关于机器摆动动力学：透视
On the machine swing dynamics: a perspective
摘要：
在应用数学和动力系统中，一种形式化的非线性随机微分方程滤波方法是库什纳设置。由于Carleman线性化的能力，非线性随机微分方程在有限闭包下可以等价地表示为一个具有增广状态的双线性随机微分方程的有限系统.有趣的是，本文的新颖之处在于将Carleman线性化嵌入到马尔可夫过程的随机演化中。为了说明马尔可夫过程的Carleman线性化，本文将Carleman线性化嵌入到非线性摆动随机微分方程中。此外，我们实现了非线性摆动方程滤波的Carleman设置。在Carleman设置中的滤波简化了算法过程。相关的增广状态解释了非线性和随机性。我们表明，在Carleman框架中的非线性随机摆动方程的过滤是更精细，以及尖锐的对比基准非线性EKF。本文提出了将Carleman嵌入到随机微分方程中去滤掉所涉及的非线性随机微分系统的有效性。这篇论文将对非线性随机动力学者探索和展开线性化嵌入技术的研究产生兴趣。
英文摘要：
A formal approach to rephrase nonlinear filtering of stochastic differential equations is the Kushner setting in applied mathematics and dynamical systems. Thanks to the ability of the Carleman linearization, the nonlinear stochastic differential equation can be equivalently expressed as a finite system of bilinear stochastic differential equations with the augmented state under the finite closure. Interestingly, the novelty of this paper is to embed the Carleman linearization into a stochastic evolution of the Markov process. To illustrate the Carleman linearization of the Markov process, this paper embeds the Carleman linearization into a nonlinear swing stochastic differential equation. Furthermore, we achieve the nonlinear swing equation filtering in the Carleman setting. Filtering in the Carleman setting has simplified algorithmic procedure. The concerning augmented state accounts for the nonlinearity as well as stochasticity. We show that filtering of the nonlinear stochastic swing equation in the Carleman framework is more refined as well as sharper in contrast to benchmark nonlinear EKF. This paper suggests the usefulness of the Carleman embedding into the stochastic differential equation to filter the concerning nonlinear stochastic differential system. This paper will be of interest to nonlinear stochastic dynamists exploring and unfolding linearization embedding techniques to their research.
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