Fundamental Limit on SISO System Identification

This paper is concerned with the fundamental limit on the identification of discrete-time SISO (Single Input Single Output) system, where the diagonal canonical form of the system is inferred from a finite number of input/output sample trajectories. Through the analysis of the Fisher information matrix used in Cramér-Rao bound, we show that the sample complexity of the identification problem using any unbiased estimator explodes superpolynomially with respect to system dimension in the average sense, assuming that the eigenvalues of the system matrix are uniformly distributed. Furthermore, we extend our result to the widely applied Ho-Kalman algorithm and prove that the algorithm is ill-conditioned for high dimensional SISO systems. Numerical results further demonstrate the conclusion of this paper.