Finding the delay that maximizes the convergence rate of a class of LTI systems

We present a framework for determining the time delay that maximizes the convergence rate of linear time-invariant (LTI) systems with a single constant delay. By exploiting the Lambert W-based representation of the characteristic roots, we prove that the functions associated with the system eigenvalues and governing the convergence rate are unimodal and intersect pairwise at most once. These properties restrict the optimal delay to a finite set of candidate points consisting of their minima and intersection points. We provide procedures for computing these points to a prescribed numerical tolerance and introduce an algorithm to identify the global optimum without exhaustively evaluating all candidates. The method is illustrated through consensus and formation-control applications and assessed against standard characteristic-root solvers for time-delayed systems.