Model-Free Geometric Fault Detection and Isolation for Nonlinear Systems Using Koopman Operator

Abstract

This paper presents a model-free fault detection and isolation (FDI) method for non-linear dynamical systems using Koopman operator theory and linear geometric technique. The key idea is to obtain a Koopman-based reduced-order model of a non-linear dynamical system and apply the linear geometric FDI method to detect and isolate faults in the system. Koopman operator is an infinite-dimensional, linear operator which lifts the nonlinear dynamic data into an infinite-dimensional space where the correlations of dynamic data behave linearly. However, due to the infinite dimensionality of this operator, an approximation of the operator is needed for practical purposes. In this work, the Koopman framework is adopted toward non-linear dynamical systems in combination with the linear geometric approach for fault detection and isolation. In order to demonstrate the efficacy of the proposed FDI solution, a mathematical nonlinear dynamical system and an experimental three-tank setup are considered. Results show a remarkable performance of the proposed geometric Koopman-based fault detection and isolation (K-FDI) technique.INDEX TERMS Model-free fault detection and isolation, Koopman operator, Extended dynamic mode decomposition, Geometric approach, Reduced-order model.

Data‐driven sensor fault detection and isolation of nonlinear systems: Deep neural‐network Koopman operator

“…In this section, a comparison study is conducted to compare the prediction performance of the proposed DNN‐based Koopman with the Koopman‐PCA in ref. [33] and show the precision and reliability of the DNN‐based Koopman predictor. For constructing this comparison, the lifting functions for Koopman‐PCA are considered to be 100 thin plate radial basis functions (RBF), where the centres of the RBF functions are chosen to be uniformly distributed on a unit box.…”

“…Thus, all new faults that are not recognized as a member of these groups cannot be diagnosed properly by this clustering method [26]. By formulating the FD problem using the Koopman operator, one can use linear model‐based FD techniques in a data‐driven manner without the need for faulty data to achieve fault isolation [33]. Since the Koopman operator is of infinite dimension, different methods such as dynamic mode decomposition (DMD) [34], extended dynamic mode decomposition [35], and their variants have been used to approximate the Koopman eigenfunctions, which lead to a finite‐dimensional representation of this operator.…”

“…[42] with the focus on fault detection of the autonomous systems using Kalman filters. A model‐free geometric method for FD of actuator faults is presented in [33], which used a Koopman‐PCA reduced‐order model to design residual generators.…”

“…First, the work in ref. [33] deals with actuator faults, while here, the method aims at faults on sensors. Besides, the method in ref.…”

“…Besides, the method in ref. [33] randomly chooses the observable functions, and the PCA technique is used to achieve a reduced‐order approximation of the Koopman operator; whereas, in the current work, a deep neural network architecture is adopted to learn a more compact closed‐form representation of the Koopman operator directly from data. Moreover, the Koopman operator obtained using the current method has a relatively lower dimension since it uses a nonlinear DNN scheme to identify the Koopman eigenfunctions rather than linear techniques such as PCA.…”