A pseudo-spectra based characterisation of the robust strong H-infinity norm of time-delay systems with real-valued and structured uncertainties

TL;DR

This paper introduces a new measure called the robust strong H-infinity norm for uncertain time-delay systems, relating it to pseudo-spectra and providing a novel computational algorithm, enhancing robustness analysis in control systems.

Contribution

It develops a relation between the robust strong H-infinity norm and pseudo-spectra, and proposes a new algorithm for its computation in uncertain delay systems.

Findings

The robust strong H-infinity norm is continuous with respect to system parameters.

A novel relation between the norm and pseudo-spectrum is established.

An algorithm for computing the norm is proposed and validated.

Abstract

This paper examines the robust (strong) H-infinity norm of a linear time-invariant system with discrete delays. The considered system is subject to real-valued, structured, Frobenius norm bounded uncertainties on the coefficient matrices. The robust H-infinity norm is the worst case value of the H-infinity norm over the realisations of the system and hence an important measure of robust performance in control engineering. However this robust H-infinity norm is a fragile measure, as for a particular realization of the uncertainties the H-infinity norm might be sensitive to arbitrarily small perturbations on the delays. Therefore, we introduce the robust strong H-infinity norm, inspired by the notion of strong stability of delay differential equations of neutral type, which takes into account both the perturbations on the system matrices and infinitesimal small delay perturbations. This quantity is a continuous function of the nominal system parameters and delays. The main contribution of this work is the introduction of a relation between this robust strong H-infinity norm and the the pseudo-spectrum of an associated singular delay eigenvalue problem. This relation is subsequently employed in a novel algorithm for computing the robust strong H-infinity norm of uncertain time-delay systems. Both the theoretical results and the algorithm are also generalized to systems with uncertainties on the delays, and systems described by a class of delay differential algebraic equations.