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

Some basic results of supervisory control theory are extended to the setting of $\omega$-languages, formal languages consisting of infinite strings. The extension permits the investigation of both liveness and safety issues in the control of discrete-event systems. A new controllability property appropriate to the infinitary setting ($\omega$-controllability) is defined; this language property captures in a natural way the limitations of available control actions. It is shown that every specification language contains a unique maximal $\omega$-controllable sublanguage, representing the least upper bound of the set of achievable closed-loop sublanguages. This supremal $\omega$-controllable sublanguage allows a simple formulation of necessary and sufficient conditions for the solvability of an infinitary supervisory control problem. The problems of effectively deciding solvability of the control problem and of effectively synthesizing appropriate supervisors are solved for the case where the plant is represented by a deterministic Buchi automaton and the specification of legal behavior by a deterministic Rabin automaton.

Supervision of Infinite Behavior of Discrete-Event Systems

The concept of observer was introduced in previous work by the authors on a hierarchical control theory of discrete-event systems (DES). It was shown that the observer property of the “causal reporter” map, which in this theory models information flow in a hierarchical DES, plays a role in ensuring that a nonblocking supervisor at a given level of a hierarchy does not cause blocking in the level below. In this paper, we investigate the following problem: Given a causal reporter map that is not an observer, how can we design an observer by modifying this map? In case the latter is represented by a finite Mealy automaton, an effective computational algorithm is developed for computing an observer with the coarsest possible equivalence kernel that is finer than that of the given map. Three examples are provided for illustration.

On the Computation of Observers in Discrete-Event Systems

A discrete event process is defined in algebraic terms and its behavior is given by an appropriate formal language.

Supervisory control of discrete event processes

In the setting of abstract automata, it is shown that a feedback regulator must incorporate a dynamic model of the exosystem ('outside world') to which the regulator is designed to respond. The result is based on general, lattice-theoretic definitions of the concepts of feedback and observability.

Towards an abstract internal model principle

A framework is introduced for passive online fault diagnosis in timed discrete-event systems (TDES). It extends the previous work of the authors (1999) on a state-based approach to fault diagnosis by incorporating timing information. This enhances the accuracy of diagnosis. In this methodology instead of directly extending the existing framework to TDES, an alternative approach is taken which, in many cases, leads to significant reduction in online computing requirements and the size of the diagnoser at the expense of more off-line design calculations.

W. M. Wonham

Papers

Supervision of Infinite Behavior of Discrete-Event Systems

On the Computation of Observers in Discrete-Event Systems

Supervisory control of discrete event processes

Towards an abstract internal model principle

Fault diagnosis in timed discrete-event systems