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% Created 2021-08-31 mar. 14:16
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% Created 2021-08-31 mar. 14:07
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% Intended LaTeX compiler: pdflatex
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\documentclass[preprint, sort&compress]{elsarticle}
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\usepackage[utf8]{inputenc}
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@ -43,13 +43,12 @@
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\begin{frontmatter}
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\begin{abstract}
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In order to obtain a better estimate of a quantity being measured, several sensors having different characteristics can be merged with a technique called ``sensor fusion''.
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The obtained ``super sensor'' can combine the benefits of the individual sensors provided that the complementary filters used in the fusion are well designed.
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Indeed, properties of the super sensor are linked to the magnitude of the complementary filters.
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Properly shaping the magnitude of complementary filters is a difficult and time-consuming task.
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The obtained ``super sensor'' can combine the benefits of the individual sensors provided that the magnitude of the complementary filters used in the fusion are well shaped.
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Properly designing complementary filters is a difficult and time-consuming task.
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In this study, we address this issue and propose a new method for designing complementary filters.
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This method uses weighting functions to specify the wanted shape of the complementary filter that are then easily obtained using the standard \(\mathcal{H}_\infty\) synthesis.
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This method uses weighting functions to specify the wanted shape of the complementary filter that are then easily obtained using the standard \(\mathcal{H}_\infty\) synthesis on a specific generalized plant.
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The proper choice of the weighting functions is discussed, and the effectiveness and simplicity of the design method is highlighted using several examples.
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Such synthesis method is further extended for the shaping of more than two complementary filters.
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Such synthesis method is further generalized to a set of more than two complementary filters.
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\end{abstract}
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\begin{keyword}
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@ -58,7 +57,7 @@ Sensor fusion \sep{} Complementary filters \sep{} \(\mathcal{H}_\infty\) synthes
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\end{frontmatter}
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\section{Introduction}
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\label{sec:orgb3159e9}
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\label{sec:org45b63f3}
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\label{sec:introduction}
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Measuring a physical quantity using sensors is always subject to several limitations.
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First, the accuracy of the measurement will be affected by several noise sources, such as the electrical noise of the conditioning electronics being used.
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@ -105,13 +104,13 @@ The synthesis method is further validated in Section~\ref{sec:application_ligo}
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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.
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\section{Sensor Fusion and Complementary Filters Requirements}
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\label{sec:orge37bf43}
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\label{sec:orgeeb9584}
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\label{sec:requirements}
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Complementary filters provides a framework for fusing signals from different sensors.
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As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
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These requirements are discussed in this section.
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\subsection{Sensor Fusion Architecture}
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\label{sec:orgd0086e9}
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\label{sec:org556a17c}
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\label{sec:sensor_fusion}
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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 \(x\).
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@ -138,7 +137,7 @@ Therefore, a pair of complementary filter needs to satisfy the following conditi
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It will soon become clear why the complementary property is important for the sensor fusion architecture.
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\subsection{Sensor Models and Sensor Normalization}
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\label{sec:orgf4072c3}
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\label{sec:org5c5b6fb}
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\label{sec:sensor_models}
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In order to study such sensor fusion architecture, a model for the sensors is required.
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@ -184,7 +183,7 @@ The super sensor output is therefore equal to:
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\end{figure}
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\subsection{Noise Sensor Filtering}
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\label{sec:org72d0e25}
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\label{sec:org4d93421}
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\label{sec:noise_filtering}
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In this section, it is supposed that all the sensors are perfectly normalized, such that:
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@ -225,7 +224,7 @@ In such case, to lower the noise of the super sensor, the norm \(|H_1(j\omega)|\
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Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
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\subsection{Sensor Fusion Robustness}
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\label{sec:org6b3c1f3}
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\label{sec:org0ab9090}
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\label{sec:fusion_robustness}
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In practical systems the sensor normalization is not perfect and condition \eqref{eq:perfect_dynamics} is not verified.
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@ -289,7 +288,7 @@ As it is generally desired to limit the maximum phase added by the super sensor,
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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.
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\section{Complementary Filters Shaping}
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\label{sec:org25756ab}
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\label{sec:orgdf903f0}
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\label{sec:hinf_method}
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As shown in Section~\ref{sec:requirements}, the noise and robustness of the super sensor are a function of the complementary filters norms.
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Therefore, a complementary filters synthesis method that allows to shape their norms would be of great use.
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@ -297,7 +296,7 @@ In this section, such synthesis is proposed by writing the synthesis objective a
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As weighting functions are used to represent the wanted complementary filters shapes during the synthesis, the proper design of weighting functions is discussed.
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Finally, the synthesis method is validated on an simple example.
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\subsection{Synthesis Objective}
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\label{sec:org6c44b50}
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\label{sec:org2467206}
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\label{sec:synthesis_objective}
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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}.
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@ -314,7 +313,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
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\(W_1(s)\) 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.
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\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
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\label{sec:org1538346}
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\label{sec:org172468e}
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\label{sec:hinf_synthesis}
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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.
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@ -370,7 +369,7 @@ There might be solutions were the objectives~\eqref{eq:hinf_cond_h1} and~\eqref{
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In practice, this is however not an found to be an issue.
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\subsection{Weighting Functions Design}
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\label{sec:org11b4246}
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\label{sec:org3b7e958}
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\label{sec:hinf_weighting_func}
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Weighting functions are used during the synthesis to specify the maximum allowed norms of the complementary filters.
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@ -419,7 +418,7 @@ An example of the obtained magnitude of a weighting function generated using \eq
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\end{figure}
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\subsection{Validation of the proposed synthesis method}
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\label{sec:orgb9a4dc3}
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\label{sec:org13db233}
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\label{sec:hinf_example}
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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:
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@ -490,7 +489,7 @@ This simple example illustrates the fact that the proposed methodology for compl
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A more complex real life example is taken up in the next section.
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\section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
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\label{sec:org157e8c9}
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\label{sec:orgb11268e}
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\label{sec:application_ligo}
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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}.
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@ -513,7 +512,7 @@ After synthesis, the obtained FIR filters were found to be compliant with the re
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However they are of very high order so their implementation is quite complex.
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In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated on the same set of requirements.
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\subsection{Complementary Filters Specifications}
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\label{sec:org872bc34}
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\label{sec:org100bb02}
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\label{sec:ligo_specifications}
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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}):
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@ -534,7 +533,7 @@ They are physically represented in Fig.~\ref{fig:fir_filter_ligo} as well as the
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\end{figure}
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\subsection{Weighting Functions Design}
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\label{sec:org97ac8c9}
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\label{sec:orgba233a6}
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\label{sec:ligo_weights}
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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.
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@ -551,7 +550,7 @@ The magnitudes of the weighting functions are shown in Fig.~\ref{fig:ligo_weight
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\end{figure}
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\subsection{\(\mathcal{H}_\infty\) Synthesis of the complementary filters}
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\label{sec:orgf148a21}
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\label{sec:org705f5c2}
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\label{sec:ligo_results}
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The proposed \(\mathcal{H}_\infty\) synthesis is performed on the generalized plant shown in Fig.~\ref{fig:h_infinity_robust_fusion_plant}.
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@ -567,10 +566,10 @@ This confirms the effectiveness of the proposed synthesis method even when the c
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\end{figure}
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\section{Discussion}
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\label{sec:orgbc3d67c}
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\label{sec:org2d41692}
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\label{sec:discussion}
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\subsection{``Closed-Loop'' complementary filters}
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\label{sec:org84a8225}
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\label{sec:org2106ef4}
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\label{sec:closed_loop_complementary_filters}
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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}.
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This is for instance presented in \cite{mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas}.
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@ -667,7 +666,7 @@ The obtained ``closed-loop'' complementary filters are indeed equal to the ones
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\end{figure}
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\subsection{Synthesis of more than two Complementary Filters}
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\label{sec:org862b3a1}
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\label{sec:org5b39592}
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\label{sec:hinf_three_comp_filters}
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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}.
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@ -771,7 +770,7 @@ A set of \(n\) complementary filters can be shaped using the generalized plant \
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\end{equation}
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\section{Conclusion}
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\label{sec:org35fb45f}
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\label{sec:org369c466}
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\label{sec:conclusion}
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Sensors measuring a physical quantities are always subject to limitations both in terms of bandwidth or accuracy.
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@ -785,12 +784,12 @@ Links with ``closed-loop'' complementary filters where highlighted, and the prop
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Future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion.
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\section*{Acknowledgment}
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\label{sec:org7a5f4b3}
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\label{sec:orgd71260d}
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This research benefited from a FRIA grant from the French Community of Belgium.
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This paper is based on a paper previously presented at the ICCMA conference~\cite{dehaeze19_compl_filter_shapin_using_synth}.
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\section*{Data Availability}
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\label{sec:org767c106}
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\label{sec:org30a4627}
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Matlab \cite{matlab20} was used for this study.
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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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