Time-delayed feedback stabilisation of nonlinear potential systems

Abstract

Mechanical systems with nonlinear potential forces and delayed feedback are studied. It is assumed that, in the absence of control, the trivial equilibrium positions of considered systems are stable, but they are not attracting ones. An approach for the constructing of nonlinear controllers providing the asymptotic stability of the equilibrium positions is proposed. By the use of the Lyapunov direct method and the Razumikhin approach, it is proved that for the corresponding closed-loop systems the asymptotic stability can be guaranteed even in the cases when delay is unknown and time-varying. Moreover, estimates for solutions of closed-loop systems are found. An example and the results of a computer simulation are presented to demonstrate the effectiveness of the proposed approach.

Keywords:

• potential system
• nonlinear control
• delay feedback
• asymptotic stability
• Lyapunov function

Acknowledgements

This work was partially supported by the Saint Petersburg State University, project nos. 9.38.674.2013 and 9.37.157.2014, and by the Russian Foundation for Basic Research, grant nos. 13-01-00347-a and 13-01-00376-a.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the Saint Petersburg State University [project no. 9.38.674.2013], [project no. 9.37.157.2014]; the Russian Foundation for Basic Research [grant no. 13-01-00347-a], [grant no. 13-01-00376-a].