Non-C2 Lie Bracket Averaging for Nonsmooth Extremum Seekers

Abstract

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.

Constrained extremum seeking stabilization of systems not affine in control

“…Then, ‐SPUAS stability guarantees that after choosing k α , by choosing sufficiently large ω ⋆ , that the trajectory of the actual system reaches and stays arbitrarily close to x ⋆ ( t ). The choice of V in this case can be thought of as a Lyapunov‐type function, as the derivative of V satisfies: please refer to previous studies for more details.…”

“…Trajectory x(t) of(36) shown alongside the trajectoryx(t) of average system(37).The system behaves as expected despite our controller design being based on the approximation of the nonlinearity. The control effort u(t) shows strong initial phase modulation before a steady state is reached.…”

“…In Scheinker and Krstic, extremum seeking for control (ESC) was introduced, to stabilize unknown, open‐loop unstable systems by using the extremum seeker itself as a high‐frequency feedback control, an approach related to vibrational control . Extremum seeking for control has been applied for the optimization of high‐voltage converter modulators and nonsmooth ESC, in which oscillation disappears as equilibrium is approached, has been developed …”

“…Extremum seeking for control has been applied for the optimization of high-voltage converter modulators 35 and nonsmooth ESC, in which oscillation disappears as equilibrium is approached, has been developed. 36 In Scheinker and Krstic, 37 a new, constrained extremum seeking form of ESC was developed in which the control efforts and parameter update rates have analytically known bounds. The bounded ESC approach is especially useful for digital implementation and for extremely noisy systems, was recently demonstrated in hardware, 38 and was recently extended from smoothly varying sinusoidal-like functions to a much larger class of functions.…”

Extremum seeking for parameter identification, implementation for electron beam property prediction

“…The new form of ES has been demonstrated as a tool for the stabilization of open-loop unstable, uncertain systems [23], including a pendulum's vertical equilibrium point [24]. Also, a modified, non-differentiable form has been developed, in which has the benefit that the system's control efforts settle to zero as equilibrium is approached [25].…”