Uniform Asymptotic Stability and Slow Convergence in Adaptive Systems

We examine convergence properties of errors in a class of adaptive systems that arises for scalar plants. We show that these adaptive systems are at best uniformly asymptotically stable in the large, and possess an infinite region where the trajectories move arbitrarily slowly, i.e. stick. We show that these properties are also exhibited by adaptive systems with closed-loop reference models which have been demonstrated to exhibit improved transient performance. Despite such transient behavior, we show that the slow convergence can still occur and has the potential to be slower than classic open-loop reference model adaptive systems.