank P (λ) < normal rank P (z)= n + min{m, r} ,

i.e., λ is a Smith zero of (2.1). The remaining part of the proof follows the same lines as the proof of Proposition 2.1 (i). c) In view of b), we only need to show that any zero of G(z) in the sense of Definition 2.2 is also a Desoer-Schulman zero of G(z). We make use of the relation (2.7). Now, if G(z) has full normal column rank, then m ≤ r and normal rank P (z) = n + m , i.e., P(z) has full normal column rank. The remaining part of the proof follows immediately from Corollary 2.4. d) The proof follows immediately from c) and from appropriate definitions of zeros. ◊ From the above discussion it follows that each zero of a transfer-function matrix defined as the point where the underlying system matrix loses its normal rank admits the physical interpretation given in Lemma 2.8 (when G(z) is asymptotically stable – see Lemma 2.6).

2.7 Algebraic Criteria of Degeneracy Proposition 2.2. [83] The system (2.1) (strictly proper or proper) is degenerate if and only if − B  normal rank P (z) < n + rank   . D

(2.8)

− B Proof. Let rank   = m' ( m' ≤ m ). D − B  (i) Suppose first that m' = m (i.e.,   has the full column rank m). D The necessity of the condition (2.8) follows from Corollary 2.4. In fact, Corollary 2.4 tells us that if the system matrix P (z) of (2.1) has full normal column rank, then the system is nondegenerate (or, equivalently, if (2.1) is degenerate, then normal rank P (z) < n + m ). For the proof of sufficiency suppose that (2.8) holds. Then for any λ ∈C

46


rank P (λ) ≤ normal rank P (z) < n + m .

(2.9)

− B From (2.9) and from the assumption rank   = m it follows that for any D given complex number λ the equation  x  0 P (λ )   =   u  0

(2.10)

x o  with n + m unknowns has a solution   with x o ≠ 0 . This means that  g  the system is degenerate. (ii) Suppose now that m' < m and assume (without loss of generality) − B  that the first m' columns of   are linearly independent. The subD − B' − B  matrix of   composed of these columns is denoted by   , i.e.,  D'  D − B  − B' − B' ' . Consider the system S( A, B' , C, D' ) and its system  D  =  D' D' '     zI − A − B' . The sets of invariant zeros for systems matrix P ' (z) =  D'   C S(A,B,C,D) (2.1) and S( A, B' , C, D' ) coincide, i.e., ZIS( A , B, C, D) = ZIS( A, B ', C, D') .

(2.11)

The proof of (2.11) follows from Definition 2.1 (i) and from the relation − B  − B' I Im   = Im   . We are to show that λ ∈ Z S( A,B,C, D) if and only if D D '     λ ∈ ZIS( A,B',C, D' ) . Suppose first that λ ∈ ZIS( A,B,C, D) , i.e., via Definition

2.1 (i), there exist x o ≠ 0 and g ∈ C m such that λx o − Ax o = Bg and − B  − B' m' Cx o + Dg = 0 . Since Im   = Im  such  , we can find a g '∈ C D D '     − B  − B' that   g =  g' . Consequently, at the same λ and x o , we get the  D  D' 


47

relations λx o − Ax o = B' g ' and Cx o + D' g' = 0 , i.e., λ ∈ ZIS( A,B',C, D' ) . The proof of the converse implication proceeds along the same lines. For P (z) and P ' ( z) the following relations hold normal rank P (z) = normal rank P ' (z)

(2.12)

rank P(λ) = rank P' (λ) for any λ ∈ C .

Now, from the first part of the proof which considers the system S( A, B' , C, D' ) it follows that S( A, B' , C, D' ) is degenerate if and only if normal rank P ' (z) < n + m' . Finally, taking into account (2.11) and (2.12), we can write the following sequence of equivalent conditions: System (2.1) is degenerate ⇔ S( A, B' , C, D' ) is degenerate ⇔ normal rank P (z) = normal rank P ' (z) < n + m' . ◊ Proposition 2.3. [83] The system (2.1) (strictly proper or proper) is nondegenerate if and only if − B normal rank P (z) = n + rank   . D

(2.13)

Proof. The claim follows from Proposition 2.2 and from the fact that nor− B  mal rank of P(z) can not be larger than n + rank   . ◊ D

Propositions 2.1, 2.2 and 2.3 can be illustrated as shown in Fig. 2.3. Corollary 2.7. [83] If in the system (2.1) (with D ≠ 0 or D = 0 ) is − B r < rank   , then the system is degenerate. D Proof. The claim follows from Proposition 2.2 and from the following relations   − B − B  normal rank P(z) ≤ min n + rank  , n + r  < n + rank   . ◊ D D  

48


Fig. 2.3.

Proposition 2.4. [83] Let G (z) denote a rxm transfer-function matrix and let a strictly proper or proper system (2.1) stand for its minimal ndimensional state space realization. Then G (z) is degenerate if and only if − B  normal rank G (z) < rank   . D

(2.14)

Proof. In the proof we make use of the relation normal rank P (z) = n + normal rank G (z) (cf., (2.7)). By virtue of Definition 2.2, G (z) is degenerate if and only if (2.1) is degenerate. However, in view of Proposition 2.2 and the relation (2.7), it takes place if an only if (2.14) holds. ◊ Proposition 2.5. [83] At the notation used in Proposition 2.4, G (z) is nondegenerate if and only if − B normal rank G (z) = rank   . D

Proof. The obvious proof is omitted here. ◊

(2.15)


49

Example 2.8. Consider a minimal and asymptotically stable system (2.1) with the matrices 1 0  B= , 0 − 1

 0 1 / 4 A= , − 2 − 1 

C = [1 1] .

The Smith form of P (z) equals 1 0 0 0  0 1 0 0  ,   0 0 1 0

hence the system has no Smith zeros (i.e., Z S = ∅ ). On the other hand, since the condition (2.8) is fulfilled (see also Corollary 2.7), the system is degenerate (i.e., by virtue of Proposition 2.1 (iii), ZI = C ). ◊ Example 2.9. Consider a system (2.1) with the matrices 0 − 2 − 1 / 3 1 1    A= 0 − 1 / 3 1  , B = 1 0 , C = [0 1 2] , D = [1 0] .  0 0 0 0 − 1 

The Smith form of P (z) is equal to 1  0 0  0 

0 0 0 1 0 0 ,  2 0 0 (z + 1)(z − ) 0 3 

0 0 1 0

0 0

i.e., Z S = {− 1, 2 / 3}. On the other hand, the condition (2.8) holds (see also Corollary 2.7) and the system is degenerate (i.e., ZI = C ). ◊ Example 2.10. Consider Example 2.5. The Smith form of P(z) equals 1 0  0  0

0 1 0 0  . 0 1 0  0 0 z2  0 0

50


The condition (2.13) of Proposition 2.3 is fulfilled, hence the system is nondegenerate and ZI = Z S = {0} . By virtue of Definition 2.3, the algebraic multiplicity of the invariant zero λ = 0 is 2. ◊ In the remaining part of this section we shall discuss the invariant and Smith zeros in strictly proper systems (2.1) with the identically zero transfer-function matrix. As is well known [20, 61, 91], such systems appear in the analysis of the disturbance decoupling problem. On the other hand, as we shall see in Chapters 5 and 6, any degenerate strictly proper system (2.1) can be transformed by an appropriate recursive procedure (which preserves the maximal (A,B)-invariant subspace contained in Ker C, the set of invariant zeros as well as the zero polynomial) into a system with the identically zero transfer-function matrix. Therefore some elementary results concerning zeros of such systems are needed. The internal dynamical structure of systems with the identically zero transfer-function matrix will be discussed in Chapter 6. Proposition 2.6. If in S(A,B,C) (2.1) is G (z) ≡ 0 , then the system is degenerate. Moreover, the zero polynomial of S(A,B,C) equals det(zI ro − A r o ) , i.e., the Smith zeros of S(A,B,C) are the i.o.d. zeros of this system. Proof. Because B ≠ 0 , for any fixed λ∉σ( A ) we can find a vector g ∈ C m such that Bg ≠ 0 . Then the triple λ, x o = (λI − A) −1 Bg ≠ 0, g satisfies (2.4). This means that any complex number which is not in the spectrum of A is an invariant zero, i.e., the system is degenerate. Of

course, by virtue of Proposition 2.1 (iii), we have ZI = C . This proves the first claim of Proposition 2.6. In order to prove the second claim, consider S(A,B,C) in its Kalman form (2.2). Of course, we can do this since similarity transformations in the state space do not change the Smith form of the system matrix, nor the zero polynomial (cf., Exercise 2.8.10 (i)). The orders of diagonal submatrices of the A-matrix are equal respectively (cf., (2.3)) n ro = nr , nro = 0 , n r o = n − nr − no and n ro = no , and (2.2) assumes the form (see also Fig. 2.4)  A ro A =  0  0

where

A13 A ro 0

A14  A 34  , A ro 

B r o  B =  0  ,  0 

C = [0 0 C ro ] ,

(2.16)


51

Fig. 2.4.

 x ro  x = x ro  .  x ro 

Note that since in (2.1) we have assumed B ≠ 0 and C ≠ 0 , in (2.16) is always n ro > 0 and nro > 0 , while for n ro is in general n ro ≥ 0 (i.e., i.o.d. zeros in S(A,B,C) may not exist). The system marix for (2.16) is zI ro     

− A ro 0 0 0

− A13

− A14

zI r o − A r o 0 0

− A 34 zI ro − A ro C ro

− B ro  0  . 0   0 

(2.17)

Interchanging successively appropriate columns in (2.17), we obtain zI ro     

− A ro 0 0 0

− B ro

− A13

0 0 0

zI ro − A r o 0 0

− A14

 − A 34  . z I ro − A ro   C ro 

(2.18)

Since the pair ( A ro , B ro ) is reachable, the pencil [zI ro − A ro − B ro ] has the full row rank n ro at any λ ∈ C and, consequently, it can be transformed by elementary column operations into the Hermite column form ~ ~ 0 A ro (z) , where A ro (z) is square, of the order n ro , upper triangular ~ and unimodular, i.e., det A ro (z) = const ≠ 0 (see (v) in Appendix C.1). Analogously, since the pair ( A ro , C ro ) is observable, the pencil

[

]

52


zI ro − A ro   C  has the full column rank n ro at any λ ∈ C and, consero   quently, it can be transformed by elementary row operations into the ~  A ro ( z )  ~ Hermite row form   , where A ro (z) is square, of the order n ro ,  0  ~ upper triangular and unimodular, i.e., det A ro (z) = const ≠ 0 . In this way, by suitable elementary row and column operations, we can transform (2.18) into the form ~ 0 A r o ( z ) − A13 − A14    0 zI ro − A r o − A 34  0 . ~ (2.19) 0 0 0 A ro ( z )    0 0 0  0

The submatrix ~  A ro ( z) − A13  zI ro − A r o  0  0 0 

− A14   − A 34  ~ A ro (z)

(2.20)

of (2.19) is square of the order n ro + n r o + n ro = n (which is equal to the normal rank of the pencil (2.17)). The determinant of (2.20) is equal to (const ) det(zI ro − A r o ) . This means that the zero polynomial of (2.17) equals det(zI ro − A r o ) . ◊ Example 2.11. Consider a SISO system (2.1) with the matrices 0 1 0  A = 0 1 / 2 1 , 0 0 0

1  b = 0 , 0

c = [0 0 1] .

The system is already in its Kalman form (2.2). The Smith form of P (z) is 1 0  0  0

0

0

1

0

0 z− 0

0

1 2

0 0 0  0

2.8 Exercises

53

and the system has one Smith zero λ = 1 / 2 . On the other hand, the

transfer function g(z) equals zero identically, i.e., g (z) = c(zI − A) −1 b ≡ 0 , and, by virtue of Proposition 2.6, the system is degenerate. ◊

2.8 Exercises 2.8.1. Show that i.d., o.d. and i.o.d. zeros (including multiplicities) of a strictly proper or proper system (2.1) are invariant under similarity transformations in the state space. Find the zero polynomials of the pencils zI − A   C  and [zI − A − B ] .   Hint. Let x' = Hx , det H ≠ 0 , denote a change of coordinates in the state space. For i.d. and o.d. zeros it is sufficient to show that under such a transformation the Smith forms and, consequently, the zero polynomials of zI − A  the pencils [zI − A − B ] and   remain unchanged. To this end,  C  use the following relations

[zI − HAH

−1

]

H −1 0 − HB = H [zI − A − B]   I   0

zI − HAH −1  H 0 zI − A  −1 =   H . −1   0 I   C   CH

For i.o.d. zeros the claim follows from remarks concerning invariance of the characteristic polynomials of diagonal submatrices of the A-matrix in the Kalman form (2.2) of (2.1). Taking a suitable H (see (A8) in Appendix A.1) decompose the system into an unobservable and an observable part. In the new coordinates zI − A   C  takes the form   zI o − A o  0   0

− A12  zI o − A o  C o 

54


zI o − A o  and since the pair ( A o , C o ) is observable,   has full column  Co  rank at any z ∈ C . Applying to this pencil the Hermite row form (see ApzI − A  pendix C) deduce that the zero polynomial of   , which determines  C  all o.d. zeros (including multiplicities) of the system, is equal to χ o (z) = det(zI o − A o ) whereas deg det(zI o − A o ) = n − no = dim X o , where no stands for the rank of the observability matrix of the system and n

X o = h Ker CA l is the unobservable subspace. l =0

Choosing a suitable H decompose the system into a reachable and an unreachable part (see (A4) in Appendix A.1). In the new coordinates the pencil [zI − A − B ] equals zI r − A r  0 

− A12 zI r − A r

− Br  0 

and, after interchanging appropriate columns, it takes the form zI r − A r  0 

− Br 0

− A12  . zI r − A r 

Since the pair ( A r , B r ) is reachable, the pencil [zI r − A r − B r ] has full row rank at any z ∈ C . Applying to this pencil the Hermite column form deduce that the zero polynomial of [zI − A − B ] , which determines all input decoupling zeros (including multiplicities) of the system, equals χ r (z) = det(zI r − A r ) . 2.8.2. Consider a strictly proper or proper SISO system S(A,b,c,d) (2.1) (i.e., m = 1, r = 1 ). Prove the following statement. A number λ ∈ C is an invariant zero of the system if and only if det P (λ) = 0 , where  zI − A − b  . P (z) =  d   c

For a strictly proper SISO system S(A,b,c) (2.1) prove the following equivalence: S(A,b,c) is degenerate if and only if det P (z) ≡ 0 (or equivalently, the transfer-function equals zero identically, i.e., g (z) ≡ 0 ).

2.8 Exercises

55

2.8.3. Consider a square m-input, m-output strictly proper or proper system − B  (2.1) for which the matrix   has full column rank. Prove the followD ing statements: a) λ ∈ C is an invariant zero of the system if and only if det P (λ) = 0 ; b) the system is degenerate if and only if det P (z) ≡ 0 (or, equivalently, det G (z) ≡ 0 ).

x o  Hint. a) Let det P (λ) = 0 . Then there exists a nonzero vector   sat g 

isfying (2.4). Suppose that in this vector is x o = 0 . Then (2.4) implies x o  − B  0 that contradicts the assumption , i.e., . This g = 0 g =   is D 0      g  nonzero. Conversely, if a triple λ, x o ≠ 0, g satisfies (2.4), then, by virtue of the condition x o ≠ 0 , columns of P (λ) are linearly dependent (over C) and, consequently, det P (λ) = 0 . b) Suppose now that det P (z) ≡ 0 . Then det P (λ) = 0 for any λ ∈ C . This means that for any fixed λ ∈ C we can find a x o ≠ 0 and a g ∈ C m such that the triple λ, x o ≠ 0, g satisfies (2.4). Hence the system is degenerate. In order to prove the implication – if the system is degenerate, then det P (z) ≡ 0 – suppose that det P (z) is not identically zero. Then, however, det P (z) is a nonzero polynomial in variable z and, in view of a), the system can not be degenerate (i.e., its invariant zeros are the roots of det P (z) ). Of course, if det P (z) is a nonzero constant, the system has no invariant zeros. The remaining part of the claim b) follows from the well known relation det P (z) = det (zI − A ) det G (z) (see Appendix C). 2.8.4. Consider the system as in Example 2.1. Find all its decoupling zeros. Observe that although λ = 0 is simultaneously an o.d. and an i.d. zero of the system, it is not its i.o.d. zero. Note that this example shows that i.o.d. zeros can not be defined as those complex numbers which are simultaneously i.d. zeros and o.d. zeros. On the other hand, λ = 1 is the only Smith zero of the system (i.e., the o.d. zero λ = 0 is not a Smith zero). Hint. Taking into account relations (2.3) find first proper block partition in the Kalman form of the system. Since g (z) ≡ 0 , we have nro = 0 .

56


The ranks of the reachability and observability matrices are equal respectively to n r = 1 and no = 1 (i.e., n r = n − nr = 2 and n o = n − no = 2 ). Now, from (2.3) we obtain n ro = 1 , n ro = 1 and n ro = 1 . 0 1 0  The change of coordinates x'= H x , where H =  − 1 0 0 , trans 0 − 1 0 forms the system into its Kalman form with the matrices 0 − 1 0 A' = 0 1 1 , 0 0 0

1  b' = 0 , 0

c' = [0 0 − 1] ,

where A' ro = [0] , A' r o = [1] and A' ro = [0] . Then use Corollary 2.1. 1 0 The Smith form of P (z) is  0  0

0 1 0 0 . 0 z − 1 0  0 0 0

0

0

2.8.5. Consider a system (2.1) with the matrices 0  − 3 1 − 1 / 3  − 3 1 0 0 , D= , B= , C= A= .    − 2 / 3  0 0  0  0 0 1 0

Verify that this system is nondegenerate (it has exactly one invariant zero which coincides with the Smith zero). Check that the system matrix has not full normal rank. 1 0 Hint. The Smith form of the system matrix equals  0  0 Use Propositions 2.3 and 2.1 (ii).

0 1

0 0

0 z+ 0

0

2 3

0 0 . 0  0

2.8.6. Consider a minimal system (2.1) with the matrices (Example 2.8)  0 1 / 4 A= , − 2 − 1 

1 0  B= , 0 − 1

C = [1 1] .

2.8 Exercises

57

Show by analyzing (2.4) that this system is degenerate. Find the outputzeroing input and solution corresponding to λ = 0 . Note that Exercises 2.8.5 and 2.8.6 illustrate Remark 2.1. Hint. At any fixed 0 ≠ x1o ∈ R the triple

(i)

 1 o  xo   − x1  λ = −1 , x =  1 o  , g =  2  3  − x1   − x1o   4  o

satisfies (2.4) (i.e., λ = −1 is an invariant zero); moreover, each triple of the form (ii)

1    xo  (λ + ) x1o  1  λ ≠ −1 , x =  o  , g = 4  o   − x1   (λ − 1) x1  o

also satisfies (2.4) (i.e., any complex number λ ≠ −1 is an invariant zero of the system). According to Lemma 2.8, the output-zeroing input and solution generated by the triple 1 o   xo  x 1 λ = 0 , x =  o  , g = 4 1   o  − x1   − x1  o

for k = 0 g , take respectively the form (x o , u o (k )) , where u o (k ) =  0 for k = 1,2,...  x o for k = 0 and x(k ) =  .  for k = 1 , 2 ,... 0 

2.8.7. Consider a system (2.1) with the matrices − 2 / 3 0  , A= − 1  0

1 − 1 B= , 0 0 

C = [1 1] .

Find its Kalman canonical form. Find all invariant zeros of this system. 2.8.8. Show that transmission zeros of a strictly proper or proper system (2.1) (see Definition 2.1 (ii)) are exactly the same objects as the transmission zeros of its transfer-function matrix (see Definition 2.2).

58


2.8.9. Prove Lemma 2.3. 2.8.10. Show that the transformations considered in Lemma 2.3 do not change the Smith form of the system matrix (in consequence, they do not change the zero polynomial). Hint. Use the following relations zI − HAH −1  −1  CH

(i)

 H = 0 rxn

0 nxr   zI − A − B   H −1  I r   C D  0 mxn

0 nxm   I m 

zI − A − BV   zI − A − B   I =   C DV   C D  0 mxn 

(ii)

zI − A − B   I  UC UD = 0    rxn

(iii)

0 nxm  V 

0 nxr  zI − A − B  U   C D 

zI − A − BF − B   zI − A − B   I 0 nxm  =    C + DF D   C D  F I m  

(iv)

(v)

− HB   D 

Denote zI − A − BT(I − DT) −1 C − (B + BT(I − DT) −1 D) r r P ' (z) =  . (I r − DT) −1 C (I r − DT) −1 D  

However (cf., (ii) and (iii) in Appendix C), I m + T(I r − DT) −1 D = (I m − TD) −1 , (I r − DT) −1 D = D(I m − TD) −1 .

Denote F = T(I r − DT) −1 C . Now, we can write  zI − A − BF − B(I m − TD) −1  P ' (z) =   D(I m − TD) −1   C + DF 0 nxm   zI − A − B   I =   −1  . D  F (I m − TD)   C

2.8 Exercises

59

2.8.11. Prove relations (i) and (ii) given in Remark 2.3. 2.8.12. Prove Lemma 2.6. 2.8.13. Prove statements given in Remark 2.4 and Lemma 2.7. 2.8.14. Prove equivalences (i), (ii) and (iii) given in Remark 2.6. 2.8.15. Consider a proper rational transfer-function g(z) of degree n in its N(z) (i.e., polynomials N (z) and D(z) are relairreducible form g( z) = D( z) tively prime and deg D(z) = n ) and assume that D(z) is monic (i.e., the coefficient associated with the n-th power of z equals 1). Moreover, let S( A, b, c, d) (with d = 0 or d ≠ 0 ) be a minimal (irreducible) realization of g(z) . Show that  zI − A − b  det P (z) = det  = det (zI − A ) g (z) and N (z) = det P (z) . d   c

2.8.16. Consider a proper SISO system S(A,b,c,d) ( d ≠ 0 ). Show that a number λ ∈ C is an invariant zero of the system if and only if λ ∈σ( A − bd −1 c) . Moreover, show that (before possible cancellation of

det(zI − ( A − bd −1 c)) as well as that, after det(zI − A ) possible cancellation of common factors,

common factors) g( z) = d

g (z) = d

det (zI r o − ( A r o − b r o d −1 c ro )) . det(z I ro − A r o )

Hint. Evaluate the determinant of the system matrix P (z)

as

−1

det P (z) = d det (zI − ( A − bd c)) (cf., e.g., [18]). Then use the Kalman canonical form (2.2) of the system.

2.8.17. Consider a SISO system S(A,b,c) (2.1) for which g (z) ≠ 0 (or, equivalently, det P (z) ≠ 0 ). Let g (z) = c(zI − A ) −1 b =

cA l b cb cAb + + ...+ + ... z z2 z l +1

60


be the series expansion of the transfer-function in a neighbourhood of the N(z) point z = ∞ . Write g( z) = , where N (z) = β1 z n−1 + . . .+ β n and D(z) D(z) = z n + α1 z n−1 +. . .+ α n . Show that the following relation holds 0 0 0   1   β1   cb      cAb cb 0 .   α1  β 2   = .  . . 0 .   .  cb      n −1 cA b . cAb cb  α n −1  β n 

Observe that if for some 0 ≤ ν ≤ n − 1 is cb =. . .= cA ν −1b = 0 and cA ν b ≠ 0 , then β1 =. . .= β ν = 0 , β ν +1 = cA ν b and g(z) may be written as β z n−(ν +1) +. . . + β n , where lim z ν +1 g (z) = β ν +1 (the number g (z) = ν +1 n n −1 z →∞ +. . .+ α n z + α1 z ν + 1 is known as the relative degree of g(z) ).

2.8.18. Consider a SISO system S(A,b,c) (2.1) with its first nonzero Markov parameter cA ν b (i.e., cb =. . .= cA ν −1b = 0 and cA ν b ≠ 0 ). Define the following n x n matrix K ν := I − b(cA ν b) −1 cA ν . Show that K ν has the following properties:

(i)

K ν2 = K ν

(ii)

Σ ν := {x : K ν x = x} = Ker(cA ν ) ,

dim Σ ν = n − 1 ;

Ω ν := {x : K ν x = 0} = Im b ,

dim Ω ν = 1 ;

(i.e., K ν is projective);

C n (R n ) = Σν ⊕ Ων ;

(iii)

Kνb = 0 ,

cA ν K ν = 0 ,

cA l for 0 ≤ l ≤ ν c(K ν A ) l =  .  0 l for 1 ≥ ν + 

2.8.19. Consider a SISO system S(A,b,c) (2.1) and let cA ν b denote its first nonzero Markov parameter and let P(z) denote the system matrix. Prove that, at the notation used in Exercise 2.8.18, the following relations hold: (i)

det (zI − K ν A ) = (cA ν b) −1 z ν +1 det P (z) ,

2.8 Exercises

(ii)

g (z) = (cA ν b) z − (ν +1)

61

det( zI − K ν A ) (before possible cancelladet(zI − A )

tion of common factors). Observe that K ν A has ν + 1 zero eigenvalues (those which correspond

to the factor z ν +1 on the right-hand side of (i)) as well as that the remaining n − (ν + 1) eigenvalues of K ν A (those which are the roots of the polynomial det P (z) ) represent all invariant zeros of the system. In view

of (ii), the zero eigenvalues of K ν A corresponding to the factor z ν +1 in (i) will be treated as representing zeros at infinity of the system and of its transfer-function g(z) (cf., Exercise 2.8.21). Note that if ν = n − 1 , then K ν A is nilpotent (i.e., all its eigenvalues are equal to zero). Hint. (i) Use the equality g (z) = c(zI − A ) −1 b = z − ν c(zI − A ) −1 A ν b and appropriate identities for determinants (see Appendix C; cf., also Exercise 2.8.25). Then use Exercise 2.8.2. 2.8.20. Consider a SISO system S(A,b,c) under the assumptions of Exercise 2.8.18. Using the Kalman form (2.2) and the definition of K ν given in Exercise 2.8.18 show that if S(A,b,c) is expressed in the form (2.2), then K ν and K ν A take respectively the form I r o  0 K ν =  0   0  A ro  0 KνA =   0   0

x

0

K νro

0

0 0

I ro 0

x   x  , 0   I r o 

x

A13

K νr o A r o 0

0 A ro

0

0

x  x  , A 34   A r o 

where K νr o = I r o − b r o (c ro A νr o b r o ) −1 c r o A νr o is projective of the rank n ro − 1 ( nro is the dimension of the minimal subsystem S(A ro , b r o , c ro ) ). 2.8.21. Employing Exercise 2.8.20 show that the equality (ii) in Exercise 2.8.19 can be written as

62


g (z) = (c ro A νr o b ro ) z − (ν +1)

det (zI ro − K νr o A ro ) , det (zI r o − A r o )

where the polynomials z −(ν +1) det (zI r o − K ν A r o ) (of the degree n r o − (ν + 1) ) and det (zI r o − A r o ) (of the degree n ro ) are relatively ro

prime. Observe that K νr o A r o has ν + 1 zero eigenvalues (treated as representing zeros at infinity of g(z)) and n r o − (ν + 1) eigenvalues which represent all transmission zeros of g(z). Hint. For the minimal subsystem use the relation det (zI ro − K ν A r o ) = (c r o A νr o b r o ) −1 z ν +1 det Pro (z) , ro

where Pr o (z) denotes the system matrix for this subsystem. 2.8.22. In S(A,B,C) (2.1) let CB = ... = CA n −1B = 0 . Show that all Markov parameters are zero. n −1

Hint. Use the Cayley-Hamilton theorem, i.e., write A n = ∑ α i A i . i =0

2.8.23. Consider a square ( m = r ) strictly proper or proper system (2.1) in which the system matrix P(z) has full normal rank (i.e., det P (z) is a nonzero polynomial). Show that det P (z) is equal (up to a multiplicative nonzero constant) to the zero polynomial (which is obtained from the Smith canonical form Ψ (z) of P(z) as a product of invariant factors – see Chap. 1, Sect. 1.1). Then show that a complex number is an invariant zero of the system if and only if it is a root of the zero polynomial. Hint. The first claim follows from the relation P (z) = U (z)Ψ(z)V (z) , where U (z) and V (z) are unimodular matrices, by taking the determinants of both sides. In order to prove the second claim, it is sufficient to show that λ ∈ C is an invariant zero of the system ⇔ det P (λ ) = 0 . To this end, observe first − B  that   in P(z) has full column rank and then use Exercise 2.8.3. D

2.8 Exercises

63

2.8.24. In a strictly proper or proper system (2.1) (not necessarily square) let the system matrix P(z) have the full normal column rank n + m . Show that: λ ∈ C is an invariant zero of the system ⇔ λ is a root of the zero polynomial. − B  Hint. Observe first that the submatrix   of P(z) has full column D rank. Write P (z) = U (z)Ψ(z)V (z) , where Ψ (z) has the Smith form and U (z) and V (z) are unimodular. Let λ be an invariant zero of the system.

 x o  0  Then (cf., Definition 2.1) P (λ )  =   where x o ≠ 0 . Thus we can  g  0 write

(i)

x o  0 ( ) ( ) ( ) = λ λ λ U Ψ V  , 0    g 

x o  where matrices U (λ ) and V (λ ) are nonsingular. Denote v o = V (λ )  .  g  0  It is clear that v o ≠ 0 . Now, from (i) we obtain Ψ (λ ) v o =   . This 0  means that Ψ (λ ) has rank (over the field of complex numbers) less than n + m and, consequently, λ must be a root of the zero polynomial. Conversely, if λ is a root of the zero polynomial, then rank Ψ(λ) < n + m and, consequently, rank P(λ) < n + m . This means

 x o  0  x o  that there exists a nonzero vector   such that P (λ )  =   . Since  g  0  g  − B  o  D  has full column rank, we must have x ≠ 0 (i.e., λ is an invariant   zero of the system).

2.8.25. Consider a square mxm system S(A,B,C) (2.1). Let the first nonzero Markov parameter CA ν B , 0 ≤ ν ≤ n − 1 , be nonsingular and let P(z) denote the system matrix. Define K ν := I − B(CA ν B) −1 CA ν .

64


Show that the following relation holds det P (z) = z − m (ν +1) det (CA ν B) det( zI − K ν A) .

Hint. Consider the identity zI − A − B   I 0  zI − A − BF − B  , =  C 0  F I m   C 0  

where F is a mxn matrix. Putting F := −(CA ν B ) −1 CA ν +1 and then taking the determinant of both sides of this identity, we obtain det P (z) = det(zI − K ν A) det (C(zI − K ν A) −1 B ) .

Then use the relations C(K ν A) l = CA l for 0 ≤ l ≤ ν and C(K ν A) l = 0 for l ≥ ν + 1 (see Lemma 3.1, the proof of point (vi)) and observe that ∞

C(zI − K ν A) −1 B = ∑

l =0

C(K ν A) l B z l +1

ν

CA l B

l =0

z l +1

= ∑

=

CA ν B z ν +1

.

3 A General Solution to the Output-Zeroing Problem

In this chapter we discuss the problem of zeroing the output in an arbitrary linear discrete-time system (2.1) with a nonvanishing transfer-function matrix. In Chapter 2 we introduced and discussed the concept of invariant zeros and its relationship with the output-zeroing problem. It has been shown in particular that to any invariant zero we can assign an outputzeroing input. In this chapter we answer the following, more general, question. Find a general expression for output-zeroing inputs which in a compact form could convey information about invariant zeros and their action in the system. More precisely, we want to characterize in a simple manner all the possible real-valued inputs and initial states which produce the identically zero system response. Strictly proper ( D = 0 ) and proper ( D ≠ 0 ) systems will be discussed separately. The presented approach is based on the Moore-Penrose pseudoinverse of the first nonzero Markov parameter. For strictly proper systems (2.1) this parameter will be denoted by CA ν B .

3.1 Preliminary Characterization of Invariant Zeros We denote by M + the Moore-Penrose pseudoinverse of matrix M [5, 18]. Recall (see Appendix B) that for a given rxm real matrix M of rank p, a factorization M = M1M 2 with a rxp matrix M1 and a pxm matrix M 2 is called the skeleton factorization of M (note that in any such factorization M1 has full column rank and M 2 has full row rank). Then M + is uniquely determined (i.e., independently upon a particular choice of

matrices M1 and M 2 ) as M + = M +2 M1+ , where M1+ = (M1T M1 ) −1 M1T and M 2+ = M T2 (M 2 M T2 ) −1 . Moreover, the relations M = MM + M and M + = M + MM + are valid. In particular, if M is square and nonsingular, then M + = M −1 .


66


3.1.1 Invariant Zeros in Proper Systems

A sufficient condition for a complex number to be an invariant zero of a proper system (2.1) can be formulated as follows. Proposition 3.1. [70, 73] A number λ ∈ C is an invariant zero of a proper system (2.1) if λ ∈ σ( A − BD + C) (i.e., λ is an eigenvalue of A − BD + C )

and there exists an associated with λ eigenvector x o satisfying x o ∈ Ker(I r − DD + )C . Then x o and g = −D + Cx o constitute state-zero and input-zero direction associated with the invariant zero λ .

Proof. We are to show that the triple λ, x o ≠ 0, g determined in the proposition satisfies (2.4). However, taking g = −D + Cx o , we can write λ x o = ( A − BD + C)x o (which follows from the fact that x o is an eigen-

vector associated with λ ) as (λI − A )x o = Bg and (I r − DD + )Cx o = 0 as Cx o + Dg = 0 . ◊ A necessary condition for a number λ ∈ C to be an invariant zero of a proper system (2.1) we can formulate as follows. Proposition 3.2. [70, 73] If λ is an invariant zero of a proper system (2.1), i.e., there exists a triple λ, x o ≠ 0, g which satisfies (2.4), then for this triple the following relations hold λ x o − ( A − BD + C)x o = Bg1 ,

and

g = g1 + g 2 ,

x o ∈ Ker(I r − DD + )C

where g1 ∈ Ker D ,

g 2 = −D + Cx o ∈ Im D T .

Proof. We can write g = g1 + g 2 with g1 and g 2 defined respectively as g1 := (I m − D + D)g and g 2 := D + Dg . From the definition of g 2 and from the

second

equality

in

(2.4)

(i.e.,

Cx o + Dg = 0 )

we

obtain

Bg 2 = BD + Dg = −BD + Cx o . Now, the first equality in (2.4) (i.e.,

λx o − Ax o = Bg ) can be written as λ x o − ( A − BD + C)x o = Bg1 . The definitions of g1 , g 2 imply Dg1 = 0 and Dg 2 = Dg . Premultiplying the

second equality in (2.4) by D + , we get g 2 = −D + Cx o and, consequently,

3.1 Preliminary Characterization of Invariant Zeros

Cx o = −Dg = −Dg 2 = −D(−D + Cx o ) = DD + Cx o +

o

which

means

67

that

T

x ∈ Ker(I r − DD )C . Finally, g 2 ∈ Im D follows from orthogonal decomposition of domain and codomain of D (see Appendix B.3). ◊ 3.1.2 Invariant Zeros in Strictly Proper Systems

We introduce first a projective (idempotent) matrix that plays a crucial role in the algebraic characterization of invariant zeros and output-zeroing inputs for strictly proper systems. Definition 3.1. [63] In S(A,B,C) (2.1) let CA ν B , where 0 ≤ ν ≤ n − 1 , denote the first nonzero Markov parameter (i.e., CB = . . . = CA ν −1B = 0 and CA ν B ≠ 0 ) and let rank CA ν B = p ≤ min{m, r} . Define the following nxn matrix K ν := I − B(CA ν B) + CA ν . ◊

(3.1)

Recall that if CA ν B = H1H 2 , where H1 ∈ R r x p , H 2 ∈ R p x m , denotes a skeleton factorization of CA ν B , then (CA ν B) + = H +2 H1+ with H 1+ = (H1T H1 ) −1 H1T and H +2 = H T2 ( H 2 H T2 ) −1 . The following lemma characterizes some useful algebraic properties of K ν .

Lemma 3.1. [73, 80] The matrix K ν (3.1) has the following properties: (i)

K ν2 = K ν

(ii)

Σ ν := {x : K ν x = x} = Ker(H1T CA ν ) , dim Σ ν = n − p ;

(iii)

Ω ν := {x : K ν x = 0} = Im(BH T2 ) ,

(iv)

C n (R n ) = Σ ν ⊕ Ων ;

(v)

K ν BH T2 = 0 ,

(vi)

H T CA l for 0 ≤ l ≤ ν ; H1T C(K ν A) l =  1  for l ≥ ν + 1 0

(i.e., K ν is projective);

dim Ω ν = p ;

H1T CA ν K ν = 0 ;

68


(vii)

C(K ν A) l = CA l

for

0 ≤ l ≤ ν.

Proof. Define B':= BH T2 , C':= H1T C . Then the pxp matrix C'A ν B' is nonsingular. This implies in turn rank B' = p , rank C' = p and p ≤ n . Note: In the proof of the above implications we make use of some simple observation. Consider mappings f :X → Y, g:Y → Z and their superposition g f : X → Z . Then, if g f is onto (i.e., the image of X under g f equals Z ), then g is onto (i.e., g (Y ) = Z ). On the other hand, if g f is injective (i.e., one-to-one), then f is also injective. ◊ At the first stage of the proof we define an auxiliary matrix K ' ν := I − B'(C'A ν B ' ) + C'A ν (in which, by virtue of nonsingularity of

C' A ν B' , is (C' A ν B' ) + = (C' A ν B' ) −1 ) and then we verify that K 'ν = K ν . In fact, we can write

K ν' = I − B' (C' A ν B' ) + C' A ν = I − BH T2 (H1T CA ν BH T2 ) + H1T CA ν = I − BH T2 (H1T H1H 2 H T2 ) + H1T CA ν = I − BH T2 (H 2 H T2 ) −1 (H1T H1 ) −1 H1T CA ν = I − BH +2 H1+ CA ν = I − B(CA ν B) + CA ν = K ν .

We proceed with the proof for K 'ν and then we use the relation K'ν = K ν . (i) The simple proof is left to the reader (see Exercise 3.3.1). (v)

K ν BH T2 = K 'ν B' = 0

H1T CA ν K ν = C' A ν K 'ν = 0 .

(ii) We have Ker(H1T CA ν ) = Ker(C' A ν ) and Σ ν = {x : K 'ν x = x} . Hence x ∈ Σ ν ⇔ K 'ν x = x ⇔ B' (C'A ν B' ) −1 C'A ν x = 0 . Since B' has full column rank, the last equality is fulfilled if and only if C'A ν x = 0 .

Thus we have proved x ∈ Σ ν ⇔ x ∈ Ker(C'A ν ) . Finally, since C' A ν has full row rank, we can write dim Σ ν = dimKer(C'A ν ) = n − p . (iii) By virtue of the definition of Ω ν (i.e., Ω ν := {x: K ν x = 0} ), the following equivalences hold x ∈ Ω ν ⇔ K 'ν x = 0 ⇔ B'(C'A ν B' ) −1 C' A ν x = x .

3.1 Preliminary Characterization of Invariant Zeros

69

Thus, if x ∈ Ω ν , then x ∈ ImB' , i.e., Ω ν ⊆ ImB' = Im(BH T2 ) . However, from the proof of (v) we have K 'ν B' = 0 which means that each column of B' = b'1 . . . b' p satisfies K ' ν b'i = 0 for i = 1,..., p and, conse-

[

]

quently, belongs to Ω ν . This means in turn that any linear combination of columns of B' is in Ω ν . Thus we have Im(BH T2 ) = ImB' ⊆ Ω ν . Finally, since Ω ν is spanned by columns of B' and B' has full column rank, we have dim Ω ν = rankB' = p . (iv) This is an immediate consequence of (i) and of the definitions of Σ ν and Ω ν . (vi) It is sufficient to observe that CB = . . . = CA ν −1B = 0 implies

C'B' = . . . = C'A ν −1 B' = 0 . Then, employing the definition of K 'ν and

C'B' = . . . = C'A ν −1 B' = 0 , we see that C' (K ' ν A ) l = C'A l for 0 ≤ l ≤ ν .

For l = ν + 1 we can write C' (K ' ν A ) ν (K ' ν A ) = C' A ν K ' ν A = 0 , for, by

virtue of (v), we have C'A ν K ' ν = 0 .

(vii) In the proof we use (3.1) and CB = . . . = CA ν −1B = 0 . ◊

Remark 3.1. Since the Moore-Penrose pseudoinverse of a matrix is determined uniquely, the properties of K ν listed in Lemma 3.1 do not depend upon a particular choice of matrices H1 , H 2 in the skeleton factorization of CA ν B . ◊ In order to derive some algebraic conditions characterizing invariant zeros, we show first that any state-zero direction has to belong to the subspace Σ ν (see Lemma 3.1). Lemma 3.2. [73, 80] If a triple λ, x o ≠ 0, g satisfies (2.4), then xo ∈

ν

Ker CA l ⊆ Σ ν ⊂ C n (i.e., x o ∈ Σ ν ) and CAν Bg = −CA ν +1x o .

l =0

Proof. The proof follows by successive multiplication of the equality λ x o − Ax o = Bg from the left by C, . .. , CA ν and by using Cx o = 0 . ◊

A sufficient condition for a complex number to be an invariant zero can be formulated as follows.

70


Proposition 3.3. [73, 80] If in a strictly proper system (2.1) λ ∈ σ(K ν A) and there exists an eigenvector x o of K ν A associated with λ such that x o ∈ Ker C , then λ is an invariant zero of the system. Moreover, x o and g = −(CA ν B) + CA ν +1x o constitute the state-zero and input-zero direction associated with λ .

Proof. We shall show that the triple λ, x o ≠ 0, g determined in the proposition satisfies (2.4). Using (3.1), we can write λ x o − K ν Ax o = 0 (which follows from the fact that λ ∈ σ(K ν A ) and x o is an associated

eigenvector) as λ x o − Ax o = Bg , where g = −(CA ν B) + CA ν +1x o . The equality Cx o = 0 follows immediately from the assumptions. In this way

the triple λ, x o ≠ 0, g satisfies Definition 2.1 (i). ◊ As a necessary condition for a complex number to be an invariant zero we use the following. Proposition 3.4. [73, 80] If in a strictly proper system (2.1) a triple λ, x o ≠ 0, g satisfies (2.4), then

(i)

λ x o − K ν Ax o = Bg1 ,

K ν Ax o − Ax o = Bg 2 ,

Cx o = 0 ,

where g = g1 + g 2 , g1 ∈ KerCA ν B , g 2 ∈ Im(CA ν B) T and g1 , g 2 are uniquely determined by g . Moreover, (ii)

Bg1 ∈ Σ ν ,

Bg 2 ∈ Ω ν

and

g 2 = −(CA ν B) + CA ν +1x o .

Proof. In the proof we use the properties of K ν listed in Lemma 3.1. For g we take the decomposition g = g1 + g 2 , where g1 and g 2 are defined as g1 := (I m − (CA ν B) + CA ν B)g and g 2 := (CA ν B) + CA ν Bg . Then Bg1 = K ν Bg and Bg 2 = (I − K ν )Bg which imply K ν Bg1 = Bg1 and K ν Bg 2 = 0 . Now, λx o − Ax o = Bg can be written as

(iii)

(λI − K ν A )x o + (K ν − I) Ax o = Bg1 + Bg 2 ,

with λx o − K ν Ax o and Bg1 in Σ ν and (K ν − I ) Ax o and Bg 2 in Ω ν .

3.2 Output-Zeroing Inputs

71

The fact that λx o − K ν Ax o lies in Σ ν is a consequence of Lemma 3.1. More precisely, since x o ∈ Σ ν (see Lemma 3.2), we can write K ν (λx o − K ν Ax o ) = λ x o − K ν Ax o . On the other hand, because C n ( R n ) is a direct sum of Σ ν and Ω ν (see Lemma 3.1 (iv)), the decomposition in (iii) is unique. From the uniqueness of this decomposition we obtain the first two equalities in (i). The expression for g 2 in (ii) follows

from the equalities g 2 = (CA ν B) + CA ν Bg and CA ν Bg = −CA ν +1x o (see Lemma 3.2). Finally, taking into account the definition of g1 given at the beginning of the proof, we observe that CA ν Bg1 = 0 , i.e., g1 ∈ KerCA ν B . The proof of the relation g 2 ∈ Im(CA ν B) T is left to the reader as an exercise. ◊

3.2 Output-Zeroing Inputs In this section we will need the following result which is obtained by a slight modification of the proof of Lemma B.1 (see Appendix B). Lemma 3.3. Consider the equation (i)

Mz (k ) = b(k ) , k ∈ N ,

where M is a rxm real and constant matrix of rank p and b(.) : N → R r is a given sequence, and suppose that this equation is solvable in the class of all sequences z (.) : N → R m (i.e., there exists at least one solution). Then any solution of (i) can be expressed as z (k ) = z ∗o (k ) + z h ( k ) , where z ∗o (k ) = M + b(k ) and z h (k ) is a solution of the equation Mz (k ) = 0 .

Proof. Let M = M1M 2 , where M1 ∈ R rxp and M 2 ∈ R pxm , be a skeleton factorization of M and let z o (k ) be a solution of (i), i.e., (ii)

Mz o (k ) = M1 (M 2 z o (k )) = b(k ) for all k ∈ N .

We denote y o (k ) = M 2 z o (k ) and then we write (ii) as (iii)

M1y o (k ) = b(k ) for all k ∈ N .

72


Since M1 has full column rank, y o (k ) is the unique solution of the equation M1y (k ) = b(k ) . Now, we can discuss the equation (iv)

M 2 z ( k ) = y o (k ) ,

where y o (k ) is treated as known. The set of solutions of (iv) is not empty (since z o (k ) satisfies (iv)). Moreover, we observe that the sequence z ∗o (k ) := M +2 y o (k )

(v)

is a solution of (iv) since we have for all k ∈ N (vi)

M 2 z ∗o (k ) = M 2 M 2+ y o (k ) = M 2 M T2 (M 2 M T2 ) −1 y o (k ) = y o (k ) .

On the other hand, we can observe that (v) is also a solution of (i). In fact, by virtue of (vi) and (iii), we can write for all k ∈ N Mz∗o (k ) = M1M 2z∗o (k ) = M1y o (k ) = M1M 2z o (k ) = Mz o ( k ) = b(k ) .

Now, we shall show that z ∗o (k ) in (v) does not depend upon a particular choice of matrices M1 and M 2 in the skeleton factorization of M. To this end, we write (v) as (vii)

z ∗o (k ) = M 2+ y o (k ) = M T2 (M 2 M T2 ) −1 y o (k ) .

On the other hand, y o (k ) , as the unique solution of the equation M1y (k ) = b(k ) , can be written in the form (viii)

y o (k ) = (M1T M1 ) −1 M1T b(k ) .

Now, introducing (viii) into the right-hand side of (vii), we obtain (ix)

z ∗o (k ) = M +2 M1+ b(k ) = M + b(k ) .

In this way we have shown that if (i) is solvable, then z ∗o (k ) = M + b(k ) is a solution of (i). Of course, the difference of any two solutions of (i) is a solution of the homogeneous equation. Consequently, any fixed solution z (k ) of (i) can be expressed as z (k ) = z ∗o (k ) + (z (k ) − z ∗o (k )) = z ∗o (k ) + z h (k ) ,

where z h (k ) satisfies Mz h (k ) = 0 for all k ∈ N . ◊


73

3.2.1 Output-Zeroing Inputs for Proper Systems

A general characterization of output-zeroing inputs and the corresponding solutions is given in the following result. Proposition 3.5. [79] Let (x o , u o (k )) be an output-zeroing input for a proper system (2.1) and let x o (k ) denote the corresponding solution. Then x o ∈ Ker(I r − DD + )C and u o (k ) , k ∈ N , is of the form

u o (k ) = −D + C( A − BD + C) k x o + k −1

− D + C[ ∑ ( A − BD + C) k −1−l Bu h (l )] + u h (k ) ,

(3.2)

l =0

for some sequence u h (.) ∈ U satisfying Du h (k ) = 0 for all k ∈ N , and x o (k ) , k ∈ N , has the form k −1

x o (k ) = ( A − BD + C) k x o + ∑ ( A − BD + C) k −1− l Bu h (l ) .

(3.3)

l =0

Moreover, x o (k ) ∈ Ker (I r − DD + )C for all k ∈ N . Proof. Let (x o , u o (k )) be an output-zeroing input for (2.1) and let x o (k ) denote the corresponding solution (with the initial state x o (0) = x o ). Then for all k ∈ N we have the equalities x o (k + 1) = Ax o ( k ) + Bu o (k ) 0 = Cx o (k ) + Du o (k )

and

x o ( 0) = x o .

(3.4)

Consider the equation Du(k ) = − Cx o (k )

(3.5)

with an unknown sequence u(.) ∈ U. Since u o (k ) satisfies (3.5), therefore, by virtue of Lemma 3.3, u o (k ) can be written as u o (k ) = −D + Cx o (k ) + u h (k ) ,

(3.6)

where u h (k ) is some sequence satisfying Du h (k ) = 0 for all k ∈ N . Because (x o , u o (k )) is assumed to be known, hence, by the uniqueness of

74


solutions, x o (k ) is known and u h (k ) may be also treated as a known sequence which is uniquely determined by (3.6). Now, introducing (3.6) into the second equality of (3.4), we obtain DD + Cx o (k ) = Cx o (k ) , i.e., x o (k ) ∈ Ker(I r − DD + )C for all k ∈ N . On the other hand, introducing (3.6) into the first equality of (3.4), we obtain x o ( k + 1) = ( A − BD + C)x o (k ) + Bu h (k ),

k∈N

(3.7)

x o ( 0) = x o

and, consequently, the only sequence x o (k ) which satisfies (3.7) has the form (3.3). Finally, (3.2) follows from (3.3) and (3.6). ◊ Remark 3.2. Naturally, Proposition 3.5 does not tell us whether the outputzeroing inputs exist. However, if the set of invariant zeros is not empty, for each such zero there exists, as we know from Lemma 2.8, an outputzeroing input which in turn may be characterized as in Proposition 3.5. In fact, suppose that λ ∈ C is an invariant zero of (2.1). This means that a triple λ, x o ≠ 0, g satisfies (2.4) and, consequently (see Lemma 2.7), the ~ (k ) = λk g , when applied to (2.1) (treated now as a complex sysinput u o ~ (k ) = 0 for all x o (k ) = λk x o and ~ y (k ) = C~ x o ( k ) + Du tem) at x o , yields ~ o ~ ~ (k ) k ∈ N . We show below that the inputs Re u (k ) and Im u o

o

~ (k )) (appearing in Remark 2.4 in the output-zeroing inputs (Re x , Re u o o ~ and (Im x , Im u (k )) ) can be expressed in the form (3.2). o

o

According to Proposition 3.2, the input-zero direction g can be decomposed as g = g1 + g 2 , where g1 ∈ KerD and g 2 = −D + Cx o . Therefore we ~ (k ) as can write u o (i)

~ ( k ) = λk g = λk g + λk g = −λk D + Cx o + λk g u o 2 1 1 , k∈N , + ~ ~ (k ) = −D Cx (k ) + u o

h

~ (k ) := λk g , k ∈ N . Because u ~ (k ) = λk g and ~ where u x o (k ) = λk x o o h 1 satisfy the state equation in (2.1), i.e.,

(ii)

~ ~ (k ) for all k ∈ N and ~ x o (k + 1) = A~ x o ( k ) + Bu x o ( 0) = x o , o

hence, introducing the right-hand side of (i) into (ii), we obtain the equalities


(iii)

75

~x (k + 1) = ( A − BD + C)~x (k ) + Bu ~ (k ) , k ∈ N , ~ x o ( 0) = x o . o o h

This means, via the uniqueness of solutions (cf., Remark 2.3), that ~ x o (k ) = λk x o = ( A − BD + C) k x o

(iv)

for all k ∈ N .

k −1 ~ (l ) + ∑ ( A − BD + C) k −1−l Bu h l =0

Introducing the right-hand side of (iv) into the right-hand side of (i) and then taking the real part, we obtain the desired result, i.e., ~ (k ) = −D + C( A − BD + C) k Re x o + Re u o

(v)

k −1 ~ (l )] + Re u ~ (k ). − D + C[ ∑ ( A − BD + C) k −1−l B Re u h h l =0

~ (k )) , we proceed analogAs for the output-zeroing input (Im x o , Im u o ously. ◊

Corollary 3.1. Let (x o , u o (k )) be an output-zeroing input for a proper system (2.1) and let x o (k ) denote the corresponding solution. Then: If B(I m − D + D) = 0 , then x o (k ) = ( A − BD + C) k x o . Moreover, the pair (x o , u ∗o (k )) , where u ∗o (k) = −D + C( A − BD + C) k x o , is also

(i)

output-zeroing and yields the solution x o (k ) = ( A − BD + C) k x o . (ii)

If D has full column rank, then u o (k ) = −D + C( A − BD + C) k x o

and x o (k ) = ( A − BD + C) k x o . Proof. (i) To the state equation of (2.1) introduce the input u *o (k ) = −D + Cx o (k )

(3.8)

at the initial state x o . In other words, consider the initial value problem x(k + 1) = Ax(k ) + Bu ∗o (k ),

x ( 0) = x o ,

k∈N .

(3.9)

Introducing (3.8) into (3.9) and taking into account the first equality of (3.4), we can write x(k + 1) − x o (k + 1) = A (x(k ) − x o (k )) + ( A − BD + C)x o (k ) − x o ( k + 1).

(3.10)

76


However, in view of (3.4), the last two terms on the right-hand side of (3.10) can be written as ( A − BD + C)x o (k ) − x o (k + 1) = Ax o (k ) − x o (k + 1) + − BD + (−Du o (k )) = −B(I m − D + D)u o (k ).

(3.11)

At B(I m − D + D) = 0 from x o (0) = x o and from (3.11) it follows that x o (k ) = ( A − BD + C) k x o .

(3.12)

This ends the proof of the first claim in (i). Moreover, from (3.8) and (3.12) we infer that u ∗o (k) = −D + C( A − BD + C) k x o .

(3.13)

Now, setting z (k ) = x(k ) − x o (k ) and taking into account (3.11), we replace (3.10) by the initial value problem z (k + 1) = Az (k ) − B(I m − D + D)u o (k ),

z ( 0) = 0 ,

k∈N .

(3.14)

At B(I m − D + D) = 0 the unique solution of (3.14) is z (k ) ≡ 0 which means that the unique solution x(k ) of (3.9) satisfies, for all k ∈ N , x(k ) = x o (k ) = ( A − BD + C) k x o .

(3.15)

In order to show that at B(I m − D + D) = 0 the pair (x o , u ∗o (k )) is an output-zeroing input for (2.1), we use (3.4), (3.8) and the relations x(k ) = x o (k ) and DD + D = D , and for all k ∈ N we obtain y (k ) = Cx(k ) + Du ∗o (k ) = Cx o (k ) − D(D + Cx o (k )) = Cx o (k ) + DD + Du o (k ) = 0.

(3.16)

This ends the proof of the second claim in (i). (ii) If D has full column rank (i.e., D + D = I m ), then (3.8) is the unique solution of (3.5) and, consequently, u o (k ) = u ∗o (k ) for all k ∈ N . ◊ Remark 3.3. Although the condition B(I m − D + D) = 0 does not imply in general that u ∗o (k ) = u o (k ) for all k ∈ N , it implies, however, that the


77

inputs u o (k ) and u ∗o (k ) applied at the initial state x o affect the state equation of (2.1) in the same way. In fact, we then have Bu ∗o (k ) − Bu o (k ) = −B( D + Cx o (k )) − Bu o (k ) = BD + Du o (k ) − Bu o (k ) = −B(I m − D + D)u o (k ) = 0.

The relation Du ∗o (k ) − Du o (k ) = 0 for all k ∈ N is clear (see (3.16)). ◊ When D has full row rank, the necessary condition given by Proposition 3.5 becomes also sufficient. Proposition 3.6. [79] In a proper system (2.1) let D have full row rank. Then (x o , u o (k )) is an output-zeroing input if and only if u o (k ) has the form (3.2), where x o ∈ R n and u h (.) is an element of U satisfying Du h (k ) = 0 for all k ∈ N . Moreover, a solution corresponding to (x o , u o (k )) has the form (3.3).

Proof. The assumption rank D = r implies DD + = I r . We show first that (3.2) applied to (2.1) at the initial state x o gives a solution of the form (3.3). To this end, in view of the uniqueness of solutions, it is sufficient to check that (3.2) and (3.3) satisfy the state equation. Next, introducing (3.2) and (3.3) to the output equation, we obtain y (k ) = Du h (k ) . This shows that if x o ∈ R n and u o (k ) is as in (3.2) (at an arbitrary u h (k ) ∈ KerD ), then (x o , u o (k )) is an output-zeroing input. The converse implication follows immediately from Proposition 3.5. ◊ A more detailed characterization of the output-zeroing problem than that obtained in Corollary 3.1 (ii) is given by the following result. Proposition 3.7. [79] In a proper system (2.1) let D have full column rank. Then (x o , u o (k )) is an output-zeroing input if and only if n −1

+ + l x o ∈ S cl D := h Ker{(I r − DD )C( A − BD C) }

(3.17)

u o (k ) = −D + C( A − BD + C) k x o .

(3.18)

l =0

and

78


Moreover, the corresponding solution equals x o (k ) = ( A − BD + C) k x o

(3.19)

cl and is contained in the subspace S cl D (3.17), i.e., x o ( k ) ∈ S D for k ∈ N .

Proof. If (x o , u o (k )) is an output-zeroing input, then, as is known from Corollary 3.1 (ii), u o (k ) has the form (3.18) and x o (k ) is as in (3.19). So we need to show (3.17) and that x o (k ) ∈ Scl D for all k ∈ N . However, by assumption, employing (3.18) and (3.19), we can write (i)

0 = y (k ) = Cx o ( k ) + Du o (k ) = (I r − DD + )Cx o (k ) = (I r − DD + )C( A − BD + C) k x o

for k ∈ N .

From (3.19) and (i) we obtain for all k ∈ N (ii)  (I r − DD + )Cx o (k + 0) = (I r − DD + )Cx o (k ) = 0   (I r − DD + )Cx o (k + 1) = (I r − DD + )C( A − BD + C)x o (k ) = 0  .   .  + + + n −1  (I r − DD )Cx o (k + (n − 1)) = (I r − DD )C( A − BD C) x o (k ) = 0 ,

i.e., x o (k ) ∈ Scl D for all k ∈ N . Substituting k = 0 in (ii), one obtains (3.17). In order to prove the converse implication, we have to show that any pair (x o , u o (k )) such that x o ∈ Scl D and u o ( k ) has the form (3.18) is an output-zeroing input and produces a solution of the form (3.19). To this end, we check first that (3.18) and (3.19) satisfy the state equation of (2.1). Then we observe that the system response corresponding to the input (3.18), when applied to the system at the initial state x o , is equal to (iii)

y (k ) = (I r − DD + )C( A − BD + C) k x o .

From (iii) and (3.17) we obtain y (0) = y (1) = ... = y ( n − 1) = 0 . In order to show that y (k ) = 0 for all k ≥ n , we use the Cayley-Hamilton theorem, n −1

i.e., we write ( A − BD + C) n = ∑ α l ( A − BD + C) l . Thus we have shown l =0


79

that (x o , u o (k )) is an output-zeroing input. Finally, in order to prove that x o (k ) ∈ Scl D for all k ∈ N , we proceed analogously as in the first part of the proof (cf., (ii)). ◊

Corollary 3.2. [79] Any proper system (2.1) can be transformed, by introducing an appropriate precompensator, into a proper system in which the first nonzero Markov parameter has full column rank. Proof. In fact, suppose that in (2.1) D has not full column rank, i.e., rank D = p < m . Let D = D1D 2 , where D1 ∈ R rxp has full column rank

and D 2 ∈ R pxm has full row rank, be a skeleton factorization of D. Introduce the precompensator D T2 into (2.1), i.e., consider the p-input, r-output system x(k + 1) = Ax(k ) + B'v (k ) y (k ) = Cx(k ) + D'v (k ) ,

(i)

where B' = BD T2 , D' = DD T2 and v ∈ R p . Since D 2 D T2 is nonsingular, rank D' = rank D1 = p , i.e., D' has full column rank. This ends the proof. Furthermore, we can observe that A − B' (D' ) + C = A − BD + C which means that matrix-characterizing output-zeroing inputs for (i) is exactly the same as for the original system (2.1). In fact, for (D' ) + we can write (D' ) + = ((DD T2 ) T DD T2 ) −1 (DD T2 ) T = ((D1D 2 D T2 ) T D1D 2 D T2 ) −1 (D1D 2 D T2 ) T = (D 2 D T2 ) −1 (D1T D1 ) −1 D1T

and, consequently, A − B' (D' ) + C = A − BDT2 (D 2DT2 ) −1 (D1T D1 ) −1 D1T C = A − BD+ C . ◊

Remark 3.4. It is easy to note that if (x o , v o (k )) is an output-zeroing input for (i), then (x o , u o (k )) , where u o (k ) = D T2 v o (k ) , is an output-zeroing input for the original system (2.1). Of course, each invariant zero of (i) is also an invariant zero of (2.1) (although the converse implication is false). For instance, system (i) is never degenerate (cf., Corollary 2.5), even if such is system (2.1).

80


Finally, after introducing into a reachable system (2.1) the precompensator D T2 , reachability may be lost (see Exercise 3.3.19). ◊ Example 3.1. Consider a square proper system (2.1) of uniform rank (i.e., such that D is nonsingular). Then (x o , u o (k )) is an output-zeroing input if and only if (i)

u o (k ) = −D −1C( A − BD −1C) k x o ,

xo ∈ R n .

Moreover, the zero dynamics are governed by the initial value problem (ii)

x(k + 1) = ( A − BD −1C) x(k ),

x ( 0) ∈ R n .

In order to prove these claims, it is sufficient to check that u o (k ) of the form (i), when applied to the system at the initial state x(0) = x o , yields the solution x(k ) = ( A − BD −1C) k x o and the identically zero system response, i.e., y (k .) ≡◊ 0 3.2.2 Output-Zeroing Inputs for Strictly Proper Systems

As a necessary condition for a pair (x o , u o (k )) to be an output-zeroing input we take the following. Proposition 3.8. [79] Let (x o , u o (k )) be an output-zeroing input for a strictly proper system (2.1) and let x o (k ) denote the corresponding solν ution. Then x o ∈ Sν := f Ker CA l and u o (k ) has the form

l =0

u o ( k ) = −(CA ν B) + CA ν +1 (K ν A ) k x o + k −1

− (CA ν B) + CA ν +1[ ∑ (K ν A ) k −1−l Bu h (l )] + u h (k ) ,

(3.20)

l =0

for some u h (.) ∈ U satisfying CA ν Bu h (k ) = 0 , and x o (k ) has the form k −1

x o ( k ) = (K ν A) k x o + ∑ (K ν A ) k −1− l Bu h (l ) . l =0

(3.21)

Moreover, x o (k ) is contained in Sν , i.e., x o (k ) ∈ Sν for all k ∈ N .


81

Proof. Let (x o , u o (k )) be an output-zeroing input and let x o (k ) be the corresponding solution. Then for all k ∈ N we have x o (k + 1) = Ax o (k ) + Bu o (k ) y (k ) = Cx o (k ) = 0

Premultiplying C, CA,..., CA

successively

ν −1

the

and first

x o ( 0) = x o .

equality

in

(3.22) (3.22)

by

and using the second equality and the relations

CB = . . . = CA ν −1B = 0 , we obtain Cx o (k ) = 0 CAx o (k ) = 0

for all k ∈ N ,

.

(3.23)

. CA ν x o (k ) = 0

i.e., x o (k ) is contained in Sν . Premultiplying the first equality in (3.22) by CA ν and using the last equality in (3.23), we obtain

0 = CA ν x o (k + 1) = CA ν +1x o (k ) + CA ν Bu o (k ) for all k ∈ N , i.e., CA ν +1x o ( k ) = −CA ν Bu o (k ) for all k ∈ N .

(3.24)

Note that, by virtue of the last equality in (3.23) and the definition of K ν (3.1), x o (k ) satisfies the condition K ν x o (k ) = x o (k ) for all k ∈ N .

(3.25)

Consider now the equation (cf., (3.24)) CA ν Bu(k ) = −CA ν +1 x o (k ) ,

(3.26)

with an unknown sequence u(.) ∈ U. Because, by assumption, u o (k ) satisfies (3.26) (see (3.24)), hence, by virtue of Lemma 3.3, u o (k ) = −(CA ν B) + CA ν +1x o (k ) + u h (k ) , k ∈ N ,

(3.27)

where u h (.) ∈ U is some sequence which satisfies CA ν Bu h (k ) = 0 . Because (x o , u o (k )) and, consequently, x o (k ) are assumed to be known,

82


u h (k ) is treated as a known sequence (which is uniquely determined by (3.27)). Introducing (3.27) into the first equality of (3.22) and employing (3.1), we can write for all k ∈ N x o (k + 1) = K ν Ax o (k ) + Bu h (k ),

x o ( 0) = x o .

(3.28)

An unique sequence x o (k ) which satisfies (3.28) has the form (3.21). Introducing (3.21) into the right-hand side of (3.27), we obtain u o (k ) in the form (3.20). ◊ Remark 3.5. Note that on the assumptions of Proposition 3.8 the input (3.20) applied to the system at an arbitrary initial state x(0) ∈ R n yields the solution x(k ) = A k (x(0) − x o ) + x o (k ) , where x o (k ) is as in (3.21), and the system output y (k ) = CA k ( x(0) − x o ) (cf., Lemma 2.6). In particular, if A is stable, then x(k ) → x o (k ) and y (k ) → 0 as k → ∞ . ◊ Remark 3.6. Suppose that λ ∈ C is an invariant zero of a strictly proper system (2.1), i.e., there exist 0 ≠ x o ∈ C n and g ∈ C m such that the triple λ, x o , g satisfies (2.4). Then, as is known from Lemma 2.7, the input ~ (k ) = λk g , when applied to the system (treated now as a complex one) u o x o (k ) = λk x o and ~ y (k ) = C~ xo (k ) = 0 . at the initial condition x o , yields ~ ~ ~ We shall show now that the inputs Re u o (k ) and Im u o (k ) (appearing in the output-zeroing inputs (Re x o , u o (k )) and (Im x o , u o (k )) – see Remark 2.4) can be written as in (3.20). To this end, it is sufficient to use the necessary condition for invariant zeros given in Proposition 3.4 which tells us that if a triple λ, x o ≠ 0, g satisfies the condition (2.4), then g can be decomposed as

(i)

g = g1 + g 2 ,

g1 ∈ KerCA ν B ,

g 2 = −(CA ν B) + CA ν +1x o .

~ (k ) as Now, using (i), we can write u o

(ii)

~ ( k ) = λk g = λk g + λk g = −λk (CA ν B) + CA ν +1x o + λk g u o 2 1 1 ν + ν +1 ~ ~ = −(CA B) CA x (k ) + u (k ) , o

h

~ (k ) := λk g , k ∈ N . Because u ~ (k ) = λk g and ~ x o (k ) = λk x o where u o h 1


83

satisfy the state equation of (2.1), i.e., (iii)

~ ~ (k ) for k ∈ N and ~ x o (k + 1) = A~ x o ( k ) + Bu xo (0) = xo , o

hence, introducing the right-hand side of (ii) into (iii), we obtain the equalities (iv)

~ ~ (k ) for k ∈ N and ~ x o (k + 1) = K ν A~ x o ( k ) + Bu x o ( 0) = x o . h

This means, by virtue of the uniqueness of solutions (see Remark 2.3), that (v)

k −1 ~ ~ (l ) , k ∈ N . x o (k ) = λk x o = (K ν A ) k x o + ∑ (K ν A ) k −1− l Bu h l =0

Finally, introducing the right-hand side of (v) into the right-hand side of ~ (k ) , we obtain the (ii) and taking the real part of the resultant form of u o desired result ~ (k ) = −(CA ν B) + CA ν +1 (K A ) k Re x o + Re u ν o k −1 ~ (l )] + Re u ~ (k ). − (CA ν B) + CA ν +1[ ∑ (K ν A ) k −1−l B Re u h h

l =0

We proceed similarly with (Im x o , u o (k )) . ◊ Corollary 3.3. Let (x o , u o (k )) be an output-zeroing input for a strictly proper system (2.1) and let x o (k ) be the corresponding solution. Then: If K ν B = 0 , then x o (k ) = (K ν A) k x o . Moreover, at K ν B = 0 the pair (x o , u ∗o (k )) , where u ∗o (k ) := −(CA ν B) + CA ν +1 (K ν A ) k x o , is

(i)

also output-zeroing and yields the solution x o (k ) = (K ν A) k x o . (ii)

If CA ν B has full column rank, then

u o (k ) = −(CA ν B) + CA ν +1 (K ν A ) k x o and x o (k ) = (K ν A) k x o .

Proof. (i) Premultiplying both sides of the first equality in (3.22) by K ν and using (3.25), we obtain x o (k + 1) = K ν x o (k + 1) = K ν Ax o (k ) +K ν Bu o (k ) for all k ∈ N , i.e., x o (k + 1) = K ν Ax o ( k ) + K ν Bu o (k ) , k ∈ N , x o (0) = x o .

At K ν B = 0 from (3.29) it follows that

(3.29)

84


x o ( k ) = (K ν A) k x o .

(3.30)

This ends the proof of the first claim in (i). For the proof of the second claim, let us introduce to the state equation of (2.1) the input u ∗o (k ) := −(CA ν B) + CA ν +1 x o (k )

(3.31)

at the initial state x o . That is, consider the initial value problem x(k + 1) = Ax(k ) + Bu ∗o (k ) , x(0) = x o , k ∈ N .

(3.32)

After using (3.31) and (3.1), the problem (3.32) can be rewritten as x(k + 1) − x o (k + 1) = A(x(k ) − x o ( k )) + (K ν A x o (k ) − x o (k + 1)) x ( 0) = x o ,

,

k∈N

(3.33)

x o ( 0) = x o .

Now, setting z (k ) = x(k ) − x o (k ) and taking into account (3.29), the problem (3.33) can be replaced by z (k + 1) = A z (k ) −K ν Bu o (k ),

z ( 0) = 0 , k ∈ N .

(3.34)

At K ν B = 0 the unique solution of (3.34) is z (k ) ≡ 0 which means in turn that the unique solution x(k ) of (3.32) satisfies x(k ) = x o (k ) = (K ν A ) k x o for all k ∈ N .

(3.35)

Consequently, since from Proposition 3.8 we have x o (k ) ∈ Sν ⊆ Ker C for all k ∈ N , the pair (x o , u ∗o (k )) , where (see (3.31) and (3.30)) u ∗o (k ) = −(CA ν B) + CA ν +1 (K ν A) k x o

(3.36)

is an output-zeroing input and gives the same solution of the state equation of (2.1) as (x o , u o (k )) . This proves the second claim of (i). (ii) By virtue of (3.1), we obtain K ν B = B(I m − (CA ν B) + (CA ν B)) . If

CA ν B has full column rank, then I m − (CA ν B) + (CA ν B) = 0 , i.e., K ν B = 0 . Moreover, in this case the unique solution of (3.26) has the

form (3.31). Hence u o (k ) ≡ u ∗o (k ) , where u ∗o (k ) is as in (3.36). ◊


85

Remark 3.7. The condition K ν B = 0 does not imply in general that u ∗o (k ) = u o (k ) for all k ∈ N , although it yields x(k ) ≡ x o (k ) . The reason behind this becomes clear if we consider the relations Bu ∗o ( k ) − Bu o ( k ) = (K ν − I ) A x o (k ) − Bu o (k ) = K ν Ax o (k ) − x o (k + 1) = −K ν Bu o (k ) .

Thus, at K ν B = 0 , although in general u ∗o (k ) ≠ u o (k ) (see Example 3.2), however both these inputs applied at the initial state x(0) = x o affect the state equation of (2.1) in exactly the same way (because we have then Bu ∗o (k ) − Bu o (k ) = 0 for all k ∈ N ). ◊

Proposition 3.9. [79] In a strictly proper system (2.1) let CA ν B have full row rank. Then (x o , u o (k )) is an output-zeroing input if and only if ν

x o ∈ Sν = h Ker CA l and u o (k ) is as in (3.20) with u h (.) ∈ U satisfying l =0

u h (k ) ∈ Ker CA ν B . Moreover, the corresponding solution x o (k ) has the form (3.21) and is contained in Sν , i.e., x o (k ) ∈ S ν for all k ∈ N .

Proof. We write the skeleton factorization of CA ν B as H1H 2 , where H1 = I r , H 2 = CA ν B . We show first that u o (k ) as in (3.20), with arbit-

rarily fixed u h (k ) ∈ Ker CA ν B and x o ∈ S ν , applied to the system at the initial state x o produces a solution of the form (3.21). To this end, it is sufficient to verify that (3.20) and (3.21) satisfy the state equation of (2.1). The corresponding output equals (note that for k = 0 is y (0) = Cx o ) (i)

k −1

y (k ) = C(K ν A) k x o + ∑ C(K ν A ) k −1− l Bu h (l ) . l =0

We have to show now that y (k ) = 0 for all k ∈ N . However, by virtue of Lemma 3.1 (vi) (at H1 = I r ), we have (ii)

CA l for 0 ≤ l ≤ ν C(K ν A ) l =  .   0 for l ≥ ν + 1

86


The term C(K ν A ) k x o on the right-hand side of (i) equals the zero vector for any k ∈ N . This is easily seen from (ii) and from the assumption ν

x o ∈ Sν = h Ker CA l The second term on the right-hand side of (i) can l =0

be written as C(K ν A ) k −1 Bu h (0) + C(K ν A ) k − 2 Bu h (1) + ...

(iii)

+ C(K ν A )Bu h (k − 2) + CBu h (k − 1) .

For any 1 ≤ k ≤ ν each component in (iii) is the zero vector (it follows

from (ii) and from CB = ... = CA ν −1B = 0 ). For k = ν + 1 the expression in (iii) assumes the form (iv)

ν

∑ C(K ν A )

l =0

ν −l

Bu h (l ) = C(K ν A) ν Bu h (ν) = CA ν Bu h (ν) = 0 .

Finally, for any k > ν + 1 , by virtue of the above discussion and (ii), each component in (iii) is the zero vector. This means that y (k ) = 0 for all k ∈ N , i.e., the considered pair (x o , u o (k )) is an output-zeroing input. The converse implication follows immediately from Proposition 3.8. ◊

Proposition 3.10. [79] In a strictly proper system (2.1) let CA ν B have full column rank. Then (x o , u o (k )) is an output-zeroing input if and only if n −1

l x o ∈ Scl ν := h Ker C(K ν A )

(3.37)

l =0

and u o (k ) = −(CA ν B) + CA ν +1 (K ν A ) k x o ,

k∈N .

(3.38)

Moreover, the corresponding solution has the form x o ( k ) = (K ν A) k x o , k ∈ N

(3.39)

cl and is contained in the subspace S cl ν , i.e., x o ( k ) ∈ Sν for all k ∈ N .

Proof. Suppose first that (x o , u o (k )) is an output-zeroing input. Then, as we know from Corollary 3.3 (ii), u o (k ) has the form (3.38) and the corresponding solution is as in (3.39). Moreover, by assumption, we have


87

y (k ) = Cx o (k ) = C(K ν A ) k x o = 0 for k ∈ N .

(i)

From (3.39) and (i) we obtain the relations

(ii)

Cx o (k + 0) = Cx o (k ) = 0   Cx o (k + 1) = C(K ν A) x o (k ) = 0   .  n −1  Cx o ( k + (n − 1)) = C(K ν A ) x o (k ) = 0

for k ∈ N ,

o cl i.e., x o (k ) ∈ Scl ν for all k ∈ N . In particular, for k = 0 we have x ∈ Sν . In order to prove the converse implication, we should show that any pair (x o , u o (k )) such that x o ∈ Scl ν and u o ( k ) has the form (3.38) is an output-zeroing input. To this end, we verify first that (3.38) and (3.39) satisfy the state equation of (2.1). This means that the input (3.38) applied to the system at the initial state x o yields a solution of the form (3.39). Furthermore, the system response is equal to

y (k ) = Cx o (k ) = C(K ν A ) k x o .

(iii)

From (iii) and (3.37) we obtain y (0) = y (1) = ... = y ( n − 1) = 0 . The proof that y (k ) = 0 for k ≥ n we proceed by induction using the Cayleyn −1

Hamilton theorem (i.e., writing (K ν A ) n = ∑ α l (K ν A ) l ). Hence we have l =0

o

shown that (x , u o (k )) is an output-zeroing input. Finally, in order to prove that x o (k ) is contained in S cl ν , we proceed analogously as in the first part of the proof (cf., (ii)). ◊ Corollary 3.4. [79] Any strictly proper system (2.1) with not the identically zero transfer-function matrix can be transformed, by introducing an appropriate precompensator, into a strictly proper system in which the first nonzero Markov parameter has full column rank. Proof. In fact, assume that in (2.1) CA ν B has not full column rank, i.e., rank CA ν B = p < m . Let CA ν B = H1H 2 , where H1 ∈ R r x p has full col-

umn rank and H 2 ∈ R p x m has full row rank, be a skeleton factorization. Introduce into (2.1) the precompensator H T2 , i.e., consider the p-input, routput system

88


x(k + 1) = Ax(k ) + B' v( k ) y (k ) = Cx(k ) ,

(i)

where B' = BH T2 and v ∈ R p . The first Markov parameter CA ν B' of (i) has full column rank. It follows from the factorization of CA ν B' as CA ν BH T2 = H1 (H 2 H T2 ) = H1H ' 2 , where H ' 2 = H 2 H T2 is nonsingular. Hence rank CA ν B' = rank H1 = p . Furthermore, we can show that output-zeroing inputs for the system (i) are characterized by matrix K ν A of the original system (2.1). To this end,

we form for (i) the matrix K ' ν := I − B'(CA ν B') + CA ν (see Lemma 3.1). However, the pseudoinverse of CA ν B' can be evaluated as (CA ν B') + = (H ' 2 ) + H1+ = (H 2 H T2 ) −1 (H1T H1 ) −1 H1T and consequently, for K 'ν is K ' ν = I − BH 2 [(H 2 H T2 ) −1 (H1T H1 ) −1 H1T ]CA ν T

= I − BH +2 H1+ CA ν = K ν .

Hence K ' ν A = K ν A , as claimed. ◊ Remark 3.8. Of course, to any output-zeroing input for the system (i) there corresponds an output-zeroing input for the original system (2.1), i.e., if (x o , v o (k )) is an output-zeroing input for (i), then (x o , u o (k )) , where u o (k ) = H T2 v o (k ) , is an output-zeroing input for (2.1). Since H T2 has full column rank, the converse implication does not hold in general. Naturally, system (i) is never degenerate (see Corollary 2.5), even if the original system (2.1) is. Note also that after introducing into a reachable system (2.1) the pre-

compensator H T2 , reachability may be lost (see Exercise 3.3.20). ◊ Example 3.2. Consider the transfer-function matrix 3z 3z    (z + 1)(3z + 1) 0 ( z + 1)(3z + 1)  G (z) =   3   0 0   (3z + 2)

and its minimal realization (2.1) with the matrices


− 1 0 − 1 / 3  − 2 / 3 0  , A= 0  0 − 1 0

1 0 1 B = 0 1 0 , 1 0 1

89

1 0 0 C= . 0 1 0

The Smith form of the system matrix P(z) is 1  0 0 0  0

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0

0 0 0 0 0 0 . 0 0  z 0

According to Proposition 2.3, the system is nondegenerate and (cf., Definition 2.3) its only invariant (transmission) zero is λ = 0 (for instance, the 2 0  o   triple λ = 0, x = 0, g =  0  satisfies (2.4)). − 1 1  1 0 1 One can easily verify that CB =   , i.e., ν = 0 , and 0 1 0 

1 / 2 0 (CB) =  0 1  , 1 / 2 0 +

 0 0 0 K 0 = I − B(CB) C =  0 0 0 , − 1 0 1  0 0 0 K 0 A =  0 0 0 . 1 / 3 0 0 +

Via Proposition 3.9, any pair (x o , u o (k )) , where x o ∈ S 0 = Ker C and u o (k ) has the form (3.20), with an arbitrarily fixed u h (k ) satisfying u h (k ) ∈ Ker CB for all k ∈ N , is an output-zeroing input, while the corresponding solution x o (k ) has the form (3.21). Thus u h (k ) may be  u h ,1 (k )    0 written as u h (k ) =   , where u h ,1 (.) : N → R is an arbitrarily − u h ,1 (k )  

90


0   fixed sequence, and x =  0  , where x 3o ≠ 0 . Finally, because x o   3 Bu h (k ) = 0 for all k ∈ N , u o (k ) in (3.20) and x o (k ) in (3.21) take respectively the form o

 1 o    2 x 3   u h ,1 (0)     0 + 0   1     x 3o  − u h ,1 (0)  u o (k ) =   2   u (k )   h ,1   0    − u ( k )    h ,1       x o (k ) =   0  0     0

0   0 x o   3

for k = 0

for k = 1,2,...

for k = 0

. ◊ for k = 1,2,...

Example 3.3. Consider the transfer-function matrix 3  3z   (z + 1)(3z + 1) 0 3z + 1 G (z) =   3  0 0    3z + 2

and its minimal state space realization (2.1) with the matrices − 1 0 − 1 / 3  − 2 / 3 0  , A= 0  0 − 1 0

The Smith form of P(z) equals

1 0 1 B = 0 1 0 , 1 0 0

1 0 0 C= . 0 1 0

3.3 Exercises

1  0 0 0  0

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0

91

0 0 0 0 0 0 . 0 0  1 0 

Therefore the system has no Smith zeros. On the other hand, by virtue of Proposition 2.2 (cf., also Proposition 2.4), the system is degenerate. Matrices CB and (CB) + are as in Example 3.2, while 0 0  0  K 0 = I − B(CB) C =  0 0 0 , − 1 / 2 0 1 +

0   0 0  K0A =  0 0 0  . 1 / 6 0 − 1 / 2

Since CB has full row rank, all output-zeroing inputs and the corresponding solutions are as described in Proposition 3.9. ◊ Example 3.4. Consider a square strictly proper system (2.1) of uniform rank (i.e., such that CA ν B is nonsingular). Then (x o , u o (k )) is an output-zeroing input for the system if and only if ν

x o ∈ Sν = h Ker CA l l =0

and u o (k ) = −(CA ν B) −1 CA ν +1 (K ν A ) k x o ; K ν := I − B(CA ν B) −1 CA ν .

Moreover, the zero dynamics are governed by the initial value problem x(k + 1) = K ν Ax(k ),

x ( 0) ∈ S ν .

The form of output-zeroing inputs follows from Corollary 3.3 (ii). In order to show that any pair ( x o ∈ S ν , u o (k ) = −(CA ν B) −1 CA ν +1 (K ν A ) k x o ) is an output-zeroing input see the proof of Proposition 3.9. ◊

3.3 Exercises 3.3.1. Consider Lemma 3.1. Show that K ν2 = K ν . Hint. Write

92


(I − B(CA ν B) + CA ν ) 2 = I − 2B(CA ν B) + CA ν + B(CA ν B) + CA ν B(CA ν B) + CA ν

and next use the relation (see (B3) in Appendix B) (CA ν B) + (CA ν B)(CA ν B) + = (CA ν B) + .

3.3.2. Consider a proper system (2.1). Verify that u o (k ) as in (3.2) and x o (k ) as in (3.3) (at an arbitrarily fixed sequence u h (.) : N → R m ) satisfy

the state equation of (2.1). Check that the corresponding system response equals y (k ) = (I r − DD + )Cx o (k ) + Du h (k ) . Show that u o (k ) as in (3.2) (at an arbitrary u h (k ) ) applied to the system at an arbitrary initial state x(0) ∈ R n yields x(k ) = A k (x(0) − x o ) + x o (k ) , where x o (k ) is as in

(3.3), and y (k ) = CA k (x(0) − x o ) + (I r − DD + )Cx o (k ) + Du h (k ) . 3.3.3. Consider a strictly proper system (2.1). Check that u o (k ) as in (3.20) and x o (k ) as in (3.21) (at an arbitrarily fixed u h (.) : N → R m ) satisfy the state equation of (2.1). Show that u o (k ) as in (3.20), at an arbitrary u h (k ) , applied to the system at an arbitrary initial state x(0) ∈ R n yields x(k ) = A k (x(0) − x o ) + x o (k ) , where x o (k ) is as in (3.21), and y (k ) = CA k (x(0) − x o ) + Cx o (k ) .

3.3.4. Into a strictly proper system (2.1) introduce the control law u( k ) = v (k ) + Fx (k ) , where F = −(CA ν B) + CA ν +1 , i.e., consider the

closed-loop system

(i)

x(k + 1) = K ν Ax(k ) + Bv (k ) y (k ) = Cx(k ) ,

where K ν := I − B(CA ν B) + CA ν . Discuss the output-zeroing problem for the system (i).

3.3 Exercises

93

Hint. Observe that the first nonzero Markov parameter of (i) equals C(K ν A ) ν B = CA ν B (to this end use Lemma 3.1 (vii)). For (i) define

K νcl := I − B[C(K ν A) ν B] + C(K ν A ) ν and note that K νcl = K ν . Now, applying Proposition 3.8 to (i), we obtain the following characterization of

the problem. Let (x o , v o ( k )) be an output-zeroing input for (i) and let ν x o (k ) denote the corresponding solution. Then x o ∈ Ker C(K ν A ) l and

l =0

v o (k ) has the form v o (k ) = −[C(K ν A ) ν B] + C(K ν A ) ν +1[(K νcl (K ν A)) k x o k −1

+ ∑ (K νcl (K ν A)) k −1−l Bv h (l )] + v h (k ) ,

(ii)

l =0

where v h (k ) ∈ Ker C(K ν A ) ν B for all k ∈ N , and x o (k ) has the form k −1

(iii)

cl k o k −1 − l x o (k ) = (K cl Bv h (l ) ν (K ν A)) x + ∑ (K ν (K ν A )) l =0

and is contained in

ν

Ker C(K ν A) l .

l =0 ν

However,

Ker C(K ν A ) l = S ν =

l =0

ν

Ker CA l and (ii) and (iii) be-

l =0

come v o ( k ) = −(CA ν B) + C(K ν A) ν +1[(K ν A ) k x o k −1

(iv)

+ ∑ (K ν A ) k −1− l Bv h (l )] + v h (k ) l =0

and k −1

x o ( k ) = (K ν A) k x o + ∑ (K ν A ) k −1− l Bv h (l ) .

(v)

l =0

In this way we have obtained the following characteriztion of the problem: if (x o , v o ( k )) is an output-zeroing input for the system (i), then ν

x o ∈ Sν = h Ker CA l l =0

and

v o (k )

has

the

form

(iv),

where

94


v h (k ) ∈ Ker CA ν B for all k ∈ N ; moreover, the corresponding solution is as in (v). Note that the first component on the right-hand side of (iv) does not affect the state equation in (i). It follows from the relations (see Lemma 3.1 (vii))

(vi)

B(CA ν B) + C(K ν A ) ν +1 = B(CA ν B) + C(K ν A ) ν K ν A = ( I − K ν ) K ν A = 0.

Thus for (iv) we have Bv o (k ) = Bv h (k ) for all k ∈ N , i.e., the solution (v) depends merely on x o and v h (k ) . If CA ν B has full row rank, then (x o , v o ( k )) is an output-zeroing input ν

for (i) if and only if x o ∈ Sν = h Ker CA l and v o ( k ) = v h (k ) , where l =0

ν

v h (k ) ∈ Ker CA B . It follows from the relation C(K ν A ) ν +1 = 0 (see (ii) in the proof of Proposition 3.9) and from Proposition 3.9.

If CA ν B has full column rank, then (x o , v o ( k )) is an output-zeroing n −1

input for (i) if and only if x o ∈ h Ker C(K ν A ) l and v o (k ) ≡ 0 . The nel =0

cessity follows from the fact that B has full column rank and, consequently (see (vi)), (CA ν B) + C(K ν A) ν +1 = 0 . Thus the first component in (iv) equals identically the zero vector. Since the second component in (iv) satisfies v h (k ) ∈ Ker CA ν B and CA ν B has full column rank, we have v h (k ) ≡ 0 . For the proof of sufficiency see Proposition 3.10. 3.3.5. To a proper system (2.1) introduce the control law u = v + Fx , where F = −D + C , i.e., consider the closed-loop system (i)

x(k + 1) = ( A − BD + C)x(k ) + Bv (k ) y (k ) = (I r − DD + )Cx(k ) + Dv (k ) .

Discuss the output-zeroing problem for the system (i). Hint. In (i) denote A' = A − BD + C, B' = B, C' = (I r − DD + )C, D' = D and then to S( A ' , B' , C' , D' ) apply Proposition 3.5 (note that (I r − DD + ) 2 = I r − DD + and A'−B' (D' ) + C' = A − BD + C as well as

3.3 Exercises

95

(D' ) + C' = 0 ). In this way we obtain the following characterization of output-zeroing inputs and the corresponding solutions for (i): if (x o , v o ( k )) is an output-zeroing input, then x o ∈ Ker (I r − DD + )C and v o ( k ) = v h (k ) , where v h (k ) ∈ Ker D for all k ∈ N ; moreover, the corresponding solution equals k −1

x o (k ) = ( A − BD + C) k x o + ∑ ( A − BD + C) k −1− l Bv h (l ) l =0

and is contained in the subspace Ker (I r − DD + )C . If D has full column rank, to S( A ' , B' , C' , D' ) we apply Proposition 3.7 and then we obtain: a pair (x o , v o ( k )) is an output-zeroing input for (i) if n −1

{

and only if x o ∈ h Ker (I r − DD + )C( A − BD + C) l l =0

moreover,

the +

corresponding k o

x o (k ) = ( A − BD C) x n −1

{

solution

}

and v o (k ) ≡ 0 ;

has

the

form

and is entirely contained in the subspace

}

+ + l f Ker (I r − DD )C( A − BD C) .

l =0

If D has full row rank, then DD + = I r and (i) takes the form (ii)

x(k + 1) = ( A − BD + C)x(k ) + Bv (k ) y (k ) = Dv(k ) .

3.3.6. In the SISO system x(k + 1) = Ax(k ) + bu (k )

(i)

y(k ) = cx(k )

let cA ν b , 0 ≤ ν ≤ n − 1 , denote the first nonzero Markov parameter. Consider the problem of reproducing the reference output y ref (k ) in which we want to find all pairs consisting of an initial state x o and an input u o (k ) such that the corresponding system response y(k ) coincides with y ref (k ) , i.e., y(k ) = y ref (k ) for all k ∈ N . Show that a pair ( x o , u o ( k )) solves the problem if and only if x o satisfies the system of equations

96


 y ref (0)   c     cA    x o =  y ref (1)   .   .     ν cA   y ref (ν)

(ii)

and u o (k ) has the form u o (k ) = −(cA ν b) −1 cA ν +1 (K ν A ) k x o

(iii)

+ (cA ν b) −1 y ref (k + ν + 1) + k −1

− (cA ν b ) −1 cA ν +1[ ∑ (K ν A ) k −1−l b(cA ν b) −1 y ref (l + ν + 1)] , l =0

where K ν := I − b(cA ν b) −1 cA ν . Note that the solution corresponding to the input (iii) and to the initial state x(0) = x o has the form k −1

(iv)

(K ν A ) k x o + ∑ (K ν A ) k −1− l b(cA ν b) −1 y ref (l + ν + 1) , k ∈ N . l =0

Hint. Let a pair ( x o , u o ( k )) solve the problem and let k −1

x o ( k ) = A k x o + ∑ A k −1− l bu o (l ) , k ∈ N l =0

denote the corresponding solution. Then, by assumption, we have k −1

(v)

y ref (k ) = cA k x o + ∑ cA k −1− l bu o (l ) for all k ∈ N . l =0

Substituting k = 0,1,..., ν in (v) and using cb = ... = cA ν −1b = 0 , we obtain (ii). On the other hand, we have (vi)

x o (k + 1) = Ax o (k ) + bu o (k ) y ref (k ) = cx o (k )

for k ∈ N and x o (0) = x o .

Premultiplying successively both sides of the first equality in (vi) by c, cA,..., cA ν −1 and using cb = ... = cA ν −1b = 0 , we obtain the following equalities

3.3 Exercises

97

cx o (k + 1) = cAx o (k ) cAx o (k + 1) = cA 2 x o (k ) . .

(vii)

cA ν −1x o (k + 1) = cA ν x o (k )

which are valid for all k ∈ N . From (vii) we derive the relation cA ν x o ( k + 1) = cx o ( k + ν + 1) which holds for all k ∈ N , i.e.,

(viii)

y ref ( k + ν + 1) = cA ν x o ( k + 1)

for all k ∈ N .

Premultiplying both sides of the first identity in (vi) by cA ν and using (viii), we obtain the identity y ref ( k + ν + 1) = cA ν +1x o ( k ) + (cA ν b) u o ( k ) for all k ∈ N .

Calculating from this identity u o (k ) and introducing it into x o (k + 1) = Ax o (k ) + bu o (k ) as well as using the definition of K ν , one

obtains x o ( k + 1) = K ν Ax o ( k ) + b(cA ν b) −1 y ref ( k + ν + 1) for all k ∈ N and x o (0) = x o . The only sequence x o ( k ) that satisfies these conditions is as in (iv). Finally, substituting x o ( k ) as in (iv) into the right-hand side of (ix)

u o ( k ) = (cA ν b) −1 y ref ( k + ν + 1) − (cA ν b) −1 cA ν +1 x o ( k ) ,

we obtain (iii). In order to prove the converse implication, it is sufficient to answer the following two questions. At first, that (iii) and (iv) satisfy the state equation of (i) (this means that the input (iii), when applied to the system exactly at the initial state x(0) = x o , produces a solution of the form (iv)). By the way, one can note that the condition (ii) is not involved in this question. At second, that if (ii) is fulfilled, then the system response corresponding to (iii) and to the initial state x(0) = x o satisfies y(k ) = y ref (k ) for all k ∈ N . However, for all k ∈ N we have y(k ) = cx o (k ) = c(K ν A ) k x o

(x)

k −1

+ ∑ c(K ν A) k −1−l b(cA ν b) −1 y ref (l + ν + 1) . l =0

98


Now, using (ii) and cb = ... = cA ν −1b = 0 as well as (cf., Lemma 3.1 (vi) or Exercise 2.8.18 (iii)) cA l for 0 ≤ l ≤ ν , c( K ν A ) l =   for l ≥ ν + 1 0

(xi)

one easily checks that for each k = 0,1,..., ν we obtain from (x) the identity y(k ) = y ref (k ) . In a similar way we obtain y(ν + 1) = y ref (ν + 1) . Replacing k by k + ν + 1 in (x), we can write y(k + ν + 1) = c(K ν A) k + ν +1 x o

(xii)

k +ν

+ ∑ c(K ν A ) k + ν −l b(cA ν b) −1 y ref (l + ν + 1) l =0

which holds for any k ∈ N . Now, it is easy to verify that at any fixed k ∈ N the right-hand side in (xii) equals y ref (k + ν + 1) . In fact, in view of (xi), the first component on the right-hand side of (xii) equals zero at k +ν

any k ∈ N . In the expression ∑ c(K ν A ) k + ν −l b(cA ν b) −1 y ref (l + ν + 1) l =0

the component corresponding to l = k equals y ref (k + ν + 1) , while all the remaining components are equal to zero. Finally, observe that since the matrix standing on the left-hand side of (ii) has full row rank (why?), the set of solutions of (ii) is always (i.e., for any vector on the right-hand side of (ii)) non-empty. Note also that the output-zeroing problem is a particular case of the problem of reproducing the reference output. 3.3.7. Suppose that the system (i) in Exercise 3.3.6 is asymptotically stable. Then we can discuss the problem of tracking the reference output y ref (k ) in which, irrespectively of what the initial state is, the system output should converge asymptotically to the prescribed reference function y ref (k ) . Show that the asymptotic stability of the system implies that any pair ( x o , u o ( k )) which solves the problem of reproducing the reference output (cf., Exercise 3.3.6) solves also the problem of tracking the reference output. Hint. Observe that in Exercise 3.3.6 the input (iii) (determined by a vector x o which satisfies (ii)) applied to the system at an arbitrary initial state x (0) produces a solution of the state equation of the form

3.3 Exercises

99

A k ( x ( 0) − x o )

(xiii)

k −1

+ (K ν A ) k x o + ∑ (K ν A ) k −1− l b(cA ν b) −1 y ref (l + ν + 1) l =0

and yields the system response y( k ) = cA k ( x (0) − x o ) + y ref ( k ) .

(xiv)

3.3.8. Consider the closed-loop (state-feedback) version of the problem of reproducing the reference output in which to the system (i) in Exercise 3.3.6 we introduce the control law u (k ) = (cA ν b) −1 v(k ) + Fx (k ) , where F = −(cA ν b ) −1 cA ν +1 . Show that introducing to the closed-loop system described above the input v(k ) = y ref (k + ν + 1) at an initial state x(0) = x o satisfying (ii) we obtain the system response with the property y(k ) = y ref (k ) for any k ∈ N .

Hint. Observe first that the closed-loop system is governed by

x(k + 1) = K ν Ax(k ) + b(cA ν b) −1 v(k )

(xv)

y(k ) = cx(k )

and then use the hint to Exercise 3.3.6. Note that applying to the system (xv) the input v(k ) = y ref (k + ν + 1) at an arbitrary initial state x (0) (not necessarily satisfying the conditions (ii) in Exercise 3.3.6), we obtain y(k ) = y ref (k ) for any k ≥ ν + 1 (i.e., the system reproduces the reference output after ν + 1 steps). To this end, write the system response as k −1

y( k ) = c(K ν A ) k x(0) + ∑ c(K ν A ) k −1− l b(cA ν b) −1 y ref (l + ν + 1) l =0

and then observe that the term c(K ν A ) k x(0) vanishes for all k ≥ ν + 1 independently upon x (0) (cf., the relation (xi) in Exercise 3.3.6), while at any fixed k ≥ ν + 1 in the expression k −1

k −1− l b(cA ν b) −1 y ref (l + ν + 1) ∑ c (K ν A )

l =0

100


the component corresponding to l = k − (ν + 1) equals y ref (k ) and all the other components are equal to zero. Note also that if all invariant zeros of the system (i) in Exercise 3.3.6 lie inside the unit disc or this system has no invariant zeros (i.e., ν = n − 1 ), then the system (xv) is asymptotically stable (i.e., all eigenvalues of K ν A lie inside the unit disc). In particular, if ν = n − 1 , then K ν A is nilpotent (see Exercise 2.8.19). 3.3.9. Consider the problem of reproducing the reference output in a proper SISO system x(k + 1) = Ax(k ) + bu (k ) y(k ) = cx(k ) + du (k ) .

(i)

Show that any pair ( x o , u o ( k )) , where u o (k ) = −d −1 c( A − bd −1c) k x o +

(ii)

k −1

− d −1 c[ ∑ ( A − bd −1c) k −1−l bd −1 y ref (l )] + d −1 y ref (k ) , l =0

solves the problem and the corresponding solution of the state equation equals k −1

(iii)

( A − bd −1c) k x o + ∑ ( A − bd −1c) k −1− l bd −1 y ref (l ) . l =0

Show that any solution to the problem has the form ( x o , u o ( k )) , where u o (k ) is as in (ii). 3.3.10. Discuss the closed-loop (state-feedback) version of the problem of reproducing the reference output in which into the system (i) in Exercise 3.3.9 we introduce the control law u (k ) = d −1v(k ) + Fx(k ) with F = −d −1c . Find the transfer function of the closed-loop system.

Hint. Observe that the closed-loop system is governed by the equations x(k + 1) = ( A − bd −1c)x(k ) + bd −1 v(k ) y( k ) = v( k ) .

3.3 Exercises

101

3.3.11. In a strictly proper SISO system x(k + 1) = Ax(k ) + bu (k )

(i)

y(k ) = cx(k )

let cA ν b , 0 ≤ ν ≤ n − 1 , denote the first nonzero Markov parameter. Consider the closed-loop system obtained from (i) by introducing the control law u (k ) = (cA ν b) −1 v(k ) + Fx (k ) , where F = −(cA ν b) −1 cA ν +1 . Show that the closed-loop system response corresponding to an arbitrary initial state x(0) and to an arbitrary input v(k ) takes the form (ii)

 cA k x(0) for 0 ≤ k ≤ ν y( k ) =  .  v ( k ( 1 )) for k 1 − ν + ≥ ν + 

Hint. Discuss the closed-loop system response k −1

y(k ) = c(K ν A ) k x(0) + ∑ c(K ν A ) k −1− l b (cA ν b) −1 v(l ) . l =0

Observe also that the closed-loop system is bounded-input bounded-output (BIBO) stable and its transfer-function equals z −(ν +1) . Moreover, this system is asymptotically stable if and only if the system (i) has no invariant zeros or all its invariant zeros lie inside the unit disc. 3.3.12. Discuss the disturbance decoupling problem in which we want to find a static feedback control completely protecting the output of the closed-loop system from disturbances affecting the state. Consider the SISO system (i)

x(k + 1) = Ax(k ) + bu (k ) + b w w (k ) y(k ) = cx(k )

in which w (k ) represents undesired input (disturbance). Suppose that for some

0 ≤ ν ≤ n − 1 is

cb = ... = cA ν −1b = 0,

cA ν b ≠ 0

as well as

cb w = . . . = cA ν w −1b w = 0, cA ν w b w ≠ 0 for some 0 ≤ ν w ≤ n − 1 . Show that the problem is solvable if and only if ν w ≥ ν + 1 . Show also that if the problem is solvable, then the state feedback matrix F = −(cA ν b ) −1 cA ν +1 gives a solution to the problem.

102


Hint. To the system (i) introduce the control law u (k ) = v(k ) + Fx(k ) and suppose that some state feedback matrix F solves the problem. Then the closed-loop system output should be completely independent on the disturbance. This should hold also in the case when v(k ) = 0 for any k ∈ N . Taking into account the closed-loop system at v(k ) ≡ 0 , i.e.,

(ii)

x(k + 1) = ( A + bF )x(k ) + b w w (k ) y(k ) = cx(k ) ,

we observe that if its output k −1

(iii)

y(k ) = c( A + bF ) k x(0) + ∑ c( A + bF ) k −1− l b w w (l ) , k ∈ N , l =0

is completely independent upon w ( k ) , then in (iii) we must have k −1

k −1− l b w w (l ) = 0 for each k = 1,2,... . This means in turn ∑ c( A + bF )

l =0

that the transfer-function g w ( z) = c(zI − ( A + bF )) −1 b w of the system (ii) equals zero identically. It implies that all Markov parameters of (ii) are equal to zero, i.e., c( A + bF) l b w = 0 for each l ∈ N . From cb = . . . = cA ν −1b = 0 we obtain c( A + bF ) l = cA l for each l = 0,1,..., ν . In

particular, 0 = c( A + bF ) ν b w = cA ν b w . This means that ν w ≥ ν + 1 . Conversely, if ν w ≥ ν + 1 , then taking F = −(cA ν b ) −1 cA ν +1 we can write the closed-loop system in the form (iv)

x(k + 1) = K ν A x(k ) + bv(k ) + b w w (k ) y( k ) = cx(k ) ,

where K ν is as in Definition 3.1. Now, using cb = . . . = cA ν −1b = 0 and cb w = . . . = cA ν w −1b w = 0 as well as Lemma 3.1 (vi), one can observe that the transfer-function g w (z) (from w to y) of the system (iv) equals zero identically, i.e., ∞

g w (z) = ∑

l =0

c( K ν A ) l b w z l +1

≡0

(this means also that the subsystem x(k + 1) = K ν Ax(k ) + b w w (k ) , y(k ) = cx(k ) is degenerate). Moreover, the response of the system (iv)

3.3 Exercises

103

corresponding to an arbitrary initial state x(0) and to arbitrary input signals v(k ), w (k ) takes the form k −1

k −1

l =0

l =0

y(k ) = c(K ν A) k x(0) + ∑ c(K ν A) k −1−l bv(l ) + ∑ c(K ν A) k −1−l b w w (l ) cA k x(0) = ν  cA bv( k − (ν + 1))

for 0 ≤ k ≤ ν for k ≥ ν + 1

,

i.e., it is completely independent upon w ( k ) . Thus, if ν w ≥ ν + 1 , the problem is solvable and F = −(cA ν b ) −1 cA ν +1 can be taken as a solution. 3.3.13. Suppose that in the system (i) in Exercise 3.3.12 we have ν w = ν and the disturbance signal is measurable. Introduce into this system the control law u = v + Fx + β w , i.e., consider the closed-loop system (v)

x(k + 1) = ( A + bF )x(k ) + bv(k ) + (b w + βb)w (k )

y(k ) = cx(k ) .

Discuss the disturbance decoupling problem for the system (v). Hint. Consider F as in Exercise 3.3.12 and take β = −

cA ν b w cA ν b

.

3.3.14. Consider the disturbance decoupling problem for the SISO system x(k + 1) = Ax(k ) + bu (k ) + b w w (k ) y( k ) = cx( k ) + du ( k )

and show that the problem is always solvable (verify that the state feedback matrix F = −d −1c solves the problem). 3.3.15. Consider the disturbance decoupling problem for the system x(k + 1) = Ax(k ) + bu ( k ) + b w w (k ) y( k ) = c x( k ) + d u ( k ) + d w w ( k )

in which the disturbance signal w ( k ) is measurable. Show that the problem is always solvable (verify that the pair (F, β) , where F = −d −1c and

104


d β = − w , solves the problem). d

3.3.16. Consider a SISO system (2.1) with the matrices 0 0  − 1 / 2  A= 0 −1/ 2 0  ,  0 0 − 1 / 2

1  b = 0 , 0

c = [0 1 1] .

Show that the system response does not depend upon input signals. 3.3.17. Consider the disturbance decoupling problem for the system (i) of Exercise 3.3.12, where 1   0 1 0 1      A = 2 − 3 0 , b = 1 , c = [0 1 − 1] , b w = 0 . 0 0 1 0 0

Find a state feedback matrix F which solves the problem and stabilizes the closed-loop system (i.e., A + bF is asymptotically stable). Hint. Using Exercise 3.3.12 check first that the problem is solvable. Then choose a matrix F in such way that the transfer-function g w (z) from w to y becomes identically zero, while the polynomial det(zI − (A + bF)) has all its roots inside the unit circle. Note that in order to fulfill the first requirement we need to find a matrix F which satisfies (at the notation used in Exercise 3.3.12) the conditions c( A + bF ) l b w = 0 for l = ν + 1, .. . , n − 1 . Then all Markov parameters of g w (z) will be zero.

3.3.18. Consider a minimal system (2.1) with the matrices 0 0  0 A= 0  0 0 

1 0 0 0 0 0  0 0 1 0 0 0  1 0 0 0 0 0 , B=  0 0 0 0 1 0   0 0 0 0 1 0  0 0 0 0 0 0 

0 0 1 1 0 0 0 0  0 , C= ,  0 0 0 0 1 − 1 0    0 1

1 0 D= . 1 0

3.3 Exercises

105

1 In a skeleton factorization D = D1D 2 take D1 =   and D 2 = [1 0] . In1 troduce into the system precompensator D T2 and check that the resulting

system S( A , BD T2 , C, DD T2 ) is not reachable (cf., Remark 3.4). 3.3.19. Consider a minimal system S(A,B,C) (2.1) with the matrices 0 0 A= 0  0

0 0 0 1  0 0 1 0 , B= 0 0 0 1   0 0 0 0

0 0 1 0 0 0  , C= . 0 1 − 1 1 0  1

1 In a skeleton factorization CB = H1H 2 take H1 =   and H 2 = [1 0] . 1 Introduce into S(A,B,C) precompensator H T2 and verify that the resulting

system S( A , BH T2 , C) is not reachable (cf., Remark 3.8).

4 The Moore-Penrose Inverse of the First Markov Parameter

In this chapter we employ the Moore-Penrose pseudoinverse of the first nonzero Markov parameter for the algebraic characterization of the system zeros, i.e., invariant, transmission and decoupling zeros. Strictly proper ( D = 0 ) and proper ( D ≠ 0 ) systems are discussed separately. For a strictly proper system (2.1) as matrix-characterizing system zeros we take matrix K ν A , where K ν := I − B(CA ν B) + CA ν , CA ν B stands for the first nonzero Markov parameter and “+” means the operation of taking the Moore-Penrose pseudoinverse. For a proper system (2.1) as matrix-describing system zeros we use matrix A − BD + C . The question of determining and interpreting the system zeros is based on the Kalman canonical decomposition theorem. As we shall see, the Kalman form of these matrices, with block partition consistent with the partition of the original system, discloses all decoupling zeros. Moreover, if the first nonzero Markov parameter has full rank and the system is nondegenerate, then these matrices characterize completely also invariant zeros.

4.1 System Zeros in Strictly Proper Systems Consider a discrete-time system (2.1) of the form x(k + 1) = Ax(k ) + Bu(k ) , x ∈ Rn , u ∈ Rm , y ∈ Rr , k ∈ N , y (k ) = Cx(k )

(4.1)

where CA ν B , 0 ≤ ν ≤ n − 1 , stands for the first nonzero Markov parameter and rank CA ν B = p ≤ min{m, r} . For (4.1) consider a class of real-valued inputs of the form

u(k , x o ) := −(CA ν B) + CA ν +1 (K ν A) k x o , x o ∈ R n , k ∈ N ,

(4.2)

where K ν := I − B(CA ν B) + CA ν .


108


Lemma 4.1. [73, 80] For a given input u(k , x o ) (4.2) a solution of the state equation of (4.1) corresponding to an arbitrary initial state x(0) ∈ R n has the form x ( k ) = A k ( x ( 0) − x o ) + ( K ν A ) k x o .

(4.3)

Proof. Simple verification that u(k , x o ) (4.2) and x(k ) (4.3) satisfy the state equation of (4.1) is left to the reader. Then, the proof follows from the uniqueness of solutions. ◊ Motivated by the results of the previous chapter we shall provide in this section a more detailed discussion of the structure of the matrix K ν A .

Let CA ν B = H1H 2 denote a skeleton factorization of CA ν B . With (4.1) we associate an auxiliary square (p-input, p-output) system S( A, B d , C d ) , where B d = BH T2 , C d = H1T C , obtained from (4.1) by introducing precompensator H T2 and postcompensator H1T . The first nonzero Markov parameter C d A ν B d of S( A, B d , C d ) is nonsingular. Consequently, rank B d = rank C d = p and p ≤ n . The matrix K ν, d := I − B d (C d A ν B d ) −1 C d A ν satisfies K ν, d = K ν (cf., the proof of Lemma 3.1). By Pd (z) we denote the system matrix of S( A, B d , C d ) , i.e.,

 zI − A − B d  . Pd (z) =  0   Cd We can formulate now the following result which shows in particular that K ν A has at least p(ν + 1) zero eigenvalues.

Lemma 4.2. [73, 80] det(zI − K ν A) = det(H1T H1 ) −1 det(H 2 H T2 ) −1 z p (ν +1) det Pd (z)

Proof.

Denote

F = −(C d A ν B d ) −1 C d A ν +1

(4.4) and

zI − K ν, d A − B d  zI − A − B d   I 0  cl Pdcl (z) =  . Then  F I  = Pd (z)   Cd 0  0  p   Cd −1 and det Pd (z) = det Pdcl (z) . Denote G cl d ( z ) = C d ( zI − K ν ,d A) B d .

4.1 System Zeros in Strictly Proper Systems

109

Now, in view of Lemma 3.1 (vi) and the relation K ν, d = K ν , we obtain ∞

C d (K ν , d A ) l B d

l =0

z l +1

G cl d (z) = ∑

and det G cl d (z) =

=

Cd A ν B d

det (C d A ν B d ) z p (ν +1)

z ν +1

.

Finally, since K ν, d A = K ν A and C d A ν B d = H1T H1H 2 H T2 , the relation (4.4) follows from det Pdcl (z) = det(zI − K ν,d A ) det G dcl (z) . ◊ Remark 4.1. Another proof of (4.4) can be found in [80]. Those zero eigenvalues of K ν A which correspond in (4.4) to the factor z p ( ν +1) can be treated, by analogy to the SISO case (cf., Exercise 2.8.19), as representing zeros at infinity of (4.1). ◊ 4.1.1 Decoupling and Transmission Zeros

For further discussion of the stucture of K ν A we use the Kalman form (2.2) of (4.1). Employing (2.2) and the definition (3.1) of K ν as well as the relation CA ν B = C r o A νr o B r o , we obtain after simple calculations I ro  0 Kν =   0   0

x

0

K νro

0

0 0

I ro 0

x  x  , 0   I ro 

(4.5)

where I ro , I r o , I ro denote identity matrices of appropriate orders, and x   x  , A 34   A r o 

(4.6)

K ν := I ro − B r o (C r o A νro B ro ) + C ro A νr o

(4.7)

A ro  0 K ν A =  0   0

x

A13

ro K ν A ro

0

0 A ro

0

0

where ro

is projective of the rank n ro − p .

110

4 The Moore-Penrose Inverse of the First Markov Parameter ro

Remark 4.2. The matrix K ν A r o in (4.6) has p(ν + 1) zero eigenvalues representing zeros at infinity of (4.1) and of its transfer-function matrix G(z). This claim follows from (2.2), Lemma 4.2 and (4.6). In fact, cancelling common factors on both sides of (4.4), we obtain ro

det(zI r o − K ν A ro )

(i)

= det(H 1T H 1 ) −1 det(H 2 H T2 ) −1 z p (ν +1) det Pd (z) , ro

 zI − A ro ro where Pd (z) =  roT  H1 C ro

− B ro H T2   stands for the system matrix of 0 

S( A ro , B ro H T2 , H1T C r o ) , i.e., of the squared down minimal subsystem of

(4.1). In order to prove (i), recall that G (z) = C(zI − A ) −1 B = C ro (zI ro − A ro ) −1 B ro = G ro (z)

and for S( A, B d , C d ) , where B d = BH T2 and C d = H1T C , we have G d (z) = C d ( zI − A) −1 B d = H1T C ro (zI ro − A ro ) −1 B ro H T2 = G dro (z) .

Now, by virtue of (4.6), the left-hand side of (4.4) can be written as det(zI − K ν A) = det(zI ro − A ro ) det(zI ro − K νro A ro ) det(zI ro − A r o ) det(zI ro − A ro ) ,

while det Pd (z) on the right-hand side of (4.4) takes the form det Pd (z) = det(zI − A) det G d (z) = det(zI − A) det G dro (z) ,

where det(zI − A) = det(zI ro − A ro ) det(zI ro − A ro ) det(zI ro − A ro ) det(zI ro − A ro )

and det(zI ro − A ro ) det G dro (z) = det Pdro (z) . ◊ Comparing the state matrix in the Kalman form (2.2) with K ν A in (4.6) and taking into account Corollary 2.1, we conclude that (4.6) discloses all decoupling zeros (including multiplicities) of (4.1). Unfortunately, in the


111

ro

general case, K ν A r o in (4.6) does not disclose all transmission zeros of G(z) (see Example 4.1). However, as we shall show in the next subsection, under additional assumptions concerning the rank of the first nonzero ro

Markov parameter, K ν A r o displays all transmission zeros of the system, whereas K ν A (taken in a special form) discloses all its invariant zeros. ro

Finally, observe that the structure of K ν A r o can be simplified. In fact, I n − p 0  transforming K νro in (4.7) into R ro K νr o R r−o1 =  ro  and writing 0 p   0 12   A11 r o A ro −1  as well as applying the transformation R ro A ro R ro as  21  A ro A 22 r o  diag (I ro , R r o , I r o , I ro ) to (2.2), we get matrices K ν A (4.8) and A (4.9) A r o   0    0  0 

 A ro   0    0  0 

x

x

A11 ro

A12 ro

0

0p

0

0

A ro

0

0

x A11 ro 21 A ro

A13 A12 ro 22 A ro

0

0

A ro

0

0

x   x    A 34  A r o 

A14   x  .  A 34  A r o 

(4.8)

(4.9)

Note that after this transformation the submatrix K νro A ro in (4.6) becomes  A11 A12  −1 ro R ro K νro A ro R ro =  ro  , where A11 r o has pν zero eigenvalues 0 0 p   representing zeros at infinity of the system (4.1) and among the remaining (nr o − p) − pν eigenvalues of A11 r o may appear some, not necessarily all,

of the transmission zeros of (4.1).

112


Example 4.1. Consider a reachable and observable system (4.1) with the matrices 0 0 A= 0  0

0 0 0 1  0 0 1 0 , B= 0 0 0 1   0 0 0 0

0 0 1 0 0 0  , C= . 0 0 − 1 1 0  1

Since det P (z) = z − 1 , we infer, via Proposition 2.3, that this system is nondegenerate and, via Corollary 2.3, Exercise 2.8.23 and Definition 2.3, that its Smith and invariant zeros coincide and the only Smith (transmission) zero of the system is λ = 1 . On the other hand, the first nonzero 1 0  Markov parameter equals CB =   (i.e., ν = 0, p = 1 ). The matrix 0 0  (4.6) takes the form 0 0 0 0  0 0 1 0   K0A =  0 0 0 1    0 0 0 0  and σ(K 0 A ) does not disclose the Smith zero of the system. ◊ 4.1.2 First Markov Parameter of Full Rank

Restricting our attention to systems with the first nonzero Markov parameter of full rank and assuming nondegeneracy when necessary, we observe that each invariant zero of the system (4.1) has to be an eigenvalue of K ν A . This implies in turn that each system zero (i.e., invariant, transmission or decoupling) of (4.1) is in σ(K ν A ) . Proposition 4.2. [73, 80] In (4.1) let the first nonzero Markov parameter have full column rank, i.e., let rank CA ν B = m . Then each system zero of (4.1) is in σ(K ν A ) . Moreover, λ is an invariant zero of the system if and only if λ ∈ σ(K ν A ) and an associated with λ eigenvector of K ν A lies in Ker C . Proof. Because K ν A discloses all decoupling zeros (see (4.6) and Corollary 2.1) and each transmission zero is also an invariant zero (see


113

Lemma 2.4), in the proof of the first claim we only need to show that each invariant zero is in σ(K ν A ) . However, since Ker CA ν B = {0} , the necessary condition for invariant zeros given in Proposition 3.4 (i) implies that g1 = 0 , i.e., each invariant zero is in σ(K ν A ) . The second claim follows from Propositions 3.3 and 3.4. In fact, suppose that λ ∈ σ(K ν A ) and x o ≠ 0 denotes an eigenvector associated with λ (i.e., λx o = K ν Ax o ) such that Cx o = 0 . Then, via Proposition 3.3, λ is an invariant zero; or, more precisely, the triple λ, x o , g = −(CA ν B) + CA ν +1x o satisfies (2.4). Conversely, suppose that λ is an invariant zero, i.e., a triple λ, x o ≠ 0, g satisfies (2.4). Then, via

Proposition 3.4 (i), λx o − K ν Ax o = Bg1 , where g1 ∈ Ker CA ν B , and Cx o = 0 . Since Ker CA ν B = {0} , we have λx o − K ν Ax o = 0 and Cx o = 0 and the proof is completed. ◊

Corollary 4.1. [73, 80] Let (4.1) be expressed in its Kalman form (2.2) and let the first nonzero Markov parameter CA ν B have full column rank. Then λ ∈ C is a transmission zero of the system (equivalently, a transmission zero of its transfer-function matrix G(z)) if and only if λ ∈ σ(K νro A ro ) (see (4.6)) and an associated with λ eigenvector of ro

K ν A r o lies in Ker C r o .

Proof. Because CA ν B = C r o A νr o B r o and C r o A νr o B r o is the first nonzero Markov parameter for the minimal subsystem S( A ro , B r o , C r o ) of (4.1), we can apply immediately the second claim of Proposition 4.2 to this subsystem. ◊ From Proposition 4.2 follows some useful characterization of invariant zeros of (4.1) as output decoupling zeros of a certain closed-loop (statefeedback) system. In order to show this, consider the system S(K ν A,B,C) obtained from (4.1) by introducing the state feedback matrix F = −(CA ν B) + CA ν +1 . Then we have the following. Proposition 4.3. [80] In S( A,B,C) (4.1) let the first nonzero Markov parameter CA ν B have full column rank. Then λ ∈ C is an invariant zero of S( A,B,C) if and only if λ is an o.d. zero of S(K ν A,B,C) .

114


Proof. The assertion is an immediate consequence of the second claim of Proposition 4.2 and of the definition of o.d. zeros (see (i) in the proof of Lemma 2.1). ◊ Remark 4.3. [80] In a suitable basis the above characterization of invariant zeros of S( A,B,C) as output decoupling zeros of S(K ν A,B,C) becomes essentially simpler. Denote by n ocl the rank of the observability matrix for S(K ν A,B,C) and by S cl ν the kernel of that matrix, i.e., n −1

Sνcl := h Ker C(K ν A ) l ⊆ R n . l =0

The number of o.d zeros of S(K ν A,B,C) (including multiplicities) is equal to n − nocl = dim S cl ν (see Appendix A.1 and Exercise 2.8.1). Let x' = Hx denote a change of basis which leads to the decomposition of S(K ν A,B,C) into an unobservable and an observable part, i.e., consider a system S((K ν A )' ,B' ,C' ) with the matrices

(i)

(K ν A )' o (K ν A )' =  0 

(K ν A )'12  B ' o  , B' =    , C' = [0 C'o ] , (K ν A )'o   B' o 

x' o  x' =   ,  x' o 

dim x' o = n − nocl ,

where the subsystem S((K ν A )' o , B' o , 0) is unobservable whereas S((K ν A )'o ,B' o ,C'o ) is observable. All output decoupling zeros (including multiplicities) of S((K ν A )' ,B' ,C' ) are characterized as the roots of the polynomial (see Appendix A.1 and Exercise 2.8.1) (ii)

χ cl o ( z ) = det ( zI ' o −(K ν A)' o ) ,

where I ' o is the identity matrix of the order n − nocl . ◊ Now, at the notation used in Remark 4.3, we can formulate the following algebraic characterization of invariant zeros. Corollary 4.2. In S(A,B,C) (4.1) let the first nonzero Markov parameter CA ν B have full column rank. Then:


115

(i)

invariant zeros of S(A,B,C) are the roots of the equation

χ cl o (z)

= det(zI ' o −(K ν A )' o ) = 0 . The number of these zeros (including

multiplicities) equals the dimension of the unobservable subspace for cl cl S(K ν A,B,C) , i.e., deg χ cl o ( z ) = n − no = dim S ν ;

(ii)

χ cl o ( z) is equal to the zero polynomial of S(A,B,C), i.e., invariant

and Smith zeros of the system are exactly the same objects (including multiplicities). Proof. (i) The claim is an immediate consequence of Proposition 4.3 and Remark 4.3. (ii) For the proof use Exercise 4.4.22 and Definition 2.3. ◊ We can pass now to the analysis of system zeros of (4.1) in the case when the first nonzero Markov parameter has full row rank and the system is nondegenerate. Proposition 4.4. [73, 80] In S(A,B,C) (4.1) let the first nonzero Markov parameter have full row rank, i.e., let rank CA ν B = r , and let S(A,B,C) be nondegenerate. Then: (i)

each system zero of S(A,B,C) is in σ(K ν A ) ;

(ii)

λ ∈ C is an invariant zero of S(A,B,C) if and only if λ ∈ σ(K ν A )

and an associated with λ eigenvector of K ν A lies in Ker C . Proof. (i) Since each decoupling zero of S(A,B,C) is an eigenvalue of K ν A (see (4.6) and Corollary 2.1) and each transmission zero is an invariant zero (see Lemma 2.4), it is sufficient to show that each invariant zero of the system lies in σ(K ν A ) . To this end, we shall show that if a triple λ, x o ≠ 0, g satisfies (2.4), then λ ∈ σ(K ν A ) and x o is an associated with λ eigenvector of K ν A satisfying Cx o = 0 . Let in the skeleton factorization CA ν B = H1H 2 be H1 = I r and H 2 = CA ν B and let a triple λ, x o ≠ 0, g satisfy (2.4). By virtue of Proposition 3.4, we obtain λx o − K ν Ax o = Bg1 , where Bg1 ∈ Ker CA ν , and Cx o = 0 . Suppose now that Bg1 ≠ 0 and consider an arbitrary complex number λ1 ∉ σ(K ν A ) . Define x1 := (λ1I − K ν A) −1 Bg1 and

116


g12 := −(CA ν B) + CA ν +1x1 . The definition (3.1) of K ν enables us to write K ν Ax1 − Ax1 = Bg12 ; moreover, K ν Bg12 = 0 . Since H1 = I r , by virtue

of Lemma 3.1(vi) and (vii), we can write C( zI − K ν A) −1 B = CA ν B / z ν +1

and Cx1 = λ−1(ν +1) CA ν Bg1 = 0 . In this way we have obtained the following relations λ1x1 − K ν Ax1 = Bg1 , K ν Ax1 − Ax1 = Bg12 and Cx1 = 0 . These relations mean that λ1 is an invariant zero of the system, for the triple λ1 , x1 ≠ 0, g1 + g12 satisfies (2.4). Since λ1 has been chosen arbitrarily, the system is degenerate. This contradiction indicates that Bg1 = 0 . Thus we have proved the implication: if a triple λ, x o ≠ 0, g satisfies (2.4), then λx o − K ν Ax o = 0 and Cx o = 0 . (ii) By virtue of the proof of (i), we only need to show the implication: if for some x o ≠ 0 is λx o − K ν Ax o = 0 and Cx o = 0 , then λ is an invariant zero of S(A,B,C). However, the validity of this implication follows immediately from Proposition 3.3. ◊ Corollary 4.3. [73, 80] Suppose that a nondegenerate system S(A,B,C) (4.1) is expressed in its Kalman form (2.2) and the first nonzero Markov parameter CA ν B of (4.1) has full row rank. Then λ ∈ C is a transmission zero of S(A,B,C) (equivalently, a transmission zero of its transfer-function matrix G(z)) if and only if λ ∈ σ(K νr o A r o ) (see (4.6)) and an associated with λ eigenvector of K νr o A r o lies in Ker C ro . Proof. We observe first that the minimal subsystem S( A ro , B r o , C ro ) of (4.1) is nondegenerate (otherwise S(A,B,C) would be degenerate) and its first nonzero Markov parameter equals CA ν B . Now, the claim follows immediately from Definition 2.1 (ii) and from Proposition 4.4 (ii) (when applied to S( A r o , B r o , C r o ) ). ◊ Under the assumptions of Proposition 4.4 all invariant zeros of (4.1) can be characterized as output decoupling zeros of the closed-loop system S(K ν A,B,C) obtained from S( A,B,C) via introducing the state feedback

matrix F = −(CA ν B) + CA ν +1 (cf., Proposition 4.3).

Proposition 4.5. [80] In a nondegenerate system S( A,B,C) (4.1) let the first nonzero Markov parameter CA ν B have full row rank. Then λ ∈ C is


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an invariant zero of S( A,B,C) if and only if λ is an output decoupling zero of S(K ν A,B,C) . Proof. The claim is an immediate consequence of Proposition 4.4 (ii) and of the definition of o.d. zeros (see (i) in the proof of Lemma 2.1). ◊ Remark 4.4. [80] The discussion in Remark 4.3 concerning the decomposition of S(K ν A,B,C) into an unobservable and an observable part remains valid also for the case considered in Proposition 4.5. The only difference is that, under the assumptions of Proposition 4.5, for the unobservable subspace of S(K ν A,B,C) the following relation holds Sνcl =

n −1 l =0

Ker C(K ν A ) l =

ν

Ker CA l = S ν .

l =0

This relation follows easily from Lemma 3.1 (vi) and from the skeleton factorization CA ν B = H1H 2 , where H1 = I r and H 2 = CA ν B . ◊ Finally, taking into account Remarks 4.3 and 4.4, we can establish the following result. Corollary 4.4. In a nondegenerate system S(A,B,C) (4.1) let the first nonzero Markov parameter CA ν B have full row rank. Then: (i)

invariant zeros of S(A,B,C) are the roots of the equation

χ cl o ( z ) = det ( zI ' o −(K ν A )' o ) = 0 . The number of these zeros (including

multiplicities) is equal to the dimension of the unobservable subspace for cl cl the system S(K ν A,B,C) , i.e., deg χ cl o ( z ) = n − no = dim S ν ;

(ii)

χ cl o ( z) is equal to the zero polynomial of S(A,B,C), i.e., invariant

and Smith zeros of S(A,B,C) are exactly the same objects (including multiplicities). Proof. (i) The claim is an immediate consequence of Proposition 4.5 and Remarks 4.3 and 4.4. (ii) The proof will be given in Chapter 5, Sect. 5.1 (see the proof of Proposition 5.4 (ii)). ◊ Example 4.2. Consider a reachable and observable system (4.1) with the matrices

118


0 − 1 − 1 / 3  A= 0 − 2 / 3 0  ,  0 0 − 1

1 0 1  1 0 0  B = 0 1 0 , C =   0 1 0 1 0 1

and its transfer-function matrix 3z   (3z + 1)(z + 1) G (z) =    0

0 3 3z + 2

3z  (3z + 1)(z + 1)  .   0

From Example 3.2 (see Sect. 3.2) we know that this system is nondegenerate. Moreover, CB has full row rank (i.e., ν = 0, p = 2 ) and  0 0 0 K 0 A =  0 0 0 . 1 / 3 0 0

Applying to S(K 0 A, B, C) the change of coordinates x' = Hx , with 0 0 1  H = 1 0 0 , 0 1 0

we obtain the decomposition into an unobservable and an observable part (cf., Remarks 4.3 and 4.4) 1 0 1 0 1 / 3 0   (K 0 A)' = 0 0 0 , B' = 1 0 1 , 0 1 0 0 0 0

0 1 0 C' =  , 0 0 1

where 0 0 (K 0 A)' o = [0] and (K 0 A)' o =  . 0 0

According to Corollary 4.4, χ cl o ( z ) = z and the only invariant zero of S( A, B, C) is λ = 0 . The remaining zero eigenvalues of (K 0 A)' represent zeros at infinity of S( A, B, C) (cf., Remark 4.1). ◊


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Example 4.3. In S(A,B,C) (4.1) let 0 0 1 / 3 0   A =  0 − 1 / 3 0  , B =  − 1 ,  0  − 1 0 − 1

1 − 1 0 C= .  0 2 1

1 The matrix CB =   has full column rank (i.e., ν = 0, p = 1 ). In the  − 3 skeleton factorization of CB we take H1 = CB, H 2 = 1 . Then

(CB) + = [1 / 10 − 3 / 10] , 0 0   1  K 0 = I − B(CB) C = 1 / 10 3 / 10 − 3 / 10 , 1 / 10 − 7 / 10 7 / 10  +

0 0   1/ 3  K 0 A = 1 / 30 − 1 / 10 3 / 10  . 1 / 30 7 / 30 − 7 / 10

The system S(K 0 A, B, C) is observable, therefore, by virtue of Corollary 4.2, S(A,B,C) has no invariant zeros. On the other hand, S(A,B,C) is observable and unreachable (the reachability matrix has rank 2). Using the change of coordinates x' = Hx , with 0 − 3 / 2 1 / 2  H = 0 − 3 / 2 3 / 2 , 1 0 0 

we transform S(A,B,C) into its Kalman form S( A ' , B' , C' ) (2.2) with matrices A', B', C' of the form (i)

0 − 1 / 3 0  1 − 4 / 3 0  ,   0 0 1 / 3

1  0 ,   0

 0 − 1 / 3 1  − 3 5 / 3 0 ,  

0 − 1 / 3  where A' r o =   , A' r o = [1 / 3] . Hence, by virtue of Corollary 2.1, 1 − 4 / 3 S(A,B,C) has an input decoupling zero at 1/3. ◊

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4.2 System Zeros in Proper Systems Consider the discrete-time proper system x(k + 1) = Ax(k ) + Bu (k ) , x∈ Rn ,u∈ Rm, y ∈ Rr , k ∈ N , y (k ) = Cx(k ) + Du(k )

(4.10)

where 0 < rank D = p ≤ min{m, r} . For (4.10) consider a class of real-valued input sequences of the form v (k , x o ) := −D + C( A − BD + C) k x o , x o ∈ R n , k ∈ N .

(4.11)

Lemma 4.3. [73, 80] For a given input v (k , x o ) (4.11) a solution of the state equation of (4.10) corresponding to an arbitrary initial state x(0) ∈ R n has the form x(k ) = A k (x(0) − x o ) + ( A − BD + C) k x o ,

(4.12)

while the system response is y (k ) = CA k ( x(0) − x o ) + (I r − DD + )C( A − BD + C) k x o .

(4.13)

Proof. The proof follows by checking that v (k , x o ) in (4.11) and x(k ) in (4.12) satisfy the state equation of (4.10) and by the uniqueness of solutions. The details are left to the reader as an exercise. ◊ Let D = D1D 2 denote a skeleton factorization of D. With S(A,B,C,D) (4.10) we associate an auxiliary square (p-input, p-output) system S( A, B d , C d , D d ) , where B d = BD T2 , C d = D1T C, D d = D1T DD T2 (note that D d is nonsingular), obtained from S(A,B,C,D) by introducing pre-

compensator D T2 and postcompensator D1T . The symbol Pd (z) stands for  zI − A − B d  the system matrix of the associated system, i.e., Pd (z) =  . D d   Cd

Lemma 4.4. [73, 80] det (zI − ( A − BD + C)) = det(D1T D1 ) −1 det(D 2 D T2 ) −1 det Pd (z)

Proof. We can write

(4.14)

4.2 System Zeros in Proper Systems

I zI − A − B d   − D −1C  C  Dd    d d d

121

0  zI − ( A − B d D −1C d ) − B d  d =  I p   0 D d  

and, consequently, det Pd (z) = det D d det(zI − ( A − B d D d−1C d )) . Finally, note that D d = (D1T D1 )(D 2 D T2 ) and A − BD + C = A − B d D d−1C d . For the proof of the last relation we observe that B d D d−1C d = BD T2 (D 2 D T2 ) −1 (D1T D1 ) −1 D1T C = BD 2+ D1+ C = BD + C .

◊

4.2.1 Decoupling and Transmission Zeros

When S(A,B,C,D) (4.10) is taken in its Kalman canonical form (2.2), then A r o   0 + A − BD C =   0  0 

A12 − B r o D + C r o

A13

A ro − B ro D + C ro 0

0 A ro

0

0

A14 − B r o D + C r o   A 24 − B r o D + C r o  . A 34   A ro 

(4.15) Remark 4.5. Comparing the state matrix in the Kalman form (2.2) with A − BD + C in (4.15) and taking into account Corollary 2.1, we conclude that (4.15) discloses all decoupling zeros (including multiplicities) of the system S(A,B,C,D) in (4.10). ◊

If λ ∈ σ( A r o − B ro D + C ro ) and the corresponding eigenvector x or o satisfies (I r − DD+ )Cro xoro = 0 , then, by virtue of Proposition 3.1, λ is an invariant zero of the minimal subsystem S( A ro , B r o , C ro , D) of S(A,B,C,D). On the other hand, by virtue of Definition 2.1, λ is also a transmission zero of the transfer-function matrix G(z) of S(A,B,C,D). Of course, this λ is an invariant zero of the whole system S(A,B,C,D) as

well. Unfortunately, in the general case, A ro − B ro D + C r o does not disclose all transmission zeros of G(z) (see Example 4.4 below). However, as we shall demonstrate in the next section, under some additional assump-

122


tions, A ro − B ro D + C ro displays all transmission zeros of S(A,B,C,D), while A − BD + C , when taken in a special form, discloses all invariant zeros of this system. Example 4.4. Consider a reachable and observable system (4.10) with the matrices 0 0  0 A= 0  0 0 

1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 , 0 0 0 1 0  0 0 0 0 1 0 0 0 0 0

1 1 0 0 0 0  C= , 0 0 0 1 − 1 0 

0 0  1 B= 0  0 0 

0 0 0 , 0  0 1

1 0 D= . 1 0

Since det P (z) = −(z 3 + z + 1)(z − 1) , we infer that normal rank P (z) = 8 and then, via Proposition 2.3, that the system is nondegenerate and, via Corollary 2.3, Exercise 2.8.23 and Definition 2.3, that Smith and invariant zeros coincide, and invariant zeros of the system are exactly the roots of the polynomial det P (z) . On the other hand, the spectrum of A − BD + C 1 1 is equal to σ( A − BD + C) = {z : z 3 (z 3 + z + ) = 0} . Hence A − BD + C 2 2 does not disclose invariant zeros of S(A,B,C,D). ◊ 4.2.2 Proper Systems with Matrix D of Full Rank

We shall show below that under some additional assumptions concerning the system S(A,B,C,D) (4.10) (in particular, concerning the rank of matrix D ) all system zeros (i.e., decoupling, transmission and invariant zeros) of S(A,B,C,D) have to appear as some eigenvalues of A − BD + C . Proposition 4.6. [73, 80] In S(A,B,C,D) (4.10) let D have full column rank, i.e., let rank D = m . Then each system zero of S(A,B,C,D) is in σ( A − BD + C) . Moreover, λ is an invariant zero of S(A,B,C,D) if and

only if λ ∈ σ( A − BD + C) and an associated eigenvector x o satisfies

(I r − DD + )Cx o = 0 .


123

Proof. By virtue of Corollary 2.5, the system is nondegenerate. Because A − BD + C discloses all decoupling zeros (see Remark 4.5) and each transmission zero is also an invariant zero (see Lemma 2.4), in the proof of the first claim we only need to show that each invariant zero of

S(A,B,C,D) is in σ( A − BD + C) . Since D has full column rank, we have D + D = I m . Suppose now that λ ∈ C is an invariant zero of S(A,B,C,D),

i.e., a triple λ, x o ≠ 0, g satisfies (i)

λx o − Ax o = Bg

and

Cx o + Dg = 0 .

Premultiplying the second identity in (i) by D + , we obtain g = −D + Cx o . Introducing this g into the first identity in (i), we obtain (λI − ( A − BD + C))x o = 0 , i.e., λ ∈ σ( A − BD + C) . Note also that introducing this g into the second identity in (i), we obtain (I r − DD + )Cx o = 0 . Thus we have shown that if λ ∈ C is an invariant

zero of S(A,B,C,D), then λ is an eigenvalue of A − BD + C . Moreover, we have observed that the corresponding to λ state-zero direction x o is also an associated with λ eigenvector of A − BD + C which satisfies

(I r − DD + )Cx o = 0 . In order to prove the second claim of the proposition, it suffices to show

that if λ ∈ σ( A − BD + C) and an associated eigenvector x o satisfies (I r − DD + )Cx o = 0 , then λ is an invariant zero of the system. This implication, however, is valid by virtue of Proposition 3.1. ◊

Corollary 4.5. [73, 80] Suppose that S(A,B,C,D) (4.10) is expressed in its Kalman form (2.2) and D has full column rank. Then λ ∈ C is a transmission zero of S( A, B, C, D) (or, equivalently, a transmission zero of its transfer-function matrix G(z)) if and only if λ ∈ σ( A r o − B r o D + C r o ) and an associated with λ eigenvector of A r o − B ro D + C r o lies in the subspace Ker (I r − DD + )C r o .

Proof. The assertion follows immediately from Definition 2.1 (ii) and from Proposition 4.6 (when applied to the subsystem S( A ro , B r o , C r o ,D) ). ◊

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From Proposition 4.6 follows some useful characterization of invariant zeros of S( A, B, C, D) (4.10) as output decoupling zeros of a certain closed-loop (state feedback) system. In order to show this, consider the system S( A − BD + C, B, (I r − DD + )C, D) obtained from S( A, B, C, D) by introducing the state feedback matrix F = −D + C . Then we have. Proposition 4.7. [80] In S( A, B, C, D) (4.10) let D have full column rank. Then λ ∈ C is an invariant zero of S( A, B, C, D) if and only if λ is an output decoupling zero of S( A − BD + C, B, (I r − DD + )C, D) . Proof. The claim is an immediate consequence of Proposition 4.6 and the definition of o.d. zeros (see (i) in the proof of Lemma 2.1 (i)). ◊ Remark 4.6. [80] In a suitable basis the above characterization of invariant zeros of S(A,B,C,D) becomes essentially simpler. Denote by n ocl the rank of the observability matrix for S( A − BD + C,B,(I r − DD + )C,D) and by S cl D the kernel of that matrix, i.e., S cl D :=

n −1 l =0

Ker{(I r − DD + )C( A − BD + C) l } ⊆ R n .

The number of o.d. zeros in S( A − BD + C,B,(I r − DD + )C,D) (including multiplicities) is equal to n − nocl = dim S cl D (see Appendix A.1 and Exercise 2.8.1). Let x' = Hx denote a change of basis which leads to the decomposition of S( A − BD + C,B,(I r − DD + )C,D) into an unobservable and an observable part, i.e., consider a system S(( A − BD + C)' ,B' ,((I r − DD + )C)' ,D' ) with the matrices ( A − BD + C)' o ( A − BD + C)' =  0 

(i)

[

((I r − DD + )C)' = 0 o

B ' o  ( A − BD + C)'12   , B' =  , + ( A − BD C)'o   B' o 

]

((I r − DD + )C)'o ,

D' = D ,


x' o  x' =   ,  x' o 

125


where the subsystem S(( A − BD + C)' o , B' o , 0 o ,D' ) is unobservable, while the subsystem S(( A − BD + C)'o ,B' o ,((I r − DD + )C)' o ,D' ) is observable. Thus all output decoupling zeros (including multiplicities) of the system S(( A − BD + C)' ,B' ,((I r − DD + )C)' ,D' ) are characterized as the roots of the polynomial (see Appendix A.1 and Exercise 2.8.1) (ii)

+ χ cl o ( z) = det( zI ' o −( A − BD C)' o ) ,

where I ' o is the identity matrix of the order n − nocl . ◊ Now, at the notation used in Remark 4.6, we can formulate the following algebraic characterization of invariant zeros. Corollary 4.6. In S(A,B,C,D) (4.10) let D have full column rank. Then: (i)

invariant zeros of S(A,B,C,D) are the roots of the equation

= det(zI ' o −( A − BD + C)' o ) = 0 . The number of these zeros (including multiplicities) equals the dimension of the unobservable subspace χ cl o (z)

for the system S( A − BD + C,B,(I r − DD + )C,D) , i.e., cl cl deg χ cl o ( z) = n − no = dim S D ;

χ cl (ii) o ( z) is equal to the zero polynomial of S(A,B,C,D), i.e., Smith zeros and invariant zeros are exactly the same objects (including multiplicities).

Proof. (i) The claim is an immediate consequence of Proposition 4.7 and Remark 4.6. (ii) For the proof use Exercise 4.4.23 and Definition 2.3. ◊ We can pass now to the analysis of system zeros of S( A, B, C, D) (4.10) in the case when D has full row rank and the system is nondegenerate. Proposition 4.8. [73, 80] In S(A,B,C,D) (4.10) let D have full row rank, i.e., let rank D = r , and let the system be nondegenerate. Then: (i) each system zero of S(A,B,C,D) is in σ( A − BD + C) ; (ii) λ is an invariant zero of S(A,B,C,D) if and only if λ ∈ σ( A − BD+ C) .

126


Proof. (i) Because all decoupling zeros of all kinds (o.d., i.o.d. and i.d.) are in σ( A − BD + C) (see Remark 4.5) and each transmission zero is an invariant zero (see Lemma 2.4), we can restrict our attention to invariant zeros only. We have to show first that if λ ∈ C is an invariant zero of S(A,B,C,D), then λ is an eigenvalue of A − BD + C . Let a triple

λ, x o ≠ 0, g satisfy (2.4). To this triple we apply Proposition 3.2. We decompose g = g1 + g 2 as in Proposition 3.2 and then we suppose that Bg1 ≠ 0 . Let λ1 ∉ σ( A − BD + C) . This means that (λ1I − ( A − BD + C)) is nonsingular and, in consequence, there exists a vector 0 ≠ x1 ∈ C n such

that λ1x1 − ( A − BD + C)x1 = Bg1 . Denote g12 = −D + Cx1 . Adding the last two equalities by sides (with the latter premultiplied by B), we obtain the equality λ1x1 − Ax1 = B(g1 + g12 ) . Since g1 ∈ Ker D and DD + = I r , we can write Cx1 + D(g1 + g12 ) = Cx1 + Dg12 = Cx1 − DD + Cx1 = 0 . Hence the triple λ1 , x1 , g1 + g12 satisfies (2.4). This means that λ1 is an invariant zero of S(A,B,C,D). Since λ1 has been chosen arbitrarily, S(A,B,C,D) is degenerate. This contradiction (with the assumption of nondegeneracy) means that Bg1 = 0 . Thus, by virtue of Proposition 3.2, we can write λx o − ( A − BD + C)x o = 0 , i.e., λ is an eigenvalue of A − BD + C . (ii) In order to complete the proof, we only need to show that each

eigenvalue of A − BD + C is an invariant zero of S(A,B,C,D). To this end,

suppose that λ ∈ σ( A − BD + C) and denote by x o an eigenvector associ-

ated with λ . Moreover, denote g = −D + Cx o . Now, using DD + = I r , we can easily verify that the triple λ, x o , g satisfies (2.4). ◊ Corollary 4.7. [73, 80] Suppose that in a nondegenerate proper system S(A,B,C,D) (4.10) matrix D has full row rank. Moreover, let S(A,B,C,D) be expressed in its Kalman form (2.2). Then a number λ is a transmission zero of S(A,B,C,D) (or, equivalently, of its transfer-function matrix G (z) ) if and only if λ ∈ σ( A r o − B r o D + C ro ) . Proof. The minimal subsystem S( A ro , B r o , C ro , D) of S(A,B,C,D) is nondegenerate (otherwise, S(A,B,C,D) would be degenerate). Now, the


127

claim follows immediately from Definition 2.1 (ii) and from Proposition 4.8 (ii) when applied to S( A r o , B r o , C r o , D) . ◊ Remark 4.7. [80] Observe that if D has full row rank, the closed-loop system obtained by introducing into (4.10) the state feedback matrix F = −D + C has the form S( A − BD + C, B, 0, D) , i.e., it is completely un-

observable. This is due to the fact that I r − DD + = 0 . In consequence, at n the notation used in Remark 4.6, nocl = 0 and Scl D = R . Thus Proposition 4.8 (ii) tells us also that λ is an invariant zero of S(A,B,C,D) if and only if

λ is an output decoupling zero of S( A − BD + C, B, 0, D) , i.e., if and only

+ if λ is a root of the polynomial χ cl o ( z) = det ( zI − ( A − BD C)) . ◊

Corollary 4.8. In a nondegenerate system S(A,B,C,D) (4.10) let D have full row rank. Then: (i)

invariant zeros of S(A,B,C,D) are the roots of the equation

χ cl o (z)

= det (zI − ( A − BD + C)) = 0 . The number of these zeros (including

multiplicities) equals n ; (ii) χ cl o ( z) is equal to the zero polynomial of S(A,B,C,D), i.e., the Smith and invariant zeros are exactly the same objects (including multiplicities). Proof. (i) The proof is an immediate consequence of Proposition 4.8 (ii) and Remark 4.7. (ii) The proof will be given in Chapter 5, Sect. 5.3 (see the proof of Proposition 5.8 (ii)). ◊ Example 4.5. In S(A,B,C,D) (4.10) let 1  1 0  A = 0 1 / 2 0  , 0 0 1 / 3

1 0 1  B = 0 1 0 , 1 0 1

1 0 0 C= , 0 1 0 

− 4 / 3 0 − 4 / 3 . D= 1 0   0

Simple verification shows that this system is minimal, i.e., reachable and observable. Moreover, normal rank P(z) = 5 . By virtue of Proposition 2.3 (see (2.13)), the system is nondegenerate. In order to find out invariant

128


(transmission) zeros, we use Corollary 4.8. In a skeleton factorization D = D1D 2 we take D1 = I r and D 2 = D . Then +

D =

D T2 (D 2 D T2 ) −1

0 1   − 3 / 8 0 7 / 4    + 1 , A − BD C =  0 = 0 − 1 / 2 0  . − 3 / 8 0 3 / 4 0 1 / 3

The polynomial 1 1 2 25 + χ cl z− ) o ( z) = det( zI − ( A − BD C)) = ( z + )(z − 6 4 2 determines all invariant zeros of S(A,B,C,D). ◊

4.3 Systems with Vector Relative Degree This section contains a concise treatment of systems with the so-called vector relative degree. All proofs are omitted here (they can be found in [74] or they can be easily adopted from the continuous-time case – see [80]) Consider a square (m-input, m-output) discrete-time system S(A,B,C) (2.1) of the form x(k + 1) = Ax(k ) + Bu(k ) , x ∈ Rn , u ∈ Rm , y ∈ Rm , k ∈ N y (k ) = Cx(k )

in which

(4.16)

 c1   .  C=   .    c m 

and c s , 1 ≤ s ≤ m , stand for the consecutive rows of C. Denote by Ss ( A, B, c s ) , s = 1,..., m , the subsystems of (4.16) with m inputs and one output. The first nonzero Markov parameter of Ss ( A, B, c s ) we denote by c s A νs B , i.e., we have c s B = ... = c s A νs −1B = 0 and c s A νs B ≠ 0 .

for some integer 0 ≤ ν s ≤ n − 1 .

(4.17)

4.3 Systems with Vector Relative Degree

129

Define a square mxm matrix  c A ν1 B    1 .  L :=    .   c m A ν m B 

(4.18)

whose rows are formed from the first nonzero Markov parameters of subsequent subsystems Ss ( A, B, c s ) , s = 1,..., m . We say that the system (4.16) has vector relative degree r [20], where  ν1 + 1   .  , r=  .    ν m + 1

if and only if L is nonsingular. In the remainder of this section it is assumed that (4.16) has vector relative degree r. Remark 4.8. Recall that a square MIMO system S(A,B,C) (2.1) is said to be diagonally decoupled if its transfer-function matrix is diagonal and nonsingular. As is well known [10], nonsingularity of L is a sufficient and necessary condition for (4.16) to be diagonally decoupled by a constant state feedback and a nonsingular precompensator. ◊ Set  c A ν1    1 .   M := .  .    c m A ν m 

(4.19)

L = MB .

(4.20)

K := I − BL−1M .

(4.21)

Then

Define a nxn matrix

130


Lemma 4.5. The matrix K (4.21) has the following properties: K2 = K ;

(i) (ii)

Σ := {x : Kx = x} = Ker M ,

dim Σ = n − m ;

Ω := {x : Kx = 0} = Im B ,

dim Ω = m ;

R n (C n ) = Σ ⊕ Ω ;

(iii)

KB = 0 ,

MK = 0 ;

(iv)

c A l c s (KA) l =  s  0

for

0 ≤ l ≤ νs

for

l ≥ νs + 1

,

s = 1, ... , m . ◊

The following result characterizes invariant zeros of (4.16) as some eigenvalues of the matrix KA . Proposition 4.9. A number λ ∈ C is an invariant zero of (4.16) if and only if λ ∈ σ(KA) and there exists an associated eigenvector x o such that x o ∈ Ker C . ◊

Moreover, we have the following. Lemma 4.6. If λ ∈ C is an invariant zero of (4.16), i.e., a triple λ, x o ≠ 0, g satisfies (2.4), then Re x o ∈ S and Im x o ∈ S ,

(4.22)

where m

νs

S := h ( h Ker c s A l ) , s =1 l = 0

(4.23)

and the subspace S satisfies S ⊆ Ker M = Σ . Moreover, the input-zero direction g is determined by x o as g = −L−1MAx o . ◊

(4.24)

Remark 4.9. Proposition 4.9 can be formulated in a somewhat more detailed form. Namely, a triple λ, x o ≠ 0, g satisfies (2.4) if and only if λ is an eigenvalue of KA , x o is an associated eigenvector which lies in Ker C and g = −L−1MAx o . ◊


Lemma 4.7.

131

zI − A − B  be the system matrix for (4.16). Let P (z) =  0   C

Then det(zI − KA) = det(L−1 )z m + ( ν1 + ... + ν m ) det P (z) . ◊

(4.25)

Remark 4.10. From (4.25) it follows that det P (z) is not the identically zero polynomial. This implies (see Exercise 2.8.23) that a complex number is an invariant zero of (4.16) if and only if it is a Smith zero of (4.16) (i.e., if and only if it is a root of det P (z) ). This fact and Lemma 4.7 tell us that the number of invariant zeros of (4.16) is equal to n − m − (ν1 + ... + ν m ) . In particular, the system has no invariant zeros if and only if n = m + (ν1 + . . . + ν m ) and this is possible merely in the case when det P (z) = const ≠ 0 . ◊ Lemma 4.8. In the system (4.16) let u o (k ) = −L−1MA(KA) k x o ,

xo ∈ R n , k ∈ N

(4.26)

denote an input vector sequence. Then: (i)

the corresponding solution of the state equation which passes at

k = 0 through an arbitrarily chosen initial state x(0) ∈ R n has the form x(k ) = A k (x(0) − x o ) + (KA) k x o ,

k∈N

(4.27)

and the system response is equal to y (k ) = CA k (x(0) − x o ) + C(KA) k x o ,

(4.28)

where the s-th row, s = 1, ..., m , in the term C(KA) k x o has the form c s A k x o for 0 ≤ k ≤ ν s ;   for k ≥ ν s + 1  0

(ii)

if x o ∈ S , where S=

m

(

νs

(4.29)

Ker c s A l ) ⊆ R n (see (4.23)), and

s =1 l = 0

x(0) = x o , then for each k ∈ N we have x(k ) = (KA ) k x o ∈ S and y (k ) = C(KA) k x o = 0 . ◊

132


Remark 4.11. Note that the first component on the right-hand side of (4.27) is a solution of the homogeneous equation x(k + 1) = Ax(k ) at the initial condition x(0) − x o . The second component, i.e., (KA ) k x o , is a particular solution of the nonhomogeneous equation x(k + 1) = Ax(k ) + Bu o (k ) which passes through x o at k = 0 . ◊ Remark 4.12. From (4.28) and (4.29) it follows that if x(0) = x o , then for all k satisfying k ≥ max s =1,..., m {ν s } + 1 we have y (k ) = 0 , i.e., the system response becomes equal to zero after a finite number of steps. Due to this property, the input sequence (4.26) can be called almost output-zeroing. On the other hand, if (4.16) is asymptotically stable, then y (k ) → 0 as k → ∞ at any fixed points x(0) and x o of the state space (not necessarily

satisfying the condition x(0) = x o ), although, in general, the sequence (KA ) k x o (see (4.27)) does not need to be bounded. ◊

Remark 4.13. The point (ii) in Lemma 4.8 tells us that any pair (x o ,u o (k )) , where x o ∈ S and u o (k ) has the form (4.26), is an outputzeroing input for (4.16). We can also prove the converse implication. ◊

Lemma 4.9. Let a pair (x o ,u o (k )) be an output-zeroing input for (4.16). Then x o ∈ S and u o (k ) has the form (4.26). Moreover, the corresponding solution has the form x o (k ) = (KA) k x o and is contained in S. ◊ From Lemmas 4.8 and 4.9 we obtain the following characterization of the output-zeroing problem for decouplable systems. Proposition 4.10. A pair (x o ,u o (k )) is an output-zeroing input for the system (4.16) if and only if x o ∈ S and u o (k ) = −L−1MA (KA) k x o . Moreover, the solution of the state equation corresponding to an outputzeroing input (x o ,u o (k )) is of the form x o (k ) = (KA) k x o and is contained in the subspace S. ◊ Remark 4.14. When an output-zeroing input (x o ,u o (k )) is applied to the system (4.16), the zero dynamics are governed by the equation x(k + 1) = KAx(k ) and initial states x(0) ∈ S . ◊


133

If S(A,B,C) (4.16) is expressed in its Kalman form (2.2), then A r o  0 KA =   0   0

x

A13

K r o A ro 0

0 A ro

0

0

x  x  , A 34   A r o 

(4.30)

where K r o := I ro − B r o L−1 M r o

(4.31)

is idempotent of the rank n ro − m (note that for the minimal subsystem S( A r o , B ro , C ro ) we determine M r o via (4.19) and L r o via (4.18); moreover, L ro = L ). As the following result states, KA in (4.30) characterizes all system zeros of S(A,B,C) (4.16). Proposition 4.11. Suppose that S(A,B,C) (4.16) is expressed in its Kalman canonical form (2.2). Then: (i) the roots of the polynomial det(zI r o − A r o ) det(zI r o − A r o ) represent all o.d. zeros; (ii) the roots of the polynomial det(zI r o − A r o ) represent all i.o.d. zeros; (iii) the roots of the polynomial det(zI r o − A ro ) det (zI ro − A r o ) represent all i.d. zeros; (iv) each decoupling (i.e., i.d., i.o.d. or o.d.) zero is an invariant zero; (v) the polynomial det( zI r o − K ro A r o ) has m + (ν1 + ... + ν m ) roots equal to zero which represent zeros at infinity of S(A,B,C). The remaining n ro − m − (ν1 + ... + ν m ) roots of this polynomial represent all transmission zeros of the system (or, equivalently, of its transfer-function matrix). ◊ Remark 4.15. Observe that for the minimal subsystem S( A r o , B ro , C ro ) of (4.16) we have (cf., (4.25) and Remark 4.2) det(zI ro − K r o A ro ) = det(L−1 )z m + (ν1 + ... + ν m ) det Pr o (z) ,

where Pr o (z) stands for the system matrix of S( A r o , B ro , C ro ) . ◊

(4.32)

134


Corollary 4.9. Suppose that (4.16) is expressed in its Kalman form (2.2). Then λ ∈ C is a transmission zero of the system (or, equivalently, a transmission zero of its transfer-function matrix) if and only if λ ∈ σ(K r o A ro ) and an associated with λ eigenvector of K r o A r o lies in Ker C r o . ◊ From Proposition 4.9 follows some useful characterization of invariant zeros of the system (4.16) as output decoupling zeros of a certain closedloop system. To this end, we consider the system S(KA,BL−1 ,C) obtained from (4.16) by introducing the state feedback matrix F = −L−1MA and precompensator P = L−1 . Proposition 4.12. A number λ ∈ C is an invariant zero of the system (4.16) if and only if λ is an output decoupling zero of S(KA,BL−1 ,C) . ◊

Remark 4.16. In a suitable basis the above characterization of invariant zeros of (4.16) as output decoupling zeros of S(KA,BL−1 ,C) becomes essentially simpler. To this end, denote by nocl the rank of the observability matrix for S(KA,BL−1 ,C) and by S cl the kernel of that matrix, i.e., the unobserv-

able subspace S cl =

n −1

Ker C(KA) l for S(KA,BL−1 ,C) . The number of

l =0

output decoupling zeros (including multiplicities) of S(KA,BL−1 ,C) is equal to n − nocl = n − m − (ν1 + . . . + ν m ) = dim S cl . On the other hand, using Lemma 4.5 (iv), we can note that the following relation holds

(i)

C    C(KA)  m νs = Scl = Ker  ( Ker c s A l ) = S  s =1 l = 0  .  n −1   C(KA)

and consequently, dim S = n − m − (ν1 + . . . + ν m ) . Let x' = Hx denote a change of basis which leads to the decomposition of S(KA,BL−1 ,C) into an unobservable and an observable part, i.e., consider a system S((KA)' ,(BL−1 )' ,C' ) with the matrices


(KA )' o (KA)' =   0

(KA)'12  , (KA)'o  C' = [0 o

(ii) x' o  x' =   ,  x' o 

135

(BL−1 )' o  (BL−1 )' =  , −1  (BL )' o  C' o ] ,

dim x' o = n − nocl = n − m − (ν1 + . . . + ν m ) ,

where the subsystem S((KA)' o , (BL−1 )' o , 0 o ) is unobservable, whereas S((KA)' o ,(BL−1 )' o ,C'o ) is observable. Thus all o.d. zeros (including

multiplicities) of S((KA)' ,(BL−1 )' ,C' ) are the roots of the polynomial (see Appendix A.1) (iii)

χ cl o ( z ) = det ( zI ' o −(KA )' o ) ,

where I ' o is the identity matrix of the order n − m − (ν1 + . . . + ν m ) . In the new coordinates the output-zeroing problem for the system S( A ' , B' , C' ) , where A ' = HAH −1 , B' = HB , C' = CH −1 , becomes significantly simpler. At first, the image of the subspace S in (i) under H (i.e., S' = H(S) ) takes the form    x' o  S' = x' =   : x'o = 0 ,  x'o   

where each vector x' o ∈ R n - m - (ν 1 + ...+ ν m ) is spanned by eigenvectors and pseudoeigenvectors (or their real and imaginary parts) corresponding to the eigenvalues of (KA )' o (i.e., to invariant zeros of S(A,B,C) (4.16)). Moreover, the general form of output-zeroing inputs for S( A ' , B' , C' ) is  x' o  −1 ((KA)' o ) k x' o    , L MAH −1    ,    0  0     

whereas the zero dynamics for this system are described as x' o (k + 1) = (KA)' o x' o (k ) . ◊

136


Finally, at the notation used in Remark 4.16, we can formulate the following algebraic characterization of invariant zeros of (4.16). Corollary 4.10. Consider the system S(A,B,C) (4.16). Then: (i) invariant zeros of the system are the roots of the equation cl χ o (z) = det(zI ' o −(KA )' o ) = 0 . The number of these zeros is equal to n − m − (ν1 + ... + ν m ) ; χ cl (ii) o ( z) is equal to the zero polynomial of S(A,B,C), i.e., invariant and Smith zeros of S(A,B,C) are exactly the same objects (including multiplicities). ◊

Example 4.6. In S(A,B,C) (4.16) let 0 0  A = 0 0  0

1 0 0 1 0 0 0 0 0 0

0 0  0 0 1 −1 2  , B =  0 0 1    − 9 − 6 0 0 0

0 0 2 , C =  2 1 0 0 0 . 0 0 1 1 1    0  1

According to (4.17), c1B = [0 0], c1AB = [1 2] and c 2 B = [1 3] , i.e., 2 ν1 = 1, ν 2 = 0 and r =   . For L (4.18), M (4.19) and K (4.21) we have 1  1 0 0 0 0  0 1 0 0 0   1 2 0 2 1 0 0  , , K = 0 − 2 0 0 0 , L= M=   1 3 0 0 1 1 1  0 0 0 1 0   0 2 0 − 1 0

whereas KA and its characteristic polynomial are equal respectively to 0 0  KA = 0 0  0

0 0  0 0 1  0 − 1

1 0 0 0 1 0 0 −2 0 0 0

0 2

and

det (zI − KA) = z 3 (z + 1)(z + 2) .

From (4.25) and Remark 4.10 we conclude that the only invariant zeros of S(A,B,C) are λ = −1 and λ = −2 . ◊

4.4 Exercises

137

Example 4.7. Consider a reachable and observable system (4.16) with the matrices 0 0 A= 0  0

0 0 0 0 1 0 , 0 0 1  0 0 0

1 0 B= 0  0

0 0 , 0  1

1 0 0 0 C= . 0 − 1 0 0

According to (4.17), we obtain c1B = [1 0] , c 2 B = c 2 AB = [0 0] and 1 c 2 A 2 B = [0 − 1] , i.e., ν1 = 0, ν 2 = 2 and r =   . Matrices L (4.18), M 3 (4.19) and K (4.21) are of the form 0 0 0 0 0 1 0 0 1 0 0 0  1 0  ,  , = , K = M L= 0 0 0 − 1    0 0 1 0 0 − 1       0 0 0 0 whereas 0 0 KA =  0  0

0 0 0 0 1 0 0 0 1  0 0 0

(note that KA = A ).

Since S(KA, BL−1 , C) is observable, S(A,B,C) has no invariant zeros (see Proposition 4.12 or Corollary 4.10 (i)). ◊

4.4 Exercises 4.4.1. Show that if in S(A,B,C) (4.1) is K ν B = 0 , then each invariant zero lies in σ(K ν A ) , i.e., the system is nondegenerate. Hint. Let a triple λ, x o ≠ 0, g satisfy (2.4), i.e., let λx o − Ax o = Bg , Cx o = 0 . Premultiplying the first identity by K ν and taking into account

that K ν x o = x o (Lemmas 3.1 (ii) and 3.2), we obtain λx o − K ν Ax o = 0 , i.e., λ ∈ σ(K ν A ) .

138


4.4.2. Prove Lemma 4.1. 4.4.3. Derive relations (4.5) and (4.6). 4.4.4. Prove Lemma 4.3. 4.4.5. Derive (4.15). 4.4.6. Show that any system S(A,B,C) (2.1) with not the identically zero transfer-function matrix can be diagonally decoupled via appropriate state feedback and pre- and/or postcompensator. Hint. Consider S(A,B,C) (2.1) with CA ν B as its first nonzero Markov parameter and denote rank CA ν B = p ≤ min{m, r} . Let CA ν B = H1H 2 be a skeleton factorization. Take into account the square (p-input, p-output) system S( A + BF, BH 2+ , H1+ C) obtained from S(A,B,C) by introducing the state feedback matrix F = −(CA ν B) + CA ν +1 and then, to the closed-loop system, the precompensator H +2 and postcompensator H1+ . Verify that the transfer-function matrix of S( A + BF, BH 2+ , H1+ C) equals I p / z ν +1 (where I p stands for the identity matrix of order p). To this end,

expand the expression H1+ C(zI − K ν A ) −1 BH +2 in power series and then use Lemma 3.1 (vi). 4.4.7. Show that any proper system S(A,B,C,D) (2.1) may be diagonally decoupled via appropriate state feedback and pre and/or postcompensator. Hint. Let 0 < rank D = p ≤ min{m, r} and let D = D1D 2 be a skeleton factorization of D. Introduce to S(A,B,C,D) the state feedback matrix F = −D + C . Then, to the closed-loop system introduce the precompensator

D +2 and postcompensator D1+ . Observe that the transfer-function matrix

of the system S( A − BD + C,BD 2+ ,0,I p ) thus obtained equals I p . 4.4.8. Check (4.28) in Lemma 4.8. 4.4.9. Verify (4.30). 4.4.10. Prove Corollary 4.9. 4.4.11. Prove the relation (i) in Remark 4.16.

4.4 Exercises

139

4.4.12. Find invariant zeros of the system in Example 4.6 using Remark 4.16 and Corollary 4.10. 4.4.13. Show that if a strictly proper system (2.1) with not the identically zero transfer-function matrix is nondegenerate, then its minimal subsystem is nondegenerate. Observe that the converse implication is not true, i.e., we can meet a degenerate system with nondegenerate minimal subsystem (see Exercise 5.5.18). 4.4.14. Consider S(A,B,C) (2.1) with CA ν B as its first nonzero Markov parameter. Find the first nonzero Markov parameter of the system S(K ν A, B, C) and find for this system the K ν -matrix and the K ν A matrix. Hint. To S(K ν A, B, C) apply Definition 3.1 and Lemma 3.1. Using Lemma 3.1 (vii) show that C(K ν A ) l B = CA l B = 0 for l = 0,1,..., ν − 1

and C(K ν A ) ν B = CA ν B ≠ 0 for l = ν , i.e., the first Markov parameter in S(K ν A, B, C) equals C(K ν A ) ν B = CA ν B . Now, according to (3.1), form the K ν -matrix for S(K ν A, B, C) and observe that I − B[C(K ν A ) ν B] + C(K ν A ) ν = I − B(CA ν B) + CA ν = K ν ,

i.e., it is equal to the matrix K ν formed for S(A,B,C). As is known, S(K ν A, B, C) arises from S(A,B,C) by introducing the

state feedback matrix − (CA ν B) + CA ν +1 . Now, performing an analogous operation in S(K ν A, B, C) , it is clear that the K ν A -matrix for S(K ν A, B, C) should be understood as the state matrix of the closed-loop system obtained from S(K ν A, B, C) by introducing the state feedback matrix − [C(K ν A ) ν B] + C(K ν A ) ν +1 . However, we obtain in this way

K ν A − B[C(K ν A ) ν B] + C(K ν A ) ν +1 = (I − B(CA ν B) + CA ν )K ν A = K νK ν A = K ν A.

This means that applying to S(K ν A, B, C) the state feedback matrix

− [C(K ν A ) ν B] + C(K ν A ) ν +1 we obtain the closed-loop system which is equal to the original system, i.e., to S(K ν A, B, C) . The reason of this is clear – the state feedback just introduced does not affect the state equation

140


in the system S(K ν A, B, C) . In fact, although in the general case − [C(K ν A) ν B] + C(K ν A ) ν +1 ≠ 0 , however (see Lemma 3.1 (i) and (vii)), B[C(K ν A) ν B] + C(K ν A ) ν +1 = B(CA ν B) + C(K ν A) ν (K ν A ) = (I − K ν )K ν A = 0 .

In some cases the state feedback matrix under considerations can be equal to zero (it takes place for instance when CA ν B has full row rank). 4.4.15. Consider a proper system S( A, B, C, D) (2.1) and the closed-loop system S( A − BD + C, B, (I r − DD + )C, D) obtained by introducing the state feedback matrix

F = −D + C . Denote

A' = A − BD + C, B' = B,

C' = (I r − DD + )C, D' = D and let F ' = −(D' ) + C' . Check that F ' = 0 .

4.4.16. Discuss Exercises 4.4.6 and 4.4.7 under the condition that the first nonzero Markov parameter has full row rank. 4.4.17. Let S(A,B,C) (2.1) be square ( m = r ) of uniform rank (i.e., its first

nonzero Markov parameter CA ν B is nonsingular). Show that the observability matrix for S(K ν A, B, C) has rank m(ν + 1) . C    C(K A )  ν    and take into account that Hint. Consider the matrix  .   .   C(K ν A ) n−1    C(K ν A ) l = CA l for 0 ≤ l ≤ ν and C(K ν A ) l = 0 for l ≥ ν + 1 (see  C   CA  :R n → R m (ν +1) Lemma 3.1). Then consider the linear mapping   .   ν CA 

[

]

and take its values on columns of the matrix B AB ... A ν B . Note that CB = ... = CA ν −1B = 0 and, since CA ν B is nonsingular, B has full column rank. Finally, observe that the images of columns of

[B

]

AB ... A ν B under the considered mapping form m(ν + 1) linearly

4.4 Exercises

141

independent vectors in R m (ν +1) . This means that this mapping is “onto”  C   CA   has the full row rank m(ν + 1) . Moreover, or, what is the same,   .   ν CA  since any linear mapping transforms linearly dependent vectors into linear-

[

]

ly dependent vectors, we conclude also that B AB ... A ν B has the full column rank m(ν + 1) . 4.4.18. Consider S(A,B,C) (2.1) of uniform rank (see Exercise 4.4.17) and take into account Remark 4.3 and Corollary 4.2. Show that the zero poly' nomial of S(A,B,C) is equal to χ cl o ( z ) = det ( zI o − (K ν A ) o ) (i.e., invariant and Smith zeros of S(A,B,C) are one and the same thing).

Hint. Consider the closed-loop system S(K ν A, B, C) and let S((K ν A )' , B' , C' ) denote its decomposition into an unobservable and an observable part (see (i) in Remark 4.3). The first Markov parameter of S((K ν A )' , B' , C' ) is the same as of S(A,B,C), i.e., it equals CA ν B . Moreover, by virtue of Exercise 2.8.25 when applied to S((K ν A )' , B' , C' ) , we can write det Pcl' (z) = z − m(ν +1) det(CA ν B) det(zI o − (K ν A) 'o ) det(zI o − (K ν A ) 'o ) ,

where Pcl' (z) stands for the system matrix of S((K ν A )' , B' , C' ) . The ' transfer-function matrix of this last system equals G cl (z) =

CA ν B

. z ν +1 Taking the right coprime factorization of G 'cl (z) (see Appendix C) with N r (z) = CA ν B and D r (z) = z ν +1 I m , we obtain the characteristic poly-

nomial

of

G 'cl (z)

as

det D r (z) = z m(ν +1)

and,

consequently,

deg G 'cl (z) = m(ν + 1) . The observable subsystem S((K ν A ) 'o , B 'o , C 'o ) of S((K ν A )' , B' , C' ) has the transfer-function matrix G 'cl (z) . Since the observability matrix for S(K ν A, B, C) has rank m(ν + 1) (see Exercise

4.4.17), the order of (K ν A) 'o is n − m(ν + 1) and, consequently, the order of (K ν A) 'o is m(ν + 1) , i.e., it equals deg G 'cl ( z) . This means (see

142


Appendix C) that S((K ν A ) 'o , B 'o , C 'o ) is a minimal realization of G 'cl (z) and det(zI o − (K ν A) 'o ) = det D r (z) = z m( ν +1) . In this way we obtain the

following relation det Pcl' (z) = det(CA ν B)det(zI o − (K ν A ) 'o ) and, since det Pcl' (z) = det P(z) (where P(z) stands for the system matrix of

S(A,B,C)), we have also det P (z) = det(CA ν B) χ cl o ( z ) . Finally, the claim follows via Exercise 2.8.23. 4.4.19. Consider S( A, B, C, D) (2.1) of uniform rank (i.e., such that D is nonsingular) and take into account Remark 4.6 and Corollary 4.6. Show −1 that χ cl o ( z ) = det( zI − ( A − BD C)) is equal to the zero polynomial of S( A, B, C, D) .

Hint. The closed-loop system S( A − BD −1C, B, (I r − D −1D)C, D) is completely unobservable since its C-matrix equals zero. Write zI − ( A − BD −1C) − B  det P (z) = det Pcl ( z) = det   0 D   = det D det(zI − ( A − BD −1C)) ,

where P (z) and Pcl (z) are respectively system matrices of S( A, B, C, D) and of the closed-loop system, and then use Exercise 2.8.23. 4.4.20. Consider S(A,B,C) (2.1) with CA ν B as its first nonzero Markov parameter. Let x' = Hx denote a change of coordinates which transforms S(A,B,C) into S( A ' , B' , C' ) . Show that the diagram in Fig. 4.1 is commutative.

Fig. 4.1.

Hint. The first nonzero Markov parameter of the system S(A',B',C'), where A' = HAH −1 , B' = HB, C' = CH −1 , equals C' ( A ' ) ν B' = CA ν B . For

4.4 Exercises

143

S(A',B',C') we form K 'ν according to Definition 3.1, i.e., K ν' = I − B' (C' ( A ' ) ν B' ) + C' ( A ' ) ν .

Note that K ν and K 'ν are interrelated as K 'ν = HK ν H −1 . In fact, K 'ν = HH −1 − HB(CA ν B) + CH −1HA ν H −1 = H (I − B(CA ν B) + CA ν )H −1 = HK ν H −1 .

Applying the same change of coordinates to S(K ν A, B, C) , we obtain the

system S((K ν A )' , B' , C' ) , where (K ν A )' = HK ν AH −1 and B' = HB ,

C' = CH −1 . Finally, in order to prove the commutativity of the diagram, it

is sufficient to verify that

(K ν A )' = K 'ν A' . However, we have

K 'ν A ' = HK ν H −1HAH −1 = HK ν AH −1 = (K ν A )' .

4.4.21. Consider a proper system S( A, B, C, D) (2.1) and let x' = Hx , denote a change of coordinates which transforms S(A,B,C,D) into S( A ' , B' , C' , D' ) . Show that the diagram in Fig. 4.2 is commutative.

Fig. 4.2.

4.4.22. Consider S(A,B,C) (2.1) in which the first nonzero Markov parameter CA ν B has full column rank. Show that the polynomial (see ' Remark 4.3 and Corollary 4.2) χ cl o ( z ) = det ( zI o − (K ν A ) o ) is equal to the zero polynomial of S(A,B,C).

Hint. Recall first that if in (2.1) the first nonzero Markov parameter has full column rank, then the system matrix has full normal column rank (see the proof of Corollary 2.5; Sect. 2.6). Now, consider the closed-loop system S(K ν A, B, C) and let S((K ν A )' , B' , C' ) denote its decomposition into

144


an unobservable and an observable part (see (i) in Remark 4.3). The first nonzero Markov parameter of S((K ν A )' , B' , C' ) is the same as of

S(A,B,C), i.e., it equals CA ν B . Thus the system matrix Pcl' (z) of S((K ν A )' , B' , C' ) , where

(i)

zI o − (K ν A ) 'o  Pcl' (z) =  0  0 

' − (K ν A )12

zI o − (K ν A ) 'o C 'o

− B 'o   − B 'o  , 0  

has the full normal column rank n + m . The idea of the proof is as follows. We shall show first that the observable subsystem S((K ν A ) 'o , B 'o , C 'o ) (note that the first nonzero Markov parameter of this subsystem is also equal to CA ν B ) has no Smith

zeros. This means that the (nocl + r ) x ( nocl + m) polynomial matrix (ii)

zI − (K ν A ) 'o Pcl' o (z) :=  o C 'o 

− B 'o   0 

has the full normal column rank n ocl + m which is equal to its local rank (over C) at any point of the complex plane; i.e., the matrix (ii) does not lose its full normal column rank at any z ∈ C (recall that nocl stands for the rank of the observability matrix for S(K ν A, B, C) ). Moreover, this means also that Pcl' (z) in (i) may lose its full normal column rank exactly at the roots of the characteristic polynomial of (K ν A) 'o . Then we shall transform the matrix (ii) into its Hermite row form (see Appendix C). In order to prove that S((K ν A ) 'o , B 'o , C 'o ) has no Smith zeros, it is sufficient to show that it has no invariant zeros (since, as is known from Proposition 2.1 (i), each Smith zero is always an invariant zero). Because the first nonzero Markov parameter of S((K ν A ) 'o , B 'o , C 'o ) has full column rank, we can apply to this system Proposition 4.3. To this end, we form for S((K ν A ) 'o , B 'o , C 'o ) (according to Definition 3.1) the projective matrix K ν' ,o := I o − B 'o [C 'o ((K ν A ) 'o ) ν B 'o ] + C 'o ((K ν A) 'o ) ν ,

4.4 Exercises

145

where C 'o ((K ν A) 'o ) ν B 'o = CA ν B is the first nonzero Markov parameter of S((K ν A ) 'o , B 'o , C 'o ) , and next we consider the closed-loop system S(K 'ν,o (K ν A ) 'o , B 'o , C 'o ) . Now, from Proposition 4.3, when applied to S((K ν A ) 'o , B 'o , C 'o ) , we infer that S((K ν A ) 'o , B 'o , C 'o ) has no invariant

zeros if and only if the closed-loop system S(K 'ν,o (K ν A ) 'o , B 'o , C 'o ) is observable. However, observability of S(K 'ν,o (K ν A ) 'o , B 'o , C 'o ) follows from the relation K ν' , o (K ν A ) 'o = (K ν A ) 'o and from observability of S((K ν A ) 'o , B 'o , C 'o ) . Hence we only need to verify the relation K ν' , o (K ν A ) 'o = (K ν A ) 'o . To this end, we apply to S((K ν A )' , B' , C' ) the

following relations K 'ν (K ν A ) ' = K 'ν K 'ν A ' = K 'ν A ' = (K ν A) ' ,

(iii) where

K 'ν := I − B ' [C ' ((K ν A ) ' ) ν B ' ] + C ' ((K ν A) ' ) ν = H (I − B(CA ν B) + CA ν )H −1 = HK ν H −1

is a projective matrix formed (according to Definition 3.1) for S((K ν A )' , B' , C' ) and H stands for the change of coordinates used in Remark 4.3. Note that relations (iii) follow from Exercises 4.4.14 and 4.4.20. Now, taking matrices (K ν A ) ' , B ' and C ' in the form given in (i) in Remark 4.3, we obtain I o K ν' =  0

X12  K 'ν , o 

and then, using K 'ν (K ν A ) ' = (K ν A ) ' (cf., (iii)), we obtain I o 0 

X12  (K ν A ) 'o  K ν' , o   0

'   (K ν A )12 (K ν A) 'o =   (K ν A ) 'o   0

'  (K ν A )12  (K ν A) 'o 

which yields the desired equality K ν' , o (K ν A ) 'o = (K ν A ) 'o . In this way we have shown that Pcl' o (z) in (ii) has full normal column rank which is equal to its local rank at any point of the complex plane.

146


Now, according to the Hermite row form (see Appendix C), there exists an unimodular matrix L o (z) of the order nocl + r such that R (z) L o (z) Pcl' o (z) =  ,  0 

(iv)

where R (z) is square (of the order nocl + m ) upper triangular and nonsingular. Moreover, since Pcl' o (z) does not lose its full normal column rank at any point of the complex plane, therefore we must have det R (z) = const ≠ 0 . Writing Pcl' (z) in (i) as zI − (K A ) ' ' ν o (z) =  o Pcl 0 

X12   Pcl' o (z)

I and premultiplying it by the unimodular matrix  o 0 using (iv), we obtain

(v)

zI o − (K ν A) 'o  0   0 

X12   R (z) , 0  

zI o − (K ν A) 'o  0 

X12   R (z)

 as well as L o (z) 0

where the submatrix (vi)

is square of the order n o + m = n − nocl + m (the last r − m rows in (v) are zero). Note that (v) has been obtained from the system matrix P (z) of the original system S( A, B, C) by elementary operations. Finally, the determinant of the pencil (vi) is equal, up to a multiplicative constant, to the zero polynomial of this pencil and, consequently, to the zero polynomial of P (z) . This proves that the zero polynomial of S( A, B, C) is equal to ' χ cl o ( z ) = det ( zI o − (K ν A ) o ) .

4.4 Exercises

147

4.4.23. Consider a system S(A,B,C,D) (2.1) in which D has full column + ' rank. Show that χ cl o ( z ) = det( zI o − ( A − BD C) o ) (see Remark 4.6 and Corollary 4.6) is equal to the zero polynomial of S(A,B,C,D).

Hint. Using unimodular matrices, the system matrix P(z) of S(A,B,C,D) can be transformed into the form

(i)

 zI o − ( A − BD + C) 'o  ' Pcl 0 (z) =   0o 

' − ( A − BD + C)12

zI o − ( A − BD + C) 'o ((I r − DD + )C) 'o

− B 'o   − B 'o  , D  

' (z) is the system matrix of the system (i) in Remark 4.6 and the where Pcl

subsystem S(( A − BD + C) 'o , B 'o , ((I r − DD + )C) 'o , D) is observable. Since ' (z) has full normal column rank n + m and D has full column rank, Pcl the submatrix

zI o − ( A − BD + C) 'o  + '  ((I r − DD )C) o

(ii)

− B 'o   D 

has full normal column rank nocl + m , where nocl stands for the rank of the observability matrix for S( A − BD + C, B, (I r − DD + )C, D) . We show first that the matrix (ii) does not lose its full normal column rank at any point of the complex plane. To this end, it is sufficient to prove that S(( A − BD + C) 'o , B 'o , ((I r − DD + )C) 'o , D) has no invariant zeros. We can show this by reductio ad absurdum. Suppose that λ ∈ C is an invariant zero of S(( A − BD + C) 'o , B 'o , ((I r − DD + )C) 'o , D) , i.e., there exist cl

0 ≠ x oo ∈ C no and g ∈ C m such that

(iii)

(λI o − ( A − BD + C) 'o )x oo = B 'o g ((I r − DD + )C) 'o x oo = −Dg .

Premultiplying the second identity in (iii) by D + and taking into account that D + D = I m , we obtain g = −D + ((I r − DD + )C) 'o x oo . However, using (i) of Remark 4.6, we can write

148


0 = D + (I r − DD + )CH −1 = D + ((I r − DD + )C) '

[

= 0o

]

D + ((I r − DD + )C) 'o .

This yields D + ((I r − DD + )C) 'o = 0 o and, consequently, g = 0 . Thus (iii) takes the form (iv)

(λI o − ( A − BD + C) 'o )x oo = 0 o ((I r − DD + )C) 'o x oo = 0.

Now, the observability of S(( A − BD + C) 'o , B 'o , ((I r − DD + )C) 'o , D) and (iv) imply x oo = 0 . This contradicts the assumption that λ is an invariant zero. Finally, we transform (ii) to its Hermite row form (cf., Hint to Exercise 4.4.22).

5 Singular Value Decomposition of the First Markov Parameter

As it has been observed in Chapter 4 the first nonzero Markov parameter of a linear discrete-time system (2.1) carries some amount of information concerning invariant and Smith zeros. The approach presented there has been based on the Moore-Penrose pseudoinverse of that parameter. It has been shown that in any nondegenerate system (2.1) with the first nonzero Markov parameter of full rank all system zeros are completely characterized as some eigenvalues of a real matrix of the order of the state matrix. Furthermore, it has been shown also that in such systems the invariant zeros may be found out as output decoupling zeros of an appropriate closed-loop (state feedback) system as well as that invariant and Smith zeros are one and the same thing. Unfortunately, when the first nonzero Markov parameter in (2.1) has not full column rank, it may happen that the system is degenerate (cf., Chap. 2, Examples 2.4, 2.8 and 2.9) or, if it is nondegenerate, some of its invariant zeros may not appear in the spectrum of K ν A or A − BD + C (cf., Chap. 4, Examples 4.1 and 4.4). In order to provide a useful tool which would enable us to decide the question of degeneracy/nondegeneracy in any system (2.1) as well as to compute its invariant and Smith zeros, we develop in this chapter a method based on the singular value decomposition (SVD) (see Appendix B) of the first nonzero Markov parameter [73, 75]. The method results in a recursive procedure which in a finite number of steps reduces the original system to a system with the first nonzero Markov parameter of full column rank or to a system with the identically zero transfer-function matrix. Since, as we shall see, this procedure preserves the set of invariant zeros as well as the zero polynomial (and, consequently, the set of Smith zeros), the invariant and Smith zeros of the original system can be found at the last step. In this chapter we will need the following characterization of invariant zeros for systems with the first nonzero Markov parameter of full column rank.


150


Proposition 5.1. If in a strictly proper system S(A,B,C) (2.1) the first non-

zero Markov parameter CA ν B has full column rank, then each invariant zero of the system lies in σ(K ν A ) . Moreover, the following conditions are equivalent: (i) (ii)

λ ∈ C is an invariant zero of S(A,B,C); λ ∈ σ(K ν A ) and an associated eigenvector lies in Ker C ;

λ is an o.d. zero of the closed-loop (state feedback) system (iii) S(K ν A,B,C) ;

(iv) λ is a Smith zero of S(A,B,C). Furthermore, the zero polynomial of S(A,B,C) is equal to det( zI ' o −(K ν A )' o ) , i.e., invariant and Smith zeros of S(A,B,C) are exactly the same objects (recall that (K ν A )' o is taken from the decomposition of the system S(K ν A,B,C) into an unobservable and an observable part). Proof. The assertions are immediate consequences of Definition 2.1 (i), Propositions 4.2 and 4.3 and Corollary 4.2. ◊

Proposition 5.2. If in a proper system S(A,B,C,D) (2.1) matrix D has full column rank, then each invariant zero of the system lies in σ( A − BD + C) . Moreover, the following conditions are equivalent: (i)

λ ∈ C is an invariant zero of S(A,B,C,D);

(ii)

λ ∈ σ( A − BD + C) and an associated eigenvector lies in

Ker (I r − DD + )C ;

(iii)

λ is an o.d. zero of the closed-loop (state feedback) system

S( A − BD + C,B,(I r − DD + )C,D) ; λ is a Smith zero of S(A,B,C,D). (iv) Furthermore, the zero polynomial of S(A,B,C,D) is equal to det( zI ' o −( A − BD + C)' o ) , i.e., invariant and Smith zeros of S(A,B,C,D)

are exactly the same objects (recall that ( A − BD + C)' o is taken from the decomposition of the system S( A − BD + C,B,(I r − DD + )C,D) into an unobservable and an observable part). Proof. The assertions are immediate consequences of Definition 2.1 (i), Propositions 4.6 and 4.7 and Corollary 4.6. ◊

5.1 Invariant and Smith Zeros in Strictly Proper Systems

151

5.1 Invariant and Smith Zeros in Strictly Proper Systems In S(A,B,C) (2.1) let the first nonzero Markov parameter CA ν B , where

0 ≤ ν ≤ n − 1 , have rank p ≤ min{m, r} . Applying SVD to CA ν B , we can write (see Appendix B, Theorem B.1)

CA ν B = U Λ V T ,

where

⎡M Λ=⎢ p ⎣ 0

0⎤ 0⎥⎦

(5.1)

is rxm-dimensional, M p is pxp diagonal with positive singular values of

CA ν B and rxr U and mxm V are orthogonal matrices. Introducing into (2.1) matrices V and U T as pre- and postcompensator, we associate with S(A,B,C) a new system S( A, B, C) (see Fig. 5.1a–b) of the form

Fig. 5.1.

152


x(k + 1) = Ax(k ) + B u (k ) y ( k ) = C x( k )

, k∈N ,

(5.2)

where B = BV ,

C = UTC

(5.3)

u = V Tu,

y = UTy

(5.4)

and

we decompose as follows (see Fig. 5.1b)

[

B = Bp

⎡ Cp ⎤ ⎡ up ⎤ ⎡ yp ⎤ , y=⎢ Bm − p , C = ⎢ ⎥, u =⎢ ⎥ ⎥ ⎢⎣ Cr − p ⎥⎦ ⎣u m − p ⎦ ⎣y r − p ⎦

]

(5.5)

and B p consists of the first p columns of B , C p consists of the first p rows of C and similarly, u p consists of the first p components of u and y p consists of the first p components of y . Moreover, CA ν B = Λ (cf.; (5.1)) is the first nonzero Markov parameter of S( A, B, C) (5.2). From (5.1) and (5.5) it follows that ⎡M CAν B = ⎢ p ⎣ 0

0⎤ ⎡ C p A ν B p =⎢ 0⎥⎦ ⎢ Cr − p A ν B p ⎣

C p A ν Bm − p ⎤ ⎥ Cr − p A ν B m − p ⎥⎦

(5.6)

and, consequently, C p Aν B p = M p , Cr − p A ν B p = 0 ,

C p A ν Bm − p = 0 , Cr − p A ν B m − p = 0 .

(5.7)

Lemma 5.1. [84] The sets of invariant zeros in systems S(A,B,C) (2.1) and S( A, B, C) (5.2) coincide, i.e., ZIS( A, B,C) = ZI . Moreover, these S( A , B , C )

systems have the same zero polynomial, i.e., ψ S( A , B, C) ( z) = ψ S( A, B , C ) ( z ) ,

and, consequently, the same set of Smith zeros, i.e.,


ZSS( A, B, C) = Z S

S( A , B , C )

153

.

Proof. The claims follow immediately from Lemma 2.3 (ii) and (iii) and from Exercise 2.8.10. ◊ For S( A, B, C) (5.2) we form, according to Definition 3.1, the projective matrix K ν := I − B ( C A ν B ) + C A ν

(5.8)

which, in view of (5.1) and (5.5), can be expressed as K ν = I − B (CA ν B) + CA ν

[

=I − Bp

⎡M B m− p ⎢ p ⎣ 0

]

0⎤ 0⎥⎦

+

⎡ Cp ⎤ ν ⎥A ⎢ ⎢⎣ Cr − p ⎥⎦

(5.9)

= I − B p M −p1 C p A ν .

Remark 5.1. Observe that matrices K ν = I − B(CA ν B) + CA ν (formed for S(A,B,C)) and K ν in (5.8) (formed for S( A, B, C) ) satisfy the condition K ν = K ν = I − B p M −p1C p A ν .

(5.10)

This fact follows from the well known relation between the MoorePenrose inverse of a matrix and SVD of that matrix (see Appendix B, Lemma B.2). In case of (5.1) it takes the form (CA ν B) + = V Λ + U T . Now, taking into account that CA ν B = Λ (cf., (5.1)), K ν can be evaluated as follows K ν = I − B ( C A ν B ) + C A ν = I − BV Λ + U T CA ν = I − B(CA ν B) + CA ν = K ν . ◊

Lemma 5.2. [84] The matrix K ν = I − B p M −p1C p A ν has the properties: (i)

K ν2 = K ν ;

(ii)

Σν := {x : K ν x = x} = Ker C p A ν ,

dim Σν = n − p ;

(iii)

Ων := {x : K ν x = 0} = Im B p ,

dim Ων = p ;

154


(iv)

C n ( R n ) = Σν ⊕ Ων ;

(v)

KνB p = 0 ,

(vi)

⎧⎪C A l C p (K ν A) l = ⎨ p ⎪⎩ 0

(vii)

C (K ν A) l = C A l

(viii)

C p AνK ν = 0 ;

K ν Bm − p = Bm − p ,

for 0 ≤ l ≤ ν ; for l ≥ ν + 1

for 0 ≤ l ≤ ν ;

l

Cr − p ( K ν A ) = Cr − p A l

for 0 ≤ l ≤ ν .

Proof. The proof follows the same lines as the proof of Lemma 3.1. ◊ The following result characterizes invariant and Smith zeros of the original system S(A,B,C) (2.1) in which the first nonzero Markov parameter is not of full column rank as invariant and Smith zeros of another (closedloop) system with a smaller number of inputs (see Fig. 5.2).

Fig. 5.2.

Proposition 5.3. [84] In S(A,B,C) (2.1) let the first nonzero Markov parameter CA ν B

have rank 0 < p < m

and in S( A, B, C) (5.2) let

B m − p ≠ 0 . Then the sequence of transformations S( A, B, C) → S( A, B , C ) → S(K ν A, B m − p , C )

has the following properties:


155

it preserves the set of invariant zeros, i.e.,

(i)

ZIS( A, B,C) = ZI

S( A, B , C )

(ii)

= ZI

S(K ν A, B m− p , C )

,

it preserves the zero polynomial, i.e., ψ S( A, B, C) (z) = ψ S( A, B , C ) (z) = ψ S(K A, B , C ) (z) , ν m− p

and, consequently, the set of Smith zeros, i.e.,

ZSS( A, B,C) = ZS

S( A, B , C )

(iii)

= ZS

S(K ν A, B m− p , C )

.

Moreover, if a triple λ, x o ≠ 0, g m− p represents an invariant zero

⎡− M −p1C p A ν +1x o ⎤ λ for S(K ν A,B m− p ,C ) , then the triple λ, x , g = V ⎢ ⎥ gm − p ⎥⎦ ⎢⎣ consists of the invariant zero λ and the corresponding state-zero and input -zero direction for S(A,B,C). o

Proof. (i) In view of Lemma 5.1, it is sufficient to show the following equivalence: λ ∈ C is an invariant zero of S( A, B, C) (5.2) if and only if λ is an invariant zero of the system S(K ν A,B m − p ,C ) (see Fig. 5.2),

where K ν = I − B p M −p1 C p A ν . Suppose first that λ ∈ C is an invariant zero of S( A, B, C) , i.e., there ⎡ gp ⎤ m exist vectors 0 ≠ x o ∈ C n and g = ⎢ ⎥ ∈ C such that (cf., Definig ⎣ m− p ⎦ tion 2.1 (i))

(iv)

λx o − Ax o = B p g p + B m − p g m − p

and C x o = 0 .

Since C A ν B is the first nonzero Markov parameter for S( A, B, C) , premultiplying successively the first identity in (iv) by C p ,.. ., C p A ν −1 and taking into account the second identity in (iv), we obtain at the last step C p A ν x o = 0 (i.e., x o ∈ Σν – see Lemma 5.2 (ii)) and, consequently, K ν x o = x o . Premultiplying the first identity in (iv) by C p A ν and taking

into account the second identity in (5.7), we get g p = −M −p1C p A ν +1x o .

156


Finally, premultiplying the first identity in (iv) by K ν and using Lemma 5.2 (v), we can write (v)

λx o − K ν Ax o = B m − p g m − p

and C x o = 0 ,

i.e., λ is an invariant zero of S(K ν A,B m − p ,C ) . Thus we have proved that if λ is an invariant zero of S( A, B, C) , then λ is also an invariant zero of S(K ν A,B m − p ,C ) . Observe, however, that we have shown even more.

⎡ gp ⎤ Namely, if a triple λ, x o ≠ 0, g = ⎢ ⎥ satisfies (iv), then the triple ⎣g m − p ⎦

λ, x o , g m− p satisfies (v) and g p = −M −p1C p A ν +1x o . Conversely, suppose that a number λ is an invariant zero of the system S(K ν A,B m − p ,C ) , i.e., a triple λ, x o , g m− p satisfies (v). Now, using

K ν = I − B p M −p1C p A ν , we can write (v) as (vi)

λx o − Ax o = B p (−M −p1 C p A ν +1x o ) + B m− p g m− p Cx o = 0 ,

⎡ gp ⎤ −1 ν +1 o i.e., the triple λ, x o ≠ 0, g = ⎢ ⎥ , where g p = −M p C p A x , satg m − p ⎣ ⎦ isfies (iv) and, consequently, λ is an invariant zero of S( A, B, C) .

(ii) The fact that S(A,B,C) and S( A, B, C) have the same zero polynomial follows from Lemma 5.1. We shall show that the zero polynomial of S(A,B,C) is the same as of S(K ν A,B m − p ,C ) . In the proof we make use of the following formulas (vii)

C p (zI − K ν A ) −1 B m − p ≡ 0 ,

(viii)

C p (zI − K ν A) −1 B p = z −(ν +1) M p ,

(ix)

z ν +1 C p (zI − K ν A) −1 = C p (z ν I + z ν −1 A + . . . + zA ν −1 + A ν ).


157

The relation (vii) follows from the fact that C A ν B is the first nonzero Markov parameter of S( A, B, C) , from (5.7) and from Lemma 5.2 (vi). In fact, we can write −1

∞

C p (K ν A ) l B m − p

l =0

z l +1

C p (zI − K ν A) B m− p = ∑ ν

C p A l B m− p

l =0

z l +1

= ∑

≡ 0.

Using the same argumentation, we can write the left-hand side of (viii) as ∞

C p (zI − K ν A ) −1 B p = ∑ ν

C p Al B p

l =0

z l +1

= ∑

C p (K ν A ) l B p z l +1

l =0 ν

=

CpA B p z ν +1

=

Mp z ν +1

.

Finally, the left-hand side of (ix) can be written as z

ν +1

=z

C p (zI − K ν A)

ν +1

ν

C p Al

l =0

z l +1

∑

−1

=z

ν +1

∞

C p (K ν A ) l

l =0

z l +1

∑

.

⎡ Fp ⎤ −1 ν +1 Denote by F = ⎢ ⎥ a mxn real matrix, where Fp = −M p C p A F ⎣⎢ m − p ⎦⎥ is pxn and (m − p ) xn Fm− p equals zero, i.e., Fm − p = 0 . The relations in (x) below show that the zero polynomial of S(A,B,C) is the same as the zero polynomial of the system S(K ν A,B ,C ) . ⎡ I 0 ⎤ ⎡zI − A − B ⎤ ⎡ I 0 ⎤ ⎡ I 0 ⎤ ⎢ ⎥ ⎢0 U T ⎥ ⎢ C 0 ⎥⎦ ⎢⎣0 V ⎥⎦ ⎣ F I m ⎦ ⎦⎣ ⎣

(x)

⎡zI − A − B p ⎢ = ⎢ Cp 0 ⎢ Cr − p 0 ⎣

I Bm − p ⎤ ⎡ ⎥⎢ −1 0 ⎥ ⎢− M p C p A ν + 1 0 0 ⎥⎦ ⎢⎣

0 Ip 0

0 ⎤ ⎥ 0 ⎥ I m − p ⎥⎦

158


⎡zI − K ν A − B p ⎢ = ⎢ Cp 0 ⎢ Cr − p 0 ⎣

− Bm − p ⎤ ⎥ 0 ⎥. 0 ⎥⎦

Consider now the following (n + r ) x (n + r ) matrix

(xi)

⎡I − z ν +1B p M −p1C p ( zI − K ν A) −1 ⎢ L( z) = ⎢ − z ν +1C p (zI − K ν A) −1 ⎢ 0 ⎢⎣

B p M −p1 z ν +1I p 0

0 ⎤ ⎥ 0 ⎥. ⎥ Ir − p ⎥ ⎦

By virtue of (ix), L(z) is a polynomial matrix. In order to show that L(z) is unimodular, it is sufficient to check that its determinant is a nonzero constant. To this end, we calculate the determinat of the submatrix (xii)

⎡I − z ν +1B p M −p1C p (zI − K ν A) −1 ⎢ ⎢⎣ − z ν +1C p (zI − K ν A) −1

B p M −p1 ⎤ ⎥. z ν +1I p ⎥⎦

However, according to the well known rules (see Appendix C), this determinant can be evaluated as det[I − z ν +1B p M −p1C p ( zI − K ν A) −1 + B p M −p1z −(ν +1) I p z ν +1 C p (zI − K ν A) −1 ] det[z ν +1I p ] = (xiii)

det[I + (1 − z ν +1 ) B p M −p1 C p (zI − K ν A) −1 ] det[z ν +1I p ] = det[I p + (1 − z ν +1 ) C p (zI − K ν A) −1 B p M −p1 ] det[z ν +1I p ] = det[I p +

1 − z ν +1 z

ν +1

I p ] det[z ν +1I p ] = 1.

This yields det L(z) = 1 , i.e., L(z) is unimodular. Now, premultiplying the matrix standing on the right-hand side of (x) by L(z) as well as using (vii) and (viii), we obtain

(xiv)

⎡(zI − K ν A) + (1 − z ν +1 ) B p M −p1C p ⎢ 0 ⎢ ⎢ Cr − p ⎢⎣

0 Mp 0

− Bm − p ⎤ ⎥ 0 ⎥. 0 ⎥⎥ ⎦


159

Interchanging appropriately rows and columns in (xiv), we can write ⎡M p ⎢ ⎢ 0 ⎢ 0 ⎣

(xv)

0 (zI − K ν A) + (1 − z

0

ν +1

) B p M −p1C p

Cr − p

⎤ ⎥ − Bm − p ⎥ . 0 ⎥⎦

Since M p is nonsingular, the zero polynomial of (xv) is the same as of the matrix ⎡( zI − K ν A) + (1 − z ν +1 ) B p M −p1C p ⎢ Cr − p ⎢⎣

(xvi)

− Bm − p ⎤ ⎥. 0 ⎥⎦

Thus we have shown that the zero polynomial of S(A,B,C) is equal to the zero polynomial of (xvi). In the last part of the proof of point (ii) of the proposition we show that the zero polynomial of S(K ν A,Bm− p ,C ) equals the zero polynomial of (xvi). This, however, is easily seen because premultiplying the system matrix ⎡zI − K ν A − B m − p ⎤ ⎢ ⎥ 0 ⎥ ⎢ Cp ⎢ Cr − p 0 ⎥⎦ ⎣

of S(K ν A,B m− p ,C ) by L(z) we obtain ⎡( zI − K ν A) + (1 − z ν +1 ) B p M −p1C p ⎢ 0 ⎢ ⎢ Cr − p ⎣⎢

(xvii)

− Bm − p ⎤ ⎥ 0 ⎥. 0 ⎥⎥ ⎦

(iii) The proof of this claim is left to the reader as an exercise. ◊ Proposition 5.4. In S(A,B,C) (2.1) let m > r and let the first nonzero Markov parameter CA ν B have full row rank r. Then: (i)

S(A,B,C) is degenerate if and only if in S( A, B, C) (5.2) is

Bm −r ≠ 0 .

160


Moreover, at Bm −r ≠ 0 , λ ∈ C is an invariant zero of S(A,B,C) if and only if λ is an invariant zero of S(K ν A,B m− r ,C ) , where K ν = I − B r M r−1Cr A ν . The transfer-function matrix of S(K ν A,B m− r ,C )

equals zero identically. Furthermore, at Bm −r ≠ 0 the zero polynomial of S(A,B,C) equals the zero polynomial of S(K ν A,B m− r ,C ) and the Smith zeros of S(A,B,C) are the i.o.d. zeros of S(K ν A,Bm− r ,C ) . S(A,B,C) is nondegenerate if and only if in S( A, B, C) (5.2) is

(ii)

B m −r = 0 . Moreover, at B m −r = 0 the following conditions are equivalent: a)

λ ∈ C is an invariant zero of S(A,B,C);

b)

λ is an invariant zero of S( A, B r ,C) ;

c)

λ is an eigenvalue of K ν A such that an associated eigenvector

lies in Ker C ; d)

λ is an o.d. zero of S(K ν A,B r ,C ) ;

e)

λ is a Smith zero of S(A,B,C).

Moreover, at Bm −r = 0 the zero polynomial of S(A,B,C) equals det(zI' o −(K ν A)'o ) , i.e., invariant and Smith zeros in S(A,B,C) are exactly the same objects – including multiplicities ( (K ν A)'o is taken from the decomposition of S(K ν A,Br ,C) into an unobservable and an observable part). Furthermore, if a pair λ, x o ≠ 0 represents an o.d. zero λ of

⎡− M −r 1Cr A ν +1x o ⎤ S(K ν A,Br ,C) , then the triple λ, x o , g = V ⎢ ⎥ , where gm − r ⎢⎣ ⎥⎦

g m − r ∈ C m − r is fixed arbitrarily, represents the invariant zero λ of S(A,B,C).

Proof. Since CA ν B has full row rank, i.e., rank CA ν B = p = r , applying

SVD to CA ν B and using the notation as in (5.1) and (5.5), we can write CA ν B = U [M r

0]V T ,

and S( A,B,C ) takes the form

[

B = Br

]

B m−r ,

C = Cr

(5.11)


x(k + 1) = Ax(k ) + B r ur (k ) + B m − r um − r (k ) y (k ) = Cr x(k )

, k∈N .

161

(5.12)

(i) At Bm −r ≠ 0 the sets of invariant zeros of S(A,B,C) and S(K ν A,Bm− r ,C ) coincide (see Proposition 5.3 (i)), i.e., λ is an invariant zero of S(A,B,C) if and only if it is an invariant zero of S(K ν A,Bm− r ,C ) . In order to show that the transfer-function matrix of S(K ν A,Bm − r ,Cr ) equals zero identically, we use relations (vi) of Lemma 5.2 (at p = r ). Thus we can write C (K ν A ) l B m −r = Cr (K ν A ) l B m− r ⎧⎪C A l B m − r for 0 ≤ l ≤ ν . =⎨ r ⎪⎩ 0 for l ≥ ν + 1

(5.13)

Since Cr A l B m −r = 0 for every 0 ≤ l ≤ ν , all Markov parameters of S(K ν A,Bm− r ,C ) are equal to zero. This yields C ( zI − K ν A) −1 B m − r ≡ 0 and, consequently, by virtue of Proposition 2.6 (when applied to S(K ν A,Bm− r ,C ) ), S(K ν A,Bm− r ,C ) is degenerate. Finally, degeneracy of S(A,B,C) follows by virtue of Proposition 5.3 (i). The last claim in (i) follows immediately from Proposition 5.3 (ii) and from Proposition 2.6 when applied to S(K ν A,Bm− r ,C ) . (ii) At Bm −r = 0 , as it easily follows from Definition 2.1 (i), λ ∈ C is an invariant zero of S( A, B , C ) if and only if λ is an invariant zero of the square system S( A, B r ,C) (which arises from S( A,B,C ) by neglecting the input u m − r – see (5.12)). Since the first nonzero Markov parameter Cr A ν B r = M r of S( A, B r ,C) is nonsingular, the invariant zeros of

S( A, Br ,C)

are characterized by Proposition 5.1 when applied to

S( A, Br ,C) . Finally, the claim concerning the equivalence of the conditions a)–e) follows via Lemma 5.1 and Corollary 2.3. The zero polynomial of S(A,B,C) equals the zero polynomial of S( A, Br ,C) . This follows, at p = r and Bm −r = 0 , from the relations (x) in the proof of Proposition 5.3. Recall that S( A, B r ,C) satisfies uniform rank condition. Now, decompose S(K ν A,Br ,C) into an unobservable and

162


an observable part, and then to S( A, B r ,C) apply Exercise 4.4.18 (or Proposition 5.1). The proof of the last assertion in (ii) is left to the reader as an exercise. ◊ Let us consider a system S(A,B,C) (2.1) in which the first nonzero Markov parameter CA ν B has rank 0 < p < min{m, r} and in (5.5) is Cr − p = 0 . The system S( A,B,C ) (5.2) can be written now as

x(k + 1) = Ax(k ) + B p u p (k ) + B m − p u m − p (k ) y p ( k ) = C p x( k )

, k∈N .

(5.14)

, k∈N

(5.15)

y r − p (k ) = 0

By S( A,B ,C p ) we denote the system x(k + 1) = Ax(k ) + B p u p (k ) + B m − p u m − p (k ) y p ( k ) = C p x( k )

obtained from (5.14) by neglecting the output y r − p . Using Definition 2.1 (i), it is easy to note that the sets of invariant zeros of systems (5.14) and (5.15) coincide. Of course, these systems have the same zero polynomial and, consequently, the same set of Smith zeros. Hence we can restrict our attention to S( A,B ,C p ) (5.15). For S( A,B ,C p ) the first nonzero Markov parameter is

[

C p Aν B = C p Aν B p

][

C p A ν Bm − p = M p

0

]

and K ν := I − B ( C p A ν B ) + C p A ν = I − B p M −p1C p A ν . Since C p A ν B has full row rank, in the analysis of invariant and Smith zeros of S( A,B ,C p ) we can apply the same arguments as in the previous case, i.e., of the system (5.12). Thus we can formulate the following result. Proposition 5.5. In S(A,B,C) (2.1) let the first nonzero Markov parameter

CA ν B

have rank

Cr − p = 0 . Then:

0 < p < min{m, r}

and in

S( A,B ,C) (5.2) let


163

S(A,B,C) is degenerate if and only if in S( A,B ,C) is B m − p ≠ 0 . Moreover, at B m − p ≠ 0 , λ ∈ C is an invariant zero of S(A,B,C) if and (i)

only if λ is an invariant zero of the system S(K ν A,B m − p ,C p ) , where K ν = I − B p M −p1C p A ν . The transfer matrix of S(K ν A,B m − p ,C p ) equals zero identically. Furthermore, at B m − p ≠ 0 the zero polynomial of S(A,B,C) equals the zero polynomial of S(K ν A,B m − p ,C p ) and the Smith zeros of S(A,B,C) are the i.o.d zeros of S(K ν A,B m − p ,C p ) . (ii)

S(A,B,C) is nondegenerate if and only if in S( A,B ,C ) is

B m − p = 0 . Moreover, at B m − p = 0 the following conditions are equiv-

alent: a)


b)

λ is an invariant zero of S( A, B p ,C p ) ;

λ is an eigenvalue of K ν A such that an associated eigenvector lies in Ker C p ;

c)

d)

λ is an o.d. zero of S(K ν A,B p ,C p ) ;

e)


At B m − p = 0 the zero polynomial of S(A,B,C) is det(zI ' o −(K ν A)' o ) , i.e., Smith zeros and invariant zeros of S(A,B,C) are exactly the same objects – including multiplicities ( (K ν A)' o is taken from decomposition of S(K ν A,B p ,C p ) into an unobservable and an observable part). Furthermore, if a pair λ, x o ≠ 0 represents an o.d. zero λ of S(K ν A,B p ,C p ) , ⎡− M −p1 C p A ν +1x o ⎤ then the triple λ, x o , g = V ⎢ ⎥ , where g m − p ∈ C m − p is g m− p ⎢⎣ ⎥⎦ fixed arbitrarily, represents the invariant zero λ of S(A,B,C).

Proof. The proof follows the same lines as the proof of Proposition 5.4. ◊

Suppose now that the first nonzero Markov parameter CA ν B of

S(A,B,C) (2.1) satisfies 0 < rank CA ν B = p < min{m, r} and in S( A,B ,C) (5.2) is Cr − p ≠ 0 and B m − p = 0 .Then S(A,B ,C ) can be written as

164


x(k + 1) = Ax(k ) + B p u p (k ) + 0 u m − p (k ) y p ( k ) = C p x( k )

, k∈N .

(5.16)

y r − p ( k ) = Cr − p x( k ) By S( A, B p , C ) we denote the system obtained from (5.16) by neglecting the input u m − p . The first nonzero Markov parameter for S( A, B p , C ) is ⎡ C p A ν B p ⎤ ⎡M ⎤ ⎥ = ⎢ p⎥ CA ν B p = ⎢ ν ⎢⎣ Cr − p A B p ⎥⎦ ⎣ 0 ⎦ and K ν := I − B p ( C A ν B p ) + C A ν = I − B p M −p1C p A ν . The sets of invariant zeros of S(A,B,C ) (5.16) and S( A, B p , C ) coincide. This fact can be easily proved via Definition 2.1 (i). Moreover, these systems have the same zero polynomial and, consequently, the same set of Smith zeros. Hence we can restrict our attention to S( A, B p , C ) only. We formulate the following. Proposition 5.6. In S(A,B,C) (2.1) let the first nonzero Markov parameter

CA ν B satisfy 0 < rank CA ν B = p < min{m, r} and in S( A,B ,C) (5.2) let Cr − p ≠ 0 and let B m − p = 0 . Then S(A,B,C) is nondegenerate and the

following conditions are equivalent: a)


b)

λ is an invariant zero of S( A,B p ,C ) ;

c)

λ is an eigenvalue of K ν A such that an associated eigenvector

lies in Ker C ; d)

λ is an o.d. zero of S(K ν A,B p ,C ) , where

K ν = I − B p M −p1 C p A ν ; e)


Moreover, the zero polynomial of S(A,B,C) is det(zI ' o −(K ν A)' o ) , i.e., Smith and invariant zeros of S(A,B,C) are exactly the same objects –

5.2 A Procedure for Computing Zeros of Strictly Proper Systems

165

including multiplicities ( (K ν A)' o is taken from the decomposition of S(K ν A,B p ,C ) into an unobservable and an observable part). Further-

more, if a pair λ, x o ≠ 0 represents an o.d. zero λ of S(K ν A,B p ,C ) , ⎡− M −p1C p A ν +1x o ⎤ then the triple λ, x o , g = V ⎢ ⎥ , where g m − p ∈ C m − p is g ⎢⎣ ⎥⎦ m− p arbitrarily fixed, represents the invariant zero λ of S(A,B,C).

Proof. In S( A ,B p ,C ) the first nonzero Markov parameter C A ν B p has

full column rank, i.e., S( A,B p ,C ) is nondegenerate. As we noticed before,

ZI

= ZI

. Moreover, by virtue of Lemma 5.1, we have

ZIS( A, B, C) = ZI

. The equivalence of the conditions b), c), d), e)

S( A, B , C )

S( A, B p, C )

S( A , B , C )

follows from Proposition 5.1 when applied to S( A,B p ,C ) . In order to prove the claim concerning the zero polynomial, we apply to S( A ,B p ,C ) Exercise 4.4.22 (we use also Lemma 5.1). The proof of the last assertion of the proposition is left to the reader as an exercise. ◊

5.2 A Procedure for Computing Zeros of Strictly Proper Systems Propositions 5.1 and 5.3–5.6 as well as Proposition 2.6 suggest a simple recursive procedure for the computation of invariant and Smith zeros of an arbitrary system S(A,B,C) (2.1). Recall that if G ( z ) = C( zI − A ) −1 B ≠ 0 , by CA ν B we denote the first nonzero Markov parameter of S(A,B,C). Procedure 5.1.

CA ν B has full column rank. 1. S(A,B,C) is nondegenerate and its invariant and Smith zeros are exactly the same objects (including multiplicities) which are the o.d. zeros of S(K ν A, B, C) (see Corollary 4.2 and Proposition 5.1). 2.

CA ν B has full row rank r and m > r .

166

2a.


B m −r = 0

S(A,B,C) is nondegenerate and its invariant and Smith zeros are exactly the same objects (including multiplicities) which are the o.d. zeros of S(K ν A, B r , C ) (see Proposition 5.4 (ii)). 2b. B m −r ≠ 0 S(A,B,C) is degenerate (Proposition 5.4 (i)) and the set of its invariant zeros coincides with the set of invariant zeros of S(K ν A, B m− r , C ) (whose transfer-function matrix equals zero identically). The Smith zeros of S(A,B,C) are the i.o.d. zeros of S(K ν A, B m− r , C ) (the zero polynomial of S(A,B,C) is equal to the zero polynomial of S(K ν A, B m− r , C ) and the latter can be found out by applying Proposition 2.6 to S(K ν A, B m− r , C ) ). 0 < rank CA ν B = p < min{m, r}

3. 3a.

Cr − p = 0

3a1.

Cr − p = 0 and B m − p = 0

S(A,B,C) is nondegenerate and its invariant and Smith zeros are exactly the same objects (including multiplicities) which are the o.d. zeros of S(K ν A, B p , C p ) (see Proposition 5.5 (ii)) or, what is the same, of S(K ν A, B p , C ) .

3a2.

Cr − p = 0 and B m − p ≠ 0

S(A,B,C) is degenerate (Proposition 5.5 (i)) and the set of its invariant zeros coincides with the set of invariant zeros of S(K ν A, B m − p , C p ) (whose transfer-function matrix equals zero identically). The Smith zeros of S(A,B,C) are the i.o.d. zeros of S(K ν A, B m − p , C p ) (the zero polynomial of S(A,B,C) is equal to the zero polynomial of S(K ν A, B m − p , C p ) and the latter can be found out by applying Proposition 2.6 to S(K ν A, B m − p , C p ) ). Comment. Observe that in 3a2 the system S(K ν A, B m − p , C p ) can be re-

placed with S(K ν A, B m− p , C ) . 3b.

Cr − p ≠ 0


3b1.

167

Cr − p ≠ 0 and B m − p = 0

S(A,B,C) is nondegenerate and its invariant and Smith zeros are exactly the same objects (including multiplicities) which are the o.d. zeros of S(K ν A, B p , C ) (see Proposition 5.6). 3b2.

Cr − p ≠ 0 and B m − p ≠ 0

The question of degeneracy/nondegeneracy of S(A,B,C) can not be decided at this step. Start the next step applying Procedure 5.1 to the system S(K ν A, B m − p , C ) (see Proposition 5.3). 4. All Markov parameters of S(A,B,C) are equal to zero (or equivalently, G ( z) = C( zI − A ) −1 B ≡ 0 ). S(A,B,C) is degenerate. The Smith zeros of S(A,B,C) are its i.o.d. zeros (see Proposition 2.6). ◊ In cases 1, 2a, 2b, 3a1, 3a2, 3b1 and 4 the question of degeneracy/nondegeneracy of the system S(A,B,C) is decided at the first step. In cases 2b, 3a2 and 4 the system is degenerate, whereas in cases 1, 2a, 3a1 and 3b1 it is nondegenerate and all its invariant zeros (equivalently, Smith zeros) can be found as o.d. zeros of an appropriate closed-loop (state feedback) system. The case 3b2 (i.e., C r − p ≠ 0 and B m − p ≠ 0 ) requires a more detailed discussion. By virtue of Proposition 5.3 (i), the set of invariant zeros of S(A,B,C) coincides with the set of invariant zeros of the system S(K ν A ,B m − p ,C ) (the same holds for the zero polynomials of S(A,B,C) and S(K ν A,B m − p ,C ) – see Proposition 5.3 (ii)). Thus the task of seeking invariant and Smith zeros of S(A,B,C) can be replaced by the task of seeking invariant and Smith zeros of a new system S( A' ,B' ,C' ) with the matrices A' = K ν A , B' = B m − p , C' = C and m' = m − p inputs. In order to find the first nonzero Markov parameter of S( A' ,B' ,C' ) , we observe first that nonsingularity of C p A ν B p = M p implies C p ≠ 0 . On the other hand, by virtue of (5.7) and Lemma 5.2 (vi), we obtain C p (K ν A) l B m − p = 0 for l = 0,1,2,... (cf., the proof of Proposition 5.3 (ii)). Furthermore, by virtue of Lemma 5.2 (vii), we have C (K ν A ) l = C A l for 0 ≤ l ≤ ν . These facts imply that Markov parameters of the system S( A' ,B' ,C' ) can be written as

168


⎡ C p (K ν A ) l B m − p ⎤ ⎡ Cp ⎤ l ⎥ C'( A' ) l B' = ⎢ ⎥ (K ν A ) B m − p = ⎢ ⎢⎣ Cr − p (K ν A) l B m− p ⎥⎦ ⎣⎢ Cr − p ⎦⎥ ⎧ ⎡0⎤ for 0 ≤ l ≤ ν ⎪⎢ ⎥ ⎪ ⎣0⎦ . =⎨ 0 ⎤ ⎡ ⎪⎢ for l ≥ ν + 1 ⎪⎣ C r − p (K ν A) l B m − p ⎥⎦ ⎩

(5.17)

If all matrices in (5.17) are zero (cf., Example 5.1), then, by virtue of point 4 of Procedure 5.1 referred to S( A' ,B' ,C' ) , S( A' ,B' ,C' ) is degenerate and the same holds for S(A,B,C) (see Proposition 5.3 (i)). If this is not the case and C'( A ' ) ν ' B ' denotes the first nonzero Markov parameter of S( A' ,B' ,C' ) , then, as it follows from (5.17), we must have ν + 1 ≤ ν' ≤ n − 1 . If C'( A ' ) ν ' B ' has full column rank, the process ends at this stage (to S( A' ,B' ,C' ) we apply the point 1 of Procedure 5.1).

If C'( A ' ) ν ' B ' has not full column rank, i.e., it has rank, say p' < min{m' , r − p} (cf., (5.17)), we apply SVD to this parameter and we associate with S( A' ,B' ,C' ) , according to the rules (5.1), (5.2) and (5.5) referred to S( A' ,B' ,C' ) , an auxiliary system S( A' ,B',C') . Now, employing Procedure 5.1 referred to S( A' ,B' ,C' ) , we see that in cases 3a1, 3a2 and 3b1 the process ends at this stage (note that the cases 2a and 2b are not possible since Markov parameters in (5.17) have not full row rank). In the case 3b2 we begin the third step applying Procedure 5.1 to a new system S(A' ' ,B' ' ,C' ' ) , where A' ' = K ' ν ' A' , B' ' = B' m'− p ' and C' ' = C' (see Fig. 5.3). Now, if C' '( A ' ' ) ν '' B ' ' denotes the first nonzero Markov parameter of S( A' ' ,B' ' ,C' ' ) , then the arguments following from (5.17) (when applied to S( A' ' ,B' ' ,C' ' ) ) show that ν'+1 ≤ ν' ' ≤ n − 1 . In this way in the case 3b2 the procedure generates a strictly increasing sequence of integers ν, ν' , ν' ' , ... which is upper bounded by n-1, i.e., 0 ≤ ν < ν' < ν' ' < ... ≤ n − 1 . This means that the process ends after at most n steps. Moreover, at the last step we can meet only two possible situations: we obtain a system whose transfer-function matrix equals zero identically


169

or a system with the first nonzero Markov parameter of full column rank (in particular, nonsingular). The above discussion can be summarized as follows.

Fig. 5.3.

Corollary 5.1. (i) The question of seeking invariant and Smith zeros of an arbitrary system S(A,B,C) (2.1) can be decided at the first step (the cases 1, 2a, 2b, 3a1, 3a2, 3b1 and 4 of Procedure 5.1) or in the case 3b2 (by applying successively Procedure 5.1) after at most n steps. (ii) At each step of the recursive process generated by the point 3b2 the set of invariant zeros and the zero polynomial remain the same as at the previous step (Proposition 5.3). The set of invariant zeros and the zero polynomial of the original system S(A,B,C) can be determined as the set of invariant zeros and the zero polynomial of the system obtained at the last step. The process ends when we obtain a nondegenerate system (the case 1, 3a1 or 3b1) or a degenerate system (the case 3a2 or 4). ◊ Corollary 5.2. If S(A,B,C) (2.1) is nondegenerate, then its invariant and Smith zeros are exactly the same objects (including multiplicities) which are the o.d. zeros of a certain closed-loop (state feedback) system.

170


If S(A,B,C) is degenerate, the set of its invariant zeros equals the whole complex plane, while the Smith zeros of S(A,B,C) are the i.o.d. zeros of a certain system whose transfer-function matrix equals zero identically. Proof. The claims follow immediately from Procedure 5.1 and Corollary 5.1. ◊ Example 5.1. In S(A,B,C) (2.1) let (cf., Example 2.4)

1/ 3 0 ⎤ ⎡ 0 ⎡0 0 ⎤ ⎡− 2 − 1 0⎤ ⎥ ⎢ . A=⎢ 0 0 1 / 3 ⎥ , B = ⎢⎢0 1⎥⎥ , C = ⎢ 0 1 0 ⎥⎦ ⎣ ⎢⎣− 1 / 3 − 2 / 3 − 1 / 3⎥⎦ ⎢⎣1 0⎥⎦ The system is minimal and asymptotically stable. In order to find its invariant zeros, observe first that in SVD (5.1) of the first nonzero Markov parameter ⎡0 − 1⎤ CB = ⎢ ⎥ (i.e., in S(A,B,C) is ν = 0 and rank CB = p = 1 ) ⎣0 1 ⎦

we can take ⎡ 1 ⎢− 2 U=⎢ 1 ⎢ ⎢⎣ 2

1 ⎤ ⎥ 2⎥, 1 ⎥ 2 ⎥⎦

⎡ 2 Λ=⎢ ⎣⎢ 0

0⎤ ⎥, 0⎦⎥

⎡0 1 ⎤ VT = ⎢ ⎥. ⎣1 0 ⎦

Now, using (5.3) and (5.5), we obtain

[

B = BV = B p

⎡0 0 ⎤ ⎡ Cp ⎤ ⎡ 2 2 0⎤ B m − p = ⎢⎢1 0⎥⎥ , C = U T C = ⎢ ⎥=⎢ ⎥ Cr − p ⎦⎥ ⎢⎣ − 2 0 0 ⎥⎦ ⎢ ⎣ ⎢⎣0 1⎥⎦

]

and, in view of (5.10) at ν = 0 , K ν = I − B p M −p1C p A ν

⎡ 1 0 0⎤ = ⎢⎢− 1 0 0⎥⎥ . ⎢⎣ 0 0 1⎥⎦

Next, by virtue of point 3b2 in Procedure 5.1, we take into account the sys-


171

tem S(K ν A,B m − p ,C ) , where 1/ 3 0 ⎤ ⎡ 0 ⎡0 ⎤ ⎡ 2 ⎥ ⎢ KνA = ⎢ 0 0 ⎥ , B m − p = ⎢⎢0⎥⎥ , C = ⎢ − 1/ 3 ⎢⎣− 2 ⎢⎣− 1 / 3 − 2 / 3 − 1 / 3⎥⎦ ⎢⎣1⎥⎦

2 0

0⎤ ⎥. 0⎥⎦

Because C B m − p = 0 and (K ν A) l B m − p = (−1 / 3) l B m − p for l = 1,2,... , all Markov parameters of S(K ν A,B m − p ,C ) are zero. This means that the transfer-function matrix of S(K ν A ,B m − p ,C ) equals identically the zero matrix. Hence, via Corollary 5.1, we infer that S(K ν A,B m − p ,C ) and, consequently, the original system S(A,B,C) are degenerate. Applying Proposition 2.6 to S(K ν A,B m − p ,C ) , we verify that this system has no i.o.d. zeros and then, by virtue of Proposition 5.3, we conclude that S(A,B,C) has no Smith zeros. ◊ Example 5.2. Consider a minimal and asymptotically stable system S(A,B,C) (2.1) with the matrices ⎡0 ⎢0 A=⎢ ⎢0 ⎢ ⎣0

0 0 0⎤ 0 1 0⎥⎥ , 0 0 1⎥ ⎥ 0 0 0⎦

⎡1 ⎢0 B=⎢ ⎢0 ⎢ ⎣0

0⎤ 0⎥⎥ , 0⎥ ⎥ 1⎦

⎡1 0 0 0 ⎤ C=⎢ ⎥. ⎣1 − 1 1 0⎦

In order to find invariant zeros of the system, we observe first that in SVD (5.1) of the first nonzero Markov parameter ⎡1 0⎤ CB = ⎢ ⎥ ⎣1 0⎦

(i.e., in S(A,B,C) is ν = 0 and rank CB = p = 1 )

we can take ⎡ ⎢ U=⎢ ⎢ ⎢⎣

1 2 1 2

1 ⎤ ⎥ 2 ⎥, 1 ⎥ − 2 ⎥⎦

⎡ 2 Λ=⎢ ⎢⎣ 0

Using (5.3), we obtain for S( A, B , C ) (5.2)

0⎤ ⎥, 0⎥⎦

⎡1 0 ⎤ VT = ⎢ ⎥. ⎣0 1 ⎦

172


B = BV = B ,

⎡ ⎢ 2 C = UTC = ⎢ ⎢0 ⎢⎣

−

⎤ 2 0⎥ 2 ⎥. 2 − 0 ⎥⎥ 2 ⎦

2 2

2 2

Now, according to the point 3b2 of Procedure 5.1, we are looking for invariant zeros of the system S(K ν A,B m − p ,C ) , where for ν = 0 ⎡ ⎢0 ⎢ K ν = I − B p M −p1C p A ν = ⎢0 ⎢0 ⎢ ⎣0

1 2 1

1 2 0

−

0

1

0

0

⎤ 0⎥ 0⎥ ⎥ 0⎥ ⎥ 1⎦

(recall that B p is the first column of B and C p is the first row of C ) and

⎡ ⎢0 ⎢ K ν A = ⎢0 ⎢0 ⎢ ⎣0

0 0 0 0

1 2 1 0 0

1⎤ ⎡0 ⎤ − ⎥ ⎡ 2 ⎢0 ⎥ ⎢ 2 ⎥ 0 , B ⎢ ⎥, C =⎢ = ⎥ m− p ⎢0 ⎥ ⎢0 1 ⎥ ⎢ ⎥ ⎢⎣ ⎥ 1 ⎣ ⎦ 0 ⎦

− 2 2

⎤ 2 0⎥ 2 ⎥. 2 − 0 ⎥⎥ 2 ⎦

2 2

Denote now A ' = K ν A, B' = B m − p , C' = C . The first nonzero Markov parameter of S( A' ,B' ,C' ) is

⎡ 0 ⎤ C' A' B' = ⎢ 2 ⎥ (i.e., in S( A' ,B' ,C' ) is ν ' = 1 and p ' = rank C' A' B' = 1 ). ⎥ ⎢− ⎣ 2 ⎦ Since C' A ' B ' has full column rank, we apply to S( A' ,B' ,C' ) the point 1 of Procedure 5.1. This means that we seek invariant zeros of S( A' ,B' ,C' ) as output decoupling zeros of the closed-loop system S(K ' ν ' A ' ,B ' ,C' ) , where K 'ν ' is evaluated according to the formula K ' ν ' = I − B ' (C' ( A ' ) ν ' B' ) + C' ( A ' ) ν '

at ν ' = 1 . However, after simple calculations we obtain

5.3 Invariant and Smith Zeros in Proper Systems

⎡1 ⎢0 K 'ν' = ⎢ ⎢0 ⎢ ⎣0

0 1 0 0

0 0 1 1

0⎤ 0⎥⎥ 0⎥ ⎥ 0⎦

and

⎡ ⎢0 ⎢ K ' ν ' A ' = ⎢0 ⎢0 ⎢ ⎣0

0 0 0 0

1 2 1 0 0

173

1⎤ − ⎥ 2 0 ⎥. ⎥ 1 ⎥ ⎥ 1 ⎦

Now, decomposing S(K ' ν ' A ' ,B ' ,C' ) into an unobservable and an observable part, it is easy to verify that λ = 1 is the only output decoupling zero of S(K ' ν ' A ' ,B ' ,C' ) . Hence λ = 1 is the only invariant zero of the system S(K ' ν ' A ' ,B ' ,C' ) and, by virtue of Corollary 5.1 (ii), we conclude that it is the only invariant zero of the original system S(A,B,C). ◊

5.3 Invariant and Smith Zeros in Proper Systems In the system S( A, B, C, D) (2.1) let D has rank 0 < p ≤ min{m, r} . Write SVD of D as (see Appendix B, Theorem B.1) D = U Λ VT ,

⎡D where Λ = ⎢ p ⎣ 0

0⎤ 0⎥⎦

(5.18)

and D p is pxp diagonal and nonsingular. Introducing into S( A, B, C, D) orthogonal matrices V and U T as pre- and postcompensator, we associate with S( A, B, C, D) a new system S( A, B, C, Λ) (see Fig. 5.4) of the form

x(k + 1) = Ax(k ) + B u ( k ) y ( k ) = C x( k ) + Λ u ( k )

, k∈N ,

(5.19)

where B = BV , C = U T C and u = V T u , y = U T y . For matrices B and

C and for vectors u and y we use the same decomposition as in (5.5). Lemma 5.3. The sets of invariant zeros of the systems S(A,B,C,D) and S( A, B, C, Λ) coincide, i.e., ZIS( A, B, C, D) = ZI . Moreover, these S(A,B , C ,Λ)

systems have the same zero polynomial, i.e., ψ S( A , B , C , D ) ( z ) = ψ S( A , B , C , Λ ) ( z ) ,

and, consequently, the same set of Smith zeros, i.e.,

174


Fig. 5.4.

Z SS( A, B, C, D) = Z S

S(A,B , C ,Λ)

.

Proof. The lemma follows immediately from Lemma 2.3 (ii) and (iii) and from Exercise 2.8.10. ◊

If in S( A, B , C, Λ) (5.19) is B m − p ≠ 0 and Cr − p ≠ 0 , then invariant and Smith zeros of S(A,B,C,D) may be characterized as invariant and Smith zeros of a certain strictly proper system with a smaller number of inputs and outputs than in the original system S(A,B,C,D) (see Fig. 5.5).


175

Proposition 5.7. In S(A,B,C,D) (2.1) let 0 < rank D = p < min{m, r} and in the associated system S( A, B, C, Λ) (5.19) let B m − p ≠ 0 and let Cr − p ≠ 0 . Then the sequence of transformations

S( A, B, C, D) → S( A, B , C, Λ) → S( A − B p D −p1C p , B m − p , Cr − p ) has the following properties: (i)

it preserves the set of invariant zeros, i.e., ZIS( A , B, C, D) = ZI

S( A , B , C , Λ )

(ii)

= ZI

S( A − B p D −p1 C p , B m − p , C r − p )

,

it preserves the zero polynomial, i.e.,

ψ S( A, B, C, D) (z) = ψ S( A, B , C, Λ ) (z) = ψ S( A − B D −1 C , B , C ) (z) , p p p m− p r− p and, consequently, the set of Smith zeros, i.e., Z SS( A , B, C, D) = Z S

S( A , B , C , Λ )

= ZS

S( A − B p D −p1 C p , B m − p , C r − p )

.

Proof. (i) In view of Lemma 5.3, it is sufficient to show that S( A, B, C, Λ)

and S( A − B p D −p1 C p , B m − p , Cr − p ) (see Fig. 5.5) have the same set of invariant zeros, i.e., we have to show that λ ∈ C is an invariant zero of

S( A − B p D −p1 C p , B m − p , Cr − p ) if and only if λ is an invariant zero of

S( A, B, C, Λ) . If λ is an invariant zero of (5.19), then, by virtue of ⎡ gp ⎤ Definition 2.1 (i), there exist vectors x o ≠ 0 and g = ⎢ ⎥ such that ⎣g m − p ⎦ (iii)

λ x o − Ax o = B g

and

C x o + Λg = 0 .

However, as we know from Proposition 3.2 (when applied to the system S( A, B, C, Λ) ), the relations (iii) imply that λ x o − ( A − B Λ + C )x o = B g1 ,

x o ∈ Ker(I r − ΛΛ + )C

(5.20)

176


Fig. 5.5.

and g = g1 + g 2 , where g1 ∈ Ker Λ and g 2 = − Λ + C x o . Using (5.5), it is easy to verify that the following relations hold true

[

A − BΛ + C = A − B p

⎡D Bm − p ⎢ p ⎣ 0

]

Ker(I r − ΛΛ + )C = Ker Cr − p ,

0⎤ 0⎥⎦

+

⎡ Cp ⎤ −1 ⎢ ⎥ = A − B pD p Cp , C ⎢⎣ r − p ⎥⎦

⎡− D −p1C p x o ⎤ g 2 = −Λ + Cx o = ⎢ ⎥. 0 ⎥⎦ ⎢⎣

Now, since g1 ∈ Ker Λ ,

⎡ gp ⎤ g=⎢ ⎥ = g1 + g 2 , ⎣g m − p ⎦

⎡− D −p1C p x o ⎤ g2 = ⎢ ⎥, 0 ⎣⎢ ⎦⎥

we can write g p = −D −p1 C p x o ,

⎡ 0 ⎤ g1 = ⎢ ⎥ ⎣g m − p ⎦


177

and

[

B g1 = B p

⎡ 0 ⎤ Bm − p ⎢ ⎥ = Bm − p gm − p . ⎣g m − p ⎦

]

⎡ gp ⎤ Hence, if a triple λ, x o ≠ 0, g = ⎢ ⎥ satisfies (iii), then ⎣g m − p ⎦

λx o − ( A − B p D −p1C p )x o = B m − p g m − p (iv)

o

Cr − p x = 0

and g p = − D −p1 C p x o .

In this way we have proved that if λ is an invariant zero of S( A, B, C, Λ) , then λ is also an invariant zero of S( A − B p D −p1 C p , B m − p , Cr − p ) . In order to prove the converse implication, it is sufficient to verify that if ⎡ gp ⎤ a triple λ, x o ≠ 0, g m − p satisfies (iv) and we define g := ⎢ ⎥ , with ⎣g m − p ⎦ g p := −D −p1C p x o , then the triple λ, x o , g satisfies (iii). This verification is left to the reader as an exercise. (ii) Because of Lemma 5.3, we only need to show that the zero polynomial of S( A − B p D −p1 C p , B m− p , Cr − p ) is the same as of the system

S( A, B , C, Λ) . Premultiplying and postmultiplying the system matrix of S( A, B, C, Λ) by appropriate unimodular matrices, we can write (v) ⎡ I B D −1 0 ⎤ ⎡zI − A − B p p p ⎢ ⎥⎢ Ip 0 ⎥ ⎢ Cp Dp ⎢0 ⎢0 ⎥ ⎢ 0 I r − p ⎥ ⎣ Cr − p 0 ⎢⎣ ⎦

− B m− p ⎤ ⎡ I ⎥⎢ −1 0 ⎥ ⎢− D p C p 0 ⎥⎦ ⎢⎣ 0

0

D −p1 0

0 ⎤ ⎥ 0 ⎥ I m− p ⎥⎦

⎡zI − ( A − B p D −p1 C p ) 0 − B m− p ⎤ ⎢ ⎥ =⎢ 0 Ip 0 ⎥. ⎢ Cr − p 0 ⎥⎥ 0 ⎢⎣ ⎦ Interchanging suitably rows and columns, we transform the matrix on the right-hand side of (v) into the form

178


⎡I p ⎢ ⎢0 ⎢0 ⎣

(vi)

0

0

zI − ( A − B p D −p1C p ) Cr − p

⎤ ⎥ − B m− p ⎥ . 0 ⎥⎦

Since the zero polynomial of (vi) is equal to the zero polynomial of the submatrix ⎡zI − ( A − B p D −p1 C p ) − B m − p ⎤ ⎢ ⎥, Cr − p 0 ⎥⎦ ⎢⎣

(vii)

we obtain the desired result. ◊ Lemma 5.4. In S(A,B,C,D) (2.1) let m > r and let D have full row rank r.

⎡ gr ⎤ Then a triple λ, x o ≠ 0, g = ⎢ ⎥ represents an invariant zero λ of ⎣ g m −r ⎦ (5.19) if and only if λx o − ( A − B r D r−1 Cr ) x o = B m − r g m − r

and g r = −D r−1Cr x o . (5.21)

Proof. Applying SVD to D, we obtain D = U Λ VT ,

Λ = [D r

0 ],

[

B = Br

]

B m−r ,

C = Cr

(5.22)

and S( A, B , C, Λ) in (5.19) takes the form x(k + 1) = Ax(k ) + B r u r ( k ) + B m−r u m−r (k ) y ( k ) = C r x( k ) + D r u r ( k ) .

(5.23)

⎡ gr ⎤ Now, if a triple λ, x o ≠ 0, g = ⎢ ⎥ represents an invariant zero λ ⎣ g m −r ⎦ of S( A, B , C, Λ) in (5.23), then this triple has to fulfill (5.20). Introducing (5.22) into (5.20), we obtain ⎡D −1 ⎤ Λ+ = ⎢ r ⎥ , ⎢⎣ 0 ⎥⎦ A − BΛ + C = A − B r D −r 1Cr ,

I r − ΛΛ + = 0 ,

⎡−D r−1Cr x o ⎤ ⎡ 0 ⎤ g1 = ⎢ g = , ⎥ 2 ⎢ ⎥ 0 ⎢⎣ ⎣ g m −r ⎦ ⎦⎥


179

and (5.20) takes the form

λx o − ( A − B r D r−1Cr )x o = B m−r g m−r ,

⎡− D r−1Cr x o ⎤ g=⎢ ⎥, ⎣⎢ g m−r ⎦⎥

(5.24)

i.e., the triple under considerations must satisfy (5.21). ⎡ gr ⎤ Conversely, if a triple λ, x o ≠ 0, g = ⎢ ⎥ satisfies (5.21), then ⎣ g m−r ⎦ (5.21) can be written as λ x o − Ax o = B r g r + B m− r g m − r Cr x o + D r g r = 0 ,

(5.25)

i.e., the considered triple satisfies Definition 2.1(i) for (5.23). ◊ Remark 5.2. From Lemma 5.4 and from Definition 2.1 (i) (when applied to (5.23)) it follows, in particular, that each eigenvalue of A − B r D −r 1 C r is

an invariant zero of S( A, B, C, Λ) in (5.23). In fact, if λ ∈ σ( A − B r D r−1Cr ) , then as x o ≠ 0 in (5.21) it is sufficient to take an associated eigenvector and set g m − r = 0 (or a g m − r such that

B m − r g m − r = 0 ). ◊ In the following result in S( A, B , C, Λ) in (5.23) we discuss separately the cases B m −r = 0 and B m −r ≠ 0 . Proposition 5.8. In S(A,B,C,D) (2.1) let m > r and let D have full row rank r. Then: (i)

S(A,B,C,D) is degenerate if and only if in S( A, B, C, Λ) (5.23) is

B m −r ≠ 0 . Moreover, at the condition B m −r ≠ 0 , the zero polynomial of S(A,B,C,D) is equal to the zero polynomial of the pencil zI − ( A − B r D r−1 C r ) − B m − r .

[

(ii)

]

S(A,B,C,D) is nondegenerate if and only if in S( A, B, C, Λ) (5.23)

is B m −r = 0 . Moreover, at B m −r = 0 , a number λ ∈ C is an invariant zero of S(A,B,C,D) if and only if λ ∈ σ( A − B r D −r 1C r ) . Furthermore, at

180


B m −r = 0 ,

the

zero

polynomial

of

S(A,B,C,D)

is

equal

to

det ( zI − ( A − B r D −r 1 C r )) , i.e., invariant and Smith zeros are exactly the same objects (including multiplicities).

Proof. Of course, by virtue of Lemma 5.3, we can consider in the proof the

system S( A, B, C, Λ) (5.23) instead of S(A,B,C,D). Moreover, in (i) it is sufficient to prove the implication: B m −r ≠ 0 ⇒ S( A, B , C, Λ) is degenerate, and in (ii) – the implication: B m −r = 0 ⇒ S( A, B, C, Λ) is nondegenerate. (i) Suppose that in S( A, B, C, Λ) (5.23) is B m −r ≠ 0 . Using Lemma 5.4, we shall show that S( A, B, C, Λ) is degenerate. To this end, let λ ∉ σ( A − B r D r−1Cr ) and let g m − r be such that B m − r g m − r ≠ 0 . Denote

x o := (λI − ( A − B r D r−1Cr )) −1 B m − r g m − r ≠ 0

and g r := − D −r 1C r (λI − ( A − B r D r−1Cr )) −1 B m − r g m − r .

⎡ gr ⎤ Then the triple λ, x o ≠ 0, g = ⎢ ⎥ satisfies (5.21), i.e., λ is an in⎣g m − r ⎦ variant zero of S( A, B, C, Λ) . Hence any λ ∉ σ( A − B r D −r 1C r ) is an invariant zero of this system. Moreover, as we have noticed in Remark 5.2, each λ ∈ σ( A − B r D r−1Cr ) is an invariant zero of S( A, B, C, Λ) . This means that at B m −r ≠ 0 any complex number is an invariant zero of

S( A, B , C, Λ) (5.23). For the proof of the second claim in (i) we can write I ⎡ ⎡ I B r D −r 1 ⎤ ⎡zI − A − B r − B m−r ⎤ ⎢ −1 ⎢ ⎥⎢ ⎥ − D r Cr Dr 0 ⎦⎢ I r ⎥⎦ ⎣ Cr ⎢⎣0 ⎢⎣ 0 ⎡zI − ( A − B r D −r 1 Cr ) 0 B m−r ⎤ =⎢ ⎥ Ir 0 ⎦⎥ 0 ⎣⎢

0 0 ⎤ −1 Dr 0 ⎥⎥ I m−r ⎥⎦ 0


181

and then, interchanging appropriately rows and columns in the matrix on the right-hand side, we obtain

⎡I r ⎢0 ⎣

0 ⎤ 0 . −1 zI − ( A − B r D r Cr ) − B m − r ⎥⎦

Finally, the claim follows via Lemma 5.3. (ii) The condition B m −r = 0 implies, via Lemma 5.4 (see (5.21)), that any invariant zero of S( A, B, C, Λ) (5.23) is contained in the spectrum of A − B r D −r 1 C r . On the other hand, as we know from Remark 5.2, each

eigenvalue of A − B r D −r 1 Cr is an invariant zero of this system. Hence, at

B m −r = 0 , S( A, B, C, Λ) is nondegenerate and λ ∈ C is its invariant zero

if and only if λ ∈ σ( A − B r D r−1Cr ) . Moreover, at B m −r = 0 , the system matrix of S( A, B, C, Λ) can be transformed by elementary operations into the form

⎡I r ⎢0 ⎣

0

zI − ( A − B r D r−1Cr )

0⎤ . 0⎥⎦

Thus the zero polynomial of S( A, B, C, Λ) (5.23) and, consequently (via Lemma 5.3) of S(A,B,C,D), equals det ( zI − ( A − B r D −r 1 C r )) . Now, the claim follows via Definition 2.3. ◊ Remark 5.3. It can be easily verified via Definition 2.1 (i) that, under the condition B m −r = 0 , a number λ ∈ C is an invariant zero of

S( A, B, C, Λ) if and only if λ is an invariant zero of the system S( A, B r , Cr , D r ) (of uniform rank) obtained from (5.23) by neglecting the input u m − r . This means that the characterization of invariant zeros given in Proposition 5.8 (ii) can also be formulated as follows: λ ∈ C is an invariant zero of S(A,B,C,D) if and only if λ is an invariant zero of S( A, B r , Cr , D r ) . ◊ Remark 5.4. As we know from Proposition 5.8 (i), at B m−r ≠ 0 the zero polynomial of S(A,B,C,D) is equal to the zero polynomial of the pencil zI − ( A − B r D r−1 C r ) − B m − r . However, by a suitable change of coordi-

[

]

nates we can decompose the pair ( A − B r D −r 1 C r , B m− r ) into a reachable

182


and an unreachable part (cf., Appendix A.1), where

⎡( A − B r D r−1Cr ) 'r ( A − B r D r−1Cr ) ' = ⎢ 0 ⎢⎣

' ⎤ ( A − B r D r−1Cr )12 ⎥, ( A − B r D −r 1Cr ) 'r ⎥⎦

⎡( B )' ⎤ (B m − r ) ' = ⎢ m − r r ⎥ 0 ⎥⎦ ⎢⎣ and the pair ( ( A − B r D r−1 C r ) 'r , ( B m − r ) 'r ) is reachable. In the new coordinates the pencil

[zI − (A − B D r

−1 r Cr )

− B m−r

]

will take the form ' ⎡zI 'r − ( A − B r D −r 1Cr ) 'r − ( A − B r D −r 1Cr )12 − ( B m−r ) 'r ⎤ ⎥. ⎢ 0 zI 'r − ( A − B r D −r 1Cr ) 'r 0 ⎦⎥ ⎣⎢

Now, interchanging in the matrix above the last two columns and then using the Hermite column form to [ zI 'r − ( A − B r D r−1 C r ) 'r − ( B m − r ) 'r ], we infer that the zero polynomial of zI − ( A − B r D r−1 C r ) − B m −r is

[

]

equal to det ( zI 'r − ( A − B r D −r 1 C r ) 'r ) . Moreover, since ( B m − r ) 'r ≠ 0 , we

have deg det (zI 'r − ( A − B r D r−1 C r ) 'r ) < n . Thus the second claim in Proposition 5.8 (i) can be formulated also as follows: at B m −r ≠ 0 the zero polynomial of S(A,B,C,D) is equal to det ( zI 'r − ( A − B r D −r 1 C r ) 'r ) ; moreover, its degree is less than the dimension of the state space. ◊

We can pass now to the case when in S(A,B,C,D) is 0 < rank D = p < min{m, r} and in the associated system S( A, B , C, Λ) (5.19) is Cr − p = 0 . In (5.19) we have then ⎡D Λ=⎢ p ⎣ 0

0⎤ , 0⎥⎦

[

B = Bp

i.e., (5.19) may be written as

]

B m− p ,

⎡C ⎤ C = ⎢ p ⎥, ⎣ 0 ⎦

(5.26)


183

x(k + 1) = Ax(k ) + B p u p (k ) + B m − p u m− p (k ) y p ( k ) = C p x( k ) + D p u p ( k )

(5.27)

y r − p (k ) = 0 .

However, as it follows immediately from Definition 2.1 (i), a number λ is an invariant zero of (5.27) if and only if λ is an invariant zero of the system

⎡ u p (k ) ⎤ B m− p ⎢ ⎥ ⎣u m− p (k )⎦ ⎡ u p (k ) ⎤ y p ( k ) = C p x( k ) + D p 0 ⎢ ⎥. ⎣u m− p (k )⎦

[

]

x(k + 1) = Ax(k ) + B p

[

]

(5.28)

Of course, systems (5.27) and (5.28) have the same zero polynomial. On the other hand, the system in (5.28) corresponds exactly to the situation discussed above, i.e., of the D-matrix of full row rank. Hence we can formulate the following. Proposition 5.9. In S(A,B,C,D) (2.1) let 0 < rank D = p < min{m, r} and in the associated system S( A, B , C, Λ) (5.19) let Cr − p = 0 . Then: (i) System S(A,B,C,D) is degenerate if and only if in the system S( A, B, C, Λ) is B m − p ≠ 0 . Moreover, at B m − p ≠ 0 , the zero polynomial of S(A,B,C,D) is equal to the zero polynomial of the pencil zI − ( A − B p D -p1C p ) − B m− p .

[

]

(ii) System S(A,B,C,D) is nondegenerate if and only if in the system S( A, B, C, Λ) is B m − p = 0 . Moreover, at B m − p = 0 , a number λ ∈ C is an invariant zero of S(A,B,C,D) if and only if λ ∈ σ( A − B p D −p1 C p ) . Furthermore, at

B m − p = 0 , the zero polynomial of the system

S(A,B,C,D) equals det(zI − ( A − B p D −p1C p )) , i.e., the invariant and Smith zeros in S(A,B,C,D) are exactly the same objects (including multiplicities). Proof. The proof follows the same lines as the proof of Proposition 5.8. ◊

184


Remark 5.5. One can observe, employing Definition 2.1 (i), that at B m − p = 0 the sets of invariant zeros of systems S( A, B , C, Λ) in (5.27)

and S( A, B p , C p , D p ) (which is obtained from (5.28) by neglecting the input u m − p ) coincide. This enables us to characterize invariant zeros of S(A,B,C,D) as invariant zeros of S( A, B p , C p , D p ) (which is of uniform rank). ◊ Remark 5.6. The second claim in Proposition 5.9 (i) can be formulated as

follows. At B m − p ≠ 0 the zero polynomial of S(A,B,C,D) is equal to det ( zI 'r − ( A − B p D −p1C p ) 'r ) , where ( A − B p D −p1 C p ) 'r is taken from the decomposition of the pair ( A − B p D −p1C p , B m − p ) into a reachable and an unreachable part (cf., Remark 5.4). Moreover, the degree of the zero polynomial of S(A,B,C,D) is less than the dimension of the state space, i.e., deg det ( zI 'r − ( A − B p D −p1C p ) 'r ) < n . ◊ We end our discussion with the case B m − p = 0 and C r − p ≠ 0 . In

S( A, B , C, Λ) (5.19) we then have ⎡D Λ=⎢ p ⎣ 0

0⎤ , 0⎥⎦

[

B = Bp

]

0,

⎡ Cp ⎤ C=⎢ ⎥. ⎢⎣ Cr − p ⎥⎦

(5.29)

However, using Definition 2.1 (i), it is easy to verify that the set of invariant zeros of S( A, B, C, Λ) with matrices as in (5.29) is exactly the ⎡D ⎤ same as the set of invariant zeros of a system S( A,B p ,C ,⎢ p ⎥ ) of the ⎣ 0 ⎦ form

x(k + 1) = Ax(k ) + B p u p (k ) ⎡D ⎤ y ( k ) = C x( k ) + ⎢ p ⎥ u p ( k ) ⎣ 0 ⎦

(5.30)

which is obtained from S( A, B , C, Λ) (5.29) by neglecting the input um − p .


185

It is also easy to verify that the systems S( A, B, C, Λ) in (5.29) and ⎡D ⎤ S( A,B p ,C ,⎢ p ⎥ ) in (5.30) have the same zero polynomial. ⎣ 0 ⎦ ⎡D ⎤ Because ⎢ p ⎥ (see (5.30)) has full column rank, in order to ⎣ 0 ⎦ characterize invariant zeros of the system in (5.30) we can apply Proposition 5.2. To this end, however, we will need the following relations +

⎡D p ⎤ ⎡D p ⎤ ⎡I p ⎢ 0 ⎥⎢ 0 ⎥ =⎢ 0 ⎣ ⎦⎣ ⎦ ⎣

0⎤ , 0⎥⎦

+ 0 ⎤ ⎡ Cp ⎤ ⎡ 0 ⎤ ⎡ D p ⎤ ⎡ D p ⎤ ⎡ C p ⎤ ⎡0 (I r − ⎢ ⎥ ⎢ ⎥ ) ⎢ ⎥=⎢ ⎥=⎢ ⎥, ⎥⎢ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎢⎣ Cr − p ⎦⎥ ⎣0 I r − p ⎦ ⎣⎢ Cr − p ⎦⎥ ⎣ Cr − p ⎦

⎡D ⎤ A − Bp⎢ p⎥ ⎣ 0 ⎦

+

(5.31)

⎡ Cp ⎤ −1 ⎢ ⎥ = A − B pD p Cp . ⎢⎣ Cr − p ⎥⎦

Now, taking into account the above discussion as well as Lemma 5.3, we can formulate the following characterization of invariant zeros of the original system S(A,B,C,D). Proposition 5.10. In S(A,B,C,D) (2.1) let 0 < rank D = p < min{m, r} and in S( A, B , C, Λ) (5.19) let B m − p = 0 and let C r − p ≠ 0 . Then S(A,B,C,D) is nondegenerate and the following conditions are equivalent: a) b) c)

λ ∈ C is an invariant zero of S(A,B,C,D); ⎡D ⎤ λ is an invariant zero of S( A,B p ,C ,⎢ p ⎥ ) in (5.30); ⎣ 0 ⎦

λ ∈ σ( A − B p D −p1 C p ) and an associated eigenvector lies in

Ker Cr − p ;

d)

⎡ 0 ⎤ ⎡D p ⎤ λ is an o.d. zero of S( A − B p D −p1 C p ,B p ,⎢ ⎥ ,⎢ ⎥) ; ⎣ Cr − p ⎦ ⎣ 0 ⎦

e)

λ is a Smith zero of S(A,B,C,D).

186


Moreover, the zero polynomial of S(A,B,C,D) is equal to det(zI ' o −( A − B p D −p1C p )' o ) , i.e., the invariant and Smith zeros of S(A,B,C,D) are exactly the same objects (recall that ( A − B p D −p1 C p )' o is ⎡ 0 ⎤ ⎡D p ⎤ obtained from S( A − B p D −p1 C p ,B p ,⎢ ⎥ ,⎢ ⎥ ) by its decomposition ⎣ Cr − p ⎦ ⎣ 0 ⎦ into an unobservable and an observable part).

Proof. The proof follows by using Proposition 5.2 to the system ⎡D ⎤ S( A,B p ,C ,⎢ p ⎥ ) in (5.30). ◊ ⎣ 0 ⎦

5.4 A Procedure for Computing Zeros of Proper Systems Summarizing the results presented in the previous section, we list below all possible situations that can be met when the question of degeneracy/ nondegeneracy of an arbitrary proper system S(A,B,C,D) (2.1) is analyzed. The proposed recursive procedure enables us to compute invariant and Smith zeros of S(A,B,C,D). Procedure 5.2. 1. D has full column rank. S(A,B,C,D) is nondegenerate and its invariant and Smith zeros are exactly the same objects (including multiplicities) which are the o.d. zeros of S( A − BD + C, B, (I r − DD + )C, D) (see Proposition 5.2). D has full row rank r and m> r .

2. 2a.

B m −r = 0

S(A,B,C,D) is nondegenerate and its invariant and Smith zeros are exactly the same objects (including multiplicities) which are the eigenvalues of A − B r D −r 1 Cr (see Proposition 5.8 (ii)). 2b.

B m −r ≠ 0

S(A,B,C,D) is degenerate (see Proposition 5.8 (i)). The Smith zeros of S(A,B,C,D) are determined by the zero polynomial of the pencil zI − ( A − B r D −r 1 C r ) − B m − r (see Remark 5.4).

[

]

5.4 A Procedure for Computing Zeros of Proper Systems

187

0 < rank D = p < min{m, r}

3. 3a.

Cr − p = 0

Cr − p = 0 and B m − p = 0 S(A,B,C,D) is nondegenerate and its invariant and Smith zeros are exactly the same objects (including multiplicities) which are the eigenvalues of

3a1.

A − B p D −p1C p (see Proposition 5.9 (ii)). Cr − p = 0 and B m − p ≠ 0 S(A,B,C,D) is degenerate (see Proposition 5.9 (i)). The Smith zeros of S(A,B,C,D) are determined by the zero polynomial of the pencil

3a2.

[zI − (A − B 3b.

-1 pD pCp )

]

− B m− p (see Remark 5.6).

Cr − p ≠ 0

3b1.

C r − p ≠ 0 and B m − p = 0

S(A,B,C,D) is nondegenerate and its invariant and Smith zeros are exactly the same objects (including multiplicities) which are the o.d. zeros of the ⎡ 0 ⎤ ⎡D p ⎤ system S( A − B p D −p1 C p ,B p ,⎢ ⎥ ,⎢ ⎥ ) (see Proposition 5.10). ⎣ Cr − p ⎦ ⎣ 0 ⎦ 3b2.

C r − p ≠ 0 and B m − p ≠ 0

The question of degeneracy/nondegeneracy of S(A,B,C,D) can not be decided at this stage. Start the next step applying Procedure 5.1 to the strictly proper system S( A − B p D −p1C p , B m − p , Cr − p ) (see Proposition 5.7). ◊ Thus in the cases 1, 2a, 2b, 3a1, 3a2 and 3b1 the question of degeneracy/nondegeneracy of S(A,B,C,D) can be decided at the first step. In the cases 2b and 3a2 the system is degenerate. In the cases 1, 2a, 3a1 and 3b1 the system is nondegenerate and its invariant and Smith zeros coincide (including multiplicities). In the case 3b2 we have to use Proposition 5.7 and then Procedure 5.1 described in Section 5.2. Note that, by virtue of Propositions 5.3 and 5.7, the set of invariant zeros and the zero polynomial of S(A,B,C,D) (and, consequently, the set of its Smith zeros) are preserved along the recursive process generated by the point 3b2 of Procedure 5.2. The above discussion can be summarized as follows.

188


Corollary 5.3. (i) The question of seeking invariant and Smith zeros of an arbitrary system S(A,B,C,D) (2.1) can be decided at the first step (the cases 1, 2a, 2b, 3a1, 3a2, 3b1 of Procedure 5.2) or in the case 3b2 of Procedure 5.2, by applying Proposition 5.7 and then Procedure 5.1, after at most n + 1 steps. (ii) In the case 3b2 the set of invariant zeros and the zero polynomial of the original system S(A,B,C,D) can be determined respectively as the set of invariant zeros and the zero polynomial of a strictly proper system obtained at the last step of the recursive process following from Procedure 5.1. The process ends when we obtain a nondegenerate system (the case 1, 3a1 or 3b1 of Procedure 5.1) or a degenerate system (the case 3a2 or 4 of Procedure 5.1). Proof. The assertions of the corollary follow immediately from Procedure 5.2 and Corollary 5.1. ◊

Corollary 5.4. If S(A,B,C,D) (2.1) is nondegenerate, then its invariant and Smith zeros are exactly the same objects (including multiplicities). Proof. The claim of the corollary follows directly from Procedure 5.2 and Corollary 5.2. ◊ Example 5.3. Consider a system S(A,B,C,D) (2.1) with the matrices ⎡− 1 ⎢0 A=⎢ ⎢0 ⎢ ⎣1

1⎤ ⎡0 ⎢1 ⎥ 1 1 0⎥ , B=⎢ ⎢0 0 − 1 0⎥ ⎢ ⎥ 1 0 0⎦ ⎣0

1

0

⎡0 0 1 0 ⎤ C = ⎢⎢0 0 0 1⎥⎥ , ⎢⎣0 1 0 0⎥⎦

0 0⎤ 0 0 ⎥⎥ , 0 1⎥ ⎥ 1 0⎦

⎡1 0 0⎤ D = ⎢⎢0 0 0⎥⎥ . ⎢⎣0 0 0⎥⎦

The system is reachable and observable and it has one single Smith zero λ = −1 .


189

At the notation used in (5.18) and (5.5) we have p = 1 and in SVD of D D = Λ, D p = 1, U = V T = I 3 . Hence in S( A, B, C, Λ) (5.19) is B = B and C = C . By B p we denote the first

(see (5.18)) we can take

column of B, and by C p – the first row of C. Since B m − p ≠ 0 (the last two columns of B) as well as Cr − p ≠ 0 (the last two rows of C), in order to find out invariant and Smith zeros of S(A,B,C,D) we can make use of the point 3b2 of Procedure 5.2. To this end, we examine first a new system S( A − B p D −p1C p ,B m − p ,Cr − p ) with the matrices ⎡− 1 ⎢0 A − B p D −p1 C p = ⎢ ⎢0 ⎢ ⎣1

1 1

0 0 0 −1 1 0

1⎤ 0⎥⎥ , 0⎥ ⎥ 0⎦

⎡0 ⎢0 Bm − p = ⎢ ⎢0 ⎢ ⎣1

0⎤ 0⎥⎥ , 1⎥ ⎥ 0⎦

⎡0 0 0 1⎤ Cr − p = ⎢ ⎥. ⎣0 1 0 0 ⎦

Denoting the above matrices respectively as A ' , B' and C' , we consider, according to Procedure 5.1, the system S( A' , B' , C' ) with m ' = 2 inputs and r ' = 2 outputs. The first nonzero Markov parameter of this system is ⎡1 0 ⎤ equal to C'B' = ⎢ ⎥ , i.e., ν' = 0, p ' = 1 . In SVD (see (5.1)) of C'B ' we ⎣0 0 ⎦ take Λ = C'B ' , M p ' = 1 and U ' = V ' T = I 2 . Thus in S( A' , B ' , C ' ) (see (5.2)) we have B ' = B' , C' = C' . Now, by B ' p ' we denote the first column of B' and by C ' p ' we denote the first row of the matrix C' . Moreover, we have B ' m '− p ' ≠ 0 (the second column of B' ) and C ' r '− p ' ≠ 0 (the second row of C' ). Hence to the system S( A' , B' , C' ) we can apply the point 3b2 of Procedure 5.1 (i.e., Proposition 5.3). However, Proposition 5.3 tells us that the set of invariant zeros of the system S( A' , B' , C' ) is the same as the set of invariant zeros of the system S(K ' ν ' A ' ,B ' m '− p ' ,C ' ) with the following matrices

190


⎡1 ⎢0 −1 ν' K 'ν' = I − B ' p ' M p ' C ' p' (A' ) = ⎢ ⎢0 ⎢ ⎣0 ⎡− 1 ⎢0 K 'ν ' A' = ⎢ ⎢0 ⎢ ⎣0

1 0 1 0 0 −1 0 0

0 1 0 0

0 0 1 0

0⎤ 0⎥⎥ , 0⎥ ⎥ 0⎦

1⎤ ⎡0⎤ ⎢0⎥ ⎥ 0⎥ ⎡0 0 0 1⎤ , B ' m '− p ' = ⎢ ⎥ , C ' = ⎢ ⎥. ⎢1 ⎥ 0⎥ ⎣0 1 0 0 ⎦ ⎢ ⎥ ⎥ 0⎦ ⎣0⎦

Now, in order to find out invariant zeros of S(K ' ν ' A ' ,B ' m '− p ' ,C ' ) , we denote A' ' = K ' ν ' A' , B' ' = B ' m '− p ' and C' ' = C ' . Then we seek the first nonzero Markov parameter of the system S( A' ' ,B' ' ,C' ' ) . However, since C' ' B ' ' = 0 and ( A ' ' ) l B' ' = ( −1) l B' ' for l = 1,2,... , we infer that all Markov parameters of S( A' ' ,B' ' ,C' ' ) are zero. Hence the transfer-function matrix of S( A' ' ,B' ' ,C' ' ) equals zero identically. This means, by virtue of Proposition 2.6, that the system S( A' ' ,B' ' ,C' ' ) is degenerate. Finally, by virtue of Corollary 5.3 (ii), we conclude that the original system S(A,B,C,D) is degenerate, while its zero polynomial is equal to the zero polynomial of the system S( A' ' ,B' ' ,C' ' ) (and the latter can be easily found using Proposition 2.6). ◊

Example 5.4. Consider a reachable and observable system S(A,B,C,D) (2.1) with the matrices (cf., Example 4.4) ⎡0 ⎢0 ⎢ ⎢0 A=⎢ 0 ⎢ ⎢0 ⎢0 ⎣

1 0 0 0 0⎤ 0 1 0 0 0⎥⎥ 0 0 0 0 0⎥ , 0 0 0 1 0⎥ ⎥ 0 0 0 0 1⎥ 0 0 0 0 0⎥⎦

⎡1 1 0 0 0 0 ⎤ C=⎢ ⎥, ⎣0 0 0 1 − 1 0 ⎦

⎡0 ⎢0 ⎢ ⎢1 B=⎢ 0 ⎢ ⎢0 ⎢0 ⎣

0⎤ 0⎥⎥ 0⎥ , 0⎥ ⎥ 0⎥ 1 ⎥⎦

⎡1 0⎤ D=⎢ ⎥. ⎣1 0⎦


191

In SVD of D (see 5.18) we take ⎡ 2 Λ=⎢ ⎣⎢ 0

0⎤ ⎥, 0⎦⎥

U=

1 ⎡1 1 ⎤ ⎢ ⎥, 2 ⎣1 − 1⎦

VT = I2 .

In S( A, B , C, Λ) (5.19) matrices B and C are equal respectively B = B and 1 ⎡1 1 0 1 − 1 0⎤ C= ⎢ ⎥. 2 ⎣1 1 0 − 1 1 0⎦ Let us denote (cf., the point 3b2 in Procedure 5.2) A' = A − B p D −p1C p , B' = B m − p , C' = Cr − p , where B' is the second column of B and C' is the

second row of C . Then after simple calculations we obtain ⎡ 0 ⎢ 0 ⎢ 1 ⎢− A' = ⎢ 2 ⎢0 ⎢0 ⎢ ⎢⎣0

1 0 0 1 1 0 − 2 0 0 0 0 0 0

0 0 0 0 1 1 − 2 2 0 1 0 0 0 0

0⎤ 0 ⎥⎥ 0⎥ ⎥. 0⎥ 1⎥⎥ 0⎥⎦

Now, by virtue of Proposition 5.7, we conclude that the set of invariant zeros and the zero polynomial of the original system S(A,B,C,D) are the same as the set of invariant zeros and the zero polynomial of the SISO system S( A' , B' , C' ) . The first nozero Markov parameter of S( A' , B' , C' ) 1 equals C' A ' B' = , i.e., ν ' = 1 , p ' = 1 . In order to find invariant zeros of 2 S( A' , B' , C' ) , we form for this system a 6x6 projective matrix K ' ν ' = I − B' (C' A ' B' ) −1 C' ( A ' ) ν ' and then we compute K ' ν ' A ' . Finally, we obtain det (zI − K ' ν ' A' ) = z 2 (z − 1)(z 3 + z + 1) and then, via Exercises 2.8.19 and 2.8.23, we infer that the SISO system has two zeros at infinity (represented by the factor z 2 ), and ( z − 1)( z 3 + z + 1) constitutes the zero polynomial of S( A' , B' , C' ) . Now, by virtue of Corollary 5.3 (ii), we conclude that S(A,B,C,D) is nondegenerate and its invariant zeros (equivalently, its Smith zeros) are the roots of the polynomial (z − 1)( z 3 + z + 1) . ◊

192


5.5 Exercises 5.5.1. Consider a system S(A,B,C) with the matrices 0 −1 ⎤ ⎡− 1 / 6 ⎢ A=⎢ 0 − 1 / 6 1 / 2 ⎥⎥ , ⎢⎣ 0 − 1 / 2⎥⎦ 0

⎡1 1 ⎤ B = ⎢⎢1 0⎥⎥ , C = [0 1 2] . ⎢⎣0 0⎥⎦

Applying Proposition 5.4 (i) show that the system is degenerate. Find its Smith zeros. 5.5.2. Consider a system S(A,B,C,D) with matrices A, B, C as in Exercise 5.5.1 and D = [1 0] . Show, by using Proposition 5.8 (i), that the system is degenerate. Using Remark 5.4 find its zero polynomial. 5.5.3. Using Proposition 5.4 (ii) find invariant zeros of S(A,B,C), where 0 1 / 3⎤ ⎡− 1 / 3 ⎢ A=⎢ 0 − 2 / 3 0 ⎥⎥ , ⎢⎣ 0 0 − 1 ⎥⎦

⎡1 0 1⎤ B = ⎢⎢0 1 0⎥⎥ , ⎢⎣1 0 1⎥⎦

⎡1 0 0⎤ C=⎢ ⎥. ⎣0 1 0 ⎦

Hint. Write SVD of CB as

⎡1 / 2 0 1 / 2 ⎤ ⎥ ⎡1 0⎤ ⎡ 2 0 0⎤ ⎢ CB = ⎢ 1 0 ⎥. ⎢ ⎥⎢ 0 ⎥ ⎣0 1⎦ ⎢⎣ 0 1 0 ⎥⎦ ⎢1 / 2 0 − 1 / 2 ⎥ ⎦ ⎣ 5.5.4. In (2.1) let 0 ⎤ 0 ⎡0 ⎤ ⎡− 1 / 6 ⎡1 1 0 ⎤ ⎡1 ⎤ ⎥ ⎢ , D= ⎢ ⎥. A=⎢ 0 − 1/ 6 0 ⎥ , B = ⎢⎢1 ⎥⎥ , C = ⎢ ⎥ ⎣1 0 0⎦ ⎣0 ⎦ ⎢⎣2⎥⎦ ⎢⎣ − 1 1 / 2 − 1 / 2⎥⎦ Applying Proposition 5.2 find the zero polynomial of this system. 5.5.5. In (2.1) let ⎡− 1 / 2 0 ⎤ , A=⎢ − 1⎥⎦ ⎣ 0

⎡− 3 1⎤ B=⎢ ⎥, ⎣ 0 0⎦

⎡0 0 ⎤ C=⎢ ⎥. ⎣1 0 ⎦

5.5 Exercises

193

Applying Proposition 5.5 (ii) find invariant zeros of this system. Find its Kalman form (2.2). Hint. Write SVD of CB as ⎡0 1 ⎤ ⎡ 10 CB = ⎢ ⎥⎢ ⎣1 0⎦ ⎢⎣ 0

0⎤ ⎡− 3 / 10 ⎥⎢ 0⎦⎥ ⎣⎢ 1 / 10

1 / 10 ⎤ ⎥. 3 / 10 ⎦⎥

Observe that A, B, C are already written in the Kalman canonical form. In order to find proper block partition in (2.2) use (2.3). Verify that nr o = degG (z) = 1, nr o = 0, n r o = 1, nr o = 0 . 5.5.6. In S(A,B,C,D) (2.1) let ⎡− 1 ⎢0 ⎢ A=⎢0 ⎢ ⎢0 ⎢⎣ 1

0 1 0 1⎤ 2 0 0 1⎥⎥ 0 0 0 1⎥ , ⎥ 2 0 1 0⎥ 0 0 1 0⎥⎦

⎡0 1 0 0 0 ⎤ C = ⎢⎢0 0 0 0 1⎥⎥ , ⎢⎣0 0 1 0 0⎥⎦

⎡0 ⎢1 ⎢ B = ⎢0 ⎢0 ⎢0 ⎣

0 0⎤ 0 0⎥ ⎥ 0 0⎥ , 0 1⎥ 1 0⎥⎦

⎡1 0 0⎤ D = ⎢⎢0 0 0⎥⎥ . ⎢⎣0 0 0⎥⎦

Applying Procedure 5.2 show that S(A,B,C,D) degenerate. Find its Kalman form (2.2). Verify that the minimal subsystem is degenerate. Find the Smith zeros of S(A,B,C,D) and of its minimal subsystem. 5.5.7. Consider a system S(A,B,C) with matrices A, B, C as in Example 5.3. Applying Procedure 5.1 show that the system is nondegenerate and find its invariant zeros. 5.5.8. In (2.1) let ⎡0 1 / 3 ⎢1 0 A=⎢ ⎢0 0 ⎢ ⎣0 − 2 / 3

0 ⎤ ⎡0 ⎥ ⎢1 0 2 / 3⎥ , B=⎢ ⎢0 0 1/ 3⎥ ⎥ ⎢ 0 0 ⎦ ⎣0

0

0⎤ 0⎥⎥ ⎡1 0 0 0⎤ , C=⎢ ⎥. 0⎥ ⎣0 0 1 0 ⎦ ⎥ 1⎦

Applying Procedure 5.1 find invariant zeros of this system.

194


5.5.9. Verify equivalences (i), (ii), (iii) and (iv) in Proposition 5.1. Hint. It is sufficient to prove the following sequence of implications

(i) ⇒ (ii) ⇒ (iii) ⇒ (i) and then to show that (i) ⇔ (iv). Since CA ν B has full column rank, (CA ν B ) + CA ν B = I m and, consequently, K ν B = 0 . (i) ⇒ (ii) Let λ, x o ≠ 0, g satisfy (2.4), i.e., λx o − Ax o = Bg and

Cx o = 0 . Premultiplying the first identity by K ν and making use of the relation K ν x o = x o (see Lemmas 3.1 and 3.2) and K ν B = 0 , we obtain λx o − K ν Ax o = 0 and Cx o = 0 , i.e., λ ∈ σ(K ν A ) and x o is an eigenvector satisfying x o ∈ Ker C . (ii) ⇒ (iii ) From (ii) we have λx o − K ν Ax o = 0 and Cx o = 0 for some x o ≠ 0 , i.e., λ is an o.d. zero of S(K ν A, B, C) . (iii ) ⇒ (i) From (iii) we have λx o − K ν Ax o = 0 and Cx o = 0 for some x o ≠ 0 . Using (3.1) and setting g = −(CA ν B ) + CA ν +1x o , we can write λx o − K ν Ax o = 0 as λx o − Ax o = Bg . Finally, (i) ⇔ (iv) follows from Corollary 2.3. 5.5.10. Verify equivalences (i), (ii), (iii) and (iv) given in Proposition 5.2. Discuss the case (I r − DD + )C = 0 . Hint. Prove (i) ⇒ (ii) ⇒ (iii) ⇒ (i) and then (i) ⇔ (iv). Observe that,

under the assumptions of Proposition 5.2, if (I r − DD + )C = 0 , then each eigenvalue of A − BD + C is an invariant zero of S(A,B,C,D) (see the last claim in Proposition 5.2). 5.5.11. Applying Proposition 5.2 find invariant zeros of the system (2.1), where 0 0⎤ ⎡− 1 0 ⎢ 0 −1 0 0 ⎥⎥ , A=⎢ ⎢0 0 −1 0 ⎥ ⎥ ⎢ 0 0 − 1⎦ ⎣0

⎡1 ⎢1 B=⎢ ⎢1 ⎢ ⎣1

0⎤ 0⎥⎥ , 1⎥ ⎥ 0⎦

5.5 Exercises

⎡0 1 0 0 ⎤ C = ⎢⎢0 0 0 1⎥⎥ , ⎢⎣ 0 1 0 1⎥⎦

195

⎡1 0⎤ D = ⎢⎢0 1⎥⎥ . ⎢⎣1 1⎥⎦

5.5.12. Prove the claim (iii) in Proposition 5.3. Hint. Consider the proof of point (i) in Proposition 5.3 and take into account that g = Vg (or, equivalently, g = V T g ).

5.5.13. Prove statements given in Remark 5.3. 5.5.14. Prove statements given in Remark 5.5. 5.5.15. Show that the recursive process following from the point 3b2 of Procedure 5.1 preserves besides invariant zeros also the corresponding to them state-zero directions. Prove an analogous result for the point 3b2 of Procedure 5.2. Hint. In case of Procedure 5.1 consider the sequence of transformations S( A, B, C) → S( A, B , C ) → S(K ν A, B m − p , C ) .

In case of Procedure 5.2 consider the sequence of transformations S( A, B, C, D) → S( A, B , C, Λ) → S( A − B p D −p1C p , B m − p , Cr − p ) . 5.5.16. Show that in any proper system S(A,B,C,D) (2.1) of uniform rank (i.e., when D is nonsingular) a number λ is an invariant zero (equivalently, a Smith zero) if and only if λ is an eigenvalue of A − BD −1C . Write the zero polynomial for such system. 5.5.17. In (2.1) let 1/ 4 ⎤ ⎡ 0 A=⎢ ⎥, ⎣− 1 / 2 − 3 / 4⎦

⎡1 0 ⎤ B=⎢ ⎥, ⎣0 − 1⎦

C = [1 1] .

Show that λ = −1 / 4 is an o.d. zero which is not a Smith zero. Applying Procedure 5.1 find invariant and Smith zeros of this system.

196


5.5.18. In S(A,B,C) (2.1) let ⎡0 ⎢0 A=⎢ ⎢0 ⎢ ⎣0

1 0 0⎤ 0 0 0⎥⎥ , 0 1 0⎥ ⎥ 0 0 0⎦

⎡0 ⎢0 B=⎢ ⎢1 ⎢ ⎣0

0⎤ 1 ⎥⎥ , 0⎥ ⎥ 0⎦

C = [1 0 0 0] .

Applying Procedure 5.1 show that this system is degenerate and its only Smith zero is λ = 0 . Verify that λ = 1 is an o.d. zero. Using the Kalman form (2.2) of the system find its minimal subsystem and show that this subsystem is nondegenerate. This example indicates that, in the general case, the minimal subsystem of a degenerate system does not need to be degenerate.

6 Output-Nulling Subspaces in Strictly Proper Systems

In this chapter we shall need the concept of some invariant subspaces of the state space that are invariant under the state matrix modulo the action of controls. For a system S(A,B,C) (2.1) a subspace X ⊆ R n is called [91] (A,B)invariant if and only if there exists a mxn real matrix F such that ( A + BF )(X) ⊆ X (in [4, 41] X is called an (A,B)-controlled invariant). The set of all (A,B)-invariant subspaces is nonempty and closed under subspace addition. In this set we can distinguish a subset consisting of all those (A,B)-invariant subspaces that are contained in Ker C . This subset is also nonempty and closed under subspace addition. In this subset there exists an unique maximal element ν ∗ ( A, B, C) which is called [91] the maximal (A,B)-invariant subspace contained in Ker C (or [4, 41], the maximal output-nulling controlled invariant subspace). The maximality of ν ∗ ( A, B, C) means that any (A,B)-invariant subspace X satisfying X ⊆ Ker C must satisfy X ⊆ ν ∗ ( A, B, C) . The subspace ν ∗ ( A, B, C) can be calculated recursively with the aid of the following algorithm [91, 61] X 0 := Ker C and, for

i = 1, .. ., n , X i := Ker C ∩ A −1 (ImB + X i −1 ) .

Then X n = ν ∗ ( A, B, C) .

The symbol A −1 means the inverse image under A , where A is thought of as a mapping of R n into itself. As we shall show in this chapter, for any system S(A,B,C) (2.1) the subspace ν ∗ ( A, B, C) can be computed via Procedure 5.1. On the other hand, we shall show that for S(A,B,C) (2.1) the set of invariant zeros is empty if and only if ν ∗ ( A, B, C) is trivial (i.e., ν ∗ ( A, B, C) = {0} ). It should be


198


emphasized, however, that this equivalence does not hold if instead of invariant zeros the Smith zeros are taken into account. We shall show also that for S(A,B,C) (2.1) a solution corresponding to a given output-zeroing input is entirely contained in ν ∗ ( A, B, C) as well as that for any point x o ∈ ν ∗ ( A, B, C) there exists an output-zeroing input such that the corre-

sponding solution passes through x o . Finally, we shall discuss some basic differences between nondegenerate and degenerate systems S(A,B,C). As we already know, in nondegenerate strictly proper systems the Smith and invariant zeros are exactly the same objects which are determined by the zero polynomial. We shall show that in case of nondegeneracy the degree of this polynomial is equal to the dimension of ν ∗ ( A, B, C) , while the zero dynamics are determined fully by the Smith zeros and are independent upon control vector. On the other hand, in degenerate strictly proper systems the degree of the zero polynomial is strictly less than the dimension of ν ∗ ( A, B, C) , while the zero dynamics essentially depend upon control vector.

6.1 Systems with the Identically Zero Transfer-Function Matrix Throughout this section we discuss a system S(A,B,C) (2.1) with the identically zero transfer-function matrix. Since G (z) ≡ 0 implies that all Markov parameters are equal to zero (i.e., CA l B = 0 for l = 0, 1, 2,... ) (recall that, by assumption, in (2.1) is B ≠ 0 and C ≠ 0 ), we can write {0} ≠ Im B ⊆ X o ,

where X o stands for the unobservable subspace, i.e., X o =

n −1 l =0

Ker CA l .

From A-invariance of X o (i.e., A( X o ) ⊆ X o ) and from Im B ⊆ X o it follows that Xr ⊆ Xo ,

[

]

where X r = Im B AB ... A n −1B is the reachable subspace. Hence we can write {0} ≠ Im B ⊆ X r ⊆ X o .

6.1 Systems with the Identically Zero Transfer-Function Matrix

199

Lemma 6.1. In S(A,B,C) (2.1) let G (z) ≡ 0 . Then a pair (x o , u o (k )) is an output-zeroing input if and only if x o ∈ X o and u o (.) ∈ U. Moreover, a solution corresponding to an output-zeroing input (x o , u o (k )) is contained in X o . Proof. Let x o ∈ X o and let u o (.) ∈ U. Denote by x o (k ) a solution corresponding to (x o , u o (k )) . Then y (0) = Cx o (0) = Cx o = 0 for k = 0 , while for k = 1, 2,..., the system response can be written as k −1

y (k ) = Cx o (k ) = CA k x o + ∑ CA k −1− l Bu o (l ) . l =0

Because CA k −1− l B = 0 and X o is A-invariant, we have A k x o ∈ X o and, consequently, y (k ) = 0 for all k ∈ N , i.e., (x o , u o (k )) is an outputzeroing input. Conversely, suppose that (x o , u o (k )) , where u o (.) ∈ U, is an outputzeroing input. Then, since all Markov parameters are equal to zero, y (k ) = CA k x o = 0 for all k ∈ N and, consequently, x o ∈ X o . k −1

Finally, x o (k ) = A k x o + ∑ A k −1− l Bu o (l ) remains in X o for all l =0

k ∈ N . This fact follows from A-invariance of X o and from x o ∈ X o and Im B ⊆ X o . ◊

Remark 6.1. Recall (see Proposition 2.6) that if in S(A,B,C) (2.1) is G (z) ≡ 0 and the system is taken in its Kalman form (2.2), then the orders of diagonal submatrices of the A-matrix in (2.2) are equal to (cf., (2.3)) n r o = nr , n r o = 0, nr o = n o − n r , n r o = no , and (2.2) assumes the form (cf., (2.16)) A ro  A= 0  0 

A13 A ro 0

A14  B r o   A 34  , B =  0  , C = 0 0 C r o ,  0  A r o 

[

]

(6.1)

200


x r o    x = x ro  . x   ro 

For the system (6.1) we have X r = {x ∈ R n : x r o = 0, x ro = 0} as the reachable subspace, X o = {x ∈ R n : x ro = 0} as the unobservable subspace, while the subspace Vr o := X r ∩ X o of all reachable and unobservable states is of the form Vro = {x ∈ R n : x r o = 0, x ro = 0} . Recall also (see Proposition 2.6) that the zero polynomial of S(A,B,C) is equal to det( zI ro − A r o ) . Now, by virtue of Lemma 6.1 and X o = {x ∈ R n : x ro = 0} , the zero dynamics of (6.1) are governed by x ro (k + 1) = A ro x ro (k ) + A13 x r o (k ) + B ro u(k ) x ro (k + 1) = A ro x r o (k )

, k∈N

(6.2)

and the corresponding solutions remain in X o = {x ∈ R n : x ro = 0} which, as we shall show in the next result, equals ν ∗ ( A, B, C) . If we constrain initial states to the subspace Vro , then, in this part of X o , the zero dynamics of (6.1) are governed by the equation x ro (k + 1) = A ro x ro (k ) + B ro u(k ) , k ∈ N .

(6.3)

Since B ro ≠ 0 , these dynamics essentially depend upon control vector. Moreover, at any initial state x o ∈ Vro and any u o (.) ∈ U a solution corresponding to (x o , u o (k )) is contained in Vro , i.e., x o (k ) ∈ Vro for all k∈N . ◊ Lemma 6.2. In S(A,B,C) (2.1) let G (z) ≡ 0 . Then the maximal (A,B)invariant subspace contained in Ker C is equal to the unobservable subspace, i.e., ν ∗ ( A, B, C) = X o .

Moreover,


201

dim ν ∗ ( A, B, C) > deg det( zI ro − A ro ) ,

i.e., the degree of the zero polynomial is strictly less than the dimension of ν ∗ ( A, B, C) .

Proof. Since in S(A,B,C) all Markov parameters are zero, at any mxn F we have C( A + BF ) l = CA l for l = 0, 1, 2,... . These relations and the Cayley-Hamilton theorem show that X o is ( A + BF ) -invariant (at any F), i.e., ( A + BF )(X o ) ⊆ X o . Now, using (6.1) and observability of the pair ( A r o , C r o ) , we conclude that any (A,B)-invariant subspace X such that X ⊆ Ker C must be contained in X o . In fact, by assumption, X is ( A + BF ) -invariant at some F. Hence ( A + BF ) k (X ) ⊆ X ⊆ Ker C at any k ∈ N . This means that for each x ∈ X we have (cf., (6.1)) C( A + BF ) k x = C r o A kro x r o = 0 for all k ∈ N . Finally, via observability of ( A r o , C r o ) , it follows that x r o = 0 . x r o  Since X o = {x : x =  x r o } , this proves that X ⊆ X o .  0 

The proof of the second claim follows immediately from the relations dim ν ∗ ( A, B, C) = n o = nro + nr o > n ro = deg det(zI r o − A ro ) . ◊

Remark 6.2. Of course, it is easy to observe that the source of degeneracy of (6.1) lies in the reachable and unobservable part of this system. In fact, as we know from the proof of Proposition 2.6, if in S(A,B,C) (2.1) is G (z) ≡ 0 , then any λ ∉ σ( A ) is an invariant zero. We shall show now that if λ ∉ σ( A ) , then the corresponding to it state-zero directions (or, more precisely, their real and imaginary parts) must belong to the subspace Vr o of all reachable and unobservable states. To this end, suppose first that a triple λ, x o ≠ 0, g satisfies (2.4). Since all Markov parameters are zero, premultiplying the first equality of (2.4) (i.e., λx o − Ax o − Bg = 0 ) successsively by C, CA,..., CA n − 2 and using the second equality of (2.4) (i.e., Cx o = 0 ), we conclude that Re x o ∈ X o and Im x o ∈ X o . Now, if S(A,B,C) is taken in the form (6.1), then

202


x o   ro  o  o  x = xro ,    0   

whereas the first equality of (2.4) takes the form (i)

(λI r o − A ro )x or o − A13 x or o − B r o g = 0 ,

(λI r o − A r o )x or o = 0 .

However, the assumption λ ∉ σ(A) and the second equality in (i) imply x or o = 0 . This means that x or o    xo =  0   0   

and, consequently, Re x o ∈ Vro and Im x o ∈ Vr o . Furthermore, from the first equality in (i) we have x or o = (λI ro − A r o ) −1 B r o g and B r o g ≠ 0 (otherwise, we would have x o = 0 ). Hence we see that any invariant zero λ ∉ σ( A ) is connected via corresponding to it state-zero directions (more precisely, by their real and imaginary parts) with Vr o , as claimed. Of course, each o.d. zero which is not i.d. (i.e., each eigenvalue of A r o ) is also connected with Vr o via associated eigenvectors and pseudoeigenvectors of A (more precisely, those whose real and imaginary parts belong to Vr o ). The corresponding to such initial conditions free evolutions (these are generated by eigenvalues of A ro ) remain in Vr o . Similarly, each i.o.d. zero, i.e., each eigenvalue of A r o (recall that i.o.d zeros are the Smith zeros of S(A,B,C)), we connect with the set  x ro  Vr o := {x ∈ R : x = x ro  , x r o ≠ 0}  0  n

via those associated eigenvectors and pseudoeigenvectors of A whose real


203

and imaginary parts are in Vr o . If A r o is nonsingular, the corresponding free evolutions (generated by eigenvalues values of A r o ) remain in Vr o . Finally, ν ∗ ( A, B, C) (which, by virtue of Lemma 6.2, is equal to the unobservable subspace of the system) can be decomposed into two disjoint subsets as X o = Vro ∪ Vr o , where, in general, the set Vr o endowed with

the relative topology with respect to the natural topology of R n is not connected. ◊ Example 6.1. In S(A,B,C) (2.1) let x  − 1 / 2 0  0  1 0 , B= , C= , x =  1 A=   − 1  0 1  − 1 0 x 2 

( G (z) ≡ 0 ).

The change of basis x' = Hx , where 0 1  H= , 1 0 

transforms S(A,B,C) into S( A' , B' , C' ) (6.1), where 0  − 1 A' =  ,  0 − 1 / 2

0 1  C' =  , 0 − 1

1 B' =   , 0

A' ro = [−1] ,

 x'  x' =  1  , x'2 

A' ro = [−1 / 2] .

Since S( A' , B' , C' ) has no input-output decoupling zeros, S(A,B,C) has no Smith zeros (see Proposition 2.6). Moreover, ν ∗ ( A' , B' , C' ) = X' o = Ker C' = V' r o .

The zero dynamics of S( A' , B' , C' ) (cf., (6.2)) are of the form x '1 (k + 1) = − x '1 (k ) + u (k ) .

Naturally, for the original system S(A,B,C) we have 0 ν ∗ ( A, B, C) = H −1 (ν ∗ ( A' , B' , C' )) = Ker C = {x ∈ R 2 : x =  } x 2 

and the zero dynamics are of the form x 2 (k + 1) = − x 2 (k ) + u ( k ) . ◊

204


6.2 First Markov Parameter of Full Column Rank Throughout this section we discuss a system S(A,B,C) (2.1) in which the first nonzero Markov parameter CA ν B has full column rank. We begin by recalling some facts already established in Chapters 3 and 4. As we know from Proposition 4.3, if rank CA ν B = m , then invariant zeros of S(A,B,C) (which, via Corollary 4.2, coincide with the Smith zeros) can be characterized as o.d. zeros of the closed-loop (state feedback) system S(K ν A,B,C) , where K ν = I − B(CA ν B) + CA ν . Let n ocl stand for the rank of the observability matrix for S(K ν A,B,C) and let S cl ν denote the unobservable subspace for this system, i.e., n −1

Sνcl := h Ker C(K ν A) l ⊆ R n . l =0

The number of o.d zeros of S(K ν A,B,C) (including multiplicities) equals n − nocl = dim S cl ν . On the other hand, recall (see Lemma 4.1) that an input

(i)

u(k ) = −(CA ν B) + CA ν +1 (K ν A) k x o ,

where x o ∈ R n is arbitrarily fixed, applied to S(A,B,C) at a given initial state x(0) ∈ R n yields (ii)

x ( k ) = A k ( x ( 0) − x o ) + ( K ν A ) k x o .

Furthermore, we have the following characterization of the outputzeroing problem (see Proposition 3.10). A pair (x o ,u o (k )) is an outputzeroing input for S(A,B,C) if and only if x o ∈ Scl ν and u o ( k ) is as in (i). Moreover, a solution corresponding to (x o ,u o (k )) has the form x o (k ) = (K ν A) k x o and is contained in S cl ν . It is important to note that these output-zeroing inputs are determined

uniquely by initial states x o ∈ Scl ν . More precisely, if an initial state x o ∈ Scl ν is fixed, then the unique input that is able to keep y ( k ) = 0 for all k ∈ N is determined by (i).

6.2 First Markov Parameter of Full Column Rank

205

Remark 6.3. In a suitable basis the characterization of the output-zeroing problem given in Proposition 3.10 becomes essentially simpler. Let x' = Hx denote a change of coordinates which leads to the decomposition of S(K ν A,B,C) into an unobservable and an observable part, i.e., consider a system S((K ν A)' ,B' ,C' ) with the matrices

(iii)

(K ν A)' o (K ν A)' =  0 

(K ν A)'12  B '  , B' =  o  , C' = [0 C'o ] ,  (K ν A)'o   B' o 

x' o  x' =   ,  x' o 


where the subsystem S((K ν A)' o , B' o , 0) is unobservable, whereas the subsystem S((K ν A)'o ,B'o ,C'o ) is observable. Moreover, consider the system S( A' , B' , C' ) obtained from S(A,B,C) by the same change of coordinates. The state vector x' in S( A' , B' , C' ) we decompose as in (iii). Note that forming for S( A' , B' , C' ) the system S(K 'ν A' , B' , C' ) we obtain the system (iii). This fact follows (see Exercise 4.4.20) from the relation (K ν A)' = K ' ν A' , where K ' ν := I − B' (C' ( A' ) ν B' ) + C' ( A' ) ν .

Of course, all invariant zeros of S( A' , B' , C' ) (which are the same as of S(A,B,C)) can be found out, via Proposition 4.3 and Corollary 4.2 when applied to S( A' , B' , C' ) , as output decoupling zeros of S(K 'ν A' , B' , C' ) . Since (K ν A)' = K ' ν A' and (K ν A)' is as in (iii), the eigenvalues of (K ν A)' o represent all invariant zeros of S( A' , B' , C' ) . Recall also (see Corollary 4.2 and Exercise 4.4.22) that det(zI' o −(K ν A)' o ) is the zero polynomial of S(A,B,C) (as well as of S( A' , B' , C' ) ). For the unobservable subspace S'cl ν of S(K ' ν A' , B' , C' ) we have S'νcl =

n −1

Ker C' (K ' ν A' ) l =

l =0

 x'  Ker C' ((K ν A)' ) l = {x'∈ R n : x' =  o } l =0  0  n −1

cl cl and S'νcl is equal to the image of S cl ν under H, i.e., S' ν = H (S ν ) . ◊

Now, the output-zeroing problem for S( A' , B' , C' ) can be described as follows.

206


Lemma 6.3. A pair (x' o ,u'o ( k )) is an output-zeroing input for the system S( A' , B' , C' ) if and only if  x' o  x' o =  o   0 

 x'  (i.e., x'o ∈ S'νcl = {x'∈ R n : x' =  o } )  0 

and ((K ν A) 'o ) k x' o  o. u' o (k ) = −(C' ( A' ) ν B' ) + C' ( A' ) ν +1  0  

Moreover, the corresponding solution has the form ((K ν A) 'o ) k x' o  o x'o (k ) =  0  

and is contained in the subspace S'cl ν . Furthermore, the zero dynamics of S( A' , B' , C' ) are of the form x' o (k + 1) = (K ν A)' o x' o (k ) . ◊

Remark 6.4. Lemma 6.3 shows clearly how invariant zeros of S( A' , B' , C' ) (i.e., eigenvalues of (K ν A)' o ) are involved in output-zeroing inputs and the corresponding solutions. In particular, it shows that the component x' o (in vectors belonging to S'cl ν ) is spanned by real and imaginary parts of eigenvectors and pseudoeigenvectors associated with these eigenvalues. It is also clear that (x o ,u o (k )) is an output-zeroing input for S(A,B,C) if and only if (x' o ,u' o (k )) , where x'o = Hx o and u'o (k ) = u o (k ) , is an output-zeroing input for S( A' , B' , C' ) . Moreover, as it follows from Lemma 6.4 below (when applied to S( A' , B' , C' ) ), the subspace S'cl ν equals the maximal ( A' , B' ) -invariant subspace contained in Ker C' , i.e., ∗ S'cl ν = ν ( A ' , B' , C' ) . ◊

Lemma 6.4. In S(A,B,C) (2.1) let rank CA ν B = m . Then the maximal (A,B)-invariant subspace contained in Ker C for S(A,B,C) is equal to the

6.2 First Markov Parameter of Full Column Rank

207

unobservable subspace for the closed-loop (state feedback) system S(K ν A,B,C) , i.e., n −1

ν ∗ ( A, B, C) = h Ker C(K ν A) l . l =0

Moreover, the dimension of ν ∗ ( A, B, C) equals the degree of the zero polynomial of S(A,B,C), i.e., dim ν ∗ ( A, B, C) = deg det ( zI' o −(K ν A)' o ) .

Proof. The fact that Scl ν =

n −1 l =0

Ker C(K ν A) l is an (A,B)-invariant sub-

space contained in Ker C is obvious (since we have K ν A = A + BF , cl where F = −(CA ν B) + CA ν +1 , and (K ν A)(Scl ν ) ⊆ S ν ⊆ Ker C ). Hence we only need to show that any (A,B)-invariant subspace contained in Ker C lies in S cl ν . Let X denote such a subspace and let F be such that ( A + BF)(X) ⊆ X ⊆ Ker C . Then ( A + BF) k (X) ⊆ X ⊆ Ker C , i.e., for any fixed x ∈ X we have the identity C( A + BF ) k x = 0 for all k ∈ N . Since CA ν B is the first nonzero Markov parameter, we can write C( A + BF ) l = CA l for each 0 ≤ l ≤ ν and, consequently, we have C( A + BF ) ν +1 = CA ν ( A + BF) . However, from the aforementioned

identity we obtain, in particular, C( A + BF) ν +1 x = 0 and, consequently, (CA ν +1 + (CA ν B)F )x = 0 , that is, (CA ν B)Fx = −CA ν +1x . Because (CA ν B) + CA ν B = I m , we can write Fx = −(CA ν B) + CA ν +1x . This last relation yields, via the definition of K ν (see (3.1)), the equality ( A + BF )x = K ν Ax . Since this equality holds at any x ∈ X , we obtain (K ν A)(X) ⊆ X ⊆ Ker C and, consequently, C(K ν A) k x = 0 for all n −1

cl k ∈ N . Thus x ∈ h Ker C(K ν A) l = S cl ν , i.e., X ⊆ S ν .

l =0

By virtue of Remark 6.3, the proof of the second claim of the lemma is obvious. ◊

208


6.3 Invariant Subspaces Consider a system S(A,B,C) (2.1) with its first nonzero Markov parameter CA ν B and let (5.1) stand for SVD of CA ν B . Moreover, let us take into account the system S( A, B , C) (5.2) (associated with S(A,B,C)) as well as the notation given in (5.3), (5.4) and (5.5). We will need first the following two simple results which establish relationships between output-zeroing inputs and maximal output-nulling controlled subspaces for systems (2.1) and (5.2).

Proposition 6.1. For S(A,B,C) (2.1) and S( A, B , C) (5.2) the following relation holds ν ∗ ( A, B, C) = ν ∗ ( A, B , C ) .

Proof. According to (5.3) and (5.4), B = BV , C = U T C , u = V T u and y = U T y . If F denotes a mxn state feedback matrix for S(A,B,C), then F = V T F is a state feedback matrix for S( A, B , C) and B F = BF . More-

over, Ker C = Ker C . Suppose that X is an ( A, B ) -invariant subspace for S( A, B , C ) contained in Ker C . Then there exists a state feedback matrix F such that ( A + B F )(X) ⊆ X ⊆ Ker C . However, taking F = V F , we can write A + BF = A + BV F = A + B F . Thus X is also an (A,B)-invariant subspace for S(A,B,C) contained in Ker C . The proof of the converse implication follows the same lines. In this way we obtain the equivalence: a subspace X is ( A, B ) -invariant and

contained in Ker C if and only if X is (A,B)-invariant and contained in Ker C . This proves that ν ∗ ( A, B, C) = ν ∗ ( A, B , C ) . ◊

Proposition 6.2. A pair (x o ,u o (k )) is an output-zeroing input for S(A,B,C) (2.1) if and only if (x o ,u o (k )) , where u o (k ) := V T u o (k ) , is an output-zeroing input for S( A, B , C ) (5.2). Moreover, (x o ,u o (k )) and (x o ,u o (k )) give the same solution.

Proof. For the proof it is sufficient to note that, in view of (5.3) and (5.4),

6.3 Invariant Subspaces

209

we have Bu = B u . Hence any admissible input u(k ) affects the state equation of S(A,B,C) in exactly the same way as u (k ) = V T u(k ) affects the state equation of S( A, B , C ) . ◊ Lemma 6.5. Consider a system S(A,B,C) (2.1) and suppose that in the associated system S( A, B , C ) (5.2) is B m − p ≠ 0 . Then the sequence of transformations S( A, B, C) → S( A, B , C ) → S(K ν A, B m− p , C )

has the property (i)

ν ∗ ( A , B , C) = ν ∗ ( A , B , C ) = ν ∗ ( K ν A , B m − p , C ) .

Proof. Of course, in view of Proposition 6.1, it is sufficient to prove the following equivalence: X is an (A,B)-invariant subspace for S(A,B,C) contained in Ker C if and only if X is an (K ν A,B m − p ) -invariant subspace for S(K ν A, B m − p , C ) contained in Ker C .  Fp  For a mxn F let F = V T F and let F =   , where F p consists of  Fm − p 

the first p rows of F . Recall also that K ν = I − B p M −p1 C p A ν .

Suppose now that X is (K ν A,B m − p ) -invariant and contained in Ker C . This means that there exists a (m − p ) xn matrix Fm− p such that (K ν A + B m − p Fm − p )(X) ⊆ X ⊆ Ker C . However, K ν A + B m − p Fm − p

can be written as K ν A + B m − p Fm − p = A − B p M −p1C p A ν +1 + B m − p Fm − p = A + B F ,

− M −p1Cp A ν +1  where F :=   , i.e., we define F p := −M −p1C p A ν +1 . F m− p  

Thus we have the relation ( A + B F )(X ) ⊆ X ⊆ Ker C . From the proof of Proposition 6.1 we know that this relation implies that X is (A,B)-invariant and contained in Ker C .

210


Conversely, let X be (A,B)-invariant and contained in Ker C . Then X is also ( A, B ) -invariant and contained in Ker C (see the proof of Proposition 6.1), i.e., there exists a mxn matrix F such that ( A + B F )(X ) ⊆ X ⊆ Ker C . This yields ( A + B F ) k (X ) ⊆ X ⊆ Ker C at any k ∈ N . Thus, at any x ∈ X and at any k ∈ N , we have C ( A + B F ) k x = 0 . Since C A ν B is the first nonzero Markov parameter

of S( A, B , C ) , we can write C ( A + B F ) l = C A l for each 0 ≤ l ≤ ν . Hence C ( A + B F ) ν +1 = C A ν ( A + B F ) . On the other hand, at any x ∈ X we have the relation C ( A + B F ) ν +1 x = 0 which can be written as ( C A ν +1 + ( C A ν B ) F )x = 0 , i.e., C A ν +1x = −( C A ν B ) F x . Substituting into this last equality

 Cp   Fp  0 , F=  , B = Bp , C =   0  Fm − p   Cr − p 

M CA ν B =  p  0

[

]

Bm − p ,

we obtain  C A ν + 1x  M  p  = − p ν +1  Cr − p A x   0

0 0

 Fp  x   Fm − p 

which yields the relation F p x = −M −p1C p A ν +1x and, consequently, B p F p x = − B p M −p1C p A ν +1x . Now, for any x ∈ X we can write ( A + B p F p )x = ( A − B p M −p1C p A ν +1 )x = K ν Ax

and, consequently, ( A + B F )x = Ax + B p F p x + B m − p Fm − p x = K ν Ax + B m − p Fm − p x .

Since

( A + B F )x ∈ X ⊆ Ker C

at

any

x∈X ,

we

can

write

(K ν A + B m − p Fm − p )x ∈ X ⊆ Ker C , i.e., (K ν A + B m − p Fm − p )(X ) ⊆ X ⊆ Ker C .

In this way we have shown that if X is (A,B)-invariant and contained in Ker C , then X is also (K ν A,B m − p ) -invariant and contained in Ker C . This completes the proof. ◊


211

Lemma 6.6. In S(A,B,C) (2.1) let m > r and let CA ν B have full row rank r. Then n −1

ν

l =0

l =0

ν ∗ ( A, B, C) = h Ker C (K ν A ) l = h Ker CA l .

Proof. In the proof we employ the notation used in Proposition 5.4. Moreover, we discuss separately the cases B m − r ≠ 0 and B m − r = 0 . In the case of B m − r ≠ 0 we consider systems S(A,B,C) and S(K ν A, B m − r , C ) . To these systems we apply Lemma 6.5 which says that the maximal (A,B)-invariant subspace contained in Ker C (for S(A,B,C)) (i)

is equal to the maximal (K ν A,B m−r ) -invariant subspace contained in Ker C (for S(K ν A, B m − r , C ) ). On the other hand, as is known from

Proposition 5.4 (i), the transfer-function matrix of S(K ν A, B m − r , C ) equals zero identically. Now, by virtue of Lemma 6.2 (when applied to S(K ν A, B m − r , C ) ) and Lemma 5.2 (vi) (at p = r ), we can write n −1

ν ∗ ( A, B, C) = ν ∗ (K ν A, B m− r , C ) = h Ker C (K ν A) l l =0

ν

ν

l =0

l =0

= h Ker C A l = h Ker CA l .

(ii) At B m − r = 0 we show first that ( A, B ) -invariant subspaces contained in Ker C (for S( A, B , C ) ) are exactly the same as ( A,B r ) -invariant subspaces contained in Ker C for S( A,B r ,C ) (which is of uniform rank, i.e., its first Markov parameter C A ν B r = M r is nonsingular). To this end, it is sufficient to observe that  F  A + B F = A + B r Fr , where F =  r   Fm − r 

[

and B = B r

]

0.

This means, in turn, that ν ∗ ( A, B , C ) = ν ∗ ( A, B r , C ) . On the other hand, the maximal ( A,B r ) -invariant subspace contained in Ker C (for S( A,B r ,C ) ) can be evaluated via treating S( A,B r ,C ) as a system with the first Markov parameter of full column rank. Thus, using Lemma 6.4 and Lemma 5.2 (vi), we can write

212

6 Output-Nulling Subspaces in Strictly Proper Systems n −1

ν

ν

l =0

l =0

l =0

ν ∗ ( A, B r , C ) = h Ker C (K ν A) l = h Ker C A l = h Ker CA l .

Finally, the claim follows via Proposition 6.1. ◊ Lemma 6.7. In S(A,B,C) (2.1) let CA ν B have rank p < min{m, r} and in S( A,B ,C ) (5.2) let Cr − p = 0 . Then n −1

ν

l =0

l =0

ν ∗ ( A, B, C) = h Ker C (K ν A ) l = h Ker CA l .

Proof. We discuss separately the cases B m − p ≠ 0 and B m − p = 0 . (i)

If B m − p ≠ 0 , then, by virtue of Lemma 6.5, ν ∗ ( A, B, C) coin-

cides with ν ∗ (K ν A, B m− p , C ) (for S(K ν A, B m − p , C ) ). However, since Ker C = Ker C = Ker C p , we conclude that ν ∗ ( A, B, C) is equal to ν ∗ (K ν A, B m− p , C p ) . By virtue of Proposition 5.5, the transfer-function

matrix of S(K ν A, B m − p , C p ) equals zero identically. Since Cr − p = 0 , the same holds for the transfer-function matrix of S(K ν A, B m − p , C ) . Now, applying Lemma 6.2 to S(K ν A, B m − p , C ) , we can evaluate ν ∗ (K ν A, B m − p , C ) as the unobservable subspace for this system, i.e., n −1

ν ∗ ( A, B, C) = ν ∗ (K ν A, B m − p , C ) = h Ker C (K ν A ) l l =0

ν

ν

l =0

l =0

= h Ker C A l = h Ker CA l .

(ii) When B m − p = 0 , we find first ν ∗ ( A, B , C ) and then we use Proposition 6.1. Because A + B F = A + B p F p and Ker C = Ker C p , hence ( A,B ) -invariant subspaces contained in Ker C coincide with ( A,B p ) -invariant subspaces contained in Ker C p . This means in turn that ν ∗ ( A, B , C ) is equal to ν ∗ ( A, B p , C p ) (for the system S( A,B p ,C p )


213

which is of uniform rank). This last subspace can be evaluated, by using Lemma 6.4 to S( A,B p ,C p ) and then Lemma 5.2 (vi), as follows n −1

ν

ν

l =0

l =0

l =0

ν ∗ ( A, B p , C p ) = h Ker C p (K ν A ) l = h Ker C p A l = h KerCA l . ◊

Lemma 6.8. In S(A,B,C) (2.1) let rank CA ν B = p < min{m, r} and in S( A,B ,C ) (5.2) let Cr − p ≠ 0 and let B m − p = 0 . Then n −1

ν ∗ ( A, B, C) = h Ker C (K ν A ) l . l =0

Proof. We find first ν ∗ ( A, B , C ) and then we use Proposition 6.1. Since A + B F = A + B p F p , we observe that ν ∗ ( A, B , C ) equals ν ∗ ( A, B p , C )

(for the system S( A,B p ,C ) in which the first Markov parameter C A ν B p has full column rank). As we know from Lemma 6.4 when applied to S( A,B p ,C ) , ν ∗ ( A, B p , C ) is equal to the unobservable subspace for the n −1 system S( A,B p ,C ) , i.e., ν ∗ ( A, B p , C ) = h Ker C (K ν A ) l . ◊ l =0

We summarize the above lemmas in the following. Proposition 6.3. Consider systems S(A,B,C) (2.1) (with CA ν B as its first nonzero Markov parameter) and S( A, B ,C ) (5.2). Let ν ∗ ( A, B, C) stand for the maximal (A,B)-invariant subspace contained in Ker C for S(A,B,C). Then: 1. If CA ν B has full column rank, then ν ∗ ( A, B, C) is equal to the unobservable subspace for the closed-loop (state feedback) system S(K ν A, B, C) , i.e., n −1

ν ∗ ( A, B, C) = h Ker C(K ν A) l . l =0

2.

If CA ν B has full row rank r and m > r , then

214


2a.

If B m − r = 0 , then ν ∗ ( A, B, C) equals the unobservable

subspace for S(K ν A, B r , C ) , i.e., n −1

ν ∗ ( A, B, C) = h Ker C (K ν A ) l . l =0

2b.

If B m − r ≠ 0 , then ν ∗ ( A, B, C) equals the unobservable

subspace for S(K ν A, B m − r , C ) (whose transfer-function matrix equals zero identically), i.e., n −1

ν ∗ ( A, B, C) = h Ker C (K ν A ) l . l =0

If 0 < rank CA ν B = p < min {m, r} , then

3. 3a.

Cr − p = 0

3a1.

If Cr − p = 0 and B m − p = 0 , then ν ∗ ( A, B, C) equals the

unobservable subspace for S(K ν A, B p , C ) , i.e., n −1

ν ∗ ( A, B, C) = h Ker C (K ν A ) l . l =0

3a2.

If Cr − p = 0 and B m − p ≠ 0 , then ν ∗ ( A, B, C) equals the

unobservable subspace for S(K ν A, B m − p , C ) (whose transfer-function matrix equals zero identically), i.e., ν ∗ ( A, B, C) =

3b.

n −1 l =0

Ker C (K ν A ) l .

Cr − p ≠ 0

3b1.

If Cr − p ≠ 0 and B m − p = 0 , then ν ∗ ( A, B, C) equals the

unobservable subspace for S(K ν A, B p , C ) , i.e.,


215

n −1

ν ∗ ( A, B, C) = h Ker C (K ν A ) l . l =0

3b2.

If Cr − p ≠ 0 and B m − p ≠ 0 , then ν ∗ ( A, B, C) = ν ∗ (K ν A, B m− p , C ) .

4. If in S(A,B,C) is G (z) ≡ 0 , then ν ∗ ( A, B, C) equals the unobservable subspace for S(A,B,C), i.e., n −1

ν ∗ ( A, B, C) = h Ker CA l . ◊ l =0

It is important to note that Propositions 6.1 and 6.3 and Lemma 6.5 show that Procedure 5.1 can be used also for computing ν ∗ ( A, B, C) in an arbitrary strictly proper system (2.1). In the case 3b2 of Proposition 6.3 we begin the recursive process (analogous to that described in Procedure 5.1) applying Proposition 6.3 to the system S(K ν A, B m − p , C ) . In the remainder of this section we shall show that invariant zeros in S(A,B,C) (2.1) exist (i.e., ZI ≠ ∅ ) if and only if ν ∗ ( A, B, C) is nontrivial (i.e., ν ∗ ( A, B, C) ≠ {0} ). To this end, we need to show first that if λ ∈ C is an invariant zero of S(A,B,C), i.e., a triple λ, x o ≠ 0, g satisfies (2.4), then Re x o ∈ ν ∗ ( A, B, C) and Im x o ∈ ν ∗ ( A, B, C) . In this context we shall discuss below all the cases listed in Proposition 6.3. 4. If G (z) ≡ 0 and (2.4) is written as λx o − Ax o = Bg and Cx o = 0 , then premultiplying subsequently the first equality of (2.4) by

C, CA,..., CA n − 2 and using the second equality as well as the relation n −1

ν ∗ ( A, B, C) = h Ker CA l we obtain easily Re x o ∈ ν ∗ ( A, B, C) and l =0

o

∗

Im x ∈ ν ( A, B, C) .

1.

If CA ν B has full column rank, then K ν B = 0 and (2.4) implies

K ν x o = x o (cf., Lemmas 3.2 and 3.1 (ii)). Thus from (2.4) we obtain λx o − K ν Ax o = 0 and Cx o = 0 . This implies that Re x o and Im x o are

216


Conversely, let X be (A,B)-invariant and contained in Ker C . Then X n −1 in ν ∗ ( A, B, C) = h Ker C(K ν A) l . ν

l =0

2. If CA B has full row rank r and m > r , the desired relations follow immediately from Lemmas 3.2 and 6.6. If in S( A, B ,C ) (5.2) is Cr − p = 0 , we consider separately the cases B m − p = 0 and B m − p ≠ 0 . 3.

3a1. At B m − p = 0 from λx o − Ax o = Bg and Cx o = 0 we obtain λx o − Ax o = B g and C x o = 0 and, consequently, λx o − Ax o = B p g p

and C p x o = 0 . Now, since K ν x o = x o (see Lemmas 3.2 and 3.1 (ii)) and K ν = K ν = I − B p M −p1C p A ν as well as K ν B p = 0 (see (5.10) and

Lemma 5.2 (v)), from λx o − Ax o = B p g p we obtain λx o − K ν Ax o = 0 . Thus we have obtained λx o − K ν Ax o = 0 and C p x o = 0 . This implies in n −1

n −1

l =0

l =0

turn Re x o ∈ h Ker C p (K ν A ) l and Im x o ∈ h Ker C p (K ν A ) l . Note that since Cr − p = 0 , in the last two relations C p can be replaced with C . 3a2. At B m − p ≠ 0 from λx o − Ax o = Bg and Cx o = 0 we obtain λx o − Ax o = B p g p + B m − p g m − p and C p x o = 0 . Using the same arguments as in the case B m − p = 0 and employing K ν B m − p = B m − p , we

premultiply λx o − Ax o = B p g p + B m − p g m − p by K ν . Thus we obtain λx o − K ν Ax o = B m − p g m − p and C p x o = 0 . This means that λ is an in-

variant zero of the system S(K ν A,B m − p ,C p ) whose transfer-function matrix equals zero identically (see Proposition 5.5 (i)). Premultiplying λx o − K ν Ax o = B m − p g m − p by C p , C p (K ν A ), . .. , C p (K ν A ) n− 2 and

making use of C p x o = 0 , we obtain Re x o ∈ Im x o ∈

n −1 l =0

n −1 l =0

Ker C p (K ν A ) l and

Ker C p (K ν A ) l . Now, the claim follows immediately from


217

the proof of Lemma 6.7 (part (ii)). Finally, note that since Cr − p = 0 , C p can be replaced with C . 3b1. If in S( A, B ,C ) (5.2) is Cr − p ≠ 0 and B m − p = 0 , then from λx o − Ax o = Bg o

and

Cx o = 0 o

we obtain

λx o − Ax o = B g

and

o

o

C x = 0 . This yields λx − K ν Ax = 0 and C x = 0 . Consequently, n −1

n −1

l =0

l =0

Re x o ∈ h Ker C (K ν A ) l and Im x o ∈ h Ker C (K ν A ) l .

3b2. As we already know from Chapter 5 (see Section 5.2), when in S( A, B ,C ) (5.2) is B m − p ≠ 0 and Cr − p ≠ 0 , the question of finding out invariant zeros of the original system S(A,B,C) can not be decided at the first step. This question, however, can be decided, applying Procedure 5.1, after a finite number of steps. At each step of the recursive process following from the point 3b2 of Procedure 5.1 the set of invariant zeros remains the same as at the previous step. The process ends when we obtain a system with the first nonzero Markov parameter of full column rank or a system whose transfer-function matrix equals zero identically. Moreover, the set of invariant zeros of S(A,B,C) can be determined as the set of invariant zeros of a system obtained at the last step. Furthermore, as it follows from Proposition 5.3 (iii), the process conserves besides invariant zeros also the corresponding state-zero directions. More precisely, if a triple λ, x o ≠ 0, g represents an invariant zero λ of S(A,B,C), then the pair λ, x o remains unchanged at each stage of the process. On the other hand, Propositions 6.1 and 6.3 and Lemma 6.5 tell us that ν ∗ ( A, B, C) is also conserved along the process, i.e., it remains the same for all systems appearing at the individual stages of the process. Thus, at B m − p ≠ 0 and Cr − p ≠ 0 , the above arguments as well as the discussion in the points 1 and 4 above enable us to conclude that if λ is an invariant zero of S(A,B,C), i.e., a triple

λ, x o ≠ 0, g

satisfies

(2.4),

then

we

have

the

relations

Re x o ∈ ν ∗ ( A, B, C) and Im x o ∈ ν ∗ ( A, B, C) .

Summarizing the above discussion, we can formulate the following. Lemma 6.9. Let λ be an invariant zero of S(A,B,C) (2.1), i.e., let a triple λ, x o ≠ 0, g satisfy λx o − Ax o = Bg and Cx o = 0 . Then Re x o ∈ ν ∗ ( A, B, C) and Im x o ∈ ν ∗ ( A, B, C) . ◊

218


Corollary 6.1. The recursive process of computing invariant zeros of S(A,B,C) (2.1) following from the point 3b2 of Procedure 5.1 preserves besides invariant zeros and the associated state-zero directions also the subspace ν ∗ ( A, B, C) . ◊ Of course, from Lemma 6.9 it follows that if in S(A,B,C) (2.1) the set Z of invariant zeros is not empty, then ν ∗ ( A, B, C) is not trivial. In fact, I

suppose that ZI ≠ ∅ and let λ ∈ ZI . Then, by virtue of Definition 2.1, there exist vectors x o ≠ 0 and g such that the triple λ, x o ≠ 0, g satisfies (2.4). However, Lemma 6.9 tells us that Re x o ∈ ν ∗ ( A, B, C) and Im x o ∈ ν ∗ ( A, B, C) . Since at least one of these vectors is a nonzero vec-

tor, we can write ν ∗ ( A, B, C) ≠ {0} . Now, we are interested in assessing validity of the implication: ∗

ν ( A, B, C) ≠ {0} ⇒ ZI ≠ ∅ . We can note, however, that it is easier to

prove the equivalent implication: Z I = ∅ ⇒ ν ∗ ( A, B, C) = {0} . In the latter, since we have assumed Z I = ∅ , it is sufficient to discuss only those cases of Procedure 5.1 when S(A,B,C) is nondegenerate, i.e., the cases 1, 2a, 3a1, 3b1 and 3b2 (in this last case we consider only the situation when a system obtained at the last step of the recursive process has the first nonzero Markov parameter of full column rank). The idea of the proof is based on simple observation that in each of the cases mentioned above the invariant zeros of S(A,B,C) are the output decoupling zeros of a certain closed-loop (state feedback) system, whereas ν ∗ ( A, B, C) equals the unobservable subspace for that system (cf., Procedure 5.1 and Proposition 6.3). Hence, if there are no invariant zeros in S(A,B,C), then there are no o.d. zeros in the closed-loop system in question and, consequently, its unobservable subspace is trivial. This means in turn that ν ∗ ( A, B, C) = {0} . A more detailed analysis is given below. CA ν B has full column rank. 1. From Procedure 5.1 (1) we know that λ is an invariant zero of S(A,B,C) if and only if λ is an o.d. zero of the system S(K ν A,B,C) . On the other n −1

hand, Proposition 6.3 (1) tells us that ν ∗ ( A, B, C) = h Ker C(K ν A) l , l =0

i.e., ν ∗ ( A, B, C) equals the unobservable subspace for S(K ν A,B,C) .


219

Now, if there are no invariant zeros in S(A,B,C), then S(K ν A,B,C) is observable and, consequently, ν ∗ ( A, B, C) = {0} . 2a. CA ν B has full row rank and B m − r = 0 . Procedure 5.1 (2a) says that λ is an invariant zero of S(A,B,C) if and only if λ is an o.d. zero of S(K ν A,B r ,C ) . However, via Proposition 6.3 (2a), ν ∗ ( A, B, C) =

n −1

l =0

Ker C (K ν A ) l . The remaining part of the proof follows

the same lines as in the case discussed above. 3a1. Cr − p = 0 and B m − p = 0 . By virtue of Procedure 5.1 (3a1), λ is an invariant zero of S(A,B,C) if and only if λ is an o.d. zero of S(K ν A,B p ,C ) . On the other hand, from n −1

Proposition 6.3 (3a1) we have ν ∗ ( A, B, C) = h Ker C (K ν A ) l . Now, if l =0

S(A,B,C) has no invariant zeros, then S(K ν A,B p ,C ) is observable and, consequently, ν ∗ ( A, B, C) = {0} . 3b1. Cr − p ≠ 0 and B m − p = 0 . By virtue of Procedure 5.1 (3b1), λ is an invariant zero of S(A,B,C) if and only if λ is an o.d. zero of the system S(K ν A,B p ,C ) . On the other hand, n −1

from Proposition 6.3 (3b1) we have ν ∗ ( A, B, C) = h Ker C (K ν A ) l . l =0

Now, if the set of invariant zeros of S(A,B,C) is empty, S(K ν A,B p ,C ) is observable and, consequently, ν ∗ ( A, B, C) = {0} . 3b2. Cr − p ≠ 0 and B m − p ≠ 0 . Since, by assumption, the set of invariant zeros of S(A,B,C) is empty, at the last step of the recursive process following from Procedure 5.1 (3b2) we obtain a system (with the first nonzero Markov parameter of full column rank) without invariant zeros. However, as we already know from the discussion in the point 1 above, its maximal output-nulling controlled invariant subspace must be trivial. Because this subspace is preserved along the process (see Corollary 6.1), in the original system S(A,B,C) we must have ν ∗ ( A, B, C) = {0} . The above discussion leads us to the desired result.

220


Proposition 6.4. For a system S(A,B,C) (2.1) let ZI denote the set of its invariant zeros and let ν ∗ ( A, B, C) stand for the maximal (A,B)-invariant subspace contained in Ker C . Then ν ∗ ( A, B, C) ≠ {0} ⇔ ZI ≠ ∅ . ◊

Remark 6.5. By virtue of Proposition 2.1 (i) (see Chap. 2, Sect. 2.6), it is clear that if in Proposition 6.4 we replace ZI with Z S , then the implica-

tion Z S ≠ ∅ ⇒ ν ∗ ( A, B, C) ≠ {0} remains valid. However, in the general case, the converse implication, i.e., ν ∗ ( A, B, C) ≠ {0} ⇒ Z S ≠ ∅ , is not true (see Example 6.2 below). In other words, Proposition 6.4 does not hold if ZI is replaced with Z S . ◊

Example 6.2. Consider S(A,B,C) as in Example 5.1. In order to find ν ∗ ( A, B, C) , we apply Proposition 6.3 (3b2) to S(A,B,C) and then Proposition 6.3 (4) to the system S(K ν A,B m − p , C ) . Hence we can write ν ∗ ( A, B, C) = ν ∗ (K ν A, B m− p , C ) =

n −1 l =0

Ker C (K ν A ) l ,

where n = 3 , and matrices K ν A and C are of the form 1/ 3 0   0  KνA =  0 −1/ 3 0  , − 1 / 3 − 2 / 3 − 1 / 3

 2 C=  − 2

2 0

0 . 0

Simple verification shows that  x1  Ker C (K ν A ) = {x =  x 2  : x 1 = 0, x 2 = 0} , l =0  x 3  n −1

l

i.e., ν ∗ ( A, B, C) is nontrivial although S(A,B,C) has no Smith zeros (see Example 5.1). ◊ Example 6.3. Consider a reachable and observable system S(A,B,C) (2.1) with the matrices


1 0 A= 2  0

0 0 1 0 1 1  , 0 1 2  1 0 0

0 1 B= 0  0

0 0 , 1  0

221

0 1 0 0 C= . 0 0 0 1

The system has one single Smith zero λ = 1 – the Smith form of the system matrix equals 1 0  0  0 0 0 

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0

0 0  0 0  0 0  . 0 0  z − 1 0 0 0

1 0  In SVD (see (5.1)) of CB =   (i.e., ν = 0 , p = 1 ) we take 0 0 U = V = I 2 and M p = 1 , and, consequently, B = B and C = C . Since

B m − p ≠ 0 (the second column of B) and Cr − p ≠ 0 (the second row of C), we consider, according to Proposition 6.3 (3b2), the system S(K ν A,B m − p , C ) , where 1 0 K ν = I − B p M −p1C p A ν =  0  0 1 0 KνA =  2  0

0 0 1 0 0 0 , 0 1 2  1 0 0

0 0 0 0 0 0 , 0 1 0  0 0 1

0 0 Bm − p =   , 1    0

C =C.

In order to find Markov parameters of S(K ν A,B m − p , C ) , we note first 0 that C B m − p =   and K ν A B m − p = B m − p . This means that all ma0

222


trices C (K ν A ) l B m − p , l = 0, 1, 2,... , are zero and, consequently, the transfer-function matrix of S(K ν A,B m − p , C ) equals zero identically. By virtue of Proposition 5.3 (i), S(A,B,C) is degenerate. In order to find ν ∗ ( A, B, C) , we use the same arguments as in Example 6.2 which yield n −1

ν ∗ ( A, B, C) = ν ∗ (K ν A, B m − p , C ) = h Ker C (K ν A) l l =0

 x1  x  = {x =  2  : x 2 = 0, x 4 = 0} . x 3    x 4 

Finally, solving (2.4) for S(A,B,C), we observe that to the Smith zero λ = 1 there correspond two linearly independent state-zero directions

which span the subspace ν ∗ ( A, B, C) . To this end, it is sufficient to verify that the triples 1    0 o 0 λ = 1, x = ,g=  0 2   0

and

0    − 1 o 0 λ = 1, x = ,g=  1  0   0

satisfy (2.4). On the other hand, for any given invariant zero λ ≠ 1 the corresponding state-zero and input-zero direction are respectively of the form 0 0 xo =  o  x 3    0

and

 − x 3o  g= . ◊ o (λ − 1) x 3 

6.4 SVD and the Output-Zeroing Problem In this section we discuss the output-zeroing problem. We shall show that in any system S(A,B,C) (2.1) if (x o , u o (k )) is an output-zeroing input and x o (k ) denotes the corresponding solution, then x o (k ) ∈ ν ∗ ( A, B, C) for

6.4 SVD and the Output-Zeroing Problem

223

all k ∈ N . Furthermore, for each x o ∈ ν ∗ ( A, B, C) there exists an outputzeroing input such that the corresponding solution passes through x o . Remark 6.6. Naturally, as we already know, the above statements are valid in the cases when CA ν B has full row rank (see Proposition 3.9, Lemma

6.6 and Example 3.4), CA ν B has full column rank (see Propositions 3.10 and 6.3 (1)) and G (z) ≡ 0 (see Lemmas 6.1 and 6.2). ◊

We discuss first the relations between output-zeroing inputs and the corresponding solutions for systems S(A,B,C) and S(K ν A, B m − p , C ) (it is assumed B m − p ≠ 0 ). Since S(K ν A, B m − p , C ) has been obtained from S( A, B , C ) in (5.2) (see Fig.5.2) and relations between output-zeroing

inputs for S(A,B,C) and S( A, B , C ) are known (see Proposition 6.2), we can restrict our attention to systems S( A, B , C ) and S(K ν A, B m − p , C ) . Consider the output-zeroing problem for the system S( A, B , C ) of the form x(k + 1) = Ax(k ) + B p u p (k ) + B m − p u m − p (k )  y p (k )   C p   x( k ) = y ( k ) C − r p r p −     

(6.4)

and for the system S(K ν A, B m − p , C ) of the form x(k + 1) = K ν Ax(k ) + B m − p u m − p (k )  y p (k )   C p   x( k ) , = y  r − p (k )  Cr − p 

(6.5)

where K ν = I − B p M −p1C p A ν .  u o, p ( k )  Suppose first that the pair (x o , uo (k )) , where u o (k ) =   , is  u o, m − p ( k ) 

an output-zeroing input for (6.4) and gives a solution x o (k ) . This implies (cf., (3.26))

224


C A ν + 1x o ( k ) = − C A ν B u o ( k )

(6.6)

and (6.6) can be written as  C p A ν + 1x o ( k )  M   = − p  Cr − p A ν +1x o (k )  0

0  u o, p ( k )  .  0  u o,m − p (k )

(6.7)

Consequently, u o, p (k ) = −M −p1C p A ν +1x o (k ) .

(6.8)

− M −1C A ν +1x o (k )  p p This means that u o (k ) =   and x o (k ) satisfy the u o ,m − p ( k )  

state equation in (6.4) and yield C x o ( k ) = 0 for all k ∈ N . Introducing these sequences into (6.4) and taking K ν as in (5.10), we infer that u o,m − p (k ) and x o (k ) fulfill the state equation in (6.5) and yield C x o ( k ) = 0 for all k ∈ N . Thus we have proved the following implica u o, p ( k )  tion: if a pair (x o , u o (k )) , where u o (k ) =   , is an output u o,m − p ( k ) 

zeroing input for S( A, B , C ) (6.4) and yields a solution x o (k ) , then (x o , uo,m − p (k )) is an output-zeroing input for S(K ν A, B m − p , C ) and

yields x o (k ) as a solution of the state equation in S(K ν A, B m − p , C ) . Moreover, x o (k ) is uniquely determined by x o and u o,m − p (k ) as k −1

x o (k ) = (K ν A ) k x o + ∑ (K ν A ) k −1− l B m − p u o, m − p (l ) . l =0

(6.9)

Of course, from (6.8) and (6.9) it follows that u o, p (k ) in (6.8) is also uniquely determined by x o and u o,m − p (k ) . Suppose now that a pair (x o , uo,m − p (k )) is an output-zeroing input for S(K ν A, B m − p , C ) in (6.5) and x o (k ) stands for the corresponding sol-

ution. Then, employing the expression (5.10) for K ν and the uniqueness


225

of

solutions, it is easy to verify that the pair − 1 ν + 1 o − M C A x (k ) p p (x o , u o (k ) =   ) gives x o (k ) as a solution of the ( k ) u   o,m − p

state equation in (6.4) and yields C x o ( k ) = 0 for all k ∈ N . In this way this pair constitutes an output-zeroing input for S( A, B , C ) and produces the same solution as (x o , uo,m − p (k )) in (6.5). Thus we have proved the following implication: if (x o , uo,m − p (k )) is an output-zeroing input for S(K ν A, B m − p , C ) and x o (k ) denotes the corresponding solution, then − M −1C A ν +1x o (k ) p p (x o , u o (k ) =   ) is an output-zeroing input for u o,m − p ( k )  

S( A, B , C ) and yields x o (k ) as the corresponding solution. Moreover,

x o (k ) is uniquely determined by x o and u o,m − p (k ) (see (6.9)).

The above discussion and Proposition 6.2 enable us to formulate the following. Lemma 6.10. Consider a system S(A,B,C) (2.1) and in S( A, B , C ) (5.2) let B m − p ≠ 0 . Then (x o , u o (k )) is an output-zeroing input for S(A,B,C)  u o, p ( k )  T if and only if (x o , uo,m − p (k )) , where   = V u o (k ) , is an ( k ) u  o,m − p  output-zeroing input for S(K ν A, B m − p , C ) . Moreover, both these outputzeroing inputs give the same solution x o (k ) which is uniquely determined by x o and u o,m − p (k ) as in (6.9) (furthermore, u o, p (k ) is also uniquely determined by x o and u o,m − p (k ) – see (6.8) and (6.9)). ◊ It is important to note that Lemma 6.10 and Proposition 6.2 tell us that the zero dynamics of S(A,B,C) are preserved along the recursive process following from the point 3b2 of Procedure 5.1. Consider now the case when in S( A, B , C ) (6.4) is Cr − p = 0 . Then, as we know from Procedure 5.1 (3a), in case of B m − p ≠ 0 the original system S(A,B,C) is degenerate, whereas in case of B m − p = 0 the system

226


S(A,B,C) is nondegenerate. We discuss separately the cases B m − p ≠ 0 and B m − p = 0 . Suppose first that B m − p ≠ 0 . Assume that (x o , u o (k )) , where  u o, p ( k )  uo (k ) =   , is an output-zeroing input for (6.4) and gives a sol u o,m − p ( k )  ution x o (k ) . Then, taking into account the same arguments as in (6.6)– (6.8), we conclude that (x o , uo,m − p (k )) is an output-zeroing input for S(K ν A, B m − p , C ) in (6.5) and produces the same solution (i.e., x o (k ) )

of the state equation in S(K ν A, B m − p , C ) . Moreover, x o (k ) is uniquely determined by x o and u o,m − p (k ) as in (6.9). However, at Cr − p = 0 and B m − p ≠ 0 , as we know from Procedure 5.1 (3a2) (see Comment to 3a2), the transfer-function matrix of S(K ν A, B m − p , C ) equals zero identically. This implies in turn, by virtue of Lemma 6.1 when applied to S(K ν A, B m − p , C ) , that x o (k ) is contained in the unobservable subspace n −1 l =0

Ker C (K ν A ) l for S(K ν A, B m − p , C ) . However (see Proposition 6.3

(3a2)), this subspace is equal to ν ∗ ( A, B, C) . Finally, using Propositions 6.1 and 6.2, we infer from the above discussion that if (x o , u o (k )) is an output-zeroing input for S(A,B,C) and x o (k ) is the corresponding solution, then x o (k ) ∈ ν ∗ ( A, B, C) for all k ∈ N . Suppose

now

that

Bm − p = 0

and

let

(x o , u o (k )) ,

where

 u o, p ( k )  o uo (k ) =   , be an output-zeroing input for (6.4) and let x (k ) ( k ) u o , m − p   denote the corresponding solution. Since B m − p = 0 , the component

u o,m − p (k ) does not affect the state equation in (6.4). Thus for all k ∈ N

x o (k + 1) = Ax o (k ) + B p u o, p (k ) o

C p x (k) = 0

(6.10)


227

and x o (0) = x o . This implies (by applying Example 3.4 to S( A, B p , C p ) ) that x o (k ) = (K ν A) k x o and is contained in the subspace

ν l =0

Ker C p A l

which, as we know from the proof of point (ii) in Lemma 6.7, is equal to ν ∗ ( A, B, C) . Now, by virtue of Proposition 6.2, we conclude that if (x o , u o (k )) is an output-zeroing input for S(A,B,C) and x o (k ) is the cor-

responding solution, then x o (k ) ∈ ν ∗ ( A, B, C) for all k ∈ N . Hence we have proved the following. Lemma 6.11. Consider a system S(A,B,C) (2.1) and in the system S( A, B , C) (5.2) let Cr − p = 0 . Then, if (x o , u o (k )) is an output-zeroing

input for S(A,B,C) and x o (k ) is the corresponding solution, we have x o (k ) ∈ ν ∗ ( A, B, C) for all k ∈ N . ◊

We shall discuss now the case when in S( A, B , C) (6.4) is Cr − p ≠ 0 and B m − p = 0 .  u o, p ( k )  Let (x o , u o (k )) , where u o (k ) =   , be an output-zeroing  u o,m − p ( k ) 

input for (6.4) and let x o (k ) denote the corresponding solution. Since B m − p = 0 , the component u o,m − p (k ) does not affect the state equation

in (6.4). In consequence, (x o , u o, p ( k )) is an output-zeroing input for the system S( A, B p , C ) (which is obtained from (6.4) by neglecting the input u o,m − p ) and yields the solution x o (k ) , i.e., u o, p (k ) and x o (k ) satisfy for all k ∈ N the relations

x o (k + 1) = Ax o (k ) + B p u o, p (k ) C x o (k ) = 0

(6.11)

at the initial condition x o (0) = x o . Since in S( A, B p , C ) the first nonzero Markov parameter C A ν B p has full column rank, from Proposition 3.10 and Proposition 6.3 (1) (when applied to S( A, B p , C ) ) it follows that

228


x o (k ) = (K ν A) k x o and is contained in the subspace ν ∗ ( A, B p , C ) . On

the other hand, as we know from the proof of Lemma 6.8, ν ∗ ( A, B p , C ) = ν ∗ ( A, B , C )

and,

by

virtue

of

Proposition 6.1,

ν ∗ ( A, B , C ) = ν ∗ ( A, B, C) . Finally, using Proposition 6.2, we obtain the following.

Lemma 6.12. Consider a system S(A,B,C) (2.1) and in the system S( A, B , C) (5.2) let Cr − p ≠ 0 and let B m − p = 0 . Then, if (x o , u o (k )) is

an output-zeroing input for S(A,B,C) and x o (k ) is the corresponding solution, then x o (k ) ∈ ν ∗ ( A, B, C) for all k ∈ N . ◊ We can formulate now the desired result. Proposition 6.5. In a system S(A,B,C) (2.1) let (x o , u o (k )) be an outputzeroing input and let x o (k ) denote the corresponding solution. Then x o (k ) ∈ ν ∗ ( A, B, C) for all k ∈ N . Moreover, for each x o ∈ ν ∗ ( A, B, C) there exists an output-zeroing input such that the corresponding solution passes through x o .

Proof. In the proof it is sufficient to discuss all those cases which are listed in Proposition 6.3 (or in Procedure 5.1). 1. If CA ν B has full column rank, the assertions of the proposition follow from Propositions 3.10 and 6.3 (1). 2. If CA ν B has full row rank r and m > r , the assertions of the proposition follow from Propositions 3.9 and 6.3 (2) and from Lemma 6.6. 4. If G (z) ≡ 0 , the proof follows directly from Lemmas 6.1 and 6.2 (see also Proposition 6.3 (4)). 3b1. If in S( A, B , C) (5.2) is Cr − p ≠ 0 and B m − p = 0 , the first claim of the proposition follows from Lemma 6.12. In order to prove the second claim, suppose that x o ∈ ν ∗ ( A, B, C) . Now, from the discussion preceding Lemma 6.12, ν ∗ ( A, B, C) = ν ∗ ( A, B , C ) = ν ∗ ( A, B p , C ) . Moreover, since S( A, B p , C ) has the first nonzero Markov parameter of full column rank

and x o ∈ ν ∗ ( A, B p , C ) , hence, using Proposition 3.10 to S( A, B p , C ) , we conclude that x o determines uniquely the output-zeroing input


229

(x o , u o, p ( k )) , where u o, p (k ) = −( C A ν B p ) + C A ν +1 (K ν A) k x o . The

corresponding solution equals x o (k ) = (K ν A) k x o . Now, it is not difficult  u o, p ( k )  to note that the pair (x o , u o (k )) , where u o (k ) =   and  u o,m − p ( k )  u o, p (k ) is determined as above and u o,m − p (k ) is an arbitrarily fixed ad-

missible input, constitutes an output-zeroing input for S( A, B , C) (5.2) and yields x o (k ) as the corresponding solution. Finally, by virtue of Proposition 6.2, the pair (x o , V u o (k )) is an output-zeroing input for S(A,B,C) and gives the solution x o (k ) = (K ν A) k x o which passes through x o for k=0. 3a. If in S( A, B , C) (5.2) is Cr − p = 0 , the first claim follows immediately from Lemma 6.11. For the proof of the second claim we shall discuss separately the cases B m − p = 0 (3a1) and B m − p ≠ 0 (3a2). 3a1. Suppose first that B m − p = 0 and let x o ∈ ν ∗ ( A, B, C) . However, as is known from the proof of Lemma 6.7 (ii) and from Proposition 6.1, we have ν ∗ ( A, B, C) = ν ∗ ( A, B , C ) = ν ∗ ( A, B p , C p ) and S( A, B p , C p ) has uniform rank. Thus x o ∈ ν ∗ ( A, B p , C p ) and determines uniquely the output-zeroing

input

(x o , u o, p ( k ))

for

S( A, B p , C p ) ,

where

u o, p (k ) = −( C p A ν B p ) −1 C p A ν +1 (K ν A) k x o . The corresponding sol-

ution is x o (k ) = (K ν A) k x o . It is clear now that the pair (x o , u o (k )) ,  u o, p ( k )  where u o (k ) =   and u o, p (k ) is determined as above and  u o,m − p ( k )  u o,m − p (k ) is an arbitrarily fixed admissible input, is an output-zeroing

input for S( A, B , C) and gives x o (k ) as the corresponding solution. Now, by virtue of Proposition 6.2, (x o , V u o (k )) constitutes an output-zeroing input for S(A,B,C) which yields the solution x o (k ) = (K ν A) k x o .

3a2. Suppose now that B m − p ≠ 0 and let x o ∈ ν ∗ ( A, B, C) . By virtue

of

Lemma 6.5,

x o ∈ ν ∗ ( A, B, C) = ν ∗ ( A, B , C ) = ν ∗ (K ν A, B m− p , C ) .

230


Moreover, via the proof of Lemma 6.7 (see (i)), ν ∗ (K ν A, B m− p , C ) = ν ∗ (K ν A, B m − p , C p ) and the transfer-function matrix of the system S(K ν A, B m − p , C p ) equals zero identically. Applying Lemmas 6.1 and

6.2

to

S(K ν A, B m − p , C p )

and

taking

into

account

that

x o ∈ ν ∗ (K ν A, B m − p , C p ) , we conclude that (x o , uo,m − p (k )) , where u o,m − p (k ) is an arbitrarily fixed admissible input, is an output-zeroing

input for S(K ν A, B m − p , C p ) and the corresponding solution x o (k ) has the form (6.9). Of course, since Cr − p = 0 , the pair (x o , uo,m − p (k )) is also an output-zeroing input for S(K ν A, B m − p , C ) (and yields the same solution x o (k ) (6.9)). Finally, by virtue of the discussion preceding Lemma 6.10,

we

infer

that

the

pair

(x o , V u o (k )) ,

where

− M −1C A ν +1x o (k )  p p

uo (k ) =   and x o (k ) is determined by x o and u o ,m − p ( k )   u o,m − p (k ) as in (6.9), is an output-zeroing input for S(A,B,C) and yields x o (k ) of the form (6.9) as the corresponding solution.

3b2. If in S( A, B , C) (5.2) is Cr − p ≠ 0 and B m − p ≠ 0 , then we apply Procedure 5.1 (3b2) to S(A,B,C). As is known (see Corollary 5.1 (ii)), the recursive process following from Procedure 5.1 (3b2) ends, after a finite number of steps, when we obtain a system in which the first nonzero Markov parameter has full column rank or a system whose transferfunction matrix equals zero identically. The process conserves besides invariant and Smith zeros also the corresponding state-zero directions (see Corollary 6.1). Lemma 6.5 says that ν ∗ ( A, B, C) is preserved along the process, whereas Lemma 6.10 and Proposition 6.2 tell us that also x o (k ) is preserved along this process. Furthermore, Lemma 6.10 displays relations between output-zeroing inputs for a system at a given step and for the system at the next step. In this way, if an output-zeroing input for the original system S(A,B,C) is known, we can determine recursively the outputzeroing input for a system at the last step. We can also do this going in the opposite direction. Namely, if an output-zeroing input for the system at the last step is known, then, using recursively Lemma 6.10, we can determine

6.5 Zeros, Output-Nulling Subspaces and Zero Dynamics

231

the output-zeroing input for the original system. Moreover, in both these cases the corresponding solution remains unchanged along the process. Suppose now that (x o , u o (k )) is an output-zeroing input for S(A,B,C) and x o (k ) stands for the corresponding solution. We are to show that x o (k ) ∈ ν ∗ ( A, B, C) for all k ∈ N . However, going forwards along the process, we can determine an output-zeroing input for the system at the

last step. As we know, this output-zeroing input yields x o (k ) as the corresponding solution. On the other hand, denote by ν ∗ the maximal output-

nulling controlled invariant subspace for this last system. Then ν ∗ can be determined by virtue of Lemma 6.2 or Lemma 6.4, while from Lemma 6.1 or from Proposition 3.10 we obtain x o ( k ) ∈ ν ∗ for all k ∈ N . Finally, since ν ∗ = ν ∗ ( A, B, C) , the first claim of the proposition follows.

In order to prove the second claim, suppose that x o ∈ ν ∗ ( A, B, C) . For the system at the last step of the process we have, as mentioned above, ν ∗ = ν ∗ ( A, B, C) . Hence x o ∈ ν ∗ . Since this system has the first nonzero Markov parameter of full column rank or the identically zero transferfunction matrix, we can determine for it (using Proposition 3.10 or Lemma 6.1) an output-zeroing input such that the corresponding solution x o (k ) satisfies x o (0) = x o . Naturally, x o (k ) ∈ ν ∗ = ν ∗ ( A, B, C) for all k ∈ N . Finally, going backwards along the process, we can find out an output-zeroing input for the original system S(A,B,C) which produces x o (k ) . This completes the proof of Proposition 6.5. ◊

6.5 Zeros, Output-Nulling Subspaces and Zero Dynamics In Chapter 2 (Section 2.7) we introduced simple algebraic conditions of degeneracy/nondegeneracy (see Propositions 2.2 and 2.3). These conditions decompose the class of all systems S(A,B,C) such that B ≠ 0 and C ≠ 0 into two disjoint subclasses: of nondegenerate and degenerate systems (see Fig. 6.1). In nondegenerate systems, as we know from Corollary 5.2, the Smith and invariant zeros are exactly the same objects which are determined as the roots of the zero polynomial. Moreover, as we show below, the degree

232


of this polynomial equals the dimension of ν ∗ ( A, B, C) , while the zero dynamics are independent upon control vector. In degenerate systems, the zero polynomial determines merely the Smith zeros, while the set of invariant zeros equals the whole complex plane. Moreover, in such systems the dimension of ν ∗ ( A, B, C) is strictly larger than the degree of the zero polynomial and the zero dynamics essentially depend upon control vector. Proposition 6.6. If S(A,B,C) (2.1) is nondegenerate, then the Smith and invariant zeros of S(A,B,C) are the same objects (including multiplicities). Moreover, deg ψ S( A,B,C) (z) = dim ν ∗ ( A, B, C) and the zero dynamics, in appropriate coordinates, have the form ξ ( k + 1) = Nξ ( k ) , where the characteristic polynomial of matrix N equals the zero polynomial of S(A,B,C) and ξ belongs to ν ∗ ( A, B, C) (when

Fig. 6.1.

∗

( A , B, C)


233

Proof. As we know from Procedure 5.1, S(A,B,C) is nondegenerate if and only if one of the cases 1, 2a, 3a1 or 3b1 of this procedure holds or if at the last step of the recursive process following from the point 3b2 we obtain a system with the first nonzero Markov parameter of full column rank. In the proof we shall discuss the above cases. Using (5.1)–(5.5) and (5.8)–(5.10) and denoting  Fp  −1 ν +1 and Fm − p = 0 , F=  , where F p = −M p C p A F  m − p 

we consider the transformations

 I 0   zI − A − B   I 0   I 0  0 U T   C 0  0 V   F I m   

(i)

 zI − A − B p  =  Cp 0  Cr − p 0 

 zI − K ν A − B p  =  Cp 0  Cr − p 0 

1.

0   0  I m − p 

− Bm − p   I 0  −1 ν +1 0  − M p C p A Ip 0   0 0 − B m− p   0 . 0 

In S(A,B,C) let CA ν B have full column rank (i.e., p = m and

B p = B ) and let x ' = Hx denote a change of basis which decomposes S(K ν A, B , C ) (see Remark 6.3) into an unobservable and an observable

part, i.e., consider a system S((K ν A ) ' , B ' , C ' ) with the matrices ( K A ) ' (K ν A) ' =  ν o 0 

[

(ii) x '  x' =  o'  .  x o 

Moreover, let us write

]

'  ' (K ν A)12 ' B o  B , = , C ' = 0 Co' ,   '  (K ν A) 'o  B  o 

234


H 0  zI − K ν A − B  H −1 0  zI − (K ν A) ' =  0 I   C 0   0 I m   C' r 

(iii)

 zI o − (K ν A) 'o  = 0  0 

' − (K ν A)12 zI o − (K ν A ) 'o

Co'

− B'  0 

− B o'   − B o'  . 0  

As we know from Corollary 4.2 (see also Exercise 4.4.22), the zero polynomial of the matrix standing on the right-hand side of (iii) equals ' χ cl o ( z ) := det( zI o − ( K ν A ) o ) . It determines all output decoupling zeros of S(K ν A, B , C ) (i.e., all invariant zeros of S(A,B,C)) and, on the other ' hand, χ cl o ( z ) = det( zI o − (K ν A ) o ) is the zero polynomial of S(A,B,C),

i.e., ψ S( A ,B,C) (z) = det( zI o − (K ν A ) 'o ) . This means, as we know from Definition 2.3, that the Smith and invariant zeros in S(A,B,C) are one and the same thing. Now, by virtue of Lemma 6.4, we can write deg ψ S( A ,B,C) (z) = deg det( zI o − (K ν A) 'o ) n −1

= dim h Ker C (K ν A ) l = dim ν ∗ ( A, B, C) . l =0

Finally, denoting by S( A ' , B' , C' ) the system obtained from S(A,B,C) by the same change of basis and using Remark 6.3 and Lemma 6.3, we obtain x '  ν ∗ ( A ' , B' , C' ) = {x'∈ R n : x' =  o'  , x 'o = 0} and the zero dynamics of  x o  S( A ' , B' , C' ) in the form x 'o ( k + 1) = (K ν A) 'o x 'o ( k ) .

3a1. In this case is Cr − p = 0 and B m − p = 0 (see Procedure 5.1). Decompose the system S(K ν A, B p , C ) , in which the first Markov parameter has full column rank, into an unobservable and an observable part (cf., (ii) above). Then, as it follows from (i) and from the proof of the point ' 1 above, χ cl o ( z ) := det( zI o − ( K ν A ) o ) equals the zero polynomial of S(A,B,C) and deg χ cl o ( z ) = dim

n−1 l =0

Ker C (K ν A ) l = dim ν ∗ ( A, B, C) .


235

The desired characterization of the zero dynamics results from the following solution to the output-zeroing problem (see Exercise 6.6.8): If S(A,B,C) (2.1) is such that in (5.2) is Cr − p = 0 and B m − p = 0 , then a pair ( x o , u o ( k )) is an output-zeroing input for S(A,B,C) if and only if x o ∈ ν ∗ ( A, B, C) and 0 − M −p1 C p A ν +1 (K ν A ) k x o    u o (k ) = V  +V , 0 u m− p (k )  

where u m − p (.) : N → R m − p is arbitrary. Moreover, the solution of the state equation in S(A,B,C) corresponding to such a pair has the form 0   x o ( k ) = (K ν A) k x o (note that the component V   does not afu m − p (k ) fect this solution). Now, the remaining part of the proof follows the same lines as the proof of the point 1 above. The zero dynamics of S( A ' , B' , C' ) are of the form x 'o ( k + 1) = (K ν A) 'o x 'o ( k ) . 2a and 3b1. In these cases the proofs follow the same lines as in points 1 and 3a1 above. 3b2. In this case we assume that at the last step of the recursive process following from Procedure 5.1 (3b2) we get a system, say S( A f , B f , C f ) , ν

in which the first Markov parameter, say C f A f f B f , has full column rank. Then we use the first part of the proof of the point 1 above referred to S( A f , B f , C f ) . In this way we get a polynomial det(zI o − (K νf A f ) 'o ) which is the zero polynomial of S( A f , B f , C f ) . Moreover, deg det(zI o − (K ν f A f ) 'o ) = dim

n −1 l =0

Ker Cf (K ν f A f ) l

∗

= dim ν ( A f , B f , C f ) .

On the other hand, by virtue of Corollary 5.1 (ii), the zero polynomial of S(A,B,C) equals det(zI o − (K ν f A f ) 'o ) and, by virtue of Corollary 6.1, ν ∗ ( A, B, C) = ν ∗ ( A f , B f , C f ) . The zero dynamics of S( A 'f , B 'f , C 'f ) are

of the form x 'o ( k + 1) = (K ν f A f ) 'o x 'o (k ) . Finally, since the solutions cor-

236


responding to output-zeroing inputs are conserved along the process (see Lemma 6.10), the zero dynamics of S( A ' , B' , C' ) have the same form. ◊ Proposition 6.7. If S(A,B,C) (2.1) is degenerate, then dim ν ∗ ( A, B, C) > deg ψ S( A,B,C) (z) .

Moreover, the Smith zeros of S(A,B,C) are the i.o.d. zeros of a certain system whose transfer-function matrix equals zero identically. Furthermore, the zero dynamics of S(A,B,C) depend essentially upon control vector. Proof. As we know from Procedure 5.1, S(A,B,C) in (2.1) is degenerate if and only if one of the cases 2b, 3a2 or 4 takes place or if at the last step of the recursive process resulting from the point 3b2 of this procedure we obtain a system with the identically zero transfer-function matrix. 4. In this case in S(A,B,C) is G (z) ≡ 0 and the assertions follow immediately from Proposition 2.6, Lemma 6.2 and Remark 6.1.

2b. In this case in S(A,B,C) is m > r and the first nonzero Markov parameter CA ν B has full row rank r as well as in S( A, B , C) (5.2) is B m − r ≠ 0 . From Procedure 5.1 (2b) we know that the Smith zeros of

S(A,B,C) are the i.o.d zeros of the system S(K ν A, B m − r , C ) whose transfer-function matrix equals zero identically (the zero polynomial of S(A,B,C) is equal to the zero polynomial of S(K ν A, B m − r , C ) ). On the

other hand, by virtue of Proposition 6.3 (2b), ν ∗ ( A, B, C) equals the unobservable subspace for S(K ν A, B m − r , C ) . Now, the claims of the proposition follow directly from Proposition 2.6, Lemma 6.2 and Remark 6.1 when these results are applied to S(K ν A, B m − r , C ) . 3a2. In this case in S( A, B , C) (5.2) is Cr − p = 0 and B m − p ≠ 0 . By virtue of Procedure 5.1 (3a2), the Smith zeros of S(A,B,C) are the i.o.d. zeros of the system S(K ν A, B m − p , C ) whose transfer-function matrix equals zero identically (the zero polynomial of S(A,B,C) is equal to the zero polynomial of S(K ν A, B m − p , C ) ). On the other hand, by virtue of Proposition 6.3 (3a2), ν ∗ ( A, B, C) equals the unobservable subspace for S(K ν A, B m − p , C ) . Now, to S(K ν A, B m − p , C ) we apply Proposition 2.6,

Lemma 6.2 and Remark 6.1 and the proposition follows.

6.6 Exercises

237

3b2. In this case we assume that at the last step of the recursive process following from Procedure 5.1 we obtain a system, say S( A f , B f , C f ) , whose transfer-function matrix equals zero identically. Then the proof follows from the fact that this process preserves the set of invariant zeros, the zero polynomial, the maximal output-nulling controlled invariant subspace as well as the solutions corresponding to output-zeroing inputs. ◊

6.6 Exercises 6.6.1. Consider a strictly proper or proper system S(A,B,C,D) (2.1). Show

that the unobservable subspace X o =

n −1 l =0

Ker CA l is A-invariant, i.e.,

A( X o ) ⊆ X o .

Hint. For x ∈ X o check that Ax ∈ X o , i.e., CA l ( Ax) = 0 , l = 0, 1,..., n − 1 . Note that for l = 0, 1,..., n − 2 the relation under considerations follows immediately from the assumption x ∈ X o . For l = n − 1 use n −1

the Cayley-Hamilton theorem, i.e., write A n = ∑ α i A i . i =0

6.6.2. Let x' = Hx denote a change of coordinates in a strictly proper or proper system (2.1). Prove that X' r = H(X r ) and X' o = H(X o ) . 6.6.3. Consider S(A,B,C) (2.1) with G (z) ≡ 0 . Show that the unobservable

subspace X o is (A+BF)-invariant at any F. 6.6.4. In S(A,B,C) (2.1) let the first nonzero Markov parameter CA ν B

have full column rank. Show that Scl ν =

n −1 l =0

KerC(K ν A) l is (A,B)-invar-

iant and contained in Ker C . Hint. Take F = −(CA ν B) + CA ν +1 and verify that A + BF = K ν A .

Then show that K ν A(Sνcl ) ⊆ Scl ν . 6.6.5. Let x' = Hx denote a change of coordinates in S(A,B,C) (2.1). Show

that ν ∗ ( A' , B' , C' ) = H(ν ∗ ( A, B, C)) .

238


6.6.6. Show that S(A,B,C) (2.1) is degenerate if and only if it can be reduced with the aid of Procedure 5.1 to a system with the identically zero transfer-function matrix. 6.6.7. Prove the following result. In S(A,B,C) (2.1) let ZI = ∅ . Then (x o , u o (k )) is an output-zeroing input if and only if x o = 0 and u o (k ) ∈ KerB for all k ∈ N . Moreover, the corresponding solution sat-

isfies x o (k ) = 0 for all k ∈ N . For an illustration of this result see Example 2.2. Note that the above result is not valid if we replace the assumption ZI = ∅ with Z S = ∅ . As an appropriate example consider a system S(A,B,C) with the matrices 0 0 A= 0  0

1 0 0 0 1 0 , 0 0 1  0 0 0

0 0 B= 1  0

1 0 , 0  1

C = [1 0 0 0] .

This system has no Smith zeros (i.e., Z S = ∅ ) and Ker B = {0} . On the other hand, it is degenerate and for this reason there exist output-zeroing inputs producing nontrivial solutions of the state equation. Hint. Take into account Propositions 6.4 and 6.5.

6.6.8. Let S(A,B,C) (2.1) be such that in S( A, B , C) (5.2) is B m − p = 0

and Cr − p = 0 . Show that for S(A,B,C) the following characterization of the output-zeroing problem takes place: A pair (x o , u o (k )) is an outputzeroing input for S(A,B,C) if and only if xo ∈

ν

Ker CA l = ν ∗ ( A, B, C)

l =0

and 0 − M −p1 C p A ν +1 (K ν A) k x o    u o (k ) = V   + V , 0  u m− p ( k )   

where u m − p (.) : N → R m − p and K ν = I − B p M −p1C p A ν . Moreover,

6.6 Exercises

239

the corresponding solution has the form x o ( k ) = (K ν A) k x o

and is contained in ν ∗ ( A, B, C) . 0   Observe also that the component V   in u o (k ) does not af u m − p (k ) fect the solution of the state equation as well as that any input with this 0   property has the form V  .  u m − p (k )

Hint. By virtue of Proposition 6.2, we can equivalently discuss the problem for S( A, B , C) in (5.2). To this end, we apply to S( A, B , C) the necessary condition for output-zeroing inputs given in Proposition 3.8.

Hence, if (x o ,u o (k )) is an output-zeroing input for S( A, B , C) and x o (k ) is the corresponding solution, then u o ( k ) = −( C A ν B ) + C A ν +1 ( K ν A ) k x o + k −1

(i)

− ( C A ν B ) + C A ν +1[ ∑ (K ν A) k −1− l B u h (l )] + u h (k ) l =0

and k −1

x o (k ) = (K ν A ) k x o + ∑ (K ν A ) k −1− l B u h (l ) ,

(ii)

l =0

where x o ∈

ν

Ker C A l and u h (k ) is an admissible input satisfying

l =0

ν

C A B u h (k ) = 0 for all k ∈ N and K ν is as in (5.9). Moreover, we have x o (k ) ∈

ν

Ker C A l for all k ∈ N .

l =0

 u h , p (k )  Now, write u h (k ) as u h (k ) =   (cf., (5.5)). The condition  u h ,m − p ( k )  B m − p = 0 implies B u h (k ) = B p u h , p (k ) . On the other hand, the condi-

tion CA ν B u h ( k ) = Λ u h (k ) = 0 yields (see (5.1)) M p u h , p (k ) = 0 and,

240


consequently, u h , p (k ) = 0 for all k ∈ N . This implies B u h (k ) = 0 for all k ∈ N . Finally, using (5.1), we can write (i) and (ii) in the form (iii)

0 − M −p1C p A ν +1 (K ν A) k x o    , uo (k ) =  + u (k ) 0   m − p  

(iv)

x o ( k ) = (K ν A) k x o .

In this way we have proved the implication: if (x o ,u o (k )) is an outputzeroing input for S( A, B , C) and x o (k ) is the corresponding solution, then x o ∈

ν

Ker C A l =

l =0

ν

Ker CA l = ν ∗ ( A, B, C) (see Lemma 6.7) and

l =0

u o (k ) has the form (iii) and x o (k ) is as in (iv). In order to prove the converse, it is sufficient to verify that (iii) and (iv)

satisfy the state equation of S( A, B , C) and to show that C p x o (k ) = 0 for all k ∈ N . This last relation follows, however, from the assumption ν

ν

l =0

l =0

x o ∈ h Ker C p A l = h Ker C A l

and from C p (K ν A) l = C p A l

for

l = 0, 1,..., ν and C p (K ν A ) l = 0 for l ≥ ν + 1 . In fact, we can write C p x o (k ) = C p (K ν A) k x o = 0 for all k ∈ N . Finally, the desired result

for S(A,B,C) follows via Proposition 6.2. In order to prove the last claim of the exercise, we shall show the following equivalence 0   u(k ) ∈ Ker B for all k ∈ N ⇔ u(k ) = V  . u ( k )  m− p 

For the proof of ⇒ ) observe first that Bu(k ) = B u (k ) = B p u p (k ) . Now, since C p A ν B p = M p is nonsingular, B p has full column rank and the condition Bu(k ) = 0 implies u p (k ) = 0 . Thus, if u(k ) ∈ Ker B , then 0   u(k ) = V u (k ) = V   . The proof of ⇐ ) is obvious. u m − p ( k )

6.6 Exercises

241

6.6.9. Let S(A,B,C) (2.1) be such that in S( A, B, C) (5.2) is B m − p ≠ 0 and Cr − p = 0 . Prove the following characterization of the output-zeroing problem for S(A,B,C): A pair (x o , u o (k )) is an output-zeroing input for S(A,B,C) if and only if xo ∈

ν

KerCA l = ν ∗ ( A, B, C)

l =0

and k −1   − M −p1C p A ν +1[(K ν A ) k x o + ∑ (K ν A ) k −1− l B m − p u m − p (l )]  u o (k ) = V l =0   u m − p (k )  

where u m − p (.) : N → R m - p and K ν = I − B p M −p1C p A ν . Moreover, the corresponding to (x o , u o (k )) solution of the state equation of S(A,B,C) has the form k −1

x o (k ) = (K ν A) k x o + ∑ (K ν A) k −1−l B m − p u m − p (l ) l =0

and is contained in ν ∗ ( A, B, C) . Hint. By virtue of Proposition 6.2, we consider the problem for S( A, B , C ) . To S( A, B , C ) we apply Proposition 3.8. It says that if (x o ,u o (k )) is an output-zeroing input for S( A, B , C ) and x o (k ) is the

corresponding solution, then u o (k ) and x o (k ) take respectively the form (i) and (ii) as in the hint to Exercise 6.6.8, whereas x o ∈

ν

Ker C A l and

l =0

ν

u h (k ) is an admissible input satisfying C A B u h (k ) = 0 , and K ν is as

ν in (5.9). Moreover, x o ( k ) ∈ h Ker C A l for all k ∈ N . l =0

 u h , p (k )  In fact, if we denote u h (k ) =   (cf., (5.5)), then the condi u h ,m − p ( k ) 

242


tion C A ν B u h ( k ) = Λ u h (k ) = 0 yields (see (5.1)) M p u h , p (k ) = 0 and, consequently, u h , p (k ) = 0 for all k ∈ N . This implies in turn B u h ( k ) = B m − p u h ,m − p (k ) . Now, in view of (5.1), the relations (i) and

(ii) of Exercise 6.6.8 take respectively the form (v) k −1   − M −p1C p A ν +1[(K ν A ) k x o + ∑ (K ν A ) k −1− l B m − p u h , m − p (l )]  uo (k ) = l =0   u − p (k ) m h ,  

and k −1

x o (k ) = (K ν A ) k x o + ∑ (K ν A ) k −1− l B m − p u h , m − p (l ) .

(vi)

l =0

Hence we have proved: if (x o , u o (k )) is an output-zeroing input for S( A, B , C )

x o (k )

and

is

the

corresponding

solution,

ν

ν

l =0

l =0

x o ∈ ν ∗ ( A, B , C ) (see Lemma 6.7, where

then

l l h Ker C A = h Ker CA )

o

and u o (k ) has the form (v) and x (k ) is of the form (vi). In order to prove the converse, it is sufficient to check that (v) and (vi) satisfy the state equation of S( A, B , C ) and to show that C p x o (k ) = 0 for all k ∈ N . This last relation follows from x o ∈ l

l

ν

Ker C Α l and from

l =0 l

C p (K ν A ) = C p A for l = 0, 1,..., ν and C p (K ν A ) = 0 for l ≥ ν + 1 as

well as from the fact that C A ν B is the first nonzero Markov parameter of S( A, B , C ) (i.e., C p A l B m − p = 0 for l = 0, 1,..., ν – see (5.1) and (5.5)). Now, we can write

y p ( k ) = C p x o ( k ) = C p (K ν A ) k x o k −1

+ ∑ C p (K ν A ) k −1− l B m − p u h , m − p (l ) = 0. l =0

6.6 Exercises

243

The remaining part of the proof follows immediately from Propositions 6.1 and 6.2. 6.6.10. Let S(A,B,C) (2.1) be such that in S( A, B , C ) (5.2) is B m − p = 0 and Cr − p ≠ 0 . Prove the following characterization of the output-zeroing problem for S(A,B,C): A pair (x o , u o (k )) is an output-zeroing input for S(A,B,C) if and only if n −1

x o ∈ h Ker C(K ν A) l = ν ∗ ( A, B, C) l =0

and 0 − M −p1 C p A ν +1 (K ν A) k x o    u o (k ) = V   + V , 0    u m− p ( k ) 

where u m − p (.) : N → R m − p . Moreover, the corresponding solution has the form x o ( k ) = (K ν A) k x o

and is contained in ν ∗ ( A, B, C) . 0   Show also that the component V   in u o (k ) does not affect  u m − p (k ) the solution of the state equation of S(A,B,C) as well as that any input with 0   this property has the form V  .  u m − p (k )

Hint. Use the same argumentation as in Exercise 6.6.8. Assuming that (x o , u o (k )) is an output-zeroing input for S( A, B , C) and x o (k ) denotes the corresponding solution, from Proposition 3.8 we obtain u o (k ) and x o (k ) respectively in the form (i) and (ii) (see Exercise 6.6.8). From the

conditions B m − p = 0 and CA ν B u h (k ) = 0 it follows that B u h (k ) = 0 . Now, by virtue of (5.1), the relations (i) and (ii) take respectively the form (iii) and (iv) (see Exercise 6.6.8).

244


6.6.11. Consider a system S(A,B,C) (2.1), with CA ν B as its first nonzero Markov parameter, and the associated system S( A, B , C) (5.2). Let x' = Hx denote a change of coordinates. Show that with S( A' , B' , C' ) ,

where A' = HAH −1 , B' = HB , C' = CH −1 , we can associate the system S( A' , B ' , C ' ) , where B ' = B' V , C ' = U T C' and orthogonal matrices U and

V are the same as in SVD of CA ν B in S(A,B,C). Hint. Take into account that Markov parameters are invariant under any change of coordinates.

7 Output-Nulling Subspaces in Proper Systems

This chapter is devoted to a brief discussion of certain geometric properties of invariant zeros in proper systems S(A,B,C,D) (2.1). Recall [1, 2] that for a proper system S(A,B,C,D) (2.1) a subspace X is an output-nulling controlled invariant subspace if and only if there exists a mxn real matrix F such that ( A + BF )(X) ⊆ X ⊆ Ker(C + DF) . The set of all such subspaces is nonempty and closed under subspace addition, so that

it has a maximal element, denoted as ν ∗ ( A, B, C, D) , which is called further the maximal output-nulling controlled invariant subspace. The subspace ν ∗ ( A, B, C, D) can be computed recursively as a limit of subspaces X i defined as follows [1, 2] X 0 := R n , X i := {x ∈ R n : ∃u ∈ R m ( Ax + Bu ∈ X i −1 , Cx + Du = 0)} .

7.1 Matrix D of Full Column Rank We begin by recalling some facts already established in Chapter 4. As is known (see Corollary 4.6), if D has full column rank (i.e., rank D = m ), then invariant zeros of S(A,B,C,D) (which coincide with the Smith zeros) may be characterized as the o.d. zeros of the closed-loop (state feedback) system S( A − BD + C, B, (I r − DD + )C, D) . Let n ocl stand for the rank of the observability matrix for the system S( A − BD + C, B, (I r − DD + )C, D) and let S cl D denote the unobservable subspace for this system, i.e., n −1

n + + l S cl D := h Ker{(I r − DD )C( A − BD C) } ⊆ R . l =0

The number of o.d. zeros of S( A − BD + C, B, (I r − DD + )C, D) (including multiplicities) equals n − nocl = dim S cl D. J. Tokarzewski: Fin. Zeros in Diskr. Time Contr. Syst., LNCIS 338, pp. 245–270, 2006. © Springer-Verlag Berlin Heidelberg 2006

246


On the other hand, recall (see Lemma 4.3) that the input u(k ) = − D + C( A − BD + C) k x o ,

(i)

where x o ∈ R n is fixed arbitrarily, applied to S(A,B,C,D) at a given initial state x(0) ∈ R n yields (ii)

x(k ) = A k (x(0) − x o ) + ( A − BD + C) k x o

and the system response (iii)

y (k ) = CA k (x(0) − x o ) + (I r − DD + )C( A − BD + C) k x o .

Furthermore, we have the following characterization of the outputzeroing problem (see Proposition 3.7). A pair (x o , u o (k )) is an output-

zeroing input for S(A,B,C,D) if and only if x o ∈ Scl D and u o ( k ) has the form (i). Moreover, the corresponding to (x o , u o (k )) solution has the

form x o (k ) = ( A − BD + C) k x o and is contained in S cl D. It is important to note that these output-zeroing inputs are determined uniquely by the initial states x o ∈ Scl D . This means, that if an initial state

x o ∈ Scl D is fixed, then the unique input that is able to keep y ( k ) = 0 for all k ∈ N has the form (i). In the following result we shall show that the maximal output-nulling

controlled invariant subspace ν ∗ ( A, B, C, D) equals S cl D. Lemma 7.1. In S(A,B,C,D) (2.1) let rank D = m . Then ν ∗ ( A, B, C, D) = S cl D.

Proof. For the proof we show first that S cl D is an ( A + BF ) -invariant subspace with the property S cl D ⊆ Ker (C + DF) . Next, we show that if X is an arbitrary output-nulling controlled invariant subspace (i.e., there exists a matrix F such that ( A + BF )(X) ⊆ X ⊆ Ker (C + DF) ), then X ⊆ S cl D.

7.1 Matrix D of Full Column Rank

247

cl + Let F = − D + C . Then ( A − BD + C)(S cl D ) ⊆ S D ⊆ Ker (I r − DD )C . In

+ fact, the inclusion S cl D ⊆ Ker (I r − DD )C follows from the definition of

+ cl cl S cl D (see above). In order to prove that ( A − BD C)(S D ) ⊆ S D , it is suffi-

cl cient to show that ( A − BD + C)x ∈ S cl D for each x ∈ S D . This last fact,

however, is an immediate consequence of the definition of S cl D and of the + Cayley-Hamilton theorem (when applied to A − BD C ). Suppose now that X is an ( A + BF) -invariant subspace contained in Ker(C + DF) . From X ⊆ Ker(C + DF) it follows that (C + DF)x = 0 for each x ∈ X . Hence we can write DFx = −Cx and, consequently, since D+ D = I m ,

we

obtain

Fx = − D + Cx .

This

yields

the

relation

+

( A + BF )x = ( A − BD C)x which is valid for each x ∈ X . In this way we

have ( A − BD + C)(X ) ⊆ X . On the other hand, from Fx = − D + Cx we obtain DFx = − DD + Cx and, consequently, Cx + DFx = Cx − DD + Cx . This means

that

X ⊆ Ker(I r − DD + )C .

Thus

( A − BD + C)(X) ⊆ X ⊆ Ker(I r − DD + )C ,

we i.e.,

have in

shown other

that

words,

(I r − DD + )C( A − BD + C) k x = 0 for each x ∈ X and for all k ∈ N . This cl enables us to write x ∈ Scl D at any x ∈ X , i.e., X ⊆ S D . ◊

Corollary 7.1. In S(A,B,C,D) (2.1) let D be square and nonsingular. Then ν ∗ ( A, B, C, D) = R n . n Proof. If D is invertible, I r − DD + = 0 and, consequently, Scl D =R . ◊

As we show below, in a suitable basis Proposition 3.7 takes much simpler form. In fact, let x' = Hx denote a change of coordinates which decomposes the system S( A − BD + C, B, (I r − DD + )C, D) into an unobservable and an observable part (cf., Remark 4.6); that is, consider a new system S(( A − BD + C)' ,B' ,((I r − DD + )C)' ,D' ) with the matrices ( A − BD + C)' o ( A − BD + C)' =  0 

B ' o  ( A − BD + C)'12  ' B , =   B'  , ( A − BD + C)'o   o

248

(iv)


[

]

((I r − DD + )C)' = 0 o

((I r − DD + )C)'o ,

x' o  x' =   ,  x' o 

D' = D ,

dim x' o = n − nocl .

Moreover, consider a system S( A' , B' , C' , D' ) which is obtained from S(A,B,C,D) by the same change of coordinates. For the system in (iv) we find its unobservable subspace as (v)

n −1 x' o  + + l S' cl D = h Ker{((I r − DD )C)' (( A − BD C)' ) } = {x' : x' =  } . l =0  0 

Now, the output-zeroing problem for S( A ' , B' , C' , D' ) may be described as follows. Proposition 7.1. A pair (x'o , u' o (k )) is an output-zeroing input for the system S( A ' , B' , C' , D' ) if and only if x' o ∈ S'cl D

 x' o  (i.e., x'o =  o  )  0 

and (( A − BD +C)'o ) k x'oo  u'o (k ) = −D+ CH −1  . 0  

Furthermore, the corresponding to (x'o , u' o (k )) solution of the state equation has the form (( A − BD + C)' o ) k x'o  o x'o (k ) =  0  

and is contained in ν ∗ ( A ' , B' , C' , D' ) = S' cl D. Proof. The proof follows immediately from Proposition 3.7 when applied


249

to S( A ' , B' , C' , D' ) and from appropriate relations following from the change of coordinates. The details are left to the reader as an exercise. ◊

7.2 Invariant Subspaces Consider a proper system S(A,B,C,D) (2.1) and the associated system S( A, B , C, Λ ) (5.19) with matrices B and C and vectors u and y decomposed as in (5.5). Proposition 7.2. For systems S(A,B,C,D) (2.1) and S( A, B , C, Λ ) (5.19) the following relation takes place ν ∗ ( A, B, C, D) = ν ∗ ( A, B , C , Λ ) .

Proof. Recall that, according to (5.3) and (5.4), we have B = BV , C = U T C , u = V T u and y = U T y . If F is a state feedback matrix for

S(A,B,C,D), then F = V T F is such a matrix for S( A, B , C, Λ ) and B F = BF . Moreover, A + B F = A + BF and C + ΛF = U T (C + DF) . Hence we have the equivalence: a subspace X is (A+BF)-invariant and contained in Ker (C + DF) if and only if X is ( A + B F ) -invariant and

contained in Ker ( C + Λ F ) . ◊ The output-zeroing inputs for S(A,B,C,D) and S( A, B , C, Λ ) are interrelated as follows. Proposition 7.3. A pair (x o , u o (k )) is an output-zeroing input for S(A,B,C,D) (2.1) if and only if (x o , u o (k )) , where u o (k ) = V T u o (k ) , is an output-zeroing input for S( A, B , C, Λ ) (5.19). Moreover, (x o , u o (k )) and (x o , u o (k )) yield the same solution. Proof. The obvious proof is left to the reader as an exercise. ◊ As we already know from Chapter 5 (see Proposition 5.7) if in the system S( A, B , C, Λ ) (5.19) is B m − p ≠ 0 and Cr − p ≠ 0 , then an appropriate sequence of transformations of S(A,B,C,D) into the strictly proper

250


system S( A − B p D −p1C p , B m− p , Cr − p ) preserves the set of invariant zeros, the zero polynomial and, consequently, the set of Smith zeros. As we show below, an analogous property takes place for maximal outputnulling controlled invariant subspaces. Lemma 7.2. Consider a system S(A,B,C,D) (2.1) and suppose that in the associated system S( A, B , C, Λ) (5.19) is B m − p ≠ 0 and Cr − p ≠ 0 . Then the sequence of transformations S( A, B, C, D) → S( A, B , C, Λ ) → S( A − B p D −p1C p , B m − p , Cr − p )

has the property ν ∗ ( A, B, C, D) = ν ∗ ( A, B , C , Λ) = ν ∗ ( A − B p D −p1C p , B m− p , C r − p ) .

Proof. Of course, by virtue of Proposition 7.2, we only need to show that

ν ∗ ( A, B , C , Λ ) = ν ∗ ( A − B p D −p1C p , B m− p , C r − p ) . Consider first the

system S( A − B p D −p1C p , B m− p , Cr − p ) and suppose that a subspace X is ( A − B p D −p1C p , B m− p ) -invariant and contained in Ker Cr − p , i.e., there

exists a (m-p)xn real matrix Fm− p such that (i)

(( A − B p D −p1C p ) + B m − p Fm − p )(X ) ⊆ X ⊆ Ker Cr − p .

 Fp  However, setting F p := − D −p1C p and taking F =   , we can write  Fm − p  the left-hand side of (i) as ( A + B F )(X ) ⊆ X . On the other hand, C + Λ F can be evaluated as  C p  D p C + ΛF =  +  Cr − p   0

0  F p   C p + D p F p   0  = =  . 0  Fm− p   C r − p   Cr − p 

This means that Ker ( C + Λ F ) = Ker Cr − p . In this way we have shown the implication: if X is ( A − B p D −p1C p , B m− p ) -invariant and contained in


251

Ker Cr − p , then X satisfies

(ii)

( A + B F )(X ) ⊆ X ⊆ Ker ( C + ΛF ) .

In order to prove the converse, suppose that for some mxn real matrix  Fp  F=  a subspace X satisfies (ii). This means that for each x ∈ X  Fm − p  (iii)

( A + B F )x = ( A + B p F p + B m − p Fm − p )x ∈ X

and (iv)

C p + D p F p  0  ( C + ΛF )x =  x =   .   Cr − p 0 

From (iv) we obtain

Cr − p x = 0

and

( C p + D p F p )x = 0 , i.e.,

F p x = − D −p1C p x for each x ∈ X . Now, (iii) can be written as

(v)

( A + B F )x = ( A + B p F p + B m − p Fm − p )x = ( A − B p D −p1 C p )x + B m − p Fm − p x ∈ X .

Hence we have obtained (( A − B p D −p1C p ) + B m − p Fm − p )(X) ⊆ X ⊆ Ker Cr − p

and the proof of the desired implication is completed. Thus we have shown the following equivalence: a subspace X is

( A − B p D −p1C p , B m− p ) -invariant and contained in Ker Cr − p if and only

if X is an output-nulling controlled invariant subspace for S( A, B , C, Λ ) . This leads immediately to the desired relation ν ∗ ( A, B , C, Λ ) = ν ∗ ( A − B p D −p1C p , B m − p , Cr − p ) . ◊

252


Proposition 7.4. Consider systems S(A,B,C,D) (2.1) and S( A, B , C, Λ ) (5.19) and let ν ∗ ( A, B, C, D) stand for the maximal output-nulling controlled invariant subspace for S(A,B,C,D). Then: If D has full column rank m, then ν ∗ ( A, B, C, D) equals the unob-

1.

servable subspace for S( A − BD + C, B, (I r − DD + )C, D) , i.e., ν ∗ ( A, B, C, D) =

2.

n −1 l =0

Ker{(I r − DD + )C( A − BD + C) l } .

If D has full row rank r and m > r , then 2a. If B m − r = 0 , then ν ∗ ( A, B, C, D) = R n . 2b. If B m − r ≠ 0 , then ν ∗ ( A, B, C, D) = R n .

3.

If 0 < rank D = p < min{m, r} , then 3a.

3b.

Cr − p = 0

3a1.

If Cr − p = 0 and B m − p = 0 , then ν ∗ ( A, B, C, D) = R n .

3a2.

If Cr − p = 0 and B m − p ≠ 0 , then ν ∗ ( A, B, C, D) = R n .

Cr − p ≠ 0

If Cr − p ≠ 0 and B m − p = 0 , then ν ∗ ( A, B, C, D) equals the unobservable subspace for the system 3b1.

 0  D p  S( A − B p D −p1C p , B p ,  ,   ) , i.e.,  Cr − p   0  ν ∗ ( A, B, C, D) =

3b2.

n−1 l =0

Ker ( Cr − p ( A − B p D −p1C p ) l ) .

If Cr − p ≠ 0 and B m − p ≠ 0 , then

ν ∗ ( A, B, C, D) = ν ∗ ( A − B p D −p1C p , B m − p , Cr − p ) .


253

Proof. 1. The claim has been proved in Lemma 7.1. 2a and 2b. Consider the algorithm for computing ν ∗ ( A, B, C, D) presented at the beginning of this chapter. Since D has full row rank, for each x ∈ R n there exists an u ∈ R m such that Cx + Du = 0 . This means that

each subspace X i in the aforementioned algorithm satisfies X i = R n and,

consequently, ν ∗ ( A, B, C, D) = R n .

3a1 and 3a2. By virtue of Proposition 7.2, instead of S(A,B,C,D) we can consider the system S( A, B , C, Λ ) . For this last system we repeat the argumentation used in the proof of points 2a and 2b above. The equation C x + Λ u = 0 takes the form C p  D p  0 0   x +  u p +  um − p =   .  0  0 0   0 

Now, it is easily seen that for each x ∈ R n there exists a vector  up  m u=  ∈ R such that the pair ( x, u ) satisfies the above equation. u − m p   Thus ν ∗ ( A, B, C, D) = ν ∗ ( A, B , C , Λ ) = R n . D  3b1. We prove first that ν ∗ ( A, B , C , Λ) = ν ∗ ( A, B p , C , p  ) . Then,  0  D  using Lemma 7.1 (referred to S( A, B p , C , p  ) ) and (5.31), we shall  0  obtain the equality D  n−1 ν ∗ ( A, B p , C , p  ) = h Ker ( Cr − p ( A − B p D −p1C p ) l ) .  0  l =0

The remaining part of the proof will follow from Proposition 7.2. D  In order to show that ν ∗ ( A, B , C , Λ) = ν ∗ ( A, B p , C , p  ) , we observe  0   Fp  first that at F =   we have  Fm − p 

254


C p + D p F p  D p  A + B F = A + B p F p and C + Λ F =  =C+ Fp .  C r − p   0   Fp  Suppose now that a subspace X satisfies at some F =   the condi Fm − p 

tion ( A + B F )(X ) ⊆ X ⊆ Ker ( C + ΛF ) . Then, at the same matrix F p as D  in F , X satisfies also ( A + B p F p )(X ) ⊆ X ⊆ Ker ( C +  p  F p ) . Con 0  versely, if a subspace X satisfies, at some matrix F p , the condition  Fp  D  ( A + B p F p )(X ) ⊆ X ⊆ Ker ( C +  p  F p ) , then, taking F =   at  Fm − p   0  the same F p and an arbitrary Fm− p , one can observe that X satisfies also ( A + B F )(X ) ⊆ X ⊆ Ker ( C + ΛF ) . In this way we have shown that X is

an output-nulling controlled invariant subspace for S( A, B , C, Λ ) if and D  only if X is such a subspace for S( A, B p , C , p  ) . This implies the  0  D  identity ν ∗ ( A, B , C , Λ) = ν ∗ ( A, B p , C , p  ) .  0 

3b2. The relation in question has been proved in Lemma 7.2. ◊ Corollary 7.2. The recursive process of computing invariant zeros of a proper system S(A,B,C,D) (2.1) following from the point 3b2 of Procedure 5.2 conserves not only invariant zeros and the associated state-zero directions but also the subspace ν ∗ ( A, B, C, D) . Proof. Recall that at the first step of the process we use Proposition 5.7 which enables us to pass from S(A,B,C,D) (2.1) through S( A, B , C, Λ ) (5.19) to a strictly proper system S( A − B p D −p1C p , B m− p , Cr − p ) (see Fig. 5.5). Next, we employ recursively Proposition 5.3. Thus the claim follows immediately from Lemma 7.2 and Corollary 6.1. ◊ In order to show that invariant zeros in S(A,B,C,D) exist if and only if ν ( A, B, C, D) is nontrivial, we proceed in a similar manner as in the ∗


255

strictly proper case (see Chap. 6, Sect. 6.3). To this end, we need first the following result. Lemma 7.3. Let λ ∈ C be an invariant zero of S(A,B,C,D) (2.1), i.e., let a triple λ, x o ≠ 0, g satisfy λx o − Ax o = Bg and Cx o + Dg = 0 . Then Re x o ∈ ν ∗ ( A, B, C, D) and Im x o ∈ ν ∗ ( A, B, C, D) .

Proof. Of course, since at ν ∗ ( A, B, C, D) = R n the assertion of the lemma is apparent, it is sufficient to discuss merely the cases 1, 3b1 and 3b2 of Proposition 7.4. 1. If D has full column rank, then, as we know from the proof of Proposition 4.6, the conditions λx o − Ax o = Bg and Cx o + Dg = 0 imply λx o − ( A − BD + C) x o = 0 and (I r − DD + )Cx o = 0 (i.e., λ is an o.d. zero of S( A − BD + C, B, (I r − DD + )C, D) – see Proposition 5.2). Pre-

multiplying λx o − ( A − BD + C) x o = 0 successively by (I r − DD + )C (I r − DD + )C( A − BD + C) .

. (I r − DD + )C( A − BD + C) n −2

and taking into account that (I r − DD + )Cx o = 0 , we obtain the desired relations (cf., Proposition 7.4 (1)). 3b1. If in S( A, B , C, Λ ) is Cr − p ≠ 0 and B m − p = 0 , then, as we know from Proposition 5.10, λ is an invariant zero of S(A,B,C,D) if and  0  D p  only if λ is an o.d. zero of S( A − B p D −p1C p , B p ,  ,  ) . Hence  Cr − p   0  (cf., proof of Proposition 5.10), if λx o − Ax o = Bg and Cx o + Dg = 0 , then λx o − ( A − B p D −p1C p )x o = 0 and Cr − p x o = 0 . Next, we obtain in the same manner as in the point 1 above the following relations Re x o ∈

n −1 l =0

Ker ( Cr − p ( A − B p D −p1C p ) l )

256


and Im x o ∈

n −1 l =0

Ker ( Cr − p ( A − B p D −p1C p ) l )

which end the proof of 3b1 (cf., Proposition 7.4 (3b1)). 3b2. In this case (i.e., Cr − p ≠ 0 and B m − p ≠ 0 ) the claim follows from Proposition 5.7, Lemma 6.9 and Lemma 7.2. In fact, if a triple λ, x o ≠ 0, g satisfies (2.4) for S(A,B,C,D), then, as we know from the

proof of Proposition 5.7, the triple λ, x o ≠ 0, g m − p satisfies (2.4) for S( A − B p D −p1C p , B m− p , Cr − p ) . Now, from Lemma 6.9 (when applied to S( A − B p D −p1C p , B m− p , Cr − p ) ) and from Lemma 7.2 it follows that Re x o and Im x o belong to ν ∗ ( A, B, C, D) = ν ∗ ( A − B p D −p1C p , B m − p , Cr − p ) . ◊

Now, the implication ZI ≠ ∅ ⇒ ν ∗ ( A, B, C, D) ≠ {0} follows immediately from Lemma 7.3. In order to prove the converse, we show below that the equivalent implication ZI = ∅ ⇒ ν ∗ ( A, B, C, D) = {0} holds true. Of course, it is sufficient to discuss only those points of Procedure 5.2 (see

Sect. 5.4) for which the condition Z I = ∅ can be fulfilled, i.e., the points 1, 3b1 and 3b2. 1.

From Proposition 5.2 we know that invariant zeros of S(A,B,C,D)

are the o.d. zeros of S( A − BD + C, B, (I r − DD + )C, D) . On the other hand, by virtue of Lemma 7.1, ν ∗ ( A, B, C, D) equals the unobservable subspace for this last system. Hence, if the set of invariant zeros of S(A,B,C,D) is empty, S( A − BD + C, B, (I r − DD + )C, D) is observable (i.e., S cl D = {0} ) and, consequently, ν ∗ ( A, B, C, D) = {0} .

3b1. From Proposition 5.10 we know that invariant zeros of the system  0  D p  S(A,B,C,D) are the o.d. zeros of S( A − B p D −p1C p , B p ,  ,  ) .  Cr − p   0 

By virtue of Proposition 7.4 (3b1), the unobservable subspace for this last


257

system equals ν ∗ ( A, B, C, D) . Hence, if the system S(A,B,C,D) has no invariant zeros, ν ∗ ( A, B, C, D) is trivial. 3b2. Since in S(A,B,C,D) we assume ZI = ∅ , at the last step of the recursive process following from the point 3b2 of Procedure 5.2 (see Chap. 5, Sect. 5.4) we obtain a strictly proper system (with the first nonzero Markov parameter of full column rank) without invariant zeros. However, as we know from the discussion preceding Proposition 6.4, or simply from Proposition 6.4, the subspace ν ∗ for such system must be trivial. Now, because ν ∗ is conserved along the process (see Corollary 7.2), hence ν ∗ ( A, B, C, D) = {0} .

In this way we have proved the following. Proposition 7.5. Consider a proper system S(A,B,C,D) (2.1) and let ZI denote the set of its invariant zeros and let ν ∗ ( A, B, C, D) stand for the maximal output-nulling controlled invariant subspace for S(A,B,C,D). Then ZI = ∅

⇔

ν ∗ ( A, B, C, D) = {0} . ◊

Remark 7.1. It is easy to note (see Example 7.1 below) that Proposition 7.5

does not hold if the condition ZI = ∅ is replaced with Z S = ∅ . ◊

Example 7.1. Consider a reachable and observable system S(A,B,C,D) (2.1) with the matrices − 2 −1 / 2 − 1 1 0 0 − 2 0 − 1 1 0 1   . A= 0 0 0  B = 0 1 0 C =  D=  0 0 0 0 1 1    0 0 0 1 1 0 

The system is an irreducible realization of the transfer-function matrix z +1  z z + 2 0 z + 2 G (z) =  z +1 1  .  0  z  z2 

258


It is straightforward to verify that S(A,B,C,D) has no Smith zeros (i.e., G(z) has no Smith transmission zeros). In SVD (5.18) of D ( rank D = p = 1 , D p = 2 ) we can take    2 0 0 1 0  T U= , V = , Λ =  0 1    0 0 0    

1 2 0 1 2

1   2  1 0 . 1 0 −  2 

0

In S( A, B , C, Λ ) (5.19) we have B = BV , C = U T C . Then, by virtue of Procedure 5.2 (3b2), we consider a strictly proper system S( A ' , B' , C' ) (see also Proposition 5.7) with the matrices 1    0 − 1 −1 / 2 −1 / 2 2     −1 0 , 0 0  , B ' = B m − p = 1 A' = A − B p D p C p =  0 0 − 1   1 1 1 / 2   2   C' = Cr − p = [0 1 1] .  1  The first nonzero Markov parameter of S( A ' , B' , C' ) is C'B' = 1 −  2  (i.e., ν' = 0, p' = r ' = 1 ). For SVD of C'B' we can write

C' B' = U '[M ' r '

0]V ' T ,

where

M' r ' =

3 2

,

U ' = −1 ,

 2 − V' =  3  1   3

1   3. 2  3

With S( A ' , B' , C' ) we associate a system S( A ' , B ' , C ' ) (5.2) with the matrices


 1   6  2 B ' = B' V ' = −  3 − 1  6

1   3  1  , 3  1 −  3 

259

C ' = U 'T C' = [0 − 1 − 1] .

Next (see the discussion preceding Corollary 5.1), we pass to a system S( A ' ' , B' ' , C' ' ) , where A ' ' = K 'ν ' A ' , B' ' = B ' m'− r ' , C' ' = C ' and K ' ν ' = I 3 − B ' r ' M ' −r '1 C ' ( A ' ) ν ' . After simple calculations we obtain 1/ 3  1 1 / 3  K ' ν ' = 0 1 / 3 − 2 / 3 0 − 1 / 3 2 / 3 

and  1/ 3  − 2 / 3 − 1 / 6 − 1 / 3     A' ' = − 2 / 3 − 2 / 3 − 1 / 3 , B' ' =  1 / 3  , C' ' = [0 − 1 − 1] . − 1 / 3   2 / 3 2/3 1 / 3   

Since C' ' B' ' = 0 and C' ' ( A ' ' ) l = 0 , l = 1, 2, ... , all Markov parameters of S( A ' ' , B' ' , C' ' ) are zero. Hence the transfer-function matrix of this system equals zero identically and, via Proposition 2.6 and Corollary 5.3 (ii), S(A,B,C,D) is degenerate. On the other hand, by virtue of Proposition 6.3 (4) (when applied to S( A ' ' , B' ' , C' ' ) ) and Corollary 7.2, ν ∗ ( A, B, C, D) equals the unobservable subspace for S( A' ' , B' ' , C' ' ) . However, as it is easy to verify, the latter is of the form {x ∈ R 3 : x 2 + x 3 = 0} .

In this way we have shown that the set of Smith zeros of S(A,B,C,D) is empty, while the maximal output-nulling controlled invariant subspace of S(A,B,C,D) is nontrivial. ◊

260


7.3 SVD and the Output-Zeroing Problem In this section we shall show that in any proper system S(A,B,C,D) (2.1) if (x o , u o (k )) is an output-zeroing input and x o (k ) stands for the corre-

sponding solution, then x o (k ) ∈ ν ∗ ( A, B, C, D) for all k ∈ N . Furthermore, for each x o ∈ ν ∗ ( A, B, C, D) there exists an output-zeroing input such that the corresponding solution passes through x o . Remark 7.2. As we already know, the above statements hold in the case when D has full row rank (see Propositions 3.6 and 7.4 (2a and 2b) as well as Corollary 7.1) or when D has full column rank (see Propositions 3.7 and 7.4 (1)). ◊ In order to discuss relationships between output-zeroing inputs for

S(A,B,C,D) (2.1) and S( A − B p D −p1C p ,B m − p ,Cr − p ) (at B m − p ≠ 0 and Cr − p ≠ 0 ), recall first that S( A − B p D −p1C p ,B m − p ,Cr − p ) (see Fig. 5.5)

is obtained from S( A, B , C, Λ ) (5.19) by introducing an appropriate state feedback matrix as well as that relationships between output-zeroing inputs for S(A,B,C,D) and S( A, B , C, Λ ) are given in Proposition 7.3. Hence we can focus our attention on systems S( A − B p D −p1C p ,B m − p ,Cr − p ) and S( A, B , C, Λ ) .

Consider the output-zeroing problem for the system S( A, B , C, Λ ) (5.19) of the form x(k + 1) = Ax( k ) + B p u p (k ) + B m − p u m− p (k )  y p (k )   C p  D p  0 u p (k ) +   u m− p (k ) =  x( k ) +   y   0  0  r − p (k )  C r − p 

(7.1)

and for the system S( A − B p D −p1C p ,B m − p ,Cr − p ) which is written in the form x(k + 1) = ( A − B p D −p1C p )x(k ) + B m − p u m − p (k ) y r − p (k ) = Cr − p x(k ).

(7.2)


261

 u o, p ( k )  Suppose that (x o , u o (k )) , where u o (k ) =   , is an output u o, m − p ( k ) 

zeroing input for (7.1) and yields a solution x o (k ) . This implies that u o (k ) and x o (k ) satisfy the state equation of (7.1) and yield  y p ( k )  0  y  =   for all k ∈ N . From this last relation we obtain  r − p ( k )  0 

u o, p (k ) = −D −p1C p x o (k ) .

(7.3)

Introducing (7.3) into (7.1), we observe that u o, m − p (k ) and x o (k ) satisfy the state equation in (7.2) and yield Cr − p x o (k ) = 0 for all k ∈ N . Thus

we

have

proved

that

if

a

pair

(x o , u o (k )) ,

where

 u o, p ( k )  uo (k ) =   , is an output-zeroing input for S( A, B , C, Λ ) in  u o, m − p ( k ) 

(7.1) and gives a solution x o (k ) , then the pair (x o , u o, m − p (k )) is an output-zeroing input for S( A − B p D −p1C p ,B m − p ,Cr − p ) in (7.2) and produces the same solution (of the state equation in (7.2)). Moreover, x o (k ) is uniquely determined by x o and u o, m − p (k ) as x o ( k ) = ( A − B p D −p1C p ) k x o k −1

+ ∑ ( A − B p D −p1C p ) k −1− l B m − p u o, m − p (l ).

(7.4)

l =0

Naturally, from (7.4) and (7.3) it follows that u o, p (k ) is also uniquely determined by x o and u o, m − p (k ) . Suppose now that a pair (x o , u o, m − p (k )) is an output-zeroing input for (7.2) and x o (k ) is the corresponding solution. Then x o (k ) has the form (7.4) (i.e., is uniquely determined by x o and u o, m − p (k ) ), and Cr − p x o (k ) = 0 for all k ∈ N . Using u o, m − p (k ) and x o (k ) , we define

262


 − D −p1C p x o (k ) for S( A, B , C, Λ ) in (7.1) the input u o (k ) :=   . Now, it is  u o,m− p (k ) 

not difficult to verify that u o (k ) and x o (k ) satisfy the state equation of  y p ( k )  0  (7.1) and yield   =   for all k ∈ N . Thus from the above and  y r − p ( k )  0 

from

the

uniqueness of solutions it follows that the pair − D −p1C p x o (k ) (x o , u o (k ) =   ) constitutes an output-zeroing input for  u o,m− p (k ) 

(7.1) and produces the solution x o (k ) .

Employing the above discussion and Proposition 7.3, we can formulate now the following characterization of the relationship between outputzeroing inputs for the system S(A,B,C,D) (2.1) and for the strictly proper system S( A − B p D −p1C p ,B m − p ,Cr − p ) (7.2).

Lemma 7.4. Consider a proper system S(A,B,C,D) (2.1) and in the associated system S( A, B , C, Λ ) (5.19) let B m − p ≠ 0 and let Cr − p ≠ 0 . Then (x o , u o (k )) is an output-zeroing input for S(A,B,C,D) if and only if  u o, p ( k )  T (x o , u o, m − p (k )) , where   = V u o ( k ) , is an output-zeroing ( k ) u m − p o ,  

input for S( A − B p D −p1C p ,B m − p ,Cr − p ) (7.2). Moreover, both these output-zeroing inputs yield the same solution x o (k ) which has the form (7.4) and is uniquely determined by x o and u o, m − p (k ) . Furthermore, u o, p (k ) is also uniquely determined by x o and u o, m − p (k ) via (7.3) and

(7.4). ◊ We shall discuss now the output-zeroing problem for the case when in the system S( A, B , C, Λ ) (7.1) is Cr − p = 0 (recall that in this case we have ν ∗ ( A, B, C, D) = R n – see Proposition 7.4 (3a1 and 3a2)). Lemma 7.5. Consider a proper system S(A,B,C,D) (2.1) and in the associated system S( A, B , C, Λ ) (5.19) let Cr − p = 0 . Then:


(i)

263

In S( A, B , C, Λ ) let Cr − p = 0 and let B m − p ≠ 0 . Moreover, let

x o ∈ R n and let u o, m − p (.): N → R m - p be an arbitrarily fixed sequence.  u o, p ( k )  Then the pair (x o , u o (k )) , where u o (k ) = V   and u o, p (k ) is  u o, m − p ( k )  defined as

u o, p (k ) = −D −p1C p [( A − B p D −p1C p ) k x o +

(7.5)

k −1

+ ∑ ( A − B p D −p1 C p ) k −1−l B m − p u o,m− p (l ) , l =0

is an output-zeroing input for S(A,B,C,D) and yields the solution x o ( k ) = ( A − B p D −p1C p ) k x o k −1

+ ∑

l =0

(7.6)

( A − B p D −p1C p ) k −1− l B m − p u o, m − p (l ).

Conversely, if (x o , u o (k )) is an output-zeroing input for S(A,B,C,D) and

yields

a

solution

x o (k ) ,

then

u o (k )

has

the

form

 u o, p ( k )  u o (k ) = V   , where u o, m − p (k ) is determined uniquely by  u o, m − p ( k ) 

u o (k ) and V whereas u o, p (k ) and x o (k ) have respectively the form (7.5) and (7.6).

(ii)

In S( A, B , C, Λ ) let Cr − p = 0 and let B m − p = 0 . Moreover, let

x o ∈ R n and let u o, m − p (.): N → R m - p be an arbitrarily fixed sequence.

Furthermore, let u o, p (.): N → R p be defined as u o, p ( k ) = − D −p1C p ( A − B p D −p1C p ) k x o .

(7.7)

 u o, p ( k )  Then the pair (x o , u o (k )) , where u o (k ) = V   , constitutes an  u o, m − p ( k )  output-zeroing input for S(A,B,C,D) and yields the solution

264


x o (k ) = ( A − B p D −p1 C p ) k x o .

(7.8)

Conversely, if (x o , u o (k )) is an output-zeroing input for S(A,B,C,D) and

yields

a

solution

x o (k ) ,

then

u o (k )

has

the

form

 u o, p ( k )  u o (k ) = V   , where u o, m − p (k ) is determined uniquely by  u o, m − p ( k ) 

u o (k ) and V whereas u o, p (k ) and x o (k ) have respectively the form (7.7) and (7.8).

Proof. (i) By virtue of Proposition 7.3, it is sufficient to show that any pair (x o , u o (k )) , where x o ∈ R n and u o, m − p (k ) is arbitrary whereas u o, p (k ) is determined as in (7.5), forms an output-zeroing input for S( A, B , C, Λ ) in (5.19) and yields a solution of the form (7.6). However, S( A, B , C, Λ ) can be written as x(k + 1) = Ax(k ) + B p u p (k ) + B m − p u m − p (k ) y p ( k ) = C p x( k ) + D p u p ( k )

(7.9)

y r − p ( k ) = 0.

Moreover, as it is easy to check, u o, m − p (k ) and u o, p (k ) in (7.5) as well as x o (k ) in (7.6) satisfy the state equation in (7.9). Furthermore, u o, p (k ) and x o (k ) introduced into the second equation of (7.9) yield y p (k ) = 0 for all k ∈ N .

In order to prove the second claim of (i), suppose that (x o , u o (k )) is an

output-zeroing input for S(A,B,C,D) and yields a solution x o (k ) . Then, by virtue of Proposition 7.3, the pair (x o , u o (k )) is an output-zeroing input for S( A, B , C, Λ ) in (7.9) and gives the same solution. Since  u o, p ( k )  T  = V u o ( k ) , and u o (k ) is known, hence u o, m − p (k ) is u ( k )  o, m − p 

uniquely determined by u o (k ) and V . On the other hand, for x o (k ) ,


265

u o, p (k ) and u o, m − p (k ) the following identities hold

x o ( k + 1) = Ax o (k ) + B p uo, p (k ) + B m − p uo, m − p (k ) 0 = C p x o ( k ) + D p u o, p ( k )

(7.10)

y r − p ( k ) = 0.

From (7.10) we obtain x o ( k + 1) = ( A − B p D −p1C p )x o (k ) + B m − p u o, m − p (k ) u o, p ( k ) =

−D −p1C p x o (k ).

(7.11)

Since the initial state x o for x o (k ) is assumed to be known, the first identity in (7.11) means that x o (k ) has the form (7.6). Now, (7.5) follows from the second identity in (7.11) and from (7.6). In order to complete the  u o, p ( k )  proof, we should take u o (k ) = V  .  u o, m − p ( k )  (ii) The proof follows the same lines as the proof of (i). The details are left to the reader as an exercise. ◊ Remark 7.3. Recall that at Cr − p = 0 and B m − p ≠ 0 the original system S(A,B,C,D) is degenerate (see Proposition 5.9 (i)). On the other hand, Lemma 7.5 (i) tells us that arbitrarily fixed x o ∈ R n and u o, m − p (k ) determine uniquely an output-zeroing input and the corresponding solution for S(A,B,C,D). When Cr − p = 0 and B m − p = 0 , S(A,B,C,D) is nondegenerate (see Proposition 5.9 (ii)) and its invariant zeros coincide with eigenvalues of A − B p D −p1C p . On the other hand, Lemma 7.5 (ii) tells us that although

the component u o, m − p (k ) affects u o (k ) (in the output-zeroing input (x o , u o (k )) ), it does not affect the corresponding solution (which depends

merely upon the initial state x o ∈ R n and eigenvalues of A − B p D −p1C p ).

More precisely, we can observe that any pair of the form (x o = 0, u o (k )) , 0   where u o (k ) = V   , is an output-zeroing input for S(A,B,C,D)  u o, m − p ( k )  and produces the trivial solution of the state equation. ◊

266


Lemma 7.6. Consider a proper system S(A,B,C,D) (2.1) and in the associated system S( A, B , C, Λ ) (5.19) let Cr − p ≠ 0 and let B m − p = 0 . Suppose

that

xo ∈

n −1 l =0

Ker( Cr − p ( A − B p D −p1C p ) l )

and

u o, m − p (.): N → R m - p is an arbitrarily fixed sequence. Moreover, let u o, p (.): N → R p be defined as in (7.7). Then the pair (x o , u o (k )) , where  u o, p ( k )  u o (k ) = V   , constitutes an output-zeroing input for  u o, m − p ( k )  S(A,B,C,D) and yields a solution of the form (7.8) which is contained in n −1

l =0

Ker( Cr − p ( A − B p D −p1C p ) l ) .

Conversely, if (x o , u o (k )) is an output-zeroing input for S(A,B,C,D) and yields a solution x o (k ) , then x o ∈

n −1

l =0

Ker( Cr − p ( A − B p D −p1C p ) l )

 u o, p ( k )  and u o (k ) has the form u o (k ) = V   , where u o, m − p (k ) is  u o, m − p ( k ) 

determined uniquely by u o (k ) and V whereas u o, p (k ) and x o (k ) have respectively the form (7.7) and (7.8). Proof. The proof is left to the reader as an exercise. ◊ We end the discussion with the following. Proposition 7.6. Consider a proper system S(A,B,C,D) (2.1) and let

(x o , u o (k )) be an output-zeroing input and let x o (k ) denote the corre-

sponding solution. Then x o (k ) ∈ ν ∗ ( A, B, C, D) for all k ∈ N . Moreover, for each x o ∈ ν ∗ ( A, B, C, D) there exists an output-zeroing input such that the corresponding solution passes through x o .

Proof. In the proof we have to discuss all the cases appearing in Procedure 5.2 (see Sect. 5.4). 1. If D has full column rank, the claims of the proposition follow from Propositions 3.7 and 7.4 (1).


267

2a and 2b. If D has full row rank and m > r , the assertions of the proposition follow from Propositions 3.6 and 7.4 (2a and 2b). 3a1 and 3a2. If in S( A, B , C, Λ ) (5.19) is Cr − p = 0 , the assertions follow immediately from Lemma 7.5 and from Proposition 7.4 (3a1 and 3a2). 3b1. If in S( A, B , C, Λ ) (5.19) is Cr − p ≠ 0 and B m − p = 0 , the claims of the proposition follow immediately from Proposition 7.4 (3b1) and Lemma 7.6. 3b2. If in S( A, B , C, Λ ) (5.19) is Cr − p ≠ 0 and B m − p ≠ 0 , then to S(A,B,C,D) we apply the recursive process based on Proposition 5.7 (see also Corollary 5.3). Moreover, Corollary 7.2 says that ν ∗ ( A, B, C, D) is conserved along the process. On the other hand, Lemmas 6.10 and 7.4 tell

us that x o (k ) is also conserved along the process as well as they display the relationship between output-zeroing inputs for a system at a given step and for the system at the next step of the process. The remaining part of the proof follows closely the proof of point 3b2 in Proposition 6.5. The details of the proof are left to the reader as an exercise. ◊

7.4 Zeros, Output-Nulling Subspaces and Zero Dynamics Proposition 7.7. Let a proper system S(A,B,C,D) (2.1) be nondegenerate. Then its Smith and invariant zeros are the same objects (including multiplicities). Moreover, the degree of the zero polynomial of the system is equal to the dimension of ν ∗ ( A, B, C, D) and the zero dynamics, in appropriate coordinates, have the form ξ (k + 1) = Nξ (k ) , where det(zI − N) is equal to the zero polynomial of the system, and ξ (k ) ∈ ν ∗ ( A, B, C, D) for all k ∈ N (when ν ∗ ( A, B, C, D) is taken in the same coordinates). Proof. The proof is left to the reader as an exercise. ◊ Proposition 7.8. Let a proper system S(A,B,C,D) (2.1) be degenerate. Then the degree of the zero polynomial of S(A,B,C,D) is strictly less than the dimension of ν ∗ ( A, B, C, D) . Moreover, the zero dynamics depend essentially upon control vector. Proof. The proof is left to the reader as an exercise. ◊

268


The above results are illustrated in Fig. 7.1.

Fig. 7.1.

Example 7.2. In S(A,B,C,D) (2.1) let 0 −1  − 1 / 6 1 1    A= 0 − 1 / 6 1 / 2  , B = 1 0 , C = [0 1 2], D = [1 0],  0 0 0 0 − 1 / 2  x1  x =  x 2  ,  x 3 

u  u =  1 . u 2 

In SVD of D (cf., (5.18)) we take U = 1, V = I 2 , Λ = D, D p = 1 (i.e., p = r = 1 ). Then, according to (5.5), we have

7.5 Exercises

269

1  1    B r = 1 , B m − r = 0 , C = Cr = C and u = u , i.e., u r = u1 , u m − r = u 2 . 0 0

By virtue of Procedure 5.2 (2b), S(A,B,C,D) is degenerate. Moreover, the zero polynomial of S(A,B,C,D) equals the zero polynomial of the pencil  1 z + 6  zI − ( A − B r D r−1Cr ) − B m− r =  0  0  

[

]

1 z+ 0

7 6

3 3 2 z+

1 2

 − 1  0. 0  

Now, simple verification shows that the zero polynomial of S(A,B,C,D) 1 7 equals ψ (z) = (z + )(z + ) . On the other hand, by virtue of Proposition 2 6 ∗ 7.4 (2b), ν ( A, B, C, D) = R 3 . Finally, by virtue of Proposition 3.6, the zero dynamics have the form x(k + 1) = ( A − BD + C)x( k ) + Bu h (k ) ,

where u h (k ) ∈ Ker D for all k ∈ N . Hence they can be written as 1 x1 (k + 1) = − x1 ( k ) − x 2 ( k ) − 3x 3 ( k ) + u 2 (k ) 6 3 7 x 2 (k + 1) = − x 2 (k ) − x 3 (k ) 2 6 1 x 3 (k + 1) = − x 3 (k ). ◊ 2

7.5 Exercises 7.5.1. Complete the proof of Proposition 7.1. 7.5.2. Prove Proposition 7.3. 7.5.3. Complete the proof of Lemma 7.5 (ii).

270


7.5.4. Prove Lemma 7.6. 7.5.5. Complete the proof of Proposition 7.6. 7.5.6. Prove the following result. In a proper system S(A,B,C,D) (2.1) let Z I = ∅ . Then (x o , u o (k )) is an output-zeroing input if and only if x o = 0

and u o (k ) ∈ KerB ∩ KerD for all k ∈ N . Moreover, the corresponding solution satisfies x o ( k ) = 0 for all k ∈ N . For an illustration of this result see Example 2.3. Note also that the

above result is not valid when we replace the assumption ZI = ∅ with

Z S = ∅ (as a suitable example take into account the system of Example 7.1 which is degenerate and has no Smith zeros and Ker B ∩ Ker D = {0} ).

Hint. Take into account Propositions 7.5 and 7.6. 7.5.7. Consider a system S(A,B,C,D) (2.1) in which D has full column rank. Let S cl D =

n −1 l =0

Ker{(I r − DD + )C( A − BD + C) l } .

cl Show that ( A − BD + C)(Scl D ) ⊆ SD .

7.5.8. Prove Propositions 7.7 and 7.8.

8 Singular Systems

In this chapter we give an outline of the algebraic characterization of invariant and decoupling zeros of a generalized state space model described by the matrix 5-tuple (E,A,B,C,D), where matrix E is singular but the pencil zE − A is regular (i.e., det(zE − A) ≠ 0 ). The proposed definition of invariant zeros of singular discrete-time systems, based in a natural way on the notions of state-zero and input-zero directions introduced in [40], is strictly linked with the output zeroing problem and for this reason these zeros have a clear dynamical interpretation. This definition constitutes an immediate extension to singular systems of the notion of invariant zeros for standard linear systems that has been introduced in Chapter 2. We present below two different approaches to the question of characterizing and computing multivariable zeros of a discrete-time system S(E,A,B,C,D). The first one is based on singular value decomposition of matrix E and has been borrowed from [42]. This approach enables us to reduce this question to the task of searching multivariable zeros of an appropriate standard linear system. The second approach employs the notions of fundamental matrices and Markov parameters.

8.1 Definitions of Zeros Consider a linear discrete-time singular system S(E,A,B,C,D) of the form Ex(k + 1) = Ax(k ) + Bu (k ) , k∈N , y (k ) = Cx(k ) + Du(k )

(8.1)

where x(k ) ∈ R n , y (k ) ∈ R r , u(k ) ∈ R m , E, A, B, C, D are real matrices of appropriate dimensions and a nxn matrix E ≠ 0 is singular but det(zE − A ) ≠ 0 . We assume B ≠ 0 and C ≠ 0 . We adopt the following classical definitions of zeros. The distinct Smith zeros of (8.1) are those points of the complex plane where the system matrix


272

8 Singular Systems

 zE − A − B  P (z) =  D   C

loses its normal rank. The Smith zeros of (8.1) are defined as the roots of the so-called zero polynomial which is the product of diagonal (invariant) polynomials of the Smith canonical form of P(z) (i.e., as the Smith zeros of the pencil P(z)). The Smith zeros of the pencil [zE − A − B ] are called the input decoupling (i.d.) zeros, whereas the Smith zeros of the pencil zE − A   C  are called the output decoupling (o.d.) zeros of (8.1). If (8.1) has   no input and no output decoupling zeros, then the Smith zeros of the underlying system matrix P(z) are called the Smith transmission zeros of the system. Note that a number λ ∈ C is an output decoupling zero of the system (8.1) if and only if there exists a vector 0 ≠ x o ∈ C n such that (λE − A)x o = 0 and Cx o = 0 . Similarly, a number λ ∈ C is an input de-

coupling zero of (8.1) if and only if there exists a vector 0 ≠ w o ∈ C n

such that (w o ) ∗ ( λ E − A) = 0 and (w o ) ∗ B = 0 .

Definition 8.1. [72, 80] A number λ ∈ C is an invariant zero of (8.1) if and only if there exist vectors 0 ≠ x o ∈ C n (state-zero direction) and g ∈ C m (input-zero direction) such that the triple λ, x o , g satisfies  λE − A − B   x o   0  .  =  C D   g  0 

(8.2)

The system is called degenerate if and only if it has an infinite number of invariant zeros. ◊ The set of all invariant zeros of (8.1) will be denoted by  x o  0 Z I = {λ ∈ C : ∃ 0 ≠ x o ∈ C n ∧ ∃ g ∈ C m (P (λ)   =   )} ,  g  0

and the set of all Smith zeros by Z S = {λ ∈ C : rank P (λ) < normal rank P (z)} .

Remark 8.1. In Definition 8.1 the state-zero direction x o must be a non-

8.1 Definitions of Zeros

273

zero vector (real or complex). Otherwise, this definition becomes senseless (for any system (8.1) each complex number may serve as an invariant zero). Observe also that in the sense of Definition 8.1 each output decoupling zero is also an invariant zero. ◊ Remark 8.2. Of course, in the system (8.1) decoupling zeros are invariant under linear nonsingular coordinate transformations in the state space. Moreover, the set ZI has the same invariance properties as Z S , i.e., it is invariant under nonsingular linear transformations in the state space, nonsingular linear transformations of the inputs or outputs and constant state or output feedback to the inputs. ◊ Remark 8.3. The main reason for determining invariant zeros of (8.1) (see Definition 8.1) as the triples (complex number ( λ ), nonzero state-zero direction ( x o ≠ 0 ), input-zero direction (g)) satisfying (8.2) is that each such zero has the output-zeroing property in the following sense: if a triple ~ (k ) = λk g , k ∈ N , applied to λ, x o ≠ 0, g satisfies (8.2), then the input u the system (which is treated now as a complex one – cf., Remark 2.4) at x (0) = x o yields the solution of the state equation the initial condition ~ ~x ( k ) = λk x o and the system response ~ y (k ) ≡ 0 . The above implication can be verified immediately by showing that the sequences under consideration satisfy the equations (8.1). Moreover, if the triples λ1 , x1o ≠ 0, g1 and λ 2 , x o2 ≠ 0, g 2 satisfy (8.2), then any linear combination of the inputs ~ (k ) = λk g , u ~ (k ) = λk g , i.e., an input u ~ ( k ) = αu ~ (k ) + βu ~ (k ) , u 1 1 1 2 2 2 1 2 o ~ where α,β ∈ C , applied to (8.1) at the initial condition x (0) = α x + βx o 1

2

x (k ) = αλk1 x1o + βλk2 x o2 and the gives the solution of the state equation ~ system response ~ y (k ) ≡ 0 .

Naturally, if a triple λ = λ e jϕ , 0 ≠ x o = Re x o + jIm x o , g satisfies (8.2), then (8.2) is satisfied also for λ = λ e − jϕ , x o = Re x o − jIm x o , g , and these triples generate two real initial states and two real inputs which produce the identically zero system response. More precisely, the pair x(0) = Re x o and u(k ) = λ k (Re g coskϕ − Im g sin kϕ) yields the sol-

ution of the state equation λ k (Re x o coskϕ − Im x o sin kϕ) and gives the system response

y (k ) ≡ 0 . Similarly, the pair

x(0) = Im x o

and

274

8 Singular Systems k

u(k ) = λ (Re g sin kϕ + Im g coskϕ) produces the solution of the state k

equation λ (Re x o sin kϕ + Im x o coskϕ) and yields y (k ) ≡ 0 . ◊

8.2 Relationships between Invariant and Smith Zeros The sets Z S and ZI are interrelated as follows. Proposition 8.1. [82] (i)

If λ ∈ C is a Smith zero of (8.1), then λ is an invariant zero of

(8.1), i.e., Z S ⊆ ZI . (ii)

System (8.1) is nondegenerate if and only if Z S = ZI .

(iii)

System (8.1) is degenerate if and only if ZI = C .

Proof. The proof follows the same lines as the proof of Proposition 2.1. A full proof of Proposition 8.1 can be found in [82]. ◊ Thus each Smith zero is also an invariant zero. Moreover, ZI may be equal to Z S (then ZI may be empty or finite) or ZI may be equal to the whole complex plane. In this way the set of invariant zeros may be empty, finite or equal to C, and when the system is nondegenerate, the sets of Smith zeros and of invariant zeros coincide. Of course, Proposition 8.1 tells us also that if in the system (8.1) there exists at least one invariant zero which is not a Smith zero, the system is degenerate (for instance, if there exists an output decoupling zero which is not a Smith zero).

8.3 Sufficient and Necessary Condition for Degeneracy Lemma 8.1. If the system matrix P(z) corresponding to the system (8.1) has full normal column rank, then the system is nondegenerate, i.e., Z S = ZI . Proof. The proof follows the same lines as the proof of Corollary 2.4. ◊ Remark 8.4. Note that Lemma 8.1 tells us also that if the system (8.1) is degenerate, then normal rank P (z) < n + m . ◊ Proposition 8.2. [83] System (8.1) is degenerate if and only if

8.4 Zeros and SVD of Matrix E

− B normal rank P (z) < n + rank   . D

275

(8.3)

Proof. The proof follows the same lines as the proof of Proposition 2.2. A detailed proof of Proposition 8.2 can be found in [83]. ◊ − B Corollary 8.1. [83] If in the system (8.1) we have r < rank   , then the D system is degenerate.

Proof. The proof follows the same lines as the proof of Corollary 2.7. ◊ Proposition 8.3. [83] System (8.1) is nondegenerate if and only if − B normal rank P (z) = n + rank   . D

(8.4)

Proof. The claim follows from Proposition 8.2 and from the fact that nor− B  mal rank of P(z) can not be greater than n + rank   . ◊ D − B  Example 8.1. In a square (m-input, m-output) system (8.1) let   have D full column rank. Then:

(i) λ ∈ C is an invariant zero of the system if and only if det P (λ) = 0 ; (ii) the system is degenerate if and only if det P( z) ≡ 0 . For the proof of these claims use Exercise 2.8.3. ◊

8.4 Zeros and SVD of Matrix E Assuming that matrix E in (8.1) has rank 0 < p < n and applying SVD to this matrix, we can write M E = U Λ V T , where U, V are orthogonal and Λ =  p  0

with pxp diagonal and nonsingular M p .

0 0

(8.5)

276

8 Singular Systems

ξ  Setting ξ = V T x and taking the decomposition ξ =  1  , where ξ 2  ξ1 ∈ R p , ξ 2 ∈ R n − p , we transform (8.1) into the form ~ ~ ~ ξ1 (k + 1) = A11ξ1 (k ) + A12 ξ 2 (k ) + B1u (k ) ~ ~ ~ 0 = A 21ξ1 (k ) + A 22 ξ 2 (k ) + B 2 u(k ) ~ ~ y (k ) = C1ξ1 (k ) + C 2 ξ 2 (k ) + Du(k ) ,

(8.6)

where the decompositions A U T A V =  11  A 21

A12  , A 22 

B  UTB =  1  , B 2 

[

~ ~ CV = C1 C 2

]

follow in a natural way from (8.5) and the following notation is used ~ ~ ~ ~ A11 = M −p1A11 , A12 = M −p1 A12 , A 21 = A 21 , A 22 = A 22 , ~ ~ B1 = M −p1B1 , B 2 = B 2 .

Now, taking into account a new input and a new output vector, we can associate with (8.6) a standard linear system of the form

[

~ ~ ξ1 (k + 1) = A11ξ1 (k ) + A12 ~ ~ A A  ~ y (k ) =  ~21 ξ1 (k ) +  ~22  C2  C1 

]

~ ξ ( k )  B1  2   u(k )  ~ B 2  ξ 2 (k )  . D   u(k ) 

(8.7)

8.4.1 Invariant Zeros

Proposition 8.4. [72] If the system (8.1) is nondegenerate, then λ is its invariant zero if and only if λ is an invariant zero of the system (8.7). Proof. Let a triple λ, x o ≠ 0, g satisfy (8.2), i.e., let λEx o − Ax o = Bg

and Cx o + Dg = 0 .

(8.8)

Applying (8.5) to (8.8), premultiplying the first identity in (8.8) by U T , taking x o = Vξ o and using the notation introduced above, we can write


277

(8.8) as

[

~ ξ o  B1  2   g  ~ B 2  ξ o2  0    =  . D   g  0

~ ~ (λI p − A11 )ξ1o = A12 ~ ~  A 21  o  A 22 ξ + ~  1 ~  C1   C2

]

(8.9)

ξ o  Now, observe that if a triple λ, ξ 1o ≠ 0,  2  satisfies (8.9), then the triple  g  ξ o  λ, x o = Vξ o = V  1o  ≠ 0, g satisfies (8.8). In this way we have proved ξ 2  the following implication. (i) If λ ∈ C is an invariant zero of the system (8.7), then λ is an invariant zero of the system (8.1). In order to prove the converse, we focus our attention on the cases when o x ≠ 0 implies ξ1o ≠ 0 . Let us note first that if ξ1 = 0 , then

0 x = V   ∈ Ker E . In fact, we have in this case ξ 2  0 M Ex = U Λ V T V   = U  p  0 ξ 2 

0  0  = 0. 0 ξ 2 

0 Conversely, from Ex = 0 it follows that V T x =   . In fact, because ξ 2  ξ  M Ex = U Λ V T x = U Λ  1  = U  p  0 ξ 2 

0  ξ1  , 0 ξ 2 

hence Ex = 0 implies ξ1 = 0 . Thus we have: Ex ≠ 0 if and only if ξ1 ≠ 0 (or equivalently, Ex = 0 ⇔ ξ1 = 0 ). As we can note, the assumption det(zE − A ) ≠ 0 for almost all z ∈ C implies Ker A ∩ Ker E = {0} (see [23]). Using this fact, we can prove the following implication: if the system (8.1) is nondegenerate and a triple λ, x o ≠ 0, g satisfies (8.8), then Ex o ≠ 0 .

278

8 Singular Systems

Assume for the proof that Ex o = 0 . Then, since Ax o ≠ 0 , the identities (8.8) would be satisfied for any λ ∈ C at the same pair x o ≠ 0, g . This, however, would contradict the assumption of nondegeneracy of (8.1). Now, we can prove the desired implication: (ii) If the system (8.1) is nondegenerate and λ ∈ C is its invariant zero, then λ is an invariant zero of the system (8.7). As it follows from (8.8) and (8.9), if a triple λ, x o ≠ 0, g satisfies (8.8), ξ o  ξ o  then the triple λ, ξ1o ,  2  , where  1o  = V T x o ≠ 0 , satisfies (8.9).  g  ξ 2 

However, we must have Ex o ≠ 0 and, consequently (using (8.5)), ξ1o ≠ 0 . This means that λ is an invariant zero of (8.7). ◊ In order to show that in Proposition 8.4 the assumption of nondegeneracy of (8.1) is essential and it can not be omitted, we can consider the following example. Example 8.2. In (8.1) let 1 0 0 − 1 0 − 3 1 0 1 1 0 0     E = 0 1 0 , A =  0 − 2 0  , B = 0 1 0 , C =  . 0 1 0  0 0 0  0 1 0 1 0 − 3

Using Proposition 8.2, we check first that the system is degenerate. Note also that for any real λ the triple 0  1    λ, x = 0, g = 0  1  2 o

satisfies (8.2). On the other hand, for the system (8.7) we have − 1 0  ~ A11 =  ,  0 − 2

[A~ 12

0 0  ~  A 21     ~  = 1 0  ,  C1   0 1  

~  A 22 ~  C2

]

− 3 1 0 1 ~ B1 =  ,  0 0 1 0

 − 3 1 0 1 ~ B2     =  0 0 0 0 . D  0 0 0 0

This system, however, has no invariant zeros (Sect. 2.4, Example 2.3). ◊


279

8.4.2 Output Decoupling Zeros

Proposition 8.5. [72] A number λ is an output decoupling zero of the system (8.1) if and only if λ is an invariant zero of a standard linear system of the form ~ ~ ξ1 (k + 1) = A11ξ 1 (k ) + A12 ξ 2 (k ) ~ ~ (8.10)  A 22   A 21  ~ y ( k ) =  ~  ξ 1 ( k ) +  ~ ξ 2 ( k ) ,  C2   C1  where ξ 2 is treated as an input vector. Proof. Setting g = 0 in (8.8) and (8.9), we obtain at once the following implication: if a triple λ, ξ1o ≠ 0, ξ o2 satisfies ~ ~ (λI p − A11 ) ξ 1o = A12 ξ o2 ~  ~ A 21 ξ o +  A 22 ξ o = 0 , ~  1 ~  2   0  C2   C1 

(8.11)

then for the pair ξ o  λ, x o = V  1o  ≠ 0 ξ 2 

we have λEx o − Ax o = 0

and Cx o = 0 .

(8.12)

Thus, if λ is an invariant zero of the system (8.10), then λ is an output decoupling zero of the system (8.1). In order to prove the converse implication, we observe first that if a pair λ, x o ≠ 0 satisfies (8.12), then Ex o ≠ 0 . In fact, otherwise (i.e., assuming Ex o = 0 ) we would have (by virtue of the relation KerE ∩ Ker A = {0} ) Ax o ≠ 0 and the pair under considerations could not satisfy (8.12). In this

way we have that if a pair λ, x o ≠ 0 satisfies (8.12), then ξ1o ≠ 0 (since Ex o ≠ 0 implies ξ1o ≠ 0 ) and, consequently, the triple λ, ξ1o ≠ 0, ξ o2 satisfies the relations in (8.11). This completes the proof of the proposition. ◊

280

8 Singular Systems

8.4.3 Input Decoupling Zeros

Proposition 8.6. [72] A number λ is an input decoupling zero of the system (8.1) if and only if λ is an invariant zero of a standard linear system S(A, B, C, D) with the matrices A T  A T  T = D . , A = M −p1 A11 , B = M −p1 A T21 , C =  12  22  T  T B  B 2   1 

(8.13)

Proof. If λ is an input decoupling zero of (8.1), then, by definition, (λE T − A T )w o = 0

and B T w o = 0

(8.14)

for some 0 ≠ w o ∈ C n . Using the notation introduced for the system (8.7)  ηo  and setting η o = U T w o , where ηo =  1o  , η1o ∈ C p , η o2 ∈ C n − p , we  η 2  transform (8.14) into the form T o (λI p − M −p1 A11 )η1 − M −p1 A T21 η o2 = 0 T T   A12 o  A  o 0 η = .  T  η1 +  22 T  2 0  B1   B 2   

(8.15)

Now, treating η1o as a state-zero direction and ηo2 as an input-zero direction, it is easily seen that if a triple λ, η1o ≠ 0, η o2 satisfies (8.15) (i.e., λ is invariant zero of S(A, B, C, D) in (8.13)), then the pair  ηo  λ, w o = Uη o = U  1o  ≠ 0 satisfies (8.14) (i.e., λ is an input decoupling  η 2  zero of (8.1)). Conversely, if a pair λ, w o ≠ 0 satisfies (8.14), then the triple λ, η1o , η o2 satisfies (8.15). On the other hand, we have E T w o ≠ 0 ⇔ η1o ≠ 0 . Furthermore, because λ, w o ≠ 0 satisfies (8.14) and the pencil zE T − A T is regular (i.e., Ker E T ∩ Ker A T = {0} ), hence E T w o ≠ 0 and, consequently, η1o ≠ 0 . This means that λ is an invariant zero of the system (8.13). ◊

8.5 Markov Parameters and the Weierstrass Canonical Form

281

Example 8.3. In (8.1) let 1 0 0 0 0 − 1  0     E = 0 1 0 , A = 1 0 − 2 , B = 0 , C = [0 0 1] . 0 0 0 0 1 − 3 1

By virtue of Proposition 8.3, the system is nondegenerate and, consequently, its invariant zeros can be found, using Proposition 8.4, as the inˆ ,Bˆ ,C ˆ ,D ˆ ) (8.7) with the matrices variant zeros of a system S( A 0 ~ ˆ =A A 11 =  1 ~ ˆ =  A 21  = 0 C ~   C1  0

0 , 0 1 , 0

[

]

 − 1 0 ~ B1 =  , − 2 0 ~ ~   ˆ =  A~22 B 2  = − 3 1 . D   D   1 0  C2

~ Bˆ = A12

ˆ is invertible, all invariant zeros of As is known (cf., Chap. 4), since D ˆ − Bˆ D ˆ . Simple calculations yield ˆ −1C ˆ ,Bˆ ,C ˆ ,D ˆ ) are the eigenvalues of A S( A

ˆ − Bˆ D ˆ = 0 0 which means that the original system (8.1) has one ˆ −1C A 1 0   invariant zero λ = 0 of order 2 (cf., also Exercise 8.7.1). ◊

8.5 Markov Parameters and the Weierstrass Canonical Form It is well known [24, 25] that for a regular pencil zE − A with the index of nilpotency q there exist matrices Φ i , i = −q, − (q − 1), ..., − 1, 0, 1, 2,... , (called fundamental matrices) such that ∞

(zE − A) −1 = ∑ Φ i z − (i +1)

(8.16)

 I for i = 0 . EΦ i − AΦ i −1 = Φ i E − Φ i −1A =  0 for i ≠ 0

(8.17)

i = −q

and

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8 Singular Systems

The transfer-function matrix of the system (8.1) can then be written as ∞

G (z) = D + C(zE − A ) −1 B = D + ∑ CΦ i Bz − (i +1) , i = −q

(8.18)

where the matrices D and CΦ i B are called the Markov parameters of the system (8.1). If a regular pencil zE − A has an index of nilpotency q and deg det(zE − A ) = n1 , then there exist nonsingular matrices P and Q such that (cf., the Weierstrass-Kronecker theorem in [12, 23, 25]) zI − A1 P (zE − A )Q =  1 0 

0  , zN − I 2 

(8.19)

where N is nilpotent and satisfies N q −1 ≠ 0, N q = 0 . This is a Weierstrass canonical form of zE − A . Using (8.19), we can write (zI − A ) −1  0 1 Q −1 (zE − A) −1 P −1 =  1 . −1  (zN − I 2 )  0 

(8.20)

When zE − A is taken in its Weierstrass canonical form (8.19), we obtain (zI − A ) −1  0 1 (zE − A ) −1 =  1 −1  (zN − I 2 )  0  0 0  ( zI1 − A1 ) −1 0 =  −1  +  0 0 0 (zN − I 2 )  

(8.21)

and A A= 1 0

0 I , E= 1  I2  0

0 , N 

0  0  0 0  0 0 Φ −q = −  , q −1  ,..., Φ −i = −  i −1 ,.. . , Φ −1 = −  0 N  0 N  0 I 2 

A i I 0  A 0 Φo =  1 , Φ1 =  1 ,..., Φ i =  1    0  0 0  0 0

Recall that and

0 ,... . 0

(zN − I 2 ) −1 = −z q −1N q −1 − ... − zN − I 2

(8.22)

8.6 Invariant Zeros and the First Markov Parameter

283

∞

(zI1 − A1 ) −1 = ∑ z − (i +1) A1i . i =0

Remark 8.5. It is easy to check that matrices Φ i in (8.22) satisfy (8.16) and (8.17), i.e., they are fundamental matrices for the pencil (8.19). ◊ Remark 8.6. The transformation P (zE − A )Q of a regular pencil zE − A , where P and Q are arbitrary nxn nonsingular matrices, does not change the Markov parameters of the system (8.1) (in consequence, also the transferfunction matrix G(z) of (8.1) remains unchanged). In fact, under such a transformation the singular system (8.1) becomes a new system S(E' , A ' , B' , C' , D' ) , where x' = Q −1x and E' = PEQ, A ' = PAQ, B' = PB, C' = CQ, D' = D . Moreover, Φ'i = Q −1Φ i P −1 are fundamental matrices for S(E' , A ' , B' , C' , D' ) and C' Φ'i B' = CΦ i B . Furthermore, from the relation

P 0   zE − A − B  Q 0  P (zE − A )Q − PB   = 0 I   C D   0 I m   CQ D  r 

(8.23)

it is clear that the transformation considered changes neither the zero polynomial, nor the set of invariant zeros (i.e., a triple λ, x o ≠ 0, g satisfies (8.2) for the system (8.1) if and only if the triple λ, x' o = Q −1x o ≠ 0, g satisfies (8.2) for S(E' , A ' , B' , C' , D' ) ). ◊

8.6 Invariant Zeros and the First Markov Parameter In this section we consider the system (8.1) in its Weierstrass canonical form (moreover, we assume D = 0 ) (a suitable procedure for finding a Weierstrass canonical form of (8.1) can be found, e.g., in [25]): I 1 0 

0  x1 (k + 1)  A1 = N   x 2 (k + 1   0

 x (k )  y (k ) = [C1 C 2 ]  1  , x 2 (k )

0   x1 ( k )   B 1  + u(k ) I 2   x 2 (k ) B 2 

(8.1’)

i.e., matrices E and A and the fundamental matrices are as in (8.22) and

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8 Singular Systems

B  B =  1  , C = [C1 C 2 ] (the system (8.1’) may be viewed as a parallel B 2  connection of the subsystems S1 ( A1 , B1 , C1 ) and S 2 ( N, I 2 , B 2 , C 2 ) ). Moreover, we assume that the first nonzero Markov parameter of (8.1’) has a negative index i (see (8.18)) and we denote this parameter by CΦ − ν B , 1 ≤ ν ≤ q , i.e., CΦ −q B = CΦ −(q −1) B = ... = CΦ −(ν +1) B = 0 and CΦ − ν B ≠ 0 ,

(8.24)

and rank CΦ − ν B = p ≤ min{m, r} . Note that using (8.18), (8.22) and (8.24), we can write the transferfunction matrix of (8.1’) as G (z) = C(zE − A) −1 B = −C 2 N ν −1B 2 z ν −1 − .. − C 2 NB 2 z − C 2 B 2 + C1 (zI1 − A1 ) −1 B1 ,

(8.25)

i.e., CΦ − ν B = −C 2 N ν −1B 2 . Define the nxn matrix K − ν := I − B(CΦ − ν B ) + CΦ − ν ,

(8.26)

where “ + ” means the operation of taking the Moore-Penrose inverse. Recall that if matrices H1 , H 2 , where H1 is rxp and H 2 is pxm, give a skeleton factorization of CΦ − ν B , i.e., CΦ − ν B = H1H 2 , then (CΦ −ν B) + = H +2 H1+ , H1+ = (H1T H1 ) −1 H1T and H 2+ = H T2 ( H 2 H T2 ) −1 .

Lemma 8.2. [83] The matrix K − ν in (8.26) has the following properties: (i) (ii) (iii)

K −2 ν = K − ν

(i.e., K − ν is idempotent); Σ − ν := {x : K − ν x = x} = Ker( H1T CΦ − ν ) , Ω − ν := {x : K − ν x = 0} = Im(BH T2 ) , n

dim Σ − ν = n − p ; dim Ω − ν = p ;

n

(iv)

C (R ) = Σ − ν ⊕ Ω − ν ;

(v)

K − ν BH T2 = 0 ,

Proof. Set C' = H1T C

H1T CΦ − ν K − ν = 0 .

and B' = BH T2 . Note that the pxp matrix

C' Φ −ν B' = H1T H1H 2 H T2 is nonsingular.


285

Define K ' − ν := I − B' (C' Φ − ν B' ) −1 C' Φ − ν . Then K' − ν = K − ν . In fact, it is sufficient to observe that B' (C' Φ −ν B' ) −1 C' Φ −ν = BH T2 (H1T CΦ −ν BH T2 ) −1 H1T CΦ −ν = BH T2 (H1T H1H 2 H T2 ) −1 H1T CΦ −ν = BH T2 (H 2 H T2 ) −1 (H1T H1 ) −1 H1T CΦ −ν = BH 2+ H1+ CΦ −ν = B(CΦ −ν B) + CΦ −ν .

The remaining part of the proof proceeds for K '− ν . It follows the same lines as the proof of Lemma 3.1 and for this reason is omitted here. ◊ Remark 8.7. [83] Using (8.22) and (8.26), we can write the matrix K − ν in the form I − B1 (C 2 N ν −1B 2 ) + C 2 N ν −1  , K −ν =  1 ν −1 + ν −1    0 I 2 − B 2 (C 2 N B 2 ) C 2 N

where

K − ν, 2 := I 2 − B 2 (C 2 N ν −1B 2 ) + C 2 N ν −1

(8.27)

is projective (idem-

potent). ◊ Lemma 8.3. [83] If in the system (8.1’) a triple λ, x o ≠ 0, g satisfies (8.2), then CΦ − q x o = 0 . .

and

CΦ − ν Bg = − CΦ − ν x o .

(8.28)

CΦ − (ν +1) x o = 0 Cx o = 0

Moreover, K − ν Ex o = Εx o .

(8.29)

Proof. The identity λEx o − Ax o = Bg is multiplied successively from the left by CΦ −q ,..., CΦ −(ν +1) , and we use the relations Φ − i E = Φ − (i +1) and Φ − i A = Φ − i (cf., (8.22)) as well as (8.24). In this way we obtain

286

8 Singular Systems

CΦ −q x o = 0,..., CΦ −(ν +1) x o = 0 . Premultiplying λEx o − Ax o = Bg by CΦ − ν , we obtain CΦ − ν Bg = − CΦ − ν x o . Finally, (8.29) follows from (8.26) and from the relations Φ − ν E = Φ − (ν +1) and CΦ − (ν +1) x o = 0 . ◊

Lemma 8.4. [83] If in the system (8.1’) a triple λ, x o ≠ 0, g satisfies (8.2), then (i)

λEx o − K − ν Ax o = Bg1 , K − ν Ax o − Ax o = Bg 2 , Cx o = 0 ,

where g = g1 + g 2 , g1 ∈ Ker(CΦ − ν B) , g 2 ∈ Im(CΦ − ν B) T and g1 , g 2 are uniquely determined by g. Moreover, (ii)

Bg1 ∈ Σ − ν , Bg 2 ∈ Ω − ν and g 2 = −(CΦ − ν B) + CΦ − ν x o .

Proof. Let

g = g1 + g 2

where

g1

and

g2

are

defined

as

g1 := (I m − (CΦ − ν B) + CΦ − ν B)g and g 2 := (CΦ − ν B) + CΦ − ν Bg . Then Bg1 = K − ν Bg and Bg 2 = (I − K − ν )Bg . Thus K − ν Bg1 = Bg1 and K − ν Bg 2 = 0 (i.e., Bg1 ∈ Σ −ν and Bg 2 ∈ Ω − ν ). Now, the identity λEx o − Ax o = Bg may be written as

(iii)

(λE − K −ν A )x o + (K −ν − I ) Ax o = Bg1 + Bg 2

with the vectors (λE − K − ν A )x o and Bg1 in Σ − ν and (K −ν − I) Ax o and Bg 2 in Ω − ν . Note that, by virtue of (8.29), we have K − ν (λEx o − K − ν Ax o ) = (λEx o − K − ν Ax o ) . Moreover, we have also K − ν (K − ν − I ) Ax o = 0 . Now, from Lemma 8.2 (iv) it follows that the decomposition (iii) is unique. This proves the first two relations in (i). The expression for g 2 in (ii) follows from the definition of g 2 and

from the relation CΦ − ν Bg = − CΦ − ν x o in (8.28). Finally, CΦ − ν Bg1 = 0 follows from the definition of g1 . ◊ Remark 8.8. [83] The pencil zE − K −ν A is not regular, i.e., det(zE − K − ν A) ≡ 0 .


287

We can verify this claim by using the relation K ν = K ' − ν (see the proof of Lemma 8.2). Thus we can write det(zE − K −ν A ) = det(zE − K ' −ν A) = det((zE − A) + B' (C' Φ −ν B' ) −1 C' Φ −ν ) = det(zE − A) det[I n + (zE − A ) −1 B' (C' Φ −ν B' ) −1 C' Φ −ν ] = det(zE − A) det[I p + C' Φ −ν (zE − A ) −1 B' (C' Φ −ν B' ) −1 ] .

Now, we show the identity C' Φ −ν (zE − A ) −1 B' = − C' Φ −ν B' which will give the desired result. To this end, observe first that Φ − ν Φ i = 0 for all i ≥ 0 (see (8.22)) and Φ − ν Φ i = − Φ − (ν − i −1) for i = −q,. ..,−1 (in particular, Φ − ν Φ −1 = − Φ − ν and Φ − ν Φ − 2 = − Φ − (ν +1) ). In this way we can write −1

Φ −ν (zE − A ) −1 = ∑ Φ −ν Φ i z −(i +1) i = −q

= Φ −ν Φ −q z q −1 + ... + Φ −ν Φ − 2 z + Φ −ν Φ −1 .

Premultiply the right-hand side of the above relation by C' and postmultiply the result by B' . Now, in view of the relation Φ − i = 0 for all i ≥ q + 1 and the assumption C'Φ −q Β' = ... = C' Φ −(ν +1) B' = 0 , we obtain the desired identity. Finally, note

that

(cf., (8.26))

K − ν A = A + BF ,

where

F = −(CΦ − ν B) + CΦ − ν (observe also that Φ −ν A = Φ −ν ). ◊ 8.6.1 First Markov Parameter of Full Column Rank

Lemma 8.5. If in the system (8.1’) the first nonzero Markov parameter CΦ − ν B has full column rank, then the system matrix P(z) of (8.1’) has full normal column rank. Proof. We consider separately two cases. In the first case we assume that (8.1’) is square (i.e., m = r ) and the mxm matrix CΦ − ν B = −C 2 N ν −1B 2 is nonsingular. Then, however, since

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8 Singular Systems

det P (z) = det( zE − A ) det G (z) , we only need to show that det G (z) ≠ 0 . Using (8.25), we can write

G (z) = − C 2 N ν −1B 2 z ν −1 (I m + H (z)) ,

where H (z) = (C 2 N ν −1B 2 ) −1 C 2 N ν −2 B 2 z −1 + ... + (C 2 N ν −1B 2 ) −1 C 2 B 2 z −(ν −1) + − (C 2 N ν −1B 2 ) −1 z −(ν −1) C1 ( zI1 − A1 ) −1 B1

and lim H (z) = 0 . Thus we have det(I m + H (z)) ≠ 0 and, consequently, z→∞

det G (z) ≠ 0 , i.e., P(z) is nonsingular. In the second case it is assumed that m < r and the rxm matrix CΦ − ν B = −C 2 N ν −1B 2 has the full column rank m. To CΦ − ν B we apply SVD, that is, we write CΦ − ν B = U Λ V T , where a rxr U and a mxm V are M  orthogonal and Λ =  m  with a mxm diagonal and nonsingular M m .  0  Set

 C  B = BV = B m and C = U T C =  m  ,  Cr − m 

where Cm consists of the first m rows of C , and observe that M m = Cm Φ − ν B m . Now, we can write  zE − A − B   I 0   zE − A − B   I 0  P (z) =  . = T 0  0 V  0  0 U   C  C

On the other side, P (z) can be written as  zE − A − B m  zE − A − B m    P ( z ) =  Cm 0  , where P ' (z) =   Cm 0    Cr − m  0  

is square. In order to show that P(z) has full normal column rank, it is sufficient to observe that det P ' (z) is a nonzero polynomial. To this end, we


289

consider a square system S ' (E, A, B m , Cm ) in which the first nonzero Markov parameter M m = Cm Φ − ν B m is nonsingular. Decomposing

[

Cm = Cm,1

]

Cm,2 (with a mxn1 Cm,1 and a mxn2 Cm,2 )

as well as  B m,1  Bm =   (with a n1xm B m,1 and a n 2 xm B m, 2 ),  B m, 2 

we obtain M m = Cm Φ − ν B m = − Cm,2 N ν −1B m, 2 .

For the transfer-function matrix of S ' (E, A, B m , Cm ) we have now G ' (z) = C m (zE − A ) −1 B m = − C m, 2 N ν −1 B m,2 z ν −1 − ... − C m, 2 B m,2 + Cm,1 (zI1 − A1 ) −1 B m,1 .

Finally, proceeding analogously as in the first case, we obtain det G ' ( z) ≠ 0 and, consequently, det P ' (z) = det(zE − A ) det G ' (z) ≠ 0 . ◊ Proposition 8.7. [83] If in the system (8.1’) the first nonzero Markov parameter CΦ − ν B = − C 2 N ν −1B 2 has full column rank, then the system is

nondegenerate, i.e., Z S = ZI . Moreover, λ ∈ C is an invariant zero of the system if and only if there exists a vector x o ≠ 0 such that  λE − K −ν A  o  0    x = 0  . C    

(8.30)

Proof. The first claim follows directly from Lemmas 8.1 and 8.5. The proof of the second claim is as follows. ⇐ ) If (8.30) is satisfied for some λ ∈ C and x o ≠ 0 , then taking into

account the definition of K − ν and setting g = −(CΦ − ν B) + CΦ − ν x o , we can transform (8.30) into the form (8.2). ⇒ ) From Lemma 8.4 it follows that if CΦ − ν B has full column rank, then g1 = 0 and, consequently, λEx o − K −ν Ax o = 0 and Cx o = 0 . ◊

290

8 Singular Systems

Remark 8.9. If in (8.1’) the matrix CΦ − ν B has full column rank, then the zE − K −ν A  pencil   has full normal column rank (n). C   zE − K −ν A  In fact, suppose that normal rank   = ρ < n . This means that C    λE − K −ν A  at any fixed λ ∈ C we have rank   ≤ ρ < n , i.e., columns of C   λE − K −ν A    are linearly dependent (over C). In consequence, there C  

exists a vector x o ≠ 0 such that (8.30) holds. Thus the system is degenerate. This, however, contradicts Proposition 8.7. From the above and from Proposition 8.7 we conclude that if in the system (8.1’) the first nonzero Markov parameter CΦ − ν B has full column rank, then invariant zeros of the system are exactly those points of the zE − K −ν A  complex plane where the pencil   loses its normal column C   rank n. ◊ 8.6.2 SVD of the First Markov Parameter

In this subsection we apply SVD to the first nonzero Markov parameter of S(E,A,B,C) (8.1’) (see (8.24)), i.e., we write (recall that we have assumed 0 < rank CΦ − ν B = p ≤ min{m, r} ) CΦ − ν B = U Λ V T ,

M where Λ =  p  0

0 0

(8.31)

is rxm-dimensional, M p is a pxp diagonal matrix with positive singular values of CΦ − ν B and U and V are respectively rxr and mxm orthogonal matrices. Introducing V and U T to S(E,A,B,C) as pre- and post-compensator, we obtain an auxiliary system S(E, A, B , C ) of the form Ex(k + 1) = Ax(k ) + B u (k ) y ( k ) = C x( k ) ,

(8.32)


291

where B = BV ,

C = UTC

and

u = VTu ,

y = UTy

(8.33)

are decomposed as follows

[

B = Bp

 Cp   up   yp  Bm − p , C =  , y= , u =   u m− p  y r − p   C r − p 

]

(8.34)

and B p consists of the first p columns of B , while C p consists of the first p rows of C . Similarly, u p consists of the first p rows of vector u and y p consists of the first p components of y . It is clear (cf., (8.24), (8.31), (8.33) and (8.34)) that C Φ −ν B is the first nonzero Markov parameter of the system (8.32) as well as that  C pΦ −ν B p CΦ − ν B =   Cr − p Φ − ν B p

C p Φ − ν B m − p  M p = Cr − p Φ − ν B m − p   0

0 , 0

(8.35)

i.e., C p Φ −ν B p = M p ,

C p Φ −ν B m − p = 0 ,

C r − p Φ −ν B p = 0 ,

C r − p Φ − ν B m − p = 0.

(8.36)

Lemma 8.6. [83] The sets of invariant zeros of the systems S(E,A,B,C) in (8.1’) and S(E, A, B , C ) in (8.32) coincide. Proof. The claim follows immediately from Remark 8.2. ◊ For the system S(E, A, B , C ) in (8.32) we form the projection matrix K − ν := I − B ( C Φ − ν B ) + C Φ − ν

(8.37)

which, in view of (8.31) and (8.34), can be evaluated as

[

K −ν = I − B p

]

M B m− p  p  0

= I − B p M −p1 C p Φ −ν .

+ 0  C p    Φ −ν 0  C r − p 

(8.38)

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8 Singular Systems

Remark 8.10. The matrices K − ν in (8.26) and K − ν (8.37) satisfy the relation K − ν = K − ν . In fact, from (8.31) it follows that (CΦ − ν B) + = V Λ + U T . Moreover, from (8.31) and (8.33) we have C Φ − ν B = Λ . Now, we can write K − ν = I − B (C Φ − ν B ) + CΦ − ν = I − BΛ + CΦ − ν = I − B(CΦ − ν B) + CΦ − ν = K − ν .

(8.39)

Furthermore, the relations (8.38) and (8.36) imply K − ν B p = 0 and K − ν B m − p = B m − p . ◊

(8.40)

Lemma 8.7. [83] Suppose that the system S(E,A,B,C) in (8.1’) is such that in the auxiliary system S(E, A, B , C ) in (8.32) is B m − p = 0 . Then the following sets of invariant zeros (for appropriate systems) coincide ZIS(E,A,B,C) = ZI

S(E,A,B , C )

= ZI

S(E,A,B p, C )

,

(8.41)

where S(E, A, B p , C ) is obtained from S(E, A, B , C ) by neglecting the input u m − p . Proof. The system (8.32) has the form Ex(k + 1) = Ax(k ) + B p u p (k ) + B m− p u m − p (k ) y ( k ) = C x( k ) .

(8.42)

 gp  If B m − p = 0 and a triple λ, x o ≠ 0, g =   satisfies (8.2) (when g m − p 

applied to the system (8.32)), then the triple λ, x o ≠ 0, g p satisfies (8.2) when applied to the system S(E, A, B p , C ) of the form Ex(k + 1) = Ax(k ) + B p u p (k ) y ( k ) = C x( k ) .

(8.43)

Thus we have shown that if λ is an invariant zero of S(E, A, B , C ) , then this λ is also an invariant zero of S(E, A, B p , C ) .


293

Conversely, if a triple λ, x o ≠ 0, g p satisfies (8.2) (when applied to  gp  S(E, A, B p , C ) ), then the triple λ, x o ≠ 0, g =   , where g m − p is g m − p  arbitrary (since in (8.42) we have B m − p = 0 ), represents the invariant

zero λ for (8.32). Finally, the first equality in (8.41) follows via Lemma 8.6. ◊ Proposition 8.8. [83] Suppose that the system S(E,A,B,C) in (8.1’) is such that in the system S(E, A, B , C ) in (8.32) is B m − p = 0 . Then (8.1’) is nondegenerate. Moreover, λ ∈ C is an invariant zero of (8.1’) if and only if there exists a vector x o ≠ 0 such that  λE − K − ν A  o  0  x =  .  C 0   

(8.44)

Proof. In view of Lemma 8.7, we can consider invariant zeros of the system S(E, A, B p , C ) . The first nonzero Markov parameter of M  S(E, A, B p , C ) is equal to C Φ − ν B p =  p  and it has full column rank.  0  Now, nondegeneracy of (8.1’) follows from Proposition 8.7 (when applied to S(E, A, B p , C ) ) as well as from Lemma 8.7.

The proof of the second claim follows the same lines as the proof of the second claim in Proposition 8.7 (when applied to S(E, A, B p , C ) . ◊ Remark 8.11. Under the assumptions of Proposition 8.8, the pencil zE − K −ν A    C  

has full normal column rank (n). The proof of this claim is analogous to that given in Remark 8.9. Thus, under the assumption of Proposition 8.8, the invariant zeros of (8.1’) are those points of the complex plane where this pencil loses its full normal column rank. ◊ Example 8.4. This example shows that in Proposition 8.8 the condition B m − p = 0 is merely a sufficient condition of nondegeneracy.

294

8 Singular Systems

Consider a system (8.1’) with the matrices

(i)

1 0 0  − 1 0 0 1 0  1 0 1      , E = 0 1 0 , A =  0 − 2 0 , B = 0 1 , C =  0 1 0  0 0 0  0 1 0 0 1

0 0 0 − 1 0    where N = [0], q = 1, A1 =   , Φ −1 = − 0 0 0 . − 0 2   0 0 1 The first nonzero Markov parameter is  − 1 0 CΦ −1B =   and rank CΦ −1B = p = 1 .  0 0 1 0 In (8.31) we take U = −I 2 , V = I 2 and Λ =   . Then in (8.34) is 0 0  0 B = B and C = −C ; moreover, B m − p = 1 . 0

On the other hand, for the system (i) we have det P (z) = z and, by virtue of Example 8.1, this system is nondegenerate and it has exactly one invariant zero λ = 0 . ◊ Example 8.5. Consider a system (8.1’) with the matrices  2 − 1 0 0    A1 =  0 0 0 , B1 = 0 , C1 = [0 − 1 0], − 1 0 0 1  0 1 1 1   N = 0 0 1 , B 2 = 1 , C 2 = [1 0 1], 0 0 0 1

q = 3.

The first nonzero Markov parameter is 0 0   B1  = −C 2 N 2 B 2 = −1 . CΦ − q B = −[C1 C 2 ]   2 0 N  B 2 

The system is nondegenerate (cf., Proposition 8.7) although the subsystem S( A1 , B1 , C1 ) is degenerate (its transfer function equals zero identically). Now, using Example 8.1, we infer that invariant zeros of the system are

8.7 Exercises

295

the roots of the polynomial det P (z) = z 2 (z 2 + 2z + 2)(z − 2) (which is the zero polynomial). Thus ZI = Z S = {0, 2, − 1 + j1, − 1 − j1} . The same result can be obtained by using Proposition 8.7 (or Remark 8.9). Calculating zE − K − q A  K − q in accordance with (8.27), we obtain   as C   z − 2 1  0 z   1 0  0  0 0 0   0 0   0 −1

     0 − 1 z z + 1 0 0 − 1 z + 1 0 0  0 0  1 0 1  0 0 0 z

0 0 0

0 0 0

0 0 1

and, as it is easy to check, this matrix loses its full normal column rank exactly at the roots of the polynomial z 2 (z 2 + 2z + 2)(z − 2). ◊ Remark 8.12. It has been shown in this section that if a singular system is taken in its Weierstrass canonical form, then, under some additional conditions, its invariant zeros can be characterized as output decoupling zeros of a closed-loop state feedback system (Propositions 8.7 and 8.8). ◊

8.7 Exercises 8.7.1. Show that systems (8.1) and (8.7) have the same zero polynomial, and then conclude that if (8.1) is nondegenerate, then its invariant and Smith zeros coincide (including multiplicities). Hint. Using the notation introduced in Section 8.4, we can write U T   0

(i)

0   zE − A − B  V 0   D   0 I m  I r   C

zM p − A11 =  − A 21  C1

− A12 − A 22 C2

− B1  − B 2  . D 

Now, premultiplying the matrix standing on the right-hand side of (i) by

296

8 Singular Systems

M −p1 0  ,  I n − p + r   0

(ii) we obtain

~ zI p − A11  ~  − A 21 ~  C1 

~ − A12 ~ − A 22 ~ C2

~ − B1  ~  − B2  . D  

In order to show that in a nondegenerate system (8.1) its invariant and Smith zeros are exactly the same objects use Proposition 8.4, Definition 2.3 and Remark 2.7. zE − A  8.7.2. Show that the zero polynomial of the pencil   is the same as  C  of the standard linear system (8.10).

Hint. Using the notation introduced in Sect. 8.4, we can write (iii)

U T   0

 zM p − A11 0  zE − A  V =  − A 21   I r   C   C1

− A12  − A 22  . C 2 

Then, premultiplying the matrix standing on the right-hand side of (iii) by matrix (ii) (see Exercise 8.7.1), we obtain ~ ~  zI p − A11 − A 12  ~ ~  − − A A  22  . 21 ~ ~  C2  C1   8.7.3. Show that the zero polynomial of the pencil [zE − A − B ] is the same as of the standard linear system (8.13).

Epilogue

In Chapters 2–7 we proposed a uniform approach to the algebraic and geometric analysis of multivariable zeros, output-zeroing problem, output-nulling subspaces and zero dynamics in finite dimensional linear time-invariant MIMO discrete-time systems. The presented approach has been based on the concept of zero directions introduced by MacFarlane and Karcanias and on information contained in the first nonzero Markov parameter of a system. The results presented there indicate that for full analysis of dynamical properties of such systems the decomposition of a class of all systems of this kind into two disjoint subclasses of nondegenerate and degenerate systems is necessary. This conclusion concerns also SISO systems for which the subclass of degenerate systems consists of all strictly proper systems with the identically zero transfer function. Of course, the analysis of multivariable zeros requires finding out of all those complex numbers which are responsible for zeroing the system output and simultaneously generate nontrivial solutions of the state equation (these zeros were termed the invariant zeros). This task, however, constitutes a nonlinear problem (cf., Eq. (2.4a) in Sect. 2.1). In particular, it has been shown also, that any degenerate system has its invariant zeros everywhere in the complex plane, while in any nondegenerate system its invariant zeros coincide with the Smith zeros. Observe also that relations between sets of individual kinds of zeros (i.e., invariant, Smith’s, transmission and decoupling zeros of different kinds) are essentially distinct for the subclass of degenerate systems and for the subclass of nondegenerate systems. In consequence, in the author’s opinion, even advanced theories of linear algebra, when applied merely to the Smith zeros, will not decide the problem of characterizing multivariable zeros. Further research can be focused on dynamical characterization of zeros at infinity. Moreover, information concerning zeros included in the first nonzero Markov parameter can probably be used for a more detailed characterization of zeros in sampled data systems (see [86]–[88] for a thorough discussion of such zeros in decouplable systems).


298

Epilogue

The proposed approach may be extended to positive systems and to systems with delays. As far as singular systems are concerned, further research can be focused on characterization of individual kinds of decoupling zeros in the context of the four-fold Kalman decomposition of a singular system (cf., [26]) as well as of maximal output-nulling subspaces.

Appendix

Appendix A A.1 Reachability and Observability A linear multivariable discrete-time control system with m inputs and r outputs is usually described, in state space form, by means of a set of linear equations x(k + 1) = Ax(k ) + Bu (k ) , y (k ) = Cx(k ) + Du(k )

k ∈ N = {0,1,2,...} ,

(A1)

in which x ∈ R n denotes the state vector, u ∈ R m is the input vector and y ∈ R r is the output vector whereas A, B, C, D are real matrices of appropriate dimensions. A linear space U over R of all vector sequences u(.): N → R m is considered as the class of admissible inputs. The analysis of the interaction between input and state and between state and output is of crucial importance for understanding a large number of relevant control problems. Fundamental tools for the analysis of such interactions are the notions of reachability and observability introduced by Kalman in the 1960's (see also [30, 93]) and the corresponding decompositions of the control system into reachable/unreachable and observable/ unobservable parts. A solution corresponding to an input u(.) ∈ U and to an initial condition x(0) = x o has the form k −1

x(k ) = A k x o + ∑ A k −1− l Bu(l ) . l =0

(A2)

A state x is called reachable from a given state x o if there exist 0 < K ∈ N and u(.) ∈ U defined on the interval < 0, K > such that x = x(K ) . As is known from the theory of linear control systems [91, 32]


300

Appendix

the set of all states reachable from the state x o = 0 forms a linear subspace X r of R n of the form X r = Im[B AB . . . A n−1B] ,

(A3)

where [B AB . . . A n−1B] is called the reachability matrix for (A1). A state x which does not belong to X r is said to be unreachable. The subspace X r is called the reachable subspace for the pair (A,B). Moreover, X r can be characterized as the smallest A-invariant subspace (i.e., satisfies A x ∈ X r if x ∈ X r ) which contains Im B (i.e., is such that Bu ∈ X r for all u ∈ R m ). This means that X r is the intersection of all A-

invariant subspaces containing Im B . In particular, if X r = R n , then the system is said to be reachable. Suppose that dim X r = nr < n (or, what is the same, rank of the reachability matrix equals n r ). Then, there exists a nonsingular linear transformation H of the state space such that after the change of coordinates x' = H x matrices A, B, C assume the form A r  0 

A12  , A r 

B r   0 ,  

[C r

Cr ] ,

(A4)

with a n r xnr -dimensional A r and a n r x m -dimensional B r , and X 'r = H (X r ) (i.e., the image of X r under H) consists of vectors having the form x' = col ( x '1 , . . . , x ' nr , 0, . . . , 0) , i.e., of all vectors whose last n − n r components are zero. The pair ( A r , B r ) is a reachable pair, i.e., it

[

n −1

]

satisfies the condition rank B r A r B r . . . A r r B r = n r . The matrix H can be formed in the following way [10]. From the reachability matrix we choose nr linearly independent columns, say γ 1 , . . . , γ nr . Then we take a set, say β nr +1 , . . . , β n , of n − n r linearly independent vectors of R n which are also linearly independent upon γ 1 , . . . , γ nr and define H −1 = [ γ 1 , . . . , γ nr ; β nr +1 , . . . , β n ] .

The decomposition (A4) χ( z) = det(zI − A) of A as

splits

the

characteristic

(A5) polynomial

Appendix A

χ( z) = χ r (z)χ r (z) ,

301

(A6)

where χ r (z) = det(zI r − A r ) and χ r (z) = det(zI r − A r ) . The roots of χ r (z) are called the reachable modes (r-modes) and the roots of χ r (z) are called the unreachable modes ( r -modes) of the system (A1). The r-modes and r -modes do not depend upon a particular choice of the matrix H transforming (A1) to the form (A4). The interaction between state and output is connected with the notion of observability. To examine this interaction we define first unobservable states. A state, treated as an initial condition for the state equation of (A1), is said to be unobservable if and only if the corresponding zero-input response is identically zero. Consider a subspace X o ⊆ R n characterized by the following properties [20]: (i) X o is invariant under A, i.e., Ax ∈ X o for each x ∈ X o , X o is contained in the kernel (the null-space) of the matrix C, i.e., (ii) Cx = 0 for each x ∈ X o , (iii) X o is the largest subspace which satisfies (i) and (ii), i.e., it contains any other A-invariant subspace which is contained in Ker C . The subspace X o is called the unobservable subspace of (A1). It consists of all unobservable states of the system and is described as  C   CA  n −1  = h Ker CA l , X o = Ker   .  l =0  n −1   CA

(A7)

 C   CA   is known as the observability matrix for (A1). where   .   n −1   CA

A state x which satisfies the condition x ∉ X o is said to be observable. If X o is the trivial subspace of the state space (i.e., X o = {0} ), then the system is called observable. If dim X o = n o < n (or, what is the same, rank of the observability matrix equals no = n − n o ), then there exists a nonsingular matrix H such

302

Appendix

that after the change of coordinates x' = H x matrices A, B, C take the form A o  0 

A12  , A o 

B o  B  ,  o

[0

Co ] ,

(A8)

with a n o x n o -dimensional A o and a r x (n − n o ) -dimensional C o . The pair ( A o , C o ) is an observable pair, i.e., it satisfies the condition  Co   C A  o o  = no . rank    .  no −1  C o A o 

In the new coordinates, the elements of X o are the points of R n having last no = n − n o components equal to zero, i.e., they are of the form x' = col ( x '1 , . . . , x ' no , 0, . . . , 0) . The matrix H can be formed as follows [10]. As last no = n − n o rows of H we take n o linearly independent rows of the observability matrix. The remaining no rows of H can be chosen arbitrarily provided that H is nonsingular. The decomposition (A8) splits the characteristic polynomial of A as χ(z) = χ o (z)χ o ( z) ,

(A9)

where χ o (z) = det(zI o − A o ) and χ o (z) = det(zI o − A o ) . The roots of χ o (z) are called the unobservable modes ( o -modes) and the roots of χ o (z) are called the observable modes (o-modes) of the system. The o -modes and o-modes remain unchanged under any choice of change of coordinates which transforms (A1) to (A8). A.2 Canonical Decomposition of the State Space The subspaces X r (the subspace of all reachable states) and X o (the subspace of all unobservable states), which have been constructed in the previous section for the system (A1), enable us to create a direct sum decomposition of the whole state space R n [4, 28, 29, 93]. Let Ξ1 be the subspace defined (uniquely) by Ξ1 := X r ∩ X o .

(A10)

Appendix A

303

Define the subspaces Ξ 2 , Ξ 3 , Ξ 4 by the following conditions X r = Ξ1 ⊕ Ξ 2 ,

(A11)

X o = Ξ1 ⊕ Ξ 3 ,

(A12)

R n = Ξ1 ⊕ Ξ 2 ⊕ Ξ 3 ⊕ Ξ 4 .

(A13)

In the above decomposition the subspaces involved are constructed in such way that they have the following interpretation: Ξ1 consists of states which are reachable and unobservable, Ξ 2 consists of states which are both reachable and observable, Ξ 3 consists of states which are neither reachable nor observable, Ξ 4 consists of states which are observable but not reachable. It is important to note that the subspaces Ξ 2 , Ξ 3 , Ξ 4 are not uniquely determined. However, as long as they satisfy conditions (A11), (A12), (A13) the above statements hold. Of course, in any such decomposition the dimensions of appropriate subspaces remain unchanged. The direct sum decomposition (A13) can be used to write the equations of the system in a special, upper triangular block form. To this end, we use a basis which is the union of bases for Ξ1 , Ξ 2 , Ξ 3 , Ξ 4 . In this new basis equations (A1) take the form [28, 29]  x ro (k + 1)   A r o  x (k + 1)   0 =  ro x r o (k + 1)  0     x r o (k + 1)   0

A12

A13

A ro 0

0 A ro

0

0

A14  A 24  A 34   A r o 

 x r o ( k )  B r o   x (k )     ro  + B r o  u(k ) x r o (k )  0       x r o (k )   0 

(A14)

[

y (k ) = 0 C ro

0 C ro

]

x r o (k )  x (k )   r o  + Du(k ) . x r o (k )    x r o (k ) 

The triple (subsystem) ( A r o , B r o , C ro ) is reachable and observable and constitutes a minimal (reachable and observable) representation of the transfer-function matrix G(z) of the system (A1). Since [10]

304

Appendix

G (z) = C(zI − A ) −1 B + D = C r o (zI r o − A r o ) −1 B r o + D ,

the order n ro of A r o is equal to the degree of the characteristic polynomial of G(z). Recall [10] that the characteristic polynomial of G(z) is defined as the least common denominator of all minors of G(z) (in computing the characteristic polynomial every minor must be reduced to an irreducible one; otherwise the result may be erroneous) and its roots are called the poles of G(z). The order of A r o is denoted by nr o . The triple (subsystem) ( A ro , B r o ,0) is reachable and unobservable.

The order of A r o is denoted by n r o . The triple (subsystem) ( A r o , 0, 0) is unreachable and unobservable. The order of A r o is denoted by n r o . The triple (subsystem) ( A r o , 0, C r o ) is unreachable and observable.

For the orders of diagonal blocks of the state matrix in (A14) the following relations hold [28] n r o + nr o = n r , n o = n r o + n r o , n = n r o + nr o + n r o + nr o .

For the system (A1) the order of the state matrix (n), the degree of G(z) ( n ro ) and the ranks of the reachability and observability matrices ( nr and n o ) can be treated as known. Hence, although the Kalman canonical form (A14) of (A1) is not unique, the orders of the diagonal blocks are determined uniquely by n, n r o = deg G (z), nr , no as n r o = nr − nro , n r o = n + nr o − nr − no , n r o = no − n r o .

(A15)

The canonical form (A14) splits the characteristic polynomial of A as χ(z) = χ r o (z)χ r o (z)χ r o (z)χ r o (z) ,

(A16)

where χ r o (z) = det (zI r o − A r o ) ,

χ r o (z) = det (zI r o − A r o ) ,

χ r o (z) = det (zI r o − A r o ) ,

χ r o (z) = det (zI r o − A r o ) .

Moreover, in any canonical form (A14) of the system (A1) the characteristic polynomials χ r o (z), χ r o ( z), χ r o (z), χ r o (z) of diagonal matrices of

Appendix B

305

the A-matrix remain the same (up to a constant). In this way the following objects are well defined. The roots of χ r o (z) are called the reachable and unobservable modes ( r o -modes) of the system (A1). Analogously, we define the ro -modes of (A1) as eigenvalues of A r o (they are also poles of G(z)), the r o -modes as eigenvalues of A r o and the ro -modes as eigenvalues of A r o . Finally, recall [4] that for finding the canonical form (A14) of (A1) we apply to (A1) the change of coordinates x' = H −1x , where submatrices of H := [H1

H2

H3

H4]

satisfy Im H1 = X r ∩ X o , Im [H1

H 2 ] = X r , Im [H1

H3 ] = X o .

Appendix B B.1 The Moore-Penrose Inverse of a Matrix Let A denote a real mxn matrix of rank p ≤ min{m, n} . A factorization A = H 1H 2 ,

(B1)

where H1 ∈ R mxp , H 2 ∈ R pxn , is called the skeleton factorization of A [18]. Although the skeleton factorization (B1) is not unique, however in any such factorization we have rank H1 = rank H 2 = rank A = p . In order to obtain the skeleton factorization (B1), we can take as columns of H1 arbitrary p linearly independent columns of A . Then the j-th, j = 1, . . . , n , column of A is a linear combination of columns of H1 with coefficients  c1, j   .  . c1, j , . . . , c p, j . As the j-th column of H 2 we take the vector   .    c p, j 

The Moore-Penrose inverse (known also as the Moore-Penrose pseudoinverse) of A is defined as follows.

306

Appendix

Definition B.1. [18, 5] A nxm matrix A + is called the Moore-Penrose

inverse of a mxn matrix A if and only if A A + A = A and there exist ma-

trices U and V such that A + = UA T = A T V . ◊ For a given matrix A there exists exactly one matrix A + . Moreover, by using the skeleton factorization (B1), A + can be determined as A + = H T2 (H 2 H T2 ) −1 (H1T H1 ) −1 H1T ,

(B2)

where H T2 (H 2 H T2 ) −1 = H +2 and (H1T H1 ) −1 H1T = H1+ . Independently upon a particular choice of matrices H1 , H 2 in the skeleton factorization of A , formula (B2) uniquely determines the MoorePenrose inverse A + of A. Moreover, A + has the properties [18]: (A + ) + = A , (A T ) + = (A + ) T , A + AA + = A + ,

(B3)

( AA + ) T = AA + , ( AA + ) 2 = AA + , ( A + A ) T = A + A, ( A + A ) 2 = A + A .

Furthermore, the following relations hold: Ker A = Im (I n − A + A ) = Ker A + A , Im A = Ker (I m − AA + ) = Im AA + , rank A = n − rank (I n − A + A) , +

(B4)

rank A = m − rank (I m − AA ) , I n − A + A = 0 ⇔ A has full column rank , I m − AA + = 0 ⇔ A has full row rank .

Remark B.1. All results presented above remain valid for a complex matrix A provided that the transpose of A will be replaced with its complex conjugate transpose. ◊ Consider a system of linear equations Ax = b , where A ∈ R mxn , b ∈ R m , x ∈ R n .

(B5)

Appendix B

307

We assume that the set of solutions of (B5) is non-empty (or, equivalently, rank A = rank [ A b] = p ≤ min{m, n} ). Let x o ∈ R n denote a solution of (B5) (i.e., Ax o = b ) and let A = H1H 2 be a skeleton factorization of A . Because p ≤ m and p ≤ n , hence H1 has full column rank and H 2 has full row rank. We can treat matrices A , H1 , H 2 as appropriate mappings; i.e., A : R n → R m , H1 : R p → R m , H 2 : R n → R p . At such treatment, H 2 is surjective whereas H1 is injective.

For x o we have H1H 2 x o = H1 (H 2 x o ) = b . Let y o = H 2 x o ∈ R p . Thus H1y o = b , i.e., y o is a solution of the equation H1y = b . Since H1 is injective, y o is the unique solution of this equation. Moreover, it is easy to verify that y o = (H1T H1 ) −1 H1T b . We consider now the equation y o = H 2 x . Of course, x o is a solution of this equation. However, since H 2 is surjective, the set of all solutions of this equation is simply the inverse image under H 2 of the point y o . We denote this set by H −2 1 ({y o }) . We shall show now that the vector x ∗o := H +2 y o is a solution of the equation y o = H 2 x . In fact, since H +2 = H T2 ( H 2 H T2 ) −1 , we can write x ∗o = H T2 (H 2 H T2 ) −1 y o , and then, setting x ∗o instead of x on the right-hand side of this equation, we obtain H 2 x ∗o = H 2 (H T2 (H 2 H T2 ) −1 y o ) = (H 2 H T2 )(H 2 H T2 ) −1 y o = y o

which proves the claim. Finally, one can observe that x ∗o is also a solution of (B5). This fact follows from the relations Ax ∗o = H1H 2 x ∗o = H1 (Η 2 x ∗o ) = H1y o = b . Furthermore, the following simple calculations show that x ∗o is determined uniquely (i.e., it does not depend upon a particular choice of matrices H1 , H 2 in the skeleton factorization of A ) x ∗o = H T2 (H 2 H T2 ) −1 y o = H T2 (H 2 H T2 ) −1 (H1T H1 ) −1 H1T b = H +2 H 1+ b = A + b .

In this way we have shown that x ∗o = A + b constitutes a solution of (B5).

308

Appendix

On the other hand, the difference of any two solutions of (B5) is a solution of the homogeneous equation Ax = 0 . Naturally, each vector of the

form x ∗o + x h , where x h is a solution of Ax = 0 , is a solution of (B5). Conversely, any solution, say x o , of (B5) can be written in such form since we can express x o as x o = x ∗o + (x o − x ∗o ) , where A(x o − x ∗o ) = 0 . Hence we have proved the following. Lemma B.1.

In (B5) let rank A = rank [ A b] = p ≤ min{m, n} . Then

x ∗o = A + b constitutes a solution of (B5), i.e., A A + b = b . Moreover, a

point x o ∈ R n is a solution of (B5) if and only if x o = x ∗o + x h , where x h satisfies Ax h = 0 . ◊ It is clear that the set of all solutions of (B5) may be viewed as the hyperplane in R n which passes through the point x ∗o = A + b and is obtained by parallel shifting of the subspace Ker A := {x ∈ R n : Ax = 0} along the vector x ∗o = A + b . Among all solutions of (B5) the solution x ∗o = A + b has the smallest norm, i.e., x ∗o ≤ x o , where x o denotes an

arbitrary solution and x := x T x for x ∈ R n . B.2 Singular Value Decomposition of a Matrix Let H be a mxn real matrix. Then H T H is square of order n. Clearly H T H is symmetric and all its eigenvalues are real. Since H T H is positive semidefinite, its eigenvalues are all nonnegative. Let λ2i , i = 1,..., n, denote eigenvalues of H T H . The set {λ i ≥ 0, i = 1,..., n} is called the singular values of H. We arrange {λ i } such that λ21 ≥ λ22 ≥ ... ≥ λ2n ≥ 0 . Suppose now that the rank of H is r. Then H T H is also of rank r. Hence we have λ21 ≥ λ22 ≥ ... ≥ λ2r > 0 and λ2r +1 = ... = λ2n = 0 . Let q i , i = 1,..., n, be the orthonormal eigenvectors of H T H associated with λ2i , i = 1,..., n. Define

[

]

Q = [q1 q 2 ... q r q r +1 ... q n ] = Q1 Q 2 .

Appendix B

Denote

the

diagonal

matrices

Σ 2 = diag{λ21 , . . . , λ2r }

Σ = diag{λ1 , . . . , λ r } . Define the mxr matrix R1 by R1 := HQ1Σ

309

and −1

. The

columns of R1 are orthonormal. Let R 2 be chosen so that R = [R1 R 2 ]  Σ 0 is orthogonal. Then R T HQ =  .  0 0

Theorem B.1. (Singular Value Decomposition) [5, 9, 10, 61] Every mxn real matrix H of rank r can be written in the form (called the singular value decomposition (SVD) of H) H = U Λ VT ,

(B6)

 Σ 0 where Λ =   is mxn-dimensional, Σ = diag{λ1 , . . . , λ r } with posi 0 0 tive singular values λ1 ≥ λ 2 ≥ ... ≥ λ r > 0 of H and U = R is orthogonal

(i.e., T

U T U = UU T = I m )

V V = VV

T

and

V =Q

is

orthogonal

(i.e.,

= I n ). ◊

Although Σ is uniquely determined by H, however the orthogonal matrices U and V are not necessarily unique. One useful application of SVD is in computing the Moore-Penrose inverse of a matrix. Lemma B.2. [61] Let a mxn real matrix H of rank r be written in the form (B6). Then H+ = V Λ+UT ,

(B7)

 Σ −1 0 −1 −1 −1 where Λ + =   is nxm and Σ = diag{λ1 , . . . , λ r } . ◊ 0  0

B.3 Endomorphisms of a Linear Space over C Let F ∈ C n x n be a complex matrix and let λ ∈ σ(F ) be an eigenvalue of F of multiplicity m ≥ 1 . For any integer j = 0, 1, 2,... define the subspace V λj := {v ∈ C n : (F − λI) j v = 0} ; for j = 0 let V0λ := {0} .

(B8)

310

Appendix

Immediately from (B8) it follows that V λj ⊆ V λj +1 , j = 0,1,2,… . Denote q j = dimV λj and let p j +1 := q j +1 − q j , j = 0,1,2,... . Since dim C n < ∞ , the sequence {p i }i∞=1 is almost equal to zero; i.e., there exists a natural number jo such that p jo > 0 and p j = 0 for any j > jo .

At the above notation the following results hold. Lemma B.3. [33] (i)

F (V λj ) ⊆ V λj , j = 0,1,2,... , i.e., V λj is F-invariant;

(ii)

q jo = m ;

(iii)

0 < p jo ≤ ... ≤ p1 . ◊

Lemma B.4. [33] Let {λ1 , . . . , λ r } denote the set of all distinct eigenvalues of F and let m1 , . . . , m r denote their multiplicities. If j o (λ i ) , λ

where i = 1,..., r , is defined as above, then the subspaces V j i ( λ ) , i = 1,..., o i r, are linearly independent. ◊ Recall that linear subspaces V1 ,..., Vr are linearly independent if for arbitrary nonzero vectors v1 ∈ V1 ,..., v r ∈ Vr vanishing of their linear combination implies that all its coefficients vanish, i.e., r

∑ α i v i = 0 ⇒ α i = 0 for every i = 1,..., r.

i =1

Lemma B.5. [33] Let λ be an eigenvalue of F of multiplicity m. Then: the sequence V λj , j = 0,1,…, of F-invariant subspaces associated with λ consists of a finite number of distinct elements and satisfies the relations (i)

{0} = V0λ ⊂ V1λ ⊂ ... ⊂ V λj = V λj +1 = V λj + 2 = ... ; o o o

the sequence of integers p j , j = 1,2,... , associated with λ is nonincreasing and almost equal to zero, i.e.,

(ii)

p1 ≥ ... ≥ p jo > 0 = p jo +1 = p jo + 2 = ... ;

(iii)

for the sets {V λj } jo=1 , {p j } jo=1 the following relations hold j

j

Appendix B

dimV λj = p1 + ... + p j

and

311

dimV λj = p1 + ... + p jo = m . ◊ o

j

The set {p j } jo=1 is called the characteristic decomposition of the algebraic multiplicity m of λ . The subspace V λj is known as the invariant o subspace of F associated with λ , whereas V1λ is called the eigenspace of F corresponding to λ ( dimV1λ determines, by definition, the geometric multiplicity of λ ). Because jo ≤ m , by virtue of Lemma B.5 (i), we have Vmλ = V λj . o

Theorem B.2. [33] Let λ1 , . . . , λ r denote all distinct eigenvalues of F ∈ C n x n and let m1 , . . . , m r denote their multiplicities. Then r

λ

λ

C n = ⊕ Vm i , where Vmi = {v ∈ C n : (F − λ i I ) mi v = 0} , i = 1,…, r. ◊ i i i =1

The following result is known as the orthogonal decomposition of domain an codomain of a matrix. It shows in particular that any linear map represented by a mxn real matrix A can be made invertible by restricting its domain and codomain. The same holds for the transpose of A. Lemma B.6. [9] Consider a mxn real matrix A of rank p. Then R n = dom A = Im A T ⊕ Ker A , R m = codom A = Im A ⊕ Ker A T , rank A = rank A T = p .

the restrictions of A on Moreover, denoting by A ImAT and A T Im A Im A T and A T on Im A , respectively, we have

A ImAT : Im A T → Im A

is a bijection;

AT

is a bijection;

Im A

:Im A → Im A T

Im A T = Im A T A , Im A = Im AA T ,

Ker A = Ker A T A , Ker A T = Ker AA T ,

312

Appendix

rank AA T = rank A = rank A T = rank A T A = p . ◊

Appendix C C.1 Polynomial and Rational Matrices Denote by R[z] the ring of polynomials in one complex variable z with coefficients in the field of real numbers R and let R (z) denote the field of rational functions in the variable z with coefficients in R . The symbol A pxq stands for the set of pxq matrices with entries in the set A (for instance, R pxq , C pxq , R[z] pxq or R (z) pxq ). Let M (z) ∈ R[z]mxn and let r be an integer such that 0 ≤ r ≤ min{m, n} . The polynomial matrix M(z) is said to have the normal (determinantal) rank r if and only if there exists at least one rxr minor which is not the zero polynomial and for each q > r every qxq minor is the identically zero polynomial. It is said also that M(z) has rank r over R[z] , or simply, normal rank r. If r = m ( r = n ) , then M(z) is said to have full normal row rank (full normal column rank). For a rational matrix M (z) ∈ R (z) mxn the normal rank is defined as rank over the field R (z) . Let M (z) ∈ R[z]mxn . Then the rank of M(z) over R[z] is equal to the rank of M(z) over R (z) , i.e., M(z) has the same normal rank as a polynomial matrix and as a rational matrix. Moreover, the maximal number of linearly independent columns (rows) over R[z] equals the maximal number of linearly independent columns (rows) over the field R (z) . The maximal number of linearly independent columns (rows) over R (z) is called the normal column rank (the normal row rank). For any polynomial matrix M (z) ∈ R[z]mxn the normal rank equals the normal column rank and equals the normal row rank. Let M (z) ∈ R[z]mxn and let λ ∈ C . We say that M(z) has rank r at λ if and only if the matrix M (λ ) ∈ C mxn has rank r. The rank of a polynomial matrix M (z) ∈ R[z]mxn at a given λ ∈ C is called also the local rank at λ . For any polynomial matrix M (z) ∈ R[z]mxn its normal rank equals its local rank at λ except for at most a finite number of points λ ∈ C .

Appendix C

313

For a rational matrix M (z) ∈ R (z) mxn and for any λ ∈ C which is not a pole of M(z) the local rank of M(z) at λ is defined as the rank of M (λ ) ∈ C mxn .

Let K denote R[z] or R (z) and let P (z) ∈ K mxn and F (z) ∈ K nxm . Then the following useful relations take place [9]: (i)

(I m + P(z)F( z)) −1 ∈ K mxm ⇔ (I n + F(z)P (z)) −1 ∈ K nxn , (I m + P (z)F (z)) −1 = I m − P (z) (I n + F( z) P( z)) −1 F (z) ,

(ii)

(iii)

(I n + F (z)P(z)) −1 = I n − F (z) (I m + P (z)F (z)) −1 P( z) ,

If (I m + P(z)F (z)) −1 ∈ K mxm or (I n + F (z)P( z)) −1 ∈ K nxn , then P (z) (I n + F (z)P (z)) −1 = (I m + P (z)F (z)) −1 P( z) ∈ K mxn ; det(I n + F (z)P (z)) = det (I m + P (z)F (z)) .

Let M (z) ∈ R[z]mxn . Elementary row operations on M(z) are of three types: a) interchange two rows, b) multiply a row by an invertible element of R[z] (i.e., a nonzero constant), c) add to a given row another row multiplied by an element of R[z] . Elementary column operations are similarly defined (i.e., it is sufficient to replace in the above definition the term “row” by “column”). An operation on a polynomial matrix is said to be an elementary operation if and only if it is a row or column elementary operation. Elementary operations are used to reduce polynomial matrices to standard forms (e.g., Hermite and Smith forms [9]). Each elementary operation leaves the normal rank unchanged. Every polynomial matrix M(z) can be reduced to the Smith form by a finite number of elementary operations performed recursively on M(z). Hermite row form of a full column rank matrix. Let M (z) ∈ R[z]mxn have full column rank over R[z] . Then there exists a unimodular matrix L(z) ∈ R[z]mxm (obtained by elementary row operations) such that

314

Appendix

R (z) L ( z )M ( z ) =    0  nonsingular.

(iv)

with R (z) ∈ R[z]nxn upper triangular and

Hermite column form of a full row rank matrix. Let M (z) ∈ R[z]mxn have full row rank over R[z] . Then there exists an unimodular matrix R (z) ∈ R[z]nxn (obtained by elementary column operations) such that

(v)

M (z)R (z) = [0 C(z)]

with C(z) ∈ R[ z] mxm upper triangular

and nonsingular. A rational matrix G (z) ∈ R (z) mxn is said to be proper if G (∞) is a finite (zero or nonzero) constant matrix. G(z) is said to be strictly proper if G (∞) = 0 . As is known (Appendix A.2), the characteristic polynomial of a proper rational matrix G(z) is defined to be the least common denominator of all minors of G(z), whereas the degree of G(z) (degG(z)) is defined to be the degree of the characteristic polynomial of G(z) (it is also called the McMillan degree or the Smith-McMillan degree). A different but equivalent definition of the characteristic polynomial and the degree of a proper rational matrix G (z) ∈ R (z) mxn is as follows [10]. Let G(z) be factored as G (z) = N r (z)D r−1 (z) where the polynomial matrices D r (z) ∈ R[z]nxn and N r (z) ∈ R[z]mxn are right coprime. Then the characteristic polynomial of G(z) is defined as det D r (z) and the degree of G(z) is defined as degG (z) = deg det D r (z) . Recall

[9, 10]

N (z) ∈ R[z]

mxn

that

polynomial

matrices

D(z) ∈ R[z]nxn

and

(where D(z) is nonsingular) are right coprime if and only

 D(z)  if the (n+m)xn matrix   has full column rank for each z ∈ C . N (z) Let S(A,B,C,D) of the form (A1) be a realization of a proper rational matrix G(z) (over the field R (z) ), i.e., S(A,B,C,D) is such that G (z) = D + C(zI − A ) −1 B and A, B, C, D are constant real matrices. Then S(A,B,C,D) is said to be the irreducible (reachable and observable) realization if and only if det(zI − A ) = k (characteristic polynomial of G (z))

(where k is a nonzero constant) or equivalently,

Appendix C

315

dim A = degG (z) .

The irreducible realizations are called also the minimal realizations. Consider a proper rational matrix G (z) ∈ R (z) rxm and let S(A,B,C,D), where A, B, C, D are respectively nxn, nxm, rxn and rxm matrices, denote an irreducible realization of G(z). By P(z) denote the system matrix of S(A,B,C,D), i.e.,  zI − A − B  . P (z) :=  n D   C

Then

(vi)

In  −C(zI − A) −1 n 

0   zI n − A − B  I r   C D 

−B zI − A   zI n − A − B  = n = −1 C(zI n − A) B + D  0 G (z)  0

and consequently,

(vii)

 zI − A − B  normal rank P(z) = normal rank  n G (z)  0

= n + normal rank G (z).

In particular, if G(z) is square (i.e., m = r ), then from (vi) one obtains (viii)

det P (z) = det (zI n − A ) detG (z) .

Note that relations (vi) and (viii) remain valid without the assumption that S(A,B,C,D) is an irreducible realization of G(z) (i.e., they hold for any given realization of G(z)).

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Index

(A,B)-invariant subspace 197 admissible input 9, 299 algebraic multiplicity 41 almost output-zeroing input 132 Cayley-Hamilton theorem 78, 87 characteristic polynomial of a rational (transfer-function) matrix 304, 314 control vector 2, 79 decoupling zero 3 degenerate transfer-function matrix 12, 48 degree of a rational matrix 314 Desoer-Schulman zero 43 diagonal decoupling 129, 138 disturbance decoupling 101 dynamical interpretation of poles 35 elementary operation 313 free evolution 30 fundamental matrices 281 Hermite column form 18, 51, 314 Hermite row form 17, 52, 313 initial state 21 input decoupling zero 3, 11, 16, 272 input-output decoupling zero 4, 12, 16 input vector 299 input-zero direction 4, 11, 272 invariant factor 3 invariant zero 5, 11, 272

Kalman canonical form (decomposition) 9, 304 linear equations 299 local rank 312 Markov parameters 6, 282 maximal output-nulling controlled invariant subspace 197, 245 minimal subsystem 11 mode observable 302 reachable 301 reachable and observable 10, 305 reachable and unobservable 10, 305 unobservable 302 unreachable 301 unreachable and observable 10, 305 unreachable and unobservable 10, 305 monic polynomial 3 Moore-Penrose inverse (pseudoinverse) 6, 65, 284, 305, 309 multivariable zero 1 nondegenerate transfer-function matrix 12, 48 nonlinear problem 9, 12 nonsingular coordinate transformation in the state space 18 nonsingular transformation of the inputs 18 nonsingular transformation of the outputs 18 normal column rank 312

324

Index

normal rank 1, 312 normal row rank 312 observability 299 observability matrix 301 orthogonal decomposition 311 orthogonal matrix 309 output decoupling zero 4, 11, 16, 272 output feedback to the inputs 18 output-nulling controlled invariant subspace 245 output vector 2, 9, 299 output-zeroing input 21 output-zeroing problem 21 pole of a system 35 of a system transfer-function ma trix 10 polynomial matrix 3, 312 proper rational matrix 314 rational matrix 312 reachability 299 reachability matrix 300 reachable subspace 300 realization of a proper rational matrix 314 irreducible (minimal) 314 relative degree 60 reproducing the reference output 95 right coprime 314 set of invariant zeros 5, 12, 272 of Smith zeros 5, 36, 272 singular value decomposition (SVD) 6, 41, 149, 151, 173, 275, 290, 309 singular values 308 skeleton factorization 65, 305 Smith canonical form 3 Smith transmission zero of a system 3, 272 Smith zero 3, 271 Smith zero of a transfer-function

matrix 4 state feedback to the inputs 18 state space form of a system 299 state vector 2, 9, 299 state-zero direction 4, 11, 272 strictly proper rational matrix 314 system degenerate 5, 11, 45, 272 discrete-time 2, 9, 271, 299 minimal 24, 35 multi input multi output (MIMO) 1 nondegenerate 5, 11, 47, 274 proper 9 single input single output (SISO) 13 singular 271 strictly proper 9 with the identically zero transferfunction matrix 50, 198 system matrix 2, 271 system zero 107 tracking the reference output 98, 99 transfer-function matrix 2, 281, 303 transmission blocking 22 transmission zero of a system 11, 20 transmission zero of a transfer-function matrix 12 uniform rank system 80, 91 unimodular matrix 3 unitary matrix 15 unobservable subspace 301 vector relative degree 129 Weierstrass canonical form 282 zero at infinity 61, 62, 109, 111, 133 zero direction 4 zero dynamics 21 zero polynomial 3, 272

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