On the machine swing dynamics: a perspective

TL;DR

This paper introduces a novel Carleman linearization approach to filter nonlinear stochastic swing equations, simplifying algorithms and outperforming traditional EKF methods in accuracy.

Contribution

It embeds Carleman linearization into stochastic differential equations, enabling refined filtering of nonlinear stochastic systems with a more straightforward algorithmic process.

Findings

Carleman linearization effectively simplifies nonlinear stochastic filtering.

Filtering in the Carleman setting outperforms benchmark EKF.

The approach provides a more refined and sharper filtering method.

Abstract

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.