Bifurcation analysis of drift instabilities in adaptive control

Bifurcation theory is used to develop a partial catalogue of the dynamic characteristics of a model-reference adaptive control system. Three pathways leading to estimator instability are identified. The first follows a route whereby a sign change leads to a reversal of the gradient direction and infinite linear drift. The second instability results from using high control/adaptive gain and leads through period doubling bifurcations to global instability. Both of these instabilities can be avoided by tuning or by simple algorithmic modifications. The third bifurcation instability is of the Hopf-type, complicated by a number of nonlocal phenomena, and leads through a sequence of global bifurcations to parameter drift and bursting in a bounded regime. This instability, which is strongly connected to the presence of a degenerate set and a period two attractor, cannot be avoided by simple tuning. It is due to the unmodeled dynamics and poor signal-to-noise ratio.>