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

We propose a top-down approach, called supervisor localization, to distributed control of large-scale discrete-event systems (DES). The essence of this approach is to allocate, or localize, ‘external’ supervisory control action to individual plant components as their ‘internal’ control strategies. To cope with state explosion of large DES, we develop a decomposition-aggregation procedure that employs a flexible decentralized and hierarchical architecture which reduces computational effort in localization. This procedure is demonstrated in detail on a benchmark Production Cell that has state size of order 108.

Supervisor localization for large-scale discrete-event systems

An approach to the design of modular supervisory control strategies based on a framework for the modeling and supervisory control of discrete-event systems is presented. As for centralized supervision, the approach allows the consideration of logic-based specifications, control-enforcement-related constraints, and temporal and utility optimality behavioral specifications. It yields modular supervisory control strategies that are least restrictive within given specifications, and which are correct by construction and offers all the advantages of distributed control. Under given conditions, modular supervision based on the techniques presented is equivalent to centralized supervision based on the techniques presented by B.A. Brandin et al. (1992) and by B.A. Brandin and W.M. Wonham (1993). A small manufacturing cell example illustrates the results presented. >

Manufacturing cell supervisory control-a modular timed discrete-event system approach

State tree structures (STS) are an adaptation of statecharts to supervisory control theory. STSLib is a C++ library that we have developed to support the symbolic analysis and synthesis of STS. This paper presents a short introduction to the library, and then applies the library to two benchmarks: (1) cat and mouse tower (CMT), (2) dining philosophers (DP). We demonstrate that STSLib can design optimal nonblocking supervisors for systems of state size up to 10626, and the resulting controllers are tractable and readily comprehensible.

STSLib and its application to two benchmarks

Optimal regulator problem, where a scalar control function is found, which minimizes the functional for quadratic performance index

Optimal bang-bang control with quadratic performance index

This tutorial paper deals with the Internal Model Principle (IMP) from different perspectives. The goal is to start from the principle as introduced and commonly used in the control theory and then enlarge the vision to other fields where “internal models” play a role. The biology and neuroscience fields are specifically targeted in the paper. The paper ends by presenting an “abstract” theory of IMP applicable to a large class of systems.

W. M. Wonham

Papers

Supervisor localization for large-scale discrete-event systems

Manufacturing cell supervisory control-a modular timed discrete-event system approach

STSLib and its application to two benchmarks

Optimal bang-bang control with quadratic performance index

Internal Models in Control, Biology and Neuroscience