Containment control for coupled harmonic oscillators with multiple leaders under directed topology

Abstract

This paper investigates the problem of containment control for coupled harmonic oscillators with multiple leaders under directed topology. Using tools from matrix, graph and stability theories, necessary and sufficient conditions are obtained for coupled harmonic oscillators under continuous-time and sampled-data-based protocols, respectively. When the continuous-time protocol is used, it is proved that every follower will ultimately converge to the convex hull spanned by the leaders if and only if there exists at least one leader that has a directed path to that follower at any time. When the sampled-data-based protocol is used, it is shown that the containment can be achieved if and only if: (1) an appropriate sampling period is chosen and (2) for every follower, there exists at least one leader that has a directed path to that follower at any time. And we also give the containment conditions for coupled harmonic oscillators under undirected topology as a special case. Finally, numerical simulations are presented to illustrate the theoretical findings.

Keywords:

• containment
• harmonic oscillator
• multiple leaders
• sampled-data
• multi-agent systems

Acknowledgements

The authors would like to thank the editor, Eric Rogers, and the anonymous referees for their helpful comments and suggestions.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China [grant number 61374139], [grant number 61034006], [grant number 61104140], [grant number 61174075], [grant number 61473129]; the Program for New Century Excellent Talents in University from Chinese Ministry of Education [grant number NCET-12-0215]; the Program for Changjiang Scholars and Innovative Research Team in University [grant number IRT1245]; Natural Science Foundation of Ministry of Education in Hunan Province [grant number 12C0077].