Determination of zeros and zero directions of linear time-invariant systems by matrix generalized inverses

Abstract

Consider the linear control system whose state-space equations are dx/dl = Ax + Bu, y = Cx + Du where x, u and y are, respectively, the state, input and output vectors, and A, B, C, D are constant matrices. Problem : find constant scalars r and constant vectors v, w such that an input of the form v exp (rt) 1(t) will yield a state of the form x = w exp (rt) and output y ≡ 0, t ≥ 0. The method developed is this: the output equation is solved for x and the solution is substituted into the state equation. The resulting equation must satisfy a consistency condition involving r, in order for a solution of the problem to exist. Matrix generalized inverses are employed both to derive consistency conditions and to obtain solutions. Several examples illustrate a variety of conditions.