Finite Sample Analysis of Open-loop Subspace Identification Methods

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.