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Research Papers

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

[+] Author and Article Information
Alexander Scheinker

Particle Accelerator RF Control Group,
Los Alamos National Laboratory,
Los Alamos, NM 87545
e-mail: alexscheinker@gmail.com

Miroslav Krstić

Professor
Mechanical and Aerospace Engineering,
University of California, San Diego,
La Jolla, CA 92093
e-mail: krstic@ucsd.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 5, 2012; final manuscript received September 9, 2013; published online October 3, 2013. Assoc. Editor: Prashant Mehta.

J. Dyn. Sys., Meas., Control 136(1), 011010 (Oct 03, 2013) (10 pages) Paper No: DS-12-1409; doi: 10.1115/1.4025457 History: Received December 05, 2012; Revised September 09, 2013

## 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.

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## Figures

Fig. 1

In application of both r = 0.5 and r = 0 controls, initially the trajectories of (70) are very similar and so are the control efforts. The advantage of the r = 0.5 controller is evident as the trajectories approach the origin. For r = 0, the αω cos(ωt) term persists, which shows up in both a persistent control effort and in the system’s inability to settle at the origin, which for |x(t)|≪1 settles to a steady state of x(t)≈(α/ω) sin(ωt), in the r = 0.5 case both |x(t)| and control effort settle to zero.

Fig. 2

As the trajectory of system (70) with the r = (1/2) controller approaches the origin the control effort also approaches zero

Fig. 3

The trajectory (x(t),y(t)) is shown superimposed on a hill with height given by z(x,y) = 15-109.8(34x2+xy2+34y2)

Fig. 4

Initially, the trajectories are similar for both r = 0 and r = 0.5. Once (x(t),y(t)) is near the origin the steady state oscillations due to the r = 0.5 controller’s input are greatly reduced relative to those due to the r = 0 controller.

Fig. 5

This figure shows the parametric view of the simulation of system (82) under both control laws. A zoom in on the last half second of the simulation clearly shows the benefit of the new controller with r = 12. Even with the nonsmooth controllers, the system may never settle to equilibrium at the origin, this would only be achieved if x(t) and y(t) reached the origin simultaneously.

Fig. 6

The initial effort of controller u is greater than that of u˜ due to the 2 k term, which dominates for large values of |(x,y)|. The advantages of controller u are seen in the great reduction of control effort once the system has reached a neighborhood of the origin as seen here in the last 3 s of simulation.

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