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@ -1,4 +1,4 @@
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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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||||
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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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||||
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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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||||
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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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@ -1,4 +1,4 @@
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#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis - Matlab Computation
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#+TITLE: A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis - Matlab Computation
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:DRAWER:
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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@ -34,23 +34,12 @@
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:END:
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* Introduction :ignore:
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In this document, the design of complementary filters is studied.
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One use of complementary filter is described below:
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#+begin_quote
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The basic idea of a complementary filter involves taking two or more sensors, filtering out unreliable frequencies for each sensor, and combining the filtered outputs to get a better estimate throughout the entire bandwidth of the system.
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To achieve this, the sensors included in the filter should complement one another by performing better over specific parts of the system bandwidth.
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#+end_quote
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This document is divided into several sections:
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- in section [[#sec:h_inf_synthesis_complementary_filters]], the $\mathcal{H}_\infty$ synthesis is used for generating two complementary filters
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- in section [[sec:three_comp_filters]], a method using the $\mathcal{H}_\infty$ synthesis is proposed to shape three of more complementary filters
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- in section [[sec:comp_filters_ligo]], the $\mathcal{H}_\infty$ synthesis is used and compared with FIR complementary filters used for LIGO
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#+begin_note
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Add the Matlab code use to obtain the results presented in the paper are accessible [[file:matlab.zip][here]] and presented below.
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#+end_note
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* H-Infinity synthesis of complementary filters
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/h_inf_synthesis_complementary_filters.m
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@ -155,6 +144,7 @@ xlabel('Frequency [Hz]'); ylabel('Magnitude');
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hold off;
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xlim([freqs(1), freqs(end)]);
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ylim([5e-4, 20]);
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yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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@ -176,23 +166,28 @@ W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G
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#+begin_src matlab :exports none
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figure;
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tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile();
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hold on;
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set(gca,'ColorOrderIndex',1)
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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xlabel('Frequency [Hz]', 'FontSize', 10); ylabel('Magnitude', 'FontSize', 10);
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hold off;
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xlim([freqs(1), freqs(end)]);
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ylim([1e-4, 20]);
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xticks([0.1, 1, 10, 100, 1000]);
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leg = legend('location', 'southeast', 'FontSize', 8);
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ylim([8e-4, 20]);
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yticks([1e-3, 1e-2, 1e-1, 1, 1e1]);
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yticklabels({'', '$10^{-2}$', '', '$10^0$', ''});
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ax1.FontSize = 9;
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leg = legend('location', 'south', 'FontSize', 8);
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leg.ItemTokenSize(1) = 18;
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/weights_W1_W2.pdf', 'width', 'wide', 'height', 'normal');
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exportFig('figs/weights_W1_W2.pdf', 'width', 'half', 'height', 350);
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#+end_src
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#+name: fig:weights_W1_W2
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@ -279,22 +274,21 @@ tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2, 1]);
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hold on;
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set(gca,'ColorOrderIndex',1)
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
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set(gca,'ColorOrderIndex',2)
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||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
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||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
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||||
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
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||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylim([1e-4, 20]);
|
||||
yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
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||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude');
|
||||
ylim([8e-4, 20]);
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yticks([1e-3, 1e-2, 1e-1, 1, 1e1]);
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yticklabels({'', '$10^{-2}$', '', '$10^0$', ''})
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||||
leg = legend('location', 'south', 'FontSize', 8, 'NumColumns', 2);
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||||
leg.ItemTokenSize(1) = 18;
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% Phase
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@ -305,16 +299,18 @@ plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
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||||
hold off;
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||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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||||
set(gca, 'XScale', 'log');
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||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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yticks([-180:90:180]);
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||||
ylim([-180, 200])
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||||
yticklabels({'-180', '', '0', '', '180'})
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||||
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||||
linkaxes([ax1,ax2],'x');
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||||
xlim([freqs(1), freqs(end)]);
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||||
#+end_src
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||||
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||||
#+begin_src matlab :tangle no :exports results :results file replace
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||||
exportFig('figs/hinf_filters_results.pdf', 'width', 'wide', 'height', 600);
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||||
exportFig('figs/hinf_filters_results.pdf', 'width', 700, 'height', 450);
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||||
#+end_src
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||||
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#+name: fig:hinf_filters_results
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@ -1352,7 +1348,7 @@ P = [ W1 0 1;
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||||
#+end_src
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||||
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||||
#+begin_src matlab :results output replace :exports both
|
||||
[L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on');
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||||
[L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
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||||
#+end_src
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||||
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||||
#+begin_src matlab
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||||
@ -1368,20 +1364,80 @@ zpk(H2)
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||||
#+RESULTS:
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||||
#+begin_example
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||||
zpk(H1)
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||||
|
||||
ans =
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||||
|
||||
(s+2.115e07) (s+153.6) (s+4.613) (s^2 + 6.858s + 12.03)
|
||||
--------------------------------------------------------
|
||||
(s+2.117e07) (s^2 + 102.1s + 2732) (s^2 + 69.43s + 3271)
|
||||
(s+3.842)^3 (s+153.6) (s+1.289e05)
|
||||
-------------------------------------------------------
|
||||
(s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272)
|
||||
|
||||
Continuous-time zero/pole/gain model.
|
||||
zpk(H2)
|
||||
|
||||
ans =
|
||||
|
||||
20455 (s+3425) (s+3318) (s^2 + 46.58s + 813.2)
|
||||
--------------------------------------------------------
|
||||
(s+2.117e07) (s^2 + 102.1s + 2732) (s^2 + 69.43s + 3271)
|
||||
125.61 (s+3358)^2 (s^2 + 46.61s + 813.8)
|
||||
-------------------------------------------------------
|
||||
(s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272)
|
||||
|
||||
Continuous-time zero/pole/gain model.
|
||||
#+end_example
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
|
||||
plot(freqs, abs(squeeze(freqresp(L, freqs, 'Hz'))), 'k--', 'DisplayName', '$|L|$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude');
|
||||
ylim([1e-3, 1e3]);
|
||||
yticks([1e-3, 1e-2, 1e-1, 1, 1e1, 1e2, 1e3]);
|
||||
yticklabels({'', '$10^{-2}$', '', '$10^0$', '', '$10^2$', ''});
|
||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
yticks([-180:90:180]);
|
||||
ylim([-180, 200])
|
||||
yticklabels({'-180', '', '0', '', '180'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/hinf_filters_results_mixed_sensitivity.pdf', 'width', 700 , 'height', 600);
|
||||
#+end_src
|
||||
|
||||
#+name: fig:hinf_filters_results_mixed_sensitivity
|
||||
#+caption:
|
||||
#+RESULTS:
|
||||
[[file:figs/hinf_filters_results_mixed_sensitivity.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(-2, 4, 1000);
|
||||
|
||||
@ -2592,10 +2648,14 @@ bibliographystyle:unsrt
|
||||
bibliography:ref.bib
|
||||
|
||||
* Functions
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
<<sec:functions>>
|
||||
|
||||
** =generateWF=: Generate Weighting Functions
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/src/generateWF.m
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
<<sec:generateWF>>
|
||||
|
||||
@ -2620,7 +2680,7 @@ function [W] = generateWF(args)
|
||||
% - w0 - Frequency at which |W(j w0)| = Gc [rad/s]
|
||||
%
|
||||
% Outputs:
|
||||
% - W - Generated Weight
|
||||
% - W - Generated Weighting Function
|
||||
#+end_src
|
||||
|
||||
*** Optional Parameters
|
||||
@ -2628,6 +2688,7 @@ function [W] = generateWF(args)
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
%% Argument validation
|
||||
arguments
|
||||
args.n (1,1) double {mustBeInteger, mustBePositive} = 1
|
||||
args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
||||
@ -2635,7 +2696,19 @@ arguments
|
||||
args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
args.w0 (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
end
|
||||
#+end_src
|
||||
|
||||
Verification that the parameters $G_0$, $G_c$ and $G_\infty$ are satisfy condition eqref:eq:cond_formula_1 or eqref:eq:cond_formula_2.
|
||||
#+name: eq:condition_params_formula
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
G_0 < 1 < G_\infty \text{ and } G_0 < G_c < G_\infty \label{eq:cond_formula_1}\\
|
||||
G_\infty < 1 < G_0 \text{ and } G_\infty < G_c < G_0 \label{eq:cond_formula_2}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
#+begin_src matlab
|
||||
% Verification of correct relation between G0, Gc and Ginf
|
||||
mustBeBetween(args.G0, args.Gc, args.Ginf);
|
||||
#+end_src
|
||||
|
||||
@ -2644,10 +2717,22 @@ mustBeBetween(args.G0, args.Gc, args.Ginf);
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
%% Initialize the Laplace variable
|
||||
s = zpk('s');
|
||||
#+end_src
|
||||
|
||||
The weighting function formula use is:
|
||||
#+name: eq:weight_formula
|
||||
\begin{equation}
|
||||
W(s) = \left( \frac{
|
||||
\hfill{} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
|
||||
}{
|
||||
\left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
|
||||
}\right)^n
|
||||
\end{equation}
|
||||
|
||||
#+begin_src matlab
|
||||
%% Create the weighting function according to formula
|
||||
W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
|
||||
(args.G0/args.Gc)^(1/args.n))/...
|
||||
((1/args.Ginf)^(1/args.n)*(1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
|
||||
@ -2660,7 +2745,7 @@ W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
% Custom validation function
|
||||
%% Custom validation function
|
||||
function mustBeBetween(a,b,c)
|
||||
if ~((a > b && b > c) || (c > b && b > a))
|
||||
eid = 'createWeight:inputError';
|
||||
@ -2672,7 +2757,6 @@ function mustBeBetween(a,b,c)
|
||||
** =generateCF=: Generate Complementary Filters
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/src/generateCF.m
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
<<sec:generateCF>>
|
||||
|
||||
@ -2687,7 +2771,7 @@ This Matlab function is accessible [[file:matlab/src/generateCF.m][here]].
|
||||
function [H1, H2] = generateCF(W1, W2, args)
|
||||
% createWeight -
|
||||
%
|
||||
% Syntax: [W] = generateCF(args)
|
||||
% Syntax: [H1, H2] = generateCF(W1, W2, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - W1 - Weighting Function for H1
|
||||
@ -2706,6 +2790,7 @@ function [H1, H2] = generateCF(W1, W2, args)
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
%% Argument validation
|
||||
arguments
|
||||
W1
|
||||
W2
|
||||
@ -2719,15 +2804,18 @@ end
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
%% The generalized plant is defined
|
||||
P = [W1 -W1;
|
||||
0 W2;
|
||||
1 0];
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results output replace :exports both
|
||||
%% The standard H-infinity synthesis is performed
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', args.method, 'DISPLAY', args.display);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% H1 is defined as the complementary of H2
|
||||
H1 = 1 - H2;
|
||||
#+end_src
|
||||
@ -1,7 +1,7 @@
|
||||
function [H1, H2] = generateCF(W1, W2, args)
|
||||
% createWeight -
|
||||
%
|
||||
% Syntax: [W] = generateCF(args)
|
||||
% Syntax: [H1, H2] = generateCF(W1, W2, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - W1 - Weighting Function for H1
|
||||
@ -14,6 +14,7 @@ function [H1, H2] = generateCF(W1, W2, args)
|
||||
% - H1 - Generated H1 Filter
|
||||
% - H2 - Generated H2 Filter
|
||||
|
||||
%% Argument validation
|
||||
arguments
|
||||
W1
|
||||
W2
|
||||
@ -21,10 +22,13 @@ arguments
|
||||
args.display char {mustBeMember(args.display,{'on', 'off'})} = 'on'
|
||||
end
|
||||
|
||||
%% The generalized plant is defined
|
||||
P = [W1 -W1;
|
||||
0 W2;
|
||||
1 0];
|
||||
|
||||
%% The standard H-infinity synthesis is performed
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', args.method, 'DISPLAY', args.display);
|
||||
|
||||
%% H1 is defined as the complementary of H2
|
||||
H1 = 1 - H2;
|
||||
|
||||
@ -11,8 +11,9 @@ function [W] = generateWF(args)
|
||||
% - w0 - Frequency at which |W(j w0)| = Gc [rad/s]
|
||||
%
|
||||
% Outputs:
|
||||
% - W - Generated Weight
|
||||
% - W - Generated Weighting Function
|
||||
|
||||
%% Argument validation
|
||||
arguments
|
||||
args.n (1,1) double {mustBeInteger, mustBePositive} = 1
|
||||
args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
||||
@ -21,16 +22,19 @@ arguments
|
||||
args.w0 (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
end
|
||||
|
||||
% Verification of correct relation between G0, Gc and Ginf
|
||||
mustBeBetween(args.G0, args.Gc, args.Ginf);
|
||||
|
||||
%% Initialize the Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
%% Create the weighting function according to formula
|
||||
W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
|
||||
(args.G0/args.Gc)^(1/args.n))/...
|
||||
((1/args.Ginf)^(1/args.n)*(1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
|
||||
(1/args.Gc)^(1/args.n)))^args.n;
|
||||
|
||||
% Custom validation function
|
||||
%% Custom validation function
|
||||
function mustBeBetween(a,b,c)
|
||||
if ~((a > b && b > c) || (c > b && b > a))
|
||||
eid = 'createWeight:inputError';
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user