Linear Multivariable Control

The aim of this volume is to provide a detailed account of the basic theory of polynomial and fractional representation methods for what is currently known as algebraic analysis and synthesis of linear multivariable control systems. The text is also designed to provide a self-contained exposition of the mathematical theory in such a way so that results and techniques of the "state space approaches" for regular and singular systems appear as special cases of a general theory covering the wider class of PMDs of linear systems. Chapter 1 deals with the algebraic structure and properties of rational matrices and rational vector spaces and their associations to the state-space minimal realization theory of proper rational matrices. This background is used in chapter 2 to introduce polynomial matrix models of linear multivariable systems and examine the various definitions and consequences of system equivalence, (finite) decoupling zeros, and (finite) poles and zeros of a multivariable system and its associated transfer function matrix. The pole-zero structure of rational matrices at the point of infinity, equivalence of rational matrices at infinity, co-primeness at infinity and the related subject of minimal realization theory for polynomial matrices are discussed in chapter 3. With chapter 3 as background, chapter 4 deals with the solution of general linear matrix differential equations corresponding to PMDs of linear systems. Problems related in "impulsive behaviour" of solutions of general homogeneous matrix differential equations due to "zeros at infinity" and "infinite elementary divisors" of the associated characteristic matrix A(s) are also examined here in detail. Finally, and based on the above results, the concepts of "Controllability and Reachability of regular and generalized state space or singular systems are examined as special cases of a general theory covering general PMDs of linear systems. The algebra of "proper and stable" rational functions and matrices is examined in chapter 5. As an application of these ideas, the problem of stabilizing single input-single output systems by means of dynamic compensation is briefly examined here. The problem of internally stabilizing a linear multivariable system which is described by a PMD is examined in chapter 6 by making use of proper and stable fractional representations examined in chapter 5. Finally the algebraic design problems of simultaneous internal stabilization and assymtotic tracking, disturbance rejection and diagonal decoupling by the use dynamic compensation are examined in chapter 7. The book is addressed at final year undergraduate mathematicians and engineers who already have a knowledge of at least the elements of state space control theory and elementary matrix theory. It is also intended for graduate students of the British MSc and research level mathematicians or engineers who would like to become familiar with current advances in multivariable control using modern algebraic approaches.