Multidimensional linear systems : factorisation and stabilisation
This thesis is concerned with various problems associated with the factorisation and feedback stabilisation of multidimensional linear discrete systems, which may be represented by rational matrices in several variables. Some factorisation techniques for polynomial and rational matrices in several variables have been explored and applied to the study of feedback stabilisation of multidimensional linear systems. The work presented here may be divided into two parts. The first part (Chapters 2 and 3) is concerned with two-dimensional systems, while the second part (Chapters 4 and 5) deals with three- and higher-dimensional systems. The emphasis of the first part is placed on the development of constructive algorithms for several kinds of factorisation of polynomial and rational matrices in two variables. In Chapter 2, an algorithm for obtaining primitive factorisation of polynomial matrices in two variables is developed, which is then followed by an algorithm for the decomposition of a rational matrix in two variables into factor coprime matrix fraction descriptions. Chapter 3 presents a procedure for the analysis and compensator design of two-dimensional feedback systems. A constructive algorithm for solving a Diophantine-type equation in two variables is derived. A necessary and sufficient condition for the feedback stabilisability of two-dimensional systems is obtained. The complete set of stabilising compensators for a given two-dimensional plant is then characterised. The role played by the matrix fraction description approach in the study of three- and higher dimensional systems, particularly with respect to the feedback stabilisation of these systems, is then investigated in detail in the second part. Chapter 4 deals with various kinds of factorisations for polynomial and rational matrices in three or more variables. For example, a criterion for the existence of primitive factorisation of a class of polynomial matrices in three or more variables is derived. By introducing a new concept: generating polynomials, it is shown that a direct generalisation of several existing results in two-dimensional systems theory to their higher-dimensional counterparts is not possible. In chapter 5, applying the generating polynomials, we obtain a stability test and a necessary and sufficient condition for feedback stabilisability of three- and higher-dimensional systems.