Control System Synthesis

Book description

This book introduces the so-called "stable factorization approach" to the synthesis of feedback controllers for linear control systems. The key to this approach is to view the multi-input, multi-output (MIMO) plant for which one wishes to design a controller as a matrix over the fraction field F associated with a commutative ring with identity, denoted by R, which also has no divisors of zero. In this setting, the set of single-input, single-output (SISO) stable control systems is precisely the ring R, while the set of stable MIMO control systems is the set of matrices whose elements all belong to R. The set of unstable, meaning not necessarily stable, control systems is then taken to be the field of fractions F associated with R in the SISO case, and the set of matrices with elements in F in the MIMO case. The central notion introduced in the book is that, in most situations of practical interest, every matrix P whose elements belong to F can be "factored" as a "ratio" of two matrices N,D whose elements belong to R, in such a way that N,D are coprime. In the familiar case where the ring R corresponds to the set of bounded-input, bounded-output (BIBO)-stable rational transfer functions, coprimeness is equivalent to two functions not having any common zeros in the closed right half-plane including infinity. However, the notion of coprimeness extends readily to discrete-time systems, distributed-parameter systems in both the continuous- as well as discrete-time domains, and to multi-dimensional systems. Thus the stable factorization approach enables one to capture all these situations within a common framework. The key result in the stable factorization approach is the parametrization of all controllers that stabilize a given plant. It is shown that the set of all stabilizing controllers can be parametrized by a single parameter R, whose elements all belong to R. Moreover, every transfer matrix in the closed-loop system is an affine function of the design parameter R. Thus problems of reliable stabilization, disturbance rejection, robust stabilization etc. can all be formulated in terms of choosing an appropriate R. This is a reprint of the book Control System Synthesis: A Factorization Approach originally published by M.I.T. Press in 1985.

Table of contents

  1. Preface
  2. Preface for the Original Edition
  3. Filtering and Sensitivity Minimization
    1. Problem Statement
    2. Some Facts about Hardy Spaces (1/2)
    3. Some Facts about Hardy Spaces (2/2)
    4. Filtering
    5. Sensitivity Minimization: Scalar Case (1/2)
    6. Sensitivity Minimization: Scalar Case (2/2)
    7. Sensitivity Minimization: Fat Plant Case (1/4)
    8. Sensitivity Minimization: Fat Plant Case (2/4)
    9. Sensitivity Minimization: Fat Plant Case (3/4)
    10. Sensitivity Minimization: Fat Plant Case (4/4)
    11. Sensitivity Minimization: General Case (1/2)
    12. Sensitivity Minimization: General Case (2/2)
    13. Two-Parameter Compensator (1/2)
    14. Two-Parameter Compensator (2/2)
  4. Robustness
    1. Introduction
    2. Graph Topology (1/4)
    3. Graph Topology (2/4)
    4. Graph Topology (3/4)
    5. Graph Topology (4/4)
    6. Graph Metric (1/2)
    7. Graph Metric (2/2)
    8. Robustness Conditions (1/4)
    9. Robustness Conditions (2/4)
    10. Robustness Conditions (3/4)
    11. Robustness Conditions (4/4)
    12. Robust and Optimal Regulation (1/4)
    13. Robust and Optimal Regulation (2/4)
    14. Robust and Optimal Regulation (3/4)
    15. Robust and Optimal Regulation (4/4)
    16. Genericity (1/2)
    17. Genericity (2/2)
  5. Extensions to General Settings
    1. Coprime Factorizations over a Commutative Ring
    2. Coprime Factorizations for Distributed Systems
    3. Controller Synthesis over a General Ring
  6. Algebraic Preliminaries
    1. Rings, Fields and Ideals
    2. Rings and Fields of Fractions
    3. Principal Ideal Domains
    4. Euclidean Domains
  7. Preliminaries on Matrix Rings
    1. Matrices and Determinants
    2. Canonical Forms (1/2)
    3. Canonical Forms (2/2)
  8. Topological Preliminaries
    1. Topological Spaces
    2. Topological Rings and Normed Algebras
  9. Bibliography (1/2)
  10. Bibliography (2/2)
  11. Author's Biography
  12. Index

Product information

  • Title: Control System Synthesis
  • Author(s): Mathukumalli Vidyasagar
  • Release date: June 2011
  • Publisher(s): Morgan & Claypool Publishers
  • ISBN: 9781608456635