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 nonminimum phase index and its application to interacting multivariable control systems

In this paper, the problem of implementing the Miller-Davison (1989) self-tuning integral robust servomechanism for the case of constant references and disturbances is considered and examined using an experimental multi-input multi-output apparatus; cases involving both known and unknown estimates of the steady state gain matrix T are considered. In order to improve the speed of response, proportional feedback has been added in both instances, and this has yielded a superior transient response.

http://ieeexplore.ieee.org/iel4/5816/15505/00735118.pdf

Control of unknown systems using switching controllers: an experimental study

Cooperative control” is a term which is used to capture those problem areas in which some type of repetition of identical or non-identical subsystems, which are interconnected together, occurs. Such systems are often found in nature, i.e. in the motion of clusters of birds, fish, insects, etc. moving together, in the cell structure of mammals and life-forms, and also in the man-made systems such as in transportation systems. In such systems, a decentralized control configuration is often applied to control the overall system, so that some common objective is achieved. It is perhaps too early to be able to list the fundamental problems arising in this problem area; there is however one important application area in which an understanding of the behaviour of the near-identical objects is desirable, and that is with respect to disease processes associated with the abnormal behaviour of certain organs of the human body. This paper gives a review of such typical example systems for the case of: • “ fish-like” objects interacting in water. • vehicles which have the front wheel drive / steering interacting in a string formation. • analyzing the behaviour of intestinal slow-wave patterns which occur in the human body.

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Cooperative Control of Large Systems

This paper considers the problem of stabilizing a linear time-invariant multivariable system by using local feedback controllers and some limited information exchange among local stations. The problem of achieving a given degree of stability with minimum transmission cost is solved.

Minimization of transmission cost in decentralized control systems

In this technical note we study the servomechanism problem for positive open-loop stable LTI systems under constant tracking signals and constant disturbances. In particular, we show that in general the MIMO servomechanism problem for positive systems is not solvable, but is solvable for various classes of MIMO systems, and for the class of SISO systems in which the disturbances are small compared to the tracking signal. In particular, we give existence results as to what subclass of reference and disturbance signals can be considered, for the case when either the mathematical model of the plant is known or unknown, and we provide the controller structure that solves the problem for the case of measurable or unmeasurable disturbance signals.

Edward J. Davison

Papers

A nonminimum phase index and its application to interacting multivariable control systems

Control of unknown systems using switching controllers: an experimental study

Cooperative Control of Large Systems

Minimization of transmission cost in decentralized control systems

The Multivariable Servomechanism Problem for Positive LTI Systems