Finite Sample Analysis of Subspace Identification for Stochastic Systems

The subspace identification method (SIM) has become a widely adopted approach for the identification of discrete-time linear time-invariant (LTI) systems. In this paper, we derive finite sample high-probability error bounds for the system matrices $A,C$, the Kalman filter gain $K$ and the estimation of system poles. Specifically, we demonstrate that, ignoring the logarithmic factors, for an $n$-dimensional LTI system with no external inputs, the estimation error of these matrices decreases at a rate of at least $\mathcal{O}(\sqrt{1/N})$, while the estimation error of the system poles decays at a rate of at least $\mathcal{O}(N^{-1/2n})$, where $N$ represents the number of sample trajectories. Furthermore, we reveal that achieving a constant estimation error requires a super-polynomial sample size in $n/m$, where $n/m$ denotes the state-to-output dimension ratio. Finally, numerical experiments are conducted to validate the non-asymptotic results.