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 deals with the design of controllers for large-scale linear time-invariant multi-input multi-output systems, such as those that often arise in process control. We focus on two standard controller objectives: (i) asymptotic regulation subject to unmeasurable constant disturbances, and/or (ii) asymptotic tracking of constant set-points. In either case, standard output-feedback controller design methodologies typically result in controllers that have order at least as large as that of the plant; for large-scale systems, the controller order can consequently be impractically large. The purpose of this paper is to introduce, for the subclass of plants that are open-loop stable, a low-order three-term (i.e., PID) multivariable control design approach that is practical to compute numerically, even for large-scale systems. A design algorithm and existence results to construct such a controller are given, and the approach is applied to several examples. Remarkably, at least for the examples considered, the three-term controller's performance is quite similar to that achieved by the standard (much higher order) controller that solves the servomechanism problem

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Multivariable three-term optimal controller design for large-scale systems

Optimal transient response shaping in model predictive control

Performance Limitations of Non-Minimum Phase Systems

Brief Limiting performance of optimal linear discrete filters

The stability of a system described by an n th order differential equation y^{(n)} + a_{n-1}y^{(n-1)} + . . . + a_{1}y + a_{0} = 0 where a_{i}=a_{i}(t, y, \dot{y}, . . . , y^{(n-1)}), i=0, 1, . . . , n - 1 , is considered. It is shown that if the roots of the characteristic equation of the system are always contained in a circle on the complex plane with center (-z, 0), z > 0 , and radius Ω such that \frac{z}{\Omega} > 1 + nC_{[n/2]} where [n/2] = nearest integer \geq n/2 and nC_{m} = n!/m!(n-m)! , where n and m are integers, then the system is uniformly asymptotically stable in the sense of Liapunov.

Edward J. Davison

Papers

Multivariable three-term optimal controller design for large-scale systems

Optimal transient response shaping in model predictive control

Performance Limitations of Non-Minimum Phase Systems

Brief Limiting performance of optimal linear discrete filters

The stability of an nth-order nonlinear time-varying differential system