Non-$C2$ Lie Bracket Averaging for Nonsmooth Extremum Seekers

A drawback of extremum seeking-based control is the introduction of a high frequency oscillation into a system's dynamics, which prevents even stable systems from settling at their equilibrium points. In this paper, we develop extremum seeking-based controllers whose control efforts, unlike that of traditional extremum seeking-based schemes, vanish as the system approaches equilibrium. Because the controllers that we develop are not differentiable at the origin, in proving a form of stability of our control scheme we start with a more general problem and extend the semiglobal practical stability result of Moreau and Aeyels to develop a relationship between systems and their averages even for systems which are nondifferentiable at a point. More specifically, in order to apply the practical stability results to our control scheme, we extend the Lie bracket averaging result of Kurzweil, Jarnik, Sussmann, Liu, Gurvits, and Li to non-C² functions. We then improve on our previous results on model-independent semiglobal exponential practical stabilization for linear time-varying single-input systems under the assumption that the time-varying input vector, which is otherwise unknown, satisfies a persistency of excitation condition over a sufficiently short window.