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

Recently we developed supervisor localization, a top-down approach to distributed control of discrete-event systems in the Ramadge-Wonham supervisory control framework. Its essence is the decomposition of monolithic (global) control action into local control strategies for the individual agents. In this paper, we establish a counterpart supervisor localization theory in the framework of State Tree Structures, known to be efficient for control design of very large systems. As before, we prove that the collective localized control behavior is identical to the monolithic optimal (i.e. maximally permissive) and nonblocking controlled behavior. Further, we propose a new and more efficient localization algorithm which exploits BDD computation.

Supervisor localization of discrete-event systems based on State Tree Structures

In this paper we present an approach for blocking detection in multiprocess DES. In the proposed approach, potential blocking states are first identified by examining shared transitions in the system components and then tested for their reachability.

Blocking detection in discrete event systems

Algebraic methods in linear multivariable control

In this paper we present a hierarchical method that breaks up a plant 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 (DES) is non-blocking. 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.

Hierarchical interface-based non-blocking verification

Discrete-event systems (DES) can be found as essential integrated subsystems in electrical power systems. The continuous trajectory of the system state can be interrupted by discrete control actions and uncontrolled disturbances. Under-load tap-changing transformers (ULTC) which obviously have discrete-event behavior are widely used in transmission systems to take care of instantaneous variations in the load conditions in substations. ULTC may be controlled either automatically or manually. The paper discusses synthesis of supervisory controller of ULTC as a discrete-event system. Different parts of the ULTC as well as its control logic (specification) have been modeled as DES. Supervisory controllers are designed for the tap-changer in Automatic and Auto/Manual modes of operation. It is shown that the specifications are controllable and the closed loop control system is non-blocking. Protective system designers in electrical powers systems can use the proposed approach to verify their control logic for ULTC.

W. M. Wonham

Papers

Supervisor localization of discrete-event systems based on State Tree Structures

Blocking detection in discrete event systems

Algebraic methods in linear multivariable control

Hierarchical interface-based non-blocking verification

Discrete-event system modeling and supervisory control for under-load tap-changing transformers