  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.
