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

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

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.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Krstić, M., and Wang, H., 2000, “Stability of Extremum Seeking Feedback for General Dynamic Systems,” Automatica, 36, pp. 595–601. [CrossRef]
Tan, Y., Nešić, D., and Mareels, I., 2006, “On Non-Local Stability Properties of Extremum Seeking Control,” Automatica, 42, pp. 889–903. [CrossRef]
DeHaan, D., and Guay, M., 2005, “Extremum-Seeking Control of State-Constrained Nonlinear Systems,” Automatica, 31, pp. 1567–1574. [CrossRef]
Kvaternik, K., and Pavel, L., 2012, “An Analytic Framework for Decentralized Extremum Seeking Control,” American Control Conference 2012, Jun., Toronto, Canada.
Biyik, E., and Arcak, M., 2008, “Gradient Climbing in Formation Via Extremum Seeking and Passivity-Based Coordination Rules,” Asian J. Control, 10, pp. 201–211. [CrossRef]
Dürr, H., Stanković, M., Ebenbauer, C., and Johansson, K., 2013, “Lie Bracket Approximation of Extremum Seeking Systems,” Automatica, 49, pp. 1538–1552. [CrossRef]
Gurvits, L., 1992, “Averaging Approach to Nonholonomic Motion Planning,” Proceedings IEEE Conference Robotics and Automation, May 12–14, Nice, France.
Gurvits, L., and Li, Z., 1992, “Smooth Time-Periodic Solutions for Non-Holonomic Motion Planning,” NYU, Technical Report No. TR-598.
Gurvits, L., and Li, Z., 1993, Nonholonomic Motion Planning, “The Springer International Series in Engineering and Computer Science,” Vol. 192, Z.Li and J. F.Canny, eds., Springer, New York.
Moreau, L., and Aeyels, D., 2000, “Practical Stability and Stabilization,” IEEE Trans. Autom. Control, 45, pp. 1554–1558. [CrossRef]
Scheinker, A., and Krstić, M., 2012, “A Universal Extremum Seeking-Based Stabilizer for Unknown LTV Systems With Unknown Control Directions,” Proc. ACC, Montreal, Canada.
Scheinker, A., and Krstić, M., 2012, “Extremum Seeking-Based Tracking for Unknown Systems With Unknowns Control Directions,” Proceedings of the 51st Conference on Decision and Control (CDC), Maui, HI.
Scheinker, A., and Krstić, M., “Maximum-Seeking for CLFs: Universal Semiglobally Stabilizing Feedback Under Unknown Control Directions,” IEEE Trans. Autom. Control (to be published).
Ariyur, K. B., and Krstić, M., 2003, Real-Time Optimization by Extremum-Seeking Control, Wiley-Interscience, Hoboken, NJ.
Zhang, C., 2012, Advances in Industrial Control: Extremum-Seeking Control and Applications, Springer, New York.
Choi, J.-Y., Krstić, M., Ariyur, K. B., and Lee, J. S., 2002, “Extremum Seeking Control for Discrete-Time Systems,” IEEE Trans. Autom. Control, 47, pp. 318–323. [CrossRef]
Rotea, M. A., 2000, “Analysis of Multivariable Extremum Seeking Algorithms,” Proceedings of the 2000 American Control Conference.
Wang, H.-H., and Krstić, M., 2000, “Extremum Seeking for Limit Cycle Minimization,” IEEE Trans. Autom. Control, 45, pp. 2432–2437. [CrossRef]
Cochran, J., and Krstić, M., 2009, “Nonholonomic Source Seeking With Tuning of Angular Velocity,” IEEE Trans. Autom. Control, 54(4), pp. 717–731. [CrossRef]
Cochran, J., Kanso, E., Kelly, S. D., Xiong, H., and Krstić, M., 2009, “Source Seeking for Two Nonholonomic Models of Fish Locomotion,” IEEE Trans. Rob. Autom., 25(5), pp. 1166–1176. [CrossRef]
Zhang, C., Arnold, D., Ghods, N., Siranosian, A., and Krstić, M., 2007, “Source Seeking With Nonholonomic Unicycle Without Position Measurement and With Tuning of Forward Velocity,” Syst. Control Lett., 56, pp. 245–252. [CrossRef]
Henning, L., Becker, R., Feuerbach, G., Muminovic, R., Brunn, A., Nitsche, W., and King, R., 2008, “Extensions of Adaptive Slope-Seeking for Active Flow Control,” J. Syst. Control Eng., 222, pp. 309–322.
Kim, K., Kasnakoglu, C., Serrani, A., and Samimy, M., 2009, “Extremum-Seeking Control of Subsonic Cavity Flow,” AIAA J., 47, pp. 195–205. [CrossRef]
Wiederhold, O., Neuhaus, L., King, R., Niese, W., Enghardt, L., Noack, B. R., and Swoboda, M., 2009, “Extensions of Extremum-Seeking Control to Improve the Aerodynamic Performance of Axial Turbomachines,” Proceedings of the 39th AIAA Fluid Dynamics Conference, AIAA 2009-4175, San Antonio, TX.
Li, Y., Rotea, M. A., Chiu, G., Mongeau, L., and Paek, I., 2005, “Extremum Seeking Control of Tunable Thermoacoustic Cooler,” IEEE Trans. Control Syst. Technol., 13, pp. 527–536. [CrossRef]
Li, P., Li, Y., and Seem, J. E., 2009, “Extremum Seeking Control for Efficient and Reliable Operation of Air-Side Economizers,” Proceedings of 2009 American Control Conference.
Creaby, J., Li, Y., and Seem, J. E., 2009, “Maximizing Wind Turbine Energy Capture Using Multivariable Extremum Seeking Control,” Wind Eng., 33, pp. 361–387. [CrossRef]
Lei, P., Li, Y., Chen, Q., and Seem, J. E., 2011, “Sequential ESC-Based Global MPPT Control for Photovoltaic Array With Variable Shading,” IEEE Trans. Sustainable Energy, 2, pp.348–358. [CrossRef]
Brunton, S., Rowley, C., Kulkarni, S., and Clarkson, C., 2010, “Maximum Power Point Tracking for Photovoltaic Optimization Using Ripple-Based Extremum Seeking Control,” IEEE Trans. Power Electron., 25, pp. 2531–2540. [CrossRef]
Peterson, K., and Stefanopoulou, A., 2004, “Extremum Seeking Control for Soft Landing of an Electromechanical Valve Actuator,” Automatica, 29, pp. 1063–1069. [CrossRef]
Zhang, X. T., Dawson, D. M., Dixon, W. E., and Xian, B., 2006, “Extremum-Seeking Nonlinear Controllers for a Human Exercise Machine,” IEEE/ASME Trans. Mechatron., 11, pp. 233–240. [CrossRef]
Carnevale, D., Astolfi, A., Centioli, C., Podda, S., Vitale, V., and Zaccarian, L., 2009, “A New Extremum Seeking Technique and Its Application to Maximize RF Heating on FTU,” Fusing Eng. Des., 84, pp. 554–558. [CrossRef]
Killingsworth, N., and Krstić, M., 2006, “PID Tuning Using Extremum Seeking,” IEEE Control Syst. Mag., 26(1), pp. 70–79. [CrossRef]
Ou, Y., Xu, C., Schuster, E., Luce, T., Ferron, J., Walker, M., and Humphreys, D., 2008, “Design and Simulation of Extremum-Seeking Open-Loop Optimal Control of Current Profile in the DIII-D Tokamak,” Plasma Phys. Controlled Nucl. Fusion Res., 50, pp.1–24. [CrossRef]
Schuster, E., Xu, C., Torres, N., Morinaga, E., Allen, C., and Krstić, M., 2007, “Beam Matching Adaptive Control Via Extremum Seeking,” Nucl. Instrum. Methods Phys. Res. A, 581, pp. 799–815. [CrossRef]
Wang, H., Yeung, S., and Krstic, M., 2000, “Experimental Application of Extremum Seeking on an Axial-Flow Compressor,” IEEE Trans. Control Syst. Technol., 8, pp. 300–309. [CrossRef]
Ren, B., Frihauf, P., Rafac, R., and Krstić, M., 2012, “Laser Pulse Shaping Via Extremum Seeking,” Control Eng. Pract., 20, pp. 674–683. Available at http://flyingv.ucsd.edu/papers/PDF/158.pdf [CrossRef]
Sussmann, H. J., and Liu, W., 1991, “Limits of Highly Oscillatory Controls and the Approximation of General Paths by Admissible Trajectories,” Proceedings of IEEE Conference on Decision and Control, Dec., Brighton, UK.
Sussmann, H. J., 1992, “New Differential Geometric Methods in Nonholonomic Path Finding,” Prog. Syst. Control Theory, 12, pp. 365–384.
Kurzweil, J., and Jarnik, J., 1988, “Iterated Lie Brackets in Limit Processes in Ordinary Differential Equations,” Results Math., 14, pp. 125–137. [CrossRef]


Grahic Jump Location
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.

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
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)

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In