Matrices, polynomials, and linear time-variant systems

Some recent developments in the applications of matrices to problems arising in linear systems theory are described. It is shown how companion form matrices can be used to provide a unified framework for dealing with the qualitative analysis of polynomials, including such problems as determination of greatest common divisors. Relationships to classical theorems involving bigradients and to controllability are discussed. When applied to the determinantal stability criteria of Hurwitz and others, the companion matrix approach results in minors of half the original orders. The problem of minimal realization of a transfer function matrix is dealt with in terms of polynomial matrices using methods due to Rosenbrock, and links with the results on scalar polynomials are demonstrated. Some applications of Lyapunov theory to systems in state-space form are briefly reviewed.