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.

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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.

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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 general description of the decentralized control problem , with particular emphasis on large scale systems, is made. In this problem, constraints on the structure of the information flow between the manipulated inputs of the system and measured outputs of the system are imposed. Typically, it is desired to find a controller for this system (subject to the above constraints), so that stability of the resultant closed loop system is obtained, and so that regulation/tracking of the outputs of the system occurs, independent of any input disturbances occurring in the system. A motivation for dealing with the decentralized problem is made, classical ways of solving the problem are described, and recent results obtained on the problem are surveyed. Some numerical examples are included, in particular, a power system consisting of three interconnected synchronous machines; it is shown in this power system that there is no real advantage is using a (more complex) centralized control system over the coventional (and more simple) decentralized control system which is normally applied.

Decentralized Stabilization and Regulation in Large Multivariable Systems

The LQ optimal regulator problem with cheap control is investigated for plants of the form x = Ax + Bu, y = Cx + Du, where D ± 0 makes the plant not strictly proper. It is found that perfect regulation of the output with closed-loop stability is achieved if and only if the plant is detectable, is minimum phase and there exist independent inputs equal to or greater in number to the number of independent outputs.

The optimal LQ regulator with cheap control for not strictly proper systems

Consider a linear time-invariant (LTI) plant which is not completely specified, but instead belongs to a finite set of known plants, say {P i : i ∈ p}. Our objective is to design a controller which provides “good” tracking and disturbance rejection, in a sufficiently well-defined sense, for this partially known plant. We first design a high-performance LTI controller K i for each possible P i if the pair (P i , K j ) is stable iff i = j and has no eigenvalues on the imaginary axis for any i, j ∈ p and if an upper bound on the magnitude of the unmeasurable disturbance signal is available, then it is shown that a switching mechanism can be used to find the correct LTI controller; furthermore, each LTI controller need only be tried once. This kind of problem often arises in an industrial setting, and is often approached using heuristic “gain-scheduling” techniques.

Adaptive Control of a Family of Plants

Structural modification of systems using discretization and generalized sampled-data hold functions

An adaptive controller is presented which can provide exponential Lyapunov stability for an unknown linear time-invariant (LTI) system. The only required a priori information about the plant is that the order of an LTI stabilizing compensator be known, although this can be reduced to assuming only that the plant is stabilizable and detectable at the expense of using a more complicated controller. This result extends the work of M. Fu and B.R. Barmish (1986) in which it is shown that there exists an adaptive controller which provides exponential Lyapunov stability if it is assumed that an upper bound on the plant order is known and that the plant lies in a known compact set; it is shown that adaptive stabilization is possible under very mild assumptions without 'large' state deviations. >

Edward J. Davison

Papers

Decentralized Stabilization and Regulation in Large Multivariable Systems

The optimal LQ regulator with cheap control for not strictly proper systems

Adaptive Control of a Family of Plants

Structural modification of systems using discretization and generalized sampled-data hold functions

An adaptive controller which provides Lyapunov stability