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Author SHA1 Message Date
Thomas Dehaeze
8cba417317 First LIGO review 2021-09-14 13:46:28 +02:00
Thomas Dehaeze
369454f877 Review 2021-09-08 10:56:25 +02:00
Thomas Dehaeze
6a65ed51a5 Add comments by Mohit and Christophe 2021-09-02 10:04:30 +02:00
Thomas Dehaeze
572e1d5abd Update title 2021-09-02 10:04:16 +02:00
Thomas Dehaeze
504f0d7935 First review 2021-09-02 10:02:12 +02:00
Thomas Dehaeze
d424d56143 Remove Mohit's folder 2021-09-01 17:05:20 +02:00
Thomas Dehaeze
f34ae9b8c4 Update main page 2021-09-01 17:04:48 +02:00
Thomas Dehaeze
4568a119a2 Update bibtex for source code 2021-09-01 17:03:27 +02:00
Thomas Dehaeze
4769d4b57f Last update before submission 2021-09-01 17:00:12 +02:00
Thomas Dehaeze
616652f67a Update matlab scripts 2021-09-01 16:50:18 +02:00
Thomas Dehaeze
946ae5f977 Update journal file 2021-09-01 15:00:34 +02:00
Thomas Dehaeze
b1517f1c35 Update license date 2021-09-01 15:00:22 +02:00
Thomas Dehaeze
5592f7a37d Completely rework matlab files 2021-09-01 15:00:04 +02:00
Thomas Dehaeze
c073d945d1 Add main bib file 2021-09-01 11:36:08 +02:00
Thomas Dehaeze
0d33958b17 Update links to files 2021-09-01 11:36:05 +02:00
Thomas Dehaeze
4836b9e254 Update bib file for matlab 2021-09-01 11:35:32 +02:00
Thomas Dehaeze
2b5757bb09 Update tangled files 2021-09-01 11:35:25 +02:00
Thomas Dehaeze
c520f7903e Update all matlab figures 2021-09-01 11:35:13 +02:00
Thomas Dehaeze
10e9c17a60 Delete unused figures 2021-09-01 11:34:32 +02:00
Thomas Dehaeze
6ce027cf8a Reworked tikz figures 2021-09-01 10:16:26 +02:00
103 changed files with 9153 additions and 19376 deletions
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2
LICENSE.txt
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@@ -1,6 +1,6 @@
MIT License
Copyright (c) 2020 Dehaeze Thomas
Copyright (c) 2021 Dehaeze Thomas
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal

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dehaeze21_desig_compl_filte.bib Normal file
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@article{dehaeze21_new_method_desig_compl_filter,
author = {Dehaeze, Thomas and Vermat, Mohit and Collette, Christophe},
title = {A New Method of Designing Complementary Filters for Sensor
Fusion Using the $\mathcal{H}_\infty$ Synthesis},
journal = {Mechanical Systems and Signal Processing},
year = 2021,
month = {Nov},
}

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@@ -3,9 +3,9 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2021-06-21 lun. 11:42 -->
<!-- 2021-09-02 jeu. 10:02 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis</title>
<title>A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis</title>
<meta name="author" content="Thomas Dehaeze" />
<meta name="generator" content="Org Mode" />
<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
@@ -33,8 +33,8 @@
<a accesskey="h" href="../index.html"> UP </a>
|
<a accesskey="H" href="../index.html"> HOME </a>
</div><div id="content">
<h1 class="title">Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis
</div><div id="content" class="content">
<h1 class="title">A new method of designing complementary filters for sensor fusion using the \(\mathcal{H}_\infty\) synthesis
<br />
<span class="subtitle">Dehaeze Thomas, Verma Mohit and Collette Christophe</span>
</h1>
@@ -44,33 +44,88 @@
</p>
<p>
For many applications, large bandwidth and dynamic ranges are requiring to use several sensors, whose signals are combined using complementary filters.
This paper presents a method for designing these complementary filters using \(\mathcal{H}_\infty\) synthesis that allows to shape the filter norms.
This method is shown to be easily applicable for the synthesis of complex complementary filters.
In order to obtain a better estimate of a quantity being measured, several sensors having different characteristics can be merged with a technique called &ldquo;sensor fusion&rdquo;.
The obtained &ldquo;super sensor&rdquo; can combine the benefits of the individual sensors provided that the complementary filters used in the fusion are well designed.
Indeed, properties of the super sensor are linked to the magnitude of the complementary filters.
Properly shaping the magnitude of complementary filters is a difficult and time-consuming task.
In this study, we address this issue and propose a new method for designing complementary filters.
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.
The proper choice of the weighting functions is discussed, and the effectiveness and simplicity of the design method is highlighted using several examples.
Such synthesis method is further extended for the shaping of more than two complementary filters.
</p>
</blockquote>
<div id="outline-container-org16737c1" class="outline-2">
<h2 id="org16737c1">Paper (<a href="journal/journal.pdf">link</a>)</h2>
<div class="outline-text-2" id="text-org16737c1">
<div id="outline-container-org8d13d5d" class="outline-2">
<h2 id="org8d13d5d">Journal Article (<a href="journal/dehaeze21_desig_compl_filte.pdf">link</a>)</h2>
<div class="outline-text-2" id="text-org8d13d5d">
<p>
The paper has been created <a href="https://orgmode.org/">Org Mode</a> (generating <a href="https://www.latex-project.org/">LaTeX</a> code) under <a href="https://www.gnu.org/software/emacs/">Emacs</a>.
</p>
<p>
To cite this paper use the following BibTeX code.
</p>
<div class="org-src-container">
<pre class="src src-bibtex"><span class="org-function-name">@article</span>{<span class="org-constant">dehaeze21_new_method_desig_compl_filter</span>,
<span class="org-variable-name">author</span> = {Dehaeze, Thomas and Vermat, Mohit and Collette, Christophe},
<span class="org-variable-name">title</span> = {A New Method of Designing Complementary Filters for Sensor
Fusion Using the <span class="org-string">$\mathcal{H}_\infty$</span> Synthesis},
<span class="org-variable-name">journal</span> = {Mechanical Systems and Signal Processing},
<span class="org-variable-name">year</span> = 2021,
<span class="org-variable-name">month</span> = {Nov},
}
</pre>
</div>
<p>
You can also use the formatted citation below.
</p>
<blockquote>
<p>
Dehaeze, T., Vermat, M., &amp; Collette, C., A new method of designing complementary filters for sensor fusion using the \(H_\infty\) synthesis, Mechanical Systems and Signal Processing, (2021).
</p>
</blockquote>
</div>
</div>
<div id="outline-container-orgb50a720" class="outline-2">
<h2 id="orgb50a720">Matlab Scripts (<a href="matlab/index.html">link</a>)</h2>
<div class="outline-text-2" id="text-orgb50a720">
<div id="outline-container-org71c1b40" class="outline-2">
<h2 id="org71c1b40">Matlab Scripts (<a href="matlab/dehaeze21_desig_compl_filte_matlab.html">link</a>)</h2>
<div class="outline-text-2" id="text-org71c1b40">
<p>
All the <a href="https://fr.mathworks.com/">Matlab</a> code that was used for the paper are accessible so that all the results are reproducible.
The source code is hosted on <a href="https://zenodo.org/record/5361920">Zenodo</a>.
</p>
<p>
If you use this code and want to refer to it in a publication, you can use the following BibTeX reference:
</p>
<div class="org-src-container">
<pre class="src src-bibtex"><span class="org-function-name">@misc</span>{<span class="org-constant">dehaeze21_new_method_desig_compl_filter_code</span>,
<span class="org-variable-name">author</span> = {Dehaeze, Thomas},
<span class="org-variable-name">doi</span> = {<span class="org-button">10.5281/zenodo.5361943</span>},
<span class="org-variable-name">howpublished</span> = {Source code on Zenodo},
<span class="org-variable-name">month</span> = sep,
<span class="org-variable-name">title</span> = {{A New Method of Designing Complementary Filters for Sensor
Fusion Using the <span class="org-string">$\mathcal{H}_\infty$</span> Synthesis}},
<span class="org-variable-name">year</span> = 2021,
}
</pre>
</div>
<p>
You can also use the formatted citation below.
</p>
<blockquote>
<p>
Dehaeze Thomas. (2021). A New Method of Designing Complementary Filters for Sensor Fusion Using the H-Infinity Synthesis. Source code on Zenodo. 10.5281/zenodo.5361943
</p>
</blockquote>
</div>
</div>
<div id="outline-container-org06a9b71" class="outline-2">
<h2 id="org06a9b71">Tikz Figures (<a href="tikz/index.html">link</a>)</h2>
<div class="outline-text-2" id="text-org06a9b71">
<div id="outline-container-org1b73cc9" class="outline-2">
<h2 id="org1b73cc9">Tikz Figures (<a href="tikz/dehaeze21_desig_compl_filte_tikz.html">link</a>)</h2>
<div class="outline-text-2" id="text-org1b73cc9">
<p>
All the figures for the paper have been generated using <a href="https://sourceforge.net/projects/pgf/">TikZ</a>.
</p>

72
index.org
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@@ -1,4 +1,4 @@
#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis
#+TITLE: A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis
:DRAWER:
#+SUBTITLE: Dehaeze Thomas, Verma Mohit and Collette Christophe
@@ -13,39 +13,71 @@
:END:
#+begin_quote
*Abstract*:
*Abstract*:
For many applications, large bandwidth and dynamic ranges are requiring to use several sensors, whose signals are combined using complementary filters.
This paper presents a method for designing these complementary filters using $\mathcal{H}_\infty$ synthesis that allows to shape the filter norms.
This method is shown to be easily applicable for the synthesis of complex complementary filters.
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".
The obtained "super sensor" can combine the benefits of the individual sensors provided that the complementary filters used in the fusion are well designed.
Indeed, properties of the super sensor are linked to the magnitude of the complementary filters.
Properly shaping the magnitude of complementary filters is a difficult and time-consuming task.
In this study, we address this issue and propose a new method for designing complementary filters.
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.
The proper choice of the weighting functions is discussed, and the effectiveness and simplicity of the design method is highlighted using several examples.
Such synthesis method is further extended for the shaping of more than two complementary filters.
#+end_quote
* Paper ([[file:journal/journal.pdf][link]])
* Journal Article ([[file:journal/dehaeze21_desig_compl_filte.pdf][link]])
:PROPERTIES:
:UNNUMBERED: t
:END:
The paper has been created [[https://orgmode.org/][Org Mode]] (generating [[https://www.latex-project.org/][LaTeX]] code) under [[https://www.gnu.org/software/emacs/][Emacs]].
* Matlab Scripts ([[file:matlab/index.org][link]])
To cite this paper use the following BibTeX code.
#+begin_src bibtex
@article{dehaeze21_new_method_desig_compl_filter,
author = {Dehaeze, Thomas and Vermat, Mohit and Collette, Christophe},
title = {A New Method of Designing Complementary Filters for Sensor
Fusion Using the $\mathcal{H}_\infty$ Synthesis},
journal = {Mechanical Systems and Signal Processing},
year = 2021,
month = {Nov},
}
#+end_src
You can also use the formatted citation below.
#+begin_quote
Dehaeze, T., Vermat, M., & Collette, C., A new method of designing complementary filters for sensor fusion using the $H_\infty$ synthesis, Mechanical Systems and Signal Processing, (2021).
#+end_quote
* Matlab Scripts ([[file:matlab/dehaeze21_desig_compl_filte_matlab.org][link]])
:PROPERTIES:
:UNNUMBERED: t
:END:
All the [[https://fr.mathworks.com/][Matlab]] code that was used for the paper are accessible so that all the results are reproducible.
* Tikz Figures ([[file:tikz/index.org][link]])
All the [[https://fr.mathworks.com/][Matlab]] code that was used for the paper are accessible so that all the results are reproducible.
The source code is hosted on [[https://zenodo.org/record/5361920][Zenodo]].
If you use this code and want to refer to it in a publication, you can use the following BibTeX reference:
#+begin_src bibtex
@misc{dehaeze21_new_method_desig_compl_filter_code,
author = {Dehaeze, Thomas},
doi = {10.5281/zenodo.5361943},
howpublished = {Source code on Zenodo},
month = sep,
title = {{A New Method of Designing Complementary Filters for Sensor
Fusion Using the $\mathcal{H}_\infty$ Synthesis}},
year = 2021,
}
#+end_src
You can also use the formatted citation below.
#+begin_quote
Dehaeze Thomas. (2021). A New Method of Designing Complementary Filters for Sensor Fusion Using the H-Infinity Synthesis. Source code on Zenodo. 10.5281/zenodo.5361943
#+end_quote
* Tikz Figures ([[file:tikz/dehaeze21_desig_compl_filte_tikz.org][link]])
:PROPERTIES:
:UNNUMBERED: t
:END:
All the figures for the paper have been generated using [[https://sourceforge.net/projects/pgf/][TikZ]].
* Cite this paper :noexport:
:PROPERTIES:
:UNNUMBERED: t
:END:
To cite this paper use the following bibtex code.
#+begin_src bibtex
#+end_src
You can also use the formatted citation below.
#+begin_quote
#+end_quote

437
journal/dehaeze21_desig_compl_filte.org
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@@ -1,4 +1,4 @@
#+TITLE: A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis
#+TITLE: Designing complementary filters for sensor fusion using $\mathcal{H}_\infty$ synthesis
:DRAWER:
#+LATEX_CLASS: elsarticle
#+LATEX_CLASS_OPTIONS: [preprint, sort&compress]
@@ -12,8 +12,8 @@
#+LATEX_HEADER_EXTRA: \author[a2,a4]{Christophe Collette}
#+LATEX_HEADER_EXTRA: \cortext[cor1]{Corresponding author. Email Address: thomas.dehaeze@esrf.fr}
#+LATEX_HEADER_EXTRA: \address[a1]{European Synchrotron Radiation Facility, Grenoble, France}
#+LATEX_HEADER_EXTRA: \address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
#+LATEX_HEADER_EXTRA: \address[a1]{European Synchrotron Radiation Facility, 38000 Grenoble, France}
#+LATEX_HEADER_EXTRA: \address[a2]{University of Li\`{e}ge, PML, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
#+LATEX_HEADER_EXTRA: \address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
#+LATEX_HEADER_EXTRA: \address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
@@ -41,7 +41,16 @@
("\\paragraph{%s}" . "\\paragraph*{%s}")
("\\subparagraph{%s}" . "\\subparagraph*{%s}"))
)
;; Remove automatic org headings
(defun my-latex-filter-removeOrgAutoLabels (text backend info)
"Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings."
(when (org-export-derived-backend-p backend 'latex)
(replace-regexp-in-string "\\\\label{sec:org[a-f0-9]+}\n" "" text)))
(add-to-list 'org-export-filter-headline-functions
'my-latex-filter-removeOrgAutoLabels)
;; Automatic delete org org-comments
(defun delete-org-comments (backend)
(loop for comment in (reverse (org-element-map (org-element-parse-buffer)
'comment 'identity))
@@ -65,13 +74,13 @@
#+begin_frontmatter
#+begin_abstract
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".
The obtained "super sensor" can combine the benefits of the individual sensors provided that the complementary filters used in the fusion are well designed.
The obtained "super sensor" combines the benefits of the individual sensors provided that the complementary filters used in the fusion are well designed.
Indeed, properties of the super sensor are linked to the magnitude of the complementary filters.
Properly shaping the magnitude of complementary filters is a difficult and time-consuming task.
In this study, we address this issue and propose a new method for designing complementary filters.
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.
In this study, this issue is addressed and a new method for designing complementary filters is proposed.
This method uses weighting functions to specify the wanted shape of the complementary filters that are then obtained using the standard $\mathcal{H}_\infty$ synthesis.
The proper choice of the weighting functions is discussed, and the effectiveness and simplicity of the design method is highlighted using several examples.
Such synthesis method is further extended for the shaping of more than two complementary filters.
Such synthesis method is further extended for the shaping of a set of more than two complementary filters.
#+end_abstract
#+begin_keyword
@@ -84,75 +93,75 @@ Sensor fusion \sep{} Complementary filters \sep{} $\mathcal{H}_\infty$ synthesis
** Introduction to Sensor Fusion :ignore:
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 is affected by several noise sources, such as electrical noise of the conditioning electronics being used.
Second, the frequency range in which the measurement is relevant is bounded by the bandwidth of the sensor.
One way to overcome these limitations is to combine several sensors using a technique called "sensor fusion"\nbsp{}cite:bendat57_optim_filter_indep_measur_two.
Fortunately, a wide variety of sensors exist, each with different characteristics.
By carefully choosing the fused sensors, a so called "super sensor" is obtain that combines benefits of individual sensors and yields a better estimate of the measured physical quantity. \par
Fortunately, a wide variety of sensors exists, each with different characteristics.
By carefully choosing the fused sensors, a so called "super sensor" is obtained that can combines benefits of the individual sensors. \par
** Advantages of Sensor Fusion :ignore:
In some situations, sensor fusion is used to increase the bandwidth of the measurement\nbsp{}cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim.
For instance, in\nbsp{}cite:shaw90_bandw_enhan_posit_measur_using_measur_accel, the bandwidth of a position sensor is increased by fusing it with an accelerometer providing the high frequency motion information.
For other applications, sensor fusion is used to obtain a estimate of the measured quantity with lower noise\nbsp{}cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur,robert12_introd_random_signal_applied_kalman.
More recently, the fusion of sensors measuring different physical quantities has been proposed to obtained interesting properties for control\nbsp{}cite:collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage.
In\nbsp{}cite:collette15_sensor_fusion_method_high_perfor, an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator to improve the stability margins of the feedback controller. \par
For instance, in\nbsp{}cite:shaw90_bandw_enhan_posit_measur_using_measur_accel the bandwidth of a position sensor is increased by fusing it with an accelerometer providing the high frequency motion information.
For other applications, sensor fusion is used to obtain an estimate of the measured quantity with lower noise\nbsp{}cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur,robert12_introd_random_signal_applied_kalman.
More recently, the fusion of sensors measuring different physical quantities has been proposed to obtain interesting properties for control\nbsp{}cite:collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage.
In\nbsp{}cite:collette15_sensor_fusion_method_high_perfor, an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator for improving the stability margins of the feedback controller. \par
** Applications :ignore:
Practical applications of sensor fusion are numerous.
It is widely used for the attitude estimation of several autonomous vehicles such as unmanned aerial vehicle\nbsp{}cite:baerveldt97_low_cost_low_weigh_attit,corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas and underwater vehicles\nbsp{}cite:pascoal99_navig_system_desig_using_time,batista10_optim_posit_veloc_navig_filter_auton_vehic.
Naturally, it is of great benefits for high performance positioning control as shown in\nbsp{}cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim,yong16_high_speed_vertic_posit_stage.
Sensor fusion was also shown to be a key technology to improve the performances of active vibration isolation systems\nbsp{}cite:tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.
This is particularly apparent for the isolation stages of gravitational wave observer\nbsp{}cite:collette15_sensor_fusion_method_high_perfor,heijningen18_low such as the ones used at the LIGO\nbsp{}cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system and at the VIRGO\nbsp{}cite:lucia18_low_frequen_optim_perfor_advan. \par
Sensor fusion was also shown to be a key technology to improve the performance of active vibration isolation systems\nbsp{}cite:tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.
Emblematic examples are the isolation stages of gravitational wave detectors\nbsp{}cite:collette15_sensor_fusion_method_high_perfor,heijningen18_low such as the ones used at the LIGO\nbsp{}cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system and at the Virgo\nbsp{}cite:lucia18_low_frequen_optim_perfor_advan. \par
** Kalman Filtering / Complementary filters :ignore:
There are mainly two ways to perform sensor fusion: either using a set of complementary filters\nbsp{}cite:anderson53_instr_approac_system_steer_comput or using Kalman filtering\nbsp{}cite:brown72_integ_navig_system_kalman_filter,odry18_kalman_filter_mobil_robot_attit_estim.
For sensor fusion applications, both methods are sharing many relationships\nbsp{}cite:brown72_integ_navig_system_kalman_filter,higgins75_compar_compl_kalman_filter,robert12_introd_random_signal_applied_kalman,becker15_compl_filter_desig_three_frequen_bands.
However, for Kalman filtering, we are forced to make assumption about the probabilistic character of the sensor noises\nbsp{}cite:robert12_introd_random_signal_applied_kalman whereas it is not the case for complementary filters.
Furthermore, the main advantages of complementary filters over Kalman filtering for sensor fusion are their very general applicability, their low computational cost\nbsp{}cite:higgins75_compar_compl_kalman_filter, and the fact that they are very intuitive as their effects can be easily interpreted in the frequency domain. \par
However, for Kalman filtering, assumptions must be made about the probabilistic character of the sensor noises\nbsp{}cite:robert12_introd_random_signal_applied_kalman whereas it is not the case with complementary filters.
Furthermore, the advantages of complementary filters over Kalman filtering for sensor fusion are their general applicability, their low computational cost\nbsp{}cite:higgins75_compar_compl_kalman_filter, and the fact that they are intuitive as their effects can be easily interpreted in the frequency domain. \par
** Design Methods of Complementary filters :ignore:
A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
For the earliest used of complementary filtering for sensor fusion, analog circuits were used to physically realize the filters\nbsp{}cite:anderson53_instr_approac_system_steer_comput.
Analog complementary filters are still used today\nbsp{}cite:yong16_high_speed_vertic_posit_stage,moore19_capac_instr_sensor_fusion_high_bandw_nanop, but most of the time they are now implemented numerically as it allows for much more flexibility. \par
In the early days of complementary filtering, analog circuits were employed to physically realize the filters\nbsp{}cite:anderson53_instr_approac_system_steer_comput.
Analog complementary filters are still used today\nbsp{}cite:yong16_high_speed_vertic_posit_stage,moore19_capac_instr_sensor_fusion_high_bandw_nanop, but most of the time they are now implemented digitally as it allows for much more flexibility. \par
Several design methods have been developed over the years to optimize complementary filters.
The easiest way to design complementary filters is to use analytical formulas.
Depending on the application, the formulas used are of first order\nbsp{}cite:corke04_inert_visual_sensin_system_small_auton_helic,yeh05_model_contr_hydraul_actuat_two,yong16_high_speed_vertic_posit_stage, second order\nbsp{}cite:baerveldt97_low_cost_low_weigh_attit,stoten01_fusion_kinet_data_using_compos_filter,jensen13_basic_uas or even higher orders\nbsp{}cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,stoten01_fusion_kinet_data_using_compos_filter,collette15_sensor_fusion_method_high_perfor,matichard15_seism_isolat_advan_ligo. \par
As the characteristics of the "super sensor" depends on the design of the complementary filters\nbsp{}cite:dehaeze19_compl_filter_shapin_using_synth, several optimization techniques were developed over the years.
Some are based on the finding the optimal parameters of analytical formulas\nbsp{}cite:jensen13_basic_uas,min15_compl_filter_desig_angle_estim,becker15_compl_filter_desig_three_frequen_bands, while other are using convex optimization tools\nbsp{}cite:hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo such as linear matrix inequalities\nbsp{}cite:pascoal99_navig_system_desig_using_time.
As the characteristics of the super sensor depends on the proper design of the complementary filters\nbsp{}cite:dehaeze19_compl_filter_shapin_using_synth, several optimization techniques have been developed.
Some are based on the finding of optimal parameters of analytical formulas\nbsp{}cite:jensen13_basic_uas,min15_compl_filter_desig_angle_estim,becker15_compl_filter_desig_three_frequen_bands, while other are using convex optimization tools\nbsp{}cite:hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo such as linear matrix inequalities\nbsp{}cite:pascoal99_navig_system_desig_using_time.
As shown in\nbsp{}cite:plummer06_optim_compl_filter_their_applic_motion_measur, the design of complementary filters can also be linked to the standard mixed-sensitivity control problem.
Therefore, all the powerful tools developed for the classical control theory can also be used for the design of complementary filters.
For instance, in\nbsp{}cite:jensen13_basic_uas the two gains of a Proportional Integral (PI) controller are optimized to minimize the noise of the super sensor. \par
** Problematic / gap in the research :ignore:
The common objective to all these complementary filters design methods is to obtain a super sensor that has desired characteristics, usually in terms of noise and dynamics.
Moreover, as reported in\nbsp{}cite:zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur, phase shifts and magnitude bumps of the "super sensors" dynamics can be observed if either the complementary filters are poorly designed or if the sensors are not well calibrated.
Therefore, the robustness of the fusion is also of concerned when designing the complementary filters.
Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to specify the wanted super sensor characteristic while ensuring good fusion robustness has been proposed. \par
The common objective of all these complementary filters design methods is to obtain a super sensor that has desired characteristics, usually in terms of noise and dynamics.
Moreover, as reported in\nbsp{}cite:zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur, phase shifts and magnitude bumps of the super sensors dynamics can be observed if either the complementary filters are poorly designed or if the sensors are not well calibrated.
Hence, the robustness of the fusion is also of concern when designing the complementary filters.
Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to specify the desired super sensor characteristic while ensuring good fusion robustness has been proposed. \par
** Describe the paper itself / the problem which is addressed :ignore:
Fortunately, both the robustness of the fusion and the super sensor characteristics can be linked to the magnitude of the complementary filters\nbsp{}cite:dehaeze19_compl_filter_shapin_using_synth.
Based on that, this paper introduces a new way to design complementary filters using the $\mathcal{H}_\infty$ synthesis which allows to shape the magnitude of the complementary filters in an easy and intuitive way. \par
Based on that, this paper introduces a new way to design complementary filters using the $\mathcal{H}_\infty$ synthesis which allows to shape the complementary filters' magnitude in an easy and intuitive way. \par
** Introduce Each part of the paper :ignore:
Section\nbsp{}ref:sec:requirements introduces the sensor fusion architecture and demonstrates how typical requirements can be linked to the complementary filters magnitudes.
In section\nbsp{}ref:sec:hinf_method, the shaping of complementary filters is written as an $\mathcal{H}_\infty$ optimization problem using weighting functions, and the simplicity of the proposed method is illustrated with an example.
Section\nbsp{}ref:sec:requirements introduces the sensor fusion architecture and demonstrates how typical requirements can be linked to the complementary filters' magnitude.
In Section\nbsp{}ref:sec:hinf_method, the shaping of complementary filters is formulated as an $\mathcal{H}_\infty$ optimization problem using weighting functions, and the simplicity of the proposed method is illustrated with an example.
The synthesis method is further validated in Section\nbsp{}ref:sec:application_ligo by designing complex complementary filters.
Section\nbsp{}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\nbsp{}ref:sec:discussion compares the proposed synthesis method with the classical mixed-sensitivity synthesis, and extends it for the shaping of more than two complementary filters.
* Sensor Fusion and Complementary Filters Requirements
<<sec:requirements>>
** Introduction :ignore:
Complementary filters provides a framework for fusing signals from different sensors.
Complementary filtering 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.
These requirements are discussed in this section.
@@ -161,16 +170,16 @@ These requirements are discussed in this section.
A general sensor fusion architecture using complementary filters is shown in Fig.\nbsp{}ref:fig:sensor_fusion_overview where several sensors (here two) are measuring the same physical quantity $x$.
The two sensors output signals $\hat{x}_1$ and $\hat{x}_2$ are estimates of $x$.
Each of these estimates are then filtered out by complementary filters and combined to form a new estimate $\hat{x}$.
These estimates are then filtered out by complementary filters and combined to form a new estimate $\hat{x}$.
The resulting sensor, termed as super sensor, can have larger bandwidth and better noise characteristics in comparison to the individual sensor.
The resulting sensor, termed as super sensor, can have larger bandwidth and better noise characteristics in comparison to the individual sensors.
This means that the super sensor provides an estimate $\hat{x}$ of $x$ which can be more accurate over a larger frequency band than the outputs of the individual sensors.
#+name: fig:sensor_fusion_overview
#+caption: Schematic of a sensor fusion architecture using complementary filters
#+caption: Schematic of a sensor fusion architecture using complementary filters.
[[file:figs/sensor_fusion_overview.pdf]]
The complementary property of filters $H_1(s)$ and $H_2(s)$ implies that the sum of their transfer functions is equal to unity.
The complementary property of filters $H_1(s)$ and $H_2(s)$ implies that the sum of their transfer functions is equal to one.
That is, unity magnitude and zero phase at all frequencies.
Therefore, a pair of complementary filter needs to satisfy the following condition:
#+name: eq:comp_filter
@@ -191,22 +200,22 @@ Before filtering the sensor outputs $\tilde{x}_i$ by the complementary filters,
This normalization consists of using an estimate $\hat{G}_i(s)$ of the sensor dynamics $G_i(s)$, and filtering the sensor output by the inverse of this estimate $\hat{G}_i^{-1}(s)$ as shown in Fig.\nbsp{}ref:fig:sensor_model_calibrated.
It is here supposed that the sensor inverse $\hat{G}_i^{-1}(s)$ is proper and stable.
This way, the units of the estimates $\hat{x}_i$ are equal to the units of the physical quantity $x$.
The sensor dynamics estimate $\hat{G}_1(s)$ can be a simple gain or a more complex transfer function.
The sensor dynamics estimate $\hat{G}_i(s)$ can be a simple gain or a more complex transfer function.
#+begin_export latex
\begin{figure}[htbp]
\begin{subfigure}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model.pdf}
\caption{\label{fig:sensor_model} Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$}
\caption{\label{fig:sensor_model} Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_calibrated.pdf}
\caption{\label{fig:sensor_model_calibrated} Normalized sensors using the inverse of an estimate $\hat{G}_i(s)$ of the sensor dynamics}
\caption{\label{fig:sensor_model_calibrated} Normalized sensors using the inverse of an estimate $\hat{G}_i(s)$ of the sensor dynamics.}
\end{subfigure}
\caption{\label{fig:sensor_models}Sensor models with and without normalization}
\caption{\label{fig:sensor_models}Sensor models with and without normalization.}
\centering
\end{figure}
#+end_export
@@ -222,7 +231,7 @@ The super sensor output is therefore equal to:
\end{equation}
#+name: fig:fusion_super_sensor
#+caption: Sensor fusion architecture with two normalized sensors
#+caption: Sensor fusion architecture with two normalized sensors.
[[file:figs/fusion_super_sensor.pdf]]
** Noise Sensor Filtering
@@ -236,23 +245,23 @@ In this section, it is supposed that all the sensors are perfectly normalized, s
The effect of a non-perfect normalization will be discussed in the next section.
Provided eqref:eq:perfect_dynamics is verified, the super sensor output $\hat{x}$ is then:
Provided\nbsp{}eqref:eq:perfect_dynamics is verified, the super sensor output $\hat{x}$ is then equal to:
#+name: eq:estimate_perfect_dyn
\begin{equation}
\hat{x} = x + H_1(s) n_1 + H_2(s) n_2
\end{equation}
From eqref:eq:estimate_perfect_dyn, the complementary filters $H_1(s)$ and $H_2(s)$ are shown to only operate on the sensor's noises.
From\nbsp{}eqref:eq:estimate_perfect_dyn, the complementary filters $H_1(s)$ and $H_2(s)$ are shown to only operate on the noise of the sensors.
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured.
This is why the two filters must be complementary.
The estimation error $\delta x$, defined as the difference between the sensor output $\hat{x}$ and the measured quantity $x$, is computed for the super sensor eqref:eq:estimate_error.
The estimation error $\delta x$, defined as the difference between the sensor output $\hat{x}$ and the measured quantity $x$, is computed for the super sensor\nbsp{}eqref:eq:estimate_error.
#+name: eq:estimate_error
\begin{equation}
\delta x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2
\end{equation}
As shown in eqref:eq:noise_filtering_psd, the Power Spectral Density (PSD) of the estimation error $\Phi_{\delta x}$ depends both on the norm of the two complementary filters and on the PSD of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$.
As shown in\nbsp{}eqref:eq:noise_filtering_psd, the Power Spectral Density (PSD) of the estimation error $\Phi_{\delta x}$ depends both on the norm of the two complementary filters and on the PSD of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$.
#+name: eq:noise_filtering_psd
\begin{equation}
\Phi_{\delta x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega)
@@ -263,17 +272,17 @@ This is the simplest form of sensor fusion with complementary filters.
However, the two sensors have usually high noise levels over distinct frequency regions.
In such case, to lower the noise of the super sensor, the norm $|H_1(j\omega)|$ has to be small when $\Phi_{n_1}(\omega)$ is larger than $\Phi_{n_2}(\omega)$ and the norm $|H_2(j\omega)|$ has to be small when $\Phi_{n_2}(\omega)$ is larger than $\Phi_{n_1}(\omega)$.
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
Hence, by properly shaping the norm of the complementary filters, it is possible to reduce the noise of the super sensor.
** Sensor Fusion Robustness
<<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\nbsp{}eqref:eq:perfect_dynamics is not verified.
In order to study such imperfection, a multiplicative input uncertainty is added to the sensor dynamics (Fig.\nbsp{}ref:fig:sensor_model_uncertainty).
The nominal model is taken as the estimated model for the normalization $\hat{G}_i(s)$, $\Delta_i$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $w_i(s)$ is a weighting transfer function representing the magnitude of the uncertainty.
The nominal model is the estimated model used for the normalization $\hat{G}_i(s)$, $\Delta_i(s)$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $w_i(s)$ is a weighting transfer function representing the magnitude of the uncertainty.
The weight $w_i(s)$ is chosen such that the real sensor dynamics $G(j\omega)$ is contained in the uncertain region represented by a circle in the complex plane, centered on $1$ and with a radius equal to $|w_i(j\omega)|$.
The weight $w_i(s)$ is chosen such that the real sensor dynamics $G_i(j\omega)$ is contained in the uncertain region represented by a circle in the complex plane, centered on $1$ and with a radius equal to $|w_i(j\omega)|$.
As the nominal sensor dynamics is taken as the normalized filter, the normalized sensor can be further simplified as shown in Fig.\nbsp{}ref:fig:sensor_model_uncertainty_simplified.
@@ -282,41 +291,40 @@ As the nominal sensor dynamics is taken as the normalized filter, the normalized
\begin{subfigure}[b]{0.59\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_uncertainty.pdf}
\caption{\label{fig:sensor_model_uncertainty} Sensor with multiplicative input uncertainty}
\caption{\label{fig:sensor_model_uncertainty} Sensor with multiplicative input uncertainty.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.39\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_uncertainty_simplified.pdf}
\caption{\label{fig:sensor_model_uncertainty_simplified} Simplified sensor model}
\caption{\label{fig:sensor_model_uncertainty_simplified} Simplified sensor model.}
\end{subfigure}
\hfill
\caption{\label{fig:sensor_models_uncertainty}Sensor models with dynamical uncertainty}
\caption{\label{fig:sensor_models_uncertainty}Sensor models with dynamical uncertainty.}
\centering
\end{figure}
#+end_export
The sensor fusion architecture with two sensor models including dynamical uncertainty is shown in Fig.\nbsp{}ref:fig:sensor_fusion_dynamic_uncertainty.
The sensor fusion architecture with the sensor models including dynamical uncertainty is shown in Fig.\nbsp{}ref:fig:sensor_fusion_dynamic_uncertainty.
#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty.
[[file:figs/sensor_fusion_dynamic_uncertainty.pdf]]
The super sensor dynamics eqref:eq:super_sensor_dyn_uncertainty is no longer equal to $1$ and now depends on the sensor dynamical uncertainty weights $w_i(s)$ as well as on the complementary filters $H_i(s)$.
The super sensor dynamics\nbsp{}eqref:eq:super_sensor_dyn_uncertainty is no longer equal to $1$ and now depends on the sensor dynamical uncertainty weights $w_i(s)$ as well as on the complementary filters $H_i(s)$.
#+name: eq:super_sensor_dyn_uncertainty
\begin{equation}
\frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s)
\end{equation}
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ as shown in Fig.\nbsp{}ref:fig:uncertainty_set_super_sensor.
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ (Fig.\nbsp{}ref:fig:uncertainty_set_super_sensor).
#+name: fig:uncertainty_set_super_sensor
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency $\omega$ is here omitted.
[[file:figs/uncertainty_set_super_sensor.pdf]]
The super sensor dynamical uncertainty (i.e. the robustness of the fusion) clearly depends on the complementary filters norms.
For instance, the phase $\Delta\phi(\omega)$ added by the super sensor dynamics at frequency $\omega$ is bounded by $\Delta\phi_{\text{max}}(\omega)$ which can be found by drawing a tangent from the origin to the uncertainty circle of the super sensor (Fig.\nbsp{}ref:fig:uncertainty_set_super_sensor).
Therefore, the phase uncertainty of the super sensor dynamics depends on the Complementary filters norms eqref:eq:max_phase_uncertainty.
The super sensor dynamical uncertainty, and hence the robustness of the fusion, clearly depends on the complementary filters' norm.
For instance, the phase $\Delta\phi(\omega)$ added by the super sensor dynamics at frequency $\omega$ is bounded by $\Delta\phi_{\text{max}}(\omega)$ which can be found by drawing a tangent from the origin to the uncertainty circle of the super sensor (Fig.\nbsp{}ref:fig:uncertainty_set_super_sensor) and that is mathematically described by\nbsp{}eqref:eq:max_phase_uncertainty.
#+name: eq:max_phase_uncertainty
\begin{equation}
@@ -330,17 +338,17 @@ Typically, the norm of the complementary filter $|H_i(j\omega)|$ should be made
<<sec:hinf_method>>
** Introduction :ignore:
As shown in Section\nbsp{}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.
As shown in Section\nbsp{}ref:sec:requirements, the noise and robustness of the super sensor are a function of the complementary filters' norm.
Therefore, a synthesis method of complementary filters that allows to shape their norm would be of great use.
In this section, such synthesis is proposed by writing the synthesis objective as a standard $\mathcal{H}_\infty$ optimization problem.
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' shape during the synthesis, their proper design is discussed.
Finally, the synthesis method is validated on an simple example.
** Synthesis Objective
<<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.
This is equivalent as to finding proper and stable transfer functions $H_1(s)$ and $H_2(s)$ such that conditions eqref:eq:hinf_cond_complementarity, eqref:eq:hinf_cond_h1 and eqref:eq:hinf_cond_h2 are satisfied.
The synthesis objective is to shape the norm of two filters $H_1(s)$ and $H_2(s)$ while ensuring their complementary property\nbsp{}eqref:eq:comp_filter.
This is equivalent as to finding proper and stable transfer functions $H_1(s)$ and $H_2(s)$ such that conditions\nbsp{}eqref:eq:hinf_cond_complementarity, eqref:eq:hinf_cond_h1 and\nbsp{}eqref:eq:hinf_cond_h2 are satisfied.
#+name: eq:comp_filter_problem_form
\begin{subequations}
@@ -350,14 +358,14 @@ This is equivalent as to finding proper and stable transfer functions $H_1(s)$ a
& |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2}
\end{align}
\end{subequations}
$W_1(s)$ 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 norm of the complementary filters during the synthesis.
** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis
<<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.
Consider the generalized plant $P(s)$ shown in Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_plant and mathematically described by eqref:eq:generalized_plant.
Consider the generalized plant $P(s)$ shown in Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_plant and mathematically described by\nbsp{}eqref:eq:generalized_plant.
#+name: eq:generalized_plant
\begin{equation}
@@ -369,28 +377,28 @@ Consider the generalized plant $P(s)$ shown in Fig.\nbsp{}ref:fig:h_infinity_rob
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/h_infinity_robust_fusion_plant.pdf}
\caption{\label{fig:h_infinity_robust_fusion_plant} Generalized plant}
\caption{\label{fig:h_infinity_robust_fusion_plant} Generalized plant.}
\vfill
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/h_infinity_robust_fusion_fb.pdf}
\caption{\label{fig:h_infinity_robust_fusion_fb} Generalized plant with the synthesized filter}
\caption{\label{fig:h_infinity_robust_fusion_fb} Generalized plant with the synthesized filter.}
\end{subfigure}
\hfill
\caption{\label{fig:h_infinity_robust_fusion} Architecture for the $\mathcal{H}_\infty$ synthesis of complementary filters}
\caption{\label{fig:h_infinity_robust_fusion} Architecture for the $\mathcal{H}_\infty$ synthesis of complementary filters.}
\centering
\end{figure}
#+end_export
Applying the standard $\mathcal{H}_\infty$ synthesis on the generalized plant $P(s)$ is then equivalent as finding a stable filter $H_2(s)$ which based on $v$, generates a signal $u$ such that the $\mathcal{H}_\infty$ norm of the system in Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_fb from $w$ to $[z_1, \ z_2]$ is less than one eqref:eq:hinf_syn_obj.
Applying the standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P(s)$ is then equivalent as finding a stable filter $H_2(s)$ which based on $v$, generates a signal $u$ such that the $\mathcal{H}_\infty$ norm of the system in Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_fb from $w$ to $[z_1, \ z_2]$ is less than one\nbsp{}eqref:eq:hinf_syn_obj.
#+name: eq:hinf_syn_obj
\begin{equation}
\left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
By then defining $H_1(s)$ to be the complementary of $H_2(s)$ eqref:eq:definition_H1, the $\mathcal{H}_\infty$ synthesis objective becomes equivalent to eqref:eq:hinf_problem which ensure that eqref:eq:hinf_cond_h1 and eqref:eq:hinf_cond_h2 are satisfied.
By then defining $H_1(s)$ to be the complementary of $H_2(s)$ eqref:eq:definition_H1, the $\mathcal{H}_\infty$ synthesis objective becomes equivalent to\nbsp{}eqref:eq:hinf_problem which ensures that\nbsp{}eqref:eq:hinf_cond_h1 and\nbsp{}eqref:eq:hinf_cond_h2 are satisfied.
#+name: eq:definition_H1
\begin{equation}
@@ -402,26 +410,25 @@ By then defining $H_1(s)$ to be the complementary of $H_2(s)$ eqref:eq:definitio
\left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
Therefore, applying the $\mathcal{H}_\infty$ synthesis on the standard plant $P(s)$ eqref:eq:generalized_plant will generate two filters $H_2(s)$ and $H_1(s) \triangleq 1 - H_2(s)$ that are complementary eqref:eq:comp_filter_problem_form and such that there norms are bellow specified bounds [[eqref:eq:hinf_cond_h1]],eqref:eq:hinf_cond_h2.
Therefore, applying the $\mathcal{H}_\infty$ synthesis to the standard plant $P(s)$ eqref:eq:generalized_plant will generate two filters $H_2(s)$ and $H_1(s) \triangleq 1 - H_2(s)$ that are complementary\nbsp{}eqref:eq:comp_filter_problem_form and such that there norms are bellow specified bounds [[eqref:eq:hinf_cond_h1]],\nbsp{}eqref:eq:hinf_cond_h2.
Note that there is not an equivalence between the $\mathcal{H}_\infty$ norm condition\nbsp{}eqref:eq:hinf_problem and the initial synthesis objectives\nbsp{}eqref:eq:hinf_cond_h1 and\nbsp{}eqref:eq:hinf_cond_h2, but only an implication.
Hence, the optimization may be a little bit conservative with respect to the "set" of filters on which it is performed.
There might be solutions were the objectives\nbsp{}eqref:eq:hinf_cond_h1 and\nbsp{}eqref:eq:hinf_cond_h2 are valid but where the $\mathcal{H}_\infty$ norm\nbsp{}eqref:eq:hinf_problem is larger than one.
Note that there is only an implication between the $\mathcal{H}_\infty$ norm condition\nbsp{}eqref:eq:hinf_problem and the initial synthesis objectives\nbsp{}eqref:eq:hinf_cond_h1 and\nbsp{}eqref:eq:hinf_cond_h2 and not an equivalence.
Hence, the optimization may be a little bit conservative with respect to the set of filters on which it is performed, see\nbsp{}[[cite:skogestad07_multiv_feedb_contr][Chap. 2.8.3]].
In practice, this is however not an found to be an issue.
** Weighting Functions Design
<<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 complementary filters' norm.
The proper design of these weighting functions is of primary importance for the success of the presented $\mathcal{H}_\infty$ synthesis of complementary filters.
First, only proper and stable transfer functions should be used.
Second, the order of the weighting functions should stay reasonably small in order to reduce the computational costs associated with the solving of the optimization problem and for the physical implementation of the filters (the order of the synthesized filters being equal to the sum of the weighting functions orders).
Third, one should not forget the fundamental limitations imposed by the complementary property eqref:eq:comp_filter.
Second, the order of the weighting functions should stay reasonably small in order to reduce the computational costs associated with the solving of the optimization problem and for the physical implementation of the filters (the synthesized filters' order being equal to the sum of the weighting functions' order).
Third, one should not forget the fundamental limitations imposed by the complementary property\nbsp{}eqref:eq:comp_filter.
This implies for instance that $|H_1(j\omega)|$ and $|H_2(j\omega)|$ cannot be made small at the same frequency.
When designing complementary filters, it is usually desired to specify their slopes, their "blending" frequency and their maximum gains at low and high frequency.
To help with the design of the weighting functions such that the above specification can be easily expressed, the formula eqref:eq:weight_formula is proposed.
To easily express these specifications, formula\nbsp{}eqref:eq:weight_formula is proposed to help with the design of weighting functions.
#+name: eq:weight_formula
\begin{equation}
@@ -432,13 +439,13 @@ To help with the design of the weighting functions such that the above specifica
}\right)^n
\end{equation}
The parameters in formula eqref:eq:weight_formula are:
- $G_0 = lim_{\omega \to 0} |W(j\omega)|$: the low frequency gain
- $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$: the high frequency gain
The parameters in formula\nbsp{}eqref:eq:weight_formula are:
- $G_0 = \lim_{\omega \to 0} |W(j\omega)|$: the low frequency gain
- $G_\infty = \lim_{\omega \to \infty} |W(j\omega)|$: the high frequency gain
- $G_c = |W(j\omega_c)|$: the gain at a specific frequency $\omega_c$ in $\si{rad/s}$.
- $n$: the slope between high and low frequency. It also corresponds to the order of the weighting function.
The parameters $G_0$, $G_c$ and $G_\infty$ should either satisfy condition eqref:eq:cond_formula_1 or eqref:eq:cond_formula_2.
The parameters $G_0$, $G_c$ and $G_\infty$ should either satisfy\nbsp{}eqref:eq:cond_formula_1 or\nbsp{}eqref:eq:cond_formula_2.
#+name: eq:condition_params_formula
\begin{subequations}
\begin{align}
@@ -447,22 +454,25 @@ The parameters $G_0$, $G_c$ and $G_\infty$ should either satisfy condition eqref
\end{align}
\end{subequations}
An example of the obtained magnitude of a weighting function generated using eqref:eq:weight_formula is shown in Fig.\nbsp{}ref:fig:weight_formula.
The typical magnitude of a weighting function generated using\nbsp{}eqref:eq:weight_formula is shown in Fig.\nbsp{}ref:fig:weight_formula.
#+name: fig:weight_formula
#+caption: Magnitude of a weighting function generated using the proposed formula eqref:eq:weight_formula, $G_0 = 1e^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$
#+caption: Magnitude of a weighting function generated using formula\nbsp{}eqref:eq:weight_formula, $G_0 = 1e^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$.
[[file:figs/weight_formula.pdf]]
** Validation of the proposed synthesis method
<<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 blending frequency is around $\SI{10}{Hz}$
- the slope of $|H_1(j\omega)|$ is $+2$ below $\SI{10}{Hz}$, its low frequency gain is $10^{-3}$
- the slope of $|H_2(j\omega)|$ is $-3$ above $\SI{10}{Hz}$, its high frequency gain is $10^{-3}$
The proposed methodology for the design of complementary filters is now applied on a simple example.
Let's suppose two complementary filters $H_1(s)$ and $H_2(s)$ have to be designed such that:
- the blending frequency is around $\SI{10}{Hz}$.
- the slope of $|H_1(j\omega)|$ is $+2$ below $\SI{10}{Hz}$.
Its low frequency gain is $10^{-3}$.
- the slope of $|H_2(j\omega)|$ is $-3$ above $\SI{10}{Hz}$.
Its high frequency gain is $10^{-3}$.
The first step is to translate the above requirements into the design of the weighting functions.
The proposed formula eqref:eq:weight_formula is here used for such purpose.
The first step is to translate the above requirements by properly designing the weighting functions.
The proposed formula\nbsp{}eqref:eq:weight_formula is here used for such purpose.
Parameters used are summarized in Table\nbsp{}ref:tab:weights_params.
The inverse magnitudes of the designed weighting functions, which are representing the maximum allowed norms of the complementary filters, are shown by the dashed lines in Fig.\nbsp{}ref:fig:weights_W1_W2.
@@ -470,31 +480,31 @@ The inverse magnitudes of the designed weighting functions, which are representi
\begin{figure}
\begin{minipage}[b]{0.49\linewidth}
\centering
\begin{tabularx}{0.65\linewidth}{ccc}
\begin{tabularx}{0.60\linewidth}{ccc}
\toprule
Parameters & \(W_1(s)\) & \(W_2(s)\)\\
Parameters & \(W_1(s)\) & \(W_2(s)\)\\
\midrule
\(G_0\) & \(0.1\) & \(1000\)\\
\(G_\infty\) & \(1000\) & \(0.1\)\\
\(G_0\) & \(0.1\) & \(1000\)\\
\(G_\infty\) & \(1000\) & \(0.1\)\\
\(\omega_c\) & \(2\pi\cdot10\) & \(2\pi\cdot10\)\\
\(G_c\) & \(0.45\) & \(0.45\)\\
\(n\) & \(2\) & \(3\)\\
\(G_c\) & \(0.45\) & \(0.45\)\\
\(n\) & \(2\) & \(3\)\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:weights_params}Parameters used for weighting functions \(W_1(s)\) and \(W_2(s)\) using \eqref{eq:weight_formula}}
\captionof{table}{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\) using~\eqref{eq:weight_formula}.}
\end{minipage}
\hfill
\begin{minipage}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/weights_W1_W2.pdf}
\caption{\label{fig:weights_W1_W2}Inverse magnitude of the weighting functions}
\caption{\label{fig:weights_W1_W2}Inverse magnitude of the weighting functions.}
\end{minipage}
\end{figure}
#+end_export
The $\mathcal{H}_\infty$ synthesis is then applied to the generalized plant of Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_plant on efficiently solved in Matlab\nbsp{}cite:matlab20 using the Robust Control Toolbox.
The standard $\mathcal{H}_\infty$ synthesis is then applied to the generalized plant of Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_plant and efficiently solved using Matlab\nbsp{}cite:matlab20.
The filter $H_2(s)$ that minimizes the $\mathcal{H}_\infty$ norm between $w$ and $[z_1,\ z_2]^T$ is obtained.
The $\mathcal{H}_\infty$ norm is here found to be close to one eqref:eq:hinf_synthesis_result which indicates that the synthesis is successful: the complementary filters norms are below the maximum specified upper bounds.
The $\mathcal{H}_\infty$ norm is here found to be close to one\nbsp{}eqref:eq:hinf_synthesis_result which indicates that the synthesis is successful: the complementary filters norms are below the maximum specified upper bounds.
This is confirmed by the bode plots of the obtained complementary filters in Fig.\nbsp{}ref:fig:hinf_filters_results.
#+name: eq:hinf_synthesis_result
@@ -502,8 +512,8 @@ This is confirmed by the bode plots of the obtained complementary filters in Fig
\left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \approx 1
\end{equation}
Their transfer functions in the Laplace domain are given in eqref:eq:hinf_synthesis_result_tf.
As expected, the obtained filters are of order $5$, that is the sum of the weighting functions orders.
The transfer functions in the Laplace domain of the complementary filters are given in\nbsp{}eqref:eq:hinf_synthesis_result_tf.
As expected, the obtained filters are of order $5$, that is the sum of the weighting functions' order.
#+name: eq:hinf_synthesis_result_tf
\begin{subequations}
@@ -515,7 +525,7 @@ As expected, the obtained filters are of order $5$, that is the sum of the weigh
#+name: fig:hinf_filters_results
#+caption: Bode plot of the obtained complementary filters
#+caption: Bode plot of the obtained complementary filters.
[[file:figs/hinf_filters_results.pdf]]
This simple example illustrates the fact that the proposed methodology for complementary filters shaping is easy to use and effective.
@@ -525,39 +535,39 @@ A more complex real life example is taken up in the next section.
<<sec:application_ligo>>
** Introduction :ignore:
Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO\nbsp{}cite:matichard15_seism_isolat_advan_ligo,hua05_low_ligo, the VIRGO\nbsp{}cite:lucia18_low_frequen_optim_perfor_advan,heijningen18_low and the KAGRA [[cite:sekiguchi16_study_low_frequen_vibrat_isolat_system][Chap. 5]].
Sensor fusion using complementary filters are widely used in the active vibration isolation systems at gravitational wave detectors, such as at the LIGO\nbsp{}cite:matichard15_seism_isolat_advan_ligo,hua05_low_ligo, the Virgo\nbsp{}cite:lucia18_low_frequen_optim_perfor_advan,heijningen18_low and the KAGRA [[cite:sekiguchi16_study_low_frequen_vibrat_isolat_system][Chap. 5]].
In the first isolation stage at the LIGO, two sets of complementary filters are used and included in a feedback loop\nbsp{}cite:hua04_low_ligo.
In the first isolation stage at the LIGO, two sets of complementary filters are used to form a super sensor that is incorporated in a feedback loop\nbsp{}cite:hua04_low_ligo.
A set of complementary filters ($L_2,H_2$) is first used to fuse a seismometer and a geophone.
Then, another set of complementary filters ($L_1,H_1$) is used to merge the output of the first "inertial super sensor" with a position sensor.
A simplified block diagram of the sensor fusion architecture is shown in Fig.\nbsp{}ref:fig:ligo_super_sensor_architecture.
#+name: fig:ligo_super_sensor_architecture
#+caption: Simplified block diagram of the sensor blending strategy for the first stage at the LIGO\nbsp{}cite:hua04_low_ligo
#+caption: Simplified block diagram of the sensor blending strategy for the first stage at the LIGO\nbsp{}cite:hua04_low_ligo.
[[file:figs/ligo_super_sensor_architecture.pdf]]
The fusion of the position sensor at low frequency with the "inertial super sensor" at high frequency using the complementary filters ($L_1,H_1$) is done for several reasons, first of which is to give the super sensor a DC sensibility and therefore allow the feedback loop to have authority at zero frequency.
The requirements on those filters are very tight and thus their design is complex and should be expressed as an optimization problem.
The fusion of the position sensor at low frequency with the "inertial super sensor" at high frequency using the complementary filters ($L_1,H_1$) is done for several reasons, first of which is to give the super sensor a DC sensibility that allows the feedback loop to have authority at zero frequency.
The requirements on those filters are stringent and thus their design is complex and should be expressed as an optimization problem.
The approach used in\nbsp{}cite:hua05_low_ligo is to use FIR complementary filters and to write the synthesis as a convex optimization problem.
After synthesis, the obtained FIR filters were found to be compliant with the requirements.
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.
However they are of high order so their implementation is quite complex.
In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated by using the same set of requirements.
** Complementary Filters Specifications
<<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\nbsp{}cite:hua04_polyp_fir_compl_filter_contr_system):
- From $0$ to $\SI{0.008}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less or equal to $8 \times 10^{-4}$
- Between $\SI{0.008}{Hz}$ to $\SI{0.04}{Hz}$, the filter $L_1(s)$ should attenuate the input signal proportional to frequency cubed
- Between $\SI{0.04}{Hz}$ to $\SI{0.1}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less than $3$
- Above $\SI{0.1}{Hz}$, the magnitude $|H_1(j\omega)|$ should be less than $0.045$
- Below $\SI{0.008}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less than $8 \times 10^{-4}$.
- From $\SI{0.008}{Hz}$ to $\SI{0.04}{Hz}$, the filter $L_1(s)$ should attenuate the input signal proportional to frequency cubed.
- From $\SI{0.04}{Hz}$ to $\SI{0.1}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less than $3$.
- Above $\SI{0.1}{Hz}$, the magnitude $|H_1(j\omega)|$ should be less than $0.045$.
These specifications are therefore upper bounds on the complementary filters' magnitudes.
These specifications are therefore upper bounds on the complementary filters' magnitude.
They are physically represented in Fig.\nbsp{}ref:fig:fir_filter_ligo as well as the obtained magnitude of the FIR filters in\nbsp{}cite:hua05_low_ligo.
#+name: fig:fir_filter_ligo
#+caption: Specifications and Bode plot of the obtained FIR complementary filters in\nbsp{}cite:hua05_low_ligo. The filters are here obtained using the SeDuMi Matlab toolbox\nbsp{}cite:sturm99_using_sedum
#+caption: Specifications and Bode plot of the obtained FIR complementary filters in\nbsp{}cite:hua05_low_ligo. The filters are here obtained using the SeDuMi Matlab toolbox\nbsp{}cite:sturm99_using_sedum.
[[file:figs/fir_filter_ligo.pdf]]
** Weighting Functions Design
@@ -567,11 +577,11 @@ The weighting functions should be designed such that their inverse magnitude is
However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters.
A Type I Chebyshev filter of order $20$ is used for the weighting transfer function $W_L(s)$ corresponding to the low pass filter.
For the one corresponding to the high pass filter $W_H(s)$, a $7^{\text{th}}$ order transfer function is designed.
The magnitudes of the weighting functions are shown in Fig.\nbsp{}ref:fig:ligo_weights.
For the one corresponding to the high pass filter $W_H(s)$, a $7^{\text{th}}$ order transfer function is manually designed.
The inverse magnitudes of the weighting functions are shown in Fig.\nbsp{}ref:fig:ligo_weights.
#+name: fig:ligo_weights
#+caption: Specifications and weighting functions inverse magnitudes
#+caption: Specifications and weighting functions' inverse magnitude.
[[file:figs/ligo_weights.pdf]]
** $\mathcal{H}_\infty$ Synthesis of the complementary filters
@@ -580,54 +590,53 @@ The magnitudes of the weighting functions are shown in Fig.\nbsp{}ref:fig:ligo_w
The proposed $\mathcal{H}_\infty$ synthesis is performed on the generalized plant shown in Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_plant.
After optimization, the $\mathcal{H}_\infty$ norm from $w$ to $[z_1,\ z_2]^T$ is found close to one indication successful synthesis.
In Fig.\nbsp{}ref:fig:comp_fir_ligo_hinf, the bode plot of the obtained complementary filters are compared with the FIR filters of order 512 obtained in\nbsp{}cite:hua05_low_ligo.
Even though the complementary filters using the $\mathcal{H}_\infty$ synthesis are of much lower order (order 27), they are found to be very close to the FIR filters.
Even though the complementary filters using the $\mathcal{H}_\infty$ synthesis are of much lower order (order 27), they are found to be close to the FIR filters.
This confirms the effectiveness of the proposed synthesis method even when the complementary filters are subject to complex requirements.
#+name: fig:comp_fir_ligo_hinf
#+caption: Comparison of the FIR filters (dashed) designed in\nbsp{}cite:hua05_low_ligo with the filters obtained with $\mathcal{H}_\infty$ synthesis (solid)
#+caption: Comparison of the FIR filters (dashed) designed in\nbsp{}cite:hua05_low_ligo with the filters obtained with $\mathcal{H}_\infty$ synthesis (solid).
[[file:figs/comp_fir_ligo_hinf.pdf]]
* Discussion
<<sec:discussion>>
** Introduction :ignore:
** "Closed-Loop" complementary filters
<<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.\nbsp{}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 idea is discussed in\nbsp{}cite:mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas.
#+name: fig:feedback_sensor_fusion
#+caption: "Closed-Loop" complementary filters
#+caption: "Closed-Loop" complementary filters.
[[file:figs/feedback_sensor_fusion.pdf]]
Consider the feedback architecture of Fig.\nbsp{}ref:fig:feedback_sensor_fusion, with two inputs $\hat{x}_1$ and $\hat{x}_2$, and one output $\hat{x}$.
The output $\hat{x}$ is linked to the inputs by eqref:eq:closed_loop_complementary_filters.
The output $\hat{x}$ is linked to the inputs by\nbsp{}eqref:eq:closed_loop_complementary_filters.
#+name: eq:closed_loop_complementary_filters
\begin{equation}
\hat{x} = \underbrace{\frac{1}{1 + L(s)}}_{S(s)} \hat{x}_1 + \underbrace{\frac{L(s)}{1 + L(s)}}_{T(s)} \hat{x}_2
\end{equation}
As for any classical feedback architecture, we have that the sum of the sensitivity and complementary sensitivity transfer function is equal to one\nbsp{}eqref:eq:sensitivity_sum.
As for any classical feedback architecture, we have that the sum of the sensitivity transfer function $S(s)$ and complementary sensitivity transfer function $T_(s)$ is equal to one\nbsp{}eqref:eq:sensitivity_sum.
#+name: eq:sensitivity_sum
\begin{equation}
T(s) + S(s) = 1
S(s) + T(s) = 1
\end{equation}
Therefore, provided that the closed-loop system is stable, the closed-loop system in Fig.\nbsp{}ref:fig:feedback_sensor_fusion is corresponding to two complementary filters.
Therefore, provided that the the closed-loop system in Fig.\nbsp{}ref:fig:feedback_sensor_fusion is stable, it can be used as a set of two complementary filters.
Two sensors can then be merged as shown in Fig.\nbsp{}ref:fig:feedback_sensor_fusion_arch.
#+name: fig:feedback_sensor_fusion_arch
#+caption: Classical feedback architecture used for sensor fusion
#+caption: Classical feedback architecture used for sensor fusion.
[[file:figs/feedback_sensor_fusion_arch.pdf]]
One of the main advantage of implementing and designing complementary filters using the feedback architecture of Fig.\nbsp{}ref:fig:feedback_sensor_fusion is that all the tools of the linear control theory can be applied for the design of the filters.
If one want to shape both $\frac{\hat{x}}{\hat{x}_1}(s) = S(s)$ and $\frac{\hat{x}}{\hat{x}_2}(s) = T(s)$, the $\mathcal{H}_\infty$ mixed-sensitivity synthesis can be easily applied.
To do so, weighting functions $W_1(s)$ and $W_2(s)$ are added to respectively shape $S(s)$ and $T(s)$ (Fig.\nbsp{}ref:fig:feedback_synthesis_architecture).
Then the system is re-organized to form the generalized plant $P_L(s)$ shown in Fig.\nbsp{}ref:fig:feedback_synthesis_architecture_generalized_plant.
The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can finally be performed by applying the $\mathcal{H}_\infty$ synthesis to the generalized plant $P_L(s)$ which is described by eqref:eq:generalized_plant_mixed_sensitivity.
Then the system is rearranged to form the generalized plant $P_L(s)$ shown in Fig.\nbsp{}ref:fig:feedback_synthesis_architecture_generalized_plant.
The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can finally be performed by applying the standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P_L(s)$ which is described by\nbsp{}eqref:eq:generalized_plant_mixed_sensitivity.
#+name: eq:generalized_plant_mixed_sensitivity
\begin{equation}
@@ -637,63 +646,62 @@ The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can finally be performed by
\end{bmatrix}
\end{equation}
The output of the synthesis is a filter $L(s)$ such that the "closed-loop" $\mathcal{H}_\infty$ norm from $[w_1,\ w_2]$ to $z$ is less than one eqref:eq:comp_filters_feedback_obj.
The output of the synthesis is a filter $L(s)$ such that the "closed-loop" $\mathcal{H}_\infty$ norm from $[w_1,\ w_2]$ to $z$ of the system in Fig.\nbsp{}ref:fig:feedback_sensor_fusion is less than one\nbsp{}eqref:eq:comp_filters_feedback_obj.
#+name: eq:comp_filters_feedback_obj
\begin{equation}
\left\| \begin{matrix} \frac{z}{w_1} \\ \frac{z}{w_2} \end{matrix} \right\|_\infty = \left\| \begin{matrix} \frac{1}{1 + L(s)} W_1(s) \\ \frac{L(s)}{1 + L(s)} W_2(s) \end{matrix} \right\|_\infty \le 1
\end{equation}
If the synthesis is successful, two complementary filters are obtained with their magnitudes bounded by the inverse magnitudes of the weighting functions.
The sensor fusion can then be implemented as shown in Fig.\nbsp{}ref:fig:feedback_sensor_fusion_arch using the feedback architecture or more classically as shown in Fig.\nbsp{}ref:fig:sensor_fusion_overview by defining the two complementary filters as in\nbsp{}eqref:eq:comp_filters_feedback.
If the synthesis is successful, the transfer functions from $\hat{x}_1$ to $\hat{x}$ and from $\hat{x}_2$ to $\hat{x}$ have their magnitude bounded by the inverse magnitude of the corresponding weighting functions.
The sensor fusion can then be implemented using the feedback architecture in Fig.\nbsp{}ref:fig:feedback_sensor_fusion_arch or more classically as shown in Fig.\nbsp{}ref:fig:sensor_fusion_overview by defining the two complementary filters using\nbsp{}eqref:eq:comp_filters_feedback.
The two architectures are equivalent regarding their inputs/outputs relationships.
#+name: eq:comp_filters_feedback
\begin{equation}
H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)}
\end{equation}
The two architecture are equivalent regarding their inputs/outputs relationships.
#+begin_export latex
\begin{figure}[htbp]
\begin{subfigure}[t]{0.6\linewidth}
\centering
\includegraphics[scale=1]{figs/feedback_synthesis_architecture.pdf}
\caption{\label{fig:feedback_synthesis_architecture} Feedback architecture with included weights}
\caption{\label{fig:feedback_synthesis_architecture} Feedback architecture with included weights.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.4\linewidth}
\centering
\includegraphics[scale=1]{figs/feedback_synthesis_architecture_generalized_plant.pdf}
\caption{\label{fig:feedback_synthesis_architecture_generalized_plant} Generalized plant}
\caption{\label{fig:feedback_synthesis_architecture_generalized_plant} Generalized plant.}
\end{subfigure}
\hfill
\caption{\label{fig:h_inf_mixed_sensitivity_synthesis} $\mathcal{H}_\infty$ mixed-sensitivity synthesis}
\caption{\label{fig:h_inf_mixed_sensitivity_synthesis} $\mathcal{H}_\infty$ mixed-sensitivity synthesis.}
\centering
\end{figure}
#+end_export
As an example, two "closed-loop" complementary filters are designed using the $\mathcal{H}_\infty$ mixed-sensitivity synthesis.
The weighting functions are designed using formula\nbsp{}eqref:eq:weight_formula with parameters shown in Table\nbsp{}ref:tab:weights_params.
After synthesis, a filter $L(s)$ is obtained, its magnitude is shown in Fig.\nbsp{}ref:fig:hinf_filters_results_mixed_sensitivity by the dashed line.
After synthesis, a filter $L(s)$ is obtained whose magnitude is shown in Fig.\nbsp{}ref:fig:hinf_filters_results_mixed_sensitivity by the black dashed line.
The "closed-loop" complementary filters are compared with the inverse magnitude of the weighting functions in Fig.\nbsp{}ref:fig:hinf_filters_results_mixed_sensitivity confirming that the synthesis is successful.
The obtained "closed-loop" complementary filters are indeed equal to the ones obtained in Section\nbsp{}ref:sec:hinf_example.
#+name: fig:hinf_filters_results_mixed_sensitivity
#+caption: Bode plot of the obtained complementary filters after $\mathcal{H}_\infty$ mixed-sensitivity synthesis
#+caption: Bode plot of the obtained complementary filters after $\mathcal{H}_\infty$ mixed-sensitivity synthesis.
[[file:figs/hinf_filters_results_mixed_sensitivity.pdf]]
** Synthesis of more than two Complementary Filters
** Synthesis of a set of three complementary filters
<<sec:hinf_three_comp_filters>>
Some applications may require to merge more than two sensors\nbsp{}cite:stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands.
For instance at the LIGO\nbsp{}cite:matichard15_seism_isolat_advan_ligo, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor (Fig.\nbsp{}ref:fig:ligo_super_sensor_architecture).\par
For instance at the LIGO, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor\nbsp{}cite:matichard15_seism_isolat_advan_ligo (Fig.\nbsp{}ref:fig:ligo_super_sensor_architecture).\par
When merging $n>2$ sensors using complementary filters, two architectures can be used as shown in Fig.\nbsp{}ref:fig:sensor_fusion_three.
The fusion can either be done in a "sequential" way where $n-1$ sets of two complementary filters are used (Fig.\nbsp{}ref:fig:sensor_fusion_three_sequential), or in a "parallel" way where one set of $n$ complementary filters is used (Fig.\nbsp{}ref:fig:sensor_fusion_three_parallel).
In the first case, typical sensor fusion synthesis techniques can be used.
However, when a parallel architecture is used, a new synthesis method for a set of more than two complementary filters is required as only simple analytical formulas have been proposed in the literature\nbsp{cite:stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands}.
However, when a parallel architecture is used, a new synthesis method for a set of more than two complementary filters is required as only simple analytical formulas have been proposed in the literature\nbsp{}cite:stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands.
A generalization of the proposed synthesis method of complementary filters is presented in this section. \par
#+begin_export latex
@@ -701,15 +709,15 @@ A generalization of the proposed synthesis method of complementary filters is pr
\begin{subfigure}[b]{0.59\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_sequential.pdf}
\caption{\label{fig:sensor_fusion_three_sequential}Sequential fusion}
\caption{\label{fig:sensor_fusion_three_sequential}Sequential fusion.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.39\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_parallel.pdf}
\caption{\label{fig:sensor_fusion_three_parallel}Parallel fusion}
\caption{\label{fig:sensor_fusion_three_parallel}Parallel fusion.}
\end{subfigure}
\caption{\label{fig:sensor_fusion_three}Possible sensor fusion architecture when more than two sensors are to be merged}
\caption{\label{fig:sensor_fusion_three}Possible sensor fusion architecture when more than two sensors are to be merged.}
\centering
\end{figure}
#+end_export
@@ -718,62 +726,16 @@ The synthesis objective is to compute a set of $n$ stable transfer functions $[H
#+name: eq:hinf_problem_gen
\begin{subequations}
\begin{align}
& \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\
& \sum_{i=1}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\
& \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:hinf_cond_perf_gen}
\end{align}
\end{subequations}
where $[W_1(s),\ W_2(s),\ \dots,\ W_n(s)]$ are weighting transfer functions that are chosen to specify the maximum wanted norms of the complementary filters during the synthesis.
Such synthesis objective is very close to the one described in Section\nbsp{}ref:sec:synthesis_objective, and indeed the proposed synthesis method is a generalization of the one presented in Section\nbsp{}ref:sec:hinf_synthesis. \par
$[W_1(s),\ W_2(s),\ \dots,\ W_n(s)]$ are weighting transfer functions that are chosen to specify the maximum complementary filters' norm during the synthesis.
Before presenting the generalized synthesis method, the case with three sensors is presented.
Consider the generalized plant $P_3(s)$ shown in Fig.\nbsp{}ref:fig:comp_filter_three_hinf_gen_plant which is also described by\nbsp{}eqref:eq:generalized_plant_three_filters.
Such synthesis objective is closely related to the one described in Section\nbsp{}ref:sec:synthesis_objective, and indeed the proposed synthesis method is a generalization of the one presented in Section\nbsp{}ref:sec:hinf_synthesis. \par
#+name: eq:generalized_plant_three_filters
\begin{equation}
\begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix}
\end{equation}
#+begin_export latex
\begin{figure}[htbp]
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_gen_plant.pdf}
\caption{\label{fig:comp_filter_three_hinf_gen_plant} Generalized plant}
\vfill
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_fb.pdf}
\caption{\label{fig:comp_filter_three_hinf_fb} Generalized plant with the synthesized filter}
\end{subfigure}
\hfill
\caption{\label{fig:comp_filter_three_hinf} Architecture for the $\mathcal{H}_\infty$ synthesis of three complementary filters}
\centering
\end{figure}
#+end_export
Applying the $\mathcal{H}_\infty$ synthesis on the generalized plant $P_3(s)$ is equivalent as to find two stable filters $[H_2(s),\ H_3(s)]$ (shown in Fig.\nbsp{}ref:fig:comp_filter_three_hinf_fb) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2, \ z_3]$ is less than one\nbsp{}eqref:eq:hinf_syn_obj_three.
#+name: eq:hinf_syn_obj_three
\begin{equation}
\left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
\end{equation}
By defining $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\infty$ synthesis solves the design problem eqref:eq:hinf_problem_gen with $n=3$. \par
An example is given to validate the method where three sensors are used in different frequency bands.
For instance a displacement sensor from DC up to $\SI{1}{Hz}$, a geophone from $1$ to $\SI{10}{Hz}$ and an accelerometer above $\SI{10}{Hz}$.
Three weighting functions are designed using formula\nbsp{}eqref:eq:weight_formula and their inverse magnitudes are shown in Fig.\nbsp{}ref:fig:three_complementary_filters_results (dashed curves).
The $\mathcal{H}_\infty$ synthesis is performed on the generalized plant $P_3(s)$ and the bode plot of the obtained complementary filters are shown in Fig.\nbsp{}ref:fig:three_complementary_filters_results (solid lines). \par
#+name: fig:three_complementary_filters_results
#+caption: Bode plot of the inverse weighting functions and of the three complementary filters obtained using the $\mathcal{H}_\infty$ synthesis
[[file:figs/three_complementary_filters_results.pdf]]
Even though there might not be any practical application for a set of more than 3 complementary filters, it can still be designed using the same procedure.
A set of $n$ complementary filters can be shaped using the generalized plant $P_n(s)$ described by\nbsp{}eqref:eq:generalized_plant_n_filters.
A set of $n$ complementary filters can be shaped by applying the standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P_n(s)$ described by\nbsp{}eqref:eq:generalized_plant_n_filters.
#+name: eq:generalized_plant_n_filters
\begin{equation}
@@ -788,17 +750,77 @@ A set of $n$ complementary filters can be shaped using the generalized plant $P_
\end{bmatrix}
\end{equation}
If the synthesis if successful, a set of $n-1$ filters $[H_2(s),\ H_3(s),\ \dots,\ H_n(s)]$ are obtained such that\nbsp{}eqref:eq:hinf_syn_obj_gen is verified.
#+name: eq:hinf_syn_obj_gen
\begin{equation}
\left\|\begin{matrix} \left(1 - \left[ H_2(s) + H_3(s) + \dots + H_n(s) \right]\right) W_1(s) \\ H_2(s) W_2(s) \\ \vdots \\ H_n(s) W_n(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
$H_1(s)$ is then defined using\nbsp{}eqref:eq:h1_comp_h2_hn which is ensuring the complementary property for the set of $n$ filters\nbsp{}eqref:eq:hinf_cond_compl_gen.
Condition\nbsp{}eqref:eq:hinf_cond_perf_gen is satisfied thanks to\nbsp{}eqref:eq:hinf_syn_obj_gen.
#+name: eq:h1_comp_h2_hn
\begin{equation}
H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) + \dots + H_n(s) \big]
\end{equation}
An example is given to validate the proposed method for the synthesis of a set of three complementary filters.
The sensors to be merged are a displacement sensor from DC up to $\SI{1}{Hz}$, a geophone from $1$ to $\SI{10}{Hz}$ and an accelerometer above $\SI{10}{Hz}$.
Three weighting functions are designed using formula\nbsp{}eqref:eq:weight_formula and their inverse magnitude are shown in Fig.\nbsp{}ref:fig:three_complementary_filters_results (dashed curves).
Consider the generalized plant $P_3(s)$ shown in Fig.\nbsp{}ref:fig:comp_filter_three_hinf_gen_plant which is also described by\nbsp{}eqref:eq:generalized_plant_three_filters.
#+name: eq:generalized_plant_three_filters
\begin{equation}
\begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix}
\end{equation}
#+begin_export latex
\begin{figure}[htbp]
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_gen_plant.pdf}
\caption{\label{fig:comp_filter_three_hinf_gen_plant} Generalized plant.}
\vfill
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_fb.pdf}
\caption{\label{fig:comp_filter_three_hinf_fb} Generalized plant with the synthesized filter.}
\end{subfigure}
\hfill
\caption{\label{fig:comp_filter_three_hinf} Architecture for the $\mathcal{H}_\infty$ synthesis of three complementary filters.}
\centering
\end{figure}
#+end_export
The standard $\mathcal{H}_\infty$ synthesis is performed on the generalized plant $P_3(s)$.
Two filters $H_2(s)$ and $H_3(s)$ are obtained such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer from $w$ to $[z_1,\ z_2,\ z_3]$ of the system in Fig.\nbsp{}ref:fig:comp_filter_three_hinf_fb is less than one.
Filter $H_1(s)$ is defined using\nbsp{}eqref:eq:h1_compl_h2_h3 thus ensuring the complementary property of the obtained set of filters.
#+name: eq:h1_compl_h2_h3
\begin{equation}
H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) \big]
\end{equation}
Figure\nbsp{}ref:fig:three_complementary_filters_results displays the three synthesized complementary filters (solid lines) which confirms that the synthesis is successful.\par
#+name: fig:three_complementary_filters_results
#+caption: Bode plot of the inverse weighting functions and of the three complementary filters obtained using the $\mathcal{H}_\infty$ synthesis.
[[file:figs/three_complementary_filters_results.pdf]]
* Conclusion
<<sec:conclusion>>
Sensors measuring a physical quantities are always subject to limitations both in terms of bandwidth or accuracy.
Complementary filters can be used to fuse multiple sensors with different characteristics in order to combine their benefits and yield a better estimate of the measured physical quantity.
The sensor fusion robustness and the obtained super sensor noise can be linked to the magnitude of the complementary filters.
In this paper, a synthesis method that enables the shaping of the complementary filters norms has been proposed.
Several example were used to emphasize the simplicity and the effectiveness of the proposed method.
Links with "closed-loop" complementary filters where highlighted, and the proposed method was generalized for the design of a set of more than two complementary filters.
A new method for designing complementary filters using the $\mathcal{H}_\infty$ synthesis has been proposed.
It allows to shape the magnitude of the filters by the use of weighting functions during the synthesis.
This is very valuable in practice as the characteristics of the super sensor are linked to the complementary filters' magnitude.
Therefore typical sensor fusion objectives can be translated into requirements on the magnitudes of the filters.
Several examples were used to emphasize the simplicity and the effectiveness of the proposed method.
However, the shaping of the complementary filters' magnitude does not allow to directly optimize the super sensor noise and dynamical characteristics.
Future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion.
* Acknowledgment
@@ -807,21 +829,20 @@ Future work will aim at developing a complementary filter synthesis method that
:END:
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\nbsp{}cite:dehaeze19_compl_filter_shapin_using_synth.
This paper has been assigned the LIGO document number LIGO-P2100328.
* Data Availability
:PROPERTIES:
:UNNUMBERED: t
:END:
Matlab cite:matlab20 was used for this study.
Matlab\nbsp{}cite:matlab20 was used for this study.
The source code is available under a MIT License and archived in Zenodo\nbsp{}cite:dehaeze21_new_method_desig_compl_filter_code.
* Bibliography :ignore:
\bibliographystyle{elsarticle-num}
\bibliography{ref}
* Local Variables :noexport:
# Local Variables:
# org-latex-packages-alist: nil

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@@ -1,4 +1,4 @@
% Created 2021-08-31 mar. 14:07
% Created 2021-09-14 mar. 13:38
% Intended LaTeX compiler: pdflatex
\documentclass[preprint, sort&compress]{elsarticle}
\usepackage[utf8]{inputenc}
@@ -22,8 +22,8 @@
\author[a3,a4]{Mohit Verma}
\author[a2,a4]{Christophe Collette}
\cortext[cor1]{Corresponding author. Email Address: thomas.dehaeze@esrf.fr}
\address[a1]{European Synchrotron Radiation Facility, Grenoble, France}
\address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
\address[a1]{European Synchrotron Radiation Facility, 38000 Grenoble, France}
\address[a2]{University of Li\`{e}ge, PML, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
\address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
\address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
\usepackage{tabularx}
@@ -34,7 +34,7 @@
\hypersetup{colorlinks=true}
\setlength{\parskip}{1em}
\date{}
\title{A new method of designing complementary filters for sensor fusion using the \(\mathcal{H}_\infty\) synthesis}
\title{Designing complementary filters for sensor fusion using \(\mathcal{H}_\infty\) synthesis}
\begin{document}
@@ -43,12 +43,13 @@
\begin{frontmatter}
\begin{abstract}
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''.
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.
Properly designing complementary filters is a difficult and time-consuming task.
In this study, we address this issue and propose a new method for designing complementary filters.
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.
The obtained ``super sensor'' combines the benefits of the individual sensors provided that the complementary filters used in the fusion are well designed.
Indeed, properties of the super sensor are linked to the magnitude of the complementary filters.
Properly shaping the magnitude of complementary filters is a difficult and time-consuming task.
In this study, this issue is addressed and a new method for designing complementary filters is proposed.
This method uses weighting functions to specify the wanted shape of the complementary filters that are then obtained using the standard \(\mathcal{H}_\infty\) synthesis.
The proper choice of the weighting functions is discussed, and the effectiveness and simplicity of the design method is highlighted using several examples.
Such synthesis method is further generalized to a set of more than two complementary filters.
Such synthesis method is further extended for the shaping of a set of more than two complementary filters.
\end{abstract}
\begin{keyword}
@@ -57,76 +58,73 @@ Sensor fusion \sep{} Complementary filters \sep{} \(\mathcal{H}_\infty\) synthes
\end{frontmatter}
\section{Introduction}
\label{sec:org45b63f3}
\label{sec:introduction}
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 is affected by several noise sources, such as electrical noise of the conditioning electronics being used.
Second, the frequency range in which the measurement is relevant is bounded by the bandwidth of the sensor.
One way to overcome these limitations is to combine several sensors using a technique called ``sensor fusion''~\cite{bendat57_optim_filter_indep_measur_two}.
Fortunately, a wide variety of sensors exist, each with different characteristics.
By carefully choosing the fused sensors, a so called ``super sensor'' is obtain that combines benefits of individual sensors and yields a better estimate of the measured physical quantity. \par
Fortunately, a wide variety of sensors exists, each with different characteristics.
By carefully choosing the fused sensors, a so called ``super sensor'' is obtained that can combines benefits of the individual sensors. \par
In some situations, sensor fusion is used to increase the bandwidth of the measurement~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim}.
For instance, in~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}, the bandwidth of a position sensor is increased by fusing it with an accelerometer providing the high frequency motion information.
For other applications, sensor fusion is used to obtain a estimate of the measured quantity with lower noise~\cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur,robert12_introd_random_signal_applied_kalman}.
More recently, the fusion of sensors measuring different physical quantities has been proposed to obtained interesting properties for control~\cite{collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage}.
In~\cite{collette15_sensor_fusion_method_high_perfor}, an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator to improve the stability margins of the feedback controller. \par
For instance, in~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel} the bandwidth of a position sensor is increased by fusing it with an accelerometer providing the high frequency motion information.
For other applications, sensor fusion is used to obtain an estimate of the measured quantity with lower noise~\cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur,robert12_introd_random_signal_applied_kalman}.
More recently, the fusion of sensors measuring different physical quantities has been proposed to obtain interesting properties for control~\cite{collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage}.
In~\cite{collette15_sensor_fusion_method_high_perfor}, an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator for improving the stability margins of the feedback controller. \par
Practical applications of sensor fusion are numerous.
It is widely used for the attitude estimation of several autonomous vehicles such as unmanned aerial vehicle~\cite{baerveldt97_low_cost_low_weigh_attit,corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas} and underwater vehicles~\cite{pascoal99_navig_system_desig_using_time,batista10_optim_posit_veloc_navig_filter_auton_vehic}.
Naturally, it is of great benefits for high performance positioning control as shown in~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim,yong16_high_speed_vertic_posit_stage}.
Sensor fusion was also shown to be a key technology to improve the performances of active vibration isolation systems~\cite{tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip}.
This is particularly apparent for the isolation stages of gravitational wave observer~\cite{collette15_sensor_fusion_method_high_perfor,heijningen18_low} such as the ones used at the LIGO~\cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system} and at the VIRGO~\cite{lucia18_low_frequen_optim_perfor_advan}. \par
Sensor fusion was also shown to be a key technology to improve the performance of active vibration isolation systems~\cite{tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip}.
Emblematic examples are the isolation stages of gravitational wave detectors~\cite{collette15_sensor_fusion_method_high_perfor,heijningen18_low} such as the ones used at the LIGO~\cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system} and at the Virgo~\cite{lucia18_low_frequen_optim_perfor_advan}. \par
There are mainly two ways to perform sensor fusion: either using a set of complementary filters~\cite{anderson53_instr_approac_system_steer_comput} or using Kalman filtering~\cite{brown72_integ_navig_system_kalman_filter,odry18_kalman_filter_mobil_robot_attit_estim}.
For sensor fusion applications, both methods are sharing many relationships~\cite{brown72_integ_navig_system_kalman_filter,higgins75_compar_compl_kalman_filter,robert12_introd_random_signal_applied_kalman,becker15_compl_filter_desig_three_frequen_bands}.
However, for Kalman filtering, we are forced to make assumption about the probabilistic character of the sensor noises~\cite{robert12_introd_random_signal_applied_kalman} whereas it is not the case for complementary filters.
Furthermore, the main advantages of complementary filters over Kalman filtering for sensor fusion are their very general applicability, their low computational cost~\cite{higgins75_compar_compl_kalman_filter}, and the fact that they are very intuitive as their effects can be easily interpreted in the frequency domain. \par
However, for Kalman filtering, assumptions must be made about the probabilistic character of the sensor noises~\cite{robert12_introd_random_signal_applied_kalman} whereas it is not the case with complementary filters.
Furthermore, the advantages of complementary filters over Kalman filtering for sensor fusion are their general applicability, their low computational cost~\cite{higgins75_compar_compl_kalman_filter}, and the fact that they are intuitive as their effects can be easily interpreted in the frequency domain. \par
A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
For the earliest used of complementary filtering for sensor fusion, analog circuits were used to physically realize the filters~\cite{anderson53_instr_approac_system_steer_comput}.
Analog complementary filters are still used today~\cite{yong16_high_speed_vertic_posit_stage,moore19_capac_instr_sensor_fusion_high_bandw_nanop}, but most of the time they are now implemented numerically as it allows for much more flexibility. \par
In the early days of complementary filtering, analog circuits were employed to physically realize the filters~\cite{anderson53_instr_approac_system_steer_comput}.
Analog complementary filters are still used today~\cite{yong16_high_speed_vertic_posit_stage,moore19_capac_instr_sensor_fusion_high_bandw_nanop}, but most of the time they are now implemented digitally as it allows for much more flexibility. \par
Several design methods have been developed over the years to optimize complementary filters.
The easiest way to design complementary filters is to use analytical formulas.
Depending on the application, the formulas used are of first order~\cite{corke04_inert_visual_sensin_system_small_auton_helic,yeh05_model_contr_hydraul_actuat_two,yong16_high_speed_vertic_posit_stage}, second order~\cite{baerveldt97_low_cost_low_weigh_attit,stoten01_fusion_kinet_data_using_compos_filter,jensen13_basic_uas} or even higher orders~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,stoten01_fusion_kinet_data_using_compos_filter,collette15_sensor_fusion_method_high_perfor,matichard15_seism_isolat_advan_ligo}. \par
As the characteristics of the ``super sensor'' depends on the design of the complementary filters~\cite{dehaeze19_compl_filter_shapin_using_synth}, several optimization techniques were developed over the years.
Some are based on the finding the optimal parameters of analytical formulas~\cite{jensen13_basic_uas,min15_compl_filter_desig_angle_estim,becker15_compl_filter_desig_three_frequen_bands}, while other are using convex optimization tools~\cite{hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo} such as linear matrix inequalities~\cite{pascoal99_navig_system_desig_using_time}.
As the characteristics of the super sensor depends on the proper design of the complementary filters~\cite{dehaeze19_compl_filter_shapin_using_synth}, several optimization techniques have been developed.
Some are based on the finding of optimal parameters of analytical formulas~\cite{jensen13_basic_uas,min15_compl_filter_desig_angle_estim,becker15_compl_filter_desig_three_frequen_bands}, while other are using convex optimization tools~\cite{hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo} such as linear matrix inequalities~\cite{pascoal99_navig_system_desig_using_time}.
As shown in~\cite{plummer06_optim_compl_filter_their_applic_motion_measur}, the design of complementary filters can also be linked to the standard mixed-sensitivity control problem.
Therefore, all the powerful tools developed for the classical control theory can also be used for the design of complementary filters.
For instance, in~\cite{jensen13_basic_uas} the two gains of a Proportional Integral (PI) controller are optimized to minimize the noise of the super sensor. \par
The common objective to all these complementary filters design methods is to obtain a super sensor that has desired characteristics, usually in terms of noise and dynamics.
Moreover, as reported in~\cite{zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur}, phase shifts and magnitude bumps of the ``super sensors'' dynamics can be observed if either the complementary filters are poorly designed or if the sensors are not well calibrated.
Therefore, the robustness of the fusion is also of concerned when designing the complementary filters.
Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to specify the wanted super sensor characteristic while ensuring good fusion robustness has been proposed. \par
The common objective of all these complementary filters design methods is to obtain a super sensor that has desired characteristics, usually in terms of noise and dynamics.
Moreover, as reported in~\cite{zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur}, phase shifts and magnitude bumps of the super sensors dynamics can be observed if either the complementary filters are poorly designed or if the sensors are not well calibrated.
Hence, the robustness of the fusion is also of concern when designing the complementary filters.
Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to specify the desired super sensor characteristic while ensuring good fusion robustness has been proposed. \par
Fortunately, both the robustness of the fusion and the super sensor characteristics can be linked to the magnitude of the complementary filters~\cite{dehaeze19_compl_filter_shapin_using_synth}.
Based on that, this paper introduces a new way to design complementary filters using the \(\mathcal{H}_\infty\) synthesis which allows to shape the magnitude of the complementary filters in an easy and intuitive way. \par
Section~\ref{sec:requirements} introduces the sensor fusion architecture and demonstrates how typical requirements can be linked to the complementary filters magnitudes.
In section~\ref{sec:hinf_method}, the shaping of complementary filters is written as an \(\mathcal{H}_\infty\) optimization problem using weighting functions, and the simplicity of the proposed method is illustrated with an example.
Based on that, this paper introduces a new way to design complementary filters using the \(\mathcal{H}_\infty\) synthesis which allows to shape the complementary filters' magnitude in an easy and intuitive way. \par
Section~\ref{sec:requirements} introduces the sensor fusion architecture and demonstrates how typical requirements can be linked to the complementary filters' magnitude.
In Section~\ref{sec:hinf_method}, the shaping of complementary filters is formulated as an \(\mathcal{H}_\infty\) optimization problem using weighting functions, and the simplicity of the proposed method is illustrated with an example.
The synthesis method is further validated in Section~\ref{sec:application_ligo} by designing complex 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~\ref{sec:discussion} compares the proposed synthesis method with the classical mixed-sensitivity synthesis, and extends it for the shaping of more than two complementary filters.
\section{Sensor Fusion and Complementary Filters Requirements}
\label{sec:orgeeb9584}
\label{sec:requirements}
Complementary filters provides a framework for fusing signals from different sensors.
Complementary filtering 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.
These requirements are discussed in this section.
\subsection{Sensor Fusion Architecture}
\label{sec:org556a17c}
\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 \(x\).
The two sensors output signals \(\hat{x}_1\) and \(\hat{x}_2\) are estimates of \(x\).
Each of these estimates are then filtered out by complementary filters and combined to form a new estimate \(\hat{x}\).
These estimates are then filtered out by complementary filters and combined to form a new estimate \(\hat{x}\).
The resulting sensor, termed as super sensor, can have larger bandwidth and better noise characteristics in comparison to the individual sensor.
The resulting sensor, termed as super sensor, can have larger bandwidth and better noise characteristics in comparison to the individual sensors.
This means that the super sensor provides an estimate \(\hat{x}\) of \(x\) which can be more accurate over a larger frequency band than the outputs of the individual sensors.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensor_fusion_overview.pdf}
\caption{\label{fig:sensor_fusion_overview}Schematic of a sensor fusion architecture using complementary filters}
\caption{\label{fig:sensor_fusion_overview}Schematic of a sensor fusion architecture using complementary filters.}
\end{figure}
The complementary property of filters \(H_1(s)\) and \(H_2(s)\) implies that the sum of their transfer functions is equal to unity.
The complementary property of filters \(H_1(s)\) and \(H_2(s)\) implies that the sum of their transfer functions is equal to one.
That is, unity magnitude and zero phase at all frequencies.
Therefore, a pair of complementary filter needs to satisfy the following condition:
\begin{equation}
@@ -137,7 +135,6 @@ 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.
\subsection{Sensor Models and Sensor Normalization}
\label{sec:org5c5b6fb}
\label{sec:sensor_models}
In order to study such sensor fusion architecture, a model for the sensors is required.
@@ -148,21 +145,21 @@ Before filtering the sensor outputs \(\tilde{x}_i\) by the complementary filters
This normalization consists of using an estimate \(\hat{G}_i(s)\) of the sensor dynamics \(G_i(s)\), and filtering the sensor output by the inverse of this estimate \(\hat{G}_i^{-1}(s)\) as shown in Fig.~\ref{fig:sensor_model_calibrated}.
It is here supposed that the sensor inverse \(\hat{G}_i^{-1}(s)\) is proper and stable.
This way, the units of the estimates \(\hat{x}_i\) are equal to the units of the physical quantity \(x\).
The sensor dynamics estimate \(\hat{G}_1(s)\) can be a simple gain or a more complex transfer function.
The sensor dynamics estimate \(\hat{G}_i(s)\) can be a simple gain or a more complex transfer function.
\begin{figure}[htbp]
\begin{subfigure}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model.pdf}
\caption{\label{fig:sensor_model} Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$}
\caption{\label{fig:sensor_model} Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_calibrated.pdf}
\caption{\label{fig:sensor_model_calibrated} Normalized sensors using the inverse of an estimate $\hat{G}_i(s)$ of the sensor dynamics}
\caption{\label{fig:sensor_model_calibrated} Normalized sensors using the inverse of an estimate $\hat{G}_i(s)$ of the sensor dynamics.}
\end{subfigure}
\caption{\label{fig:sensor_models}Sensor models with and without normalization}
\caption{\label{fig:sensor_models}Sensor models with and without normalization.}
\centering
\end{figure}
@@ -179,11 +176,10 @@ The super sensor output is therefore equal to:
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/fusion_super_sensor.pdf}
\caption{\label{fig:fusion_super_sensor}Sensor fusion architecture with two normalized sensors}
\caption{\label{fig:fusion_super_sensor}Sensor fusion architecture with two normalized sensors.}
\end{figure}
\subsection{Noise Sensor Filtering}
\label{sec:org4d93421}
\label{sec:noise_filtering}
In this section, it is supposed that all the sensors are perfectly normalized, such that:
@@ -194,23 +190,23 @@ In this section, it is supposed that all the sensors are perfectly normalized, s
The effect of a non-perfect normalization will be discussed in the next section.
Provided \eqref{eq:perfect_dynamics} is verified, the super sensor output \(\hat{x}\) is then:
Provided~\eqref{eq:perfect_dynamics} is verified, the super sensor output \(\hat{x}\) is then equal to:
\begin{equation}
\label{eq:estimate_perfect_dyn}
\hat{x} = x + H_1(s) n_1 + H_2(s) n_2
\end{equation}
From \eqref{eq:estimate_perfect_dyn}, the complementary filters \(H_1(s)\) and \(H_2(s)\) are shown to only operate on the sensor's noises.
From~\eqref{eq:estimate_perfect_dyn}, the complementary filters \(H_1(s)\) and \(H_2(s)\) are shown to only operate on the noise of the sensors.
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured.
This is why the two filters must be complementary.
The estimation error \(\delta x\), defined as the difference between the sensor output \(\hat{x}\) and the measured quantity \(x\), is computed for the super sensor \eqref{eq:estimate_error}.
The estimation error \(\delta x\), defined as the difference between the sensor output \(\hat{x}\) and the measured quantity \(x\), is computed for the super sensor~\eqref{eq:estimate_error}.
\begin{equation}
\label{eq:estimate_error}
\delta x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2
\end{equation}
As shown in \eqref{eq:noise_filtering_psd}, the Power Spectral Density (PSD) of the estimation error \(\Phi_{\delta x}\) depends both on the norm of the two complementary filters and on the PSD of the noise sources \(\Phi_{n_1}\) and \(\Phi_{n_2}\).
As shown in~\eqref{eq:noise_filtering_psd}, the Power Spectral Density (PSD) of the estimation error \(\Phi_{\delta x}\) depends both on the norm of the two complementary filters and on the PSD of the noise sources \(\Phi_{n_1}\) and \(\Phi_{n_2}\).
\begin{equation}
\label{eq:noise_filtering_psd}
\Phi_{\delta x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega)
@@ -221,18 +217,17 @@ This is the simplest form of sensor fusion with complementary filters.
However, the two sensors have usually high noise levels over distinct frequency regions.
In such case, to lower the noise of the super sensor, the norm \(|H_1(j\omega)|\) has to be small when \(\Phi_{n_1}(\omega)\) is larger than \(\Phi_{n_2}(\omega)\) and the norm \(|H_2(j\omega)|\) has to be small when \(\Phi_{n_2}(\omega)\) is larger than \(\Phi_{n_1}(\omega)\).
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
Hence, by properly shaping the norm of the complementary filters, it is possible to reduce the noise of the super sensor.
\subsection{Sensor Fusion Robustness}
\label{sec:org0ab9090}
\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.
In order to study such imperfection, a multiplicative input uncertainty is added to the sensor dynamics (Fig.~\ref{fig:sensor_model_uncertainty}).
The nominal model is taken as the estimated model for the normalization \(\hat{G}_i(s)\), \(\Delta_i\) is any stable transfer function satisfying \(|\Delta_i(j\omega)| \le 1,\ \forall\omega\), and \(w_i(s)\) is a weighting transfer function representing the magnitude of the uncertainty.
The nominal model is the estimated model used for the normalization \(\hat{G}_i(s)\), \(\Delta_i(s)\) is any stable transfer function satisfying \(|\Delta_i(j\omega)| \le 1,\ \forall\omega\), and \(w_i(s)\) is a weighting transfer function representing the magnitude of the uncertainty.
The weight \(w_i(s)\) is chosen such that the real sensor dynamics \(G(j\omega)\) is contained in the uncertain region represented by a circle in the complex plane, centered on \(1\) and with a radius equal to \(|w_i(j\omega)|\).
The weight \(w_i(s)\) is chosen such that the real sensor dynamics \(G_i(j\omega)\) is contained in the uncertain region represented by a circle in the complex plane, centered on \(1\) and with a radius equal to \(|w_i(j\omega)|\).
As the nominal sensor dynamics is taken as the normalized filter, the normalized sensor can be further simplified as shown in Fig.~\ref{fig:sensor_model_uncertainty_simplified}.
@@ -240,34 +235,34 @@ As the nominal sensor dynamics is taken as the normalized filter, the normalized
\begin{subfigure}[b]{0.59\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_uncertainty.pdf}
\caption{\label{fig:sensor_model_uncertainty} Sensor with multiplicative input uncertainty}
\caption{\label{fig:sensor_model_uncertainty} Sensor with multiplicative input uncertainty.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.39\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_uncertainty_simplified.pdf}
\caption{\label{fig:sensor_model_uncertainty_simplified} Simplified sensor model}
\caption{\label{fig:sensor_model_uncertainty_simplified} Simplified sensor model.}
\end{subfigure}
\hfill
\caption{\label{fig:sensor_models_uncertainty}Sensor models with dynamical uncertainty}
\caption{\label{fig:sensor_models_uncertainty}Sensor models with dynamical uncertainty.}
\centering
\end{figure}
The sensor fusion architecture with two sensor models including dynamical uncertainty is shown in Fig.~\ref{fig:sensor_fusion_dynamic_uncertainty}.
The sensor fusion architecture with the sensor models including dynamical uncertainty is shown in Fig.~\ref{fig:sensor_fusion_dynamic_uncertainty}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensor_fusion_dynamic_uncertainty.pdf}
\caption{\label{fig:sensor_fusion_dynamic_uncertainty}Sensor fusion architecture with sensor dynamics uncertainty}
\caption{\label{fig:sensor_fusion_dynamic_uncertainty}Sensor fusion architecture with sensor dynamics uncertainty.}
\end{figure}
The super sensor dynamics \eqref{eq:super_sensor_dyn_uncertainty} is no longer equal to \(1\) and now depends on the sensor dynamical uncertainty weights \(w_i(s)\) as well as on the complementary filters \(H_i(s)\).
The super sensor dynamics~\eqref{eq:super_sensor_dyn_uncertainty} is no longer equal to \(1\) and now depends on the sensor dynamical uncertainty weights \(w_i(s)\) as well as on the complementary filters \(H_i(s)\).
\begin{equation}
\label{eq:super_sensor_dyn_uncertainty}
\frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s)
\end{equation}
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on \(1\) with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) as shown in Fig.~\ref{fig:uncertainty_set_super_sensor}.
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on \(1\) with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) (Fig.~\ref{fig:uncertainty_set_super_sensor}).
\begin{figure}[htbp]
\centering
@@ -275,9 +270,8 @@ The dynamical uncertainty of the super sensor can be graphically represented in
\caption{\label{fig:uncertainty_set_super_sensor}Uncertainty region of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency \(\omega\) is here omitted.}
\end{figure}
The super sensor dynamical uncertainty (i.e. the robustness of the fusion) clearly depends on the complementary filters norms.
For instance, the phase \(\Delta\phi(\omega)\) added by the super sensor dynamics at frequency \(\omega\) is bounded by \(\Delta\phi_{\text{max}}(\omega)\) which can be found by drawing a tangent from the origin to the uncertainty circle of the super sensor (Fig.~\ref{fig:uncertainty_set_super_sensor}).
Therefore, the phase uncertainty of the super sensor dynamics depends on the Complementary filters norms \eqref{eq:max_phase_uncertainty}.
The super sensor dynamical uncertainty, and hence the robustness of the fusion, clearly depends on the complementary filters' norm.
For instance, the phase \(\Delta\phi(\omega)\) added by the super sensor dynamics at frequency \(\omega\) is bounded by \(\Delta\phi_{\text{max}}(\omega)\) which can be found by drawing a tangent from the origin to the uncertainty circle of the super sensor (Fig.~\ref{fig:uncertainty_set_super_sensor}) and that is mathematically described by~\eqref{eq:max_phase_uncertainty}.
\begin{equation}
\label{eq:max_phase_uncertainty}
@@ -288,19 +282,17 @@ 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.
\section{Complementary Filters Shaping}
\label{sec:orgdf903f0}
\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.
Therefore, a complementary filters synthesis method that allows to shape their norms would be of great use.
As shown in Section~\ref{sec:requirements}, the noise and robustness of the super sensor are a function of the complementary filters' norm.
Therefore, a synthesis method of complementary filters that allows to shape their norm would be of great use.
In this section, such synthesis is proposed by writing the synthesis objective as a standard \(\mathcal{H}_\infty\) optimization problem.
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' shape during the synthesis, their proper design is discussed.
Finally, the synthesis method is validated on an simple example.
\subsection{Synthesis Objective}
\label{sec:org2467206}
\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}.
This is equivalent as to finding proper and stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions \eqref{eq:hinf_cond_complementarity}, \eqref{eq:hinf_cond_h1} and \eqref{eq:hinf_cond_h2} are satisfied.
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}.
This is equivalent as to finding proper and stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions~\eqref{eq:hinf_cond_complementarity}, \eqref{eq:hinf_cond_h1} and~\eqref{eq:hinf_cond_h2} are satisfied.
\begin{subequations}
\label{eq:comp_filter_problem_form}
@@ -310,15 +302,14 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
& |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2}
\end{align}
\end{subequations}
\(W_1(s)\) 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 norm of the complementary filters during the synthesis.
\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
\label{sec:org172468e}
\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.
Consider the generalized plant \(P(s)\) shown in Fig.~\ref{fig:h_infinity_robust_fusion_plant} and mathematically described by \eqref{eq:generalized_plant}.
Consider the generalized plant \(P(s)\) shown in Fig.~\ref{fig:h_infinity_robust_fusion_plant} and mathematically described by~\eqref{eq:generalized_plant}.
\begin{equation}
\label{eq:generalized_plant}
@@ -329,27 +320,27 @@ Consider the generalized plant \(P(s)\) shown in Fig.~\ref{fig:h_infinity_robust
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/h_infinity_robust_fusion_plant.pdf}
\caption{\label{fig:h_infinity_robust_fusion_plant} Generalized plant}
\caption{\label{fig:h_infinity_robust_fusion_plant} Generalized plant.}
\vfill
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/h_infinity_robust_fusion_fb.pdf}
\caption{\label{fig:h_infinity_robust_fusion_fb} Generalized plant with the synthesized filter}
\caption{\label{fig:h_infinity_robust_fusion_fb} Generalized plant with the synthesized filter.}
\end{subfigure}
\hfill
\caption{\label{fig:h_infinity_robust_fusion} Architecture for the $\mathcal{H}_\infty$ synthesis of complementary filters}
\caption{\label{fig:h_infinity_robust_fusion} Architecture for the $\mathcal{H}_\infty$ synthesis of complementary filters.}
\centering
\end{figure}
Applying the standard \(\mathcal{H}_\infty\) synthesis on the generalized plant \(P(s)\) is then equivalent as finding a stable filter \(H_2(s)\) which based on \(v\), generates a signal \(u\) such that the \(\mathcal{H}_\infty\) norm of the system in Fig.~\ref{fig:h_infinity_robust_fusion_fb} from \(w\) to \([z_1, \ z_2]\) is less than one \eqref{eq:hinf_syn_obj}.
Applying the standard \(\mathcal{H}_\infty\) synthesis to the generalized plant \(P(s)\) is then equivalent as finding a stable filter \(H_2(s)\) which based on \(v\), generates a signal \(u\) such that the \(\mathcal{H}_\infty\) norm of the system in Fig.~\ref{fig:h_infinity_robust_fusion_fb} from \(w\) to \([z_1, \ z_2]\) is less than one~\eqref{eq:hinf_syn_obj}.
\begin{equation}
\label{eq:hinf_syn_obj}
\left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
By then defining \(H_1(s)\) to be the complementary of \(H_2(s)\) \eqref{eq:definition_H1}, the \(\mathcal{H}_\infty\) synthesis objective becomes equivalent to \eqref{eq:hinf_problem} which ensure that \eqref{eq:hinf_cond_h1} and \eqref{eq:hinf_cond_h2} are satisfied.
By then defining \(H_1(s)\) to be the complementary of \(H_2(s)\) \eqref{eq:definition_H1}, the \(\mathcal{H}_\infty\) synthesis objective becomes equivalent to~\eqref{eq:hinf_problem} which ensures that~\eqref{eq:hinf_cond_h1} and~\eqref{eq:hinf_cond_h2} are satisfied.
\begin{equation}
\label{eq:definition_H1}
@@ -361,27 +352,25 @@ By then defining \(H_1(s)\) to be the complementary of \(H_2(s)\) \eqref{eq:defi
\left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
Therefore, applying the \(\mathcal{H}_\infty\) synthesis on the standard plant \(P(s)\) \eqref{eq:generalized_plant} will generate two filters \(H_2(s)\) and \(H_1(s) \triangleq 1 - H_2(s)\) that are complementary \eqref{eq:comp_filter_problem_form} and such that there norms are bellow specified bounds \eqref{eq:hinf_cond_h1},\eqref{eq:hinf_cond_h2}.
Therefore, applying the \(\mathcal{H}_\infty\) synthesis to the standard plant \(P(s)\) \eqref{eq:generalized_plant} will generate two filters \(H_2(s)\) and \(H_1(s) \triangleq 1 - H_2(s)\) that are complementary~\eqref{eq:comp_filter_problem_form} and such that there norms are bellow specified bounds \eqref{eq:hinf_cond_h1},~\eqref{eq:hinf_cond_h2}.
Note that there is not an equivalence between the \(\mathcal{H}_\infty\) norm condition~\eqref{eq:hinf_problem} and the initial synthesis objectives~\eqref{eq:hinf_cond_h1} and~\eqref{eq:hinf_cond_h2}, but only an implication.
Hence, the optimization may be a little bit conservative with respect to the ``set'' of filters on which it is performed.
There might be solutions were the objectives~\eqref{eq:hinf_cond_h1} and~\eqref{eq:hinf_cond_h2} are valid but where the \(\mathcal{H}_\infty\) norm~\eqref{eq:hinf_problem} is larger than one.
Note that there is only an implication between the \(\mathcal{H}_\infty\) norm condition~\eqref{eq:hinf_problem} and the initial synthesis objectives~\eqref{eq:hinf_cond_h1} and~\eqref{eq:hinf_cond_h2} and not an equivalence.
Hence, the optimization may be a little bit conservative with respect to the set of filters on which it is performed, see~\cite[Chap. 2.8.3]{skogestad07_multiv_feedb_contr}.
In practice, this is however not an found to be an issue.
\subsection{Weighting Functions Design}
\label{sec:org3b7e958}
\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 complementary filters' norm.
The proper design of these weighting functions is of primary importance for the success of the presented \(\mathcal{H}_\infty\) synthesis of complementary filters.
First, only proper and stable transfer functions should be used.
Second, the order of the weighting functions should stay reasonably small in order to reduce the computational costs associated with the solving of the optimization problem and for the physical implementation of the filters (the order of the synthesized filters being equal to the sum of the weighting functions orders).
Third, one should not forget the fundamental limitations imposed by the complementary property \eqref{eq:comp_filter}.
Second, the order of the weighting functions should stay reasonably small in order to reduce the computational costs associated with the solving of the optimization problem and for the physical implementation of the filters (the synthesized filters' order being equal to the sum of the weighting functions' order).
Third, one should not forget the fundamental limitations imposed by the complementary property~\eqref{eq:comp_filter}.
This implies for instance that \(|H_1(j\omega)|\) and \(|H_2(j\omega)|\) cannot be made small at the same frequency.
When designing complementary filters, it is usually desired to specify their slopes, their ``blending'' frequency and their maximum gains at low and high frequency.
To help with the design of the weighting functions such that the above specification can be easily expressed, the formula \eqref{eq:weight_formula} is proposed.
To easily express these specifications, formula~\eqref{eq:weight_formula} is proposed to help with the design of weighting functions.
\begin{equation}
\label{eq:weight_formula}
@@ -392,15 +381,15 @@ To help with the design of the weighting functions such that the above specifica
}\right)^n
\end{equation}
The parameters in formula \eqref{eq:weight_formula} are:
The parameters in formula~\eqref{eq:weight_formula} are:
\begin{itemize}
\item \(G_0 = lim_{\omega \to 0} |W(j\omega)|\): the low frequency gain
\item \(G_\infty = lim_{\omega \to \infty} |W(j\omega)|\): the high frequency gain
\item \(G_0 = \lim_{\omega \to 0} |W(j\omega)|\): the low frequency gain
\item \(G_\infty = \lim_{\omega \to \infty} |W(j\omega)|\): the high frequency gain
\item \(G_c = |W(j\omega_c)|\): the gain at a specific frequency \(\omega_c\) in \(\si{rad/s}\).
\item \(n\): the slope between high and low frequency. It also corresponds to the order of the weighting function.
\end{itemize}
The parameters \(G_0\), \(G_c\) and \(G_\infty\) should either satisfy condition \eqref{eq:cond_formula_1} or \eqref{eq:cond_formula_2}.
The parameters \(G_0\), \(G_c\) and \(G_\infty\) should either satisfy~\eqref{eq:cond_formula_1} or~\eqref{eq:cond_formula_2}.
\begin{subequations}
\label{eq:condition_params_formula}
\begin{align}
@@ -409,57 +398,59 @@ The parameters \(G_0\), \(G_c\) and \(G_\infty\) should either satisfy condition
\end{align}
\end{subequations}
An example of the obtained magnitude of a weighting function generated using \eqref{eq:weight_formula} is shown in Fig.~\ref{fig:weight_formula}.
The typical magnitude of a weighting function generated using~\eqref{eq:weight_formula} is shown in Fig.~\ref{fig:weight_formula}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/weight_formula.pdf}
\caption{\label{fig:weight_formula}Magnitude of a weighting function generated using the proposed formula \eqref{eq:weight_formula}, \(G_0 = 1e^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\)}
\caption{\label{fig:weight_formula}Magnitude of a weighting function generated using formula~\eqref{eq:weight_formula}, \(G_0 = 1e^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\).}
\end{figure}
\subsection{Validation of the proposed synthesis method}
\label{sec:org13db233}
\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.
Let's suppose two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
\begin{itemize}
\item the blending frequency is around \(\SI{10}{Hz}\)
\item the slope of \(|H_1(j\omega)|\) is \(+2\) below \(\SI{10}{Hz}\), its low frequency gain is \(10^{-3}\)
\item the slope of \(|H_2(j\omega)|\) is \(-3\) above \(\SI{10}{Hz}\), its high frequency gain is \(10^{-3}\)
\item the blending frequency is around \(\SI{10}{Hz}\).
\item the slope of \(|H_1(j\omega)|\) is \(+2\) below \(\SI{10}{Hz}\).
Its low frequency gain is \(10^{-3}\).
\item the slope of \(|H_2(j\omega)|\) is \(-3\) above \(\SI{10}{Hz}\).
Its high frequency gain is \(10^{-3}\).
\end{itemize}
The first step is to translate the above requirements into the design of the weighting functions.
The proposed formula \eqref{eq:weight_formula} is here used for such purpose.
The first step is to translate the above requirements by properly designing the weighting functions.
The proposed formula~\eqref{eq:weight_formula} is here used for such purpose.
Parameters used are summarized in Table~\ref{tab:weights_params}.
The inverse magnitudes of the designed weighting functions, which are representing the maximum allowed norms of the complementary filters, are shown by the dashed lines in Fig.~\ref{fig:weights_W1_W2}.
\begin{figure}
\begin{minipage}[b]{0.49\linewidth}
\centering
\begin{tabularx}{0.65\linewidth}{ccc}
\begin{tabularx}{0.60\linewidth}{ccc}
\toprule
Parameters & \(W_1(s)\) & \(W_2(s)\)\\
Parameters & \(W_1(s)\) & \(W_2(s)\)\\
\midrule
\(G_0\) & \(0.1\) & \(1000\)\\
\(G_\infty\) & \(1000\) & \(0.1\)\\
\(G_0\) & \(0.1\) & \(1000\)\\
\(G_\infty\) & \(1000\) & \(0.1\)\\
\(\omega_c\) & \(2\pi\cdot10\) & \(2\pi\cdot10\)\\
\(G_c\) & \(0.45\) & \(0.45\)\\
\(n\) & \(2\) & \(3\)\\
\(G_c\) & \(0.45\) & \(0.45\)\\
\(n\) & \(2\) & \(3\)\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:weights_params}Parameters used for weighting functions \(W_1(s)\) and \(W_2(s)\) using \eqref{eq:weight_formula}}
\captionof{table}{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\) using~\eqref{eq:weight_formula}.}
\end{minipage}
\hfill
\begin{minipage}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/weights_W1_W2.pdf}
\caption{\label{fig:weights_W1_W2}Inverse magnitude of the weighting functions}
\caption{\label{fig:weights_W1_W2}Inverse magnitude of the weighting functions.}
\end{minipage}
\end{figure}
The \(\mathcal{H}_\infty\) synthesis is then applied to the generalized plant of Fig.~\ref{fig:h_infinity_robust_fusion_plant} on efficiently solved in Matlab~\cite{matlab20} using the Robust Control Toolbox.
The standard \(\mathcal{H}_\infty\) synthesis is then applied to the generalized plant of Fig.~\ref{fig:h_infinity_robust_fusion_plant} and efficiently solved using Matlab~\cite{matlab20}.
The filter \(H_2(s)\) that minimizes the \(\mathcal{H}_\infty\) norm between \(w\) and \([z_1,\ z_2]^T\) is obtained.
The \(\mathcal{H}_\infty\) norm is here found to be close to one \eqref{eq:hinf_synthesis_result} which indicates that the synthesis is successful: the complementary filters norms are below the maximum specified upper bounds.
The \(\mathcal{H}_\infty\) norm is here found to be close to one~\eqref{eq:hinf_synthesis_result} which indicates that the synthesis is successful: the complementary filters norms are below the maximum specified upper bounds.
This is confirmed by the bode plots of the obtained complementary filters in Fig.~\ref{fig:hinf_filters_results}.
\begin{equation}
@@ -467,8 +458,8 @@ This is confirmed by the bode plots of the obtained complementary filters in Fig
\left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \approx 1
\end{equation}
Their transfer functions in the Laplace domain are given in \eqref{eq:hinf_synthesis_result_tf}.
As expected, the obtained filters are of order \(5\), that is the sum of the weighting functions orders.
The transfer functions in the Laplace domain of the complementary filters are given in~\eqref{eq:hinf_synthesis_result_tf}.
As expected, the obtained filters are of order \(5\), that is the sum of the weighting functions' order.
\begin{subequations}
\label{eq:hinf_synthesis_result_tf}
@@ -482,18 +473,17 @@ As expected, the obtained filters are of order \(5\), that is the sum of the wei
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/hinf_filters_results.pdf}
\caption{\label{fig:hinf_filters_results}Bode plot of the obtained complementary filters}
\caption{\label{fig:hinf_filters_results}Bode plot of the obtained complementary filters.}
\end{figure}
This simple example illustrates the fact that the proposed methodology for complementary filters shaping is easy to use and effective.
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}
\label{sec:orgb11268e}
\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 the active vibration isolation systems at gravitational wave detectors, such as 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}.
In the first isolation stage at the LIGO, two sets of complementary filters are used and included in a feedback loop~\cite{hua04_low_ligo}.
In the first isolation stage at the LIGO, two sets of complementary filters are used to form a super sensor that is incorporated in a feedback loop~\cite{hua04_low_ligo}.
A set of complementary filters (\(L_2,H_2\)) is first used to fuse a seismometer and a geophone.
Then, another set of complementary filters (\(L_1,H_1\)) is used to merge the output of the first ``inertial super sensor'' with a position sensor.
A simplified block diagram of the sensor fusion architecture is shown in Fig.~\ref{fig:ligo_super_sensor_architecture}.
@@ -501,115 +491,111 @@ A simplified block diagram of the sensor fusion architecture is shown in Fig.~\r
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/ligo_super_sensor_architecture.pdf}
\caption{\label{fig:ligo_super_sensor_architecture}Simplified block diagram of the sensor blending strategy for the first stage at the LIGO~\cite{hua04_low_ligo}}
\caption{\label{fig:ligo_super_sensor_architecture}Simplified block diagram of the sensor blending strategy for the first stage at the LIGO~\cite{hua04_low_ligo}.}
\end{figure}
The fusion of the position sensor at low frequency with the ``inertial super sensor'' at high frequency using the complementary filters (\(L_1,H_1\)) is done for several reasons, first of which is to give the super sensor a DC sensibility and therefore allow the feedback loop to have authority at zero frequency.
The requirements on those filters are very tight and thus their design is complex and should be expressed as an optimization problem.
The fusion of the position sensor at low frequency with the ``inertial super sensor'' at high frequency using the complementary filters (\(L_1,H_1\)) is done for several reasons, first of which is to give the super sensor a DC sensibility that allows the feedback loop to have authority at zero frequency.
The requirements on those filters are stringent and thus their design is complex and should be expressed as an optimization problem.
The approach used in~\cite{hua05_low_ligo} is to use FIR complementary filters and to write the synthesis as a convex optimization problem.
After synthesis, the obtained FIR filters were found to be compliant with the requirements.
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.
However they are of high order so their implementation is quite complex.
In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated by using the same set of requirements.
\subsection{Complementary Filters Specifications}
\label{sec:org100bb02}
\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}):
\begin{itemize}
\item From \(0\) to \(\SI{0.008}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less or equal to \(8 \times 10^{-4}\)
\item Between \(\SI{0.008}{Hz}\) to \(\SI{0.04}{Hz}\), the filter \(L_1(s)\) should attenuate the input signal proportional to frequency cubed
\item Between \(\SI{0.04}{Hz}\) to \(\SI{0.1}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less than \(3\)
\item Above \(\SI{0.1}{Hz}\), the magnitude \(|H_1(j\omega)|\) should be less than \(0.045\)
\item Below \(\SI{0.008}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less than \(8 \times 10^{-4}\).
\item From \(\SI{0.008}{Hz}\) to \(\SI{0.04}{Hz}\), the filter \(L_1(s)\) should attenuate the input signal proportional to frequency cubed.
\item From \(\SI{0.04}{Hz}\) to \(\SI{0.1}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less than \(3\).
\item Above \(\SI{0.1}{Hz}\), the magnitude \(|H_1(j\omega)|\) should be less than \(0.045\).
\end{itemize}
These specifications are therefore upper bounds on the complementary filters' magnitudes.
These specifications are therefore upper bounds on the complementary filters' magnitude.
They are physically represented in Fig.~\ref{fig:fir_filter_ligo} as well as the obtained magnitude of the FIR filters in~\cite{hua05_low_ligo}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/fir_filter_ligo.pdf}
\caption{\label{fig:fir_filter_ligo}Specifications and Bode plot of the obtained FIR complementary filters in~\cite{hua05_low_ligo}. The filters are here obtained using the SeDuMi Matlab toolbox~\cite{sturm99_using_sedum}}
\caption{\label{fig:fir_filter_ligo}Specifications and Bode plot of the obtained FIR complementary filters in~\cite{hua05_low_ligo}. The filters are here obtained using the SeDuMi Matlab toolbox~\cite{sturm99_using_sedum}.}
\end{figure}
\subsection{Weighting Functions Design}
\label{sec:orgba233a6}
\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.
However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters.
A Type I Chebyshev filter of order \(20\) is used for the weighting transfer function \(W_L(s)\) corresponding to the low pass filter.
For the one corresponding to the high pass filter \(W_H(s)\), a \(7^{\text{th}}\) order transfer function is designed.
The magnitudes of the weighting functions are shown in Fig.~\ref{fig:ligo_weights}.
For the one corresponding to the high pass filter \(W_H(s)\), a \(7^{\text{th}}\) order transfer function is manually designed.
The inverse magnitudes of the weighting functions are shown in Fig.~\ref{fig:ligo_weights}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/ligo_weights.pdf}
\caption{\label{fig:ligo_weights}Specifications and weighting functions inverse magnitudes}
\caption{\label{fig:ligo_weights}Specifications and weighting functions' inverse magnitude.}
\end{figure}
\subsection{\(\mathcal{H}_\infty\) Synthesis of the complementary filters}
\label{sec:org705f5c2}
\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}.
After optimization, the \(\mathcal{H}_\infty\) norm from \(w\) to \([z_1,\ z_2]^T\) is found close to one indication successful synthesis.
In Fig.~\ref{fig:comp_fir_ligo_hinf}, the bode plot of the obtained complementary filters are compared with the FIR filters of order 512 obtained in~\cite{hua05_low_ligo}.
Even though the complementary filters using the \(\mathcal{H}_\infty\) synthesis are of much lower order (order 27), they are found to be very close to the FIR filters.
Even though the complementary filters using the \(\mathcal{H}_\infty\) synthesis are of much lower order (order 27), they are found to be close to the FIR filters.
This confirms the effectiveness of the proposed synthesis method even when the complementary filters are subject to complex requirements.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/comp_fir_ligo_hinf.pdf}
\caption{\label{fig:comp_fir_ligo_hinf}Comparison of the FIR filters (dashed) designed in~\cite{hua05_low_ligo} with the filters obtained with \(\mathcal{H}_\infty\) synthesis (solid)}
\caption{\label{fig:comp_fir_ligo_hinf}Comparison of the FIR filters (dashed) designed in~\cite{hua05_low_ligo} with the filters obtained with \(\mathcal{H}_\infty\) synthesis (solid).}
\end{figure}
\section{Discussion}
\label{sec:org2d41692}
\label{sec:discussion}
\subsection{``Closed-Loop'' complementary filters}
\label{sec:org2106ef4}
\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}.
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 idea is discussed in~\cite{mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/feedback_sensor_fusion.pdf}
\caption{\label{fig:feedback_sensor_fusion}``Closed-Loop'' complementary filters}
\caption{\label{fig:feedback_sensor_fusion}``Closed-Loop'' complementary filters.}
\end{figure}
Consider the feedback architecture of Fig.~\ref{fig:feedback_sensor_fusion}, with two inputs \(\hat{x}_1\) and \(\hat{x}_2\), and one output \(\hat{x}\).
The output \(\hat{x}\) is linked to the inputs by \eqref{eq:closed_loop_complementary_filters}.
The output \(\hat{x}\) is linked to the inputs by~\eqref{eq:closed_loop_complementary_filters}.
\begin{equation}
\label{eq:closed_loop_complementary_filters}
\hat{x} = \underbrace{\frac{1}{1 + L(s)}}_{S(s)} \hat{x}_1 + \underbrace{\frac{L(s)}{1 + L(s)}}_{T(s)} \hat{x}_2
\end{equation}
As for any classical feedback architecture, we have that the sum of the sensitivity and complementary sensitivity transfer function is equal to one~\eqref{eq:sensitivity_sum}.
As for any classical feedback architecture, we have that the sum of the sensitivity transfer function \(S(s)\) and complementary sensitivity transfer function \(T_(s)\) is equal to one~\eqref{eq:sensitivity_sum}.
\begin{equation}
\label{eq:sensitivity_sum}
T(s) + S(s) = 1
S(s) + T(s) = 1
\end{equation}
Therefore, provided that the closed-loop system is stable, the closed-loop system in Fig.~\ref{fig:feedback_sensor_fusion} is corresponding to two complementary filters.
Therefore, provided that the the closed-loop system in Fig.~\ref{fig:feedback_sensor_fusion} is stable, it can be used as a set of two complementary filters.
Two sensors can then be merged as shown in Fig.~\ref{fig:feedback_sensor_fusion_arch}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/feedback_sensor_fusion_arch.pdf}
\caption{\label{fig:feedback_sensor_fusion_arch}Classical feedback architecture used for sensor fusion}
\caption{\label{fig:feedback_sensor_fusion_arch}Classical feedback architecture used for sensor fusion.}
\end{figure}
One of the main advantage of implementing and designing complementary filters using the feedback architecture of Fig.~\ref{fig:feedback_sensor_fusion} is that all the tools of the linear control theory can be applied for the design of the filters.
If one want to shape both \(\frac{\hat{x}}{\hat{x}_1}(s) = S(s)\) and \(\frac{\hat{x}}{\hat{x}_2}(s) = T(s)\), the \(\mathcal{H}_\infty\) mixed-sensitivity synthesis can be easily applied.
To do so, weighting functions \(W_1(s)\) and \(W_2(s)\) are added to respectively shape \(S(s)\) and \(T(s)\) (Fig.~\ref{fig:feedback_synthesis_architecture}).
Then the system is re-organized to form the generalized plant \(P_L(s)\) shown in Fig.~\ref{fig:feedback_synthesis_architecture_generalized_plant}.
The \(\mathcal{H}_\infty\) mixed-sensitivity synthesis can finally be performed by applying the \(\mathcal{H}_\infty\) synthesis to the generalized plant \(P_L(s)\) which is described by \eqref{eq:generalized_plant_mixed_sensitivity}.
Then the system is rearranged to form the generalized plant \(P_L(s)\) shown in Fig.~\ref{fig:feedback_synthesis_architecture_generalized_plant}.
The \(\mathcal{H}_\infty\) mixed-sensitivity synthesis can finally be performed by applying the standard \(\mathcal{H}_\infty\) synthesis to the generalized plant \(P_L(s)\) which is described by~\eqref{eq:generalized_plant_mixed_sensitivity}.
\begin{equation}
\label{eq:generalized_plant_mixed_sensitivity}
@@ -619,79 +605,77 @@ The \(\mathcal{H}_\infty\) mixed-sensitivity synthesis can finally be performed
\end{bmatrix}
\end{equation}
The output of the synthesis is a filter \(L(s)\) such that the ``closed-loop'' \(\mathcal{H}_\infty\) norm from \([w_1,\ w_2]\) to \(z\) is less than one \eqref{eq:comp_filters_feedback_obj}.
The output of the synthesis is a filter \(L(s)\) such that the ``closed-loop'' \(\mathcal{H}_\infty\) norm from \([w_1,\ w_2]\) to \(z\) of the system in Fig.~\ref{fig:feedback_sensor_fusion} is less than one~\eqref{eq:comp_filters_feedback_obj}.
\begin{equation}
\label{eq:comp_filters_feedback_obj}
\left\| \begin{matrix} \frac{z}{w_1} \\ \frac{z}{w_2} \end{matrix} \right\|_\infty = \left\| \begin{matrix} \frac{1}{1 + L(s)} W_1(s) \\ \frac{L(s)}{1 + L(s)} W_2(s) \end{matrix} \right\|_\infty \le 1
\end{equation}
If the synthesis is successful, two complementary filters are obtained with their magnitudes bounded by the inverse magnitudes of the weighting functions.
The sensor fusion can then be implemented as shown in Fig.~\ref{fig:feedback_sensor_fusion_arch} using the feedback architecture or more classically as shown in Fig.~\ref{fig:sensor_fusion_overview} by defining the two complementary filters as in~\eqref{eq:comp_filters_feedback}.
If the synthesis is successful, the transfer functions from \(\hat{x}_1\) to \(\hat{x}\) and from \(\hat{x}_2\) to \(\hat{x}\) have their magnitude bounded by the inverse magnitude of the corresponding weighting functions.
The sensor fusion can then be implemented using the feedback architecture in Fig.~\ref{fig:feedback_sensor_fusion_arch} or more classically as shown in Fig.~\ref{fig:sensor_fusion_overview} by defining the two complementary filters using~\eqref{eq:comp_filters_feedback}.
The two architectures are equivalent regarding their inputs/outputs relationships.
\begin{equation}
\label{eq:comp_filters_feedback}
H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)}
\end{equation}
The two architecture are equivalent regarding their inputs/outputs relationships.
\begin{figure}[htbp]
\begin{subfigure}[t]{0.6\linewidth}
\centering
\includegraphics[scale=1]{figs/feedback_synthesis_architecture.pdf}
\caption{\label{fig:feedback_synthesis_architecture} Feedback architecture with included weights}
\caption{\label{fig:feedback_synthesis_architecture} Feedback architecture with included weights.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.4\linewidth}
\centering
\includegraphics[scale=1]{figs/feedback_synthesis_architecture_generalized_plant.pdf}
\caption{\label{fig:feedback_synthesis_architecture_generalized_plant} Generalized plant}
\caption{\label{fig:feedback_synthesis_architecture_generalized_plant} Generalized plant.}
\end{subfigure}
\hfill
\caption{\label{fig:h_inf_mixed_sensitivity_synthesis} $\mathcal{H}_\infty$ mixed-sensitivity synthesis}
\caption{\label{fig:h_inf_mixed_sensitivity_synthesis} $\mathcal{H}_\infty$ mixed-sensitivity synthesis.}
\centering
\end{figure}
As an example, two ``closed-loop'' complementary filters are designed using the \(\mathcal{H}_\infty\) mixed-sensitivity synthesis.
The weighting functions are designed using formula~\eqref{eq:weight_formula} with parameters shown in Table~\ref{tab:weights_params}.
After synthesis, a filter \(L(s)\) is obtained, its magnitude is shown in Fig.~\ref{fig:hinf_filters_results_mixed_sensitivity} by the dashed line.
After synthesis, a filter \(L(s)\) is obtained whose magnitude is shown in Fig.~\ref{fig:hinf_filters_results_mixed_sensitivity} by the black dashed line.
The ``closed-loop'' complementary filters are compared with the inverse magnitude of the weighting functions in Fig.~\ref{fig:hinf_filters_results_mixed_sensitivity} confirming that the synthesis is successful.
The obtained ``closed-loop'' complementary filters are indeed equal to the ones obtained in Section~\ref{sec:hinf_example}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/hinf_filters_results_mixed_sensitivity.pdf}
\caption{\label{fig:hinf_filters_results_mixed_sensitivity}Bode plot of the obtained complementary filters after \(\mathcal{H}_\infty\) mixed-sensitivity synthesis}
\caption{\label{fig:hinf_filters_results_mixed_sensitivity}Bode plot of the obtained complementary filters after \(\mathcal{H}_\infty\) mixed-sensitivity synthesis.}
\end{figure}
\subsection{Synthesis of more than two Complementary Filters}
\label{sec:org5b39592}
\subsection{Synthesis of a set of three complementary 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}.
For instance at the LIGO~\cite{matichard15_seism_isolat_advan_ligo}, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor (Fig.~\ref{fig:ligo_super_sensor_architecture}).\par
For instance at the LIGO, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor~\cite{matichard15_seism_isolat_advan_ligo} (Fig.~\ref{fig:ligo_super_sensor_architecture}).\par
When merging \(n>2\) sensors using complementary filters, two architectures can be used as shown in Fig.~\ref{fig:sensor_fusion_three}.
The fusion can either be done in a ``sequential'' way where \(n-1\) sets of two complementary filters are used (Fig.~\ref{fig:sensor_fusion_three_sequential}), or in a ``parallel'' way where one set of \(n\) complementary filters is used (Fig.~\ref{fig:sensor_fusion_three_parallel}).
In the first case, typical sensor fusion synthesis techniques can be used.
However, when a parallel architecture is used, a new synthesis method for a set of more than two complementary filters is required as only simple analytical formulas have been proposed in the literature~\{\cite{stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands}\}.
However, when a parallel architecture is used, a new synthesis method for a set of more than two complementary filters is required as only simple analytical formulas have been proposed in the literature~\cite{stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands}.
A generalization of the proposed synthesis method of complementary filters is presented in this section. \par
\begin{figure}[htbp]
\begin{subfigure}[b]{0.59\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_sequential.pdf}
\caption{\label{fig:sensor_fusion_three_sequential}Sequential fusion}
\caption{\label{fig:sensor_fusion_three_sequential}Sequential fusion.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.39\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_parallel.pdf}
\caption{\label{fig:sensor_fusion_three_parallel}Parallel fusion}
\caption{\label{fig:sensor_fusion_three_parallel}Parallel fusion.}
\end{subfigure}
\caption{\label{fig:sensor_fusion_three}Possible sensor fusion architecture when more than two sensors are to be merged}
\caption{\label{fig:sensor_fusion_three}Possible sensor fusion architecture when more than two sensors are to be merged.}
\centering
\end{figure}
@@ -699,62 +683,16 @@ The synthesis objective is to compute a set of \(n\) stable transfer functions \
\begin{subequations}
\label{eq:hinf_problem_gen}
\begin{align}
& \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\
& \sum_{i=1}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\
& \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:hinf_cond_perf_gen}
\end{align}
\end{subequations}
where \([W_1(s),\ W_2(s),\ \dots,\ W_n(s)]\) are weighting transfer functions that are chosen to specify the maximum wanted norms of the complementary filters during the synthesis.
Such synthesis objective is very close to the one described in Section~\ref{sec:synthesis_objective}, and indeed the proposed synthesis method is a generalization of the one presented in Section~\ref{sec:hinf_synthesis}. \par
\([W_1(s),\ W_2(s),\ \dots,\ W_n(s)]\) are weighting transfer functions that are chosen to specify the maximum complementary filters' norm during the synthesis.
Before presenting the generalized synthesis method, the case with three sensors is presented.
Consider the generalized plant \(P_3(s)\) shown in Fig.~\ref{fig:comp_filter_three_hinf_gen_plant} which is also described by~\eqref{eq:generalized_plant_three_filters}.
Such synthesis objective is closely related to the one described in Section~\ref{sec:synthesis_objective}, and indeed the proposed synthesis method is a generalization of the one presented in Section~\ref{sec:hinf_synthesis}. \par
\begin{equation}
\label{eq:generalized_plant_three_filters}
\begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix}
\end{equation}
\begin{figure}[htbp]
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_gen_plant.pdf}
\caption{\label{fig:comp_filter_three_hinf_gen_plant} Generalized plant}
\vfill
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_fb.pdf}
\caption{\label{fig:comp_filter_three_hinf_fb} Generalized plant with the synthesized filter}
\end{subfigure}
\hfill
\caption{\label{fig:comp_filter_three_hinf} Architecture for the $\mathcal{H}_\infty$ synthesis of three complementary filters}
\centering
\end{figure}
Applying the \(\mathcal{H}_\infty\) synthesis on the generalized plant \(P_3(s)\) is equivalent as to find two stable filters \([H_2(s),\ H_3(s)]\) (shown in Fig.~\ref{fig:comp_filter_three_hinf_fb}) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2, \ z_3]\) is less than one~\eqref{eq:hinf_syn_obj_three}.
\begin{equation}
\label{eq:hinf_syn_obj_three}
\left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
\end{equation}
By defining \(H_1(s) \triangleq 1 - H_2(s) - H_3(s)\), the proposed \(\mathcal{H}_\infty\) synthesis solves the design problem \eqref{eq:hinf_problem_gen} with \(n=3\). \par
An example is given to validate the method where three sensors are used in different frequency bands.
For instance a displacement sensor from DC up to \(\SI{1}{Hz}\), a geophone from \(1\) to \(\SI{10}{Hz}\) and an accelerometer above \(\SI{10}{Hz}\).
Three weighting functions are designed using formula~\eqref{eq:weight_formula} and their inverse magnitudes are shown in Fig.~\ref{fig:three_complementary_filters_results} (dashed curves).
The \(\mathcal{H}_\infty\) synthesis is performed on the generalized plant \(P_3(s)\) and the bode plot of the obtained complementary filters are shown in Fig.~\ref{fig:three_complementary_filters_results} (solid lines). \par
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/three_complementary_filters_results.pdf}
\caption{\label{fig:three_complementary_filters_results}Bode plot of the inverse weighting functions and of the three complementary filters obtained using the \(\mathcal{H}_\infty\) synthesis}
\end{figure}
Even though there might not be any practical application for a set of more than 3 complementary filters, it can still be designed using the same procedure.
A set of \(n\) complementary filters can be shaped using the generalized plant \(P_n(s)\) described by~\eqref{eq:generalized_plant_n_filters}.
A set of \(n\) complementary filters can be shaped by applying the standard \(\mathcal{H}_\infty\) synthesis to the generalized plant \(P_n(s)\) described by~\eqref{eq:generalized_plant_n_filters}.
\begin{equation}
\label{eq:generalized_plant_n_filters}
@@ -769,28 +707,85 @@ A set of \(n\) complementary filters can be shaped using the generalized plant \
\end{bmatrix}
\end{equation}
If the synthesis if successful, a set of \(n-1\) filters \([H_2(s),\ H_3(s),\ \dots,\ H_n(s)]\) are obtained such that~\eqref{eq:hinf_syn_obj_gen} is verified.
\begin{equation}
\label{eq:hinf_syn_obj_gen}
\left\|\begin{matrix} \left(1 - \left[ H_2(s) + H_3(s) + \dots + H_n(s) \right]\right) W_1(s) \\ H_2(s) W_2(s) \\ \vdots \\ H_n(s) W_n(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
\(H_1(s)\) is then defined using~\eqref{eq:h1_comp_h2_hn} which is ensuring the complementary property for the set of \(n\) filters~\eqref{eq:hinf_cond_compl_gen}.
Condition~\eqref{eq:hinf_cond_perf_gen} is satisfied thanks to~\eqref{eq:hinf_syn_obj_gen}.
\begin{equation}
\label{eq:h1_comp_h2_hn}
H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) + \dots + H_n(s) \big]
\end{equation}
An example is given to validate the proposed method for the synthesis of a set of three complementary filters.
The sensors to be merged are a displacement sensor from DC up to \(\SI{1}{Hz}\), a geophone from \(1\) to \(\SI{10}{Hz}\) and an accelerometer above \(\SI{10}{Hz}\).
Three weighting functions are designed using formula~\eqref{eq:weight_formula} and their inverse magnitude are shown in Fig.~\ref{fig:three_complementary_filters_results} (dashed curves).
Consider the generalized plant \(P_3(s)\) shown in Fig.~\ref{fig:comp_filter_three_hinf_gen_plant} which is also described by~\eqref{eq:generalized_plant_three_filters}.
\begin{equation}
\label{eq:generalized_plant_three_filters}
\begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix}
\end{equation}
\begin{figure}[htbp]
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_gen_plant.pdf}
\caption{\label{fig:comp_filter_three_hinf_gen_plant} Generalized plant.}
\vfill
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_fb.pdf}
\caption{\label{fig:comp_filter_three_hinf_fb} Generalized plant with the synthesized filter.}
\end{subfigure}
\hfill
\caption{\label{fig:comp_filter_three_hinf} Architecture for the $\mathcal{H}_\infty$ synthesis of three complementary filters.}
\centering
\end{figure}
The standard \(\mathcal{H}_\infty\) synthesis is performed on the generalized plant \(P_3(s)\).
Two filters \(H_2(s)\) and \(H_3(s)\) are obtained such that the \(\mathcal{H}_\infty\) norm of the closed-loop transfer from \(w\) to \([z_1,\ z_2,\ z_3]\) of the system in Fig.~\ref{fig:comp_filter_three_hinf_fb} is less than one.
Filter \(H_1(s)\) is defined using~\eqref{eq:h1_compl_h2_h3} thus ensuring the complementary property of the obtained set of filters.
\begin{equation}
\label{eq:h1_compl_h2_h3}
H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) \big]
\end{equation}
Figure~\ref{fig:three_complementary_filters_results} displays the three synthesized complementary filters (solid lines) which confirms that the synthesis is successful.\par
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/three_complementary_filters_results.pdf}
\caption{\label{fig:three_complementary_filters_results}Bode plot of the inverse weighting functions and of the three complementary filters obtained using the \(\mathcal{H}_\infty\) synthesis.}
\end{figure}
\section{Conclusion}
\label{sec:org369c466}
\label{sec:conclusion}
Sensors measuring a physical quantities are always subject to limitations both in terms of bandwidth or accuracy.
Complementary filters can be used to fuse multiple sensors with different characteristics in order to combine their benefits and yield a better estimate of the measured physical quantity.
The sensor fusion robustness and the obtained super sensor noise can be linked to the magnitude of the complementary filters.
In this paper, a synthesis method that enables the shaping of the complementary filters norms has been proposed.
Several example were used to emphasize the simplicity and the effectiveness of the proposed method.
Links with ``closed-loop'' complementary filters where highlighted, and the proposed method was generalized for the design of a set of more than two complementary filters.
A new method for designing complementary filters using the \(\mathcal{H}_\infty\) synthesis has been proposed.
It allows to shape the magnitude of the filters by the use of weighting functions during the synthesis.
This is very valuable in practice as the characteristics of the super sensor are linked to the complementary filters' magnitude.
Therefore typical sensor fusion objectives can be translated into requirements on the magnitudes of the filters.
Several examples were used to emphasize the simplicity and the effectiveness of the proposed method.
However, the shaping of the complementary filters' magnitude does not allow to directly optimize the super sensor noise and dynamical characteristics.
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}
\label{sec:orgd71260d}
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 has been assigned the LIGO document number LIGO-P2100328.
\section*{Data Availability}
\label{sec:org30a4627}
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}.
\bibliographystyle{elsarticle-num}

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\begin{tikzpicture}
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journal/ref.bib
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@@ -14,7 +14,7 @@
doi = {10.1117/12.552518},
school = {stanford university},
title = {Low frequency vibration isolation and alignment system for
advanced {LIGO}},
{Advanced LIGO}},
year = 2005,
}
@@ -31,7 +31,7 @@
@article{matichard15_seism_isolat_advan_ligo,
author = {Matichard, F and Lantz, B and Mittleman, R and Mason, K and
Kissel, J and others},
title = {Seismic Isolation of Advanced {LIGO}: Review of Strategy,
title = {Seismic Isolation of {Advanced LIGO}: Review of Strategy,
Instrumentation and Performance},
journal = {Classical and Quantum Gravity},
volume = 32,
@@ -285,7 +285,7 @@
Giaime, Joseph A and Hammond, Giles Dominic and Hardham, C and
Hennessy, Mike and How, Jonathan P and Lantz, Brian T and
Macinnis, M and others},
title = {Low-frequency active vibration isolation for advanced {LIGO}},
title = {Low-frequency active vibration isolation for {Advanced LIGO}},
booktitle = {Gravitational Wave and Particle Astrophysics Detectors},
year = 2004,
volume = 5500,
@@ -303,10 +303,10 @@
}
@phdthesis{lucia18_low_frequen_optim_perfor_advan,
author = {Trozzo Lucia},
author = {L. Trozzo},
school = {University of Siena},
title = {Low Frequency Optimization and Performance of Advanced
Virgo Seismic Isolation System},
title = {Low Frequency Optimization and Performance of {Advanced
Virgo} Seismic Isolation System},
year = 2018,
}
@@ -378,11 +378,21 @@
}
@misc{dehaeze21_new_method_desig_compl_filter_code,
author = {Thomas Dehaeze},
doi = {10.5281/zenodo.3894342},
howpublished = {Source Code on Zonodo},
month = 09,
title = {A New Method of Designing Complementary Filters for Sensor
Fusion using $\mathcal{H}_\infty$ Synthesis},
author = {Dehaeze, Thomas},
doi = {10.5281/zenodo.5361943},
howpublished = {Source code on Zenodo},
month = sep,
title = {{A New Method of Designing Complementary Filters for Sensor
Fusion Using the $\mathcal{H}_\infty$ Synthesis}},
year = 2021,
}
@book{skogestad07_multiv_feedb_contr,
author = {Skogestad, Sigurd and Postlethwaite, Ian},
title = {Multivariable Feedback Control: Analysis and Design -
Second Edition},
year = 2007,
publisher = {John Wiley},
isbn = {978-0470011683},
note = {isbn:978-0470011683},
}

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#!/bin/env perl
# Shebang is only to get syntax highlighting right across GitLab, GitHub and IDEs.
# This file is not meant to be run, but read by `latexmk`.
# ======================================================================================
# Perl `latexmk` configuration file
# ======================================================================================
# ======================================================================================
# PDF Generation/Building/Compilation
# ======================================================================================
@default_files=('dehaeze21_desig_compl_filte_matlab.tex');
# PDF-generating modes are:
# 1: pdflatex, as specified by $pdflatex variable (still largely in use)
# 2: postscript conversion, as specified by the $ps2pdf variable (useless)
# 3: dvi conversion, as specified by the $dvipdf variable (useless)
# 4: lualatex, as specified by the $lualatex variable (best)
# 5: xelatex, as specified by the $xelatex variable (second best)
$pdf_mode = 1;
# Treat undefined references and citations as well as multiply defined references as
# ERRORS instead of WARNINGS.
# This is only checked in the *last* run, since naturally, there are undefined references
# in initial runs.
# This setting is potentially annoying when debugging/editing, but highly desirable
# in the CI pipeline, where such a warning should result in a failed pipeline, since the
# final document is incomplete/corrupted.
#
# However, I could not eradicate all warnings, so that `latexmk` currently fails with
# this option enabled.
# Specifically, `microtype` fails together with `fontawesome`/`fontawesome5`, see:
# https://tex.stackexchange.com/a/547514/120853
# The fix in that answer did not help.
# Setting `verbose=silent` to mute `microtype` warnings did not work.
# Switching between `fontawesome` and `fontawesome5` did not help.
$warnings_as_errors = 0;
# Show used CPU time. Looks like: https://tex.stackexchange.com/a/312224/120853
$show_time = 1;
# Default is 5; we seem to need more owed to the complexity of the document.
# Actual documents probably don't need this many since they won't use all features,
# plus won't be compiling from cold each time.
$max_repeat=7;
# --shell-escape option (execution of code outside of latex) is required for the
#'svg' package.
# It converts raw SVG files to the PDF+PDF_TEX combo using InkScape.
#
# SyncTeX allows to jump between source (code) and output (PDF) in IDEs with support
# (many have it). A value of `1` is enabled (gzipped), `-1` is enabled but uncompressed,
# `0` is off.
# Testing in VSCode w/ LaTeX Workshop only worked for the compressed version.
# Adjust this as needed. Of course, only relevant for local use, no effect on a remote
# CI pipeline (except for slower compilation, probably).
#
# %O and %S will forward Options and the Source file, respectively, given to latexmk.
#
# `set_tex_cmds` applies to all *latex commands (latex, xelatex, lualatex, ...), so
# no need to specify these each. This allows to simply change `$pdf_mode` to get a
# different engine. Check if this works with `latexmk --commands`.
set_tex_cmds("--shell-escape -interaction=nonstopmode --synctex=1 %O %S");
# Use default pdf viewer
$pdf_previewer = 'zathura';
# option 2 is same as 1 (run biber when necessary), but also deletes the
# regeneratable bbl-file in a clenaup (`latexmk -c`). Do not use if original
# bib file is not available!
$bibtex_use = 2; # default: 1
# Change default `biber` call, help catch errors faster/clearer. See
# https://web.archive.org/web/20200526101657/https://www.semipol.de/2018/06/12/latex-best-practices.html#database-entries
$biber = "biber --validate-datamodel %O %S";
# ======================================================================================
# Auxiliary Files
# ======================================================================================
# Let latexmk know about generated files, so they can be used to detect if a
# rerun is required, or be deleted in a cleanup.
# loe: List of Examples (KOMAScript)
# lol: List of Listings (`listings` and `minted` packages)
# run.xml: biber runs
# glg: glossaries log
# glstex: generated from glossaries-extra
push @generated_exts, 'loe', 'lol', 'run.xml', 'glg', 'glstex';
# Also delete the *.glstex files from package glossaries-extra. Problem is,
# that that package generates files of the form "basename-digit.glstex" if
# multiple glossaries are present. Latexmk looks for "basename.glstex" and so
# does not find those. For that purpose, use wildcard.
# Also delete files generated by gnuplot/pgfplots contour plots
# (.dat, .script, .table).
$clean_ext = "%R-*.glstex %R_contourtmp*.*";

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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Initialize Frequency Vector
freqs = logspace(-1, 3, 1000);
%% Weighting Function Design
% Parameters
n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2;
% Formulas
W = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
%% Magnitude of the weighting function with parameters
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(W, freqs, 'Hz'))), 'k-');
plot([1e-3 1e0], [G0 G0], 'k--', 'LineWidth', 1)
text(1e0, G0, '$\quad G_0$')
plot([1e1 1e3], [G1 G1], 'k--', 'LineWidth', 1)
text(1e1,G1,'$G_{\infty}\quad$','HorizontalAlignment', 'right')
plot([w0/2/pi w0/2/pi], [1 2*Gc], 'k--', 'LineWidth', 1)
text(w0/2/pi,1,'$\omega_c$','VerticalAlignment', 'top', 'HorizontalAlignment', 'center')
plot([w0/2/pi/2 2*w0/2/pi], [Gc Gc], 'k--', 'LineWidth', 1)
text(w0/2/pi/2, Gc, '$G_c \quad$','HorizontalAlignment', 'right')
text(w0/5/pi/2, abs(evalfr(W, j*w0/5)), 'Slope: $n \quad$', 'HorizontalAlignment', 'right')
text(w0/2/pi, abs(evalfr(W, j*w0)), '$\bullet$', 'HorizontalAlignment', 'center')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([5e-4, 20]);
yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
%% Design of the Weighting Functions
W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45);
W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
%% Plot of the Weighting function magnitude
figure;
tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile();
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, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]', 'FontSize', 10); ylabel('Magnitude', 'FontSize', 10);
hold off;
xlim([freqs(1), freqs(end)]);
xticks([0.1, 1, 10, 100, 1000]);
ylim([8e-4, 20]);
yticks([1e-3, 1e-2, 1e-1, 1, 1e1]);
yticklabels({'', '$10^{-2}$', '', '$10^0$', ''});
ax1.FontSize = 9;
leg = legend('location', 'south', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
%% Generalized Plant
P = [W1 -W1;
0 W2;
1 0];
%% H-Infinity Synthesis
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
%% Define H1 to be the complementary of H2
H1 = 1 - H2;
%% Bode plot of the complementary filters
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', '$H_2$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude');
ylim([8e-4, 20]);
yticks([1e-3, 1e-2, 1e-1, 1, 1e1]);
yticklabels({'', '$10^{-2}$', '', '$10^0$', ''})
leg = legend('location', 'south', 'FontSize', 8, 'NumColumns', 2);
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)]);

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matlab/matlab/2_ligo_complementary_filters.m Normal file
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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Initialize Frequency Vector
freqs = logspace(-3, 0, 1000);
%% Upper bounds for the complementary filters
figure;
hold on;
set(gca,'ColorOrderIndex',1)
plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
set(gca,'ColorOrderIndex',1)
plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',1)
plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2)
plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([1e-4, 10]);
leg = legend('location', 'southeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
%% Initialized CVX
cvx_startup;
cvx_solver sedumi;
%% Frequency vectors
w1 = 0:4.06e-4:0.008;
w2 = 0.008:4.06e-4:0.04;
w3 = 0.04:8.12e-4:0.1;
w4 = 0.1:8.12e-4:0.83;
%% Filter order
n = 512;
%% Initialization of filter responses
A1 = [ones(length(w1),1), cos(kron(w1'.*(2*pi),[1:n-1]))];
A2 = [ones(length(w2),1), cos(kron(w2'.*(2*pi),[1:n-1]))];
A3 = [ones(length(w3),1), cos(kron(w3'.*(2*pi),[1:n-1]))];
A4 = [ones(length(w4),1), cos(kron(w4'.*(2*pi),[1:n-1]))];
B1 = [zeros(length(w1),1), sin(kron(w1'.*(2*pi),[1:n-1]))];
B2 = [zeros(length(w2),1), sin(kron(w2'.*(2*pi),[1:n-1]))];
B3 = [zeros(length(w3),1), sin(kron(w3'.*(2*pi),[1:n-1]))];
B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))];
%% Convex optimization
cvx_begin
variable y(n+1,1)
% t
maximize(-y(1))
for i = 1:length(w1)
norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
end
for i = 1:length(w2)
norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
end
for i = 1:length(w3)
norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
end
for i = 1:length(w4)
norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
end
cvx_end
h = y(2:end);
%% Combine the frequency vectors to form the obtained filter
w = [w1 w2 w3 w4];
H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
%% Bode plot of the obtained complementary filters
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
set(gca,'ColorOrderIndex',1)
plot(w, abs(1-H), '-', 'DisplayName', '$L_1$');
plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
set(gca,'ColorOrderIndex',2)
plot(w, abs(H), '-', 'DisplayName', '$H_1$');
plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude');
hold off;
ylim([5e-3, 10]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 16;
% Phase
ax2 = nexttile;
hold on;
plot(w, 180/pi*unwrap(angle(1-H)), '-');
plot(w, 180/pi*unwrap(angle(H)), '-');
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
yticks([-360:180:180]); ylim([-380, 200]);
linkaxes([ax1,ax2],'x');
xlim([1e-3, 1]);
%% Design of the weight for the high pass filter
w1 = 2*pi*0.008; x1 = 0.35;
w2 = 2*pi*0.04; x2 = 0.5;
w3 = 2*pi*0.05; x3 = 0.5;
% Slope of +3 from w1
wH = 0.008*(s^2/w1^2 + 2*x1/w1*s + 1)*(s/w1 + 1);
% Little bump from w2 to w3
wH = wH*(s^2/w2^2 + 2*x2/w2*s + 1)/(s^2/w3^2 + 2*x3/w3*s + 1);
% No Slope at high frequencies
wH = wH/(s^2/w3^2 + 2*x3/w3*s + 1)/(s/w3 + 1);
% Little bump between w2 and w3
w0 = 2*pi*0.045; xi = 0.1; A = 2; n = 1;
wH = wH*((s^2 + 2*w0*xi*A^(1/n)*s + w0^2)/(s^2 + 2*w0*xi*s + w0^2))^n;
wH = 1/wH;
wH = minreal(ss(wH));
%% Design of the weight for the low pass filter
n = 20; % Filter order
Rp = 1; % Peak to peak passband ripple
Wp = 2*pi*0.102; % Edge frequency
% Chebyshev Type I filter design
[b,a] = cheby1(n, Rp, Wp, 'high', 's');
wL = 0.04*tf(a, b);
wL = 1/wL;
wL = minreal(ss(wL));
%% Magnitude of the designed Weights and initial specifications
figure;
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|W_L|^{-1}$');
plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|W_H|^{-1}$');
plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([5e-3, 10]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 16;
%% Generalized plant for the H-infinity Synthesis
P = [0 wL;
wH -wH;
1 0];
%% Standard H-Infinity synthesis
[Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
%% High pass filter as the complementary of the low pass filter
Hh = 1 - Hl;
%% Minimum realization of the filters
Hh = minreal(Hh);
Hl = minreal(Hl);
%% Bode plot of the obtained filters and comparison with the upper bounds
figure;
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'DisplayName', '$L_1^\prime$');
plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$H_1^\prime$');
plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([5e-3, 10]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 16;
%% Comparison of the complementary filters obtained with H-infinity and with CVX
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', ...
'DisplayName', '$L_1(s)$ - $\mathcal{H}_\infty$');
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', ...
'DisplayName', '$H_1(s)$ - $\mathcal{H}_\infty$');
set(gca,'ColorOrderIndex',1);
plot(w, abs(1-H), '--', ...
'DisplayName', '$L_1(s)$ - FIR');
set(gca,'ColorOrderIndex',2);
plot(w, abs(H), '--', ...
'DisplayName', '$H_1(s)$ - FIR');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude');
set(gca, 'XTickLabel',[]);
ylim([5e-3, 10]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 16;
% Phase
ax2 = nexttile;
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hl, freqs, 'Hz')))), '-');
set(gca,'ColorOrderIndex',2);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hh, freqs, 'Hz')))), '-');
set(gca,'ColorOrderIndex',1);
plot(w, 180/pi*unwrap(angle(1-H)), '--');
set(gca,'ColorOrderIndex',2);
plot(w, 180/pi*unwrap(angle(H)), '--');
set(gca, 'XScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks([-360:180:180]); ylim([-380, 200]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);

68
matlab/matlab/3_closed_loop_complementary_filters.m Normal file
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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Initialize Frequency Vector
freqs = logspace(-1, 3, 1000);
%% Design of the Weighting Functions
W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45);
W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
%% Generalized plant for "closed-loop" complementary filter synthesis
P = [ W1 0 1;
-W1 W2 -1];
%% Standard H-Infinity Synthesis
[L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
%% Complementary filters
H1 = inv(1 + L);
H2 = 1 - H1;
%% Bode plot of the obtained Complementary filters with upper-bounds
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)]);

37
matlab/matlab/three_comp_filters.m → matlab/matlab/4_three_complementary_filters.m
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@@ -4,21 +4,14 @@ clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
freqs = logspace(-2, 4, 1000);
addpath('./src');
% Weights
% First we define the weights.
n = 2; w0 = 2*pi*1; G0 = 1/10; G1 = 1000; Gc = 1/2;
W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
freqs = logspace(-2, 3, 1000);
%% Design of the Weighting Functions
W1 = generateWF('n', 2, 'w0', 2*pi*1, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.5);
W2 = 0.22*(1 + s/2/pi/1)^2/(sqrt(1e-4) + s/2/pi/1)^2*(1 + s/2/pi/10)^2/(1 + s/2/pi/1000)^2;
W3 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.5);
n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 1/2;
W3 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
%% Inverse magnitude of the weighting functions
figure;
hold on;
set(gca,'ColorOrderIndex',1)
@@ -34,27 +27,23 @@ xlim([freqs(1), freqs(end)]); ylim([2e-4, 1.3e1])
leg = legend('location', 'northeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
% H-Infinity Synthesis
% Then we create the generalized plant =P=.
%% Generalized plant for the synthesis of 3 complementary filters
P = [W1 -W1 -W1;
0 W2 0 ;
0 0 W3;
1 0 0];
% And we do the $\mathcal{H}_\infty$ synthesis.
%% Standard H-Infinity Synthesis
[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
% Obtained Complementary Filters
% The obtained filters are:
%% Synthesized H2 and H3 filters
H2 = tf(H(1));
H3 = tf(H(2));
%% H1 is defined as the complementary filter of H2 and H3
H1 = 1 - H2 - H3;
%% Bode plot of the obtained complementary filters
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
@@ -79,7 +68,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude');
set(gca, 'XTickLabel',[]);
ylim([1e-4, 20]);
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
leg = legend('location', 'northeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
% Phase
@@ -94,7 +83,7 @@ plot(freqs, 180/pi*phase(squeeze(freqresp(H3, freqs, 'Hz'))));
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
yticks([-360:90:360]); ylim([-270, 270]);
yticks([-180:90:180]); ylim([-220, 220]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);

410
matlab/matlab/comp_filters_ligo.m
View File

@@ -1,410 +0,0 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
freqs = logspace(-3, 0, 1000);
addpath('./src');
% Specifications
% The specifications for the filters are:
% 1. From $0$ to $0.008\text{ Hz}$,the magnitude of the filter’s transfer function should be less than or equal to $8 \times 10^{-3}$
% 2. From $0.008\text{ Hz}$ to $0.04\text{ Hz}$, it attenuates the input signal proportional to frequency cubed
% 3. Between $0.04\text{ Hz}$ and $0.1\text{ Hz}$, the magnitude of the transfer function should be less than 3
% 4. Above $0.1\text{ Hz}$, the maximum of the magnitude of the complement filter should be as close to zero as possible. In our system, we would like to have the magnitude of the complementary filter to be less than $0.1$. As the filters obtained in cite:hua05_low_ligo have a magnitude of $0.045$, we will set that as our requirement
% The specifications are translated in upper bounds of the complementary filters are shown on figure [[fig:ligo_specifications]].
figure;
hold on;
set(gca,'ColorOrderIndex',1)
plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
set(gca,'ColorOrderIndex',1)
plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',1)
plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2)
plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([1e-4, 10]);
leg = legend('location', 'southeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
% FIR Filter
% We here try to implement the FIR complementary filter synthesis as explained in cite:hua05_low_ligo.
% For that, we use the [[http://cvxr.com/cvx/][CVX matlab Toolbox]].
% We setup the CVX toolbox and use the =SeDuMi= solver.
cvx_startup;
cvx_solver sedumi;
% We define the frequency vectors on which we will constrain the norm of the FIR filter.
w1 = 0:4.06e-4:0.008;
w2 = 0.008:4.06e-4:0.04;
w3 = 0.04:8.12e-4:0.1;
w4 = 0.1:8.12e-4:0.83;
% We then define the order of the FIR filter.
n = 512;
A1 = [ones(length(w1),1), cos(kron(w1'.*(2*pi),[1:n-1]))];
A2 = [ones(length(w2),1), cos(kron(w2'.*(2*pi),[1:n-1]))];
A3 = [ones(length(w3),1), cos(kron(w3'.*(2*pi),[1:n-1]))];
A4 = [ones(length(w4),1), cos(kron(w4'.*(2*pi),[1:n-1]))];
B1 = [zeros(length(w1),1), sin(kron(w1'.*(2*pi),[1:n-1]))];
B2 = [zeros(length(w2),1), sin(kron(w2'.*(2*pi),[1:n-1]))];
B3 = [zeros(length(w3),1), sin(kron(w3'.*(2*pi),[1:n-1]))];
B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))];
% We run the convex optimization.
cvx_begin
variable y(n+1,1)
% t
maximize(-y(1))
for i = 1:length(w1)
norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
end
for i = 1:length(w2)
norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
end
for i = 1:length(w3)
norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
end
for i = 1:length(w4)
norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
end
cvx_end
h = y(2:end);
% #+RESULTS:
% #+begin_example
% cvx_begin
% variable y(n+1,1)
% % t
% maximize(-y(1))
% for i = 1:length(w1)
% norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
% end
% for i = 1:length(w2)
% norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
% end
% for i = 1:length(w3)
% norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
% end
% for i = 1:length(w4)
% norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
% end
% cvx_end
% Calling SeDuMi 1.34: 4291 variables, 1586 equality constraints
% For improved efficiency, SeDuMi is solving the dual problem.
% ------------------------------------------------------------
% SeDuMi 1.34 (beta) by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
% Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
% eqs m = 1586, order n = 3220, dim = 4292, blocks = 1073
% nnz(A) = 1100727 + 0, nnz(ADA) = 1364794, nnz(L) = 683190
% it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
% 0 : 4.11E+02 0.000
% 1 : -2.58E+00 1.25E+02 0.000 0.3049 0.9000 0.9000 4.87 1 1 3.0E+02
% 2 : -2.36E+00 3.90E+01 0.000 0.3118 0.9000 0.9000 1.83 1 1 6.6E+01
% 3 : -1.69E+00 1.31E+01 0.000 0.3354 0.9000 0.9000 1.76 1 1 1.5E+01
% 4 : -8.60E-01 7.10E+00 0.000 0.5424 0.9000 0.9000 2.48 1 1 4.8E+00
% 5 : -4.91E-01 5.44E+00 0.000 0.7661 0.9000 0.9000 3.12 1 1 2.5E+00
% 6 : -2.96E-01 3.88E+00 0.000 0.7140 0.9000 0.9000 2.62 1 1 1.4E+00
% 7 : -1.98E-01 2.82E+00 0.000 0.7271 0.9000 0.9000 2.14 1 1 8.5E-01
% 8 : -1.39E-01 2.00E+00 0.000 0.7092 0.9000 0.9000 1.78 1 1 5.4E-01
% 9 : -9.99E-02 1.30E+00 0.000 0.6494 0.9000 0.9000 1.51 1 1 3.3E-01
% 10 : -7.57E-02 8.03E-01 0.000 0.6175 0.9000 0.9000 1.31 1 1 2.0E-01
% 11 : -5.99E-02 4.22E-01 0.000 0.5257 0.9000 0.9000 1.17 1 1 1.0E-01
% 12 : -5.28E-02 2.45E-01 0.000 0.5808 0.9000 0.9000 1.08 1 1 5.9E-02
% 13 : -4.82E-02 1.28E-01 0.000 0.5218 0.9000 0.9000 1.05 1 1 3.1E-02
% 14 : -4.56E-02 5.65E-02 0.000 0.4417 0.9045 0.9000 1.02 1 1 1.4E-02
% 15 : -4.43E-02 2.41E-02 0.000 0.4265 0.9004 0.9000 1.01 1 1 6.0E-03
% 16 : -4.37E-02 8.90E-03 0.000 0.3690 0.9070 0.9000 1.00 1 1 2.3E-03
% 17 : -4.35E-02 3.24E-03 0.000 0.3641 0.9164 0.9000 1.00 1 1 9.5E-04
% 18 : -4.34E-02 1.55E-03 0.000 0.4788 0.9086 0.9000 1.00 1 1 4.7E-04
% 19 : -4.34E-02 8.77E-04 0.000 0.5653 0.9169 0.9000 1.00 1 1 2.8E-04
% 20 : -4.34E-02 5.05E-04 0.000 0.5754 0.9034 0.9000 1.00 1 1 1.6E-04
% 21 : -4.34E-02 2.94E-04 0.000 0.5829 0.9136 0.9000 1.00 1 1 9.9E-05
% 22 : -4.34E-02 1.63E-04 0.015 0.5548 0.9000 0.0000 1.00 1 1 6.6E-05
% 23 : -4.33E-02 9.42E-05 0.000 0.5774 0.9053 0.9000 1.00 1 1 3.9E-05
% 24 : -4.33E-02 6.27E-05 0.000 0.6658 0.9148 0.9000 1.00 1 1 2.6E-05
% 25 : -4.33E-02 3.75E-05 0.000 0.5972 0.9187 0.9000 1.00 1 1 1.6E-05
% 26 : -4.33E-02 1.89E-05 0.000 0.5041 0.9117 0.9000 1.00 1 1 8.6E-06
% 27 : -4.33E-02 9.72E-06 0.000 0.5149 0.9050 0.9000 1.00 1 1 4.5E-06
% 28 : -4.33E-02 2.94E-06 0.000 0.3021 0.9194 0.9000 1.00 1 1 1.5E-06
% 29 : -4.33E-02 9.73E-07 0.000 0.3312 0.9189 0.9000 1.00 2 2 5.3E-07
% 30 : -4.33E-02 2.82E-07 0.000 0.2895 0.9063 0.9000 1.00 2 2 1.6E-07
% 31 : -4.33E-02 8.05E-08 0.000 0.2859 0.9049 0.9000 1.00 2 2 4.7E-08
% 32 : -4.33E-02 1.43E-08 0.000 0.1772 0.9059 0.9000 1.00 2 2 8.8E-09
% iter seconds digits c*x b*y
% 32 49.4 6.8 -4.3334083581e-02 -4.3334090214e-02
% |Ax-b| = 3.7e-09, [Ay-c]_+ = 1.1E-10, |x|= 1.0e+00, |y|= 2.6e+00
% Detailed timing (sec)
% Pre IPM Post
% 3.902E+00 4.576E+01 1.035E-02
% Max-norms: ||b||=1, ||c|| = 3,
% Cholesky |add|=0, |skip| = 0, ||L.L|| = 4.26267.
% ------------------------------------------------------------
% Status: Solved
% Optimal value (cvx_optval): -0.0433341
% h = y(2:end);
% #+end_example
% Finally, we compute the filter response over the frequency vector defined and the result is shown on figure [[fig:fir_filter_ligo]] which is very close to the filters obtain in cite:hua05_low_ligo.
w = [w1 w2 w3 w4];
H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
plot(w, abs(H), 'k-');
plot(w, abs(1-H), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude');
set(gca, 'XTickLabel',[]);
ylim([5e-3, 5]);
% Phase
ax2 = nexttile;
hold on;
plot(w, 180/pi*unwrap(angle(H)), 'k-');
plot(w, 180/pi*unwrap(angle(1-H)), 'k--');
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
yticks([-450:90:180]); ylim([-450, 200]);
linkaxes([ax1,ax2],'x');
xlim([1e-3, 1]);
% Weights
% We design weights that will be used for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
% These weights will determine the order of the obtained filters.
% Here are the requirements on the filters:
% - reasonable order
% - to be as close as possible to the specified upper bounds
% - stable minimum phase
% The bode plot of the weights is shown on figure [[fig:ligo_weights]].
w1 = 2*pi*0.008; x1 = 0.35;
w2 = 2*pi*0.04; x2 = 0.5;
w3 = 2*pi*0.05; x3 = 0.5;
% Slope of +3 from w1
wH = 0.008*(s^2/w1^2 + 2*x1/w1*s + 1)*(s/w1 + 1);
% Little bump from w2 to w3
wH = wH*(s^2/w2^2 + 2*x2/w2*s + 1)/(s^2/w3^2 + 2*x3/w3*s + 1);
% No Slope at high frequencies
wH = wH/(s^2/w3^2 + 2*x3/w3*s + 1)/(s/w3 + 1);
% Little bump between w2 and w3
w0 = 2*pi*0.045; xi = 0.1; A = 2; n = 1;
wH = wH*((s^2 + 2*w0*xi*A^(1/n)*s + w0^2)/(s^2 + 2*w0*xi*s + w0^2))^n;
wH = 1/wH;
wH = minreal(ss(wH));
n = 20; Rp = 1; Wp = 2*pi*0.102;
[b,a] = cheby1(n, Rp, Wp, 'high', 's');
wL = 0.04*tf(a, b);
wL = 1/wL;
wL = minreal(ss(wL));
figure;
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|w_H|^{-1}$');
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|w_L|^{-1}$');
plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec.');
plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
plot([0.1, 10], [0.045, 0.045], 'k--', 'HandleVisibility', 'off');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([1e-3, 10]);
leg = legend('location', 'southeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
% H-Infinity Synthesis
% We define the generalized plant as shown on figure [[fig:h_infinity_robst_fusion]].
P = [0 wL;
wH -wH;
1 0];
% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
[Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
% #+RESULTS:
% #+begin_example
% [Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
% Resetting value of Gamma min based on D_11, D_12, D_21 terms
% Test bounds: 0.3276 < gamma <= 1.8063
% gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
% 1.806 1.4e-02 -1.7e-16 3.6e-03 -4.8e-12 0.0000 p
% 1.067 1.3e-02 -4.2e-14 3.6e-03 -1.9e-12 0.0000 p
% 0.697 1.3e-02 -3.0e-01# 3.6e-03 -3.5e-11 0.0000 f
% 0.882 1.3e-02 -9.5e-01# 3.6e-03 -1.2e-34 0.0000 f
% 0.975 1.3e-02 -2.7e+00# 3.6e-03 -1.6e-12 0.0000 f
% 1.021 1.3e-02 -8.7e+00# 3.6e-03 -4.5e-16 0.0000 f
% 1.044 1.3e-02 -6.5e-14 3.6e-03 -3.0e-15 0.0000 p
% 1.032 1.3e-02 -1.8e+01# 3.6e-03 0.0e+00 0.0000 f
% 1.038 1.3e-02 -3.8e+01# 3.6e-03 0.0e+00 0.0000 f
% 1.041 1.3e-02 -8.3e+01# 3.6e-03 -2.9e-33 0.0000 f
% 1.042 1.3e-02 -1.9e+02# 3.6e-03 -3.4e-11 0.0000 f
% 1.043 1.3e-02 -5.3e+02# 3.6e-03 -7.5e-13 0.0000 f
% Gamma value achieved: 1.0439
% #+end_example
% The high pass filter is defined as $H_H = 1 - H_L$.
Hh = 1 - Hl;
Hh = minreal(Hh);
Hl = minreal(Hl);
% The size of the filters is shown below.
size(Hh), size(Hl)
% #+RESULTS:
% #+begin_example
% size(Hh), size(Hl)
% State-space model with 1 outputs, 1 inputs, and 27 states.
% State-space model with 1 outputs, 1 inputs, and 27 states.
% #+end_example
% The bode plot of the obtained filters as shown on figure [[fig:hinf_synthesis_ligo_results]].
figure;
hold on;
set(gca,'ColorOrderIndex',1);
plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
set(gca,'ColorOrderIndex',1);
plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',1);
plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2);
plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$H_H$');
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'DisplayName', '$H_L$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([1e-3, 10]);
leg = legend('location', 'southeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
% Compare FIR and H-Infinity Filters
% Let's now compare the FIR filters designed in cite:hua05_low_ligo and the one obtained with the $\mathcal{H}_\infty$ synthesis on figure [[fig:comp_fir_ligo_hinf]].
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', ...
'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$');
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', ...
'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$');
set(gca,'ColorOrderIndex',1);
plot(w, abs(H), '--', ...
'DisplayName', '$H_H(s)$ - FIR');
set(gca,'ColorOrderIndex',2);
plot(w, abs(1-H), '--', ...
'DisplayName', '$H_L(s)$ - FIR');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude');
set(gca, 'XTickLabel',[]);
ylim([5e-3, 10]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 16;
% Phase
ax2 = nexttile;
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hh, freqs, 'Hz')))), '-');
set(gca,'ColorOrderIndex',2);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hl, freqs, 'Hz')))), '-');
set(gca,'ColorOrderIndex',1);
plot(w, 180/pi*unwrap(angle(H)), '--');
set(gca,'ColorOrderIndex',2);
plot(w, 180/pi*unwrap(angle(1-H)), '--');
set(gca, 'XScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks([-450:90:180]); ylim([-450, 200]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);

0
matlab/matlab/src/generateCF.m → matlab/matlab/generateCF.m
View File

0
matlab/matlab/src/generateWF.m → matlab/matlab/generateWF.m
View File

177
matlab/matlab/h_inf_synthesis_complementary_filters.m
View File

@@ -1,177 +0,0 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
freqs = logspace(-1, 3, 1000);
addpath('./src');
% Design of Weighting Function
% A formula is proposed to help the design of the weighting functions:
% \begin{equation}
% W(s) = \left( \frac{
% \frac{1}{\omega_0} \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_0} \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}
% The parameters permits to specify:
% - the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$
% - the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$
% - the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
% - the absolute slope between high and low frequency: $n$
% The general shape of a weighting function generated using the formula is shown in figure [[fig:weight_formula]].
n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2;
W = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(W, freqs, 'Hz'))), 'k-');
plot([1e-3 1e0], [G0 G0], 'k--', 'LineWidth', 1)
text(1e0, G0, '$\quad G_0$')
plot([1e1 1e3], [G1 G1], 'k--', 'LineWidth', 1)
text(1e1,G1,'$G_{\infty}\quad$','HorizontalAlignment', 'right')
plot([w0/2/pi w0/2/pi], [1 2*Gc], 'k--', 'LineWidth', 1)
text(w0/2/pi,1,'$\omega_c$','VerticalAlignment', 'top', 'HorizontalAlignment', 'center')
plot([w0/2/pi/2 2*w0/2/pi], [Gc Gc], 'k--', 'LineWidth', 1)
text(w0/2/pi/2, Gc, '$G_c \quad$','HorizontalAlignment', 'right')
text(w0/5/pi/2, abs(evalfr(W, j*w0/5)), 'Slope: $n \quad$', 'HorizontalAlignment', 'right')
text(w0/2/pi, abs(evalfr(W, j*w0)), '$\bullet$', 'HorizontalAlignment', 'center')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([5e-4, 20]);
% #+name: fig:weight_formula
% #+caption: Gain of the Weighting Function formula
% #+RESULTS:
% [[file:figs/weight_formula.png]]
n = 2; w0 = 2*pi*10; G0 = 1/10; G1 = 1000; Gc = 0.45;
W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 0.45;
W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
figure;
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, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([1e-4, 20]);
xticks([0.1, 1, 10, 100, 1000]);
leg = legend('location', 'southeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
% H-Infinity Synthesis
% We define the generalized plant $P$ on matlab.
P = [W1 -W1;
0 W2;
1 0];
% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
% #+RESULTS:
% #+begin_example
% [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
% Test bounds: 0.3223 <= gamma <= 1000
% gamma X>=0 Y>=0 rho(XY)<1 p/f
% 1.795e+01 1.4e-07 0.0e+00 1.481e-16 p
% 2.406e+00 1.4e-07 0.0e+00 3.604e-15 p
% 8.806e-01 -3.1e+02 # -1.4e-16 7.370e-19 f
% 1.456e+00 1.4e-07 0.0e+00 1.499e-18 p
% 1.132e+00 1.4e-07 0.0e+00 8.587e-15 p
% 9.985e-01 1.4e-07 0.0e+00 2.331e-13 p
% 9.377e-01 -7.7e+02 # -6.6e-17 3.744e-14 f
% 9.676e-01 -2.0e+03 # -5.7e-17 1.046e-13 f
% 9.829e-01 -6.6e+03 # -1.1e-16 2.949e-13 f
% 9.907e-01 1.4e-07 0.0e+00 2.374e-19 p
% 9.868e-01 -1.6e+04 # -6.4e-17 5.331e-14 f
% 9.887e-01 -5.1e+04 # -1.5e-17 2.703e-19 f
% 9.897e-01 1.4e-07 0.0e+00 1.583e-11 p
% Limiting gains...
% 9.897e-01 1.5e-07 0.0e+00 1.183e-12 p
% 9.897e-01 6.9e-07 0.0e+00 1.365e-12 p
% Best performance (actual): 0.9897
% #+end_example
% We then define the high pass filter $H_1 = 1 - H_2$. The bode plot of both $H_1$ and $H_2$ is shown on figure [[fig:hinf_filters_results]].
H1 = 1 - H2;
% Obtained Complementary Filters
% The obtained complementary filters are shown on figure [[fig:hinf_filters_results]].
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', '$w_1$');
set(gca,'ColorOrderIndex',2)
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
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', '$H_2$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude');
set(gca, 'XTickLabel',[]);
ylim([1e-4, 20]);
yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
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;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
yticks([-180:90:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);

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Sensor fusion , Optimal filters , $\mathcal{H}_\infty$ synthesis ,
Vibration isolation , Precision
* Introduction
The sensors used for measuring physical quantity often works well within
a limited frequency range called as the bandwidth of the sensor. The
signals recorded by the sensor beyond its bandwidth are often corrupt
with noise and are not reliable. Many dynamical systems require
measurements over a wide frequency range. Very often a variety of
sensors are utilized to sense the same quantity. These sensors have
different operational bandwidth and are reliable only in a particular
frequency range. The signals from the different sensors are fused
together in order to get the reliable measurement of the physical
quantity over wider frequency band. The combining of signals from
various sensor is called sensor
fusion \cite{hua04_polyp_fir_compl_filter_contr_system}. The resulting
sensor is referred as "super sensor" since it can have better noise
characteristics and can operate over a wider frequency band as compared
to the individual sensor used for
merging \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}.
Sensor fusion is most commonly employed in the navigation systems to
accurately measure the position of a vehicle. The GPS sensors, which are
accurate in low frequency band, are merged with the high-frequency
accelerometers. Zimmermann and
Sulzer \cite{zimmermann92_high_bandw_orien_measur_contr} used sensor
fusion to measure the orientation of a robot. They merged inclinometer
and accelerometers for accurate angular measurements over large
frequency band.
Corke \cite{corke04_inert_visual_sensin_system_small_auton_helic} merged
inertial measurement unit with the stereo vision system for measurement
of attitude, height and velocity of an unmanned helicopter. Min and
Jeung \cite{min15_compl_filter_desig_angle_estim} used accelerometer and
gyroscope for angle estimations. Baerveldt and Klang \cite{baerveldt97}
used an inclinometer and a gyroscope to measure the orientation of the
autonomous helicopter. The measurement of the 3D orientation using a
gyroscope and an accelerometer was demonstrated by Roberts et
al. \cite{roberts03_low}. Cao et
al. \cite{cao20_adapt_compl_filter_based_post} used sensor fusion to
obtain the lateral and longitudinal velocities of the autonomous
vehicle.
Sensor fusion is also used for enhancing the working range of the active
isolation system. For example, the active vibration isolation system at
the Laser Interferometer Gravitational-Wave Observatory
(LIGO) \cite{matichard15_seism_isolat_advan_ligo} utilizes sensor
fusion. The position sensors, seismometer and geophones are used for
measuring the motion of the LIGO platform in different frequency
bands \cite{hua05_low_ligo}. Tjepkema et
al. \cite{tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip}
used sensor fusion to isolate precision equipment from the ground
motion. The feedback from the accelerometer was used for active
isolation at low frequency while force sensor was used at high
frequency. Various configurations of sensor fusion for active vibration
isolation systems are discussed by Collette and
Matichard \cite{collette15_sensor_fusion_method_high_perfor}. Ma and
Ghasemi-Nejhad \cite{ma04_frequen_weigh_adapt_contr_simul} used laser
sensor and piezoelectric patches for simultaneous tracking and vibration
control in smart structures. Recently, Verma et
al. \cite{verma21_virtual_sensor_fusion_high_precis_contr} presented
virtual sensor fusion for high precision control where the signals from
a physical sensor are fused with a sensor simulated virtually.
Fusing signals from different sensors can typically be done using Kalman
filtering \cite{odry18_kalman_filter_mobil_robot_attit_estim, ren19_integ_gnss_hub_motion_estim, faria19_sensor_fusion_rotat_motion_recon, liu18_innov_infor_fusion_method_with, abdel15_const_low_cost_gps_filter, biondi17_attit_recov_from_featur_track}
or complementary
filters \cite{brown72_integ_navig_system_kalman_filter}. A set of
filters is said to be complementary if the sum of their transfer
functions is equal to one at all frequencies. When two filters are
complementary, usually one is a low pass filter while the other is an
high pass filter. The complementary filters are designed in such a way
that their magnitude is close to one in the bandwidth of the sensor they
are combined with. This enables to measure the physical quantity over
larger bandwidth. There are two different categories of complementary
filters --- frequency domain complementary filters and state space
complementary filters. Earliest application of the the frequency domain
complementary filters was seen in Anderson and
Fritze \cite{anderson53_instr_approac_system_steer_comput}. A simple RC
circuit was used to physically realize the complementary filters.
Frequency domain complementary filters were also used
in \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel, zimmermann92_high_bandw_orien_measur_contr, baerveldt97, roberts03_low}.
State space complementary filter finds application in tracking
orientation of the flexible links in a
robot \cite{bachmann03_desig_marg_dof, salcudean91_global_conver_angul_veloc_obser, mahony08_nonlin_compl_filter_special_orthog_group}
and are particularly useful for multi-input multi-output systems.
Pascoal et al. \cite{pascoal00_navig_system_desig_using_time} presented
complementary filters which can adapt with time for navigation system
capable of estimating position and velocity using GPS and SONAR sensors.
The noise characteristics of the super sensor are governed by the norms
of the complementary filters. Therefore, the proper design of the
complementary filters for sensor fusion is of immense importance. The
design of complementary filters is a complex task as they need to tuned
as per the specification of the sensor. In many applications, analytical
formulas of first and second order complementary filters are
used \cite{corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas}.
These filters are easy to tune and simple to implement using an analog
circuit \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop,cite:yong16_high_speed_vertic_posit_stage}.
However, these low order complementary filters are not optimal, and high
order complementary filters can lead to better
fusion \cite{jensen13_basic_uas,shaw90_bandw_enhan_posit_measur_using_measur_accel}.
Several design techniques have been proposed to design higher order
complementary filters.
Pascoal \cite{pascoal00_navig_system_desig_using_time} used linear
matrix inequalities (LMIs) \cite{boyd94_linear} for the design of time
varying complementary filters. LMIs were also used by Hua et
al. \cite{hua04_polyp_fir_compl_filter_contr_system} to design finite
impulse response (FIR) filters for the active vibration isolation system
at LIGO.
Plummer \cite{plummer06_optim_compl_filter_their_applic_motion_measur}
proposed an optimal design method using the $\mathcal{H}_{\infty}$
synthesis and weighting functions representing the measurement noise of
the sensors.
Although various methods have been presented in the literature for the
design of complementary filters, there is a lack of general and simple
framework that allows to shape the norm of complementary filters. Such a
method would prove to be very useful as the noise of the "supper sensor"
and its dynamical characteristics depend on the norm of the filters.
This paper presents such a framework based on the $\mathcal{H}_\infty$
norm minimization. The proposed method is quite general and can be
easily extended to a case where more than two complementary filters
needs to be designed. The organization of this paper is as follows.
Section [[*Complementary filters requirements][2]] presents the design
requirements of ideal complementary filters. It also demonstrates how
the noise and robustness characteristics of the "super sensor" can be
transformed into upper bounds on the norm of the complementary filters.
The framework for the design of complementary filters is detailed in
Section [[*Design formulation using $\mathcal{H}_\infty$ synthesis][3]].
This is followed by the application of the design method to
complementary filter design for the active vibration isolation at LIGO
in Section [[*Application: Complementary Filter Design for Active
Vibration Isolation of LIGO][4]]. Finally, concluding remarks are
presented in Section [[*Concluding remarks][5]].
* Complementary filters requirements
** Introduction :ignore:
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. These
requirements are discussed in this section.
** Complementary characteristics
Consider a case where two different sensors are used for measuring the
same quantity, $x$ in different frequency range. The inherent dynamics
of the sensors is represented by transfer functions $G_1(s)$ and
$G_2(s)$. The two sensor also have uncorrelated noise characteristics
given by $n_1$ and $n_2$. The signals from these two sensors are fused
using complementary filters $H_1(s)$ and $H_2(s)$. The architecture of
sensor fusion using complementary filters is shown in
Figure [[#fig:fusion_super_sensor][1]]. The resulting sensor, termed as
"super sensor", can have larger bandwidth and better noise
characteristics in comparison to the individual sensor. This means that
the super sensor provides an estimate $\hat{x}$ of $x$ which can be more
accurate over a larger frequency band than the outputs of the individual
sensors. Based on Figure [[#fig:fusion_super_sensor][1]], the estimate
of the physical quantity as measured by the super sensor can be written
as $$\label{eq:comp_filter_estimate}
\hat{x} = \left(G_1 H_1 + G_2 H_2\right) x + H_1 n_1 + H_2 n_2$$
#+caption: [fig:fusion_super_sensor]Block diagram of sensor fusion with
complementary filters
[[file:figs/fusion_super_sensor.pdf]]
The complementary property of filters $H_1(s)$ and $H_2(s)$ simply
implies that the summation of their transfer functions is equal to
unity. That is, unity magnitude and zero phase at all frequencies
(filters which satisfies only the magnitude condition are referred as
"magnitude complementary filter pair"). Therefore, a pair of strict
complementary filter needs to satisfy the following condition.
$$\label{eq:comp_filter}
H_1(s) + H_2(s) = 1$$
** Noise characterization
In order to compute the noise characteristics associated with the
estimate $\hat{x}$, it is first assumed that the dynamics of the
individual sensors are perfect: $$\label{eq:perfect_dynamics}
G_1(s) = G_2(s) = 1$$
The output of the super sensor, $\hat{x}$, based on the block diagram
shown in Figure [[#fig:fusion_super_sensor][1]] can be written as
$$\label{eq:estimate_perfect_dyn}
\hat{x} = x + H_1 n_1 + H_2 n_2$$
The complementary are operating only on the noise component of the
individual sensor. Thus, this sensor fusion architecture permits to
filter the noise of both sensors without introducing any distortion in
the physical quantity to be measured. The estimation error, $\delta x$,
of the super sensor can be written as $$\label{eq:estimate_error}
\delta x \triangleq \hat{x} - x = H_1 n_1 + H_2 n_2$$
The power spectral density (PSD) of the super sensor's estimation error
is given by $$\label{eq:noise_filtering_psd}
\Phi_{\delta x} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2}$$
where, $\Phi_{\delta x}$ is the PSD of estimation error, $\Phi_{n_1}$
and $\Phi_{n_2}$ are the PSDs of the noise associated with the
individual sensor.
It can be seen that the estimation error's PSD
depends on the PSD of the noise in individual sensor as well as the norm
of the complementary filters.
Therefore, by properly shaping the norm of
the complementary filters, it is possible to minimize the noise of the
super sensor noise.
** Robustness requirements
In the previous subsection, the inherent sensor dynamics were ignored.
However in the real system, the sensor dynamics is not equal to unity.
In such cases, the output of the sensor is normalized using a filter
whose transfer function is equal to the inverse of the sensor dynamics.
There are two major concerns in using inversion. First being the sensors
may not have been calibrated properly and the actual sensor dynamics is
not exactly compensated by the inverse filter. The second problem is
that the inversion of sensor dynamics can result in an improper transfer
function and hence may not be physically realizable. We here suppose
that the sensor dynamics can be inverted using a proper and stable
transfer function $\hat{G}_i(s)$. However, we suppose there exists a
normalization error since $\hat{G}_i^{-1}(s) G_i(s) \neq 1$. This
normalization error can be represented using frequency dependent
multiplicative uncertainty
(Figure [[#fig:sensor_fusion_dynamic_uncertainty][2]]). In
Figure [[#fig:sensor_fusion_dynamic_uncertainty][2]], $\Delta_i(s)$
satisfies $\|\Delta_i(s)\|_\infty \le 1$ and $|w_i(s)|$ is a frequency
dependent weighting function that represents the uncertainty
corresponding to the normalization error.
#+caption: [fig:sensor_fusion_dynamic_uncertainty] Representation of
normalization error in sensor fusion using multiplicative uncertainty
[[file:figs/sensor_fusion_dynamic_uncertainty.pdf]]
Based on Figure [[#fig:sensor_fusion_dynamic_uncertainty][2]], the super
sensor dynamics can be written as
$$\label{eq:super_sensor_dyn_uncertainty}
\frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s)$$
The dynamics of the super sensor now depends on the weighting functions
($w_1(s),w_2(s)$) and the complementary filters ($H_1(s),H_2(s)$).
The robust stability of the fusion can be studied graphically (refer
Figure [[#fig:uncertainty_set_super_sensor][3]]). The frequency response of the fusion output is plotted in a complex plane. The unity transfer function leads to a point $(1,0)$ located on the real axis. The uncertainty associated with first sensor at a particular frequency is represented by a circle with the center at (1,0) and radius $|w_1H_1|$. The uncertainty associated with the second is also represented using a circle centered at any point on the circle representing uncertainty associated with the first sensor and radius equal to $|w_2H_2|$. Therefore, the overall uncertainty of the fusion is represented with a circle centered at (1,0) and radius equal to $|w_1H_1|+|w_2H_2|$.
Mathematically, the maximum phase difference at frequency
$\omega$ that can result from fusion is given by
$$\label{eq:max_phase_uncertainty}
\Delta\phi(\omega) = \arcsin\left( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \right)$$
#+caption: [fig:uncertainty_set_super_sensor]Robustness analysis of
sensor fusion in the complex plane. The uncertainty associated with the
super sensor dynamics are represented with a solid circle while those
associated with individual sensors are represented with dashed circles.
[[file:figs/uncertainty_set_super_sensor.pdf]]
A constraint on the maximum phase difference at a given frequency can be
incorporated in the design problem using the following equation:
$$\label{eq:max_uncertainty_super_sensor}
\max_\omega \big( \left|w_1 H_1\right| + \left|w_2 H_2\right|\big) < \sin\left( \Delta \phi_\text{max} \right)$$
where $\Delta \phi_\text{max}$ is the maximum allowable phase
difference. It can also be inferred from the above equation that the
magnitude of the complementary filter ($|H_i|$) should be tuned to a
smaller value at the frequencies where the magnitude of weighing
transfer functions ($|w_i|$) representing sensor uncertainty is large.
* Design formulation using $\mathcal{H}_\infty$ synthesis
** Introduction :ignore:
In this section, the shaping of complementary filters is expressed as an optimal $\mathcal{H}_{\infty}$ synthesis problem.
The synthesis goal is to shape the frequency response of the filters such that they satisfy the design requirements presented in Section [[*Complementary filters
requirements][2]].
** Synthesis problem formulation
The first step is to formulate the filter design problem as a
generalized plant-controller structure \cite{boyd91_linear}. The
generalized plant and controller structure for complementary filters
design is shown in Figure [[#fig:h_infinity_robust_fusion][4]]. In the
figure, $P(s)$ is the generalized plant, $u$ is the "control input", $v$
is the "measured output" and $H_2(s)$ is the controller (filter) to be
designed. The regulated outputs of the generalized plant, $z_1$ and
$z_2$, are given by $$\begin{split}
z_1 &= W_1(s)(1-H_2(s)) w = W_1(s) H_1(s) w \text{ by defining } H_{1}(s) \triangleq 1 - H_{2}(s)\\
z_2 &= W_2(s) H_2(s) w
\end{split}$$ where $w$ is the "exogenous input" to the plant,
$W_1(s), W_2(s)$ are the weighting functions for shaping the
complementary filters.
The dynamics of the generalized plant can be written as
$$\label{eq:generalized_plant}
\begin{bmatrix} z_1 \\ z_2 \\ v \end{bmatrix} = P(s) \begin{bmatrix} w\\u \end{bmatrix}; \quad P(s) = \begin{bmatrix}W_1(s) & -W_1(s) \\ 0 & W_2(s) \\ 1 & 0 \end{bmatrix}$$
The weighting functions are chosen based on the specifications and
requirements set for the complementary filters (discussed in
Section [[*Design of weighting functions][3.2]]). The objective of the
optimization is to design a filter $H_2(s)$ such that the following
conditions are satisfied $$\label{eq:comp_filter_problem_form}
\begin{split}
\left| \frac{z_{1}}{w} \right| &= |1-H_2(s)| \le \frac{1}{|W_1(s)|} \\
\left| \frac{z_{1}}{w} \right| &= |H_2(s)| \le \frac{1}{|W_2(s)|}
\end{split}, \quad \forall \omega \in \mathbb{R}^{+}$$
#+caption: [fig:h_infinity_robust_fusion]Generalized plant controller
structure for the design of complementary filters
[[file:figs/h_infinity_robust_fusion.pdf]]
Based on Figure [[#fig:h_infinity_robust_fusion][4]], the
$\mathcal{H}_{\infty}$ synthesis problem for the complementary filters
can be stated as
#+begin_quote
Find a stable transfer function, $H_2(s)$, which takes measured
output, $v$, as input and generates a control input, $u$, such that
the $\mathcal{H}_\infty$ norm of the generalized plant from exogenous
input, $w$, to the regulated output, ${[z_1,z_2]}^T$ is less than
unity.
#+end_quote
Mathematically, the synthesis objective can be written as
$$\begin{split}
&\left\|\begin{matrix} \left[1 - H_2(s)\right] W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 \\
\Longleftrightarrow & \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1; \quad H_1(s) \triangleq 1 - H_2(s)
\end{split}
\label{eq:hinf_syn_obj}$$ The above optimization problem can be
efficiently solved in Matlab \cite{MATLAB2009} using Riccati formulae,
linear matrix inequality based method or maximum entropy method.
** Design of weighting functions
The choice of weighting function governs the shape of the designed
complementary filters. Therefore, it is very important that the design
specifications are appropriately transformed into the weighting
functions. The choice of weighting functions is also constrained by the
following factors
1. Only proper and stable transfer functions can be used as weighting
functions
2. As the order of the designed filter is equal to the sum of the orders
of the weighting functions, the order of the weighting function needs
to be reasonably small to ensure the physical implementation of the
designed complementary filters. This also reduces the computational
cost of the optimization problem.
3. The complementary property of the filter imposes a fundamental
limitations on the weighting functions. The imposes a restriction
that the magnitude of the filters $H_1(s)$ and $H_2(s)$ cannot be
made small simultaneously at the same frequency.
The specifications of the complementary filters are typically expressed
using the following parameters --- low frequency gain, high frequency
gain, slope (order of the filter) and the crossover frequency. We
propose a weighting function that allows to translate the above
requirements by setting simple parameters: $$\label{eq:weight_formula}
W(s) = \displaystyle\left( \frac{
\hfill{} \displaystyle\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\displaystyle\frac{G_0}{G_c}\right)^{\displaystyle\frac{2}{n}}}{1 - {\left(\displaystyle\frac{G_c}{G_\infty}\right)}^{\displaystyle\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\displaystyle\frac{1}{n}}
}{
\left(\displaystyle\frac{1}{G_\infty}\right)^{\displaystyle\frac{1}{n}}\displaystyle \frac{1}{\omega_0} \sqrt{\displaystyle\frac{1 - \left(\displaystyle\frac{G_0}{G_c}\right)^{\displaystyle\frac{2}{n}}}{1 - \left(\displaystyle\frac{G_c}{G_\infty}\right)^{\displaystyle\frac{2}{n}}}} s + \left(\displaystyle\frac{1}{G_c}\right)^{\displaystyle\frac{1}{n}}
}\right)^n$$ where, $G_0 = \lim_{\omega \to 0} |W(j\omega)|$ is the
low frequency gain, $G_\infty = \lim_{\omega \to \infty} |W(j\omega)|$
is the high frequency gain, $\omega_c$ is the crossover frequency,
$G_c = |W(j\omega_c)|$ is the absolute gain at the crossover frequency
and $n$ is the order of the filter. As an illustration, the magnitude of
the frequency response of the weighting function with the parameters
$G_0 = 0.001$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$,
$n = 3$ and having high pass characteristics is shown in
Figure [[#fig:weight_formula][5]].
#+caption: [fig:weight_formula]Magnitude of the frequency response of
the weighting function obtained using
equation [[#eq:weight_formula][[eq:weight_formula]]] with the parameters
$G_0 = 0.001$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$,
$n = 3$
[[file:figs/weight_formula.pdf]]
** Verification
The proposed methodology for the design of complementary filters is now
applied on the following example.
*Example.* Design complementary filters with the merging frequency
around 10 Hz. The low pass filter should have DC gain of 0.001 and slope
-2 above the merging frequency. The high pass filter should have a slope
of 3 below the merging frequency and 0.001 high frequency gain.
The first step is to design the weighting functions that translate the
above requirements. To do so,
equation [[#eq:weight_formula][[eq:weight_formula]]] is used. The
parameters corresponding to $W_1(s)$ and $W_2(s)$ are listed in
Table [[#tab:weights_params][1]]. The obtained transfer functions of the
weighting functions are $$\begin{split}
W_1(s) &= \dfrac{1000 (s+34.55)^2}{(s+3455)^2}\\
W_2(s) &= \dfrac{0.1 (s+87.43)^3}{(s+4.058)^3}
\end{split}$$
Using these weighting functions, the generalized plant is evaluated
using equation [[#eq:generalized_plant][[eq:generalized_plant]]]. The
optimal complementary filters are obtained by solving the optimization
problem given by equation [[#eq:hinf_syn_obj][[eq:hinf_syn_obj]]]. The
complementary filters obtained after optimization are $$\begin{split}
H_1(s) &= \frac{10^{-8} (s+6.6\times 10^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3\times 10^3) (s^2 + 72s + 3580)}\\
H_2(s) &= \frac{(s+6.6\times 10^4) (s+160) (s+4)^3}{(s+6.6\times 10^4) (s^2 + 106 s + 3\times 10^3) (s^2 + 72s + 3580)}
\end{split}$$
The obtained complementary filters are of order 5 which corresponds to
the sum of the orders of the weighting functions used. The frequency
responses of the designed complementary filters are shown in
Figure [[#fig:hinf_filters_results][6]]. It can be seen that the
designed filters fulfills all the design specifications and hence
demonstrates the effectiveness of the designed methodology (more complex
real life example is taken up in Section [[*Application: Complementary
Filter Design for Active Vibration Isolation of LIGO][4]]).
<<tab:weights_params>>
| Parameter | $W_1(s)$ | $W_2(s)$ |
|------------------------+----------+----------|
| $n$ | $2$ | $3$ |
| $G_c$ | $0.5$ | $0.5$ |
| $G_\infty$ | $1000$ | $0.1$ |
| $G_0$ | $0.1$ | $1000$ |
| $\omega_c$ [$\si{Hz}$] | $11$ | $10$ |
#+caption: [tab:weights_params]Parameters used for $W_1(s)$ and $W_2(s)$
#+caption: [fig:hinf_filters_results]Bode plot of the weighting
functions and designed complementary filters
[[file:figs/hinf_filters_results.pdf]]
** TODO Extension to set of three complementary filters
In certain applications, more than two sensors are used to measure the
same quantity and can be merged together to form a "super sensor". In
such case, a set of three (or more) complementary filters is required.
It is here shown that the proposed method can be generalized for the
design of a set of arbitrary number of complementary filters. The
control objective is now to design of a set of $n$ complementary filters
($H_i(s), i=1,\cdots,n$) which satisfy the following conditions
$$\label{eq:hinf_problem_gen}
\begin{split}
&\sum_{i=0}^n H_i(s) = 1 \\
&\left| H_i(s) \right| < \frac{1}{\left| W_i(s) \right|}
\end{split}$$ Here, we extend the method to a case of three
complementary filters. The generalized plant controller setup for this
case is shown in Figure [[#fig:comp_filter_three_hinf][7]]. The
synthesis objective is to design filters $H_2(s)$ and $H_3(s)$ such that
the $\mathcal{H}_\infty$ norm from exogenous input $w$ to regulated
output vector $[z_1,z_2,z_3]^T$ is less than unity. That is,
$$\label{eq:hinf_syn_obj_three}
\begin{split}
&\left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1\\
\equiv &\left\| \begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1; \quad H_1(s) \triangleq 1 - H_2(s) - H_3(s)
\end{split}$$
#+caption: [fig:comp_filter_three_hinf]Generalized plant controller
setup for designing a set of three complementary filters using
$\mathcal{H}_\infty$ synthesis
[[file:figs/comp_filter_three_hinf.pdf]]
To validate this synthesis method, let's take an example where 3 sensors
are merged together. The three sensors are working in the following
three frequency range --- below 1 Hz, between 1Hz to 10Hz and above
10Hz. The weighting functions used for the synthesis are $$\begin{split}
W_1(s) &= \dfrac{1000 (s+3.141)^2}{(s+314.1)^2}\\
W_2(s) &= \dfrac{2200 (s+62.83)^2 (s+6.283)^2}{(s+6283)^2 (s+0.06283)^2}\\
W_3(s) &= \dfrac{0.1 (s+87.43)^3}{(s+4.058)^3}
\end{split}$$ The complementary filters are obtained by solving the
optimization problem given by
equation [[#eq:hinf_syn_obj_three][[eq:hinf_syn_obj_three]]]. The
frequency response of the designed filters and the weighting functions
are shown in Figure [[#fig:three_complementary_filters_results][8]].
#+caption: [fig:three_complementary_filters_results]Bode plot of the
weighting functions and designed set of three complementary filters
[[file:figs/three_complementary_filters_results.pdf]]
* Application: Complementary Filter Design for Active Vibration
** Introduction :ignore:
Isolation of LIGO Gravitational waves can help in detection various astrophysical events occurring in our universe. This can also pave a path to validate theories built around the existence of gravitational waves. However, the detection of these waves is an arduous task owing to the extraordinary small strain experienced by the earth due to gravitational waves. Various methods have been proposed for their detection, out of which laser interferometers are the most popular ones. Laser interferometers offers large projection range and high displacement sensitivity. Among the existing detector, Laser interferometer gravitation-wave observatory (LIGO) is the most sensitive operational detector. LIGO consists of two longs arms, referred as beam tubes, that are placed orthogonal to each other. The arms of the LIGO accommodates a Michleson interferometer with a cavity (Fabry-Perot). The mirrors at the extremity of the cavity serve
as inertial test masses which responds to the strain induced due to the
gravitational waves. The optics of the LIGO are suspended like a
pendulum. The schematics of the LIGO are shown in
Figure [[#ligo_schematics][9]].
#+caption: [ligo_schematics]Schematics of LIGO
[[file:figs/ligo.png]]
The isolation of the terrestrial interferometers is necessary in order
to isolate the motion of the suspended inertial masses from the seismic
ground motion. The vibration isolation system attenuates the effect of
disturbances on the motion of the suspended masses in the gravitational
wave band. The other sources that can affect the sensitivity of the LIGO
are thermal noise, human activities, tidal motion etc. The suspensions
of the LIGO needs to serve two purpose --- isolation and alignment. The
alignment is also important in order to ensure that the interferometer
beam is targeted at the center of the suspended mass. The current
vibration isolation system for LIGO has seven different stages. In the
first stage, hydraulically actuated external pre-isolators are used for
attenuating large ground motions. Two stages of active electromagnetic
isolation system are placed next to pre-isolators. This is followed by
four stages of pendulum based passive isolation system.
In the active isolation stage of LIGO, different sensors are used to
sense the same physical signal in different frequency range. For
example, seismometers are used to sense the position of the platform in
the frequency band 0.5--10 Hz while geophones are employed above 10 Hz.
The signals recorded from different sensor are fused using complementary
filters \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}.
The stringent requirements on these filters complicate their design.
Hua \cite{hua05_low_ligo} proposed complementary FIR filters which were
synthesized using convex optimization. The designed FIR filters were
found to be compliant with the design specifications. However, the order
of the designed filter was very high, which limits its application to a
practical system. In this section, we demonstrate the design of
complementary filters with the same specification using the proposed
method based on $\mathcal{H}_\infty$- synthesis.
** Design specifications
The design specification of the complementary filters (as listed out
in \cite{hua05_low_ligo}) are as follows:
1. In the frequency range $0$-$\SI{0.008}{Hz}$: the high pass filter's
magnitude should be less than $8 \times 10^{-4}$.
2. For frequency range $\SI{0.008}{Hz}$-$\SI{0.04}{Hz}$: slope of the
high pass filter is equal to three.
3. Between $\SI{0.04}{Hz}$-$\SI{0.1}{Hz}$ frequency range: the high pass
filter's magnitude should be less than $3$.
4. For frequencies above $\SI{0.1}{Hz}$: the low pass filter's magnitude
should be less than $0.045$.
The specification of the complementary filters are shown graphically by
dashed black lines in Figure [[#fig:ligo_weights][10]].
** Weighting Functions Design
As the synthesis objective of the complementary filters is described by
Eq. [[#eq:hinf_problem_gen][[eq:hinf_problem_gen]]], it is clear that
the weighting functions should be chosen such that their inverse
magnitude represent the maximum allowed norm of the complementary
filters. This can be done manually using by combining poles and zeros or
using useful formulas such as
Eq. [[#eq:weight_formula][[eq:weight_formula]]]. It is important to note
that the order of the filters should be kept reasonable small in order
to keep the computational cost of the optimization reasonable. This will
also ensure that the designed filters are realizable in the physical
world. The transfer function representing weights should also be stable
and minimum phase.
The weighting function corresponding to the low pass filter, $w_L(s)$,
is here taken as Type I Chebyshev filter. The order of the weighting
function for low pass filter is set as 20.
The weighting function for the high pass filter, $w_H(s)$, is designed
in such a way that its magnitude response is as close as possible to the
design specifications. This was achieved using a combination of high-,
low- and band-pass filters in the particular frequency band. The overall
order of the weighting function for high pass filter is 7.
The magnitude responses of the inverse of the designed weighting
functions and their comparison with the specifications are shown in
Figure [[#fig:ligo_weights][10]]. It can be seen that the inverse of the
designed weights, shown in solid blue line for high pass filter and
solid red line for low pass filter, are close to the specifications
shown in black dotted line.
#+caption: [fig:ligo_weights]Specifications and weighting functions
magnitudes
[[file:figs/ligo_weights.pdf]]
** $\mathcal{H}_\infty$ synthesis of complementary filters
The complementary filters are designed using $\mathcal{H}_\infty$
synthesis based on the architecture shown in
Figure [[#fig:h_infinity_robust_fusion][4]]. The generalized plant is
obtained by substituting the transfer functions of the designed
weighting functions in
equation [[#eq:generalized_plant][[eq:generalized_plant]]], i.e.,
$W_1(s)=w_H(s)$ and $W_2(s)=w_L(s)$. The objective of the
$\mathcal{H}_\infty$ synthesis is given by
equation [[#eq:hinf_syn_obj][[eq:hinf_syn_obj]]]. The optimization
problem in solved in Matlab \cite{MATLAB2009} using Ricatti method. The
frequency responses of the designed optimal complementary filters are
shown in Figure [[#fig:comp_fir_ligo_hinf][11]]. The order of the
filters obtained using $\mathcal{H}_\infty$ optimization is $27$.
Now, we compare the designed complementary filters with the FIR filters
designed by Hua \cite{hua05_low_ligo}. The transfer function of the FIR
filter, $G(\omega)$, is the Fourier transform of its coefficients,
$g(n)$. That is, $$G(\omega) = \sum_n g(n)e^{-j2n\pi\omega}$$ The FIR
filter synthesis was formulated as a convex optimization problem. The
objective of the optimization problem was to find the filter's
coefficients such that their norm are below the specified upper bounds.
The optimization problem was solved using
SeDuMi \cite{sturm99_using_sedum} and the obtained order for the FIR
filters is $512$. The bode plot of the FIR filters are shown with dotted
lines in Figure [[#fig:comp_fir_ligo_hinf][11]]. It can be seen that
frequency responses of the designed complementary filters matches quite
well with those of the FIR filters. The designed complementary filters
are of much lower order and can be implemented with less computational
cost can the FIR filters. The proposed methodology for the design of
complementary filters can be effectively employed to obtain physically
realizable filters.
#+caption: [fig:comp_fir_ligo_hinf]Bode plot of the filters designed
using $\mathcal{H}_\infty$ synthesis and FIR
filters \cite{hua05_low_ligo}
[[file:figs/comp_fir_ligo_hinf.pdf]]
* Concluding remarks
The measurements from the sensors are reliable only within its
bandwidth. The signals from different sensors are usually fused in order
to measure a physical quantity over larger bandwidth. The sensor
obtained after fusion is called as super sensor as it has superior noise
characteristics and wider bandwidth. Complementary filters are used for
the combining the signals from different sensors. A new framework based
on $\mathcal{H}_\infty$ synthesis has been presented in this paper to
aid the design of complementary filters. The method presented allows to
shape the complementary filters based on the design specifications. The
task of filter design is posed as an $\mathcal{H}_{\infty}$ synthesis
problem. The design specifications of the systems are transformed in the
form of weighting functions. These weighting functions are used in the
optimization problem to constraint the filter response in a frequency
band. The method has also been demonstrated for designing a set of three
complementary filters. The design frame is general, simple to implement
and can easily be extended to difference scenarios of sensor fusion. The
effectiveness of the method is demonstrated for a real life application
where complementary filters are designed for active vibration isolation
of Laser Interferometer Gravitational-Wave Observatory (LIGO). The
filters designed with the proposed method have been with compared with
the finite impulse response (FIR) filters. It was found that the filters
designed using $\mathcal{H}_\infty$ have lower order compared to FIR
filters. The designed filters are physically realizable and have lesser
computational cost compared to FIR filters. The proposed method can be
effectively used to shape complementary filters based on design
specifications. The method can be further be extended for the design of
robust complementary filters with desired noise characteristics
considering uncertainties in the sensor dynamics. This is the focus of
our future research.
* Acknowledgment
:PROPERTIES:
:CLASS: unnumbered
:END:
The authors would like to acknowledge the help received from the French
Community of Belgium for funding the FRIA Grant of Thomas Dehaeze (Grant
No. FC 31597).

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\documentclass[sort&compress, review,12pt,times]{elsarticle}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{grffile}
\usepackage{rotating}
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\usepackage{capt-of}
\usepackage{hyperref}
\usepackage{bm}
\usepackage{array}
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\usepackage{algorithmic}
\usepackage{textcomp}
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\usepackage{cases}
\usepackage{tabularx,siunitx,booktabs}
\usepackage{algorithmic}
\newcounter{example}[section]
\newenvironment{example}[1][]{\refstepcounter{example}\par\medskip
\noindent \textbf{Example. #1} \rmfamily}{\medskip}
\usepackage[top=3cm, bottom=3cm, left=3cm, right=3cm]{geometry}
\journal{Mechanical Systems and Signal Processing}
\author[a1,a2]{Thomas Dehaeze\corref{cor1}}
\author[a3,a4]{Mohit Verma}
\author[a2,a4]{Christophe Collette}
\cortext[cor1]{Corresponding author. Email Address: dehaeze.thomas@gmail.com}
\address[a1]{European Synchrotron Radiation Facility, Grenoble, France}
\address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
\address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
\address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
\date{}
\title{A new method of designing complementary filters for sensor fusion using \(\mathcal{H}_\infty\) synthesis}
\begin{document}
\begin{frontmatter}
\begin{abstract}
Sensor have limited bandwidth and are accurate only in a certain frequency band.
In many applications, the signals of different sensor are fused together in order to either enhance the stability or improve the operational bandwidth of the system.
The sensor signals can be fused using complementary filters.
The tuning of complementary filters is a complex task and is the subject of this paper.
The filters needs to meet design specifications while satisfying the complementary property.
This paper presents a framework to shape the norm of complementary filters using the $\mathcal{H}_\infty$ norm minimization.
The design specifications are imposed as constraints in the optimization problem by appropriate selection of weighting functions.
The proposed method is quite general and easily extendable to cases where more than two sensors are fused.
Finally, the proposed method is applied to the design of complementary filter design for active vibration isolation of the Laser Interferometer Gravitation-wave Observatory (LIGO).
\end{abstract}
\begin{keyword}
Sensor fusion \sep{} Optimal filters \sep{} $\mathcal{H}_\infty$ synthesis \sep{} Vibration isolation \sep{} Precision
\end{keyword}
\end{frontmatter}
\section{Introduction}\label{sec:introduction}
The sensors used for measuring physical quantity often works well within a limited frequency range called as the bandwidth of the sensor.
The signals recorded by the sensor beyond its bandwidth are often corrupt with noise and are not reliable.
Many dynamical systems require measurements over a wide frequency range.
Very often a variety of sensors are utilized to sense the same quantity.
These sensors have different operational bandwidth and are reliable only in a particular frequency range.
The signals from the different sensors are fused together in order to get the reliable measurement of the physical quantity over wider frequency band.
The combining of signals from various sensor is called sensor fusion~\cite{hua04_polyp_fir_compl_filter_contr_system}.
The resulting sensor is referred as ``super sensor'' since it can have better noise characteristics and can operate over a wider frequency band as compared to the individual sensor used for merging~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}.
Sensor fusion is most commonly employed in the navigation systems to accurately measure the position of a vehicle.
The GPS sensors, which are accurate in low frequency band, are merged with the high-frequency accelerometers.
Zimmermann and Sulzer~\cite{zimmermann92_high_bandw_orien_measur_contr} used sensor fusion to measure the orientation of a robot.
They merged inclinometer and accelerometers for accurate angular measurements over large frequency band.
Corke~\cite{corke04_inert_visual_sensin_system_small_auton_helic} merged inertial measurement unit with the stereo vision system for measurement of attitude, height and velocity of an unmanned helicopter.
Min and Jeung~\cite{min15_compl_filter_desig_angle_estim} used accelerometer and gyroscope for angle estimations.
Baerveldt and Klang~\cite{baerveldt97} used an inclinometer and a gyroscope to measure the orientation of the autonomous helicopter.
The measurement of the 3D orientation using a gyroscope and an accelerometer was demonstrated by Roberts et al.~\cite{roberts03_low}.
Cao et al.~\cite{cao20_adapt_compl_filter_based_post} used sensor fusion to obtain the lateral and longitudinal velocities of the autonomous vehicle.
Sensor fusion is also used for enhancing the working range of the active isolation system.
For example, the active vibration isolation system at the Laser Interferometer Gravitational-Wave Observatory (LIGO)~\cite{matichard15_seism_isolat_advan_ligo} utilizes sensor fusion.
The position sensors, seismometer and geophones are used for measuring the motion of the LIGO platform in different frequency bands~\cite{hua05_low_ligo}.
Tjepkema et al.~\cite{tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip} used sensor fusion to isolate precision equipment from the ground motion.
The feedback from the accelerometer was used for active isolation at low frequency while force sensor was used at high frequency.
Various configurations of sensor fusion for active vibration isolation systems are discussed by Collette and Matichard~\cite{collette15_sensor_fusion_method_high_perfor}.
Ma and Ghasemi-Nejhad~\cite{ma04_frequen_weigh_adapt_contr_simul} used laser sensor and piezoelectric patches for simultaneous tracking and vibration control in smart structures.
Recently, Verma et al.~\cite{verma21_virtual_sensor_fusion_high_precis_contr} presented virtual sensor fusion for high precision control where the signals from a physical sensor are fused with a sensor simulated virtually.
Fusing signals from different sensors can typically be done using Kalman filtering~\cite{odry18_kalman_filter_mobil_robot_attit_estim, ren19_integ_gnss_hub_motion_estim, faria19_sensor_fusion_rotat_motion_recon, liu18_innov_infor_fusion_method_with, abdel15_const_low_cost_gps_filter, biondi17_attit_recov_from_featur_track} or complementary filters~\cite{brown72_integ_navig_system_kalman_filter}.
A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
When two filters are complementary, usually one is a low pass filter while the other is an high pass filter.
The complementary filters are designed in such a way that their magnitude is close to one in the bandwidth of the sensor they are combined with.
This enables to measure the physical quantity over larger bandwidth.
There are two different categories of complementary filters --- frequency domain complementary filters and state space complementary filters.
Earliest application of the the frequency domain complementary filters was seen in Anderson and Fritze~\cite{anderson53_instr_approac_system_steer_comput}.
A simple RC circuit was used to physically realize the complementary filters.
Frequency domain complementary filters were also used in~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel, zimmermann92_high_bandw_orien_measur_contr, baerveldt97, roberts03_low}.
State space complementary filter finds application in tracking orientation of the flexible links in a robot~\cite{bachmann03_desig_marg_dof, salcudean91_global_conver_angul_veloc_obser, mahony08_nonlin_compl_filter_special_orthog_group} and are particularly useful for multi-input multi-output systems.
Pascoal et al.~\cite{pascoal00_navig_system_desig_using_time} presented complementary filters which can adapt with time for navigation system capable of estimating position and velocity using GPS and SONAR sensors.
The noise characteristics of the super sensor are governed by the norms of the complementary filters.
Therefore, the proper design of the complementary filters for sensor fusion is of immense importance.
The design of complementary filters is a complex task as they need to tuned as per the specification of the sensor.
In many applications, analytical formulas of first and second order complementary filters are used~\cite{corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas}.
These filters are easy to tune and simple to implement using an analog circuit~\cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop,cite:yong16_high_speed_vertic_posit_stage}.
However, these low order complementary filters are not optimal, and high order complementary filters can lead to better fusion~\cite{jensen13_basic_uas,shaw90_bandw_enhan_posit_measur_using_measur_accel}.
Several design techniques have been proposed to design higher order complementary filters.
Pascoal~\cite{pascoal00_navig_system_desig_using_time} used linear matrix inequalities (LMIs)~\cite{boyd94_linear} for the design of time varying complementary filters.
LMIs were also used by Hua et al.~\cite{hua04_polyp_fir_compl_filter_contr_system} to design finite impulse response (FIR) filters for the active vibration isolation system at LIGO.
Plummer~\cite{plummer06_optim_compl_filter_their_applic_motion_measur} proposed an optimal design method using the $\mathcal{H}_{\infty}$ synthesis and weighting functions representing the measurement noise of the sensors.
Although various methods have been presented in the literature for the design of complementary filters, there is a lack of general and simple framework that allows to shape the norm of complementary filters.
Such a method would prove to be very useful as the noise of the ``supper sensor'' and its dynamical characteristics depend on the norm of the filters.
This paper presents such a framework based on the $\mathcal{H}_\infty$ norm minimization.
The proposed method is quite general and can be easily extended to a case where more than two complementary filters needs to be designed.
The organization of this paper is as follows.
Section~\ref{sec:requirements} presents the design requirements of ideal complementary filters.
It also demonstrates how the noise and robustness characteristics of the ``super sensor'' can be transformed into upper bounds on the norm of the complementary filters.
The framework for the design of complementary filters is detailed in Section~\ref{sec:hinf_method}.
This is followed by the application of the design method to complementary filter design for the active vibration isolation at LIGO in Section~\ref{sec:application_ligo}.
Finally, concluding remarks are presented in Section~\ref{sec:conclusion}.
\section{Complementary filters requirements}\label{sec:requirements}
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.
These requirements are discussed in this section.
\subsection{Complementary characteristics}\label{sec:sensor_fusion}
Consider a case where two different sensors are used for measuring the same quantity, $x$ in different frequency range.
The inherent dynamics of the sensors is represented by transfer functions \(G_1(s)\) and \(G_2(s)\).
The two sensor also have uncorrelated noise characteristics given by \(n_1\) and \(n_2\).
The signals from these two sensors are fused using complementary filters \(H_1(s)\) and \(H_2(s)\).
The architecture of sensor fusion using complementary filters is shown in Figure~\ref{fig:fusion_super_sensor}.
The resulting sensor, termed as ``super sensor'', can have larger bandwidth and better noise characteristics in comparison to the individual sensor.
This means that the super sensor provides an estimate \(\hat{x}\) of \(x\) which can be more accurate over a larger frequency band than the outputs of the individual sensors.
Based on Figure~\ref{fig:fusion_super_sensor}, the estimate of the physical quantity as measured by the super sensor can be written as
\begin{equation}
\label{eq:comp_filter_estimate}
\hat{x} = \left(G_1 H_1 + G_2 H_2\right) x + H_1 n_1 + H_2 n_2
\end{equation}
\begin{figure}
\centering
\includegraphics[scale=1]{figs/fusion_super_sensor.pdf}
\caption{\label{fig:fusion_super_sensor}Block diagram of sensor fusion with complementary filters}
\end{figure}
The complementary property of filters \(H_1(s)\) and \(H_2(s)\) simply implies that the summation of their transfer functions is equal to unity.
That is, unity magnitude and zero phase at all frequencies (filters which satisfies only the magnitude condition are referred as ``magnitude complementary filter pair'').
Therefore, a pair of strict complementary filter needs to satisfy the following condition.
\begin{equation}
\label{eq:comp_filter}
H_1(s) + H_2(s) = 1
\end{equation}
\subsection{Noise characterization}\label{sec:noise_filtering}
In order to compute the noise characteristics associated with the estimate $\hat{x}$, it is first assumed that the dynamics of the individual sensors are perfect:
\begin{equation}
\label{eq:perfect_dynamics}
G_1(s) = G_2(s) = 1
\end{equation}
The output of the super sensor, \(\hat{x}\), based on the block diagram shown in Figure~\ref{fig:fusion_super_sensor} can be written as
\begin{equation}
\label{eq:estimate_perfect_dyn}
\hat{x} = x + H_1 n_1 + H_2 n_2
\end{equation}
The complementary are operating only on the noise component of the individual sensor.
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured.
The estimation error, \(\delta x\), of the super sensor can be written as
\begin{equation}
\label{eq:estimate_error}
\delta x \triangleq \hat{x} - x = H_1 n_1 + H_2 n_2
\end{equation}
The power spectral density (PSD) of the super sensor's estimation error is given by
\begin{equation}
\label{eq:noise_filtering_psd}
\Phi_{\delta x} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2}
\end{equation}
where, $\Phi_{\delta x}$ is the PSD of estimation error, $\Phi_{n_1}$ and $\Phi_{n_2}$ are the PSDs of the noise associated with the individual sensor.
It can be seen that the estimation error's PSD depends on the PSD of the noise in individual sensor as well as the norm of the complementary filters.
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
\subsection{Robustness requirements}\label{sec:fusion_robustness}
In the previous subsection, the inherent sensor dynamics were ignored.
However in the real system, the sensor dynamics is not equal to unity.
In such cases, the output of the sensor is normalized using a filter whose transfer function is equal to the inverse of the sensor dynamics.
There are two major concerns in using inversion.
First being the sensors may not have been calibrated properly and the actual sensor dynamics is not exactly compensated by the inverse filter.
The second problem is that the inversion of sensor dynamics can result in an improper transfer function and hence may not be physically realizable.
We here suppose that the sensor dynamics can be inverted using a proper and stable transfer function \(\hat{G}_i(s)\).
However, we suppose there exists a normalization error since \(\hat{G}_i^{-1}(s) G_i(s) \neq 1\).
This normalization error can be represented using frequency dependent multiplicative uncertainty (Figure~\ref{fig:sensor_fusion_dynamic_uncertainty}).
In Figure~\ref{fig:sensor_fusion_dynamic_uncertainty}, \(\Delta_i(s)\) satisfies \(\|\Delta_i(s)\|_\infty \le 1\) and \(|w_i(s)|\) is a frequency dependent weighting function that represents the uncertainty corresponding to the normalization error.
\begin{figure}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_dynamic_uncertainty.pdf}
\caption{\label{fig:sensor_fusion_dynamic_uncertainty} Representation of normalization error in sensor fusion using multiplicative uncertainty}
\end{figure}
Based on Figure~\ref{fig:sensor_fusion_dynamic_uncertainty}, the super sensor dynamics can be written as
\begin{equation}
\label{eq:super_sensor_dyn_uncertainty}
\frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s)
\end{equation}
The dynamics of the super sensor now depends on the weighting functions ($w_1(s),w_2(s)$) and the complementary filters ($H_1(s),H_2(s)$).
The robust stability of the fusion can be studied graphically (refer Figure~\ref{fig:uncertainty_set_super_sensor}).
The frequency response of the fusion output is plotted in a complex plane.
The unity transfer function leads to a point $(1,0)$ located on the real axis.
The uncertainty associated with first sensor at a particular frequency is represented by a circle with the center at (1,0) and radius $|w_1H_1|$.
The uncertainty associated with the second is also represented using a circle centered at any point on the circle representing uncertainty associated with the first sensor and radius equal to $|w_2H_2|$.
Therefore, the overall uncertainty of the fusion is represented with a circle centered at (1,0) and radius equal to $|w_1H_1|+|w_2H_2|$.
The maximum phase difference that can result from the fusion is found by drawing a tangent from the origin to the uncertainty circle of super sensor.
Mathematically, the maximum phase difference at frequency $\omega$ that can result from fusion is given by
\begin{equation}
\label{eq:max_phase_uncertainty}
\Delta\phi(\omega) = \arcsin\left( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \right)
\end{equation}
\begin{figure}
\centering
\includegraphics[scale=1]{figs/uncertainty_set_super_sensor.pdf}
\caption{\label{fig:uncertainty_set_super_sensor}Robustness analysis of sensor fusion in the complex plane. The uncertainty associated with the super sensor dynamics are represented with a solid circle while those associated with individual sensors are represented with dashed circles.}
\end{figure}
A constraint on the maximum phase difference at a given frequency can be incorporated in the design problem using the following equation:
\begin{equation}
\label{eq:max_uncertainty_super_sensor}
\max_\omega \big( \left|w_1 H_1\right| + \left|w_2 H_2\right|\big) < \sin\left( \Delta \phi_\text{max} \right)
\end{equation}
where \(\Delta \phi_\text{max}\) is the maximum allowable phase difference.
It can also be inferred from the above equation that the magnitude of the complementary filter (\(|H_i|\)) should be tuned to a smaller value at the frequencies where the magnitude of weighing transfer functions (\(|w_i|\)) representing sensor uncertainty is large.
\section{Design formulation using \(\mathcal{H}_\infty\) synthesis}\label{sec:hinf_method}
In this section, the shaping of complementary filters is expressed as an optimal $\mathcal{H}_{\infty}$ synthesis problem.
The synthesis goal is to shape the frequency response of the filters such that they satisfy the design requirements presented in Section~\ref{sec:requirements}.
\subsection{Synthesis problem formulation}\label{sec:hinf_synthesis}
The first step is to formulate the filter design problem as a generalized plant-controller structure~\cite{boyd91_linear}.
The generalized plant and controller structure for complementary filters design is shown in Figure~\ref{fig:h_infinity_robust_fusion}.
In the figure, $P(s)$ is the generalized plant, $u$ is the ``control input'', $v$ is the ``measured output'' and $H_2(s)$ is the controller (filter) to be designed.
The regulated outputs of the generalized plant, $z_1$ and $z_2$, are given by
\begin{equation}
\begin{split}
z_1 &= W_1(s)(1-H_2(s)) w = W_1(s) H_1(s) w \text{ by defining } H_{1}(s) \triangleq 1 - H_{2}(s)\\
z_2 &= W_2(s) H_2(s) w
\end{split}
\end{equation}
where $w$ is the ``exogenous input'' to the plant, $W_1(s), W_2(s)$ are the weighting functions for shaping the complementary filters.
The dynamics of the generalized plant can be written as
\begin{equation}
\label{eq:generalized_plant}
\begin{bmatrix} z_1 \\ z_2 \\ v \end{bmatrix} = P(s) \begin{bmatrix} w\\u \end{bmatrix}; \quad P(s) = \begin{bmatrix}W_1(s) & -W_1(s) \\ 0 & W_2(s) \\ 1 & 0 \end{bmatrix}
\end{equation}
The weighting functions are chosen based on the specifications and requirements set for the complementary filters (discussed in Section~\ref{sec:hinf_weighting_func}).
The objective of the optimization is to design a filter $H_2(s)$ such that the following conditions are satisfied
\begin{equation}
\label{eq:comp_filter_problem_form}
\begin{split}
\left| \frac{z_{1}}{w} \right| &= |1-H_2(s)| \le \frac{1}{|W_1(s)|} \\
\left| \frac{z_{1}}{w} \right| &= |H_2(s)| \le \frac{1}{|W_2(s)|}
\end{split}, \quad \forall \omega \in \mathbb{R}^{+}
\end{equation}
\begin{figure}
\centering
\includegraphics[scale=1]{figs/h_infinity_robust_fusion.pdf}
\caption{\label{fig:h_infinity_robust_fusion}Generalized plant controller structure for the design of complementary filters}
\end{figure}
Based on Figure~\ref{fig:h_infinity_robust_fusion}, the $\mathcal{H}_{\infty}$ synthesis problem for the complementary filters can be stated as
\begin{quote}
Find a stable transfer function, $H_2(s)$, which takes measured output, $v$, as input and generates a control input, $u$, such that the $\mathcal{H}_\infty$ norm of the generalized plant from exogenous input, $w$, to the regulated output, ${[z_1,z_2]}^T$ is less than unity.
\end{quote}
Mathematically, the synthesis objective can be written as
\begin{equation}
\begin{split}
&\left\|\begin{matrix} \left[1 - H_2(s)\right] W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 \\
\Longleftrightarrow & \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1; \quad H_1(s) \triangleq 1 - H_2(s)
\end{split}
\label{eq:hinf_syn_obj}
\end{equation}
The above optimization problem can be efficiently solved in \textsc{Matlab}~\cite{MATLAB2009} using Riccati formulae, linear matrix inequality based method or maximum entropy method.
\subsection{Design of weighting functions}\label{sec:hinf_weighting_func}
The choice of weighting function governs the shape of the designed complementary filters.
Therefore, it is very important that the design specifications are appropriately transformed into the weighting functions.
The choice of weighting functions is also constrained by the following factors
\begin{enumerate}
\item[(a)] Only proper and stable transfer functions can be used as weighting functions
\item[(b)] As the order of the designed filter is equal to the sum of the orders of the weighting functions, the order of the weighting function needs to be reasonably small to ensure the physical implementation of the designed complementary filters.
This also reduces the computational cost of the optimization problem.
\item[(c)] The complementary property of the filter imposes a fundamental limitations on the weighting functions.
The imposes a restriction that the magnitude of the filters $H_1(s)$ and $H_2(s)$ cannot be made small simultaneously at the same frequency.
\end{enumerate}
The specifications of the complementary filters are typically expressed using the following parameters --- low frequency gain, high frequency gain, slope (order of the filter) and the crossover frequency.
We propose a weighting function that allows to translate the above requirements by setting simple parameters:
\begin{equation}
\label{eq:weight_formula}
W(s) = \displaystyle\left( \frac{
\hfill{} \displaystyle\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\displaystyle\frac{G_0}{G_c}\right)^{\displaystyle\frac{2}{n}}}{1 - {\left(\displaystyle\frac{G_c}{G_\infty}\right)}^{\displaystyle\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\displaystyle\frac{1}{n}}
}{
\left(\displaystyle\frac{1}{G_\infty}\right)^{\displaystyle\frac{1}{n}}\displaystyle \frac{1}{\omega_0} \sqrt{\displaystyle\frac{1 - \left(\displaystyle\frac{G_0}{G_c}\right)^{\displaystyle\frac{2}{n}}}{1 - \left(\displaystyle\frac{G_c}{G_\infty}\right)^{\displaystyle\frac{2}{n}}}} s + \left(\displaystyle\frac{1}{G_c}\right)^{\displaystyle\frac{1}{n}}
}\right)^n
\end{equation}
where, \(G_0 = \lim_{\omega \to 0} |W(j\omega)|\) is the low frequency gain, \(G_\infty = \lim_{\omega \to \infty} |W(j\omega)|\) is the high frequency gain, $\omega_c$ is the crossover frequency, \(G_c = |W(j\omega_c)|\) is the absolute gain at the crossover frequency and $n$ is the order of the filter.
As an illustration, the magnitude of the frequency response of the weighting function with the parameters \(G_0 = 0.001\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\) and having high pass characteristics is shown in Figure~\ref{fig:weight_formula}.
\begin{figure}
\centering
\includegraphics[scale=1]{figs/weight_formula.pdf}
\caption{\label{fig:weight_formula}Magnitude of the frequency response of the weighting function obtained using equation~\eqref{eq:weight_formula} with the parameters \(G_0 = 0.001\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\)}
\end{figure}
\subsection{Verification}\label{sec:hinf_example}
The proposed methodology for the design of complementary filters is now applied on the following example.
\begin{example}
Design complementary filters with the merging frequency around 10~Hz.
The low pass filter should have DC gain of 0.001 and slope -2 above the merging frequency.
The high pass filter should have a slope of 3 below the merging frequency and 0.001 high frequency gain.
\end{example}
\noindent The first step is to design the weighting functions that translate the above requirements.
To do so, equation~\eqref{eq:weight_formula} is used.
The parameters corresponding to $W_1(s)$ and $W_2(s)$ are listed in Table~\ref{tab:weights_params}.
The obtained transfer functions of the weighting functions are
\begin{equation}
\begin{split}
W_1(s) &= \dfrac{1000 (s+34.55)^2}{(s+3455)^2}\\
W_2(s) &= \dfrac{0.1 (s+87.43)^3}{(s+4.058)^3}
\end{split}
\end{equation}
Using these weighting functions, the generalized plant is evaluated using equation~\eqref{eq:generalized_plant}.
The optimal complementary filters are obtained by solving the optimization problem given by equation~\eqref{eq:hinf_syn_obj}.
The complementary filters obtained after optimization are
\begin{equation}
\begin{split}
H_1(s) &= \frac{10^{-8} (s+6.6\times 10^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3\times 10^3) (s^2 + 72s + 3580)}\\
H_2(s) &= \frac{(s+6.6\times 10^4) (s+160) (s+4)^3}{(s+6.6\times 10^4) (s^2 + 106 s + 3\times 10^3) (s^2 + 72s + 3580)}
\end{split}
\end{equation}
The obtained complementary filters are of order 5 which corresponds to the sum of the orders of the weighting functions used.
The frequency responses of the designed complementary filters are shown in Figure~\ref{fig:hinf_filters_results}.
It can be seen that the designed filters fulfills all the design specifications and hence demonstrates the effectiveness of the designed methodology (more complex real life example is taken up in Section~\ref{sec:application_ligo}).
\begin{table}
\caption{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\)}
\centering
\begin{tabularx}{0.5\linewidth}{Xcc}
\toprule
Parameter & \(W_1(s)\) & \(W_2(s)\)\\
\midrule
\(n\) & \(2\) & \(3\)\\
\(G_c\) & \(0.5\) & \(0.5\)\\
\(G_\infty\) & \(1000\) & \(0.1\)\\
\(G_0\) & \(0.1\) & \(1000\)\\
\(\omega_c\) [\(\si{Hz}\)] & \(11\) & \(10\)\\
\bottomrule
\end{tabularx}
\end{table}
\begin{figure}
\centering
\includegraphics[scale=1]{figs/hinf_filters_results.pdf}
\caption{\label{fig:hinf_filters_results}Bode plot of the weighting functions and designed complementary filters}
\end{figure}
\subsection{Extension to set of three complementary filters}\label{sec:hinf_three_comp_filters}
In certain applications, more than two sensors are used to measure the same quantity and can be merged together to form a ``super sensor''.
In such case, a set of three (or more) complementary filters is required.
It is here shown that the proposed method can be generalized for the design of a set of arbitrary number of complementary filters.
The control objective is now to design of a set of $n$ complementary filters ($H_i(s), i=1,\cdots,n$) which satisfy the following conditions
\begin{equation}
\label{eq:hinf_problem_gen}
\begin{split}
&\sum_{i=0}^n H_i(s) = 1 \\
&\left| H_i(s) \right| < \frac{1}{\left| W_i(s) \right|}
\end{split}
\end{equation}
Here, we extend the method to a case of three complementary filters.
The generalized plant controller setup for this case is shown in Figure~\ref{fig:comp_filter_three_hinf}.
The synthesis objective is to design filters $H_2(s)$ and $H_3(s)$ such that the $\mathcal{H}_\infty$ norm from exogenous input $w$ to regulated output vector $[z_1,z_2,z_3]^T$ is less than unity.
That is,
\begin{equation}
\label{eq:hinf_syn_obj_three}
\begin{split}
&\left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1\\
\equiv &\left\| \begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1; \quad H_1(s) \triangleq 1 - H_2(s) - H_3(s)
\end{split}
\end{equation}
\begin{figure}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf.pdf}
\caption{\label{fig:comp_filter_three_hinf}Generalized plant controller setup for designing a set of three complementary filters using \(\mathcal{H}_\infty\) synthesis}
\end{figure}
To validate this synthesis method, let's take an example where 3 sensors are merged together.
The three sensors are working in the following three frequency range --- below 1~Hz, between 1Hz to 10Hz and above 10Hz.
The weighting functions used for the synthesis are
\begin{equation}
\begin{split}
W_1(s) &= \dfrac{1000 (s+3.141)^2}{(s+314.1)^2}\\
W_2(s) &= \dfrac{2200 (s+62.83)^2 (s+6.283)^2}{(s+6283)^2 (s+0.06283)^2}\\
W_3(s) &= \dfrac{0.1 (s+87.43)^3}{(s+4.058)^3}
\end{split}
\end{equation}
The complementary filters are obtained by solving the optimization problem given by equation~\eqref{eq:hinf_syn_obj_three}.
The frequency response of the designed filters and the weighting functions are shown in Figure~\ref{fig:three_complementary_filters_results}.
\begin{figure}
\centering
\includegraphics[scale=1]{figs/three_complementary_filters_results.pdf}
\caption{\label{fig:three_complementary_filters_results}Bode plot of the weighting functions and designed set of three complementary filters}
\end{figure}
\section{Application: Complementary Filter Design for Active Vibration Isolation of LIGO}\label{sec:application_ligo}
Gravitational waves can help in detection various astrophysical events occurring in our universe.
This can also pave a path to validate theories built around the existence of gravitational waves.
However, the detection of these waves is an arduous task owing to the extraordinary small strain experienced by the earth due to gravitational waves.
Various methods have been proposed for their detection, out of which laser interferometers are the most popular ones.
Laser interferometers offers large projection range and high displacement sensitivity.
Among the existing detector, Laser interferometer gravitation-wave observatory (LIGO) is the most sensitive operational detector.
LIGO consists of two longs arms, referred as beam tubes, that are placed orthogonal to each other.
The arms of the LIGO accommodates a Michleson interferometer with a cavity (Fabry-Perot).
The mirrors at the extremity of the cavity serve as inertial test masses which responds to the strain induced due to the gravitational waves.
The optics of the LIGO are suspended like a pendulum.
The schematics of the LIGO are shown in Figure~\ref{ligo_schematics}.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figs/ligo.png}
\caption{\label{ligo_schematics}Schematics of LIGO}
\end{figure}
The isolation of the terrestrial interferometers is necessary in order to isolate the motion of the suspended inertial masses from the seismic ground motion.
The vibration isolation system attenuates the effect of disturbances on the motion of the suspended masses in the gravitational wave band.
The other sources that can affect the sensitivity of the LIGO are thermal noise, human activities, tidal motion etc.
The suspensions of the LIGO needs to serve two purpose --- isolation and alignment.
The alignment is also important in order to ensure that the interferometer beam is targeted at the center of the suspended mass.
The current vibration isolation system for LIGO has seven different stages.
In the first stage, hydraulically actuated external pre-isolators are used for attenuating large ground motions.
Two stages of active electromagnetic isolation system are placed next to pre-isolators.
This is followed by four stages of pendulum based passive isolation system.
In the active isolation stage of LIGO, different sensors are used to sense the same physical signal in different frequency range.
For example, seismometers are used to sense the position of the platform in the frequency band 0.5--10~Hz while geophones are employed above 10~Hz.
The signals recorded from different sensor are fused using complementary filters~\cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}.
The stringent requirements on these filters complicate their design.
Hua~\cite{hua05_low_ligo} proposed complementary FIR filters which were synthesized using convex optimization.
The designed FIR filters were found to be compliant with the design specifications.
However, the order of the designed filter was very high, which limits its application to a practical system.
In this section, we demonstrate the design of complementary filters with the same specification using the proposed method based on $\mathcal{H}_\infty$- synthesis.
\subsection{Design specifications}\label{sec:ligo_specifications}
The design specification of the complementary filters (as listed out in~\cite{hua05_low_ligo}) are as follows:
\begin{enumerate}
\item[(a)] In the frequency range \(0\)-\(\SI{0.008}{Hz}\): the high pass filter's magnitude should be less than \(8 \times 10^{-4}\).
\item[(b)] For frequency range \(\SI{0.008}{Hz}\)-\(\SI{0.04}{Hz}\): slope of the high pass filter is equal to three.
\item[(c)] Between \(\SI{0.04}{Hz}\)-\(\SI{0.1}{Hz}\) frequency range: the high pass filter's magnitude should be less than \(3\).
\item[(d)] For frequencies above \(\SI{0.1}{Hz}\): the low pass filter's magnitude should be less than \(0.045\).
\end{enumerate}
The specification of the complementary filters are shown graphically by dashed black lines in Figure~\ref{fig:ligo_weights}.
\subsection{Weighting Functions Design}\label{sec:ligo_weights}
As the synthesis objective of the complementary filters is described by Eq.~\eqref{eq:hinf_problem_gen}, it is clear that the weighting functions should be chosen such that their inverse magnitude represent the maximum allowed norm of the complementary filters.
This can be done manually using by combining poles and zeros or using useful formulas such as Eq.~\eqref{eq:weight_formula}.
It is important to note that the order of the filters should be kept reasonable small in order to keep the computational cost of the optimization reasonable.
This will also ensure that the designed filters are realizable in the physical world.
The transfer function representing weights should also be stable and minimum phase.
The weighting function corresponding to the low pass filter, \(w_L(s)\), is here taken as Type I Chebyshev filter.
The order of the weighting function for low pass filter is set as 20.
The weighting function for the high pass filter, \(w_H(s)\), is designed in such a way that its magnitude response is as close as possible to the design specifications.
This was achieved using a combination of high-, low- and band-pass filters in the particular frequency band.
The overall order of the weighting function for high pass filter is 7.
The magnitude responses of the inverse of the designed weighting functions and their comparison with the specifications are shown in Figure~\ref{fig:ligo_weights}.
It can be seen that the inverse of the designed weights, shown in solid blue line for high pass filter and solid red line for low pass filter, are close to the specifications shown in black dotted line.
\begin{figure}
\centering
\includegraphics[scale=1]{figs/ligo_weights.pdf}
\caption{\label{fig:ligo_weights}Specifications and weighting functions magnitudes}
\end{figure}
\subsection{\(\mathcal{H}_\infty\) synthesis of complementary filters}\label{sec:ligo_results}
The complementary filters are designed using \(\mathcal{H}_\infty\) synthesis based on the architecture shown in Figure~\ref{fig:h_infinity_robust_fusion}.
The generalized plant is obtained by substituting the transfer functions of the designed weighting functions in equation~\eqref{eq:generalized_plant}, i.e., $W_1(s)=w_H(s)$ and $W_2(s)=w_L(s)$.
The objective of the \(\mathcal{H}_\infty\) synthesis is given by equation~\eqref{eq:hinf_syn_obj}.
The optimization problem in solved in \textsc{Matlab}~\cite{MATLAB2009} using Ricatti method.
The frequency responses of the designed optimal complementary filters are shown in Figure~\ref{fig:comp_fir_ligo_hinf}.
The order of the filters obtained using \(\mathcal{H}_\infty\) optimization is \(27\).
Now, we compare the designed complementary filters with the FIR filters designed by Hua~\cite{hua05_low_ligo}.
The transfer function of the FIR filter, $G(\omega)$, is the Fourier transform of its coefficients, $g(n)$.
That is,
\begin{equation}
G(\omega) = \sum_n g(n)e^{-j2n\pi\omega}
\end{equation}
The FIR filter synthesis was formulated as a convex optimization problem.
The objective of the optimization problem was to find the filter's coefficients such that their norm are below the specified upper bounds.
The optimization problem was solved using SeDuMi~\cite{sturm99_using_sedum} and the obtained order for the FIR filters is \(512\).
The bode plot of the FIR filters are shown with dotted lines in Figure~\ref{fig:comp_fir_ligo_hinf}.
It can be seen that frequency responses of the designed complementary filters matches quite well with those of the FIR filters.
The designed complementary filters are of much lower order and can be implemented with less computational cost can the FIR filters.
The proposed methodology for the design of complementary filters can be effectively employed to obtain physically realizable filters.
\begin{figure}
\centering
\includegraphics[scale=1]{figs/comp_fir_ligo_hinf.pdf}
\caption{\label{fig:comp_fir_ligo_hinf}Bode plot of the filters designed using $\mathcal{H}_\infty$ synthesis and FIR filters~\cite{hua05_low_ligo}}
\end{figure}
\section{Concluding remarks}\label{sec:conclusion}
The measurements from the sensors are reliable only within its bandwidth.
The signals from different sensors are usually fused in order to measure a physical quantity over larger bandwidth.
The sensor obtained after fusion is called as super sensor as it has superior noise characteristics and wider bandwidth.
Complementary filters are used for the combining the signals from different sensors.
A new framework based on $\mathcal{H}_\infty$ synthesis has been presented in this paper to aid the design of complementary filters.
The method presented allows to shape the complementary filters based on the design specifications.
The task of filter design is posed as an $\mathcal{H}_{\infty}$ synthesis problem.
The design specifications of the systems are transformed in the form of weighting functions.
These weighting functions are used in the optimization problem to constraint the filter response in a frequency band.
The method has also been demonstrated for designing a set of three complementary filters.
The design frame is general, simple to implement and can easily be extended to difference scenarios of sensor fusion.
The effectiveness of the method is demonstrated for a real life application where complementary filters are designed for active vibration isolation of Laser Interferometer Gravitational-Wave Observatory (LIGO).
The filters designed with the proposed method have been with compared with the finite impulse response (FIR) filters.
It was found that the filters designed using $\mathcal{H}_\infty$ have lower order compared to FIR filters.
The designed filters are physically realizable and have lesser computational cost compared to FIR filters.
The proposed method can be effectively used to shape complementary filters based on design specifications.
The method can be further be extended for the design of robust complementary filters with desired noise characteristics considering uncertainties in the sensor dynamics.
This is the focus of our future research.
\section*{Acknowledgment}
The authors would like to acknowledge the help received from the French Community of Belgium for funding the FRIA Grant of Thomas Dehaeze (Grant No.~FC 31597).
\bibliographystyle{elsarticle-num}
\bibliography{ref}
\end{document}

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