The operation of dynamic systems is often accompanied by abrupt and random changes in their configurations, which will dramatically change the stationary probability density function of their response. In this article, an effective procedure is proposed to reshape the stationary probability density function of nonlinear stochastic systems against abrupt changes. Based on the Markov jump theory, such a system is formulated as a continuous system with discrete Markov jump parameters. The limiting averaging principle is then applied to suppress the rapidly varying Markov jump process to generate a probability-weighted system. Then, the approximate expression of the stationary probability density function of the system is obtained, based on which the reshaping control law can be designed, which has two parts: (i) the first part (conservative part) is designed to make the reshaped system and the undisturbed system have the same Hamiltonian; (ii) the second (dissipative part) is designed so that the stationary probability density function of the reshaped system is the same as that of undisturbed system. The proposed law is exactly analytical and no online measurement is required. The application and effectiveness of the proposed procedure are demonstrated by using an example of three degrees-of-freedom nonlinear stochastic system subjected to abrupt changes.

中文翻译：

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改变非线性随机系统对突然变化的概率密度函数

动态系统的操作通常会伴随其配置的突然和随机变化，这将极大地改变其响应的平稳概率密度函数。在本文中，提出了一种有效的方法来针对非线性变化重塑非线性随机系统的平稳概率密度函数。基于马尔可夫跳跃理论，这种系统被公式化为具有离散马尔可夫跳跃参数的连续系统。然后应用极限平均原理来抑制快速变化的马尔可夫跳跃过程，以生成概率加权系统。然后，获得系统平稳概率密度函数的近似表达式，并据此设计重塑控制律，该律包括两个部分：（i）第一部分（保守部分）设计成使重塑系统和原状系统具有相同的哈密顿量；（ii）设计第二个（耗散部分），以使重塑系统的平稳概率密度函数与未扰动系统的平稳概率密度函数相同。拟议的法律完全是分析性的，不需要在线测量。以一个经历了突变的三自由度非线性随机系统为例，说明了该方法的应用和有效性。拟议的法律完全是分析性的，不需要在线测量。以一个经历了突变的三自由度非线性随机系统为例，说明了该方法的应用和有效性。拟议的法律完全是分析性的，不需要在线测量。以一个经历了突变的三自由度非线性随机系统为例，说明了该方法的应用和有效性。