用规范形式解决线性系统辨识的参数不可辨识问题
Canonical Bayesian Linear System Identification
- 将系统规范形式嵌入贝叶斯框架，消除参数歧义
- 在有限数据下仍能生成稳定、可解释的后验分布
- 适合需要可靠不确定性估计的控制系统建模
标准贝叶斯方法在时不变线性系统辨识中受限于参数不可辨识性，导致后验分布复杂且多峰，使推断效率低下。本文通过在贝叶斯框架内引入线性系统的规范形式，严格证明该最小参数化可完整捕获所有不变系统动态（如传递函数、特征值、输出预测分布），同时解决辨识问题。该方法支持有意义的结构感知先验（如通过特征值强制稳定性），并满足 Bernstein--von Mises 定理条件，重建贝叶斯与频率学派大样本渐近之间的联系。使用现代 MCMC 方法的大量仿真表明，相比标准参数化，规范形式在计算效率、后验可解释性及不确定性估计方面均有显著优势，尤其在数据有限时表现更稳健。
原文摘要 · Abstract (English)
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.
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