Robust Control Theory in Hilbert Space

This book presents an operator theoretic approach to robust control analysis for linear time-varying systems. It emphasizes the conceptual similarity with the H control theory for time-invariant systems and at the same time clarifies the major difficulties confronted in the time varying case. The necessary operator theory is developed from first principles and the book is 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 stabilization and the existance of co-prime factorizations is described. Uniform optimal control problems are formulated as model-matching problems and are reduced to four block problems. Robustness is considered both from the point of view of fractional representations and the "time varying gap" metric, and the relationship between these types of uncertainties is clarified. The book closes with the solution of the orthogonal embedding problem for time varying contractive systems. This book will be useful to both mathematicians interested in the potential applications of operator theory incontrol and control engineers who wish to deal with some of the more mathematically sophisticated extension of their work. TOC:Basic Hilbert Space Theory.- Geometry of Hilbert Space.- Basic Operator Theory.- Banach Algebras.- Exercises. Operator Theoretic Preliminaries. Functional Calculus for Selfadjoint Operators. Polar Decompositions. Weak Convergence in H. Operator Topologies. The Minimum Modulus of an Operator. Exercises.- A Distance Formula and Some Consequences. Operator Matrices. Multiplication Operators and Nehari's Theorem. Nest Algebras. Exercises.