On Moment Matching for Stochastic Systems

Abstract

In this paper we study the problem of model reduction by moment matching for stochastic systems. We characterize the mathematical object which generalizes the notion of moment to stochastic differential equations and we find a class of models which achieve moment matching. However, differently from the deterministic case, these reduced-order models cannot be considered "simpler" because of the high computational cost paid to determine the moment. To overcome this difficulty, we relax the moment matching problem in two different ways and we present two classes of reduced-order models which, approximately matching the stochastic moment, are computationally tractable.

Output Regulation of Linear Stochastic Systems

“…In the following results we derive the steady-state response of the state of the closed-loop system (34). These will be used later to characterise the properties of the steady-state tracking error.…”

“…Corollary 2. Consider system (34). Assume that there exist matrices K and L such that condition (S O A ) is satisfied.…”

“…dΦ t = (Adt + F dW t )Φ t . Then, the following lemma (the proof of which can be found in [34]) characterises the steady-state response of a linear stochastic system of the form (1) when u t = 0. Lemma 1.…”

“…Lemma 1. [34] Consider the interconnection of system (1) and the signal generator (2) with u t = 0. Suppose that Assumption 1 holds and that system (1) is asymptotically stable.…”