Robust Control Theory in Hilbert Space

by Feintuch, Avraham

• ISBN13: 9780387982915
• ISBN10: 0387982914
• Format: Hardcover
• Copyright: 1998-01-01
• Publisher: Springer Verlag

Summary

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 in control and control engineers who wish to deal with some of the more mathematically sophisticated extension of their work.

Table of Contents

Preface	vii
1 Basic Hilbert Space Theory	1	(18)
1.1 Geometry of Hilbert Space	1	(8)
1.2 Basic Operator Theory	9	(3)
1.3 Banach Algebras	12	(6)
1.4 Exercises	18	(1)
2 Operator Theoretic Preliminaries	19	(14)
2.1 Functional Calculus for Self-Adjoint Operators	19	(3)
2.2 Polar Decompositions	22	(2)
2.3 Weak Convergence in H	24	(3)
2.4 Operator Topologies	27	(2)
2.5 The Minimum Modulus of an Operator	29	(1)
2.6 Exercises	30	(3)
3 A Distance Formula and Some Consequences	33	(22)
3.1 Operator Matrices	33	(5)
3.2 Multiplication Operators and Nehari's Theorem	38	(9)
3.3 Nest Algebras	47	(5)
3.4 Exercises	52	(2)
3.5 References, Notes, and Remarks	54	(1)
4 Factorization Theorems	55	(22)
4.1 Inner-Outer Factorization in H(2)	55	(3)
4.2 Inner-Outer Factorizations in Nest Algebras	58	(5)
4.3 Cholesky Factorization	63	(4)
4.4 J-Spectral Factorizations	67	(5)
4.5 The Entropy Operator	72	(1)
4.6 Exercises	73	(1)
4.7 References, Notes, and Remarks	74	(3)
5 Linear Systems	77	(10)
5.1 The Extended Space of a Hilbert Space	77	(3)
5.2 Causality	80	(2)
5.3 Unbounded Linear Operators on H	82	(2)
5.4 Stable Systems	84	(1)
5.5 Exercises	85	(1)
5.6 References, Notes, and Remarks	86	(1)
6 Stabilization	87	(30)
6.1 Feedback Systems	87	(4)
6.2 Graph Representations	91	(2)
6.3 Representations and Strong Representations	93	(5)
6.4 Stabilization and Strong Representations	98	(7)
6.5 Construction of Coprime Factorizations from State Space Models	105	(3)
6.6 Strong and Simultaneous Stabilization	108	(5)
6.7 Exercises	113	(1)
6.8 References, Notes, and Remarks	114	(3)
7 Uniform Optimal Control	117	(26)
7.1 Sensitivity Minimization	117	(2)
7.2 The Scalar Commutant Lifting Theorem	119	(3)
7.3 Uniform Optimal Control for Time-Varying Systems	122	(6)
7.4 Solving the Model-Matching Problem	128	(4)
7.5 Maximizing the Entropy Operator	132	(3)
7.6 Application to Time-Varying Sampled-Data Systems	135	(5)
7.7 Exercises	140	(1)
7.8 References, Notes, and Remarks	141	(2)
8 Robustness of Time-Varying Systems	143	(22)
8.1 Proper Representation Uncertainty	143	(5)
8.2 Maximal Stabilizable Operator Balls	148	(7)
8.3 Uniform Robust Stability	155	(8)
8.4 Exercises	163	(1)
8.5 References, Notes, and Remarks	163	(2)
9 The Gap Metric and Internal Stability	165	(22)
9.1 Directed Gaps and Gaps	165	(3)
9.2 Gap Balls	168	(4)
9.3 Properties of the Gap Metric	172	(3)
9.4 The Gap Metric and Robustness	175	(1)
9.5 Stability and the Gap Metric	176	(6)
9.6 Exercises	182	(2)
9.7 References, Notes, and Remarks	184	(3)
10 Robust Stabilization in the Gap Metric	187	(20)
10.1 The Gap and the 2-block Problem	187	(3)
10.2 Maximally Stabilizing Controllers	190	(4)
10.3 More on XXX(L(1), L(2))	194	(3)
10.4 The Gap Metric and Coprime Factor Perturbations	197	(7)
10.5 Exercises	204	(1)
10.6 References, Notes, and Remarks	204	(3)
11 Orthogonal Embedding of Time-Varying Systems	207	(10)
11.1 The Time-Invariant Case	207	(2)
11.2 Causality on H + H	209	(1)
11.3 Isometric 4-block Extensions	210	(1)
11.4 Strict Contractions	211	(2)
11.5 Special Cases and Examples	213	(2)
11.6 Exercises	215	(1)
11.7 References, Notes, and Remarks	216	(1)
Index	217