  This paper deals with the convergence time analysis of a class of fixed-time
stable systems with the aim to provide a new non-conservative upper bound for
its settling time. Our contribution is fourfold. First, we revisit the
well-known class of fixed-time stable systems, given in (Polyakov et al.,2012},
while showing the conservatism of the classical upper estimate of the settling
time. Second, we provide the smallest constant that uniformly upper bounds the
settling time of any trajectory of the system under consideration. Third,
introducing a slight modification of the previous class of fixed-time systems,
we propose a new predefined-time convergent algorithm where the least upper
bound of the settling time is set a priori as a parameter of the system. At
last, predefined-time controllers for first order and second order systems are
introduced. Some simulation results highlight the performance of the proposed
scheme in terms of settling time estimation compared to existing methods.
