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)

The author of the above [ibid., vol. AC-13. pp. 99-102, Feb. 1968] has called the following to the attention of the Editor. Equations are corrected as follows: on page 99 (the first equation in the abstract), page 100 equation (1) and equation (12), page 101 equation (16) and (25).

https://ieeexplore.ieee.org/iel5/9/24101/01098887.pdf

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

The problem of identifying a SISO discrete linear time invariant system of known order n, where n is large, and with a finite input/output data record, is considered in this paper. It is shown that for the class of least squares identification techniques, there is a fundamental difficulty in carring out such an identification. In particular, it is shown that generically the condition number associated with the parameter estimation problem monotonically increases as the order n increases, and depends on the controllability and observability measures of the system. Moreover, it is shown that some classes of "large scale problems" are intrinsically difficult to correctly identify. An example of a class of large flexible space structures problems is given to show the type of difficulties which may arise for large systems.

Numerical Problems in Identification of Large Scale Systems

In this chapter, a technique is presented to design a compensator for an LTI system to solve the robust servomechanism problem in both centralized and decentralized forms. Robust servomechanism problem is a fundamental problem in control theory, where the objective is to design a controller so that the output signal tracks a reference input and rejects a disturbance input with no steady-state error when the parameters of the system are subject to change. It is assumed that the reference and disturbance inputs can be expressed as a combination of exponentials which are not decaying. This includes sinusoidal signals, ramp signals as well as constant signals (which are commonly used test signals in control). The structure of the controller in both centralized and decentralized forms depends on the characteristics of the reference and disturbance signals.

Robust Servomechanism Problem

Up to this point in the book, we have assumed that the plant model has been given, and the objective is to carry out system analysis and controller design. In many situation, however, a good model is not available. The two typical solutions to this problem are (i) to model the plant and identify the parameters and (ii) use an adaptive controller. In both cases certain plant sructural information is typically needed, such as the plant order, which may be available in simple cases but not readily so in more complicated situations. In this chapter we examine a more direct approach, wherein the goal is to design a controller for the plant without the intermediate step of obtaining a detailed model; it is a generalization of the classical approach to tuning a three-term PID controller used for single-input single-output systems. The method requires the plant to be stable and it assumes that (i) a finite number of experiments can be carried out on the plant, and (ii) it is possible to carry out on-line tuning of the controller parameters. While the method is presented for the case of step tracking, the approach can be generalized.

Edward J. Davison

Papers

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

Numerical Problems in Identification of Large Scale Systems

Robust Servomechanism Problem

On Tuning Regulators

Robustness computation of real stability radii for linear time-invariant discrete time-delay systems