Abstract The moment equations technique together with modified cumulant-neglect closure techniques is developed for a non-linear dynamic system subjected to a random train of impulses driven by an Erlang renewal counting process. The original non-Markov problem is converted into a Markov one by recasting the excitation process with the aid of an auxiliary, pure-jump stochastic process characterized by a Markov chain. Hence the conversion is carried out at the expense of augmentation of the state space of the dynamic system by Markov states of the auxiliary jump process. The problem is characterized by the set of joint probability density — discrete distribution function, which are the joint probabilities of the original state vector and of the Markov chain being in the particular j th state. Accordingly the statistical moments of the state variables are defined as integrals with respect to the mixed-type, probability density — discrete distribution function. The differential equations for moments are obtained with the aid of the forward integro-differential Chapman–Kolmogorov operator (Iwankiewicz, 2014). Two modified closure techniques are developed. The first one is the result of application of cumulant-neglect closure directly to unconditional moments. The second modified closure approximation technique is based on the representation of the joint probability density — discrete distribution function by conditioning it on two mutually exclusive and exhaustive events: that the Markov chain is in the j th state while the system is at rest (it is in the j th state for the first time) and that the Markov chain is in the j th state while the system is not at rest (it is in the j th state for any subsequent time). Thus the joint probability density function consists of a Dirac-delta spike and of the continuous part (cf Iwankiewicz et al. (1990)). The cumulant-neglect closure approximations are first formulated for the conditional moments resulting from the continuous part of the probability density function and next for the unconditional moments. As an example of a non-linear system the oscillator with cubic restoring force term is considered. The equations for moments up to the fourth-order are derived. Hence the modified cumulant-neglect closure approximations are derived for redundant fifth- and sixth order moments, both centralized and ordinary. The developed moment equations with modified cumulant-neglect closure techniques are verified against Monte Carlo simulations.

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非线性动力系统在更新脉冲过程激励下的矩方程和累积忽略闭合技术

摘要 矩方程技术和修正的累积-忽略闭包技术是为非线性动态系统开发的，该系统受到由 Erlang 更新计数过程驱动的随机脉冲序列的影响。通过借助以马尔可夫链为特征的辅助纯跳跃随机过程重铸激励过程，将原始非马尔可夫问题转换为马尔可夫问题。因此，转换是以通过辅助跳跃过程的马尔可夫状态增加动态系统的状态空间为代价来进行的。该问题的特征在于一组联合概率密度-离散分布函数，即原始状态向量和马尔可夫链处于特定第j个状态的联合概率。因此，状态变量的统计矩被定义为关于混合型概率密度离散分布函数的积分。借助前向积分微分 Chapman-Kolmogorov 算子 (Iwankiewicz, 2014) 获得矩的微分方程。开发了两种改进的闭合技术。第一个是将累积-忽略闭包直接应用于无条件矩的结果。第二种改进的闭包逼近技术基于联合概率密度的表示 - 离散分布函数，通过对两个互斥且详尽的事件进行调节：马尔可夫链在系统静止时处于第 j 个状态（它第一次处于第 j 个状态），而马尔科夫链在系统不静止时处于第 j 个状态（它是在任何后续时间处于第 j 个状态）。因此，联合概率密度函数由 Dirac-delta 尖峰和连续部分组成（参见 Iwankiewicz 等人（1990））。累积-忽略闭包近似首先为由概率密度函数的连续部分产生的条件矩制定，然后为无条件矩制定。作为非线性系统的一个例子，考虑了具有三次恢复力项的振荡器。推导出四阶矩的方程。因此，针对集中式和普通的冗余五阶和六阶矩导出了修正的累积量-忽略闭包近似值。使用改进的累积量-忽略闭合技术开发的矩方程通过蒙特卡罗模拟进行验证。