Data Driven Estimation of Stochastic Switched Linear Systems of Unknown Order
Tuhin Sarkar Alexander Rakhlin Munther Dahleh TS,AR,MD are with Massachusetts Institute of Technology, Cambridge, MA 02139 (email: tsarkar,rakhlin,
Abstract
We address the problem of learning the parameters of a mean square stable switched linear systems (SLS) with unknown latent space dimension, or order, from its noisy input–output data. In particular, we focus on learning a good lower order approximation of the underlying model allowed by finite data. Motivated by subspace-based algorithms in system theory, we construct a Hankel-like matrix from finite noisy data using ordinary least squares. Such a formulation circumvents the non-convexities that arise in system identification, and allows for accurate estimation of the underlying SLS as data size increases. Since the model order is unknown, the key idea of our approach is model order selection based on purely data dependent quantities. We construct Hankel-like matrices from data of dimension obtained from the order selection procedure. By exploiting tools from theory of model reduction for SLS, we obtain suitable approximations via singular value decomposition (SVD) and show that the system parameter estimates are close to a balanced truncated realization of the underlying system with high probability.
中文速览
均方稳定切换线性系统（switched linear systems, SLS）在控制、时序分析等领域广泛应用，但当系统阶数未知时，如何从有限噪声数据中可靠地辨识其参数是一个尚未解决的难题。本文提出一种基于子空间方法的数据驱动算法：先用普通最小二乘法从观测的输入-输出-切换序列构造类Hankel矩阵，再设计一套纯数据驱动的模型阶数选择准则，在估计误差与截断误差之间取得平衡，从而确定合适的有限维近似阶数；最后对所构造的有限时间Hankel估计量做奇异值分解（SVD），借助平衡截断（balanced truncation）理论提取系统参数。理论分析表明，随着样本量增大，Hankel矩阵估计误差以$\widetilde{\mathcal{O}}(N_S^{-1/\Delta_s})$的速率收敛，所得参数估计以高概率逼近真实系统的低阶平衡截断实现，且恢复精度仅依赖目标阶次对应的Hankel奇异值而非更小的奇异值。这是首个在模型阶数未知条件下给出SLS辨识有限样本复杂度保证的工作，为切换系统的数据驱动建模与控制提供了坚实的理论基础。
原文 arXiv:1909.04617;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1909.04617v2