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 considers an extension of the tuning regulator design method of [1]-[3] to include a large class of nonlinear multivariable systems. In particular, it is desired to solve the servomechanism problem for an unknown nonlinear plant, which is assumed to be open loop asymptotically stable, for the class of constant set-points and disturbances. In this case, three classes of nonlinear controllers, which are gain-scheduling controllers, are described: (i) error attenuation, (ii) feedforward and (iii) robust controllers, and sufficient conditions by which the resultant closed loop system gives global asymptotic stability and regulation are obtained. The sufficient conditions obtained simplify to the necessary and sufficient conditions [1] associated with the case when the plant is assumed to be described by a linear time-invariant model.

Control of Unknown Nonlinear System using Gain-Scheduling Tuning Regulators

A limitation in achieving effective control for disturbance rejection/set point tracking in a multivariable plant is often associated with the operating range of the control input signals associated with the plant, i.e. control input saturation, and maximum rates of input change constraints. This paper proposes a LTI controller, called an "unsaturated controller", for a stable linear time invariant plant, which has the property that it solves the robust servomechanism problem for control disturbances/set-points, and maximizes the operating range of the controller for the plant, subject to any given maximum rate of change constraints, which may be imposed on the control inputs of the plant. Some application studies are included to illustrate the behaviour of the controller.

The Robust Servomechanism Problem with Input Saturation Constraint

The minimal realization of a class of decentralized systems (single-input, single-output channels, distinct eigenvalues) is considered. It is shown that "parameter independent" eigenvalues may be easily identified and discarded from the state-space model. On the other hand, the "parameter dependent" fixed modes are identified in the transfer-function-matrix and minimization is done before going to a state space description.

http://ieeexplore.ieee.org/iel5/4788052/4788053/04788202.pdf

Minimal Realization of a Class of Decentralized Systems

Decentralized control using local models for large-scale systems.

In this paper, the concept of restricted real perturbation values of a complex matrix triplet is introduced, and a formula for computing lower bounds of these values is presented. Restricted real perturbation values are a generalization of the real perturbation values introduced in [1], which is a key concept in evaluating various robustness radii found in the control literature, such as the real controllability/ observability radius, the real decentralized fixed-mode radius, the real minimum-phase radius, etc. The generalization to restricted real perturbation values is needed for an extension of these radii to account for more general system perturbation structures. As an example, we will use the results of this paper to compute the true value of the structured real controllability radius of the multi-link inverted pendulum system. Also, we will numerically investigate cases of when the provided lower bounds are achievable.

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Edward J. Davison

Papers

Control of Unknown Nonlinear System using Gain-Scheduling Tuning Regulators

The Robust Servomechanism Problem with Input Saturation Constraint

Minimal Realization of a Class of Decentralized Systems

Decentralized control using local models for large-scale systems.

Restricted real perturbation values with applications to the structured real controllability radius of LTI systems