An arbitrary-order predefined-time exact differentiator for signals with exponential growth bound

Constructing differentiation algorithms with a fixed-time convergence and a predefined Upper Bound on their Settling Time (\textit{UBST}), i.e., predefined-time differentiators, is attracting attention for solving estimation and control problems under time constraints. However, existing methods are limited to signals having an $n$-th Lipschitz derivative. Here, we introduce a general methodology to design $n$-th order predefined-time differentiators for a broader class of signals: for signals, whose $(n+1)$-th derivative is bounded by a function with bounded logarithmic derivative, i.e., whose $(n+1)$-th derivative grows at most exponentially. Our approach is based on a class of time-varying gains known as Time-Base Generators (\textit{TBG}). The only assumption to construct the differentiator is that the class of signals to be differentiated $n$-times have a $(n+1)$-th derivative bounded by a known function with a known bound for its $(n+1)$-th logarithmic derivative. We show how our methodology achieves an \textit{UBST} equal to the predefined time, better transient responses with smaller error peaks than autonomous predefined-time differentiators, and a \textit{TBG} gain that is bounded at the settling time instant.