Control theory-informed machine learning aided stable kernel representation for nonlinear system monitoring

This paper proposes a monitoring scheme for nonlinear dynamic systems characterized by uncertainties and partially known models. We first establish a monitoring framework grounded in nonlinear control theory, specifically utilizing the normalized stable kernel representation (SKR). By analyzing the associated Hamiltonian system, we identify the conditions under which the normalized SKR can be constructed, ensuring optimal uncertainty estimation via the solution of a corresponding Hamilton-Jacobi Equation (HJE). These conditions further lead to the formulation of residual-based monitoring indicators. Subsequently, we present a data-driven implementation of this framework that integrates control-theoretic principles with machine learning, termed control theory-informed machine learning (CTIML). In this approach, neural networks are employed to approximate the HJE solution and the corresponding observer gain. These networks are trained using fault-free operational data, incorporating loss terms designed to enforce the satisfaction of the HJE and related theoretical properties, thus preserving crucial system characteristics like the lossless property. The resulting CTIML-based normalized SKR generates residuals and HJE satisfaction metrics used for online monitoring. Finally, the efficacy of the proposed methodology is validated through a case study.