A robust adaptive minimum variance regulator

In this paper we investigate the twin issues of robustness as well as performance in the adaptive control of linear, stochastic systems. Towards this end we present an adaptive controller which provides optimal performance under ideal conditions, while providing stable behavior when the idealness assumptions are violated. Specifically, our adaptive controller has the following properties: (i) When the system under control is of the order assumed, with delay also as assumed, is of minimum phase, and the disturbance satisfies a positive real condition on its spectrum, then the mean-square tracking error is optimal. (ii) When the positive realness condition on the disturbance is violated, then the system remains mean square stable. (iii) When the system violates the "ideal" conditions and is merely in a graph-topological neighborhood of systems satisfying (i), then the adaptive controller still provides mean square stability. (iv) When the adaptation gain is made non-vanishing, then stability is still preserved.