Data-driven Fault Diagnosis of Nonlinear Systems With Parameter Uncertainty Using Deep Koopman Operator and Weighted Window Extended Dynamic Mode Decomposition

Abstract

In this study, we propose a data-driven fault diagnosis method for nonlinear systems with parameter uncertainty using Koopman operator. The Koopman operator is an infinite-dimensional linear operator that transforms a nonlinear dynamical system to a high-dimensional linear system. Using this property, we obtain an equivalent linear system to detect and identify the fault situation by analyzing the system matrices A and B. In this paper, a deep Koopman operator is proposed to find an observable function automatically by leveraging the capability of deep neural networks. A weighted window extended dynamic mode decomposition (WW-EDMD) is used to obtain the Koopman operator through a recursive procedure reducing computation time and memory usage. A forgetting factor is also implemented to enhance the fault detection ability, giving a higher weight to the latest data. To detect a loss of effectiveness (LoE) fault under a parameter uncertainty, the equivalent linear model is updated at each time, and if the norm of the input matrix B is less than the designed threshold, the LoE fault is detected and identified. The results of the numerical simulation show that the proposed method has a better fault detection capability than the method using window extended dynamic mode decomposition that only updates the matrix B under parameter variation.