Chain-Scattering Approach to H-Infinity Control

H. Kimura

The University of Tokyo, Japan
December 1996 * 256 pages * Hardcover * $64.50 * 0-8176-3787-7
Series: Systems & Control: Foundations & Applications

The advent of H-infinity control was a truly remarkable innovation in multivariable control theory. It eliminated the classical/modern dichotomy that had been a major source of long-standing skepticism about the applicability of modern control theory, by amalgamating the 'philosophy' of classical design with 'computation' based on the state-space problem setting. It enhanced the application by deepening the theory mathematically and logically, not by weakening it as was done by the reformers of modern control theory in the early 70's.

However, very few practical design engineers are familiar with the theory, even though several theoretical frameworks have been proposed, namely interpolation theory, matrix dilation, differential games, approximation theory, linear matrix inequalities, etc. But none of these frameworks have proven to be a natural, simple, and comprehensive exposition of H-infinity control theory that is accessible to practical engineers and demonstrably the most natural control strategy to achieve the control objectives.

The purpose of this book is to provide a natural theoretical framework that is understandable with little mathematical background. The notion of chain-scattering, well-known in classical circuit theory but new to control theorists, plays a fundamental role in this book. It captures an essential feature of the control systems design, reducing it to a J-lossless factorization, which leads us naturally to the idea of H-infinity control. The J-lossless conjugation, an essentially new notion in linear system theory, then provides a powerful tool for computing this factorization. Thus, the chain-scattering representation, the J-lossless factorization, and the J-lossless conjugation are the three key notions that provide the thread of development in this book. The book is completely self-contained and requires little mathematical background other than some familiarity with linear algebra. It will be useful to practicing engineers in control system design and as a text for a graduate course in H-infinity control and its applications.


* Impacts of H-infinity Control
* Theoretical Background

Elements of Linear System Theory
* State Space Description of Linear Systems
* Controllability and Observability
* State Feedback and Output Insertion
* Stability of Linear Systems

Norms and Factorizations
* Norms of Signals and Systems
* Hamiltonians and Riccati Equations

C-Scattering Representations of Plants
* Algebra of Chain-Scattering Representation
* State-Space Forms of Chain-Scattering Representation
* Dualization
* J-Lossess and (J,J')-Lossless Systems
* Dual (J,J')-Lossless Systems
* Feedback and Terminations

J-Lossless Conjugation and Interpolation
* J-Lossless Conjugation
* Connections to Classical Interpolation Problems
* Sequential Structure of J-Lossless Conjugation

J-Lossless Factorization
* (J,J)-Lossless Factorization and its Dual
* (J,J)-Lossless Factorization by J-Lossless Conjugation
* (J,J)-Lossless Factorization in State Space
* Dual (J,J)-Lossless Factorization in State Space
* Hamiltonian Matrices

H-infinity Control via (J,J)-Lossless Factorization
* Formulation of H-infinity Control
* Chain-Scattering Representations of Plants and H-infinity Control
* Solvability Conditions for Two-Block Cases
* Plant Augmentations and Chain-Scattering Representations

Statespace Solutions to H-infinity Control Problems
* Problem Formulation and Plant Augmentation
* Solution to H-infinity Control Problems for Augmented Plants
* Maximum Augmentations
* State-Space Solutions
* Some Special Cases

Structure of H-infinity Control
* Stability Properties
* Closed-Loop Structures of H-infinity Control
* Examples

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