Stability of extremum seeking feedback for general nonlinear dynamic systems

While the mainstream methods of adaptive control (both linear and nonlinear ) deal only with regulation to known set points or reference trajectories, in many applications the set point should be selected to achieve a maximum of an uncertain reference-to-output equilibrium map. The techniques of the so-called "extremum control" or "self-optimizing control" developed for thi s problem in the 1950-1960s have long gone out of fashion in the theoretica l control literature because of the difficulties that arise in a rigorous a nalytical treatment. In this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis. Our scheme is much more general that th e existing extremum control results which represent the plant as a static n onlinear map possibly cascaded with a linear dynamic block - we allow the p lant to be a general nonlinear dynamic system (possibly non-affine in contr ol and open-loop unstable) whose reference-to-output equilibrium map has a maximum, and whose equilibria are locally exponentially stabilizable. (C) 2 000 Elsevier Science Ltd. All rights reserved.