On the Approximation of Toeplitz Operators for Nonparametric $\mathcal{H}_\infty$-norm Estimation

TL;DR

This paper provides sharp non-asymptotic bounds for approximating the $ ext{H}_ ext{infty}$-norm of a stable SISO LTI system using Toeplitz operator truncations, highlighting differences with parametric FIR identification.

Contribution

It establishes precise bounds on the size of Toeplitz matrix truncations necessary for accurate $ ext{H}_ ext{infty}$-norm estimation and compares nonparametric and parametric methods.

Findings

Sharp bounds on Toeplitz truncation length for norm approximation

Demonstration of cases where nonparametric methods are worse than parametric

Construction of FIR filters illustrating the bounds' sharpness

Abstract

Given a stable SISO LTI system $G$, we investigate the problem of estimating the $\mathcal{H}_\infty$-norm of $G$, denoted $||G||_\infty$, when $G$ is only accessible via noisy observations. Wahlberg et al. recently proposed a nonparametric algorithm based on the power method for estimating the top eigenvalue of a matrix. In particular, by applying a clever time-reversal trick, Wahlberg et al. implement the power method on the top left $n \times n$ corner $T_n$ of the Toeplitz (convolution) operator associated with $G$. In this paper, we prove sharp non-asymptotic bounds on the necessary length $n$ needed so that $||T_n||$ is an $\varepsilon$-additive approximation of $||G||_\infty$. Furthermore, in the process of demonstrating the sharpness of our bounds, we construct a simple family of finite impulse response (FIR) filters where the number of timesteps needed for the power method is arbitrarily worse than the number of timesteps needed for parametric FIR identification via least-squares to achieve the same $\varepsilon$-additive approximation.