Bandwidth (computing)

Computer network

Mathematics

Economic growth

Euler's formula

Euler method

Partial differential equation

Algorithm

Differentiator

Explicit and implicit methods

Mathematical analysis

Economics

Exact differential equation

Convergence (economics)

Computer science

Discretization

Backward Euler method

First-order partial differential equation

International Journal of Robust and Nonlinear Control

A complete review of the known differentiators and their time‐discretizations have been addressed in this study. To resolve the drawbacks of the explicit (forward Euler) discretization, which is commonly utilized in sliding‐mode‐based differentiators, implicit time discretization methods are proposed to handle the set‐valued functions. The proposed schemes are supported by some analytical results to show their crucial properties, for example, finite‐time convergence, exactness, invariant sliding‐surface, chattering elimination, insensitivity to the gains during the sliding‐phase, and the well‐posedness. The causal implementation of the proposed implicit schemes has been addressed clearly and supported by flowcharts. Semi‐implicit schemes are also presented to provide a compromise between the performance and the ease of implementation. Finally, an exhaustive comparison is made using numerical experiments among 25 different state‐of‐the‐art differentiators to evaluate the behaviors of the differentiators facing the noise and initial error. Realistic conditions are considered in the simulations to provide useful information based on practical conditions. General conclusions are that implicit discretizations can supersede explicit and semi‐implicit ones.

Time‐discretizations of differentiators: Design of implicit algorithms and comparative analysis