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 this problem in the 1950-1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical 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 the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block - we allow the plant to be a general nonlinear dynamic system (possibly non-affine in control and open-loop unstable) whose reference-to-output equilibrium map has a maximum, and whose equilibria are locally exponentially stabilizable.
Krstić, M., & Wang, H. H. (2000). Stability of extremum seeking feedback for general nonlinear dynamic systems. Automatica, 36(4), 595–601. https://doi.org/10.1016/S0005-1098(99)00183-1