Defining a Pseudo-Metric Topology on Linear Dynamic Systems

This paper defines a (pseudo) metric topology on the space of stable discrete linear time-invariant (LTI) dynamic systems. The article seeks to present a solution to the problem of comparing two LTI systems or, more formally, to find a metric for the space of stable discrete LTI dynamic systems. To this effect, by comparing the performance of two Kalman filters designed for two dynamic systems, a distance-like pseudo-norm between two systems is developed. The defined metric topology can be exploited to select the closest model, among several possible models P_s, all of which are known, to an observed data sequence modeled as P_★. Numerical simulations are provided illustrating the efficacy of the metric derived.