Adaptive Control of Systems Subject to a Class of Random Parameter Variations and Disturbances

Abstract

The adaptive control of linear discrete time parameter systems is studied for the case where both the (unobserved) disturbances and the (unknown) parameters are random. The class of disturbance processes considered is a generalization of the usual white noise process. The random parameters are permitted to be (convergent) martingale processes evolving within the set of parameters corresponding to (time varying) inverse stable systems whose moving average noise process satisfies a (time varying) positive real condition. The main result of this paper generalizes those found in [1–4] for constant parameter systems. Specifically, we show that a stochastic gradient parameter identification algorithm generating ${\hat \theta _N},N = 1,2,..,$ combined with a minimum variance feedback controller (designed using $[{\hat \theta _N}$), results in a closed loop system that is stable, and whose performance asymptotically approaches that of the system regulated by a controller designed knowing the true system parameter $\mathop \theta \limits^ \circ$.