Outliers and Random Noises in System Identification: a Compressed Sensing Approach

TL;DR

This paper introduces a robust system identification method using compressed sensing techniques with Toeplitz matrices, effectively correcting sparse outliers and random noise, and providing performance guarantees.

Contribution

It develops a novel approach applying Toeplitz structured matrices for sparse error correction in system identification, with proven error correction thresholds and asymptotic accuracy.

Findings

Toeplitz matrices can correct a significant percentage of outliers.

Estimation error approaches zero with increasing data when noise density is non-vanishing.

No probabilistic assumptions are needed on outliers.

Abstract

In this paper, we consider robust system identification under sparse outliers and random noises. In this problem, system parameters are observed through a Toeplitz matrix. All observations are subject to random noises and a few are corrupted with outliers. We reduce this problem of system identification to a sparse error correcting problem using a Toeplitz structured real-numbered coding matrix. We prove the performance guarantee of Toeplitz structured matrix in sparse error correction. Thresholds on the percentage of correctable errors for Toeplitz structured matrices are established. When both outliers and observation noise are present, we have shown that the estimation error goes to 0 asymptotically as long as the probability density function for observation noise is not "vanishing" around 0. No probabilistic assumptions are imposed on the outliers.