Differential Operator Representations

Abstract

The primary goal of this chapter is to present an in-depth study of the third common method for representing the dynamical behavior of linear systems, namely the differential operator representation. We begin by considering the various transfer relations between any two of the three primary methods used to describe dynamical system behavior, noting that certain of the (six) relations have already been given in the earlier chapters. In particular, in Section 5.2, we mathematically define a notion of “equivalence” between state-space systems and the more general class of differential operator systems. We note that this notion of equivalence implies a number of desirable necessary conditions for equivalence. For example, system order and mode or pole locations are preserved as well as the ability to set initial conditions on either equivalent system. We further note that our definition of equivalence reduces to the standard definition 3.4.4 when both systems are in state-space form. We then present an algorithm for obtaining equivalent state-space representations of any systems whose dynamics are expressed in the more general differential operator form. Paramount to the development are the notions of row proper and column proper polynomial matrices which were introduced in Section 2.5, and the structure theorem which was introduced in Section 4.3.