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Linear System Theory

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Table of Contents

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(*).- 2.1.1 Definitions.- 2.1.2 Structure ofR(*).- 2.1.3 State Transition Matrix.- 2.1.4 State Transition Map and Response Map.- 2.1.5 Impulse Response Matrix.- 2.1.6 Adjoint Equations.- 2.1.7 Linear-Quadratic Optimization.- 2.2 Applications.- 2.2.1 Variational Equation.- 2.2.2 Control Correction Example.- 2.2.3 Optimization Example.- 2.2.4 Periodically Varying Differential Equations.- 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].- 3.2.1 Definition.- 3.2.2 State Transition Matrix.- 3.2.3 The State Transition and Response Map of R.- 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.- 7.2.1 Stability of x = A(t)x.- 7.2.2 Bounded Trajectories and Regulation.- 7.2.3 Response to T-Periodic Inputs.- 7.2.4 Periodically Varying System with Periodic Input.- 7.2.5 Slightly Nonlinear Systems.- 7d Stability: The Discrete-Time Case.- 7d.1 I/O Stability.- 7d.2 State Related Stability Concepts.- 7d.2.1 Stability of x(k+1) = A(k)x(k).- 7d.2.2 Bounded Trajectories and Regulation.- 7d.2.3 Response to q-Periodic Inputs.- 8 Controllability and Observability.- 8.1 Controllability and Observability of Dynamical Systems.- 8.2 Controllability of the Pair (A(*),B(*)).- 8.2.1 Controllability of the Pair (A(*),B(*)).- 8.2.2 The Cost of Control.- 8.2.3 Stabilization by Linear State Feedback.- 8.3 Observability of the Pair (C(*),A(*)).- 8.4 Duality.- 8.5 Linear Time-Invariant Systems.- 8.5.1 Observability Properties of the Pair (C,A).- 8.5.2 Controllability of the Pair (A,B).- 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.2.1 Controllability of the Pair (A(*),B(*)).- 8d.2.2 The Cost of Control.- 8d.3 Observability of the Pair (C(*),A(*)).- 8d.4 Duality.- 8d.5 Linear Time-Invariant Systems.- 8d.5.1 Observability of the Pair (C,A).- 8d.5.2 Reachability and Controllability of the Pair(A,B).- 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.- 11.1.1 State Space Analysis.- 11.1.2 Special Case:R1andR2have no Unstable Hidden Modes.- 11.1.3 The Discrete-Time Case.- 11.2 Nyquist Criterion.- 11.2.1 The Nyquist Criterion.- 11.2.2 Remarks on the Nyquist Criterion.- 11.2.3 Proof of Nyquist Criterion.- 11.2.4 The Discrete-Time Case.- 11.3 Robustness.- 11.3.1 Robustness With Respect to Plant Perturbations.- 11.3.2 Robustness With Respect to Exogenous Disturbances.- 11.3.3 Robust Regulation.- 11.3.4 Bandwidth-Robustness Tradeoff.- 11.3.5 The Discrete-Time Case.- 11.4 Kharitonov's Theorem.- 11.4.1 Hurwitz Polynomials.- 11.4.2 Kharitonov's Theorem.- 11.5 Robust Stability Under Structured Perturbations.- 11.5.1 General Robustness Theorem.- 11.5.2 Special Case: Affine Maps and Convexity.- 11.5.3 The Discrete Time Case.- 11.6 Stability Under Arbitrary Additive Plant Perturbations.- 11.7 Transmission Zeros.- 11.7.1 Single-Input Single-Output Case.- 11.7.2 Multi-Input Multi-Output Case: Assumptions and Definitions.- 11.7.3 Characterization of the Zeros.- 11.7.4 Application to Unity Feedback Systems.- 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.

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