Polynomial and Rational Matrices : Applications in Dynamical Systems Theory (Communications and Control Engineering)

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Polynomial and Rational Matrices : Applications in Dynamical Systems Theory (Communications and Control Engineering)

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  • 製本 Hardcover:ハードカバー版/ページ数 503 p./サイズ 10 illus.
  • 言語 ENG
  • 商品コード 9781846286049
  • DDC分類 629.83120151

Full Description


This book reviews new results in the application of polynomial and rational matrices to continuous- and discrete-time systems. It provides the reader with rigorous and in-depth mathematical analysis of the uses of polynomial and rational matrices in the study of dynamical systems. It also throws new light on the problems of positive realization, minimum-energy control, reachability, and asymptotic and robust stability.

Table of Contents

Notation                                           xv
1 Polynomial Matrices 1
1.1 Polynomials 1
1.2 Basic Notions and Basic Operations on 5
Polynomial Matrices
1.3 Division of Polynomial Matrices 9
1.4 Generalized Bezoute Theorem and the 16
Cayley幽amilton Theorem
1.5 Elementary Operations on Polynomial 20
Matrices
1.6 Linear Independence, Space Basis and 23
Rank of Polynomial Matrices
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 32
Smith Canonical Form
1.9 Elementary Divisors and Zeros of 37
Polynomial Matrices
1.9.1 Elementary Divisors 37
1.9.2 Zeros of Polynomial Matrices 39
1.10 Similarity and Equivalence of First 42
Degree Polynomial Matrices
1.11 Computation of the Frobenius and Jordan 45
Canonical Forms of Matrices
1.11.1 Computation of the Frobenius 45
Canonical Form of a Square Matrix
1.11.2 Computation of the Jordan Canonical 47
Form of a Square Matrix
1.12 Computation of Similarity Transformation 49
Matrices
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 59
Diagonalisation of Matrices
1.13.1 Matrices of Simple Structure 59
1.13.2 Diagonalisation of Matrices of 61
Simple Structure
1.13.3 Diagonalisation of an Arbitrary 65
Square Matrix by the Use of a Matrix with
Variable Elements
1.14 Simple Matrices and Cyclic Matrices 67
1.14.1 Simple Polynomial Matrices 67
1.14.2 Cyclic Matrices 69
1.15 Pairs of Polynomial Matrices 75
1.15.1 Greatest Common Divisors and Lowest 75
Common Multiplicities of Polynomial Matrices
1.15.2 Computation of Greatest Common 77
Divisors of a Polynomial Matrix
1.15.3 Computation of Greatest Common 78
Divisors and Smallest Common Multiplicities
of Polynomial Matrices
1.15.4 Relatively Prime Polynomial Matrices 84
and the Generalised Bezoute Identity
1.15.5 Generalised Bezoute Identity 86
1.16 Decomposition of Regular Pencils of 87
Matrices
1.16.1 Strictly Equivalent Pencils 87
1.16.2 Weierstrass Decomposition of Regular 92
Pencils
1.17 Decomposition of Singular Pencils of 95
Matrices
1.17.1 Weierstrass揖ronecker Theorem 95
1.17.2 Kronecker Indices of Singular 102
Pencils and Strict Equivalence of Singular
Pencils
2 Rational Functions and Matrices 107
2.1 Basic Definitions and Operations on 107
Rational Functions
2.2 Decomposition of a Rational Function into 116
a Sum of Rational Functions
2.3 Basic Definitions and Operations on 124
Rational Matrices
2.4 Decomposition of Rational Matrices into a 128
Sum of Rational Matrices
2.5 The Inverse Matrix of a Polynomial Matrix 132
and Its Reducibility
2.6 Fraction Description of Rational Matrices 136
and the McMillan Canonical Form
2.6.1 Fractional Forms of Rational Matrices 136
2.6.2 Relatively Prime Factorization of 146
Rational Matrices
2.6.3 Conversion of a Rational Matrix into 152
the McMillan Canonical Form
2.7 Synthesis of Regulators 155
2.7.1 System Matrices and the General 155
Problem of Synthesis of Regulators
2.7.2 Set of Regulators Guaranteeing Given 159
Characteristic Polynomials of a Closed-loop
System
3 Normal Matrices and Systems 163
3.1 Normal Matrices 163
3.1.1 Definition of the Normal Matrix 163
3.1.2 Normality of the Matrix [Is  A]-ケ 164
for a Cyclic Matrix
3.1.3 Rational Normal Matrices 168
3.2 Fraction Description of Normal Matrices 170
3.3 Sum and Product of Normal Matrices and 175
Normal Inverse Matrices
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
3.4.1 Decomposition of Normal Matrices into 182
the Sum of Normal Matrices
3.4.2 Structural Decomposition of Normal 185
Matrices
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 200
Systems
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 210
General Case
4 The Problem of Realization 219
4.1 Basic Notions and Problem Formulation 219
4.2 Existence of Minimal and Cyclic 220
Realisations
4.2.1 Existence of Minimal Realisations 220
4.2.2 Existence of Cyclic Realisations 224
4.3 Computation of Cyclic Realisations 226
4.3.1 Computation of a Realisation with the 226
Matrix A in the Frobenius Canonical Form
4.3.2 Computation of a Cyclic Realisation 232
with Matrix A in the Jordan Canonical Form
4.4 Structural Stability and Computation of 244
the Normal Transfer Matrix
4.4.1 Structural Controllability of Cyclic 244
Matrices
4.4.2 Structural Stability of Cyclic 245
Realisation
4.4.3 Impact of the Coefficients of the 247
Transfer Function on the System Description
4.4.4 Computation of the Normal Transfer 249
Matrix on the Basis of Its Approximation
5 Singular and Cyclic Normal Systems 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
5.3 Computation of Equivalent Standard 272
Systems for Linear Singular Systems
5.3.1 Discrete-time Systems and Basic 272
Notions
5.3.2 Computation of Fundamental Matrices 276
5.3.3 Equivalent Standard Systems 279
5.3.4 Continuous-time Systems 282
5.4 Electrical Circuits as Examples of 285
Singular Systems
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 291
System Decomposition
5.5.2 Conclusions and Theorems Following 295
from System Decomposition
5.6 Decomposition of Singular Systems 298
5.6.1 Weierstrass訪ronecker Decomposition 298
5.6.2 Basic Theorems 299
5.7 Structural Decomposition of a Transfer 305
Matrix of a Singular System
5.7.1 Irreducible Transfer Matrices 305
5.7.2 Fundamental Theorem and Decomposition 306
Procedure
6 Matrix Polynomial Equations, and Rational and 313
Algebraic Matrix Equations
6.1 Unilateral Polynomial Equations with Two 313
Variables
6.1.1 Computation of Particular Solutions 313
to Polynomial Equations
6.1.2 Computation of General Solutions to 319
Polynomial Equations
6.1.3 Computation of Minimal Degree 322
Solutions to Polynomial Matrix Equations
6.2 Bilateral Polynomial Matrix Equations 325
with Two Unknowns
6.2.1 Existence of Solutions 325
6.2.2 Computation of Solutions 328
6.3 Rational Solutions to Polynomial Matrix 332
Equations
6.3.1 Computation of Rational Solutions 332
6.3.2 Existence of Rational Solutions of 333
Polynomial Matrix Equations
6.3.3 Computation of Rational Solutions to 334
Polynomial Matrix Equations
6.4 Polynomial Matrix Equations 336
6.4.1 Existence of Solutions 336
6.4.2 Computation of Solutions 337
6.5 The Kronecker Product and Its Applications 340
6.5.1 The Kronecker Product of Matrices and 340
Its Properties
6.5.2 Applications of the Kronecker Product 343
to the Formulation of Matrix Equations
6.5.3 Eigenvalues of Matrix Polynomials 345
6.6 The Sylvester Equation and Its 347
Generalization
6.6.1 Existence of Solutions 347
6.6.2 Methods of Solving the Sylvester 349
Equation
6.6.3 Generalization of the Sylvester 357
Equation
6.7 Algebraic Matrix Equations with Two 358
Unknowns
6.7.1 Existence of Solutions 358
6.7.2 Computation of Solutions 360
6.8 Lyapunov Equations 361
6.8.1 Solutions to Lyapunov Equations 361
6.8.2 Lyapunov Equations with a Positive 363
Semidefinite Matrix
7 The Realisation Problem and Perfect Observers 367
of Singular Systems
7.1 Computation of Minimal Realisations for 367
Singular Linear Systems
7.1.1 Problem Formulation 367
7.1.2 Problem Solution 369
7.2 Full- and Reduced-order Perfect Observers 376
7.2.1 Reduced-order Observers 378
7.2.2 Perfect Observers for Standard Systems 384
7.3 Functional Observers 392
7.4 Perfect Observers for 2D Systems 396
7.5 Perfect Observers for Systems with 400
Unknown Inputs
7.5.1 Problem Formulation 400
7.5.2 Problem Solution 402
7.6 Reduced-order Perfect Observers for 2D 409
Systems with Unknown Inputs
7.6.1 Problem Formulation 408
7.6.2 Problem Solution 411
8 Positive Linear Systems with Delays 421
8.1 Positive Discrete-time and 421
Continuous-time Systems
8.1.1 Discrete-time Systems 421
8.1.2 Continuous-time Systems 424
8.2 Stability of Positive Linear 425
Discrete-time Systems with Delays
8.2.1 Asymptotic Stability 425
8.2.2 Stability of Systems with Pure Delays 432
8.2.3 Robust Stability of Interval Systems 434
8.3 Reachability and Minimum Energy Control 437
8.3.2 Minimum Energy Control 442
8.4 Realisation Problem for Positive 446
Discrete-time Systems
8.4.1 Problem Formulation 446
8.4.2 Problem Solution 447
8.5 Realisation Problem for Positive 456
Continuous-time Systems with Delays
8.5.1 Problem Formulation 456
8.5.2 Problem Solution 457
8.6 Positive Realisations for Singular 463
Multi-variable Discrete-time Systems with
Delays
8.6.1 Problem Formulation 463
8.6.1 Problem Solution 466
A Selected Problems of Controllability and 473
Observability of Linear Systems
A.1 Reachability 473
A.2. Controllability 477
A.3 Observability 480
A.4 Reconstructability 483
A.5 Dual System 485
6 Stabilizability and Detectability 485
References 487
Index 501