Exponential Convergence Analysis of Serial Differentiators for Robust and Accurate Estimation of High-Order Derivatives

This paper presents a rigorous mathematical analysis of the higher-order serial differentiator (HOSD), originally proposed by the authors for estimating time derivatives of time-varying signals. Unlike previous studies that partially relied on numerical simulations to demonstrate asymptotic convergence of the estimation error, this work provides a complete analytical proof that both first- and higher-order derivative estimates converge exponentially fast to an arbitrarily small neighborhood of zero. Furthermore, it is analytically shown that replacing the discontinuous sign function in the differentiator dynamics with a smooth saturation function effectively mitigates chattering while preserving convergence to a small neighborhood around zero. The proposed analysis significantly strengthens the theoretical foundation of the HOSD and enhances its robustness and practical applicability in real-time control and estimation systems.