# Ordered-Bipartite Consensus of Multi-Agent Systems under Finite Time Control
## 1 Introduction  
No matter what the background of the times is, the exploration of unknown regions and the monitoring of dangerous areas always stand at the forefront, which is why research work in the field of robotics is keeping up with the times. In the current period, intelligent agents have been born along with rapid development. Some intelligent agents are already utilized in civilian life and military reserves. Compared to well-designed robots, multi-agent systems are adaptive and self-organizing because they are formed by a considerable number of simple individuals that can perform distributed perception and behavior. Therefore, the failure of individual behavior does not cause the overall behavior of this multi-agent system to fail. This reflects that the multi-agent system has better fault tolerance and higher robustness. In addition, the systems respond to complex tasks by choosing the number and the type of individuals. The control approach of the multi-agent system is to implement the code with a central processor. Therefore, the combination of information technologies makes it more advantageous in terms of implementation benefits. So, as far as the present day is concerned, the study of \(MAS\) is a task that must be carried out in the course of human development [1; 2; 3; 4]. According to the different task objectives of group systems, collaborative control problems are mainly classified into three types of problems, namely, consensus problems, swarm control, and formation control. Nonetheless, research on consensus control problems has always been fundamental to solving other collaborative control problems. For example, the formation control of a \(MAS\) is to achieve consensus of relative positions. Therefore, this paper discusses problems that belong to the field of consensus control of the \(MAS\). Reviewing the scientific process at the end of the 20 th century, the convergence and rapid development of scientific fields such as computer science and asynchronous computing have given rise to the concept of consensus control [5]. [PERSON] developed a general framework for the consensus control problem with respect to integrator networks [6]. Stepping into the era of the 21 st century, the field of _MAS_ has seen a boom in research on networks. Consensus control algorithms based on graph theory have received increasing attention and have a relatively mature theory. [PERSON] extended the consensus problem to second-order systems, and obtained sufficient and necessary conditions for the stability of _MAS_ in directed networks [7]. [PERSON] et al. proposed a consensus protocol for higher-order multi-agent systems, illustrating the way in which agents achieve consensus using the relative position information shared by their surrounding neighbors [8]. With the increasing theoretical research and the increased requirements of modern industrial production in terms of scale and refinement, complex behavior control has become an important aspect of _MAS_'s theory research. Among the methods to portray the coordinated control of multi-task objectives for networked _MAS_, _GC_ and bipartite consensus are in irreplaceable positions. As a result, various approaches for _GC_ have been under development, with studies represented by [9; 10; 11] among others. Since [PERSON] pioneered a bipartite consensus model for networked systems in a structurally balanced network topology in 2013 [12], various sequential works have been carried out. Recent research in [13; 14; 15] gave discussions on the consensus problems for cooperative-competitive networks. Inspired by these works, [PERSON] provided a novel symmetric positive definite matrix to derive the gain selection criteria in the leader-following case for networked Lagrangian systems [16]. [PERSON] further introduced an acyclic partition and obtained the explicit expressions of the final states in the leaderless case [17]. However, once the _MAS_'s consensus problems are mentioned, its convergence rate is further considered.