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 presents the fundamental concepts of a generalized central power management system and a decentralized, robust control strategy for autonomous mode of operation of a microgrid that includes multiple distributed energy resource (DER) units. DER units are divided into voltage-controlled and power-controlled DER units. The frequency of each DER unit is determined by its independent internal oscillator and all oscillators are synchronized by a common time-reference signal based on a global positioning system (GPS). The power management system (PMS) specifies the power and voltage set points for the local controllers of each DER unit. A linear, time-invariant, multivariable, robust, decentralized control system is designed to track the setpoints. Each control agent guarantees fast tracking, zero steady state error, and robust performance despite uncertainties of the microgrid parameters, topology, and the operating point. Existence conditions, control design procedures, eigenanalysis and robust stability analysis of the closed-loop system, and performance of the control strategy based on digital time-domain simulation studies in PSCAD/EMTDC platform, are reported. Performance of the control system is also verified based on hardware-in-the-loop (HIL) studies in the RTDS environment.

A Generalized Decentralized Robust Control of Islanded Microgrids

A new algorithm for the design of linear multivariable systems is presented. This algorithm constructs a set of minimal-order (McMillan degree) solutions to a proper rational matrix equation. The algorithm has direct application in exact model matching, inverse systems, and dynamic observers. In particular, application of the algorithm is made to the construction of a minimal-order inverse system.

A minimization algorithm for the design of linear multivariable systems

An adaptive controller that can provide exponential Lyapunov stability for an unknown linear time-invariant (LTI) system is presented. 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 (see ibid., vol.AC-31, p.1097-1103, Dec. 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 shows that adaptive stabilization is possible under very mild assumptions without large state deviations. >

An adaptive controller which provides Lyapunov stability

A numerically stable and fast computational method is given for the solution of the matrix Ricatti differential equation with finite terminal time.

Edward J. Davison

Papers

A Generalized Decentralized Robust Control of Islanded Microgrids

A minimization algorithm for the design of linear multivariable systems

An adaptive controller which provides Lyapunov stability

The numerical solution of the matrix Riccati differential equation

Paper: Connectability and structural controllability of composite systems