The differential operator approach to linear system analysis and design

Abstract

The time domain, state-space approach to control systems analysis and design has dominated the control literature since the early 1960's. More recently, however, there has been a resurgent interest in frequency domain methods. This is due in large part to the development of an alternative (to the state-space approach) time domain technique, namely the differential operator approach for the analysis and synthesis of linear multivariable systems. More specifically, a significant number of investigators have now noted and employed the fact that the (pxm) transfer matrix, T(s), of a linear, time-invariant, multivariable system can be factored as R(s) P-1R(s) [orP-1Q(s) Q(s)]. with R(s) and PR(s) [PQ(s) and Q(s)] relatively right (left) prime polynomial matrices in the Laplace operator's and PR(s) [PQ(s)] column (row) proper. Furthermore, such a factorization directly implies a minimal (i.e. both controllable and observable) time domain realization of T(s) in “differential operator form”, namely PR(D) z(t) = u(t); y(t) = R(D) z(t) [orPQ(D) y(t) = Q(D) u(t)], where PR(D) andR(D) [PQ(D) andQ(D)] are polynomial matrices in the differential operator D = d/dt, withPR(D) [PQ(D)] nonsingular. It might be noted that a differential operator description of the dynamical behavior of a physical system often follows as a direct result of applying well-known physical laws to describe the performance of the system. The primary purpose of this paper is to delineate and unify a number of the more important differential operator techniques which have recently been developed and to compare them to analogous, but perhaps more familiar state-space procedures.