Quasi-Exponential Convergence of Higher-Order Serial Differentiators Under Bounded Input Perturbation

This paper studies Higher-Order Serial Differentiators (HOSDs) under bounded input perturbations and develops rigorous event-based convergence bounds for their estimation errors. The analysis is formulated in terms of quasi-exponential convergence (QEC), which is used here as an event-driven characterization of transient decay for switching-based differentiators. First, for a single-stage estimator subject to bounded measurement perturbations, we prove that the derivative estimation error converges to an explicit gain-dependent steady-state tube, thereby quantifying the noise–bandwidth trade-off. Second, we establish a QEC transfer property for cascaded $n$ -stage HOSDs: if the perturbation entering one stage satisfies a QEC-type bound, then the next stage inherits a corresponding practical QEC bound under a standard nonaccumulating switching condition. Third, in the nominal perturbation-free case, we provide a refined event-based proof of convergence to zero, with the transient behavior quantified within the same QEC framework. These results provide explicit and gain-tunable bounds on both transient performance and steady-state accuracy, offering practical guidance for serial differentiator design under perturbations.