AlgDiff

AlgDiff: A Python class that provides all necessary tools for the design, analysis, and discretization of algebraic differentiators. An interface to Matlab is also provided. This implementation was released as part of the survey [1]. A tutorial for algebraic differentiators and this package can be found in [6].

The toolbox is licensed under the BSD-3-Clause License, which is suitable for both academic and industrial/commercial purposes.

This code has been created for research purposes at the Chair of Systems Theory and Control Engineering of Saarland University, Germany. We apply algebraic differentiators to solve different problems related to control theory and signal processing: Parameter estimation, feedback control, fault detection and fault tolerant control, model-free control ...

Table of Contents

• Motivation
• On algebraic differentiators
• GUI
• Prerequisites for the implementation
• Installation
• How to use the implementation
• Troubleshooting
• Documentation
• Questions & Contact
• License
• References

Motivation

Estimating the derivatives of noisy signals is of paramount importance in many fields of engineering and applied mathematics. It is, however, a longstanding ill-posed and challenging problem, in the sense that a small error in measurement data can induce a significant error in the estimated derivatives.

Figure 1 shows the results of the numerical estimation of the first time derivative of a noisy signal based on an algebraic differentiator on the one hand and the simple difference quotient rule on the other. This simulation shows the excellent performance of this numerical differentiation approach. | | |:--:| | Figure 1. Numerical differentiation of a noisy signal using a simple difference quotient on the one hand and an algebraic differentiator on the other |

On algebraic differentiators

Algebraic differentiators have been derived and discussed in the systems and control theory community. The initial works based on differential-algebraic methods have been developed by Mboup, Join, and Fliess in [2]. These numerical, non-asymptotic approximation approaches for higher-order derivatives of noisy signals are well suited for real-time embedded systems. A historical overview and a detailed discussion of these differentiators and their time-domain and frequency-domain properties are given in the survey [1].

The approximation-theoretic derivation recalled in the survey [1] permits the interpretation of the estimation process by the following three steps illustrated in the figure below stemming from [1]:

1. Projection: At time $t$, the sough $n$-th order time derivative $y^{(n)}$ over the interval $I{T}(t)$ is projected onto the space of polynomials of degree $\mathrm{N}$. This yields the polynomial $p\mathrm{N}$ depicted in the left and middle part of Figure 2.
2. Evaluation: The polynomial $p\mathrm{N}$ is evaluated at $t-\deltat$, which gives an estimate ${\hat{y}^{(n)}(t)=p{N}(t-\deltat)}$ for the derivative $y^{(n)}(t)$ as depicted in the central part of Figure 2. Choosing the delay to be the largest root of a special Jacobi polynomial increases the approximation order by 1 with a minimal delay. Alternatively, a delay-free estimation or even a prediction of the future derivative might be selected, at the cost of a reduced accuracy.
3. Repetition: The first two steps are repeated at each discrete time instant $t_i$ while keeping the parameters of the differentiator constant. This yields the estimate $\hat{y}^{(n)}$ depicted in the right part of the Figure 2.

| | |:--:| | Figure 2. Three-step process of the estimation of the $n$-th order derivative ${y^{(n)}:t\mapsto y^{(n)}(t)}$ of a signal $y:t\mapsto y(t)$ using algebraic differentiators (figure from [1]) |

Algebraic differentiators can be interpreted as linear time-invariant filters with a finite-duration impulse response. Figure 3 visualizes the online estimation process of the first derivative of a noisy signal. The filter window, the buffered signal, and the filter kernel can be clearly seen.

| | |:--:| | Figure 3. Visualization of the online estimation of the first derivative a noisy signal using an algebraic differentiator.|

These filters can be approximated as lowpass filters with a known cutoff frequency and a stopband slope. Figure 4 presents the amplitude and phase spectra of two exemplary filters. The lowpass approximation is also shown. | | |:--:| | Figure 4. Amplitude and phase spectra of two different filters and the corresponding lowpass approximation of the amplitude spectrum |

See [1], [3], [4], and [5] for more details on the parametrization of these differentiators.

GUI

Since Version 1.1 a GUI is provided. Executable files for Linux and Windows operating systems are provided and do not require the installation of additional software. The binary files of the GUIs for different operating systems can be downloaded from the latest release page. Neither Python not Matlab have to be installed to start designing algebraic differentiators, get discrete filter coefficients, and estimate derivatives. This GUI can be used to plot relevant data (impulse and step responses, amplitude and phase spectra, estimated derivatives, ...), display relevant properties of the differentiators (estimation delay, cutoff frequency, window length, discretization effects, ...), and load measured signals for the estimation of their derivatives without a single line of code. Relevant properties, signals, spectra, and discrete filter coefficients can be exported for further processing. For testing the import of measurement data, a file has been provided in the folder DataForGUI.

| | |:--:| | Figure 5. GUI for the interactive design, analysis, and use of algebraic differentiators |

Prerequisites for the implementation

The code is implemented in Python 3. To use all functionalities the required packages are given in the requirements.txt file. The examples implemented in Python are written in jupyter notebooks and require the packages jupyterlatexenvs for the generation of useful documentations and matplotlib for the creation of plots. The functions in the toolbox can also be used in Matlab for which different examples are also included. Check the Matlab documentation for more details on the compatibility of your Matlab version with Python.

Installation

To use AlgDiff using pip run the following in the command line:

$ pip install AlgDiff

How to use the implementation

The contribution of this implementation is an easy to use framework for the design and discretization of algebraic differentiators to achieve desired filter characteristics, i.e., to specify the cutoff frequency and the stopband slope. The file algebraicDifferentiator.py implements the class AlgebraicDifferentiator. This class contains all necessary functions for the design, analysis, and discretization of the differentiators.

Different examples are provided as jupyter notebooks and Matlab code in the following: * A quick start in a jupyter notebook available also as an HTML file * A detailed jupyter notebook available also as an HTML file * The simultaneous elimination of a harmonic disturbance and approximation of derivatives is demonstrated in the jupyter notebook available also as an HTML file * A quick start in Matlab * A Matlab code with several examples

Troubleshooting

A list of known issues and fixes is also provided.

Documentation

A detailed documentation for all the functions is also provided in the webpage of the project.

Questions & Contact

Feel free to contact A. Othmane in case of suggestions or questions via email by amine.othmane [at] uni-saarland [dot] de.

License

BSD 3-Clause "New" or "Revised" License, see License-file.

References

M. Mboup, C. Join, and M. Fliess, "Numerical differentiation with annihilators in noisy environment", Numerical Algorithms, 50 (4), 439–467, 2009, https://doi.org/10.1007/s11075-008-9236-1

M. Mboup and S. Riachy, "Frequency-domain analysis and tuning of the algebraic differentiators," Int. J. Control , 91 (9), 2073–2081, 2018, https://doi.org/10.1080/00207179.2017.1421776

A. Othmane and J. Rudolph "AlgDiff: an open source toolbox for the design, analysis and discretisation of algebraic differentiators", at - Automatisierungstechnik https://doi.org/10.1515/auto-2023-0035 https://arxiv.org/abs/2308.04888