1 Basic Hilbert Space Theory.- 2 Operator Theoretic Preliminaries.- 3 A Distance Formula and Some Consequences.- 4 Factorization Theorems.- 5 Linear Systems.- 6 Stabilization.- 7 Uniform Optimal Control.- 8 Robustness of Time-Varying Systems.- 9 The Gap Metric and Internal Stability.- 10 Robust Stabilization in the Gap Metric.- 11 Orthogonal Embedding of Time-Varying Systems.
An operator theoretic approach to robust control analysis for linear time-varying systems, with the emphasis on the conceptual similarity with the H control theory for time-invariant systems. It clarifies the major difficulties confronted in the time varying case and all the necessary operator theory is developed from first principles, making the book as self-contained as possible. After presenting the necessary results from the theories of Toeplitz operators and nest algebras, linear systems are defined as input-output operators and the relationship between stabilisation and the existence of co-prime factorisations is described. Uniform optimal control problems are formulated as model-matching problems and are reduced to four block problems, while robustness is considered both from the point of view of fractional representations and the "time varying gap" metric, as is the relationship between these types of uncertainties. The book closes with the solution of the orthogonal embedding problem for time-varying contractive systems. As such, this book is useful to both mathematicians and to control engineers.
- The necessary operator theory is developed from first principles - The book is as self-contained as possible - All competition is aimed at the engineering community; this book is aimed at mathematicians interested in the potential applications and extensions of their work