A study on fault diagnosis in nonlinear dynamic systems with uncertainties

TL;DR

This paper develops a unified framework for fault diagnosis in nonlinear dynamic systems using stable image and kernel representations, projection techniques, and Bregman divergences within Hamiltonian system models, accommodating model-based, data-driven, and machine learning approaches.

Contribution

It introduces a novel fault diagnosis framework employing Hamiltonian systems and Bregman divergences, unifying various strategies and addressing non-Euclidean data spaces.

Findings

Effective fault detection using projection onto nominal system manifolds.

Bregman divergence with Hamiltonian functions enhances performance-oriented fault detection.

Kernel-based methods enable fault estimation and uncertainty analysis.

Abstract

In this draft, fault diagnosis in nonlinear dynamic systems is addressed. The objective of this work is to establish a framework, in which not only model-based but also data-driven and machine learning based fault diagnosis strategies can be uniformly handled. Instead of the well-established input-output and the associated state space models, stable image and kernel representations are adopted in our work as the basic process model forms. Based on it, the nominal system dynamics can then be modelled as a lower-dimensional manifold embedded in the process data space. To achieve a reliable fault detection as a classification problem, projection technique is a capable tool. For nonlinear dynamic systems, we propose to construct projection systems in the well-established framework of Hamiltonian systems and by means of the normalised image and kernel representations. For nonlinear dynamic systems, process data form a non-Euclidean space. Consequently, the norm-based distance defined in Hilbert space is not suitable to measure the distance from a data vector to the manifold of the nominal dynamics. To deal with this issue, we propose to use a Bregman divergence, a measure of difference between two points in a space, as a solution. Moreover, for our purpose of achieving a performance-oriented fault detection, the Bregman divergences adopted in our work are defined by Hamiltonian functions. This scheme not only enables to realise the performance-oriented fault detection, but also uncovers the information geometric aspect of our work. The last part of our work is devoted to the kernel representation based fault detection and uncertainty estimation that can be equivalently used for fault estimation. It is demonstrated that the projection onto the manifold of uncertainty data, together with the correspondingly defined Bregman divergence, is also capable for fault detection.