On the steady state behavior of stochastic prosesses with applications to ARMAX systems

The question of the existence of steady state solutions to the closed loop system equations for stochastic systems, and in particular, ARMAX systems under non-linear feedback control is addressed. Using the ergodic theory of time invariant Markov processes, it is shown that corresponding to a large class of control laws there exist shift invariant measures for the joint disturbance-input-output processes. This permits an asymptotic analysis of (1) averages of functions of the state process; (2) the probability distributions governing the state process; (3) loss functions on the system in steady state.