A path integration method for solving Markovian jump stochastic dynamical systems is presented. The Markovian jump process and the state vector of the system are combined into a new augmented state vector. The randomness of the Markovian jump is modeled by a stochastic process, which is merged with the stochastic perturbations to form the stochastic source affecting the evolution of the augmented system. Then a probability mapping is constructed, mapping the probability space of the stochastic source to the short-time transition probability density function of the augmented system. By solving this probability mapping, the short-time transition probability density function of the path integration method is obtained, thus a path integration method is customized for Markovian jump stochastic systems. Finally, a hydraulic relief valve system is used as an application example to demonstrate the availability of the path integration algorithm. The results suggest that the pre-compression parameters and coefficient of restitution of the plug can induce stochastic P-bifurcation phenomena. When the above two parameters are small, the system becomes unstable with the parameters taken in this example. Monte Carlo simulations prove that the proposed method effectively captures the transient and stationary responses of Markovian jump systems.

中文翻译：

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具有马尔可夫跳跃的随机系统的路径积分解决方案


提出了一种求解马尔可夫跳跃随机动力系统的路径积分方法。将马尔可夫跳跃过程和系统的状态向量组合成新的增广状态向量。马尔可夫跳跃的随机性通过随机过程建模，该过程与随机扰动合并形成影响增强系统演化的随机源。然后构造概率映射，将随机源的概率空间映射到增强系统的短时转移概率密度函数。通过求解该概率映射，得到路径积分方法的短时转移概率密度函数，从而为马尔可夫跳跃随机系统定制了路径积分方法。最后以液压溢流阀系统作为应用实例，验证了路径积分算法的可用性。结果表明，塞子的预压缩参数和恢复系数可以引起随机 P 分岔现象。当上述两个参数较小时，采用本例中采用的参数系统会变得不稳定。蒙特卡罗模拟证明该方法有效地捕获了马尔可夫跳跃系统的瞬态和稳态响应。