There is an increasing interest in designing differentiators, which converge
exactly before a prespecified time regardless of the initial conditions, i.e.,
which are fixed-time convergent with a predefined Upper Bound of their Settling
Time (UBST), due to their ability to solve estimation and control problems with
time constraints. However, for the class of signals with a known bound of their
(n+1)-th time derivative, the existing design methodologies are either only
available for first-order differentiators, yielding a very conservative UBST,
or result in gains that tend to infinity at the convergence time. Here, we
introduce a new methodology based on time-varying gains to design
arbitrary-order exact differentiators with a predefined UBST. This UBST is a
priori set as one parameter of the algorithm. Our approach guarantees that the
UBST can be set arbitrarily tight, and we also provide sufficient conditions to
obtain exact convergence while maintaining bounded time-varying gains.
Additionally, we provide necessary and sufficient conditions such that our
approach yields error dynamics with a uniformly Lyapunov stable equilibrium.
Our results show how time-varying gains offer a general and flexible
methodology to design algorithms with a predefined UBST