In this paper, the convergence issue of the Particle Swarm Optimization (PSO) algorithm is investigated. Most of the models of PSO algorithms are time-invariant linear models with the assumption the local and global best solutions do not change, i.e., the stagnation assumption. However, in this paper, a discrete-time switched linear model is introduced to study the stability and convergence of the PSO algorithm without the stagnation assumption. By considering the updates of local best positions and global best solutions, a sequence of state transform matrixes is generated during the searching process. The semistability of the proposed switched linear system is studied. The conditions of the convergence in mean and convergence in probability are derived by using the recently developed results in paracontraction. Moreover, numerical examples are provided to verify the results proposed in this paper.

本文研究了粒子群算法（PSO）的收敛性问题。大多数PSO算法模型都是时不变线性模型，其假设是本地和全局最佳解不变，即停滞假设。然而，在本文中，引入了离散时间切换线性模型来研究没有停滞假设的PSO算法的稳定性和收敛性。通过考虑局部最佳位置和全局最佳解的更新，在搜索过程中会生成一系列状态变换矩阵。研究了所提出的线性切换系统的半稳定性。均值收敛和概率收敛的条件是通过使用最近开发的超收缩结果得出的。而且，