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This paper examines the robust (strong) H-infinity norm of a lineartime-invariant system with discrete delays. The considered system is subject toreal-valued, structured, Frobenius norm bounded uncertainties on thecoefficient matrices. The robust H-infinity norm is the worst case value of theH-infinity norm over the realisations of the system and hence an importantmeasure of robust performance in control engineering. However this robustH-infinity norm is a fragile measure, as for a particular realization of theuncertainties the H-infinity norm might be sensitive to arbitrarily smallperturbations on the delays. Therefore, we introduce the robust strongH-infinity norm, inspired by the notion of strong stability of delaydifferential equations of neutral type, which takes into account both theperturbations on the system matrices and infinitesimal small delayperturbations. This quantity is a continuous function of the nominal systemparameters and delays. The main contribution of this work is the introductionof a relation between this robust strong H-infinity norm and the thepseudo-spectrum of an associated singular delay eigenvalue problem. Thisrelation is subsequently employed in a novel algorithm for computing the robuststrong H-infinity norm of uncertain time-delay systems. Both the theoreticalresults and the algorithm are also generalized to systems with uncertainties onthe delays, and systems described by a class of delay differential algebraicequations.

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