Normal forms, bifurcations and limit dynamics in adaptive control systems

Adaptive controllers are used in systems where one or more parameters are unknown. Such controllers are designed to stabilize the system using an estimate for the unknown parameters that is adapted automatically as part of the stabilization. One drawback in adaptive control design is the possibility that the closed-loop limit system is not stable. The worst situation is the existence of a destabilized limit system attracting a large open subset of initial conditions. These situations lie behind bad behaviour of the closed-loop adaptive control system. The main issue in this paper is to identify and characterize the occurrence of such bad behaviour in the adaptive stabilization of first- and second-order systems with one unknown parameter. We develop normal forms for all possible cases and find the conditions that lead to bad behaviour. In this context we discuss a number of bifurcation-like phenomena.