Robust Control Theory in Hilbert Space

1. Front Matter

   Pages i-xv

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2. Basic Hilbert Space Theory

   • Avraham Feintuch

   Pages 1-18

3. Operator Theoretic Preliminaries

   • Avraham Feintuch

   Pages 19-31

4. A Distance Formula and Some Consequences

   • Avraham Feintuch

   Pages 33-54

5. Factorization Theorems

   • Avraham Feintuch

   Pages 55-75

6. Linear Systems

   • Avraham Feintuch

   Pages 77-86

7. Stabilization

   • Avraham Feintuch

   Pages 87-115

8. Uniform Optimal Control

   • Avraham Feintuch

   Pages 117-142

9. Robustness of Time-Varying Systems

   • Avraham Feintuch

   Pages 143-164

10. The Gap Metric and Internal Stability

    • Avraham Feintuch

    Pages 165-185

11. Robust Stabilization in the Gap Metric

    • Avraham Feintuch

    Pages 187-205

12. Orthogonal Embedding of Time-Varying Systems

    • Avraham Feintuch

    Pages 207-216

13. Back Matter

    Pages 217-225

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Motivation The latest texts on linear systems for engineering students have begun incorpo­ rating chapters on robust control using the state space approach to HOC control for linear finite dimensional time-invariant systems. While the pedagogical and computational advantages of this approach are not to be underestimated, there are, in my opinion, some disadvantages. Among these disadvantages is the narrow viewpoint that arises from the amputation of the finite dimensional time-invariant case from the much more general theory that had been developed using frequency domain methods. The frequency domain, which occupied center stage for most of the develop­ ments of HOC control theory, presents a natural context for analysis and controller synthesis for time-invariant linear systems, whether of finite or infinite dimen­ sions. A fundamental role was played in this theory by operator theoretic methods, especially the theory of Toeplitz and skew-Toeplitz operators. The recent lecture notes of Foias, Ozbay, and Tannenbaum [3] display the power of this theory by constructing robust controllers for the problem of a flexible beam. Although controller synthesis depends heavily on the special computational ad­ vantages of time-invariant systems and the relationship between HOC optimization and classical interpolation methods, it turns out that the analysis is possible without the assumption that the systems are time-invariant.

• Hilbert space
• Operator theory
• control
• control theory
• optimal control
• robust control
• stability
• stabilization

• Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

  Avraham Feintuch

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