Non-asymptotic Error Analysis of Subspace Identification for Deterministic Systems

The subspace identification method (SIM) has been extensively employed in the identification of discrete-time multiple-input multiple-output (MIMO) linear time-invariant (LTI) systems. This paper focuses on the analysis of perturbation errors for the system matrices in state-space models and the corresponding system poles, under two unified SIMs, based on a single finite-length input/output sample trajectory. Specifically, we derive non-asymptotic upper bounds on these errors, providing a unified perspective across various SIM variants. Furthermore, we prove that SIMs become ill-conditioned for MIMO systems when the state-to-output dimensionality ratio $n/m$ is large, regardless of system parameters. Finally, numerical experiments are conducted to validate the non-asymptotic results and the ill-conditionedness of SIMs.