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arXiv 2501.16639eess.SYcs.SY
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开环子空间辨识方法的有限样本分析

Finite Sample Analysis of Open-loop Subspace Identification Methods

Jiabao He, Ingvar Ziemann, Cristian R. Rojas, S. Joe Qin, Håkan Hjalmarsson

AI总结:

本文针对开环子空间辨识方法(SIMs)提出有限样本分析,通过三步过程利用 ARX 模型与加权 SVD 推导系统矩阵误差界限,证明了所需数据量并推广至多种 SIMs 变体。

摘要:

子空间辨识方法(SIMs)以其针对 MIMO 系统的简单参数化和稳健的数值特性而闻名。然而,对 SIMs 进行全面的统计分析仍然是一个开放问题。遵循 SIMs 中通常使用的三步过程,本研究提出了开环 SIMs 的有限样本分析。在步骤 1 中,我们从一个简约的 SIM 开始。利用对单个 ARX 模型的最新分析,我们获得了由一组 ARX 模型构造的类 Hankel 矩阵的联合误差界限。步骤 2 涉及通过加权奇异值分解(SVD)进行模型降阶,其中我们使用 SVD 的稳健性结果分别获得扩展可控性和可观测性矩阵的误差界限。最后的步骤 3 侧重于推导系统矩阵的误差界限,其中研究了两种不同的实现算法,即 MOESP 类型和 CVA 类型。我们的结果不仅与经典的渐近结果一致,而且还展示了需要多少数据才能以高概率保证所需的误差界限。所提出的方法推广了相关的有限样本分析,并广泛适用于 SIMs 的许多变体。

英文摘要:

Subspace identification methods (SIMs) are known for their simple parameterization for MIMO systems and robust numerical properties. However, a comprehensive statistical analysis of SIMs remains an open problem. Following a three-step procedure generally used in SIMs, this work presents a finite sample analysis for open-loop SIMs. In Step 1 we begin with a parsimonious SIM. Leveraging a recent analysis of an individual ARX model, we obtain a union error bound for a Hankel-like matrix constructed from a bank of ARX models. Step 2 involves model reduction via weighted singular value decomposition (SVD), where we use robustness results for SVD to obtain error bounds on extended controllability and observability matrices, respectively. The final Step 3 focuses on deriving error bounds for system matrices, where two different realization algorithms, the MOESP type and the CVA type, are studied. Our results not only agree with classical asymptotic results, but also show how much data is needed to guarantee a desired error bound with high probability. The proposed method generalizes related finite sample analyses and applies broadly to many variants of SIMs.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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