<i>p</i> Components of Cluster-Lag Consensus for Second-Order Multiagent Systems With Adaptive Controller on Cooperative–Competitive Networks
Yi Wang, Haiyu Song, Guoyuan Chen, Zhongjun Ma, Jinde Cao
Abstract
The consensus tracking problem means that a group of followers tracks the desired trajectory with local communication. In this article, partial components of cluster consensus have been considered. In this scenario, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> components of the followers in different clusters track the leader at different lag times, while <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> components of each agent in the same cluster reach a consensus, which is called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> components of cluster-lag (PCCL) consensus. By using a seminorm <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\|x_{i}\|_{2,p}$ </tex-math></inline-formula> and a Lyapunov–Krasovskii functional, PCCL consensus for second-order multiagent systems with homogeneous nonlinear systems on cooperative–competitive networks has been considered. For the case that the communication network graph is undirected, a decentralized adaptive controller, which is based on the exchanged neighbors’ information from the same cluster, is designed such that all the agents reach PCCL consensus. For the directed graph case, an adaptive protocol based on the intracoupling strength is constructed for each cluster to achieve PCCL consensus. Finally, two simulation examples are illustrated to show the effectiveness of the proposed control protocols.