# Ordered-Bipartite Consensus of Multi-Agent Systems under Finite Time Control
## 1 Introduction
On the one hand, infinite time convergence is not feasible in practical problems. On the other hand, a predictable convergence time is expected for _MAS_ with multi-task objectives. Therefore, finite-time control methods and fixed-time control methods are proposed to increase the convergence rate of _MAS_. The finite-time/fixed-time control of _MAS_ uses symbolic functions to solve the consensus problem. Due to the advantage of the high convergence rate of finite-time protocols, finite-time control of the _MAS_ has evolved since its introduction. In recent related research, finite-time control ideas have been applied to various types of control for _MAS_. The work in [18] derived a criterion and provided a sufficient condition to ensure fixed-time stabilization of switched positive nonlinear systems; the article [19] solved finite-time leader-follower output consensus problems by using the mismatched disturbance observer. By using hierarchical non-singular terminal sliding mode, finite-time observer-based consensus was designed for underactuated USVs to deal with the trajectory tracking tasks [20]. This method was also applicable to earth-observation tasks for multiple airships [21]. The existing studies on the single group or single bipartite consensus will not be suitable for a _MAS_ to complete complex behavior goals. For such problems, _GC_ control of a _MAS_ is feasible. However, those complex behaviors require massive information interaction in complete consensus control. Thus, the control protocols for _MAS_ reaching complex behavior goals should consider the problem of handling redundant information, which is also an essential concern for _GC_. Thus, this paper proposes the acyclic _GC_ and acyclic _GBC_ control protocols for a _MAS_. Featured by graded (or step-wise) multi-objective tasks, cooperative-competitive relations are required in hierarchical situations. For instance, the unmanned management system perceives order as a fundamental requirement. In addition to achieving complex behavior consensus, the _MAS_ needs to improve its convergence rate. From this point, both the acyclic _GC_ and acyclic _GBC_ control protocols are given based on finite-time control. Motivated by those points, this paper will continue to explore the _GC_ problems as well as _GBC_ problems for _MAS_ with finite-time control. In particular, the discussion covered in this paper is concerned with cooperative-competitive networks with acyclic partition topology. The main innovations are summarized as follows:  
1. This work deals with acyclic topology for its inter-group communications problems. Combining with the cooperative-competitive topology further, the _MAS_ performs _GC_ behavior in per-group symmetry. In this way, the inter-group problems of signed, directed interaction are also solved. 2. The full discussion revolves around the finite-time control of _GC_ coordination and _GBC_ coordination. With these considerations, the smoothness of finite-time control protocols in signed digraphs obtains novel aspects. 3. This paper takes the cooperative-competitive mechanism to the information interactions pattern in the first demonstration. The work is also specialized in its mathematical model by constructing a suitable signed graph with an acyclic partition. At this point, the corresponding conditions on algebraic graph theory are summarized. Indeed, based on these properties, the _GC_ will be verified by the mathematical induction method, and the final _GC_ takes an ordered and symmetric result.