Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

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Linear Matrix Inequalities in System and Control Theory

This paper studies the control of a class of discrete event processes, i.e. processes that are discrete, asynchronous and possibly nondeter-ministic. The controlled process is described as the generator of a formal language, while the controller, or supervisor, is constructed from the grammar of a specified target language that incorporates the desired closed-loop system behavior. The existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language. Two examples are provided.

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Supervisory control of a class of discrete event processes

Geared primarily to an audience consisting of mathematically advanced undergraduate or beginning graduate students, this text may additionally 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. The minimal mathematical background required is a working knowledge of linear algebra and differential equations. The book covers what constitutes the common core of control theory and is unique in its emphasis on foundational aspects. While covering a wide range of topics written in a standard theorem/proof style, it also develops the necessary techniques from scratch. In this second edition, new chapters and sections have been added, dealing with time optimal control of linear systems, variational and numerical approaches to nonlinear control, nonlinear controllability via Lie-algebraic methods, and controllability of recurrent nets and of linear systems with bounded controls.

Mathematical Control Theory: Deterministic Finite Dimensional Systems

A long-run equilibrium theory of turnover is presented and is shown to explain the important regularities that have been observed by empirical investigators. A worker's productivity in a particular job is not known ex ante and becomes known more precisely as the worker's job tenure increases. Turnover is generated by the existence of a nondegenerate distribution of the worker's productivity across different. The nondegeneracy is caused by the assumed variation in the quality of the worker-employer match.

Job Matching and the Theory of Turnover

A discrete event system (DES) is a dynamic system that evolves in accordance with the abrupt occurrence, at possibly unknown irregular intervals, of physical events. Such systems arise in a variety of contexts ranging from computer operating systems to the control of complex multimode processes. A control theory for the logical aspects of such DESs is surveyed. The focus is on the qualitative aspects of control, but computation and the related issue of computational complexity are also considered. Automata and formal language models for DESs are surveyed. >

The control of discrete event systems

A problem in the control of automata on infinite strings is defined and analyzed. The key to the investigation is the development of a fixpoint characterization of the "controllability subset" of a deterministic Rabin automaton, the set of states from which the automaton can be controlled to the satisfaction of its own acceptance condition. The fixpoint representation allows straightforward computation of the controllability subset and the construction of a suitable state-feedback control for the automaton. The results have applications to control synthesis, automaton synthesis, and decision procedures for logical satisfiability; in particular, they represent a direct, efficient and natural solution to Church's problem, the construction of winning strategies for two-player zero-sum $\omega$-regular games of perfect information, and the emptiness problem for automata on infinite trees.

Control of Infinite Behavior of Finite Automata

men p(&), ~, A, V) is a ~~~l~~~ algebra, which is isomorphic to the Boolean algebra (2", -, n , U ), with

http://ieeexplore.ieee.org/iel4/9/6026/00233154.pdf

Control of Vector Discrete-Event Systems

This paper extends the authors' previous work on observability of discrete-event systems by taking time into consideration. In a timed discrete-event system, events must occur within their respective lower and upper time bounds. A supervisor can disable, enable, or force some events to achieve a given control objective. The authors assume that the supervisor does not observe all events, which is often the case in practice. The authors generalize the concept of observability to timed discrete-event systems and show that it characterizes the existence condition for a supervisor. The authors also generalize normality, a stronger version of observability, to timed discrete-event systems, which has nice properties that are absent in observability. The authors then derive conditions under which observability and normality are equivalent. The authors propose two methods to synthesize a supervisor, a direct approach and an indirect approach. An example is given to illustrate the results. >

Supervisory control of timed discrete-event systems under partial observation

For the multivariable control system described by \[\dot x = Ax + Bu,\quad y = Cx,\quad z = Dx,\] constructive necessary and sufficient conditions are given for the existence of state feedback $ u = Fx$ such that (i) $\ker F \supset \ker C$ (observability constraint), (ii) $D\exp [t(A + BF)] \to 0$ as $t \to \infty $ (output regulation), and (iii) any unstable modes of $A + BF$ are either uncontrollable or unobservable at y (internal stability). It is assumed that $\ker C$ is A-invariant, or equivalently that an observer or dynamic compensator is utilized. A common application is treated, and sensitivity is considered for a simple example.

W. M. Wonham

Papers

Control of Infinite Behavior of Finite Automata

Control of Vector Discrete-Event Systems

Supervisory control of timed discrete-event systems under partial observation

Regulation and Internal Stabilization in Linear Multivariable Systems

Optimal stochastic control