Quasi-Exponential Convergence of Continuous-Time Higher-Order Serial Differentiators

This paper revisits the convergence characterization of the continuous-time higher-order serial differentiator (HOSD) and clarifies that the rigorously provable property is \emph{quasi-exponential convergence} (QEC) rather than conventional exponential convergence. A complete time-domain QEC proof is given for the first stage by combining an event-to-event contraction on the switching (zero-crossing) sequence with an inter-event monotonicity argument. For higher-order stages, where each input is an internally generated switching signal, we establish a technically complete induction proof using a piecewise extension of the first-stage result, a QEC-to-QEC implication for derivatives under bounded (piecewise) second derivatives, and closure of QEC under summation. Under standard bounded-derivative assumptions and a nonaccumulating (no-Zeno) switching condition, every derivative estimation error of the HOSD is shown to converge to zero quasi-exponentially.