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.1 Rokni-Lamooki, Townley & Osinga Bifurcations and limit dynamics 1 Preliminaries and motivationAdaptive control is an established approach to the control of (parametrically) uncertain systems. Loosely speaking, there are two approaches to adaptive control: in one approach the adaptive controller tries to 'learn' the unknown parameters, the learning is guaranteed under suitable assumptions and the adaptive controller converges to a non-adaptive limit controller which is then itself guaranteed to be stabilizing; see for example [10,11]. In a second so-called direct approach there is no attempt to learn the unknown parameters but instead the aim is simply to stabilize the system's dynamics. Again under suitable assumptions the adaptive controller converges to a limiting closed-loop system. However, there is no guarantee that the limit controller is stabilizing. In fact, the possibility that such destabilizing limit controllers can be the limit for a large open set of initial conditions, i.e. the phenomenon is generic, is of great concern. Such situations give rise to undesirable closed-loop behaviour of the adaptive control system. In this paper we aim to understand and characterize the occurence of such undesirable behaviour. From the viewpoint of dynamical systems theory, we are classifying normal forms for two-and three-dimensional systems. However, there is an added complication because the adaptive part of the controller causes the generic case to have one zero eigenvalue and there are also further degeneracies. This leads us to consider a number of different concepts of bifurcations.This paper focusses on the class of adaptive back-stepping control for systems in strict feedback form. Let us, therefore, begin by explaining these notions. As a simple example, we consider a chain of two integrators given by ẋ = f 1 (x, y), y = f 2 (x, y, u),where f 1 (0, 0) = 0 and f 2 (0, 0, 0) = 0. The goal is to find a control u as a function of x and y such that the origin becomes asymptotically stable. The idea of back-stepping is to stabilize this system using feedback via two steps: in the first step we consider the ...
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