Anisotropic Norm Bounded Real Lemma for Linear Discrete Time Varying Systems

TL;DR

This paper introduces an anisotropic norm bounded real lemma for linear discrete time varying systems, providing a state-space criterion to evaluate worst-case disturbance attenuation under statistical uncertainty in noise distribution.

Contribution

It extends the classical Bounded Real Lemma to account for anisotropic noise with uncertain distribution, incorporating an entropy-based measure of noise deviation.

Findings

Provides a state-space inequality involving Riccati equations

Extends H-infinity control theory to anisotropic noise

Offers a criterion for worst-case disturbance attenuation

Abstract

We consider a finite horizon linear discrete time varying system whose input is a random noise with an imprecisely known probability law. The statistical uncertainty is described by a nonnegative parameter a which constrains the anisotropy of the noise as an entropy theoretic measure of deviation of the actual noise distribution from Gaussian white noise laws with scalar covariance matrices. The worst-case disturbance attenuation capabilities of the system with respect to the statistically uncertain random inputs are quantified by the a-anisotropic norm which is an appropriately constrained operator norm of the system. We establish an anisotropic norm bounded real lemma which provides a state-space criterion for the a-anisotropic norm of the system not to exceed a given threshold. The criterion is organized as an inequality on the determinants of matrices associated with a difference Riccati equation and extends the Bounded Real Lemma of the H-infinity-control theory. We also provide a necessary background on the anisotropy-based robust performance analysis.