0
Research Papers

A Universal Feedback Controller for Discontinuous Dynamical Systems Using Nonsmooth Control Lyapunov Functions

[+] Author and Article Information
Teymur Sadikhov

School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: tsadikhov@gatech.edu

Wassim M. Haddad

Professor
School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: wm.haddad@aerospace.gatech.edu

Alternatively, we can consider Krasovskii solutions of Eq. (2) wherein the possible misbehavior of the derivative of the state on null measure sets is not ignored; that is, K[Fφ](x) is replaced with K[Fφ](x)=δ>0co¯{Fφ(Bδ(x))} and where Fφ is assumed to be locally bounded.

The assumption that LGV(x) is single-valued is necessary. Specifically, as will be seen later in the paper, the requirement that there exists z¯LGV(x) such that, for all uRm,max[LGV(x)u]=z¯u holds if and only if LGV(x) is a singleton. To see this, let q,rLGV(x), with q≠r, and assume, ad absurdum, that the required z¯ exists. Then, either q-z¯0 or r-z¯0. Assume q-z¯0 and let uT=q-z¯. Then, qu-z¯u=(q-z¯)u=(q-z¯)(q-z¯)T=q-z¯22>0. Hence, qu>z¯u, which leads to a contradiction.

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 5, 2013; final manuscript received September 9, 2014; published online November 7, 2014. Assoc. Editor: Yang Shi.

J. Dyn. Sys., Meas., Control 137(4), 041005 (Apr 01, 2015) (6 pages) Paper No: DS-13-1432; doi: 10.1115/1.4028593 History: Received November 05, 2013; Revised September 09, 2014

The consideration of nonsmooth Lyapunov functions for proving stability of feedback discontinuous systems is an important extension to classical stability theory since there exist nonsmooth dynamical systems whose equilibria cannot be proved to be stable using standard continuously differentiable Lyapunov function theory. For dynamical systems with continuously differentiable flows, the concept of smooth control Lyapunov functions was developed by Artstein to show the existence of a feedback stabilizing controller. A constructive feedback control law based on a universal construction of smooth control Lyapunov functions was given by Sontag. Even though a stabilizing continuous feedback controller guarantees the existence of a smooth control Lyapunov function, many systems that possess smooth control Lyapunov functions do not necessarily admit a continuous stabilizing feedback controller. However, the existence of a control Lyapunov function allows for the design of a stabilizing feedback controller that admits Filippov and Krasovskii closed-loop system solutions. In this paper, we develop a constructive feedback control law for discontinuous dynamical systems based on the existence of a nonsmooth control Lyapunov function defined in the sense of generalized Clarke gradients and set-valued Lie derivatives.

FIGURES IN THIS ARTICLE
<>
Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Brogliato, B., 1999, Nonsmooth Mechanics, 2nd ed., Springer, Berlin, Germany.
Pfeiffer, F., and Glocker, C., 1996, Multibody Dynamics With Unilateral Contacts, Wiley, NY.
Pereira, G. A. S., Campos, M. F. M., and Kumar, V., 2004, “Decentralized Algorithms for Multi-Robot Manipulation Via Caging,” Int. J. Rob. Res., 23(7,8), pp. 783–795. [CrossRef]
Olfati-Saber, R., and Murray, R. M., 2004, “Consensus Problems in Networks of Agents With Switching Topology and Time-Delays,” IEEE Trans. Autom. Control, 49(9), pp. 1520–1533. [CrossRef]
Cortes, J., 2008, “Discontinuous Dynamical Systems: A Tutorial on Solutions, Nonsmooth Analysis, and Stability,” IEEE Control Syst. Mag., 28(3), pp. 36–73. [CrossRef]
Shevitz, D., and Paden, B., 1994, “Lyapunov Stability Theory of Nonsmooth Systems,” IEEE Trans. Autom. Control, 39(9), pp. 1910–1914. [CrossRef]
Artstein, Z., 1983, “Stabilization With Relaxed Controls,” Nonlinear Anal. Theory Methods Appl., 7(11), pp. 1163–1173. [CrossRef]
Sontag, E. D., 1989, “A ‘Universal’ Construction of Artstein's Theorem on Nonlinear Stabilization,” Syst. Control Lett., 13(2), pp. 117–123. [CrossRef]
Rifford, L., 2001, “On the Existence of Nonsmooth Control-Lyapunov Functions in the Sense of Generalized Gradients,” ESAIM Control Optim. Calculus Variations, 6, pp. 593–611. [CrossRef]
Rifford, L., 2002, “Semiconcave Control-Lyapunov Functions and Stabilizing Feedbacks,” SIAM J. Control Optim., 41(3), pp. 659–681. [CrossRef]
Hirschorn, R., 2008, “Lower Bounded Control-Lyapunov Functions,” Commun. Inf. Syst., 8(4), pp. 399–412.
Rifford, L., 2000, “Existence of Lipschitz and Semiconcave Control-Lyapunov Functions,” SIAM J. Control Optim., 39(4), pp. 1043–1064. [CrossRef]
Clarke, F. H., 1983, Optimization and Nonsmooth Analysis, Wiley, NY.
Bacciotti, A., and Ceragioli, F., 1999, “Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions,” ESAIM Control Optim. Calculus Variations, 4, pp. 361–376. [CrossRef]
Filippov, A. F., 1988, Differential Equations With Discontinuous Right-Hand Sides, Kluwer, Dordrecht, The Netherlands.
Aubin, J. P., and Cellina, A., 1984, Differential Inclusions, Springer, Berlin, Germany.
Teel, A., Panteley, E., and Loria, A., 2002, “Integral Characterization of Uniform Asymptotic and Exponential Stability With Applications,” Math. Control Signal Syst., 15, pp. 177–201. [CrossRef]
Evans, L. C., 2002, Partial Differential Equations, American Mathematical Society, Providence, RI.
Cortés, J., and Bullo, F., 2005, “Coordination and Geometric Optimization Via Distributed Dynamical Systems,” SIAM J. Control Optim., 44(5), pp. 1543–1574. [CrossRef]
Haddad, W. M., and Chellaboina, V., 2008, Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach, Princeton University Press, Princeton, NJ.
Hui, Q., Haddad, W. M., and Bhat, S. P., 2009, “Semistability, Finite-Time Stability, Differential Inclusions, and Discontinuous Dynamical Systems Having a Continuum of Equilibria,” IEEE Trans. Autom. Control, 54(10), pp. 2465–2470. [CrossRef]
Paden, B. E., and Sastry, S. S., 1987, “A Calculus for Computing Filippov's Differential Inclusion With Application to the Variable Structure Control of Robot Manipulators,” IEEE Trans. Circuit Syst., 34(1), pp. 73–82. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Phase portrait of the open-loop nonsmooth harmonic oscillator

Grahic Jump Location
Fig. 2

Phase portrait of the closed-loop nonsmooth harmonic oscillator

Grahic Jump Location
Fig. 3

State trajectories of the closed-loop system versus time

Grahic Jump Location
Fig. 4

Control trajectories of the closed-loop system versus time

Grahic Jump Location
Fig. 5

Phase portrait of the open-loop system

Grahic Jump Location
Fig. 6

Phase portrait of the closed-loop system

Grahic Jump Location
Fig. 7

State trajectories of the closed-loop system versus time

Grahic Jump Location
Fig. 8

Control trajectories of the closed-loop system versus time

Tables

Errata

Discussions

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