System zeros and the weakly unobservable subspace

Abstract

In this chapter we first give a brief review of some elementary material on polynomial matrices and the Smith form. We then continue our study of the system ∑ given by the equations

1(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t)

and introduce an important polynomial matrix associated with this system, the system matrix of ∑. Using the system matrix, we introduce the concepts of transmission polynomials, and zeros of the system ∑. Next, we discuss the weakly unobservable subspace, and the controllable weakly unobservable subspace associated with ∑. The weakly unobservable subspace is used to give a geometric characterization of the property of strong observability. We conclude this chapter with a characterization of the transmission polynomials and the zeros of ∑ in terms of the weakly unobservable and controllable weakly unobservable subspace.