Equivalence Notions for Discrete-Time Stochastic Linear Control Systems

TL;DR

This paper introduces new notions of equivalence for discrete-time stochastic linear control systems, providing geometric conditions for verification and methods for model reduction, with connections to stochastic realization and reachability.

Contribution

It defines stochastic bisimulation and external behavior equivalence for stochastic linear systems, with geometric criteria and model reduction techniques.

Findings

Derived geometric necessary and sufficient conditions for equivalence notions.

Proposed Kalman-like decomposition for model reduction.

Established links with stochastic realization theory and reachability.

Abstract

In this paper we propose definitions of equivalence via stochastic bisimulation and of equivalence of stochastic external behavior for the class of discrete-time stochastic linear control systems with possibly degenerate normally distributed disturbances. The first notion is inspired by the notion of probabilistic bisimulation for probabilistic chains while the second one by the notion of equivalence of external behavior for (nonstochastic) behavioral systems. Geometric necessary and sufficient conditions for checking these notions are derived. Model reduction via Kalman-like decomposition is also proposed. Connections with stochastic linear realization theory and stochastic reachability are established.