Rational matrix structure

Recent work1-9, has brought out the importance of a closer examination of the pole/zero and vector-space structure of rational matrices G(s). Results developed by several people are brought together and and examined here from a unified point of view that yields some new insights and results. Motivated by certain basic ideas from valuation theory that were introduced into system theory by Forney2, we begin in Sec. 2 by providing a concise definition of pole/zero structure of G(s) that moreover treates the structure at s = ? on an equal footing with the structure at finite s. As an example of the application of our definition, we demonstrate that the excess of poles over zeros of G(s) is actually a property of the column and row range-spaces of G(s). Application to the characterization of solutions of the so-called ´minimal design problem´1-4 is also noted. In Sec. 3 we consider how the pole/zero structure of G(s) is related to that of its columns taken separately. We are thus led to the notion of column-reducedness of G(s) at some value q of its argument s; this is shown to be a significant generalization of the well-known concept of column-properness of a polynomial matrix, see Wolovich10. In particular, the pole/zero structure of G(s) at s = q is that of its columns taken separately if and only if G(s) is column-reduced at q. We go on to consider the determination of pole/zero structure of a rational matrix at s = q by transformation to column-reduced form at q. The transformation is described in terms of its effect on the coefficients in the Laurent expansion of G(s) at q. Connection is thereby made with results of Dewilde and coworkers11-12, on extraction of pole/zero structure from Laurent expansions. That such an extraction procedure implicitly underlies several well-known alogirhtms in system theory (e.g., the Jordan-form realization algorithms of Kuo13, Funahashi14, and others, on the ´structure algorithm´ of Silverman15, for system inversion) is pointed- out. In Sec. 4 we briefly consider some small extensions of Forney´s results2 on polynomial bases for rational vector spaces. In particular, we consider the structure of certain rational bases for such spaces.