A note on reachable set bounding for delayed systems with polytopic uncertainties

The problem of reachable set estimation for linear delayed systems subject to polytopic uncertainties is revisited in this paper. The maximal Lyapunov–Krasovskii functional, combined with the free-weighting matrix technique, is utilized to derive a refined condition for a non-ellipsoidal reachable set bound. One of our discoveries is that the number of Lyapunov matrices can be chosen to be greater than the number of vertices for the uncertain polytope. Moreover, the choice of appropriate Lyapunov–Krasovskii functional candidate and the introduction of Leibniz–Newton formula lead to decoupling of the system matrices and the Lyapunov matrices. The useful term which was ignored in our previous result is retained. These treatments bring much tighter bound of the reachable set than the existing ones. Finally, the negligence in our previous paper is pointed out.