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 the problem of finding a robust servomechanism controller which achieves exact asymptotic tracking and regulation for a given class of reference and disturbance signals, and arbitrarily good approximate asymptotic regulation for other classes of reference and disturbance input signals. Necessary and sufficient conditions for the existence of such a controller, and a characterization of controllers which accomplish this are given. An explicit algorithm for such controllers is also given; in particular, it is shown that the design of such a controller can always be accomplished by using the cheap control design method as used in [1] for the multivariable robust servomechanism problem. A degenerate type of robust servomechanism controller (called a high gain servomechanism controller) consisting of only static feedback gains, is also given, which has the property that it produces arbitrarily good approximate regulation for all tracking/disturbance signals; thus this controller shows that dynamics in the feedback controller are not essential for achieving good approximate error regulation. Some numerical examples are included to illustrate the results.

Approximate error regulation - The robust servomechanism problem for unknown input signals

It is shown that the design procedure proposed to solve the “perfect robust servomechanism” problem [1] has the property that it results in acceptable perturbation bounds with respect to both plant and controller uncertainty.

Optimal controllers which ‘work’ in presence of plant & controller uncertainty

In this chapter, the stability of decentralized control systems is investigated, which is shown to be dependent on the notion of decentralized fixed modes (DFMs). Different techniques are provided to characterize the DFMs of a general proper finite-dimensional LTI system. The first technique is an efficient numerical method which can be easily utilized on MATLAB. It basically uses the random number generator to check if any of the eigenvalues of the system remains fixed w.r.t. a generic constant gain matrix of the desired decentralized structure. The other techniques are analytical and need to verify the rank of a set of matrices obtained from the system matrices. One of the techniques requires checking the transmission zeros of a set of systems. The technique is initially given for a decentralized control system with SISO subsystems and is then extended to the systems with MIMO subsystems. The main characteristic of this method is that it requires the transmission zero of every (virtual) system in a set to be checked, which implies that the ranks of all of the corresponding matrices need to be verified. The other method, on the other hand, requires the ranks of a different set of matrices to be checked. The main difference of this method with the former one is that if any of the rank conditions does not hold, the corresponding eigenvalue is a DFM and there is no need to check the remaining matrices. Furthermore, a numerical method is provided which unlike the one described earlier does not require the choice of generic gain matrices. Finally, a procedure is introduced to find a LTI decentralized controller of proper structure to locate the poles of the closed-loop system in desired locations in the complex plane. The procedure uses centralized design techniques to find the local controllers.

Decentralized Control Systems

Definitions of gain margin tolerance and time lag tolerance for a multivariable linear time-invariant system are given which correspond to a generalization of the notions of gain and phase margin of classical control system theory. It is shown that a plant in state space description has a specified gain margin tolerance with respect to a given control configuration if and only if a certain augmented state space description of the plant remains stable when used in the same control configuration. Applications of these results are then made to obtain existence conditions to solve the stabilization problem and the robust servomechanism problem with gain margin and time lag tolerance constraints. In particular, it is shown that: (i) Generically, there exists no controller to stabilize an unstable multivariable system so that the resultant closed loop system possesses a non zero gain margin. (ii) For stable plants, there always exists a controller to solve the robust servomechanism problem so that it has an arbitrary specified gain margin/time lag tolerance constraint if and only if there exists a solution to the robust servomechanism problem for the plant. A controller synthesis method is then given which solves the robust servomechanism problem so that it has a specified gain margin/time lag tolerance. Various examples of the proposed algorithm are given to illustrate the type of results that may be obtained.

Edward J. Davison

Papers

Approximate error regulation - The robust servomechanism problem for unknown input signals

Optimal controllers which ‘work’ in presence of plant & controller uncertainty

Decentralized Control Systems

Solution of the robust servomechanism problem subject to specified gain margin and time lag tolerance constraints

The Future of Control Theory for Complex Systems