# Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies
## 1 Introduction  
Multi-agent systems (MASs), with entities possessing flexibility and autonomy, have been extensively studied to analyze and simulate complex systems in the past few years [1; 2; 3; 4]. As an essential category of distributed artificial intelligence, MASs have been applied in the field of control engineering, such as the formation of spacecraft systems [5], robot systems [6; 7], underwater vehicle systems [8], and unmanned aerial vehicles [9], etc. Tracking approaches for MASs have been widely used in areas of collaborative target tracking [10], convoy protection [11] and defensive navigation [12]. During the multi-agent tracking process, where some tracking agents aim to track the target agent, the communication topology between agents may change or even be unconnected due to obstacles, communication limitations and unknown disturbances. In terms of the case where the communication topology of the agents is connected, the whole interactive multi-agent tracking system is stable, while the unconnected topologies of the agents lead to the intermittent existence of unstable subsystems. In this paper, the division of the stable subsystem and the unstable subsystem results from the change of communication topology. Thus, it is not based on the system of the agent itself, but on the whole system between multiple agents. The time-varying topologies of the agents are generally described as switching topologies. The switching topology is an essential but challenging issue in MASs practical applications. The process of MASs switching topology is usually complicated because many kinds of scenarios must be considered when designing the switching rules [13; 14; 15; 16]. Up to now, some researchers have focused on switching topologies of multi-agent systems, and numerous valuable research results were obtained with different models and control strategies [17; 18; 19; 20; 21; 22; 23]. To be more specific, consensus problems for networks of dynamic agents with switching topologies were addressed in [24; 25; 26; 27]. Multi-agent systems with Lipschitz-type node dynamics was studied in [28], which obtained sufficient conditions for consensus tracking under weakly connected and switched topologies. The authors in [29] achieved exponential tracking of heterogeneous second-order time-varying nonlinear multi-agent systems under directed switching topology through constructing the topology-dependent Lyapunov function and designing appropriate time-varying regulation factors. In [30], a coordinated formation control task was accomplished with an iterative update algorithm based on the nearest neighbor relative distances under the condition that the topology has a spanning tree over iteration intervals with finite lengths. The core of the approaches in [31; 32; 33] was the achievements of the switching controllability in discrete-time multi-agent systems with time-delays on undirected networks. But, these results only considered multi-agent systems which switch among stable subsystems, in other words, the switching topologies were all-connected. The complexity is significantly increased if there are unstable subsystems in multi-agent switching systems [34]. The work in [35] focused on a class of cyclic switched nonlinear systems, and achieved the finite-time stability by making the most of the trade-off among impulsive dynamics, initial conditions, and the dynamics of each mode. The authors in [36] discussed the case that switching subsystems were unstable, designed the dwell-time and proposed an event-triggered controller so as to achieve global stability. The multi-agent relay tracking problems with changing the number of agents, jumping of tracking errors, and communication time delays were solved in [37]. However, these studies were based on the premise that the sequence of topologies follows a strict cyclic period. Based on the above discussions, this paper proposes a quasi-cyclic method that allows the topologies of multi-agent tracking systems with unstable subsystems to switch in a less conservative way.