Online identification of nonlinear time-varying systems with uncertain information

Digital twins (DTs), serving as the core enablers for real-time monitoring and predictive maintenance of complex cyber-physical systems, impose critical requirements on their virtual models: high predictive accuracy, strong interpretability, and online adaptive capability. However, existing techniques struggle to meet these demands simultaneously: Bayesian methods excel in uncertainty quantification but lack model interpretability, while interpretable symbolic identification methods (e.g., SINDy) are constrained by their offline, batch-processing nature, which make real-time updates challenging. To bridge this semantic and computational gap, this paper proposes a novel Bayesian Regression-based Symbolic Learning (BRSL) framework. The framework formulates online symbolic discovery as a unified probabilistic state-space model. By incorporating sparse horseshoe priors, model selection is transformed into a Bayesian inference task, enabling simultaneous system identification and uncertainty quantification. Furthermore, we derive an online recursive algorithm with a forgetting factor and establish precise recursive conditions that guarantee the well-posedness of the posterior distribution. These conditions also function as real-time monitors for data utility, enhancing algorithmic robustness. Additionally, a rigorous convergence analysis is provided, demonstrating the convergence of parameter estimates under persistent excitation conditions. Case studies validate the effectiveness of the proposed framework in achieving interpretable, probabilistic prediction and online learning.