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.