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 Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

The use of statistical techniques to build approximations of expensive computer analysis codes pervades much of today’s engineering design. These statistical approximations, or metamodels, are used to replace the actual expensive computer analyses, facilitating multidisciplinary, multiobjective optimization and concept exploration. In this paper, we review several of these techniques, including design of experiments, response surface methodology, Taguchi methods, neural networks, inductive learning and kriging. We survey their existing application in engineering design, and then address the dangers of applying traditional statistical techniques to approximate deterministic computer analysis codes. We conclude with recommendations for the appropriate use of statistical approximation techniques in given situations, and how common pitfalls can be avoided.

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Metamodels for Computer-Based Engineering Design: Survey and Recommendations

The complex structured singular value

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

The presence of uncertain parameters in a state space or frequency domain description of a linear, time-invariant system manifests itself as variability in the coefficients of the characteristic polynomial. If the family of all such polynomials is polytopic in coefficient space, we show that the root locations of the entire family can be completely determined by examining only the roots of the polynomials contained in the exposed edges of the polytope. These procedures are computationally tractable, and this criterion improves upon the presently available stability tests for uncertain systems, being less conservative and explicitly determining all root locations. Equally important is the fact that the results are also applicable to discrete-time systems.

Root locations of an entire polytope of polynomials: It suffices to check the edges

Contents: Examples for Modelling of Plants with Uncertain Parameters.- Control System Structures.- Analysis and Design.- Classical Stability Tests Applied to Uncertain Polynomials.- Testing Sets.- Value Set Construction.- The Stability Radius.- Single-Loop Feedback Structures.- Gamma-Stability.- Robustness of Sampled-Data Control Systems.- Parameter Space Design.- Design by Optimizing a Vector Performance Index.- The Four-Wheel Car Steering Model.- Polynomials and Polynomial Equations.

Robust Control: Systems With Uncertain Physical Parameters

By assuming controller structures in Chapter 2 we have generated several examples of closed-loop characteristic polynomials p(s, q,k), where the vector k contains the free design parameters in the fixed controller structure and q contains the uncertain plant parameters in a given operating domain Q, i.e. q ∈Q.

Analysis and Design

In [1], Kharitonov gave an elegant and simple stability criterion for continuous-time systems. This note reports on similar results for discrete-time systems.

Some discrete-time counterparts to Kharitonov's stability criterion for uncertain systems

The presence of uncertain parameters in either a state space or frequency domain description of a linear, time-invariant system manifests itself as variations in the coefficients of the characteristic polynomial. If the family of all such polynomials is polytopic in coefficient space, we'll show that the root locations of the entire family can be completely determined by examining only the roots of the polynomials contained in the exposed edges of the polytope. These results are computationally feasible and this crterion goes beyond the presently available stability tests for uncertain systems by being less conservative in all cases and by explicitly determining all root locations. Equally important is the fact that these results are also applicable to discrete-time systems

Andrew C. Bartlett

Papers

Root locations of an entire polytope of polynomials: It suffices to check the edges

Robust Control: Systems With Uncertain Physical Parameters

Analysis and Design

Some discrete-time counterparts to Kharitonov's stability criterion for uncertain systems

Root Locations of an Entire Polytope of Polynomials: It Suffices to Check the Edges