Equivalence Notions for Discrete-Time Stochastic Linear Control Systems
  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. 
Giordano Pola, Costanzo Manes, Arjan J. van der Schaft and Maria   Domenica Di Benedetto
math.OC