Elsevier S & T

Numerical Methods for Linear Control Systems

Datta, Biswa

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Table of contents
  • Contentsvii
  • Prefacexxiii
  • Acknowledgmentsxxix
  • About the Authorxxxi
  • List of Algorithmsxxxiii
  • Notations and Symbolsxxxvii
  • CHAPTER 1. INTRODUCTION AND OVERVIEW1
  • 1.1 Linear and Numerical Linear Algebra (Chapter 2 and Chapters 3 and 4)2
  • 1.2 System Responses (Chapter 5)3
  • 1.3 Controllability and Observability problems (Chapter 6)4
  • 1.4 Stability and Inertia (Chapter 7)5
  • 1.5 Lyapunov, Sylvester, and Algebraic Riccati Equations (Chapters 8 and 13)6
  • 1.6 Realization and Identification (Chapter 9)8
  • 1.7 Feedback Stabilization and Eigenvalue Assignment (Chapters 10 and 11)9
  • 1.8 State Estimation (Chapter 12)10
  • 1.9 Internal Balancing and Model Reduction (Chapter 14)11
  • 1.10 Nearness to Uncontrollability and Instability (Chapters 6 and 7) and Robust Stability and Stabi12
  • 1.11 Sensitivity and Condition Numbers of Control Problems13
  • 1.12 H∞ -Control (Chapter 10)14
  • 1.13 Software for Control Problems15
  • References15
  • PART I: REVIEW OF LINEAR AND NUMERICAL LINEAR ALGEBRA17
  • CHAPTER 2. A REVIEW OF SOME BASIC CONCEPTS AND RESULTS FROM THEORETICAL LINEAR ALGEBRA19
  • 2.1 Introduction19
  • 2.2 Orthogonality of Vectors and Subspaces19
  • 2.3 Matrices20
  • 2.4 Some Special Matrices23
  • 2.5 Vector and Matrix Norms27
  • 2.6 Norm Invariant Properties Under Unitary Matrix Multiplication30
  • 2.7 Kronecker Product, Kronecker Sum, and Vec Operation31
  • 2.8 Chapter Notes and Further Reading31
  • References32
  • CHAPTER 3. SOME FUNDAMENTAL TOOLS AND CONCEPTS FROM NUMERICAL LINEAR ALGEBRA33
  • 3.1 Introduction33
  • 3.2 Floating Point Numbers and Errors in Computations34
  • 3.3 Conditioning, Efficiency, Stability, and Accuracy37
  • 3.4 LU Factorization45
  • 3.5 Numerical Solution of the Linear System Ax=b53
  • 3.6 The QR Factorization56
  • 3.7 Orthonormal Bases and Orthogonal Projections Using QR Factorization63
  • 3.8 The Least-Squares Problem64
  • 3.9 The Singular Value Decomposition (SVD)67
  • 3.10 Summary and Review75
  • 3.11 Chapter Notes and Further Reading78
  • References78
  • CHAPTER 4. CANONICAL FORMS OBTAINED VIA ORTHOGONAL TRANSFORMATIONS79
  • 4.1 Importance and Significance of Using Orthogonal Transformations79
  • 4.2 Hessenberg Reduction of a Matrix81
  • 4.3 The Real Schur Form of A: The QR Iteration Method83
  • 4.4 Computing the Singular Value Decomposition (SVD)91
  • 4.5 The Generalized Real Schur Form: The QZ algorithm94
  • 4.6 Computing of the Eigenvectors of the Pencil A – λB99
  • 4.7 Summary and Review101
  • 4.8 Chapter Notes and Further Reading102
  • References103
  • PART II: CONTROL SYSTEMS ANALYSIS105
  • CHAPTER 5. LINEAR STATE-SPACE MODELS AND SOLUTIONS OF THE STATE EQUATIONS107
  • 5.1 Introduction107
  • 5.2 State-Space Representations of Control Systems108
  • 5.3 Solutions of a Continuous-Time System: System Responses122
  • 5.4 State-Space Solution of the Discrete-Time System139
  • 5.5 Transfer Function and Frequency Response140
  • 5.6 Some Selected Software146
  • 5.7 Summary and Review149
  • 5.8 Chapter Notes and Further Reading151
  • Exercises151
  • References156
  • CHAPTER 6. CONTROLLABILITY, OBSERVABILITY, AND DISTANCE TO UNCONTROLLABILITY159
  • 6.1 Introduction159
  • 6.2 Controllability: Definitions and Basic Results160
  • 6.3 Observability: Definitions and Basic Results165
  • 6.4 Decompositions of Uncontrollable and Unobservable Systems167
  • 6.5 Controller- and Observer-Canonical Forms169
  • 6.6 Numerical Difficulties with theoretical criteria of controllability and observability171
  • 6.7 A Numerically Effective Test of Controllability173
  • 6.8 A Numerically Effective Test of Observability183
  • 6.9 Distance to an Uncontrollable System183
  • 6.10 Distance to Uncontrollability and the Singular values of the Controllability Matrix190
  • 6.11 Some Selected Software192
  • 6.12 Summary and Review193
  • 6.13 Chapter Notes and Further Reading194
  • Exercises194
  • References198
  • CHAPTER 7. STABILITY, INERTIA, AND ROBUST STABILITY201
  • 7.1 Introduction201
  • 7.2 Stability of a Continuous-time System202
  • 7.3 Stability of a Discrete-time System213
  • 7.4 Some Inertia Theorems215
  • 7.5 Determining the Stability and Inertia of a Nonsymmetric Matrix218
  • 7.6 Distance to an Unstable System223
  • 7.7 Robust Stability230
  • 7.8 The Structured Stability Radius232
  • 7.9 Some Selected Software235
  • 7.10 Summary and Review235
  • 7.11 Chapter Notes and Further Reading237
  • Exercises238
  • References241
  • CHAPTER 8. NUMERICAL SOLUTIONS AND CONDITIONING OF LYAPUNOV AND SYLVESTER EQUATIONS245
  • 8.1 Introduction245
  • 8.2 The Existence and Uniqueness of Solutions247
  • 8.3 Perturbation Analysis and the Condition Numbers249
  • 8.4 Analytical Methods for the Lyapunov Equations: Explicit Expressions for Solutions262
  • 8.5 Numerical Methods for the Lyapunov and Sylvester Equations263
  • 8.6 Direct Computations of the Cholesky Factors of Symmetric Positive Definite Solutions of Lyapunov284
  • 8.7 Comparisions of Different Methods and Conclusions293
  • 8.8 Some Selected Software293
  • 8.9 Summary and Review296
  • 8.10 Chapter Notes and Further Reading297
  • Exercises298
  • References301
  • PART III: CONTROL SYSTEMS DESIGN305
  • CHAPTER 9. REALIZATION AND SUBSPACE IDENTIFICATION307
  • 9.1 Introduction307
  • 9.2 State-Space Realizations of a Transfer Function308
  • 9.3 Computing Minimal Realizations from Markov Parameters314
  • 9.4 Subspace Identification Algorithms324
  • 9.5 Some Selected Software334
  • 9.6 Summary and Review335
  • 9.7 Chapter Notes and Further Reading337
  • Exercises337
  • References340
  • CHAPTER 10. FEEDBACK STABILIZATION, EIGENVALUE ASSIGNMENT, AND OPTIMAL CONTROL343
  • 10.1 Introduction343
  • 10.2 State-Feedback Stabilization345
  • 10.3 Detectability353
  • 10.4 Eigenvalue and Eigenstructure Assignment Problems354
  • 10.5 The Quadratic Optimization Problems363
  • 10.6 H∞-Control Problems373
  • 10.7 The Complex Stability Radius and Riccati Equation386
  • 10.8 Some Selected Software391
  • 10.9 Summary and Review393
  • 10.10 Chapter Notes and Further Reading397
  • Exercises398
  • References401
  • CHAPTER 11. NUMERICAL METHODS AND CONDITIONING OF THE EIGENVALUE ASSIGNMENT PROBLEMS405
  • 11.1 Introduction405
  • 11.2 Numerical Methods for the Single-input Eigenvalue Assignment Problem407
  • 11.3 Numerical Methods for the Multi-input Eigenvalue Assignment Problem421
  • 11.4 Conditioning of the Feedback Problem439
  • 11.5 Conditioning of the Closed-loop Eigenvalues443
  • 11.6 Robust Eigenvalue Assignment445
  • 11.7 Comparison of Efficiency and Stability" the Single-input EVA Problem452
  • 11.8 Comparison of Efficiency and Stability: the Multi-input EVA Problem453
  • 11.9 Comparative Discussion of Various Methods and Recommendation453
  • 11.10 Some Selected Software455
  • 11.11 Summary and Review456
  • 11.12 Chapter Notes and Further Reading459
  • Exercises460
  • References464
  • CHAPTER 12. STATE ESTIMATION: OBSERVER AND THE KALMAN FILTER469
  • 12.1 Introduction469
  • 12.2 State Estimation via Eigenvalue Assignment470
  • 12.3 State Estimation via Sylvester Equation471
  • 12.4 Reduced-order State Estimation474
  • 12.5 Combined State Feedback and Observer Design482
  • 12.6 Characterization of Nonsingular Solutions of the Sylvester Equation483
  • 12.7 Numerical Solutions of the Sylvester-Observer Equation485
  • 12.8 Numerical Solutions of a Constrained Sylvester- observer Equation496
  • 12.9 Optimal State Estimation: The Kalman Filter499
  • 12.10 The Linear Quadratic Gaussian Problem505
  • 12.11 Some Selected Software509
  • 12.12 Summary and Review510
  • 12.13 Chapter Notes and Further Reading513
  • Exercises514
  • References516
  • CHAPTER 13. NUMERICAL SOLUTIONS AND CONDITIONING OF ALGEBRAIC RICCATI EQUATIONS519
  • 13.1 Introduction519
  • 13.2 The Existence and Uniqueness of the Stabilizing Solution of the CARE521
  • 13.3 The Existence and Uniqueness of the Stabilizing Solution of the DARE529
  • 13.4 Conditioning of the Riccati Equations530
  • 13.5 Computational Methods for Riccati Equations539
  • 13.6 The Schur and Inverse-Free Generalized Schur Methods for the Descriptor Riccati Equations579
  • 13.7 Conclusions and Table of Comparisons581
  • 13.8 Some Selected Software583
  • 13.9 Summary and Review585
  • 13.10 Chapter Notes and Further Reading588
  • Exercises591
  • References593
  • CHAPTER 14. INTERNAL BALANCING AND MODEL REDUCTION601
  • 14.1 Introduction601
  • 14.2 Internal Balancing for Continuous-time Systems602
  • 14.3 Internal Balancing for Discrete-time Systems609
  • 14.4 Model Reduction611
  • 14.5 Hankel-Norm Approximations623
  • 14.6 Model Reduction of an Unstable System633
  • 14.7 Frequency-Weighted Model Reduction633
  • 14.8 Summary and Comparisons of Model Reduction Procedures635
  • 14.9 Some Selected Software636
  • 14.10 Summary and Review638
  • 14.11 Chapter Notes and Further Reading640
  • Exercises640
  • References640
  • PART IV: SPECIAL TOPICS647
  • CHAPTER 15. LARGE-SCALE MATRIX COMPUTATIONS IN CONTROL: KRYLOV SUBSPACE METHODS649
  • 15.1 Introduction649
  • 15.2 The Arnoldi and Block Arnoldi Methods650
  • 15.3 Scopes of using the Krylov Subspace Methods in Control653
  • 15.4 Arnoldi Methods for Lyapunov, Sylvester, and Algebraic Riccati Equations653
  • 15.5 Arnoldi Method for Partial Eigenvalue Assignment659
  • 15.6 Lanczos and Arnoldi Methods for Model Reduction659
  • 15.7 Chapter Notes and Further Reading662
  • Research Problems662
  • References663
  • APPENDIX A. SOME EXISTING SOFTWARE FOR CONTROL SYSTEMS DESIGN AND ANALYSIS669
  • A.1 MATLAB CONTROL SYSTEM TOOLBOX669
  • A.2 MATCONTROL669
  • A.3 Control System Professional„Advanced Numerical Methods (CSP-ANM)670
  • A.4 SLICOT670
  • A.5 MATRIX671
  • A.6 System Identification Software671
  • References672
  • APPENDIX B. MATCONTROL AND LISTING OF MATCONTROL FILES673
  • B. 1 About Matcontrol673
  • B.2 Chapterwise Listing of Matcontrol Files674
  • APPENDIX C. CASE STUDY: CONTROL OF A 9-STATE AMMONIA REACTOR679
  • C.1 Introduction679
  • C.2 Testing the controllability680
  • C.3 Testing the Observability680
  • C.4 Testing the Stability681
  • C.5 Lyapunov Stabilization681
  • C.6 Pole-Placement Design682
  • C.7 The LQR and LQG Designs682
  • C.8 State-Estimation(observer): Kalman estimator versus Sylvester Estimator685
  • C.9 System Identification and Model Reduction686
  • References688
  • Index689
  • Limited Warranty696
Book details
  • Vendor Elsevier S & T
  • SKU 9780122035906
  • ISBN-13 9780080537887
  • Author Datta, Biswa
  • Category Technology & Engineering
  • Subject Automation

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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