Survey on algebraic numerical differentiation: historical developments, parametrization, examples, and applications

Abstract

The historical developments, parametrization, and numerous applications of algebraic numerical differentiation methods initiated by Fliess, Mboup, Join, and Sira-Ramírez are presented. These numerical non-asymptotic approximation approaches for higher-order derivatives of noisy signals are suited for real-time embedded systems. The different derivation approaches are reviewed according to their historical developments. Links to numerous established numerical differentiation algorithms like the Savitzky-Golay filters are discussed. Recent tuning approaches that are reviewed facilitate a parametrization for a good estimation accuracy without relying on trial-and-error approaches, which is essential for industrial applications. The use of these guidelines is demonstrated using two concrete examples: Identification of system parameters from noisy measurements and numerical inversion of the dynamics of analogue anti-aliasing filters. Moreover, an extensive literature survey with various applications of these methods by the control community in all engineering areas is presented. The problems solved include, but are not limited to, parameter estimation, state reconstruction, feedback control, fault diagnosis, anomaly detection, fault-tolerant control, and model-free control. A MATLAB and Python toolbox implementing all necessary functions for the design, analysis, and discretization of the filters is made available, for which a link is provided. Its use is demonstrated in a short appendix.

Keywords:

• Numerical differentiation
• algebraic differentiators
• orthogonal polynomials
• linear filtering
• confluent hypergeometric function
• FIR filters

Data availability statement

A Python package implementing all functions required for the design, analysis, and discretization of algebraic differentiators can be found in Othmane (2021). An interface to MATLAB is also provided. More details on the toolbox can be found in the Appendix A. The toolbox is released under the BSD-3-Clause License.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Amine Othmane

Amine Othmane received the Bachelor of Science degree in Mechatronics from Saarland University, Saarbrücken, Germany, in 2015 and the Master of Science degree in Engineering Cybernetics from the University of Stuttgart, Stuttgart, Germany, in 2018. During his studies, he spent three months at the University of Alberta, Edmonton, Canada, and a semester at the Norwegian University of Science and Technology, Trondheim, Norway. Since 2018 he has been a Doctoral Student with the Chair of Systems Theory and Control Engineering at Saarland University, Saarbrücken, Germany, and the Laboratory of Signals and Systems at Université Paris-Saclay, CNRS, CentraleSupélec, Gif-sur-Yvette, France, within a joint doctoral project (cotutelle). His research interests include state and parameter estimation, numerical differentiation, and model-free control.

Lothar Kiltz

Lothar Kiltz received the diploma in systems and electrical engineering and the doctorate degree in mechatronics engineering from Saarland University, Saarbrücken, Germany. In 2015, he joined the corporate research and development department of ZF Friedrichshafen AG, Friedrichshafen, Germany, where he has been the head of the corporate expert team of control engineering since 2020. His research interests include modelling, analysis, optimisation, and automatic control of automotive systems.

Joachim Rudolph

Joachim Rudolph received the Doctorat degree from the Université Paris XI, Orsay, France, in 1991, and the Dr.-Ing. habil. degree from Technische Universität Dresden (TU Dresden), Dresden, Germany, in 2003. He has been a Privatdozent with TU Dresden, and was appointed a apl. Professor in 2008. Since 2009, he has been a Full Professor and the Head of the Chair of Systems Theory and Control Engineering with Saarland University, Saarbrücken, Germany. His current research interests include controller and observer design for nonlinear and infinite dimensional systems, algebraic systems theory, and the solution of demanding practical control problems.