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Mathematical Control Theory

Mathematics is playing an ever more important role in the physical and biologi- cal sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modem as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and rein- force the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematics Sci- ences (AMS) series, which will focus on advanced textbooks and research-level monographs. v Preface This textbook introduces the basic concepts and results of mathematical control and system theory. Based on courses that I have taught during the last 15 years, it presents its subject in a self-contained and elementary fashion. It is geared primarily to an audience consisting of mathematically mature advanced undergraduate or beginning graduate students. In addi- tion, it can be used by engineering students interested in a rigorous, proof- oriented systems course that goes beyond the classical frequency-domain material and more applied courses.
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Table of Contents

1 Introduction.- 1.1 What Is Mathematical Control Theory?.- 1.2 Proportional-Derivative Control.- 1.3 Digital Control.- 1.4 Feedback Versus Precomputed Control.- 1.5 State-Space and Spectrum Assignment.- 1.6 Outputs and Dynamic Feedback.- 1.7 Dealing with Nonlinearity.- 1.8 A Brief Historical Background.- 1.9 Some Topics Not Covered.- 2 Systems.- 2.1 Basic Definitions.- 2.2 I/O Behaviors.- 2.3 Discrete-Time.- 2.4 Linear Discrete-Time Systems.- 2.5 Smooth Discrete-Time Systems.- 2.6 Continuous-Time.- 2.7 Linear Continuous-Time Systems.- 2.8 Linearizations Compute Differentials.- 2.9 More on Differentiability*.- 2.10 Sampling.- 2.11 Volterra Expansions*.- 2.12 Notes and Comments.- 3 Reachability and Controllability.- 3.1 Basic Reachability Notions.- 3.2 Time-Invariant Systems.- 3.3 Controllable Pairs of Matrices.- 3.4 Controllability Under Sampling.- 3.5 More on Linear Controllability.- 3.6 First-Order Local Controllability.- 3.7 Piecewise Constant Controls.- 3.8 Notes and Comments.- 4 Feedback and Stabilization.- 4.1 Constant Linear Feedback.- 4.2 Feedback Equivalence*.- 4.3 Disturbance Rejection and Invariance*.- 4.4 Stability and Other Asymptotic Notions.- 4.5 Unstable and Stable Modes*.- 4.6 Lyapunov's Direct Method.- 4.7 Linearization Principle for Stability.- 4.8 More on Smooth Stabilizability*.- 4.9 Notes and Comments.- 5 Outputs.- 5.1 Basic Observability Notions.- 5.2 Time-Invariant Systems.- 5.3 Continuous-Time Linear Systems.- 5.4 Linearization Principle for Observability.- 5.5 Realization Theory for Linear Systems.- 5.6 Recursion and Partial Realization.- 5.7 Rationality and Realizability.- 5.8 Abstract Realization Theory*.- 5.9 Notes and Comments.- 6 Observers and Dynamic Feedback.- 6.1 Observers and Detectability.- 6.2 Dynamic Feedback.- 6.3 External Stability for Linear Systems.- 6.4 Frequency-Domain Considerations.- 6.5 Parameterization of Stabilizers.- 6.6 Notes and Comments.- 7 Optimal Control.- 7.1 An Optimal Control Problem.- 7.2 Dynamic Programming.- 7.3 The Continuous-Time Case.- 7.4 Linear Systems with Quadratic Cost.- 7.5 Infinite-Time Problems.- 7.6 Tracking.- 7.7 (Deterministic) Kalman Filtering.- 7.8 Notes and Comments.- Appendixes.- A Linear Algebra.- A.1 Operator Norms.- A.2 Singular Values.- A.3 Jordan Forms and Matrix Functions.- A.4 Continuity of Eigenvalues.- B Differentials.- B.1 Finite Dimensional Mappings.- B.2 Maps Between Normed Spaces.- C Ordinary Differential Equations.- C.1 Review of Lebesgue Measure Theory.- C.2 Initial-Value Problems.- C.3 Existence and Uniqueness Theorem.- C.4 Continuous Dependence.- C.5 Linear Differential Equations.- C.6 Stability of Linear Equations.

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