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

The increasing use of computer control systems in safety-critical real-time systems has led to a need for methods to ensure the correct operation of real-time control systems. Through an example, this paper introduces the use of algebraic equivalence to verify the correct operation of such systems. A controller is considered verified if its implementation is proven to be equivalent to its specification. Real-time systems are modeled using a modified version of Ostroff's (1989, 1990) timed transition models (TTM's), which is introduced along with our adaptation of Milner's (1980, 1989) observation equivalence to TTM's. A set of "behavior preserving" transformations is then developed, shown to be consistent for proving observation equivalence, and applied to solve an industrial real-time controller software verification problem. Finally the incompleteness of a given set of transformations is briefly discussed. >

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Equivalence preserving transformations for timed transition models

The timed discrete-event system framework admits disablement and forcing as means of control, as well as the treatment of hard temporal behavioural constraints. The notion of modular supervisory control of discrete-event systems proposed by Wonham-Ramadge (1988) is extended to this framework. The approach consists of dividing the overall supervisory task into two or more subtasks, with the resulting individual subsupervisors being run concurrently to implement a solution of the original control problem, if necessary under the assumption of partial observation. The design of the corresponding subsupervisors deals with issues of conflict, blocking, optimality and correctness. The results presented can be used in the development of control software for interacting concurrent processes. Based on them, experimental work (hardware and software) has been carried out on the supervision of manufacturing systems. >

Modular supervisory control of timed discrete-event systems

We present a hierarchical method that decomposes a system into two subsystems, and restricts the interaction of the subsystems by means of an interface. We present a definition for an interface, and define a set of interface consistency properties that can be used to verify if a discrete-event system is nonblocking and controllable. Each clause of the definition can be verified using only one of the two subsystems; thus the complete system model never needs to be constructed, offering significant savings in computational effort. Additionally, the development of clean interfaces facilitates re-use of the component subsystems.

Hierarchical interface-based supervisory control: serial case

In this paper, we describe the manufacturing testbed we have built to investigate the implementation of RW supervisors on programmable logic controllers (PLC). We discuss the modeling of the testbed and the design of its controllers. Finally, we present several theorems for verifying controllability and nonblocking on large systems.

Discrete event systems modeling and control of a manufacturing testbed

A framework is proposed for investigating the verification and synthesis of controllers for real-time discrete event systems. In the framework, plants and controllers are modelled with extended state machines (ESMs), and a real-time temporal logic (RTL) is used for specification of required plant behaviour and for verifying that controllers satisfy their specifications. Controllers are implemented as tasks in a real-time distributed programming language. This paper discusses the issues of modelling and specification.

W. M. Wonham

Papers

Equivalence preserving transformations for timed transition models

Modular supervisory control of timed discrete-event systems

Hierarchical interface-based supervisory control: serial case

Discrete event systems modeling and control of a manufacturing testbed

State machines, temporal logic and control: A framework for discrete event systems