Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness

Diederich Hinrichsen and Anthony J. Pritchard

Publisher:

Springer Verlag

Publication Date:

2005

Number of Pages:

804

Format:

Hardcover

Series:

Texts in Applied Mathematics 48

Price:

79.95

ISBN:

3-540-44125-5

Category:

Monograph

MAA Review
Table of Contents

[Reviewed by

William J. Satzer

, on

05/25/2005

]

Mathematical Systems Theory I (subtitled "Modelling, State Space Analysis, Stability and Robustness") provides a detailed and rigorous mathematical development of finite-dimensional, time-invariant linear systems. One way to think of this book is to see it as the rigorous mathematical foundation underlying the "linear systems" course common to many engineering curricula. A second volume is evidently in the works; it will concentrate on control theory.

The intended audience for this text is advanced undergraduates and first or second year graduate students. The authors suggest that several courses based on their book are possible depending on the selection of chapters. Indeed, the content ranges from rigorous proofs of basic results in linear systems theory to new results on the research frontier. In that sense, the book also serves as a valuable research reference.

The first two chapters — discussing mathematical modeling and basic state space theory — are introductory, and nicely self-contained. The chapter on modeling includes a broad collection of examples ranging from population dynamics and economics to switching networks in digital systems. The remaining three chapters make up more than 70% of the text; this is significantly more demanding material and is intended to prepare the reader for research in systems theory. Chapter 3 focuses on stability theory, emphasizing Lyapunov stability. System perturbations are discussed in Chapter 4; both polynomial and matrix perturbation techniques are presented. The last chapter, on uncertain systems, includes new tools developed by the authors to deal with model uncertainty (imperfectly known parameters or dynamics and incomplete models).

It is a little disappointing not to see any mention of the interaction of systems theory with data — as, for example, in the standard Kalman filter. Students approaching systems theory from the mathematical side are often surprised by the difficulties that real data can introduce. It would have useful to see some discussion of this both in the introductory chapters and in the chapter on uncertain systems.

Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Preface vii
1 MathematicalModels 1
1.1 Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 TranslationalMechanical Systems . . . . . . . . . . . . . . . . . . . 13
1.3.2 Mechanical Systems with Rotational Elements . . . . . . . . . . . . 18
1.3.3 The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.4 Electromagnetism and Electrical Systems . . . . . . . . . . . . . . . . . . . 39
1.4.1 Maxwell's Equations and the Elements of Electrical Circuits . . . . . 39
1.4.2 Electrical Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.4.3 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.5 Digital Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.5.1 Combinational Switching Networks . . . . . . . . . . . . . . . . . . . 59
1.5.2 Sequential Switching Networks . . . . . . . . . . . . . . . . . . . . . 62
1.5.3 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 68
1.6 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1.6.1 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2 Introduction to State Space Theory 73
2.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.1.1 The General Concept of a Dynamical System . . . . . . . . . . . . . 74
2.1.2 Differentiable Dynamical Systems . . . . . . . . . . . . . . . . . . . . 83
2.1.3 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.1.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.1.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.2.1 General Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.2.2 Free Motions of Time-Invariant Linear Differential Systems . . . . . 104
2.2.3 FreeMotions of Time-Invariant Linear Difference Systems . . . . . . 113
2.2.4 Infinite Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . 115
2.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.2.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.3 Linear Systems: Input-Output Behaviour . . . . . . . . . . . . . . . . . . . 124
xii Contents
2.3.1 Input-Output Behaviour in Time Domain . . . . . . . . . . . . . . . 124
2.3.2 Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.3.3 Relationship Between Input-Output Operators and
TransferMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.3.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 153
2.4 Transformations and Interconnections . . . . . . . . . . . . . . . . . . . . . 154
2.4.1 Morphisms and Standard Constructions . . . . . . . . . . . . . . . . 154
2.4.2 Composite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
2.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
2.4.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 167
2.5 Sampling and Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 168
2.5.1 A/D- and D/A-Conversion of Signals . . . . . . . . . . . . . . . . . . 169
2.5.2 The Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 171
2.5.3 Sampling Continuous Time Systems . . . . . . . . . . . . . . . . . . 175
2.5.4 Approximation of Continuous Systems by Discrete Systems . . . . . 177
2.5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
2.5.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 192
3 Stability Theory 193
3.1 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
3.1.1 Local Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.1.2 Stability Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
3.1.3 Limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
3.1.4 Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
3.1.5 Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
3.1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
3.1.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 215
3.2 Liapunov's Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
3.2.1 General Definitions and Results . . . . . . . . . . . . . . . . . . . . . 217
3.2.2 Time-Varying Finite Dimensional Systems . . . . . . . . . . . . . . 229
3.2.3 Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . 235
3.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
3.2.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 251
3.3 Linearization and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
3.3.1 Stability Criteria for Time-Varying Linear Systems . . . . . . . . . . 254
3.3.2 Time-Invariant Systems: Spectral Stability Criteria . . . . . . . . . 263
3.3.3 Numerical Stability of Discretization Methods . . . . . . . . . . . . . 268
3.3.4 Liapunov Functions for Time-Varying Linear Systems . . . . . . . . 272
3.3.5 Liapunov Functions for Time-Invariant Linear Systems . . . . . . . . 282
3.3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
3.3.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 295
3.4 Stability Criteria for Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 296
3.4.1 Stability Criteria and the Argument Principle . . . . . . . . . . . . . 297
3.4.2 Characterization of Stability via the Cauchy Index . . . . . . . . . . 308
3.4.3 Hermite Forms and BÃ¯ezoutiants . . . . . . . . . . . . . . . . . . . . . 313
3.4.4 HankelMatrices and Rational Functions . . . . . . . . . . . . . . . . 320
3.4.5 Applications to Stability . . . . . . . . . . . . . . . . . . . . . . . . . 334
3.4.6 Schur Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
Contents xiii
3.4.7 Algebraic Stability Domains and Linear Matrix Equations . . . . . . 357
3.4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
3.4.9 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 366
4 Perturbation Theory 369
4.1 Perturbation of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
4.1.1 Dependence of the Roots on the Coefficient Vector . . . . . . . . . . 370
4.1.2 Polynomials with Holomorphic Coefficients . . . . . . . . . . . . . . 376
4.1.3 The Sets of Hurwitz and Schur Polynomials . . . . . . . . . . . . . . 384
4.1.4 Kharitonov's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 389
4.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
4.1.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 396
4.2 Perturbation ofMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
4.2.1 Continuity and Analyticity of Eigenvalues . . . . . . . . . . . . . . . 398
4.2.2 Estimates for Eigenvalues and Growth Rates . . . . . . . . . . . . . 404
4.2.3 Smoothness of Eigenprojections and Eigenvectors . . . . . . . . . . . 409
4.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
4.2.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 429
4.3 The Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . 431
4.3.1 Singular Values and Singular Vectors . . . . . . . . . . . . . . . . . . 431
4.3.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . 435
4.3.3 Matrices Depending on a Real Parameter . . . . . . . . . . . . . . . 439
4.3.4 Relations between Eigenvalues and Singular Values . . . . . . . . . . 444
4.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
4.3.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 448
4.4 Structured Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
4.4.1 Elements of Ã¦-Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 449
4.4.2 Ã¦-Values for Real Full-Block Perturbations . . . . . . . . . . . . . . 465
4.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
4.4.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 481
4.5 Computational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
4.5.1 Condition Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
4.5.2 Matrix Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 492
4.5.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
4.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
4.5.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 515
5 Uncertain Systems 517
5.1 Models of Uncertainty and Tools for their Analysis . . . . . . . . . . . . . . 520
5.1.1 General Definitions and Basic Properties . . . . . . . . . . . . . . . . 520
5.1.2 Perturbation Structures . . . . . . . . . . . . . . . . . . . . . . . . . 530
5.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
5.1.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 542
5.2 Spectral Value Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
5.2.1 General Definitions and Results . . . . . . . . . . . . . . . . . . . . . 544
5.2.2 Complex Full-Block Perturbations . . . . . . . . . . . . . . . . . . . 556
5.2.3 Real Full-Block Perturbations . . . . . . . . . . . . . . . . . . . . . . 561
5.2.4 The Unstructured Case (Pseudospectra) . . . . . . . . . . . . . . . . 569
5.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
xiv Contents
5.2.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 583
5.3 Stability Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
5.3.1 General Definitions and Results . . . . . . . . . . . . . . . . . . . . . 586
5.3.2 Complex Full-Block Perturbations . . . . . . . . . . . . . . . . . . . 591
5.3.3 Real Full-Block Perturbations . . . . . . . . . . . . . . . . . . . . . . 596
5.3.4 Hamiltonian Characterization of the Complex Stability Radius . . . 602
5.3.5 The Unstructured Case . . . . . . . . . . . . . . . . . . . . . . . . . 609
5.3.6 Dependence on SystemData . . . . . . . . . . . . . . . . . . . . . . 614
5.3.7 Stability Radii and the Cayley Transformation . . . . . . . . . . . . 617
5.3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
5.3.9 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 624
5.4 Root Sets and Stability Radii of Polynomials . . . . . . . . . . . . . . . . . 625
5.4.1 General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
5.4.2 Complex Perturbation Structures . . . . . . . . . . . . . . . . . . . . 633
5.4.3 Real Perturbation Structures . . . . . . . . . . . . . . . . . . . . . . 637
5.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
5.4.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 646
5.5 Transient Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
5.5.1 Transient Bounds and Initial Growth Rate . . . . . . . . . . . . . . 648
5.5.2 Contractions and Estimates of the Transient Bound . . . . . . . . . 658
5.5.3 Spectral Value Sets and Transient Behaviour . . . . . . . . . . . . . 669
5.5.4 Robustness of (M,Ã¡)-Stability . . . . . . . . . . . . . . . . . . . . . 675
5.5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
5.5.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 684
5.6 More General Perturbation Classes . . . . . . . . . . . . . . . . . . . . . . . 686
5.6.1 The Perturbation Classes . . . . . . . . . . . . . . . . . . . . . . . . 687
5.6.2 Stability Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696
5.6.3 The Aizerman Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 701
5.6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709
5.6.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 711
Appendix 715
A.1 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
A.1.1 Norms of Vectors andMatrices . . . . . . . . . . . . . . . . . . . . . 715
A.1.2 Spectra and Determinants . . . . . . . . . . . . . . . . . . . . . . . . 719
A.1.3 Real Representation of ComplexMatrices . . . . . . . . . . . . . . . 720
A.1.4 Direct Sums and Kronecker Products . . . . . . . . . . . . . . . . . 720
A.1.5 HermitianMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
A.2 Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
A.2.1 Topological Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 724
A.2.2 Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
A.2.3 Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 727
A.2.4 Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 729
A.2.5 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . 732
A.2.6 Maximum Principle and Subharmonic Functions . . . . . . . . . . . 733
A.3 Convolutions and Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 735
A.3.1 Sequences: Convolution and z-Transforms . . . . . . . . . . . . . . . 735
A.3.2 Lebesgue Spaces, Convolution of Functions, Laplace Transforms . . 739
A.3.3 Fourier Series and Fourier Transforms . . . . . . . . . . . . . . . . . 744
Contents xv
A.3.4 Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750
A.4 Linear Operators and Linear Forms . . . . . . . . . . . . . . . . . . . . . . . 753
A.4.1 Summability and Generalized Fourier Series . . . . . . . . . . . . . . 753
A.4.2 Linear Operators on Banach Spaces . . . . . . . . . . . . . . . . . . 754
A.4.3 Linear Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . 757
A.4.4 Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
References 763
Glossary 789
Index 795

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Systems Theory