On the Equivalency and the Reduction of Linear Discrete Systems with Noisy Observations

This paper is concerned with the problem of finding a reduced form of a linear discrete system with noisy observations, where the reduced form of the system is a system with minimum dimensional state space which is equivalent to the given system. For systems initially in their zero states, two systems are defined to be equivalent in the sense that the responses of these systems to the same input sequence have the same mean and the same variance, and a reduction algorithm for determining a reduced form of a given system is presented. For systems initially in their arbitrary states, several other concepts of equivalence of the systems are introduced, and a reduction algorithm for determining a reduced form of a given system under each equivalent definition is presented. The relations between these equivalent concepts and the properties of these reduced forms are examined. Reduction algorithms presented in this paper do not require additional restrictions on system parameters and are readily programmable for computer execution.