# Impulsive Multiple-Bipartite Consensus Control for Networked Second-Order Multi-Agent Systems
## 1 Introduction  
The consensus problem of multi-agent systems (MASs) enjoys valuable and potential significance in light of its physical background and engineering applications, e.g., in the deployment of distributed artificial intelligence, the smart scheduling of transportation, and the control of autonomous multiple robots and application of mobile sensor networks [1, 2, 3, 4, 5, 6]. Consensus for MASs has three main categories of theoretical studies: biological community consensus theory, pure basic theory and various applications of consensuses [7]. As one of the important emerging behaviors, the consensus problem requires that all the agents in the network reach an agreement on a common value or state [8, 9, 10], and consensus can be seen as the most fundamental point of many practical complex control objectives. For the consensus problem of MASs, the mathematical model of the consensus problem formulation, the agents' dynamics and the network structure have been widely extended. Various forms of consensus have been investigated in recent decades. For instance, a quantized consensus requires that a network achieve coordinate behavior using the sampling information control method and quantized data in advance [11, 12, 13]. Bipartite consensus [14] (Altafini model) requires that all agents converge to a value of the same amplitude but a different direction [15, 16, 17]. The scaled consensus, which can be regarded as the generalized scenario of bipartite consensus, requires that the agents reach assigned proportions rather than a common value [18, 19, 20]. Cluster (group) consensus, which divides the corresponding agent set into separate groups, drives all agents in the same group to reach a complete consensus [21, 22, 23]. With regard to the agents' dynamics, the single-integrator dynamic model [24], double-integrator dynamic model [25, 26] and high-order dynamic model [27] were extensively investigated. Additionally, several types of the network structure, namely fixed type [28], switching type [29] and stochastic type [30], have been involved in theaforementioned literature. Notably, the references mentioned above mainly focused on the complete consensus or single bipartite consensus [11; 12; 13; 14; 15; 16; 17; 18; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. However, with the development of modern industrial applications, the complexity level of the control objectives is becoming increasingly higher; thus, complete consensus and single bipartite consensus cannot meet the modeling requirements in many application scenarios. Based on this, group-bipartite consensus is introduced to describe the compound tasks of the single integral MASs in [31] with a continuous control protocol. On the other hand, the impulsive control strategy is a basic type of discontinuous control method. Owing to the advantage of simplicity, effectiveness, and robustness [32; 33], the impulsive control technique [8], was deployed in the coordination control of MASs. For instance, [PERSON] et al. [34] studied the robust impulsive synchronization of uncertain dynamical networks with bounded coupling functions. Via Lyapunov functionals and some analysis techniques, [PERSON] et al. [35] derived some sufficient conditions for pinning the synchronization of stochastic impulsive discrete-time networks in mean square, and the developed approaches were applied to a scale-free network composed of discrete-time neural networks. In the reference [36], the synchronization of complex dynamical networks (CDNs) with system delay and multiple coupling delays was studied via impulsive distributed control. It should be noted that the aforementioned literature is all focused on designing single-order distributed impulsive control protocol algorithms for MASs in networks. To the best of our knowledge, the impulsive control strategy and second-order node dynamics which are more widely used have not been involved.