随机激励的非线性Markov跳变系统的稳态响应

大量实际工程问题需要用同时包含连续和离散变量的Markov跳变系统来描述.本文介绍了一类随机激励的单自由度(强)非线性Markov跳变系统的稳态响应的研究方法.首先,基于随机平均法导出具有Markov跳变参数的平均It随机微分方程,原系统方程的维数得到降低.接着,根据跳变过程原理,建立Fokker-Planck-Kolmogorov(FPK)方程组,方程组中的方程与系统的结构状态一一对应且互相耦合.求解该FPK方程组,得到Markov跳变系统的稳态随机响应及其统计量.最后,以一个高斯白噪声激励的Markov跳变Duffing振子为例,计算得到不同跳变规律下系统的稳态响应.研究结果表明,Markov跳变系统的稳态响应可以看作是各结构状态子系统稳态响应的加权和,加权值由跳变规律决定.

Stationary response of stochastically excited nonlinear Markovian jump system

Many practical problems should be described by nonlinear Markov jump systems involving both continuous and discrete variables. In this paper, the stationary response of stochastically excited single degree of freedom (strongly) nonlinear system with Markovian jump parameters is studied. Firstly, the averaged It? differential equation with Markovian jump is derived based on the stochastic averaging method. Then, according to the Markovian jump principle, the finite set of (Fokker Planck Kolmogorov) FPK equations are formulated. The FPK equations coupled with each other through the absorptive terms and reductive terms. The stationary response and its statistics of the Markovian jump system can be obtained by solving the FPK equations numerically. Finally, as an example, the responses of a Markovian jump Duffing oscillator subjected to Gaussian white noise are studied. Numerical results show that the stationary response of the jump system can be regard as a weighted sum of the responses of no jump system, and the weighted value is determined by the jump rules.