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Online identification of nonlinear time-varying systems with uncertain information

He Ren, Gaowei Yan, Hang Liu, Lifeng Cao, Zhijun Zhao, Gang Dang

TL;DR

BRSL addresses online identification of nonlinear time-varying systems with interpretable symbolic models and quantified uncertainty. It formulates online symbolic learning as a Bayesian regression problem with sparse horseshoe priors, yielding Gaussian posteriors and automatic model selection. A recursive forgetting-factor scheme with precise recursive conditions ensures posterior well-posedness and robustness, with convergence guarantees under persistent excitation. Numerical case studies, including synthetic sparse data and a Lorenz-based system, validate interpretable probabilistic predictions and robust online learning for digital-twin applications.

Abstract

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.

Online identification of nonlinear time-varying systems with uncertain information

TL;DR

BRSL addresses online identification of nonlinear time-varying systems with interpretable symbolic models and quantified uncertainty. It formulates online symbolic learning as a Bayesian regression problem with sparse horseshoe priors, yielding Gaussian posteriors and automatic model selection. A recursive forgetting-factor scheme with precise recursive conditions ensures posterior well-posedness and robustness, with convergence guarantees under persistent excitation. Numerical case studies, including synthetic sparse data and a Lorenz-based system, validate interpretable probabilistic predictions and robust online learning for digital-twin applications.

Abstract

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.
Paper Structure (10 sections, 1 theorem, 57 equations, 8 figures)

This paper contains 10 sections, 1 theorem, 57 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Methodology
  4. Bayesian Regression-Based Symbolic Learning
  5. Recursive condition of BRSL
  6. Convergence
  7. Numerical simulations
  8. Case study 1
  9. Case study 2
  10. Conclusions

Key Result

Lemma 1

Under the conditions of Theorem thm3, there exists a constant ${{\alpha }_{2}}>0$ such that then the signal $\boldsymbol{\Lambda }\left( t \right)$ is said to satisfy the PE condition.

Figures (8)

  • Figure 1: Absolute error statistics for non-zero parameters
  • Figure 2: Absolute error statistics for zero-valued parameters
  • Figure 3: The prediction of BRSL before and after the change of working conditions
  • Figure 4: The convergence of the means and variances in the recursive process
  • Figure 5: SHAP analysis of feature contributions for $x_1$
  • ...and 3 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Remark 4