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)

This paper studies the decentralized control of a platoon of identical vehicles when each control agent is assumed to only have knowledge of the distance between itself and its immediate forward neighbor. In particular, it is desired to solve the decentralized robust servomechanism problem (RSP), so that the vehicles' separation distances are regulated to specified set points, independent of the lead vehicle's velocity and such that the system is string stable. It is shown that for a large class of identical decentralized controllers, namely those decentralized controllers which solve the RSP and which have stable stabilizing compensators, e.g. a 3-term controller, that it is impossible to solve the above problem. This gives motivation to consider nonidentical decentralized controllers for the platoon vehicle problem, and it is shown in this case that it is possible to solve the above problem. A number of examples are included, including examples which have a large number of vehicles in a platoon, i.e. N=2000.

Decentralized control of a large platoon of vehicles using non-identical controllers

The optimal output feedback control of a synchronous machine, consisting of constant output feedback gains from only those states which are physically available for measurement, is considered in this paper. An integral quadratic performance index of outputs and controller inputs is minimized for impulse type disturbances occurring in the system. This gives a physically realizable control system which optimally "stabilizes" the excitation and input power to the synchronous machine. Numerical examples are included to show the application of the theory to a thermal and a hydraulic machine equipped with static exciters, and connected to an infinite bus. It is found that it is desirable to control a synchronous machine using both exciter and input power control gates with both power angle and speed fed back. A reduction in the value of the index of performance by a factor as large as 66 may be achieved using this control configuration instead of the conventional control configuration.

The Optimal Output Feedback Control of a Synchronous Machine

A self-tuning regulator is presented which consists of a multivariable integral compensator together with a tuning mechanism which tunes the parameters of this compensator. It is shown that if the reference and disturbance are constant signals, then the controller and the plant states remain bounded for all t, and asymptotic error regulation occurs. It is also shown that the controller can be modified slightly so that it is noise-tolerant. A number of simulation examples are induced to illustrate the behaviour of the controller when applied to unknown plants. >

The self-tuning robust servomechanism problem

The concept of stability radius is generalized to matrix pairs. A matrix pair is said to be stable if its generalized eigenvalues are located in the open left half of the complex plane. The stability radius of a matrix pair (A, B) is defined to be the norm of the smallest perturbation Delta A such that (A+ Delta A, B) is unstable. The purpose is to estimate the stability radius of a given matrix pair. Depending on whether the matrices under consideration are complex or real, the problem can be classified into two cases. The complex case is easy and a complete solution is provided. The real case is more difficult, and only a partial solution is given. >

Edward J. Davison

Papers

Decentralized control of a large platoon of vehicles using non-identical controllers

The Optimal Output Feedback Control of a Synchronous Machine

The self-tuning robust servomechanism problem

The stability robustness of generalized eigenvalues

Robust Control of a General Servomechanism Problem: The Servo Compensator