Control Systems Synthesis: A Factorization Approach, Part I

Mathukumalli Vidyasagar

May 2022 · Springer Nature

About this ebook

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: Introduction / Proper Stable Rational Functions / Scalar Systems: An Introduction / Matrix Rings / Stabilization

About the author

Mathukumalli Vidyasagar was born in Guntur,India in 1947.He received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Wisconsin in Madison, in 1965, 1967, and 1969, respectively. From the next twenty years he taught mostly in Canada, before returning to his native India in 1989. Over the next twenty years, he built up two research organizations from scratch, first the Centre for Artificial Intelligence and Robotics under the Ministry of Defence, Government of India, and later the Advanced Technology Center in Tata Consultancy Services (TCS), India’s largest software company. After retiring from TCS in 2009, he joined the University of Texas at Dallas as the Cecil & Ida Green Chair in Systems Biology Science, and he became the Founding Head of the Bioengineering Department. His current research interests are stochastic processes and stochastic modeling, and their application to problems in computational biology. He has received a number of awards in recognition of his research, including the 2008 IEEE Control Systems Award.