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

The problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency-domain results on H/sup infinity / optimization. A complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given. Relations between these robust stabilization problems and H/sup infinity / control theory are explored. It is also shown that in a number of cases, if a robust stabilization problem can be solved via Lyapunov methods, then it can be also be solved via H/sup infinity / control theory-based methods. >

Robust stabilization of uncertain linear systems: quadratic stabilizability and H/sup infinity / control theory

An LQG (linear quadratic Gaussian) control-design problem involving a constraint on H/sup infinity / disturbance attenuation is considered. The H/sup infinity / performance constraint is embedded within the optimization process by replacing the covariance Lyapunov equation by a Riccati equation whose solution leads to an upper bound on L/sub 2/ performance. In contrast to the pair of separated Riccati equations of standard LQG theory, the H/sup infinity /-constrained gains are given by a coupled system of three modified Riccati equations. The coupling illustrates the breakdown of the separation principle for the H/sup infinity /-constrained problem. Both full- and reduced-order design problems are considered with an H/sup infinity / attenuation constraint involving both state and control variables. An algorithm is developed for the full-order design problem and illustrative numerical results are given. >

LQG control with an H/sup infinity / performance bound: a Riccati equation approach

The authors present linear matrix inequality (LMI) conditions for output feedback control problems. The results are based on sufficient conditions because they are dependent on the particular state-space representation used for describing the system. Nevertheless, the conditions are not sensitive to a certain class of state-space transformations, and if the control problem is feasible then there exists some state-space transformation leading the conditions to be necessary and sufficient for the problem. The authors approach can be used for designing decentralized controllers and is easily extended to H/sub 2/, H/sub /spl infin// and mixed H/sub 2//H/sub /spl infin// problems via standard LMI techniques. The continuous- and discrete-time cases are considered and numerical examples are given to illustrate the results.

Sufficient LMI conditions for output feedback control problems

This paper investigates the use of generalized sampled-data hold functions (GSHF) in the control of linear time-invariant systems. The idea of GSHF is to periodically sample the output of the system, and generate the control by means of a hold function applied to the resulting sequence. The hold function is chosen based on the dynamics of the system to be controlled. This method appears to have several advantages over dynamic controllers: it has the efficacy of state feedback without the requirement of state estimation; it provides the control system designer with substantially more freedom; and it requires few on-line computations. This paper focuses on four questions: pole assignment, specific behavior, noise sensitivity, and robustness. Among the problems solved are: simultaneous arbitrary pole assignment for a finite number of systems by a single GSHF controller, exact model matching, decoupling, and optimal noise rejection. Examples are given.

Control of linear systems using generalized sampled-data hold functions

Given a dynamical system whose description includes time-varying uncertain parameters, it is often desirable to design an output feedback controller leading to asymptotic stability of a given equilibrium point. When designing such a controller, one may consider static (i.e., memoryless) or dynamic compensation. In this paper, we show that solvability of various output feedback design problems is implied by existence of a solution to a certain constrained Lyapunov problem (CLP). The CLP can be stated in purely algebraic terms. Once the CLP is described, we provide necessary and sufficient conditions for its solution to exist. Subsequently, we consider application of the CLP to a number of robust stabilization problems involving static output feedback and observer-based feedback.

The constrained Lyapunov problem and its application to robust output feedback stabilization

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The constrained Lyapunov problem and its application to robust output feedback stabilization