ℋ∞ control design for time-delay linear systems: A rational transfer function based approach

The paper by R. H. Korogui, A. R. Fioravanti, and J. C. Geromel focuses on the H∞ control design for time-delay linear systems. The results are based on the use of a comparison system obtained via Rekasius’ substitution. As it is known from the delay systems’ literature, this transformation allows to approximate the non-rational transfer functions by rational ones. Moreover the two transfer functions coincide on the imaginary axis which is essential to detect the exact crossings of the poles from the left-hand side to the right-hand side of the complex plane or reversed. As pointed out by the authors this property is instrumental for stability analysis of time-delay systems. Based on the fact that the H∞ norm of an analytic transfer function is defined as the supremum of its maximal singular value on the imaginary axis, the matching of the rational and non-rational transfer functions will help to deal with the H∞ norm of time-delay linear systems. By recalling Theorem 1 and Corollary 1 provided in their paper [4], the authors discuss how the H∞ norm of the comparison function is a lower bound of the H∞ norm of the original time-delay transfer function and in addition that there exists, under a few classical conditions, a subinterval ensuring the bounding (12) for the H∞ norm of the time-delay