This paper investigates the stochastic stability of discrete time positive Markov jump nonlinear systems (PMJNS). Some definitions on stochastic stability for discrete time PMJNS are introduced first. Then, using the multiply max-separable Lyapunov function method, some stochastic stability criterions of discrete time PMJNS are provided, and some corresponding criterions are also provided for discrete time positive Markov jump linear systems (PMJLS). Different from previous conclusions that require subsystems to be stable or marginally stable, the obtained results allow some subsystems to be unstable. Based on the proposed criterions, the stochastic stability behavior of discrete time positive Markov jump systems can be obtained just from the algebraic properties of the system function and the probability characteristics of the Markov chain. To illustrate the main results, two simulation examples are provided at the end.

本文研究了离散时间正马尔可夫跳跃非线性系统（PMJNS）的随机稳定性。首先介绍了离散时间PMJNS随机稳定性的一些定义。然后，利用乘法最大可分离Lyapunov函数方法，给出了离散时间PMJNS的随机稳定性判据，并为离散时间正马尔可夫跳跃线性系统（PMJLS）提供了相应的判据。与之前要求子系统稳定或边际稳定的结论不同，所得结果允许部分子系统不稳定。基于所提出的准则，仅根据系统函数的代数性质和马尔可夫链的概率特性就可以得到离散时间正马尔可夫跳跃系统的随机稳定性行为。为了说明主要结果，最后提供了两个模拟示例。