Abstract
This paper investigates the stabilizability and bipartite containment control problem of general linear multi-agent systems over signed directed graphs, where the negative edges indicate the antagonistic interactions between agents. By employing the proposed bipartite containment control protocols over signed directed graphs, and combining the linear system stabilization theory and the signed Laplacian matrix, the necessary condition for the stabilization of multi-agent systems with multiple leaders is discussed. According to this necessary condition, a leader-follower matching method is presented to establish leader-follower multi-agent system networks. Based on the leader-follower topology established above and the designed state feedback gain, the stability of multi-agent systems is proved by using the system error and the algebraic Riccati equation. It is shown that the followers can gradually converge into the convex hull spanned by the states and the sign-inverted states of leaders, and the bipartite containment control of multi-agent systems can be realized. The simulation results verify the feasibility of the theoretical analysis.
Highlights
During the past decades, the researches on distributed cooperative control of multi-agent systems had received extensive attention, such as consensus, flocking, formation, formation-containment, multi-target tracking and so on [1]–[6]
STABILIZATION CONDITION FOR MULTI-AGENT SYSTEMS Considering the continuous-time agent linear dynamics system, the dynamic of each agent is described by xi (t) = Axi (t) + Bui (t), i = 1, . . . , n, (1)
In order to represent the influence of the loss of structurally balanced independent strongly connected component (SBiSCC) on the stability of whole system, a set of the maximum matching edge sets is determined as 1+ → 3−, 3+ → 5−, 4+ → 7−, 5+ → 1−, 6+ → 12−, 8+ → 2−, 9+ → 10−, 10+ → 13−, 11+ → 4−, 13+ → 9−, the nonmatching nodes determined by this maximum matching edge set are 6, 8 and 11, which are selected controlled follower nodes
Summary
The researches on distributed cooperative control of multi-agent systems had received extensive attention, such as consensus, flocking, formation, formation-containment, multi-target tracking and so on [1]–[6]. The necessary and sufficient conditions for multi-agent systems to achieve containment were given in [16] On this basis, Cao [17] studied the containment control problem with stationary leaders and dynamic leaders under fixed and switching topologies. For the multi-leader first-order multi-agent systems, based on the researches of [36], the bipartite containment control was studied in [39]. This paper copes with the situation that the linear multi-leader systems over signed networks can realize bipartite containment by selecting the least controlled followers. The characteristics of the exact and least controlled followers to achieve containment control of general linear multi-agent systems over arbitrary signed directed graphs are discussed, which cannot be dealt with well through the bipartite graph maximum matching algorithm mentioned in [30]. Path from vi to vj, and there exists a path from vj to vi, we call G a strongly connected graph
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