Finite Sample Performance Analysis of MIMO Systems Identification

TL;DR

This paper analyzes the finite sample identification performance of MIMO systems, revealing ill-conditioning issues in common algorithms and establishing fundamental limits on pole estimation accuracy as system dimensions grow.

Contribution

It demonstrates the ill-conditioning of Ho-Kalman and MOESP algorithms for large MIMO systems and derives a fundamental sample complexity limit based on the Cramér-Rao bound.

Findings

Ho-Kalman and MOESP algorithms become ill-conditioned for large n/m or n/p ratios.

The sample complexity for unbiased pole estimation explodes superpolynomially with system size.

Numerical results confirm the theoretical ill-conditioning and fundamental limits.

Abstract

This paper is concerned with the finite sample identification performance of an n dimensional discrete-time Multiple-Input Multiple-Output (MIMO) Linear Time-Invariant system, with p inputs and m outputs. We prove that the widely-used Ho-Kalman algorithm and Multivariable Output Error State Space (MOESP) algorithm are ill-conditioned for MIMO systems when n/m or n/p is large. Moreover, by analyzing the Cra\'mer-Rao bound, we derive a fundamental limit for identifying the real and stable (or marginally stable) poles of MIMO system and prove that the sample complexity for any unbiased pole estimation algorithm to reach a certain level of accuracy explodes superpolynomially with respect to n/(pm). Numerical results are provided to illustrate the ill-conditionedness of Ho-Kalman algorithm and MOESP algorithm as well as the fundamental limit on identification.