High gain adaptive control revisited: first and second order case

Adaptive controllers based on high gain feedback suffer from lack of robustness with respect to bounded disturbances. Existing modifications prevent the feedback gain from drifting away, but at the same time introduce solutions that, even in the absence of disturbances, do not converge to zero. We investigate a further modification that maintains the robustness and rules out undesirable solutions when disturbances are not present. We restrict ourselves to first and second order systems