Mathematical Control Theory

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 ExternalStability 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.