Sontag E.D. Mathematical Control Theory: Deterministic Finite-Dimensional Systems

Second edition. — Springer-Verlag, 1998. — XVI, 531 p. — (Texts in Applied Mathematics, 6). — ISBN: 0-387-984895.This textbook introduces the core concepts and results of Control and System Theory. Unique in its emphasis on foundational aspects, it takes a "hybrid" approach in which basic results are derived for discrete and continuous time scales, and discrete and continuous state variables. Primarily geared towards mathematically advanced undergraduate or graduate students, it may also be suitable for a second engineering course in control which goes beyond the classical frequency domain and state-space material. The choice of topics, together with detailed end-of-chapter links to the bibliography, makes it an excellent research reference as well.
The Second Edition constitutes a substantial revision and extension of the First Edition, mainly adding or expanding upon advanced material, including: Lie-algebraic accessibility theory, feedback linearization, controllability of neural networks, reachability under input constraints, topics in nonlinear feedback design (such as backstepping, damping, control-Lyapunov functions, and topological obstructions to stabilization), and introductions to the calculus of variations, the maximum principle, numerical optimal control, and linear time-optimal control.
Also covered, as in the First Edition, are notions of systems and automata theory, and the algebraic theory of linear systems, including controllability, observability, feedback equivalence, and minimality; stability via Lyapunov, as well as input/output methods; linear-quadratic optimal control; observers and dynamic feedback; Kalman filtering via deterministic optimal observation; parametrization of stabilizing controllers, and facts about frequency domain such as the Nyquist criterion.Preface to the Second Edition.
Preface to the First Edition.What Is Mathematical Control Theory?
Proportional-Derivative Control.
Digital Control.
Feedback Versus Precomputed Control.
State-Space and Spectrum Assignment.
Outputs and Dynamic Feedback.
Dealing with Nonlinearity.
A Brief Historical Background.
Some Topics Not Covered.
Systems.
Basic Definitions.
I/O Behaviors.
Discrete-Time.
Linear Discrete-Time Systems.
Smooth Discrete-Time Systems.
Continuous-Time.
Linear Continuous-Time Systems.
Linearizations Compute Differentials.
More on Differentiability.
Sampling.
Volterra Expansions.
Notes and Comments.
Reachability and Controllability.
Basic Reachability Notions.
Time-Invariant Systems.
Controllable Pairs of Matrices.
Controllability Under Sampling.
More on Linear Controllability.
Bounded Controls.
First-Order Local Controllability.
Controllability of Recurrent Nets.
Piecewise Constant Controls.
Notes and Comments.
Nonlinear Controllability.
Lie Brackets.
Lie Algebras and Flows.
Accessibility Rank Condition.
Ad, Distributions, and Frobenius’ Theorem.
Necessity of Accessibility Rank Condition.
Additional Problems.
Notes and Comments.
Feedback and Stabilization.
Constant Linear Feedback.
Feedback Equivalence.
Feedback Linearization.
Disturbance Rejection and Invariance.
Stability and Other Asymptotic Notions.
Unstable and Stable Modes.
Lyapunov and Control-Lyapunov Functions.
Linearization Principle for Stability.
Introduction to Nonlinear Stabilization.
Notes and Comments.
Outputs.
Basic Observability Notions.
Time-Invariant Systems.
Continuous-Time Linear Systems.
Linearization Principle for Observability.
Realization Theory for Linear Systems.
Recursion and Partial Realization.
Rationality and Realizability.
Abstract Realization Theory.
Notes and Comments.
Observers and Dynamic Feedback.
Observers and Detectability.
Dynamic Feedback.
External Stability for Linear Systems.
Frequency-Domain Considerations.
Parametrization of Stabilizers.
Notes and Comments.
Optimality: Value Function.
Dynamic Programming.
Linear Systems with Quadratic Cost.
Tracking and Kalman Filtering.
Infinite-Time (Steady-State) Problem.
Nonlinear Stabilizing Optimal Controls.
Notes and Comments.
Optimality: Multipliers.
Review of Smooth Dependence.
Unconstrained Controls.
Excursion into the Calculus of Variations.
Gradient-Based Numerical Methods.
Constrained Controls: Minimum Principle.
Notes and Comments.
Optimality: Minimum-Time for Linear Systems.
Existence Results.
Maximum Principle for Time-Optimality.
Applications of the Maximum Principle.
Remarks on the Maximum Principle.
Additional Exercises.
Notes and Comments.
Appendixes.
Linear Algebra.
Differentials.
Ordinary Differential Equations.List of Symbols.