Finite Sample Analysis of System Poles for Ho-Kalman Algorithm

The Ho-Kalman algorithm has been widely employed for the identification of discrete-time linear time-invariant (LTI) systems. In this paper, we investigate the pole estimation error for the Ho-Kalman algorithm based on finite input/output sample data. Building upon prior works, we derive finite sample error bounds for system pole estimation in both single-trajectory and multiple-trajectory scenarios. Specifically, we prove that, with high probability, the estimation error for an $n$-dimensional system decreases at a rate of at least $\mathcal{O}(T^{-1/2n})$ in the single-trajectory setting with trajectory length $T$, and at a rate of at least $\mathcal{O}(N^{-1/2n})$ in the multiple-trajectory setting with $N$ independent trajectories. Furthermore, we reveal that in both settings, achieving a constant estimation error requires a super-polynomial sample size in $ \max\{n/m, n/p\} $, where $n/m$ and $n/p$ denote the state-to-output and state-to-input dimension ratios, respectively. Finally, numerical experiments are conducted to validate the non-asymptotic results of system pole estimation.