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.

/pdf/essentials-of-robust-control-3kaks6yk3h.pdf

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)

Necessary and sufficient conditions for a linear multivariable feedback composite system to be completely controllable and completely observable are derived. The present derivation also provides a simple proof of the Hsu-Chen Theorem.

https://ieeexplore.ieee.org/iel5/9/24129/01100201.pdf

On the controllability and observability of composite systems

Discrete-time control of continuous systems with approximate decentralized fixed modes

The problem of constructing a controller which gives a desirable transient response, e.g., a time response which is smooth, ripple-free, fast, and with negligible interaction effects, is considered in this paper for the servomechanism problem for continuous linear time invariant (LTI) systems. It is shown that this may be accomplished by constructing a controller which minimizes the following cheap control performance index:

Optimal Transient Response Shaping of the Servomechanism Problem

The problem of designing realistic decentralized controllers to solve the robust decentralized servomechanism problem [1] is considered in this paper. In particular, it is desired to find a decentralized controller for a plant to solve the robust servomechanism problem so that closed loop stability and asymptotic regulation occur, and also so that other desirable properties of the controlled system, such as fast response, low-interaction, integrity, tolerance to plant variations, constraints on gain magnitudes etc. occur. The method of design is based on extending the centralized design method of [2] to deal with the decentralized case. A number of examples are included to illustrate the design method.

The Design of Decentralized Controllers for the Robust Servomechanism Problem using Parameter Optimization Methods

A graphic procedure is presented which allows the describing function technique to be extended to a single-loop feedback system with two nonlinearities. The graphic technique is very simple and immediately allows qualitative answers, or quantitative answers subject to the usual errors and restrictions of the describing function technique, to be obtained regarding the presence of limit cycles, regions of stability, instability, etc. The method essentially is as follows. A plot of G_{1}(j\omega) G_{2}(j_\omega) in Fig. 1 vs. ω is made, and the point of intersection of G_{1}(j\omega) G_{2}(j\omega) with the negative real axis is noted, for example, at G_{1}(j\omega^{\ast}) G_{2}(j\omega^{\ast}) =-1/\Gamma, \Gamma > 0 . By plotting |G_{d_{1}}(A1)| vs. A 1 in the second quadrant, and |G_{d_{2}}(A_{2})| vs. A 2 in the fourth quadrant, it is possible to plot a curve (relating |G_{d_{1}}| vs. |G_{d_{1}}|) in the first quadrant. If this curve intersects |G_{d_{1}}| |G_{d_{2}}| = \Gamma , a limit cycle exists in the system. If no intersection takes place, then no limit cycle exists in the system.

Edward J. Davison

Papers

On the controllability and observability of composite systems

Discrete-time control of continuous systems with approximate decentralized fixed modes

Optimal Transient Response Shaping of the Servomechanism Problem

The Design of Decentralized Controllers for the Robust Servomechanism Problem using Parameter Optimization Methods

Application of the describing function technique in a single-loop feedback system with two nonlinearities