Polynomial and Rational Matrices: Applications in Dynamical Systems Theory

513 Pages·2007·3.922 MB·English

by  Tadeusz Kaczorek (auth.)

Communications and Control Engineering
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Polynomial and
Rational Matrices
Applications in Dynamical Systems Theory
123
TadeuszKaczorek,Prof.drhab.inz˙.
InstituteofControlandIndustrialElectronics
FacultyofElectricalEngineering
WarsawUniversityofTechnology
00-662Warsaw
ul.Koszykowa75m.19
Poland
SeriesEditors
E.D.Sontag·M.Thoma·A.Isidori·J.H.vanSchuppen
BritishLibraryCataloguinginPublicationData
Kaczorek,T.(Tadeusz),1932-
Polynomialandrationalmatrices:applicationsin
dynamicalsystemstheory.-(Communicationsandcontrol
engineering)
1.Automaticcontrol-Mathematics2.Electrical
engineering-Mathematics3.Matrices4.Linearsystems
5.Polynomials
I.Title
629.8’312
ISBN-13:ISBN-13:9781846286049
ISBN-10:ISBN-10:1846286042
LibraryofCongressControlNumber:2006936878
CommunicationsandControlEngineeringSeriesISSN0178-5354
ISBN 978-1-84628-604-9 e-ISBN 1-84628-605-0 Printedonacid-freepaper
©Springer-VerlagLondonLimited2007
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Preface
This monograph covers the selected applications of polynomial and rational
matrices to the theory of both continuous-time and discrete-time linear systems. It
is an extended English version of its preceding Polish edition, which was based on
the lectures delivered by the author to the Ph.D. students of the Faculty of
Electrical Engineering at Warsaw University of Technology during the academic
year 2003/2004.
The monograph consists of eight chapters, an appendix and a list of references.
Chapter 1 is devoted to polynomial matrices. It covers the following topics:
basic operations on polynomial matrices, the generalised Bézoute theorem, the
Cayley(cid:16)Hamilton theorem, elementary operations on polynomial matrices, the
choosing of a basis for a space of polynomial matrices, equivalent polynomial
matrices, reduced row matrices and reduced column matrices, the Smith canonical
form of polynomial matrices, elementary divisors and zeros of polynomial
matrices, similarity of polynomial matrices, the Frobenius and Jordan canonical
forms, cyclic matrices, pairs of polynomial matrices, the greatest common divisors
and the smallest common multiplicities of matrices, the generalised Bezoute
identity, regular and singular matrix pencil decompositions, and the
Weierstrass(cid:16)Kronecker canonical form of a matrix pencil.
Rational functions and matrices are discussed in Chap. 2. With the basic
definitions and operations on rational functions introduced at the beginning, the
following issues are subsequently addressed: decomposition into the sum of
rational functions, operations on rational matrices, the decomposition of a matrix
into the sum of rational matrices, the inverse matrix of a polynomial matrix and its
reducibility, the McMillan canonical form of rational matrices, the first
factorization of rational matrices and the application of rational matrices in the
synthesis of control systems.
Chapter 3 addresses normal matrices and systems. A rational matrix is called
normal if every non-zero minor of size 2 of the polynomial matrix of the
denominator is divisible by the minimal polynomial of this matrix. It has been
proved that a rational matrix is normal if and only if its McMillan polynomial is
equal to the smallest common denominator of all the elements of the rational
matrix. Further, the following issues are discussed: the fractional forms of normal
vi Preface
matrices, the sum and product of normal matrices, the inverse matrix of a normal
matrix, the decomposition of normal matrices into the sum of normal matrices, the
structural decomposition of normal matrices, the normalisation of matrices via
feedback and electrical circuits as examples of normal systems.
The problem of the realisation of normal matrices is addressed in Chap. 4. The
problem formulation is provided; further the following issues are discussed:
necessary and sufficient conditions for the existence of minimal and cyclic
realisations, methods of computing the realisation with the state matrix in both the
Frobenius and Jordan canonical forms, structural stability and the computation of
the normal transfer function matrix
Chapter 5 is devoted to normal singular systems. In particular it focuses on
discrete singular systems, cyclic pairs of matrices, the normal inverse matrices of
cyclic pairs, normal transfer matrices, reachability and cyclicity of singular systems,
cyclicity of feedback systems, computation of equivalent standard systems for
singular systems. It is shown that electrical circuits consisting of resistances and
inductances or resistances and capacities, together with ideal voltage (current)
sources, constitute examples of singular continuous-time systems. Both the Kalman
decomposition and the structural decomposition of the transfer matrix are generalised
to the case of singular systems.
Polynomial matrix equations, both rational and algebraic, are discussed in
Chap. 6. The chapter begins with unilateral polynomial equations with two
unknown matrices. Subsequently the following issues are addressed: the
computation of minimal degree solutions to matrix equations, bilateral polynomial
equations, the computation of rational solutions to polynomial equations, matrix
equations of the m-th order, the Kronecker product of matrices and its applications,
and the methods for computing solutions to Sylvester and Lapunov matrix
equations.
Chapter 7, the last one, is devoted to the problem of realisation and perfect
observers for linear systems. A new method for computing minimal realisation for
a given improper transfer matrix is provided together with the existence conditions;
subsequently the methods for computing full and reduced order observers, as well
as functional perfect observers, for 1D and 2D systems are given.
In Chap. 8 some new results (published and unpublished) are presented on
positive linear discrete-time and continuous-time systems with delays: asymptotic
and robust stability, reachability, minimum energy control and positive realisation
problem.
The Appendix contains some basic definitions and theorems pertaining to the
controllability and observability of linear systems.
The monograph contains some original results of the author, most of which
have already been published.
It is haped that this monograph will be of value to Ph.D. students and
researchers from the field of control theory and circuit theory. It can be also
recommended for undergraduates in electrical engineering, electronics,
mechatronics and computer engineering.
I would like to express my gratitude to Professors M. Bus(cid:225)owicz and J. Klamka,
the reviewers of the Polish version of the book, for their valuable comments and
Preface vii
suggestions, which helped to improve this monograph. I also wish to thank my
Ph.D. students, the first readers of the manuscript, for their remarks.
I wish to extend my special thanks to my Ph.D. students Maciej Twardy,
Konrad Markowski and Stefan Krzemi(cid:276)ski for their valuable help in the
preparation of this English edition.
T. Kaczorek
Contents
Notation.................................................................................................................xv
1PolynomialMatrices............................................................................................1
1.1 Polynomials...................................................................................................1
1.2 Basic Notions and Basic Operations on Polynomial Matrices.......................5
1.3 Division of Polynomial Matrices...................................................................9
1.4 Generalized Bezoute Theorem and the Cayley–Hamilton Theorem...........16
1.5 Elementary Operations on Polynomial Matrices.........................................20
1.6 Linear Independence, Space Basis and Rank of Polynomial Matrices........23
1.7. Equivalents of Polynomial Matrices...........................................................27
1.7.1 Left and Right Equivalent Matrices...................................................27
1.7.2 Row and Column Reduced Matrices..................................................30
1.8 Reduction of Polynomial Matrices to the Smith Canonical Form...............32
1.9 Elementary Divisors and Zeros of Polynomial Matrices.............................37
1.9.1 Elementary Divisors...........................................................................37
1.9.2 Zeros of Polynomial Matrices............................................................39
1.10 Similarity and Equivalence of First Degree Polynomial Matrices.............42
1.11 Computation of the Frobenius and Jordan Canonical Forms of Matrices..45
1.11.1 Computationof theFrobenius Canonical Formof a Square
Matrix..............................................................................................45
1.11.2 Computation of the Jordan Canonical Form of a Square Matrix......47
1.12 Computation of Similarity Transformation Matrices.................................49
1.12.1 Matrix Pair Method..........................................................................49
1.12.2 Elementary Operations Method........................................................54
1.12.3 Eigenvectors Method........................................................................57
1.13 Matrices of Simple Structure and Diagonalisation of Matrices.................59
1.13.1 Matrices of Simple Structure............................................................59
1.13.2 Diagonalisation of Matrices of Simple Structure.............................61
1.13.3 Diagonalisation of anArbitrarySquare Matrix by the Useofa
Matrix with Variable Elements........................................................65
1.14 Simple Matrices and Cyclic Matrices........................................................67
1.14.1 Simple Polynomial Matrices............................................................67
x Contents
1.14.2 Cyclic Matrices................................................................................69
1.15 Pairs of Polynomial Matrices.....................................................................75
1.15.1 Greatest Common DivisorsandLowest Common Multiplicities of
Polynomial Matrices........................................................................75
1.15.2 Computationof Greatest CommonDivisorsof a Polynomial
Matrix..............................................................................................77
1.15.3 Computation of Greatest Common Divisors and Smallest Common
Multiplicities of Polynomial Matrices.............................................78
1.15.4 Relatively Prime Polynomial Matrices and the Generalised Bezoute
Identity.............................................................................................84
1.15.5 Generalised Bezoute Identity...........................................................86
1.16 Decomposition of Regular Pencils of Matrices.........................................87
1.16.1 Strictly Equivalent Pencils...............................................................87
1.16.2WeierstrassDecomposition of Regular Pencils................................92
1.17 Decomposition of Singular Pencils of Matrices........................................95
1.17.1Weierstrass–KroneckerTheorem.....................................................95
1.17.2 Kronecker IndicesofSingularPencils and Strict Equivalenceof
Singular Pencils.............................................................................102
2 RationalFunctions andMatrices...................................................................107
2.1 Basic Definitions and Operations on Rational Functions..........................107
2.2 Decomposition of a Rational Function into a Sum of Rational Functions.116
2.3 Basic Definitions and Operations on Rational Matrices............................124
2.4 Decomposition of Rational Matrices into a Sum of Rational Matrices.....128
2.5 The Inverse Matrix of a Polynomial Matrix and Its Reducibility..............132
2.6 Fraction Description of Rational Matrices and the McMillan Canonical
Form..........................................................................................................136
2.6.1 Fractional Forms of Rational Matrices.............................................136
2.6.2 Relatively Prime Factorization of Rational Matrices.......................146
2.6.3Conversion of aRational Matrix intothe McMillan Canonical
Form...............................................................................................152
2.7 Synthesis of Regulators.............................................................................155
2.7.1 SystemMatrices and theGeneralProblemof Synthesis of
Regulators......................................................................................155
2.7.2 Setof RegulatorsGuaranteeing Given Characteristic Polynomials
of a Closed-loop System ...............................................................159
3 Normal MatricesandSystems........................................................................163
3.1 Normal Matrices........................................................................................163
3.1.1 Definition of the Normal Matrix......................................................163
3.1.2 Normality of the Matrix [Is – A]-1 for a Cyclic Matrix....................164
3.1.3 Rational Normal Matrices................................................................168
3.2 Fraction Description of Normal Matrices..................................................170
3.3 Sum and Product of Normal Matrices and Normal Inverse Matrices........175
3.3.1 Sum and Product of Normal Matrices..............................................175
3.3.2 The Normal Inverse Matrix..............................................................180
3.4 Decomposition of Normal Matrices...........................................................182
Contents xi
3.4.1Decomposition of Normal Matrices into theSumofNormal
Matrices.........................................................................................182
3.4.2 Structural Decomposition of Normal Matrices................................185
3.5 Normalisation of Matrices Using Feedback...............................................191
3.5.1 State-feedback .................................................................................191
3.5.2 Output-feedback...............................................................................197
3.6 Electrical Circuits as Examples of Normal Systems..................................200
3.6.1 Circuits of the Second Order............................................................200
3.6.2 Circuits of the Third Order...............................................................203
3.6.3 Circuits of the Fourth Order and the General Case..........................210
4 The Problem ofRealization ...........................................................................219
4.1 Basic Notions and Problem Formulation...................................................219
4.2 Existence of Minimal and Cyclic Realisations..........................................220
4.2.1 Existence of Minimal Realisations...................................................220
4.2.2 Existence of Cyclic Realisations......................................................224
4.3 Computation of Cyclic Realisations..........................................................226
4.3.1Computationofa RealisationwiththeMatrixAinthe Frobenius
CanonicalForm..............................................................................226
4.3.2Computationofa Cyclic Realisationwith MatrixAinthe Jordan
CanonicalForm..............................................................................232
4.4 Structural Stability and Computation of the Normal Transfer Matrix.......244
4.4.1 Structural Controllability of Cyclic Matrices...................................244
4.4.2 Structural Stability of Cyclic Realisation.........................................245
4.4.3Impact of the Coefficients oftheTransferFunction onthe System
Description.....................................................................................247
4.4.4Computationof theNormal TransferMatrix on theBasis of Its
Approximation...............................................................................249
5 Singularand Cyclic NormalSystems.............................................................255
5.1 Singular Discrete Systems and Cyclic Pairs..............................................255
5.1.1 Normal Inverse Matrix of a Cyclic Pair...........................................257
5.1.2 Normal Transfer Matrix...................................................................260
5.2 Reachability and Cyclicity.........................................................................264
5.2.1 Reachability of Singular Systems.....................................................264
5.2.2 Cyclicity of Feedback Systems........................................................267
5.3Computation of Equivalent Standard Systems for Linear Singular
Systems.....................................................................................................272
5.3.1Discrete-time Systemsand Basic Notions ......................................272
5.3.2 Computation of Fundamental Matrices............................................276
5.3.3 Equivalent Standard Systems...........................................................279
5.3.4Continuous-time Systems ...............................................................282
5.4 Electrical Circuits as Examples of Singular Systems................................285
5.4.1 RL Circuits.......................................................................................285
5.4.2 RC Circuits.......................................................................................288
5.5 Kalman Decomposition.............................................................................291
5.5.1 Basic Theorems and a Procedure for System Decomposition..........291