There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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.

/pdf/linear-matrix-inequalities-in-system-and-control-theory-2q5cf0pqm2.pdf

Linear Matrix Inequalities in System and Control Theory

Variable structure systems consist of a set of continuous subsystems together with suitable switching logic. Advantageous properties result from changing structures according to this switching logic. Design and analysis for this class of systems are surveyed in this paper.

Variable structure systems with sliding modes

1. Introduction. 2. Linear Algebra. 3. Linear Systems. 4. H2 and Ha Spaces. 5. Internal Stability. 6. Performance Specifications and Limitations. 7. Balanced Model Reduction. 8. Uncertainty and Robustness. 9. Linear Fractional Transformation. 10. m and m- Synthesis. 11. Controller Parameterization. 12. Algebraic Riccati Equations. 13. H2 Optimal Control. 14. Ha Control. 15. Controller Reduction. 16. Ha Loop Shaping. 17. Gap Metric and ...u- Gap Metric. 18. Miscellaneous Topics. Bibliography. Index.

/pdf/essentials-of-robust-control-3kaks6yk3h.pdf

Essentials of Robust Control

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)

A recent problem in adaptive control has been the task of controlling a system that cannot be characterized by a single mathematical model, and instead undergoes changes within a certain family of models. One strategy for controlling such a system has been to use switching between a fixed number of controllers to compensate for changing plant dynamics. This paper considers the problem of controlling an unknown plant contained in a family of models together with the additional task of controlling a plant whose plant dynamics shift from one member of the family to another. In particular, this paper describes a new switching controller that uses observers to improve switching transients, while maintaining an acceptable level of robustness.

/pdf/control-of-plants-which-change-using-switching-controllers-4r5qzfjrok.pdf

Control of plants which change using switching controllers

An overview of some problem associated with the robust control of industrial systems, together with some notions/design techniques which are potentially useful, are given in this paper. The paper begins by describing some representative "real system" examples (a large flexible space structure, a nuclear reactor control problem), and examines the type of problems which arise in attempting to control these systems. In this case, it is concluded that an input/output description of a plant does not by itself necessarily provide sufficient information to be able to guarantee robustness properties of a controller. The application of "gain margin" as a measure of robustness, and the parameter structural perturbation problem (which arises from "low frequency" uncertainty) are shown to be important for these classes of problems. Given that it is difficult to obtain even crude models for industrial systems, it is then suggested that a possible effective control design procedure is to use a "tuning regulator for unknown systems" approach, which does not require that a mathematical model of the system be available.

Robust control for industrial systems

Reliability of the Robust Servomechanism Controller for Decentralized Systems

An algebraic characterization of quotient decentralized fixed modes

It is well known that a linear time-invariant (LTI) system can be stabilized using decentralized LTI control if and only if the system does not possess any unstable decentralized fixed modes (DFMs). However, in industrial system application studies, it often is the case that a system has no DFMs, but may have approximate DFMs (ADFMs), which are modes that are not DFMs, but are “close” to being DFMs; in particular, such ADFMs can be divided into two types: “structured” ADFMs and “unstructured” ADFMs.In general ADFMs have the property that, although they are not fixed, they may require a “huge control energy” to shift the modes to desirable regions of the complex plane, which may be impossible to obtain. It is thus important to be able to characterize and determine what modes of a system, if any, are ADFMs, and this is the focus of the chapter. A number of industrial application problems will be used to demonstrate the effectiveness of the proposed algorithms for the calculation of ADFMs and to illustrate the properties of these ADFMs.

Edward J. Davison

Papers

Control of plants which change using switching controllers

Robust control for industrial systems

Reliability of the Robust Servomechanism Controller for Decentralized Systems

An algebraic characterization of quotient decentralized fixed modes

Characterization and Calculation of Approximate Decentralized Fixed Modes (ADFMs)