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

• Published in: International journal of control 2015-02, Vol.88 (2), p.248-255
• Main Authors: Xu, Chengjie, Zheng, Ying, Su, Housheng, Wang, Hua O.
• Format: Article
• Language: English
• Subjects: Computer simulation; Containment; Controller area network; Followers; harmonic oscillator; Harmonic oscillators; Hulls (structures); Leadership; multi-agent systems; multiple leaders; Oscillators; sampled-data; Sampling; Topology
• ISSN: 0020-7179; 1366-5820
• DOI: 10.1080/00207179.2014.944873

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.