Weighted coordinated motion for coupled harmonic oscillators with heterogeneous interactions of cooperation and competition

Abstract

ABSTRACT This paper investigates the weighted coordinated motion of coupled harmonic oscillators over weighted cooperative–competitive interactions. First, the traditional synchronisation and containment control concepts are extended to the weighted synchronisation and weighted containment control for the new cooperative–competitive networks. Then, weighted synchronisation problem and weighted containment tracking motion are respectively solved for the coupled harmonic oscillators. Compared with some existing results, the frequency parameters of coupled harmonic oscillators are unknown in this paper and some novel networked structures are introduced in order to make the harmonic oscillator followers converge to the convex hull spanned by leaders' weighted states. Particularly, if the topologies formed by every leader group and follower group are interactively balanced and every follower is weight-product balanced, or if some leader groups are not interactively balanced but satisfy some conditions, the coupled harmonic oscillators can all realise the weighted containment tracking motion. Simulation results are given to demonstrate the effectiveness of the proposed control schemes.