A redesign methodology generating predefined-time differentiators with bounded time-varying gains

There is an increasing interest in designing differentiators, which converge exactly before a prespecified time regardless of the initial conditions, that is, 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 (Formula presented.) th time derivative, the existing design methodologies yield 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 (TVG) 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 TVG. Additionally, we provide necessary and sufficient conditions such that our approach yields error dynamics with a uniformly Lyapunov stable equilibrium. Our results show how TVG offer a general and flexible methodology to design algorithms with a predefined UBST.