0
Technical Brief

Semistability Analysis of the Chaplygin Sleigh and Nonsmooth Mechanical Oscillator

[+] Author and Article Information
N. Sai Pushpak

Dynamics and Control Laboratory,
Department of Electrical Engineering,
Indian Institute of Technology Madras,
Chennai 600036, India
e-mail: saipushpak.n@gmail.com

Arun D. Mahindrakar

Dynamics and Control Laboratory,
Department of Electrical Engineering,
Indian Institute of Technology Madras,
Chennai 600036, India
e-mail: arun_dm@iitm.ac.in

Anup Ekbote

Dynamics and Control Laboratory,
Department of Electrical Engineering,
Indian Institute of Technology Madras,
Chennai 600036, India
e-mail: anupekbote@gmail.com

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 14, 2013; final manuscript received January 29, 2014; published online March 11, 2014. Assoc. Editor: Prashant Mehta.

J. Dyn. Sys., Meas., Control 136(3), 034505 (Mar 11, 2014) (4 pages) Paper No: DS-13-1198; doi: 10.1115/1.4026653 History: Received May 14, 2013; Revised January 29, 2014

An appropriate notion for stability studies of dynamical systems with a continuum of equilibria is semistability. One approach to semistability investigation is based on the nontangency between the vector field and the set of equilibria. In this work, we introduce a novel way of verifying nontangency by computing an outer estimate of the direction cone and illustrate its application through two well-known second-order systems. The stable equilibria of both the systems are shown to possess the semistability property.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Bhat, S. P., and Bernstein, D. S., 2003, “Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria,” SIAM J. Control Optim., 42, pp. 1745–1775. [CrossRef]
Campbell, S. L., and Rose, N. J., 1979, “Singular Perturbation of Autonomous Linear Systems,” SIAM J. Math. Anal., 10, pp. 542–551. [CrossRef]
Bernstein, D. S., and Bhat, S. P., 1994, “Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems,” American Control Conference, Vol. 2, IEEE, pp. 2355–2359.
Bernstein, D. S., and Bhat, S. P., 1995, “Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems,” ASME J. Mech. Des., 117, pp. 145–153. [CrossRef]
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]
Pushpak, N. S., and Mahindrakar, A. D., July 9-13, 2012, “Semistability Analysis of the Chaplygin Sleigh,” International Symposium on Mathematical Theory of Networks and Systems.
Bloch, A. M., 2003, Nonholonomic Mechanics and Control, Vol. 24, Springer-Verlag, New York.
Alvarez, J., Orlov, I., and Acho, L., 2000, “An Invariance Principle for Discontinuous Dynamic Systems With Application to a Coulomb Friction Oscillator,” ASME J. Dyn. Syst., Meas., Control, 122, pp. 687–690. [CrossRef]
Sampei, M., and Furuta, K., 1986, “On Time Scaling for Nonlinear Systems: Application to Linearization,” IEEE Trans. Autom. Control, 31(5), pp. 459–462. [CrossRef]
Filippov, A. F., and Arscott, F. M., 1988, Differential Equations With Discontinuous Righthand Sides, Springer, New York.

Figures

Grahic Jump Location
Fig. 1

Overbounding sets under the map f

Grahic Jump Location
Fig. 2

Tangent cone, direction cone, and F∧x

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