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

A survey of current continuous nonlinear multi-objective optimization (MOO) concepts and methods is presented. It consolidates and relates seemingly different terminology and methods. The methods are divided into three major categories: methods with a priori articulation of preferences, methods with a posteriori articulation of preferences, and methods with no articulation of preferences. Genetic algorithms are surveyed as well. Commentary is provided on three fronts, concerning the advantages and pitfalls of individual methods, the different classes of methods, and the field of MOO as a whole. The Characteristics of the most significant methods are summarized. Conclusions are drawn that reflect often-neglected ideas and applicability to engineering problems. It is found that no single approach is superior. Rather, the selection of a specific method depends on the type of information that is provided in the problem, the user’s preferences, the solution requirements, and the availability of software.

Survey of multi-objective optimization methods for engineering

A tutorial account of variable structure control with sliding mode is presented. The purpose is to introduce in a concise manner the fundamental theory, main results, and practical applications of this powerful control system design approach. This approach is particularly attractive for the control of nonlinear systems. Prominent characteristics such as invariance, robustness, order reduction, and control chattering are discussed in detail. Methods for coping with chattering are presented. Both linear and nonlinear systems are considered. Future research areas are suggested and an extensive list of references is included. >

Variable structure control: a survey

Survey paper: Set invariance in control

Presents a guide to sliding mode control for practicing control engineers. It offers an accurate assessment of the so-called chattering phenomenon, catalogs implementable sliding mode control design solutions, and provides a frame of reference for future sliding mode control research.

A control engineer's guide to sliding mode control

We consider a dynamic system containing uncertain elements. Only the set of possible values of these uncertainties is known. Based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state. These feedback controls are continuous functions of the state.

Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems

This paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems. The vector of uncertain parameters $q( \cdot )$ is time-varying, and its values $q(t)$ lie within a prespecified bounding set Q in $R^p $. Furthermore, no statistical description of $q( \cdot )$ is assumed, and the controller is shown to render the closed loop system “practically stable” in a so-called guaranteed sense; that is, the desired stability properties are assured no matter what admissible uncertainty $q( \cdot )$ is realized. Within the perspective of previous research in this area, this paper contains one salient feature: the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time-invariant. In contrast, in much of the previous literature (see, for example, [1], [2], [7], and [9]), a linear system is stabilized via nonlinear control. Another feature of this paper is the fact that the...

A New Class of Stabilizing Controllers for Uncertain Dynamical Systems

We consider a class of linear dynamical systems containing uncertain elements and subject to uncertain inputs, and for which either uncertain state or output is available. We construct a feedback control, utilizing measured state or estimated state, which guarantees that every system response is ultimately bounded within a certain neighborhood of the zero state. Performance resulting from use of this control is compared with that due to the use of purely linear feedback control.

On the Efficacy of Nonlinear Control in Uncertain Linear Systems

This paper describes an approach to ultimate boundedness control under hypotheses less stringent than heretofore required. Instead of imposing the so-called matching conditions of previous authors, a certain "decomposition" of the dynamics is performed. The resulting "decomposed system" has two parts-a matched portion and a mismatched portion. A certain measure of mismatch M is defined and it is shown that effective control is possible as long as M does not exceed some critical mismatch threshold M*. It is seen that this threshold depends on the choice of feedback. Hence, it becomes possible to investigate the tradeoffs between the size of the mismatch threshold and the sizes of the feedback gains which achieve the desired ultimate boundedness of solutions.

George Leitmann

Papers

Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems

A New Class of Stabilizing Controllers for Uncertain Dynamical Systems

On the Efficacy of Nonlinear Control in Uncertain Linear Systems

On ultimate boundedness control of uncertain systems in the absence of matching assumptions

Guaranteed Asymptotic Stability for Some Linear Systems With Bounded Uncertainties