This paper focuses on the convergence analysis of first-order discrete multi-agent systems (MASs) with cooperative-competitive mechanisms. Firstly, compared with the existing results, our paper uses the condition of containing a directed spanning to replace that of strong connectivity, and gets a lager range of $\epsilon$ to guarantee that $\underset{k\to +\infty }{lim}{P}^k$ exists, which greatly improves the famous Perron–Frobenius theorem in Olfati-Saber et al. (2007)[24]. Subsequently, we can divide all the agents into $m$ subgroups according to the actual demand, and give the design method of weights so that system can achieve different consensus. We further generalize the results from first-order MASs to second-order MASs. Finally, numerical examples are given to verify the effectiveness of our results.

本文重点研究具有合作竞争机制的一阶离散多智能体系统 (MAS) 的收敛性分析。首先，与已有结果相比，本文采用包含有向跨度的条件代替强连通性的条件，得到更大范围的$\epsilon$ 以保证 $\underset{克\to +\infty }{林}{磷}^{克}$存在，极大地改进了 Olfati-Saber 等人著名的 Perron-Frobenius 定理。(2007)[24]。随后，我们可以将所有代理分为$米$根据实际需求进行分组，并给出权重的设计方法，使系统能够达成不同的共识。我们进一步将一阶 MAS 的结果推广到二阶 MAS。最后，给出数值例子来验证我们结果的有效性。