Control System Synthesis : A Factorization Approach

TLDR: In this article, the stable factorization approach is introduced to the synthesis of feedback controllers for linear control systems, where the controller is designed as a matrix over a fraction field associated with a commutative ring with identity, denoted by R, which also has no divisors of zero.

Abstract: 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 MIT Press in 1985 Table of Contents: Introduction / Proper Stable Rational Functions / Scalar Systems: An Introduction / Matrix Rings / Stabilization