  Algorithms having uniform convergence with respect to their initial condition
(i.e., with fixed-time stability) are receiving increasing attention for
solving control and observer design problems under time constraints. However,
we still lack a general methodology to design these algorithms for high-order
perturbed systems when we additionally need to impose a user-defined
upper-bound on their settling time, especially for systems with perturbations.
Here, we fill this gap by introducing a methodology to redesign a class of
asymptotically, finite- and fixed-time stable systems into non-autonomous
fixed-time stable systems with a user-defined upper-bound on their settling
time. Our methodology redesigns a system by adding time-varying gains. However,
contrary to existing methods where the time-varying gains tend to infinity as
the origin is reached, we provide sufficient conditions to maintain bounded
gains. We illustrate our methodology by building fixed-time online
differentiators with user-defined upper-bound on their settling time and
bounded gains.
