Exact Delay Margin on Containment Control for Discrete-Time First-Order Multi-Agent Systems

In this paper, we investigate the containment control problem of discrete-time first-order unstable multi-agent systems with undirected fixed communication topologies and uncertain communication delays. In particular, we focus on the delay margin achieved by two kinds of containment protocols, one is $P$-type and the other is $PD$-type. We provide the exact expression of the delay margin and its explicit bound by optimizing the network coupling strengths within their allowable ranges. The results show that for a strictly unstable discrete-time first-order multi-agent system, its delay margin depends on the unstable pole location of each agent, as well as on the eigen-ratio of the network topologies. In addition, the delay margin can be enlarged by $PD$-type containment protocol with respect to $P$-type one. Finally, two numerical examples are given to illustrate the effectiveness of the results.