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 (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 (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 UBST equal to the predefined time, better transient responses with smaller error peaks than autonomous predefinedtime differentiators, and a TBG gain that is bounded at the settling time instant.