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)

In this paper, new theoretical and simulated results of a self-tuning proportional-integral-derivative (PID) robust servomechanism (for the class of constant reference and disturbance signals) are given, for cases involving both known and unknown estimates of the steady-state gain matrix /spl Tscr/.

Control of unknown MIMO systems using self-tuning PID controllers

The authors address the computational problem encountered by a novel method of stability robustness analysis previously outlined, when it is applied to large systems. An iterative procedure is developed to compute the singular values and singular vectors of certain classes of large composite matrices; such a procedure can be used to solve problems which, because of dimensionality, cannot be solved by applying the QR transformation method. The procedure is then applied to the determination of stability robustness bounds of large-state-space systems with real perturbations. Various numerical examples, including a 46th-order spacecraft system, are given to illustrate the results obtained. >

Computation of the stability robustness of large state space models with real perturbations

This paper presents a new optimal decentralized controller design method for solving the tracking and disturbance rejection problems for the class of large scale LTI systems, using only low order decentralized controllers. In particular, the class of large scale systems considered in this paper is described by the LTI model M2y + M1y + M0y = Bu + Eω, where y is the output, u is the input, ω is an unknown constant disturbance, where e = y − yref is the tracking error in the system and yref is a constant given signal. To illustrate the type of results which can be obtained using the new optimal decentralized control design method, the control of a large flexible space structure is studied and compared with the standard centralized LQR-Observer controller. In this case it is shown that the new decentralized controller obtained is orders of magnitude lower in dimension than the standard centralized LQR-Observer controller. The proposed controller also has the property that if a sensor/actuator failure occurs, the resulting system has certain fail-safe properties. The proposed optimal controller also has the property that it is strongly robust as compared to the standard centralized LQR-Observer controller. In particular, the proposed controller can have the property of being some 5 orders of magnitude more robust than the standard LQR-Observer controller. An illustration of the new decentralized controller design method is applied to a large flexible space structure (LFSS) system with 5 inputs and 5 outputs and of order 24.

Optimized decentralized control of large scale systems

http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2008/data/papers/1474.pdf

The Transmission Zero at s Radius and the Minimum Phase Radius of LTI Systems

Describes a graduate student laboratory experiment called "MARTS" (multivariable apparatus for real time control studies), which has been in use at the Systems Control Group, University of Toronto since 1985. The laboratory experiment is unique in that it uses industrial commercial process control sensors/actuators for its hardware, and thus when problems arise in controller implementation on MARTS, students cannot invoke the usual excuse that inadequate or faulty university sensor/actuator equipment is to blame. The paper describes the experimental apparatus, the control problem specifications to be carried out on the apparatus, mathematical modelling of the system, typical control experiments which have been carried out on the system using both conventional and unconventional controllers, and features which make the control laboratory experiment challenging.

Edward J. Davison

Papers

Control of unknown MIMO systems using self-tuning PID controllers

Computation of the stability robustness of large state space models with real perturbations

Optimized decentralized control of large scale systems

The Transmission Zero at s Radius and the Minimum Phase Radius of LTI Systems

Description of control laboratory multivariable experiment: "MARTS"