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.

Two methods for the systematic parametrization of real-time derivative estimators based on recent algebraic techniques are introduced. These numerical differentiators are considered as finite impulse response filters the amplitude spectrum of which is chosen such as to compensate for relevant disturbances. The first proposed parametrization method allows for the elimination of harmonic disturbances. The second one relies on an approximation of the derivative estimators as low-pass filters of arbitrarily large order with an arbitrary cutoff frequency. The practical application of the first method is demonstrated in an experimental case study, where mechanical vibrations occur.

A model-based online fault-diagnosis scheme for an electromagnetically supported plate is presented as an example of a nonlinear and open-loop unstable system. First, residuals for sensor as well as for actuator faults are generated using algebraic derivative estimators. Then, the robust detection and isolation of step-like sensor and actuator faults is presented. The performance of the proposed algorithms is illustrated by experimental results.

A nonlinear mathematical model for the dynamics of permanent magnet synchronous machines with interior magnets is discussed. The model of the current dynamics captures saturation and dependency on the rotor angle. Based on the model, a flatness-based field-oriented closed-loop controller and a feed-forward compensation of torque ripples are derived. Effectiveness and robustness of the proposed algorithms are demonstrated by simulation results.