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

Cover......Page 1
Texts in Applied Mathematics 48......Page 2
Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness......Page 4
Copyright - ISBN: 3642039405......Page 5
Preface......Page 8
Contents......Page 12
Mathematical Models......Page 18
1.1 Population Dynamics......Page 19
Modelling in general......Page 23
Population Dynamics......Page 24
1.2 Economics......Page 25
1.2.1 Notes and References......Page 29
1.3.1 Translational Mechanical Systems......Page 30
1.3.2 Mechanical Systems with Rotational Elements......Page 35
1.3.3 The Variational Method......Page 44
1.3.4 Notes and References......Page 55
Maxwell’s Equations......Page 56
The Elements of Electric Circuits......Page 61
1.4.2 Electrical Networks......Page 67
1.4.3 Notes and References......Page 72
1.5 Digital Systems......Page 73
1.5.1 Combinational Switching Networks......Page 76
1.5.2 Sequential Switching Networks......Page 79
1.5.3 Notes and References......Page 85
1.6 Heat Transfer......Page 87
1.6.1 Notes and References......Page 89
Introduction to State Space Theory......Page 90
2.1.1 The General Concept of a Dynamical System......Page 91
2.1.2 Differentiable Dynamical Systems......Page 100
2.1.3 System Properties......Page 105
2.1.4 Linearization......Page 109
2.1.5 Exercises......Page 111
2.1.6 Notes and References......Page 115
2.2.1 General Linear Systems......Page 117
2.2.2 Free Motions of Time–Invariant Linear Differential Systems......Page 121
2.2.3 Free Motions of Time–Invariant Linear Difference Systems......Page 130
2.2.4 Infinite Dimensional Systems......Page 132
2.2.5 Exercises......Page 138
2.2.6 Notes and References......Page 140
Impulse Responses......Page 141
Input-Output Operators in Time Domain......Page 148
Signal Transforms......Page 155
Transfer Matrices......Page 157
Interpretation of the Transfer Function: Response to Sinusoidal Inputs......Page 160
Frequency Responses......Page 162
2.3.3 Relationship Between Input–Output Operators andTransfer Matrices......Page 164
2.3.4 Exercises......Page 168
2.3.5 Notes and References......Page 170
2.4.1 Morphisms and Standard Constructions......Page 171
2.4.2 Composite Systems......Page 177
2.4.3 Exercises......Page 183
2.4.4 Notes and References......Page 184
2.5 Sampling and Approximation: Relations Between Continuous and Discrete Time Systems......Page 185
2.5.1 A/D- and D/A-Conversion of Signals......Page 186
2.5.2 The Sampling Theorem......Page 188
2.5.3 Sampling Continuous Time Systems......Page 192
Sample and hold method......Page 194
Euler’s method......Page 195
Single and multi-step methods......Page 199
Dependence on the control functions......Page 204
2.5.5 Exercises......Page 206
2.5.6 Notes and References......Page 209
Stability Theory......Page 210
3.1 General Definitions......Page 211
3.1.1 Local Flows......Page 212
3.1.2 Stability Definitions......Page 216
3.1.3 Limit Sets......Page 219
3.1.4 Recurrence......Page 223
3.1.5 Attractors......Page 228
3.1.6 Exercises......Page 230
3.1.7 Notes and References......Page 232
3.2.1 General Definitions and Results......Page 234
3.2.2 Time–Varying Finite Dimensional Systems......Page 246
3.2.3 Time–Invariant Systems......Page 252
3.2.4 Exercises......Page 265
3.2.5 Notes and References......Page 268
3.3 Linearization and Stability......Page 270
3.3.1 Stability Criteria for Time-Varying Linear Systems......Page 271
3.3.2 Time–Invariant Systems: Spectral Stability Criteria......Page 280
3.3.3 Numerical Stability of Discretization Methods......Page 285
3.3.4 Liapunov Functions for Time-Varying Linear Systems......Page 289
3.3.5 Liapunov Functions for Time-Invariant Linear Systems......Page 299
3.3.6 Exercises......Page 308
3.3.7 Notes and References......Page 312
3.4 Stability Criteria for Polynomials......Page 313
Complex polynomials......Page 314
Real Polynomials......Page 317
3.4.2 Characterization of Stability via the Cauchy Index......Page 325
Hermite Form......Page 330
B´ezoutiants......Page 334
3.4.4 Hankel Matrices and Rational Functions......Page 337
3.4.5 Applications to Stability......Page 351
Analytic criteria for Schur stability......Page 357
Hermitian Forms and Schur Polynomials......Page 364
3.4.7 Algebraic Stability Domains and Linear Matrix Equations......Page 374
3.4.8 Exercises......Page 378
Notes and references concerning the subsections......Page 383
4.1 Perturbation of Polynomials......Page 386
4.1.1 Dependence of the Roots on the Coefficient Vector......Page 387
4.1.2 Polynomials with Holomorphic Coefficients......Page 393
4.1.3 The Sets of Hurwitz and Schur Polynomials......Page 401
4.1.4 Kharitonov’s Theorem......Page 406
4.1.5 Exercises......Page 410
4.1.6 Notes and References......Page 413
4.2.1 Continuity and Analyticity of Eigenvalues......Page 415
4.2.2 Estimates for Eigenvalues and Growth Rates......Page 421
4.2.3 Smoothness of Eigenprojections and Eigenvectors......Page 426
4.2.5 Notes and References......Page 446
4.3.1 Singular Values and Singular Vectors......Page 448
4.3.2 Singular Value Decomposition......Page 452
4.3.3 Matrices Depending on a Real Parameter......Page 456
4.3.4 Relations between Eigenvalues and Singular Values......Page 461
4.3.5 Exercises......Page 463
4.3.6 Notes and References......Page 465
4.4.1 Elements of μ-Analysis......Page 466
4.4.2 mu-Values for Real Full-Block Perturbations......Page 482
4.4.3 Exercises......Page 497
4.4.4 Notes and References......Page 498
4.5 Computational Aspects......Page 501
4.5.1 Condition Numbers......Page 502
Condition Number for Solving Linear Matrix Equations......Page 503
Condition Number for Solving Liapunov Equations......Page 505
Condition Number for Determining the Eigenvalues of a Matrix......Page 507
Householder transformations......Page 509
Transformation to Hessenberg form......Page 510
Transformation to bidiagonal form......Page 512
QR factorization......Page 514
Schur form......Page 516
4.5.3 Algorithms......Page 518
The QR Algorithm......Page 519
Determining the eigenvalues of a matrix......Page 523
Determining the singular values of a matrix......Page 525
Solving Liapunov equations......Page 528
4.5.4 Exercises......Page 530
4.5.5 Notes and References......Page 532
Uncertain Systems......Page 534
5.1.1 General Definitions and Basic Properties......Page 537
Affine Perturbations......Page 547
Linear Fractional Perturbations......Page 554
5.1.3 Exercises......Page 557
5.1.4 Notes and References......Page 559
5.2.1 General Definitions and Results......Page 561
Multi-Block Perturbations......Page 567
Gershgorin Type Uncertainty......Page 569
Dependence on System Data......Page 572
5.2.2 Complex Full-Block Perturbations......Page 573
5.2.3 Real Full-Block Perturbations......Page 578
General Real Case......Page 584
5.2.4 The Unstructured Case (Pseudospectra)......Page 586
5.2.5 Exercises......Page 597
5.2.6 Notes and References......Page 600
5.3 Stability Radii......Page 602
5.3.1 General Definitions and Results......Page 603
Multi-Block Perturbations......Page 605
Gershgorin Type Uncertainty......Page 606
5.3.2 Complex Full-Block Perturbations......Page 608
5.3.3 Real Full-Block Perturbations......Page 613
5.3.4 Hamiltonian Characterization of the Complex Stability Radius......Page 619
5.3.5 The Unstructured Case......Page 626
5.3.6 Dependence on System Data......Page 631
5.3.7 Stability Radii and the Cayley Transformation......Page 634
5.3.8 Exercises......Page 638
5.3.9 Note and References......Page 641
5.4.1 General Formulas......Page 642
5.4.2 Complex Perturbation Structures......Page 650
5.4.3 Real Perturbation Structures......Page 654
5.4.4 Exercises......Page 661
5.4.5 Note and References......Page 663
5.5.1 Transient Bounds and Initial Growth Rate......Page 665
Initial Growth Rate......Page 670
5.5.2 Contractions and Estimates of the Transient Bound......Page 675
Estimates for Transient Bounds via Liapunov Norms......Page 680
5.5.3 Spectral Value Sets and Transient Behaviour......Page 686
5.5.4 Robustness of (M, β)-Stability......Page 692
5.5.5 Exercises......Page 697
5.5.6 Notes and References......Page 701
5.6 More General Perturbation Classes......Page 703
5.6.1 The Perturbation Classes......Page 704
5.6.2 Stability Radii......Page 713
5.6.3 The Aizerman Conjecture......Page 718
5.6.4 Exercises......Page 726
5.6.5 Notes and References......Page 728
A.1.1 Norms of Vectors and Matrices......Page 732
A.1.2 Spectra and Determinants......Page 736
A.1.4 Direct Sums and Kronecker Products......Page 737
A.1.5 Hermitian Matrices......Page 739
A.2.1 Topological Preliminaries......Page 741
A.2.2 Path Integrals......Page 742
A.2.3 Holomorphic Functions......Page 744
A.2.4 Isolated Singularities......Page 746
A.2.5 Analytic Continuation......Page 749
A.2.6 Maximum Principle and Subharmonic Functions......Page 750
A.3.1 Sequences: Convolution and z-Transforms......Page 752
A.3.2 Lebesgue Spaces, Convolution of Functions, Laplace Transforms......Page 756
A.3.3 Fourier Series and Fourier Transforms......Page 761
A.3.4 Hardy Spaces......Page 767
A.4.1 Summability and Generalized Fourier Series......Page 770
A.4.2 Linear Operators on Banach Spaces......Page 771
A.4.3 Linear Operators on Hilbert Spaces......Page 774
A.4.4 Spectral Theory......Page 776
Bibliography......Page 780
Glossary......Page 806
Index......Page 812