Output - Induced Subspaces , Invariant Directions and Interpolation in Linear Discrete - Time Stochastic

In this paper we analyze the structure of the class of discrete-time linear stochastic systems in terms of the geometric theory of stochastic realization. We discuss the role of invariant directions, zeros of spectral factors and output-induced subspaces in determining the systems-theoretical properties of the stochastic systems. A prototype interpolation problem for recovering lost state information is discussed and it is shown how it can be solved via Kalman filtering recursions tying together the state processes of a family of totally ordered splitting subspaces. 1. Introduction. It is a somewhat surprising fact that, in the discrete time case, the family of minimal state space representations x(t + 1) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (1.1)