Linear System Theory

This book is the result of our teaching over the years an undergraduate course on Linear Optimal Systems to applied mathematicians and a first-year graduate course on Linear Systems to engineers. The contents of the book bear the strong influence of the great advances in the field and of its enormous literature. However, we made no attempt to have a complete coverage. Our motivation was to write a book on linear systems that covers finite dimensional linear systems, always keeping in mind the main purpose of engineering and applied science, which is to analyze, design, and improve the performance of phy sical systems. Hence we discuss the effect of small nonlinearities, and of perturbations of feedback. It is our on the data; we face robustness issues and discuss the properties hope that the book will be a useful reference for a first-year graduate student. We assume that a typical reader with an engineering background will have gone through the conventional undergraduate single-input single-output linear systems course; an elementary course in control is not indispensable but may be useful for motivation. For readers from a mathematical curriculum we require only familiarity with techniques of linear algebra and of ordinary differential equations.

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This volume is intended for engineers in research and development and applied mathematicians. It is also designed to be a useful reference for graduate students in linear systems with interests in control. With this purpose in mind, the discrete-time case is treated in an isomorphic fashion with the continuous-time case. This volume is self-contained: four mathematical appendices develop the many specialized mathematical results needed in the main text. In the development of Linear System Theory emphasis is placed on careful and precise exposition of fundamental concepts and results. The main topics of Linear System Theory are treated systematically: the dynamics of linear time-varying and time-invariant systems; stability; controllability and observability; realizations; linear feedback and estimation; linear quadratic optimal control; finally, the last chapter develops the main results of unity-feedback MIMO systems. At various suitable places basic computational issues and robustness issues are discussed.

Contenu
1 Introduction.- 1.1 Science and Engineering.- 1.2 Physical Systems, Models, and Representations.- 1.3 Robustness.- 2 The System RepresentationR(•) = [A(•),B(•),C(•),D(•)].- 2.1 Fundamental Properties ofR(•).- 2.2 Applications.- 2d The Discrete-Time System RepresentationRd(•) = [A(•),B(•),C(•),D(•)].- 2d.1 Fundamental Properties ofRd(•).- 2d.2 Application: Periodically Varying Recursion Equations.- 3 The System RepresentationR= [A,B,C,D], Part I.- 3.1 Preliminaries.- 3.2 General Properties ofR= [A,B,C,D].- 3.3 Properties of R when A has a Basis of Eigenvectors.- 3d The Discrete-Time System Representation Rd = [A,B,C,D].- 3d.1 Preliminaries.- 3d.2 General Properties of Rd.- 3d.3 Properties of Rd when A has a Basis of Eigenvectors.- 4 The System Representation R = [A,B,C,D], Part II.- 4.1 Preliminaries.- 4.2 Minimal Polynomial.- 4.3 Decomposition Theorem.- 4.4 The Decomposition of a Linear Map.- 4.5 Jordan Form.- 4.6 Function of a Matrix.- 4.7 Spectral Mapping Theorem.- 4.8 The Linear Map X ? AX+XB.- 5 General System Concepts.- 5.1 Dynamical Systems.- 5.2 Time-Invariant Dynamical Systems.- 5.3 Linear Dynamical Systems.- 5.4 Equivalence.- 6 Sampled Data Systems.- 6.1 Relation BetweenL- and z-Transforms.- 6.2 D/A Converter.- 6.3 A/D Converter.- 6.4 Sampled-Data System.- 6.5 Example.- 7 Stability.- 7.1 I/O Stability.- 7.2 State Related Stability Concepts and Applications.- 7d Stability: The Discrete-Time Case.- 7d.1 I/O Stability.- 7d.2 State Related Stability Concepts.- 8 Controllability and Observability.- 8.1 Controllability and Observability of Dynamical Systems.- 8.2 Controllability of the Pair (A(•),B(•)).- 8.3 Observability of the Pair (C(•),A(•)).- 8.4 Duality.- 8.5 Linear Time-Invariant Systems.- 8.6 Kalman Decomposition Theorem.-8.7 Hidden Modes, Stabilizability, and Detectability.- 8.8 Balanced Representations.- 8.9 Robustness of Controllability.- 8d Controllability and Observability: The Discrete-Time Case.- 8d.1 Controllability and Observability of Dynamical Systems.- 8d.2 Reachability and Controllability of the Pair (A(•),B(•)).- 8d.3 Observability of the Pair (C(•),A(•)).- 8d.4 Duality.- 8d.5 Linear Time-Invariant Systems.- 8d.6 Kalman Decomposition Theorem.- 8d.7 Stabilizability and Detectability.- 9 Realization Theory.- 9.1 Minimal Realizations.- 9.2 Controllable Canonical Form.- 10 Linear State Feedback and Estimation.- 10.1 Linear State Feedback.- 10.2 Linear Output Injection and State Estimation.- 10.3 State Feedback of the Estimated State.- 10.4 Infinite Horizon Linear Quadratic Optimization.- 10d.4 Infinite Horizon Linear Quadratic Optimization. The Discrete-Time Case.- 11 Unity Feedback Systems.- 11.1 The Feedback System ?c.- 11.2 Nyquist Criterion.- 11.3 Robustness.- 11.4 Kharitonov's Theorem.- 11.5 Robust Stability Under Structured Perturbations.- 11.6 Stability Under Arbitrary Additive Plant Perturbations.- 11.7 Transmission Zeros.- Appendix A Linear Maps and Matrix Analysis.- A.1 Preliminary Notions.- A.2 Rings and Fields.- A.3 Linear Spaces.- A4. Linear Maps.- AS. Matrix Representation.- A.5.1 The Concept of Matrix Representation.- A.5.2 Matrix Representation and Change of Basis.- A.5.3 Range and Null Space: Rank and Nullity.- A.5.4 Echelon Forms of a Matrix.- A.6 Notmed Linear Spaces.- A.6.1 Norms.- A.6.2 Convergence.- A.6.3 Equivalent Norms.- A.6.4 The Lebesgue Spaces 1P and LP [Tay.1].- A.6.5 Continuous Linear Transformations.- A.7 The Adjoint of a Linear Map.- A.7.1 Inner Products.- A.7.2 Adjoints of Continuous Linear Maps.- A.7.3 Properties of the Adjoint.-A.7.4 The Finite Rank Operator Fundamental Lemma.- A.7.5 Singular Value Decomposition (SVD).- Appendix B Differential Equations.- BA Existence and Uniqueness of Solutions.- B.1.1 Assumptions.- B.1.2 Fundamental Theorem.- B.1.3 Construction of a Solution by Iteration.- B.1.4 The Bellman-Gronwall Inequality.- B.1.5 Uniqueness.- B.2 Initial Conditions and Parameter Perturbations.- B.3 Geometric Interpretation and Numerical Calculations.- Appendix C Laplace Transforms.- C.1 Definition of the Laplace Transform.- C.2 Properties of Laplace Transforms.- Appendix D the z-Transform.- D.1 Definition of the z-Transform.- D.2 Properties of the z-Transform.- References.- Abbreviations.- Mathematical Symbols.