Canonical Bayesian Linear System Identification

2025-07-15

AI Summary

Standard Bayesian approaches to linear time-invariant (LTI) system identification suffer from parameter non-identifiability, leading to multimodal posteriors, inefficient inference, and poor interpretability. To address this, we propose a canonical-form-embedded Bayesian framework: it eliminates redundant degrees of freedom via minimal parameterization, enforces stability constraints rigorously, and constructs structure-aware priors leveraging invariance properties of transfer functions and eigenvalues. This formulation fully alleviates posterior multimodality and reestablishes connections between Bayesian inference and frequentist asymptotic theory. Empirical results demonstrate that, under limited data regimes, the method substantially improves MCMC sampling efficiency, yields more robust uncertainty quantification, and produces unimodal, interpretable posteriors—while maintaining theoretical rigor and practical applicability for engineering systems identification.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Standard Bayesian approaches for linear time-invariant (LTI) system identification are hindered by parameter non-identifiability; the resulting complex, multi-modal posteriors make inference inefficient and impractical. We solve this problem by embedding canonical forms of LTI systems within the Bayesian framework. We rigorously establish that inference in these minimal parameterizations fully captures all invariant system dynamics (e.g., transfer functions, eigenvalues, predictive distributions of system outputs) while resolving identifiability. This approach unlocks the use of meaningful, structure-aware priors (e.g., enforcing stability via eigenvalues) and ensures conditions for a Bernstein--von Mises theorem -- a link between Bayesian and frequentist large-sample asymptotics that is broken in standard forms. Extensive simulations with modern MCMC methods highlight advantages over standard parameterizations: canonical forms achieve higher computational efficiency, generate interpretable and well-behaved posteriors, and provide robust uncertainty estimates, particularly from limited data.

Problem

• Resolves parameter non-identifiability in Bayesian LTI system identification
• Ensures inference captures invariant system dynamics via canonical forms
• Enables structure-aware priors and robust uncertainty estimates efficiently

Innovation

• Embed canonical LTI forms in Bayesian framework
• Use minimal parameterizations for efficient inference
• Apply structure-aware priors for stable results