A note about high-order semi-implicit differentiation: application to a numerical integration scheme with Taylor-based compensated error

TL;DR

The paper tackles improving real-time estimation of higher-order derivatives by formulating a third-order semi-implicit differentiator (SIHD-3) that incorporates Taylor-refinement corrections to previous semi-implicit approaches. It introduces an operator framework with estimations $z_i$, error $e_1$, and convergence controls, and then integrates SIHD-3 into an observer-based NSFD numerical scheme for solving $rac{dy}{dt}=f(y,u)$. Numerical results on a sine signal and three ODE examples show ultra-low estimation errors in noiseless settings and competitive performance against Euler and Runge–Kutta methods, with flexibility introduced by the NSFD time-step $oldsymbol{llipsi}$. The work highlights practical potential for real-time robotics and other applications, while outlining future directions in noise handling, convergence analysis, and extensions to semi-implicit schemes.

Abstract

In this brief, we discuss the implementation of a third order semi-implicit differentiator as a complement of the recent work by the author that proposes an interconnected semi-implicit Euler double differentiators algorithm through Taylor expansion refinement. The proposed algorithm is dual to the interconnected approach since it offers alternative flexibility to be tuned and to be implemented in real-time processes. In particular, an application to a numerical integration scheme is presented as the Taylor refinement can be of interest to improve the global convergence. Numerical results are presented to support the rightness of the proposed method.

Figures (7)

• Figure 1: Example of SIHD3-based differentiation of a sine function without extra noise.
• Figure 2: Example of SIHD3-based differentiation of a sine function without extra noise - estimation errors.
• Figure 3: Example of SIHD3-based differentiation of a sine function including extra noise.
• Figure 4: Example of SIHD3-based differentiation of a sine function including extra noise - estimation error.
• Figure 5: Illustration of the SIHD3 scheme convergence compared with Euler forward scheme and Runge-Kutta scheme - Example 1.