# Containment Control of First-Order Multi-Agent Systems under PI Coordination Protocol +
## 1 Introduction  
Distributed coordination control of multi-agent systems has been an active research area for its widespread potential engineering applications including cooperative surveillance, sensor networks, spacecraft formation flying, etc. The most fundamental problem studied in coordination control is consensus problem [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11] which means that each agent reaches an agreement based on the information from its relative agents. According to the different number of leaders in the multi-agent system, the consensus problem is generally divided into leaderless case [12; 13], single-leader-follower case [14; 15; 16], and multiple-leader-follower case. The containment control problem [17; 18; 19; 20; 21; 22; 23; 24] is considered as a special multiple-leader-follower consensus problem for multi-agent system with multiple leaders and followers, and it requires all followers to converge into the convex hull spanned by leaders. Many valuable results on the containment control problem of first-order multi-agent systems, which are investigated in this paper, have been obtained in recent decades. Basic results for realizing containment of continuous-time first-order multi-agent systems with stationary leaders have been given by [PERSON] in [25], and the convergence conditions under fixed and switching topologies are dependent on the topology structure. [PERSON] and their colleagues [26] investigated the containment problem of first-order multi-agent system with communication noises, and designed a time-varying gain to reduce noises. A PD-type control protocol was introduced and a parameter condition was given to guarantee the containment achievement under input delays in [PERSON]'s work [27]. [PERSON] and partners[28] provided necessary and sufficient criteria for containment convergence if the communicationdata rates are limited. [PERSON] and colleagues [29] applied the event-triggered scheme into containment control algorithms, and the convergence problem with single time delay and multiple time delays were discussed, respectively. Considering the discrete-time first-order multi-agent systems. The authors in [30] adopted the asynchronous containment control protocol and got sufficient conditions for reaching containment. Additionally, containment problem of multi-agent systems with second-order dynamics and general linear dynamics has also attracted extensive attention [31; 32; 33; 34]. According to aforementioned works, we note that general containment control protocols make each follower finally converge to different point in the convex hull spanned by leaders. General containment problem only focuses on the containment and ignores the consensus, which is called rendezvous in this paper, of the followers. If we take both containment and consensus into consideration, a different control protocol is required, and we define this problem as a containment-rendezvous problem in this paper. Actually, the average-tracking problem of reference signals is a typical containment-rendezvous problem, if we regard the reference signals as leaders. [PERSON] and [PERSON] [35] considered the average-tracking problem of first-order multi-agent systems with unmatched reference signals, and used the frequency-domain analysis method, which will be adopted in this paper, to get the convergence condition. On the basis of average-consensus algorithm, [PERSON] [36] designed a containment-rendezvous algorithm that made all followers track one point in the convex hull of first-order agents, but the results depended on the balance property of communication topology. Inspired by above works, this paper focuses on the containment-rendezvous problem of discrete-time first-order multi-agent systems under general communication topology. On the basis of the average-consensus algorithm, a proportional-integral (PI) coordination control protocol is proposed for reaching containment-rendezvous. We first analyze the protocol for the followers under general connected topology, and obtain a sufficient convergence condition.