Abstract

Extremum seeking (ES), a non-model-based optimization method whose development begun in the 1920s, has thus far remained limited to stable plants. Removing this limitation is a logical challenge because ES, at its core, is a method for stabilization – of extrema of input-output maps of systems in steady state. We introduce a framework in which ES solves the problem of stabilization of general nonlinear systems affine in control, without requiring the knowledge of the system’s input and drift vector fields. In this framework a control Lyapunov function is being minimized using ES, whereas the plant’s state assumes implicitly a role of a vector-valued integrator in the learning portion of the ES algorithm. The mathematical machinery behind the new approach is a combination of the Lie bracket averaging method of Gurvits, Sussmann, and coworkers (an alternative to the conventional integration-based Krylov-Bogolyubov averaging) and of an approximation-based semiglobal practical stability theory of Moreau and Aeyels. When applied to linear systems, the ES approach solves the classical challenge in adaptive control, posed by Morse, of stabilization of systems with unknown control directions. Unlike the approach with Nussbaum gain functions, which provides a classical solution to this problem, the ES approach achieves stability even when the control directions change rapidly with time.