Numerical methods for linear control systems: design and analysis (Final draft, March 10, 2003)

✍ Scribed by Biswa Nath Datta

• Year: 2003
• Tongue: English
• Leaves: 1010
• Category: Library

✦ Synopsis

Numerical Methods for Linear Control Systems Design and Analysis is an interdisciplinary textbook aimed at systematic descriptions and implementations of numerically-viable algorithms based on well-established, efficient and stable modern numerical linear techniques for mathematical problems arising in the design and analysis of linear control systems both for the first- and second-order models. MATLAB-based software is included for implementing all of the major algorithms from the book. * Unique coverage of modern mathematical concepts such as parallel computations, second-order systems, and large-scale solutions Background material in linear algebra, numerical linear algebra, and control theory included in text Step-by-step explanations of the algorithms and examples* Includes MATLAB-based solution software

✦ Table of Contents

Cover
......Page 1
NUMERICAL METHODS FOR LINEAR CONTROL SYSTEMS DESIGN AND ANALYSIS......Page 2
Contents......Page 3
Preface......Page 17
Contents......Page 22
1.1 Linear and Numerical Linear Algebra (Chapter 2 and Chapters 3 and 4)......Page 25
1.2 System Responses (Chapter 5)......Page 28
1.3 Controllability and Observability problems (Chapter 6)......Page 30
1.4 Stability and Inertia (Chapter 7)......Page 31
1.5 Lyapunov, Sylvester, and Algebraic Riccati Equations (Chapter 8 and Chapter 13)......Page 32
1.6 Realization and Identi cation (Chapter 9)......Page 35
1.7 Feedback Stabilization and Eigenvalue Assignment (Chapter 10 and Chapter 11)......Page 36
1.8 State Estimation (Chapter 12)......Page 38
1.9 Internal Balancing and Model Reduction (Chapter 14)......Page 40
1.11 Numerical Methods for Control Problems Modeled by Systems of Second-Order Di erential Equations (Chapter 16)......Page 41
1.12.1 Nearness to Uncontrollability and Instability......Page 43
1.12.2 Robust stability and stability Radius (Chapter 7 and Chapter 10)......Page 44
1.13 Sensitivity and Condition Numbers of Control Problems......Page 45
1.13.2 Conditioning of the Lyapunov and Sylvester Equations (Chapter 8)......Page 46
1.13.3 Conditioning of the Algebraic Riccati Equations (Chapter 13)......Page 47
1.14 H1-Control (Chapter 10)......Page 48
1.15 Software for Control Problems......Page 50
Part I Review of Linear and Numerical Linear Algebra......Page 52
Contents......Page 55
2.2 Vectors......Page 57
2.3.1 Basic Concepts......Page 58
2.3.3 Rank of a Matrix......Page 61
2.3.4 The Inverse of a Matrix......Page 62
2.3.7 Orthogonal Projection......Page 63
2.4.2 Unitary (Orthogonal) Matrix......Page 64
2.4.4 Hessenberg (Almost Triangular) Matrix......Page 65
2.4.6 Nonderogatory Matrix......Page 66
2.4.8 Positive De nite Matrix......Page 67
2.4.9 Block Matrices......Page 68
2.5.1 Vector Norms......Page 69
2.5.2 Matrix Norms......Page 70
The Frobenius Norm......Page 71
2.6 Norm Invariant Properties Under Unitary Matrix Multiplication......Page 72
2.7 Kronecker Product, Kronecker Sum and Vec Operation......Page 73
Contents......Page 81
3.2.1 Floating Point Numbers......Page 83
3.2.2 Rounding Errors......Page 84
3.2.3 Laws of Floating Point Arithmetic......Page 85
3.3.1 Algorithms and Pseudocodes......Page 86
3.3.3 Solving a Lower Triangular System......Page 87
3.3.5 The Concept of Numerical Stability......Page 88
3.3.6 Conditioning of the Problem and Perturbation Analysis......Page 90
3.3.8 Conditioning of the Linear System and Eigenvalue Problems......Page 91
3.4.1 LU Factorization using Gaussian Elimination......Page 95
3.4.2 The Cholesky Factorization......Page 100
3.4.3 LU Factorization of an Upper Hessenberg Matrix......Page 101
3.5.2 Solving Ax = b using Gaussian Elimination with Partial Pivoting......Page 103
3.5.6 Computing the Determinant of A......Page 105
3.6 The QR Factorization......Page 106
3.6.2 The Householder QR Factorization......Page 107
3.6.3 Givens Matrices......Page 110
3.6.4 The QR Factorization Using Givens Rotations......Page 112
3.7 Orthonormal Bases and Orthogonal Projections Using QR Factorization......Page 113
3.8.1 Solving the Least-Squares Problem Using Normal Equations......Page 114
3.8.2 Solving the Least-Squares Problem Using QR Factorization......Page 115
3.9 The Singular Value Decomposition (SVD)......Page 116
3.9.1 The Singular Value Decomposition and the Structure of a Matrix......Page 118
3.9.2 Orthonormal Bases and Orthogonal Projections......Page 119
3.9.3 The Rank and the Rank-De ciency of a Matrix......Page 120
3.9.4 Numerical Rank......Page 121
3.9.5 Solving the Least Squares Problem Using the Singular Value Decomposition......Page 122
3.10 Summary and Review......Page 124
3.11 Chapter Notes and Further Reading......Page 127
Contents......Page 129
4.1 Importance and Signi cance of Using Orthogonal Transformations......Page 130
4.2 Hessenberg Reduction of a Matrix......Page 132
4.2.1 Uniqueness in Hessenberg Reduction: The Implicit Q Theorem......Page 133
4.3 The Real Schur Form of A:TheQR Iteration Method......Page 134
4.3.2 The Hessenberg QR Iteration and Shift of Origin......Page 135
4.3.3 The Double Shift QR Iteration......Page 136
4.3.4 Obtaining the Real Schur Form A......Page 137
4.3.5 The Real Schur Form and Invariant Subspaces......Page 139
4.4 Power Iteration and Inverse Iteration......Page 141
4.5 Computing the Singular Value Decomposition......Page 142
4.6 The Generalized Real Schur Form: The QZ algorithm......Page 144
4.6.1 Reduction to Hessenberg-Triangular Form......Page 146
4.6.2 Reduction to the Generalized Real Schur Form......Page 148
4.7 Computing of the Eigenvectors of the Pencil A B.......Page 150
4.8 Summary and Review......Page 151
4.9 Chapter Notes and Further Reading......Page 153
Part II Control Systems Analysis......Page 155
Contents......Page 157
5.1 Introduction......Page 158
5.2.1 Continuous-Time Systems......Page 159
5.2.2 Discrete-Time Systems......Page 170
5.2.3 Descriptor Systems......Page 171
5.3 Solutions of a Continuous-Time System: System Responses......Page 174
5.3.1 Some Important Properties of the Matrix eAt......Page 178
5.3.2 Sensitivity of eAt......Page 179
5.3.3 Computational Methods for eAt......Page 180
5.3.4 Comparison of Di erent Methods for Computing the Exponential Matrix......Page 188
5.3.5 Evaluating an Integral with the Exponential Matrix......Page 189
5.4 State-Space Solution of the Discrete-Time System......Page 190
5.5.1 Transfer Function......Page 192
5.5.2 The Frequency Response Matrix and its Computation......Page 194
5.6.3 SLICOT......Page 198
5.6.4 MATRIXX......Page 199
5.7 Summary and Review......Page 200
5.8 Chapter Notes and Further Reading......Page 202
Contents......Page 213
6.1 Introduction......Page 214
6.2.1 Controllability of a Continuous-Time System......Page 215
6.2.2 Controllability of a Discrete-Time System......Page 219
6.3.1 Observability of a Continuous-time System......Page 220
6.4 Decompositions of Uncontrollable and Unobservable Systems.......Page 222
6.5 Controller and Observer Canonical Forms......Page 224
6.6 Numerical Di culties with Theoretical Criteria of Controllability and Observability......Page 226
6.7 A Numerically E ective Test of Controllability......Page 229
6.9 Distance to an Uncontrollable System......Page 238
6.9.1 Newton's and the Bisection Methods for Computing the Distance to Uncontrollability......Page 240
6.9.2 The Wicks-DeCarlo Method for Distance to Uncontrollability......Page 244
6.10 Distance to Uncontrollability and the Singular Values of the Controllability Matrix......Page 246
6.11.3 CSP-ANM......Page 247
6.12 Summary and Review......Page 248
6.13 Chapter Notes and Further Reading......Page 250
Contents......Page 259
7.1 Introduction......Page 260
7.2 Stability of a Continuous-time System......Page 261
7.2.1 Eigenvalue Criterion of Continuous-Time Stability......Page 262
7.2.2 Continuous-time Lyapunov Stability Theory......Page 263
7.2.3 Lyapunov Equations and Controllability and Observability Grammians......Page 268
7.2.4 Lyapunov Equations and the H2-Norm.......Page 269
7.3.1 Stability of a Homogeneous Discrete-Time System......Page 271
7.4.1 The Sylvester Law of Inertia......Page 274
7.4.2 The Lyapunov Inertia Theorems......Page 275
7.5 Determining the Stability and Inertia of a Nonsymmetric Matrix......Page 278
7.6 Distance to an Unstable System......Page 282
7.7 Robust Stability......Page 288
7.8 The Structured Stability Radius......Page 293
7.10 Summary and Review......Page 297
7.11 Chapter Notes and Further Reading......Page 299
Contents......Page 311
8.1 Introduction......Page 313
8.2.1 The Sylvester Equation: XA + BX = C......Page 314
8.2.3 The Discrete Lyapunov Equation: AT XA X = C......Page 315
8.3.1 Perturbation Analysis for the Sylvester Equation......Page 316
8.3.2 The Condition Number of the Sylvester Equation......Page 318
8.3.4 The Condition Number of the Lyapunov Equation......Page 319
8.3.5 Sensitivity of the Stable Lyapunov Equation......Page 320
8.3.7 Sensitivity of the Stable Discrete Lyapunov Equation......Page 323
8.3.8 Determining Ill-Conditioning from the Eigenvalues......Page 324
8.3.9 A Condition Number Estimator for the Sylvester Equation: AT X......Page 326
Remarks:......Page 327
8.4 Analytical Methods for the Lyapunov Equations: Explicit Expressions for Solutions......Page 328
8.5 Numerical Methods for the Lyapunov and Sylvester Equations.......Page 329
8.5.1 Numerical Instability of Diagonalization, Jordan Canonical Form, and Companion Form Techniques.......Page 330
8.5.2 The Schur Method for the Lyapunov Equation: XA + AT X = C......Page 331
8.5.3 The Hessenberg-Schur Method for the Sylvester Equation......Page 335
Flop-count:......Page 339
8.5.4 The Schur Method for the Discrete Lyapunov Equation......Page 340
8.5.5 Residual and Backward Error in the Schur and Hessenberg-Schur Algorithms......Page 343
8.5.6 A Hessenberg Method for the Sylvester Equation: AX + XB = C......Page 345
8.5.7 The Hessenberg-Schur Method for the Discrete Sylvester Equation......Page 349
8.6.1 Computing the Cholesky Factor of the Positive De nite Solution of the Lyapunov Equation......Page 350
8.6.2 Computing the Cholesky Factor of the Positive De nite Solution of the Discrete Lyapunov Equation......Page 355
8.7 Comparisons of Di erent Methods and Conclusions......Page 358
8.8.3 CSP-ANM......Page 359
8.8.5 MATRIXX......Page 360
8.9 Summary and Review......Page 361
8.10 Chapter Notes and Further Reading......Page 362
PART III CONTROL SYSTEMS DESIGN......Page 373
Contents......Page 375
9.1 Introduction......Page 376
9.2.1 Controllable and Observable Realizations......Page 377
9.2.2 Minimal Realization......Page 379
9.3 Computing Minimal Realizations From Markov Parameters......Page 382
9.3.1 Some Basic Properties of the Hankel Matrix of Markov Parameters......Page 383
9.3.2 An SVD Method for Minimal Realization......Page 384
9.3.3 A Modi ed SVD Method for Minimal Realization......Page 387
9.4.1 A Subspace Deterministic Model Identi cation Algorithm......Page 392
9.4.2 A Stochastic Subspace Model Identi cation Algorithm......Page 397
9.4.3 Continuous-time System Identi cation......Page 399
9.4.4 Frequency-Domain Identi cation......Page 400
9.5.4 SLICOT......Page 402
9.6 Summary and Review......Page 403
9.7 Chapter Notes and Further Reading......Page 404
Contents......Page 477
11.1 Introduction......Page 478
11.2 Numerical Methods for the Single-input Eigenvalue Assignment Problem......Page 480
11.2.1 A Recursive Algorithm for the Single-input EVA Problem......Page 483
11.2.2 An Error Analysis of the Recursive Single-Input Method......Page 488
11.2.3 The QR and RQ Implementations of Algorithm 11.2.1......Page 489
11.3 Numerical Methods for the Multi-input Eigenvalue Assignment Problem......Page 495
11.3.1 A Recursive Multi-input Eigenvalue Assignment Algorithm......Page 496
11.3.2 The Explicit QR Algorithm for the Multi-input EVA Problem......Page 499
11.3.3 The Schur Method for the Multi-input Eigenvalue Assignment Problem......Page 506
11.4.1 The Single-Input Case......Page 515
11.4.2 The Multi-input Case......Page 518
11.4.3 Absolute and Relative Condition Numbers......Page 519
11.5 Conditioning of the Closed-loop Eigenvalues......Page 521
11.6.1 Measures of Sensitivity......Page 523
11.6.2 Statement of the Robust Eigenvalue Assignment Problem......Page 524
11.6.3 A Solution Technique for the Robust Eigenvalue Assignment Problem......Page 525
Flop-Count......Page 527
Exercise 3 n . Stable 8).......Page 531
Stable......Page 532
11.9 Comparative Discussion of Various Methods and Recommendation......Page 533
11.10.3 CSP-ANM......Page 534
11.11 Summary and Review......Page 535
11.12 Chapter Notes and Further Reading......Page 538
Contents......Page 551
12.1 Introduction......Page 552
12.2 State Estimation via Eigenvalue Assignment......Page 553
12.3 State Estimation via Sylvester Equation......Page 554
12.4.1 Reduced-order State Estimation via Eigenvalue Assignment......Page 557
12.4.2 Reduced-order State Estimation via Sylvester-observer Equation......Page 562
12.5 Combined State Feedback and Observer Design......Page 564
12.6 Characterization of Nonsingular Solutions of the Sylvester Equation......Page 566
12.7 Numerical Solutions of the Sylvester-Observer Equation......Page 568
12.7.1 A Recursive Method for the Hessenberg Sylvester-Observer Equation.......Page 569
12.7.2 A Recursive Block-Triangular Algorithm for the Hessenberg SylvesterObserver Equation......Page 573
12.8 Numerical Solution of a Constrained Sylvester-observer Equation......Page 578
12.9 Optimal State Estimation: The Kalman Filter......Page 582
12.10 The Linear Quadratic Gaussian Problem......Page 587
12.11.5 MATRIXX......Page 592
12.12 Summary and Review......Page 593
12.13 Chapter Notes and Further Reading......Page 596
Contents......Page 605
13.1 Introduction......Page 607
13.2 The Existence and Uniqueness of the Stabilizing Solution of the CARE......Page 609
13.3 The Existence and Uniqueness of the Stabilizing Solution of the DARE......Page 617
13.4.1 Conditioning of the CARE......Page 618
13.4.2 Conditioning of the DARE......Page 623
13.5 Computational Methods for Riccati Equations......Page 626
13.5.1 The Invariant Subspace Methods......Page 627
13.5.2 The De ating Subspace Methods......Page 636
13.5.3 The Matrix Sign Function Methods......Page 646
Newton's Method for the Continuous-time Algebraic Riccati Equation......Page 653
Newton's method for the Discrete Algebraic Riccati Equation......Page 661
13.6.1 The Generalized Schur Method for the DCARE......Page 667
13.6.3 The Inverse-Free Generalized Schur Method for the DDARE......Page 668
13.7 Conclusions and Table of Comparisons......Page 669
13.8.3 CSP-ANM......Page 671
13.9 Summary and Review......Page 672
13.10 Chapter Notes and Further Reading......Page 676
Contents......Page 691
13.11 Introduction......Page 693
13.12 The Existence and Uniqueness of the Stabilizing Solution of the CARE......Page 695
13.13 The Existence and Uniqueness of the Stabilizing Solution of the DARE......Page 703
13.14.1 Conditioning of the CARE......Page 704
13.14.2 Conditioning of the DARE......Page 709
13.15 Computational Methods for Riccati Equations......Page 712
13.15.1 The Invariant Subspace Methods......Page 713
13.15.2 The De ating Subspace Methods......Page 722
13.15.3 The Matrix Sign Function Methods......Page 732
Newton's Method for the Continuous-time Algebraic Riccati Equation......Page 739
Newton's method for the Discrete Algebraic Riccati Equation......Page 747
13.16.1 The Generalized Schur Method for the DCARE......Page 753
13.16.3 The Inverse-Free Generalized Schur Method for the DDARE......Page 754
13.17 Conclusions and Table of Comparisons......Page 755
13.18.3 CSP-ANM......Page 757
13.19 Summary and Review......Page 758
13.20 Chapter Notes and Further Reading......Page 762
Contents......Page 779
14.1 Introduction......Page 780
14.2.1 Internal Balancing of a Minimal Realization......Page 781
14.2.2 Internal Balancing of a Nonminimal Realization......Page 786
14.3 Internal Balancing For Discrete-time Systems......Page 788
14.4 Model Reduction......Page 790
14.4.1 Model Reduction via Balanced Truncation......Page 791
14.4.2 The Schur Method for Model Reduction......Page 794
14.4.3 A Balancing-Free Square-Root Method for Model Reduction......Page 802
14.5.1 A Characterization of All Solutions to Hankel-Norm Approximation......Page 803
14.6 Model Reduction of an Unstable System......Page 812
14.7 Frequency Weighted Model Reduction......Page 813
14.8 Summary and Comparisons of Model Reduction Procedures......Page 814
14.9.5 MATRIXX......Page 816
14.10 Summary and Review......Page 817
14.11 Chapter Notes and Further Reading......Page 819
PART IV SPECIAL TOPICS......Page 829
Contents......Page 831
15.1 Introduction......Page 832
15.2 A General Discussion on Krylov Subspace Methods......Page 833
15.3.1 The Arnoldi Method......Page 835
15.3.2 Solving Ax = b using the Arnoldi Method......Page 837
15.3.3 The GMRES Method for Solving Ax = b......Page 842
15.3.4 Solving Shifted Linear Systems Using the Arnoldi Method......Page 843
15.3.5 The Block Arnoldi Method......Page 844
15.4 The Lanczos Method......Page 845
15.4.1 The Block Lanczos Method.......Page 847
15.5.1 Scopes of using the Krylov Subspace Methods in Control......Page 848
15.5.2 The Controllability and the observability problems and the Krylov Subspace Methods......Page 849
15.5.3 Arnoldi Method For Rank-one Lyapunov Equation......Page 850
15.5.4 Restarted Arnoldi Method for Lyapunov Equation......Page 855
15.5.5 Block Arnoldi Method For Discrete Lyapunov Equation......Page 856
15.5.6 Arnoldi Methods for Sylvester Equation......Page 860
15.5.7 Block Arnoldi Methods for Sylvester Equation......Page 865
15.5.8 Arnoldi-Method for Sylvester-Observer Equation (single-output case)......Page 870
15.5.9 Arnoldi-Method for Continuous-time Algebraic Riccati Equation......Page 876
15.5.10 Arnoldi Method for Partial Eigenvalue Assignment......Page 880
15.5.11 Lanczos and Arnoldi Methods for Model Reduction......Page 884
15.6 Summary and Review......Page 901
15.7 Chapter Notes and Further Reading......Page 905
Contents......Page 917
16.1 Introduction......Page 918
16.2 First-order Representations......Page 921
16.3 The Quadratic Eigenvalue Problem (QEP)......Page 923
16.3.1 Linearization of the Quadratic Eigenvalue Problem......Page 925
16.3.2 Solving the Quadratic Eigenvalue Problem.......Page 927
16.3.3 Computation of the Partial Spectrum of the Quadratic Eigenvalue Problem: Shift and Invert Strategy and the Jacobi-Davidso......Page 928
16.4 Independent Modal Control (IMSC) Approach......Page 931
16.4.2 Modal Solution of the Output Feedback Problem......Page 933
16.4.3 Engineering and Computational Di culties with the IMSC Approach......Page 934
16.4.4 Simultaneous Diagonalization of the Pencil K M:......Page 935
16.4.5 Simultaneous Triangularization Under Modal Damping......Page 936
16.5.1 Introduction......Page 937
16.5.2 Eigenvalue Criteria of Controllability and Stabilizability......Page 938
16.5.3 Stability of Second-order systems......Page 940
16.6.1 Eigenvalue Bounds for the Quadratic Pencil......Page 947
16.6.2 Orthogonality of the Eigenvectors of the Quadratic Pencil......Page 949
16.6.3 Rayleigh-Quotient Expressions for Quadratic Pencil......Page 952
16.7.1 Introduction......Page 955
16.7.2 Problem Statements......Page 956
16.7.3 A Nonmodal Solution of the Feedback Stabilization Problem......Page 959
16.7.4 A Direct and Partial Modal Approach for Partial Eigenvalue and Eigenstructure Assignment......Page 961
16.7.5 Robust Eigenvalue Assignment in Second-order System......Page 974
16.8 Summary and Review......Page 984
16.9 Chapter Notes and Further Reading......Page 989
Appendix A SOME EXISTING SOFTWARE FOR CONTROL SYSTEMS DESIGN AND ANALYSIS......Page 1001
A.3 CONTROL SYSTEM PROFESSIONAL { ADVANCED NUMERICAL METHODS (CSP-ANM).......Page 1002
A.5 MATRIXX......Page 1003
A.6.3 ADAPTX......Page 1004
Appendix B MATCONTROL AND LISTING OF MATCONTROL FILES......Page 1007