A Zero Filter Augmentation for Robust Adaptive Control of Weakly Minimum Phase Finite-Dimensional Systems

Abstract

Past work has shown that there is a stabilizing direct model reference adaptive control law with persistent disturbance rejection and robustness properties. For these results to hold, the plant must be minimum phase (all transmission zeros stable). For this paper, the plant will be a weakly minimum phase linear system, i.e. there will be a finite number of isolated unstable zeros with real part equal to zero. All other zeros will be stable. Our principal result here is that the direct adaptive controller can be compensated with a zero filter for the unstable zeros which will produce the desired robust adaptive control results even though the plant is only weakly minimum phase. The new result shows that all state tracking errors converge to a prescribed neighborhood of zero even though the plant is not truly minimum phase. The result will not require the use of the standard Barbalat Lemma which requires certain signals to be uniformly continuous. Our results are illustrated with a simple example to show how the adaptive control behaves for weakly minimum phase systems.