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% Created 2021-09-01 mer. 10:05 % Created 2021-09-01 mer. 17:02
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[preprint, sort&compress]{elsarticle} \documentclass[preprint, sort&compress]{elsarticle}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -58,7 +58,7 @@ Sensor fusion \sep{} Complementary filters \sep{} \(\mathcal{H}_\infty\) synthes
\end{frontmatter} \end{frontmatter}
\section{Introduction} \section{Introduction}
\label{sec:org759f4b3} \label{sec:org2e9b248}
\label{sec:introduction} \label{sec:introduction}
Measuring a physical quantity using sensors is always subject to several limitations. Measuring a physical quantity using sensors is always subject to several limitations.
First, the accuracy of the measurement will be affected by several noise sources, such as the electrical noise of the conditioning electronics being used. First, the accuracy of the measurement will be affected by several noise sources, such as the electrical noise of the conditioning electronics being used.
@ -105,13 +105,13 @@ The synthesis method is further validated in Section~\ref{sec:application_ligo}
Section~\ref{sec:discussion} compares the proposed synthesis method with the classical mixed-sensitivity synthesis, and extends it to the shaping of more than two complementary filters. Section~\ref{sec:discussion} compares the proposed synthesis method with the classical mixed-sensitivity synthesis, and extends it to the shaping of more than two complementary filters.
\section{Sensor Fusion and Complementary Filters Requirements} \section{Sensor Fusion and Complementary Filters Requirements}
\label{sec:org1ec960d} \label{sec:org11c2c5d}
\label{sec:requirements} \label{sec:requirements}
Complementary filters provides a framework for fusing signals from different sensors. Complementary filters provides a framework for fusing signals from different sensors.
As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements. As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
These requirements are discussed in this section. These requirements are discussed in this section.
\subsection{Sensor Fusion Architecture} \subsection{Sensor Fusion Architecture}
\label{sec:org12550f0} \label{sec:orgf188ee8}
\label{sec:sensor_fusion} \label{sec:sensor_fusion}
A general sensor fusion architecture using complementary filters is shown in Fig.~\ref{fig:sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\). A general sensor fusion architecture using complementary filters is shown in Fig.~\ref{fig:sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\).
@ -138,7 +138,7 @@ Therefore, a pair of complementary filter needs to satisfy the following conditi
It will soon become clear why the complementary property is important for the sensor fusion architecture. It will soon become clear why the complementary property is important for the sensor fusion architecture.
\subsection{Sensor Models and Sensor Normalization} \subsection{Sensor Models and Sensor Normalization}
\label{sec:org09fb715} \label{sec:orgb672f70}
\label{sec:sensor_models} \label{sec:sensor_models}
In order to study such sensor fusion architecture, a model for the sensors is required. In order to study such sensor fusion architecture, a model for the sensors is required.
@ -184,7 +184,7 @@ The super sensor output is therefore equal to:
\end{figure} \end{figure}
\subsection{Noise Sensor Filtering} \subsection{Noise Sensor Filtering}
\label{sec:org9e08a61} \label{sec:org6621cf0}
\label{sec:noise_filtering} \label{sec:noise_filtering}
In this section, it is supposed that all the sensors are perfectly normalized, such that: In this section, it is supposed that all the sensors are perfectly normalized, such that:
@ -225,7 +225,7 @@ In such case, to lower the noise of the super sensor, the norm \(|H_1(j\omega)|\
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise. Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
\subsection{Sensor Fusion Robustness} \subsection{Sensor Fusion Robustness}
\label{sec:org526930e} \label{sec:orge7e30ce}
\label{sec:fusion_robustness} \label{sec:fusion_robustness}
In practical systems the sensor normalization is not perfect and condition \eqref{eq:perfect_dynamics} is not verified. In practical systems the sensor normalization is not perfect and condition \eqref{eq:perfect_dynamics} is not verified.
@ -289,7 +289,7 @@ As it is generally desired to limit the maximum phase added by the super sensor,
Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain. Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain.
\section{Complementary Filters Shaping} \section{Complementary Filters Shaping}
\label{sec:orgfd9f96b} \label{sec:orgffe483d}
\label{sec:hinf_method} \label{sec:hinf_method}
As shown in Section~\ref{sec:requirements}, the noise and robustness of the super sensor are a function of the complementary filters norms. As shown in Section~\ref{sec:requirements}, the noise and robustness of the super sensor are a function of the complementary filters norms.
Therefore, a complementary filters synthesis method that allows to shape their norms would be of great use. Therefore, a complementary filters synthesis method that allows to shape their norms would be of great use.
@ -297,7 +297,7 @@ In this section, such synthesis is proposed by writing the synthesis objective a
As weighting functions are used to represent the wanted complementary filters shapes during the synthesis, the proper design of weighting functions is discussed. As weighting functions are used to represent the wanted complementary filters shapes during the synthesis, the proper design of weighting functions is discussed.
Finally, the synthesis method is validated on an simple example. Finally, the synthesis method is validated on an simple example.
\subsection{Synthesis Objective} \subsection{Synthesis Objective}
\label{sec:orgf199dad} \label{sec:orgc020e5d}
\label{sec:synthesis_objective} \label{sec:synthesis_objective}
The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:comp_filter}. The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:comp_filter}.
@ -314,7 +314,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
\(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are carefully chosen to specify the maximum wanted norms of the complementary filters during the synthesis. \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are carefully chosen to specify the maximum wanted norms of the complementary filters during the synthesis.
\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis} \subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
\label{sec:org6bbfe23} \label{sec:org6c86cb9}
\label{sec:hinf_synthesis} \label{sec:hinf_synthesis}
In this section, it is shown that the synthesis objective can be easily expressed as a standard \(\mathcal{H}_\infty\) optimization problem and therefore solved using convenient tools readily available. In this section, it is shown that the synthesis objective can be easily expressed as a standard \(\mathcal{H}_\infty\) optimization problem and therefore solved using convenient tools readily available.
@ -370,7 +370,7 @@ There might be solutions were the objectives~\eqref{eq:hinf_cond_h1} and~\eqref{
In practice, this is however not an found to be an issue. In practice, this is however not an found to be an issue.
\subsection{Weighting Functions Design} \subsection{Weighting Functions Design}
\label{sec:org7255223} \label{sec:orgf11126f}
\label{sec:hinf_weighting_func} \label{sec:hinf_weighting_func}
Weighting functions are used during the synthesis to specify the maximum allowed norms of the complementary filters. Weighting functions are used during the synthesis to specify the maximum allowed norms of the complementary filters.
@ -419,7 +419,7 @@ An example of the obtained magnitude of a weighting function generated using \eq
\end{figure} \end{figure}
\subsection{Validation of the proposed synthesis method} \subsection{Validation of the proposed synthesis method}
\label{sec:orgca261f8} \label{sec:org254cb4b}
\label{sec:hinf_example} \label{sec:hinf_example}
The proposed methodology for the design of complementary filters is now applied on a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that: The proposed methodology for the design of complementary filters is now applied on a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
@ -490,7 +490,7 @@ This simple example illustrates the fact that the proposed methodology for compl
A more complex real life example is taken up in the next section. A more complex real life example is taken up in the next section.
\section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO} \section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
\label{sec:org00afdda} \label{sec:org87f5cba}
\label{sec:application_ligo} \label{sec:application_ligo}
Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO~\cite{matichard15_seism_isolat_advan_ligo,hua05_low_ligo}, the VIRGO~\cite{lucia18_low_frequen_optim_perfor_advan,heijningen18_low} and the KAGRA \cite[Chap. 5]{sekiguchi16_study_low_frequen_vibrat_isolat_system}. Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO~\cite{matichard15_seism_isolat_advan_ligo,hua05_low_ligo}, the VIRGO~\cite{lucia18_low_frequen_optim_perfor_advan,heijningen18_low} and the KAGRA \cite[Chap. 5]{sekiguchi16_study_low_frequen_vibrat_isolat_system}.
@ -513,7 +513,7 @@ After synthesis, the obtained FIR filters were found to be compliant with the re
However they are of very high order so their implementation is quite complex. However they are of very high order so their implementation is quite complex.
In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated on the same set of requirements. In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated on the same set of requirements.
\subsection{Complementary Filters Specifications} \subsection{Complementary Filters Specifications}
\label{sec:org4c34a41} \label{sec:orgf53acf5}
\label{sec:ligo_specifications} \label{sec:ligo_specifications}
The specifications for the set of complementary filters (\(L_1,H_1\)) used at the LIGO are summarized below (for further details, refer to~\cite{hua04_polyp_fir_compl_filter_contr_system}): The specifications for the set of complementary filters (\(L_1,H_1\)) used at the LIGO are summarized below (for further details, refer to~\cite{hua04_polyp_fir_compl_filter_contr_system}):
@ -534,7 +534,7 @@ They are physically represented in Fig.~\ref{fig:fir_filter_ligo} as well as the
\end{figure} \end{figure}
\subsection{Weighting Functions Design} \subsection{Weighting Functions Design}
\label{sec:org2098436} \label{sec:org4b996f7}
\label{sec:ligo_weights} \label{sec:ligo_weights}
The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem. The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem.
@ -551,7 +551,7 @@ The magnitudes of the weighting functions are shown in Fig.~\ref{fig:ligo_weight
\end{figure} \end{figure}
\subsection{\(\mathcal{H}_\infty\) Synthesis of the complementary filters} \subsection{\(\mathcal{H}_\infty\) Synthesis of the complementary filters}
\label{sec:org6f6d00c} \label{sec:org64068df}
\label{sec:ligo_results} \label{sec:ligo_results}
The proposed \(\mathcal{H}_\infty\) synthesis is performed on the generalized plant shown in Fig.~\ref{fig:h_infinity_robust_fusion_plant}. The proposed \(\mathcal{H}_\infty\) synthesis is performed on the generalized plant shown in Fig.~\ref{fig:h_infinity_robust_fusion_plant}.
@ -567,10 +567,10 @@ This confirms the effectiveness of the proposed synthesis method even when the c
\end{figure} \end{figure}
\section{Discussion} \section{Discussion}
\label{sec:org268cb51} \label{sec:org36bf100}
\label{sec:discussion} \label{sec:discussion}
\subsection{``Closed-Loop'' complementary filters} \subsection{``Closed-Loop'' complementary filters}
\label{sec:org5bce6c6} \label{sec:org38ca6e5}
\label{sec:closed_loop_complementary_filters} \label{sec:closed_loop_complementary_filters}
An alternative way to implement complementary filters is by using a fundamental property of the classical feedback architecture shown in Fig.~\ref{fig:feedback_sensor_fusion}. An alternative way to implement complementary filters is by using a fundamental property of the classical feedback architecture shown in Fig.~\ref{fig:feedback_sensor_fusion}.
This is for instance presented in \cite{mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas}. This is for instance presented in \cite{mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas}.
@ -667,7 +667,7 @@ The obtained ``closed-loop'' complementary filters are indeed equal to the ones
\end{figure} \end{figure}
\subsection{Synthesis of more than two Complementary Filters} \subsection{Synthesis of more than two Complementary Filters}
\label{sec:org5cb9fba} \label{sec:org16306e2}
\label{sec:hinf_three_comp_filters} \label{sec:hinf_three_comp_filters}
Some applications may require to merge more than two sensors~\cite{stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands}. Some applications may require to merge more than two sensors~\cite{stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands}.
@ -771,7 +771,7 @@ A set of \(n\) complementary filters can be shaped using the generalized plant \
\end{equation} \end{equation}
\section{Conclusion} \section{Conclusion}
\label{sec:org85a0b4e} \label{sec:org05380ec}
\label{sec:conclusion} \label{sec:conclusion}
Sensors measuring a physical quantities are always subject to limitations both in terms of bandwidth or accuracy. Sensors measuring a physical quantities are always subject to limitations both in terms of bandwidth or accuracy.
@ -785,12 +785,12 @@ Links with ``closed-loop'' complementary filters where highlighted, and the prop
Future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion. Future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion.
\section*{Acknowledgment} \section*{Acknowledgment}
\label{sec:org5dd6e93} \label{sec:orgcfbe3a3}
This research benefited from a FRIA grant from the French Community of Belgium. This research benefited from a FRIA grant from the French Community of Belgium.
This paper is based on a paper previously presented at the ICCMA conference~\cite{dehaeze19_compl_filter_shapin_using_synth}. This paper is based on a paper previously presented at the ICCMA conference~\cite{dehaeze19_compl_filter_shapin_using_synth}.
\section*{Data Availability} \section*{Data Availability}
\label{sec:orgabe6565} \label{sec:orga0214e5}
Matlab \cite{matlab20} was used for this study. Matlab \cite{matlab20} was used for this study.
The source code is available under a MIT License and archived in Zenodo~\cite{dehaeze21_new_method_desig_compl_filter_code}. The source code is available under a MIT License and archived in Zenodo~\cite{dehaeze21_new_method_desig_compl_filter_code}.

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@ -378,11 +378,11 @@
} }
@misc{dehaeze21_new_method_desig_compl_filter_code, @misc{dehaeze21_new_method_desig_compl_filter_code,
author = {Thomas Dehaeze}, author = {Dehaeze, Thomas},
doi = {10.5281/zenodo.3894342}, doi = {10.5281/zenodo.5361943},
howpublished = {Source Code on Zonodo}, howpublished = {Source code on Zenodo},
month = 09, month = sep,
title = {A New Method of Designing Complementary Filters for Sensor title = {{A New Method of Designing Complementary Filters for Sensor
Fusion using $\mathcal{H}_\infty$ Synthesis}, Fusion Using the $\mathcal{H}_\infty$ Synthesis}},
year = 2021, year = 2021,
} }