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tdehaeze committed 2020-10-05 14:59:04 +02:00
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.gitignore
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administrative/
## Core latex/pdflatex auxiliary files:
*.aux
*.lof
*.log
*.lot
*.fls
*.out
*.toc
*.fmt
*.fot
*.cb
*.cb2
.*.lb
## Intermediate documents:
*.dvi
*.xdv
*-converted-to.*
# these rules might exclude image files for figures etc.
# *.ps
# *.eps
# *.pdf
## Generated if empty string is given at "Please type another file name for output:"
.pdf
## Bibliography auxiliary files (bibtex/biblatex/biber):
*.bbl
*.bcf
*.blg
*-blx.aux
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*.run.xml
## Build tool auxiliary files:
*.fdb_latexmk
*.synctex
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## Auxiliary and intermediate files from other packages:
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# endfloat
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TSWLatexianTemp*
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LICENSE.txt
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MIT License
Copyright (c) 2020 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
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
README.md
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# Complementary Filters Shaping Using H-Infinity Synthesis
<!-- [![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.3937394.svg)](https://doi.org/10.5281/zenodo.3937394) -->
<!-- # Paper -->
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index.html
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<head>
<!-- 2020-10-05 lun. 14:58 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis</title>
<meta name="generator" content="Org mode" />
<meta name="author" content="Thomas Dehaeze" />
<link rel="stylesheet" type="text/css" href="css/htmlize.css"/>
<link rel="stylesheet" type="text/css" href="css/readtheorg.css"/>
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<div id="content">
<h1 class="title">Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis
<br />
<span class="subtitle">Dehaeze Thomas, Verma Mohit and Collette Christophe</span>
</h1>
<blockquote>
<p>
<b>Abstract</b>:
</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.
</p>
</blockquote>
<div id="outline-container-orgf99c99d" class="outline-2">
<h2 id="orgf99c99d">Paper (<a href="paper/paper.pdf">link</a>)</h2>
<div class="outline-text-2" id="text-orgf99c99d">
<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>
</div>
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<h2 id="orga085973">Tikz Figures (<a href="tikz/index.html">link</a>)</h2>
<div class="outline-text-2" id="text-orga085973">
<p>
All the figures for the paper have been generated using <a href="https://sourceforge.net/projects/pgf/">TikZ</a>.
</p>
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index.org
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#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis
:DRAWER:
#+SUBTITLE: Dehaeze Thomas, Verma Mohit and Collette Christophe
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#+begin_quote
*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.
#+end_quote
* Paper ([[file:paper/paper.pdf][link]])
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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]]) :noexport:
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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]])
:PROPERTIES:
:UNNUMBERED: t
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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
@inproceedings{dehaeze19_compl_filter_shapin_using_synth,
author = {Dehaeze, Thomas and Vermat, Mohit and Collette Christophe},
title = {Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis},
booktitle = {7th International Conference on Control, Mechatronics and Automation (ICCMA)},
year = {2019},
language = {english},
doi = {10.1109/ICCMA46720.2019.8988642},
}
#+end_src
You can also use the formatted citation below.
#+begin_quote
Dehaeze T., Vermat M., and Collette C., "Complementary Filters Shaping Using ℋ∞ Synthesis," 2019 7th International Conference on Control, Mechatronics and Automation (ICCMA), Delft, Netherlands, 2019, pp. 459-464.
#+end_quote
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js/jquery.stickytableheaders.min.js
Vendored
+1
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@@ -0,0 +1 @@
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js/readtheorg.js
+85
View File
@@ -0,0 +1,85 @@
$(function() {
$('.note').before("<p class='admonition-title note'>Note</p>");
$('.seealso').before("<p class='admonition-title seealso'>See also</p>");
$('.warning').before("<p class='admonition-title warning'>Warning</p>");
$('.caution').before("<p class='admonition-title caution'>Caution</p>");
$('.attention').before("<p class='admonition-title attention'>Attention</p>");
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$('.important').before("<p class='admonition-title important'>Important</p>");
$('.hint').before("<p class='admonition-title hint'>Hint</p>");
$('.error').before("<p class='admonition-title error'>Error</p>");
$('.danger').before("<p class='admonition-title danger'>Danger</p>");
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$(document).on('click', "[data-toggle='wy-nav-top']", function() {
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$("[data-toggle='rst-versions']").toggleClass("shift");
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$('table').stickyTableHeaders();
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js/siunitx.js
+2
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#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis
:DRAWER:
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#+LATEX_HEADER_EXTRA: \author[1,2]{Dehaeze Thomas\corref{cor1}}
#+LATEX_HEADER_EXTRA: \author[2,3]{Verma Mohit}
#+LATEX_HEADER_EXTRA: \author[3,3]{Collette Christophe}
#+LATEX_HEADER_EXTRA: \cortext[cor1]{Corresponding author. Email Address: dehaeze.thomas@gmail.com}
#+LATEX_HEADER_EXTRA: \address[1]{European Synchrotron Radiation Facility, Grenoble, France}
#+LATEX_HEADER_EXTRA: \address[2]{Precision Mechatronics Laboratory, Univeristy of Liege, Belgium}
#+LATEX_HEADER_EXTRA: \address[3]{BEAMS Department, Free University of Brussels, Belgium}
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* Front Matter :ignore:
#+begin_frontmatter
#+begin_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.
#+end_abstract
#+begin_keyword
Complementary Filters \sep Sensor Fusion \sep H-Infinity Synthesis
#+end_keyword
#+end_frontmatter
* Introduction
<<sec:introduction>>
*** Establish the importance of the research topic :ignore:
# What are Complementary Filters
A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
These filters are used when two or more sensors are measuring the same physical quantity with different noise characteristics. Unreliable frequencies of each sensor are filtered out by the complementary filters and then combined to form a super sensor giving a better estimate of the physical quantity over a wider bandwidth.
This technique is called sensor fusion and is used in many applications.\par
*** Applications of complementary filtering :ignore:
# Improve bandwidth for UAV
In cite:zimmermann92_high_bandw_orien_measur_contr,corke04_inert_visual_sensin_system_small_auton_helic,min15_compl_filter_desig_angle_estim, various sensors (accelerometers, gyroscopes, vision sensors, etc.) are merged using complementary filters for the attitude estimation of Unmanned Aerial Vehicles (UAV).
# Improving the control robustness
In cite:collette15_sensor_fusion_method_high_perfor, several sensor fusion configurations using different types of sensors are discussed in order to increase the control bandwidth of active vibration isolation systems.
# Merging of different sensor types
Furthermore, sensor fusion is used in the isolation systems of the Laser Interferometer Gravitational-Wave Observator (LIGO) to merge inertial sensors with relative sensors
cite:matichard15_seism_isolat_advan_ligo,hua04_polyp_fir_compl_filter_contr_system. \par
*** Current design methods for complementary filters :ignore:
# Why Design of Complementary Filter is important
As the super sensor noise characteristics largely depend on the complementary filter norms, their proper design is of primary importance for sensor fusion.
# Discuss the different approach to complementary filter design
In cite:corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas,min15_compl_filter_desig_angle_estim, first and second order analytical formulas of complementary filters have been presented.
# Third Order and Higher orders
Higher order complementary filters have been used in
cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,collette15_sensor_fusion_method_high_perfor.
# Alternate Formulation
In cite:jensen13_basic_uas, the sensitivity and complementary sensitivity transfer functions of a feedback architecture have been proposed to be used as complementary filters. The design of such filters can then benefit from the classical control theory developments.
# LMI / convex Optimization
Linear Matrix Inequalities (LMIs) are used in cite:pascoal99_navig_system_desig_using_time for the synthesis of complementary filters satisfying some frequency-like performance.
# FIR Filters
Finally, a synthesis method of high order Finite Impulse Response (FIR) complementary filters using convex optimization has been developed in cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system. \par
*** Describe a gap in the research :ignore:
# There is a need for easy synthesis methods for complementary filters
Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available.
*** Describe the paper itself / the problem which is addressed :ignore:
# In this paper, we propose a synthesis method for the shaping of complementary filters using the $\mathcal{H}_\infty$ norm.\par
This paper presents a new design method of complementary filters based on $\mathcal{H}_\infty$ synthesis.
This design method permits to easily shape the norms of the generated filters.\par
*** Introduce Each part of the paper :ignore:
The section ref:sec:requirements gives a brief overview of sensor fusion using complementary filters and explains how the typical requirements for such fusion can be expressed as upper bounds on the filters norms.
In section ref:sec:hinf_method, a new design method for the shaping of complementary filters using $\mathcal{H}_\infty$ synthesis is proposed.
In section ref:sec:application_ligo, the method is used to design complex complementary filters that are used for sensor fusion at the LIGO.
Our conclusions are drawn in the final section.
* Complementary Filters Requirements
<<sec:requirements>>
** Sensor Fusion Architecture
<<sec:sensor_fusion>>
Let's consider two sensors measuring the same physical quantity $x$ with dynamics $G_1(s)$ and $G_2(s)$, and with uncorrelated noise characteristics $n_1$ and $n_2$.
The signals from both sensors are fed into two complementary filters $H_1(s)$ and $H_2(s)$ and then combined to yield an estimate $\hat{x}$ of $x$ as shown in Fig. ref:fig:fusion_super_sensor.
#+name: eq:comp_filter_estimate
\begin{equation}
\hat{x} = \left(G_1 H_1 + G_2 H_2\right) x + H_1 n_1 + H_2 n_2
\end{equation}
#+name: fig:fusion_super_sensor
#+caption: Sensor fusion architecture
#+attr_latex: :scale 1
[[file:figs/fusion_super_sensor.pdf]]
The complementary property of $H_1(s)$ and $H_2(s)$ implies that their transfer function sum is equal to one at all frequencies eqref:eq:comp_filter.
#+name: eq:comp_filter
\begin{equation}
H_1(s) + H_2(s) = 1
\end{equation}
** Noise Sensor Filtering
<<sec:noise_filtering>>
Let's first consider sensors with perfect dynamics
#+name: eq:perfect_dynamics
\begin{equation}
G_1(s) = G_2(s) = 1
\end{equation}
The estimate $\hat{x}$ is then described by
#+name: eq:estimate_perfect_dyn
\begin{equation}
\hat{x} = x + H_1 n_1 + H_2 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 noise.
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured.
Let's define the estimation error $\delta x$ by eqref:eq:estimate_error.
#+name: eq:estimate_error
\begin{equation}
\delta x \triangleq \hat{x} - x = H_1 n_1 + H_2 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 $\Phi_{n_2}$.
#+name: eq:noise_filtering_psd
\begin{equation}
\Phi_{\delta x} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2}
\end{equation}
Usually, the two sensors have high noise levels over distinct frequency regions.
In order to lower the noise of the super sensor, the value of the norm $|H_1|$ has to be lowered when $\Phi_{n_1}$ is larger than $\Phi_{n_2}$ and that of $|H_2|$ lowered when $\Phi_{n_2}$ is larger than $\Phi_{n_1}$.
** Robustness of the Fusion
<<sec:fusion_robustness>>
In practical systems the sensor dynamics is not perfect and eqref:eq:perfect_dynamics is not verified.
In such case, one can use an inversion filter $\hat{G}_i^{-1}(s)$ to normalize the sensor dynamics, where $\hat{G}_i(s)$ is an estimate of the sensor dynamics $G_i(s)$.
However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and $\hat{G}_i^{-1}(s) G_i(s) \neq 1$.
Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. ref:fig:sensor_fusion_dynamic_uncertainty where $\Delta_i$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $|w_i(s)|$ is a weight representing the magnitude of the uncertainty.
#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
#+attr_latex: :scale 1
[[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 dynamics 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 uncertainty region of the super sensor can be 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.
#+name: fig:uncertainty_set_super_sensor
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (solid circle). The contribution of both sensors 1 and 2 to the uncertainty are represented respectively by a dotted and a dashed circle
#+attr_latex: :scale 1
[[file:figs/uncertainty_set_super_sensor.pdf]]
The maximum phase added $\Delta\phi(\omega)$ by the super sensor dynamics at frequency $\omega$ is then
#+name: eq:max_phase_uncertainty
\begin{equation}
\Delta\phi(\omega) = \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big)
\end{equation}
As it is generally desired to limit the maximum phase added by the super sensor, $H_1(s)$ and $H_2(s)$ should be designed such that eqref:eq:max_uncertainty_super_sensor is satisfied.
#+name: eq:max_uncertainty_super_sensor
\begin{equation}
\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 allowed added phase.
Thus the norm of the complementary filter $|H_i|$ should be made small at frequencies where $|w_i|$ is large.
* Complementary Filters Shaping using $\mathcal{H}_\infty$ Synthesis
<<sec:hinf_method>>
** Introduction :ignore:
As shown in Sec. ref:sec:requirements, the performance and robustness of the sensor fusion architecture depends on the complementary filters norms.
Therefore, the development of a synthesis method of complementary filters that allows the shaping of their norm is necessary.
** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis
<<sec:hinf_synthesis>>
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 stable transfer functions $H_1(s)$ and $H_2(s)$ such that conditions eqref:eq:comp_filter_problem_form are satisfied.
#+name: eq:comp_filter_problem_form
\begin{subequations}
\begin{align}
& H_1(s) + H_2(s) = 1 \label{eq:hinf_cond_complementarity} \\
& |H_1(j\omega)| \le \frac{1}{|W_1(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h1} \\
& |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2}
\end{align}
\end{subequations}
where $W_1(s)$ and $W_2(s)$ are two weighting transfer functions that are chosen to shape the norms of the corresponding filters.
In order to express this optimization problem as a standard $\mathcal{H}_\infty$ problem, the architecture shown in Fig. ref:fig:h_infinity_robust_fusion is used where the generalized plant $P$ is described by eqref:eq:generalized_plant.
#+name: eq:generalized_plant
\begin{equation}
\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}
#+name: fig:h_infinity_robust_fusion
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
#+attr_latex: :scale 1
[[file:figs/h_infinity_robust_fusion.pdf]]
The $\mathcal{H}_\infty$ filter design problem is then to find a stable filter $H_2(s)$ which based on $v$, generates a signal $u$ such that the $\mathcal{H}_\infty$ norm from $w$ to $[z_1, \ z_2]$ is less than one 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}
This is equivalent to having eqref:eq:hinf_problem by defining $H_1(s)$ as the complementary filter of $H_2(s)$ eqref:eq:definition_H1.
#+name: eq:hinf_problem
\begin{equation}
\left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
#+name: eq:definition_H1
\begin{equation}
H_1(s) \triangleq 1 - H_2(s)
\end{equation}
The complementary condition eqref:eq:hinf_cond_complementarity is ensured by eqref:eq:definition_H1.
The conditions eqref:eq:hinf_cond_h1 and eqref:eq:hinf_cond_h2 on the filters shapes are satisfied by eqref:eq:hinf_problem.
Therefore, all the conditions eqref:eq:comp_filter_problem_form are satisfied using this synthesis method based on $\mathcal{H}_\infty$ synthesis, and thus it permits to shape complementary filters as desired.
** Weighting Functions Design
<<sec:hinf_weighting_func>>
The proper design of the weighting functions is of primary importance for the success of the presented complementary filters $\mathcal{H}_\infty$ synthesis.
First, only proper, stable and minimum phase transfer functions should be used.
Second, the order of the weights 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 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 time.
# Explain why we propose such weighting function
When designing complementary filters, it is usually desired to specify the slope of the filter, its crossover frequency and its gain at low and high frequency.
To help with the design of the weighting functions such that the above specification can be easily expressed, the following formula is proposed.
#+name: eq:weight_formula
\begin{equation}
W(s) = \left( \frac{
\hfill{} \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 permit 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 parameters $G_0$, $G_c$ and $G_\infty$ should either satisfy condition eqref:eq:cond_formula_1 or eqref:eq:cond_formula_2.
#+name: eq:condition_params_formula
\begin{subequations}
\begin{align}
G_0 < 1 < G_\infty \text{ and } G_0 < G_c < G_\infty \label{eq:cond_formula_1}\\
G_\infty < 1 < G_0 \text{ and } G_\infty < G_c < G_0 \label{eq:cond_formula_2}
\end{align}
\end{subequations}
The general shape of a weighting function generated using eqref:eq:weight_formula is shown in Fig. ref:fig:weight_formula.
#+name: fig:weight_formula
#+caption: Magnitude of a weighting function generated using the proposed formula eqref:eq:weight_formula, $G_0 = 1e^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$
#+attr_latex: :scale 1
[[file:figs/weight_formula.pdf]]
** Validation of the proposed synthesis method
<<sec:hinf_example>>
Let's validate the proposed design method of complementary filters with a simple example where two complementary filters $H_1(s)$ and $H_2(s)$ have to be designed such that:
- the merging frequency is around $\SI{10}{Hz}$
- the slope of $|H_1(j\omega)|$ is $-2$ above $\SI{10}{Hz}$
- the slope of $|H_2(j\omega)|$ is $+3$ below $\SI{10}{Hz}$
- the gain of both filters is equal to $10^{-3}$ away from the merging frequency
The weighting functions $W_1(s)$ and $W_2(s)$ are designed using eqref:eq:weight_formula.
The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_synthesis_results.
#+name: tab:weights_params
#+caption: Parameters used for $W_1(s)$ and $W_2(s)$
#+ATTR_LATEX: :environment tabularx :width 0.5\linewidth :align Xcc
#+ATTR_LATEX: :center t :booktabs t :float t
| Parameter | $W_1(s)$ | $W_2(s)$ |
|------------------------+----------+----------|
| $G_0$ | $0.1$ | $1000$ |
| $G_\infty$ | $1000$ | $0.1$ |
| $\omega_c$ [$\si{Hz}$] | $11$ | $10$ |
| $G_c$ | $0.5$ | $0.5$ |
| $n$ | $2$ | $3$ |
The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_synthesis_results and their transfer functions in the Laplace domain are given below.
\begin{align*}
H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\
H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}
\end{align*}
#+name: fig:hinf_synthesis_results
#+caption: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/hinf_synthesis_results.pdf]]
** Synthesis of Three Complementary Filters
<<sec:hinf_three_comp_filters>>
*** Why it is used sometimes :ignore:
Some applications may require to merge more than two sensors.
In such a case, it is necessary to design as many complementary filters as the number of sensors used.
*** Mathematical Problem :ignore:
The synthesis problem is then to compute $n$ stable transfer functions $H_i(s)$ such that eqref:eq:hinf_problem_gen is satisfied.
#+name: eq:hinf_problem_gen
\begin{subequations}
\begin{align}
& \sum_{i=0}^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}
*** H-Infinity Architecture :ignore:
The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. ref:fig:comp_filter_three_hinf.
The $\mathcal{H}_\infty$ synthesis objective applied on $P(s)$ is to design two stable filters $H_2(s)$ and $H_3(s)$ 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.
#+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}
#+name: fig:comp_filter_three_hinf
#+caption: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
#+attr_latex: :scale 1
[[file:figs/comp_filter_three_hinf.pdf]]
By choosing $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\infty$ synthesis solves the design problem eqref:eq:hinf_problem_gen. \par
*** Example of generated complementary filters :ignore:
An example is given to validate the method where three sensors are used in different frequency bands (up to $\SI{1}{Hz}$, from $1$ to $\SI{10}{Hz}$ and above $\SI{10}{Hz}$ respectively).
Three weighting functions are designed using eqref:eq:weight_formula and shown by dashed curves in Fig. ref:fig:hinf_three_synthesis_results.
The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_three_synthesis_results.
#+name: fig:hinf_three_synthesis_results
#+caption: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/hinf_three_synthesis_results.pdf]]
* Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO
<<sec:application_ligo>>
** Introduction :ignore:
Several complementary filters are used in the active isolation system at the LIGO cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system.
The requirements on those filters are very tight and thus their design is complex.
The approach used in cite:hua05_low_ligo for their design is to write the synthesis of complementary FIR filters as a convex optimization problem.
The obtained FIR filters are compliant with the requirements. However they are of very high order so their implementation is quite complex.
The effectiveness of the proposed method is demonstrated by designing complementary filters with the same requirements as the one described in cite:hua05_low_ligo.
** Complementary Filters Specifications
<<sec:ligo_specifications>>
The specifications for one pair of complementary filters used at the LIGO are summarized below (for further details, refer to cite:hua04_polyp_fir_compl_filter_contr_system) and shown in Fig. ref:fig:ligo_weights:
- From $0$ to $\SI{0.008}{Hz}$, the magnitude of the filter's transfer function should be less or equal to $8 \times 10^{-4}$
- Between $\SI{0.008}{Hz}$ to $\SI{0.04}{Hz}$, the filter should attenuate the input signal proportional to frequency cubed
- Between $\SI{0.04}{Hz}$ to $\SI{0.1}{Hz}$, the magnitude of the transfer function should be less than $3$
- Above $\SI{0.1}{Hz}$, the magnitude of the complementary filter should be less than $0.045$
** Weighting Functions Design
<<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 as 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.
#+name: fig:ligo_weights
#+caption: Specifications and weighting functions magnitudes
#+attr_latex: :scale 1
[[file:figs/ligo_weights.pdf]]
** $\mathcal{H}_\infty$ Synthesis
<<sec:ligo_results>>
$\mathcal{H}_\infty$ synthesis is performed using the architecture shown in Fig. ref:eq:generalized_plant.
The complementary filters obtained are of order $27$.
In Fig. ref:fig:comp_fir_ligo_hinf, their bode plot is compared with the FIR filters of order 512 obtained in cite:hua05_low_ligo.
They are found to be very close to each other and this shows the effectiveness of the proposed synthesis method.
#+name: fig:comp_fir_ligo_hinf
#+caption: Comparison of the FIR filters (solid) designed in cite:hua05_low_ligo with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed)
#+attr_latex: :scale 1
[[file:figs/comp_fir_ligo_hinf.pdf]]
* Conclusion
<<sec:conclusion>>
This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor.
Typical specification on the super sensor noise and on the robustness of the sensor fusion has been shown to be linked to the norm of the complementary filters.
Therefore, a synthesis method that permits the shaping of the complementary filters norms has been proposed and has been successfully applied for the design of complex filters.
Future work will aim at further developing this synthesis method for the robust and optimal synthesis of complementary filters used in sensor fusion.
* Acknowledgment
:PROPERTIES:
:UNNUMBERED: t
:END:
This research benefited from a FRIA grant from the French Community of Belgium.
* Bibliography :ignore:
\bibliographystyle{elsarticle-num}
\bibliography{ref}
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@article{collette15_sensor_fusion_method_high_perfor,
author = {Collette, Christophe and Matichard, Fabrice},
title = {Sensor Fusion Methods for High Performance Active Vibration
Isolation Systems},
journal = {Journal of Sound and Vibration},
volume = 342,
pages = {1-21},
year = 2015,
doi = {10.1016/j.jsv.2015.01.006},
}
@phdthesis{hua05_low_ligo,
author = {Hua, Wensheng},
school = {stanford university},
title = {Low frequency vibration isolation and alignment system for
advanced LIGO},
year = 2005,
}
@inproceedings{hua04_polyp_fir_compl_filter_contr_system,
author = {Hua, Wensheng and Debra, B. and Hardham, T. and Lantz, T. and Giaime, A.},
title = {Polyphase FIR Complementary Filters for Control Systems},
booktitle = {Proceedings of ASPE Spring Topical Meeting on Control of
Precision Systems},
year = 2004,
pages = {109--114},
}
@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,
Instrumentation and Performance},
journal = {Classical and Quantum Gravity},
volume = 32,
number = 18,
pages = 185003,
year = 2015,
publisher = {IOP Publishing},
}
@article{min15_compl_filter_desig_angle_estim,
author = {Min, Hyung Gi and Jeung, Eun Tae},
title = {Complementary Filter Design for Angle Estimation Using Mems
Accelerometer and Gyroscope},
journal = {Department of Control and Instrumentation, Changwon National
University, Changwon, Korea},
pages = {641--773},
year = 2015,
}
@article{corke04_inert_visual_sensin_system_small_auton_helic,
author = {Corke, Peter},
title = {An Inertial and Visual Sensing System for a Small Autonomous
Helicopter},
journal = {Journal of Robotic Systems},
volume = 21,
number = 2,
pages = {43-51},
year = 2004,
}
@inproceedings{jensen13_basic_uas,
author = {Austin Jensen and Cal Coopmans and YangQuan Chen},
title = {Basics and guidelines of complementary filters for small UAS
navigation},
booktitle = {2013 International Conference on Unmanned Aircraft Systems
(ICUAS)},
year = 2013,
month = 5,
}
@inproceedings{pascoal99_navig_system_desig_using_time,
author = {A. Pascoal and I. Kaminer and P. Oliveira},
title = {Navigation System Design Using Time-Varying Complementary
Filters},
booktitle = {Guidance, Navigation, and Control Conference and Exhibit},
year = 1999,
}
@article{zimmermann92_high_bandw_orien_measur_contr,
author = {M. Zimmermann and W. Sulzer},
title = {High Bandwidth Orientation Measurement and Control Based on
Complementary Filtering},
journal = {Robot Control 1991},
pages = {525-530},
year = 1992,
publisher = {Elsevier},
series = {Robot Control 1991},
}
@inproceedings{baerveldt97_low_cost_low_weigh_attit,
author = {A.-J. Baerveldt and R. Klang},
title = {A Low-Cost and Low-Weight Attitude Estimation System for an
Autonomous Helicopter},
booktitle = {Proceedings of IEEE International Conference on Intelligent
Engineering Systems},
year = 1997,
month = {-},
}
@article{shaw90_bandw_enhan_posit_measur_using_measur_accel,
author = {F.R. Shaw and K. Srinivasan},
title = {Bandwidth Enhancement of Position Measurements Using Measured
Acceleration},
journal = {Mechanical Systems and Signal Processing},
volume = 4,
number = 1,
pages = {23-38},
year = 1990,
}
@article{brown72_integ_navig_system_kalman_filter,
author = {R. G. Brown},
title = {Integrated Navigation Systems and Kalman Filtering: a
Perspective},
journal = {Navigation},
volume = 19,
number = 4,
pages = {355-362},
year = 1972,
}
@article{mahony08_nonlin_compl_filter_special_orthog_group,
author = {Robert Mahony and Tarek Hamel and Jean-Michel Pflimlin},
title = {Nonlinear Complementary Filters on the Special Orthogonal
Group},
journal = {IEEE Transactions on Automatic Control},
volume = 53,
number = 5,
pages = {1203-1218},
year = 2008,
}
tikz/config.html
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#+TITLE: LaTeX Configuration for Tikz Figures
:DRAWER:
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
#+HTML_HEAD: <script src="../js/jquery.min.js"></script>
#+HTML_HEAD: <script src="../js/bootstrap.min.js"></script>
#+HTML_HEAD: <script src="../js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script src="../js/readtheorg.js"></script>
#+PROPERTY: header-args:latex+ :tangle config.tex
#+PROPERTY: header-args:latex+ :exports code
:END:
* Packages
#+begin_src latex
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[french, english]{babel} % Last language is main language
\usepackage{lmodern} % Latin Modern Font
\usepackage{gensymb} % Generic symbols for both text and math mode
\usepackage{standalone} % Used to generate standalone Tikz
\usepackage{amsmath} % Main math Package
\usepackage{mathtools} % Extension package to amsmath
\usepackage{amsthm} % Typesetting theorems (AMS style)
\usepackage{amsfonts} % More fonts from the AMS
\usepackage{textcomp} % provide many text symbols
\usepackage{steinmetz} % For phase symbol
\usepackage{xstring} % Utils to manipulate strings
\usepackage{etoolbox} % Add basic if/then
\usepackage{esvect} % Beautyfull vectors
\usepackage{graphicx} % Enhanced support for graphics
\usepackage{grffile} % Used by matlab2tikz
\usepackage{microtype} % typographic tuning
\usepackage{setspace} % for line spacing, e.g. \onehalfspacing
\usepackage{tabularx} % table features
\usepackage{enumitem} % for simple list modifications
\usepackage{booktabs} % better table support
\usepackage{stackengine} %
\usepackage[load-configurations=abbreviations]{siunitx} % SI units
\sisetup{
locale = US,
detect-all,
range-phrase=--,
range-units=single
}
#+end_src
* Tikz related packages
#+begin_src latex
\usepackage{tikz} % Tikz
\usepackage{tikzscale} % Used to scale Tikz graphics
\usepackage{adjustbox} % Used to proper positioning of tikz pictures
\usepackage{circuitikz} % Draw electronic circuits
\usepackage{pgfpages} % Needed to use notes
\usepackage{pgfplots} % Used to plot functions
#+end_src
* Tikz Libraries
#+begin_src latex
\usetikzlibrary{arrows} % Arrow tip library
\usetikzlibrary{arrows.meta} % Add some arrows
\usetikzlibrary{calc} % The library allows advanced Coordinate Calculations
\usetikzlibrary{intersections} % calculate intersections of paths
\usetikzlibrary{matrix} %
\usetikzlibrary{patterns} %
\usetikzlibrary{shapes} % Defines circle and rectangle
\usetikzlibrary{shapes.geometric} % Use for the shape diamond and isosceles triangle
\usetikzlibrary{snakes} % snake=coil and snake=zigzag using segment amplitude=10pt
\usetikzlibrary{positioning} % Additional options for placing nodes
\usetikzlibrary{3d} % Plot 3D shapes
\usetikzlibrary{spy} % Creating a magnified area
\usetikzlibrary{decorations.text} % Used to make text follows a curve
\usetikzlibrary{decorations.pathmorphing} % deformation of a path
\usetikzlibrary{decorations.markings} % Used for spring and damper
\usetikzlibrary{babel} % A tiny library that make the interaction with the babel package easier
\usetikzlibrary{plotmarks} % This library defines a number of plot marks
\usetikzlibrary{fit} % Used to make rectangle as nodes by specifying two points
\usetikzlibrary{backgrounds} % Used to put things under others
#+end_src
* PGF Plot libraries and config
#+begin_src latex
\usepgfplotslibrary{patchplots}
\usepgfplotslibrary{groupplots}
\pgfplotsset{compat=newest}
\pgfplotsset{plot coordinates/math parser=false}
#+end_src
* Setup size of figures
#+begin_src latex
\newlength{\fheight}
\newlength{\fwidth}
\setlength{\fwidth}{85mm}
\setlength{\fheight}{112mm}
#+end_src
* Setup Arrows style
#+begin_src latex
\tikzset{>=Stealth}
% Setup default Linewidth
\tikzset{every path/.style={line width=1pt}}
#+end_src
* Colors
#+begin_src latex
\usepackage{xcolor}% Color extension
\definecolor{mycolor1}{RGB}{79,115,193}
\definecolor{mycolor2}{RGB}{213,91,53}
\definecolor{mycolor3}{RGB}{152,126,49}
#+end_src
* Control
** Blocks
#+begin_src latex
\tikzset{%
block/.style n args={2}{%
draw,
fill=white,
minimum width = #1,
minimum height = #2,
},
block/.default={1.2cm}{1.0cm}
}
#+end_src
** Branches
#+begin_src latex
\tikzstyle{branch}=[fill,shape=circle,minimum size=4pt,inner sep=0pt]
\tikzstyle{->top}=[-{Stealth[color=black, scale=0.8]}, draw=white, double=black, double distance=1pt, line width=1pt]
\tikzstyle{<-top}=[{stealth[color=black, scale=0.8]}-, draw=white, double=black, double distance=1pt, line width=1pt]
#+end_src
** Hand Writen Style
Usefull for schematic plots
#+begin_src latex
\tikzstyle{handwriten}=[decorate,decoration={random steps,amplitude=0.1pt,segment length=0.8pt}]
#+end_src
** DAC
#+begin_src latex
\tikzset{%
DAC/.style={%
draw,
signal,
}
}
#+end_src
** ADC
#+begin_src latex
\tikzset{%
ADC/.style={%
draw,
signal,
signal to = west,
}
}
#+end_src
** Gain
#+begin_src latex
\tikzset{%
gain right/.style={%
draw,
regular polygon,
regular polygon sides = 3,
inner sep = 2pt,
shape border rotate=-90
},
gain left/.style={%
draw,
regular polygon,
regular polygon sides = 3,
inner sep = 2pt,
shape border rotate=90
},
gain top/.style={%
draw,
regular polygon,
regular polygon sides = 3,
inner sep = 2pt,
shape border rotate=0
},
gain bottom/.style={%
draw,
regular polygon,
regular polygon sides = 3,
inner sep = 2pt,
shape border rotate=180
},
}
#+end_src
** Add / Substract / Divide / Multiply block
#+begin_src latex
\tikzset{% Add block with Circled operations
addc/.style n args={5}{%
draw,
fill=white,
circle,
outer sep = 0pt,
inner sep = 0pt,
minimum size = 2em,
execute at begin node={\LARGE $#1$},
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
\ifx#2\empty\else
node[draw, circle, outer sep=6pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
\fi
\ifx#3\empty\else
node[draw, circle, outer sep=6pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
\fi
\ifx#4\empty\else
node[draw, circle, outer sep=6pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
\fi
\ifx#5\empty\else
node[draw, circle, outer sep=6pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
\fi
}
},
addc/.default={+}{}{}{}{},
}
#+end_src
#+begin_src latex
\tikzset{% Add Block
addb/.style n args={5}{%
draw,
fill=white,
circle,
outer sep = 0pt,
inner sep = 0pt,
minimum size = 2em,
execute at begin node={\LARGE $#1$},
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
\ifx#2\empty\else
node[outer sep=2pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
\fi
\ifx#3\empty\else
node[outer sep=2pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
\fi
\ifx#4\empty\else
node[outer sep=2pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
\fi
\ifx#5\empty\else
node[outer sep=2pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
\fi
}
},
addb/.default={+}{}{}{}{},
}
#+end_src
* Plots
** Default line caps
#+begin_src latex
\pgfplotsset{
every axis plot/.append style={line join=round},
every axis plot/.append style={line cap=round},
}
#+end_src
** Grid
#+begin_src latex
\pgfplotsset{grid style={black}}
\pgfplotsset{major grid style={black!30!white}}
\pgfplotsset{minor grid style={black!10!white}}
\pgfplotsset{xmajorgrids}
\pgfplotsset{ymajorgrids}
#+end_src
** Lines
#+begin_src latex
\pgfplotsset{separate axis lines=false} % draw axis as rectangle and not as 4 lines
\pgfplotsset{every outer x axis line/.append style={black}}
\pgfplotsset{every outer y axis line/.append style={black}}
\pgfplotsset{axis background/.style={fill=white}}
\pgfplotsset{axis x line*=bottom} % solid line on the bottom with thin on the top
\pgfplotsset{axis y line*=left} % solid line on the left with thin on the right
#+end_src
** Ticks
#+begin_src latex
\pgfplotsset{every y tick label/.append style={font=\color{black}}}
\pgfplotsset{every y tick/.append style={black}}
\pgfplotsset{every x tick label/.append style={font=\color{black}}}
\pgfplotsset{every x tick/.append style={black}}
#+end_src
** Size
If =scale only axis=false= (the default), pgfplots will try to produce the desired width including labels, titles and ticks.
#+begin_src latex
\pgfplotsset{scale only axis=true}
#+end_src
** Label
Used to align all of ylabel of one figure.
#+begin_src latex
\pgfplotsset{ylabel absolute}
#+end_src
** Legend
#+begin_src latex
% https://tex.stackexchange.com/questions/54794/using-a-pgfplots-style-legend-in-a-plain-old-tikzpicture#54834
% argument #1: any options
\newenvironment{customlegend}[1][]{%
\begingroup
% inits/clears the lists (which might be populated from previous
% axes):
\csname pgfplots@init@cleared@structures\endcsname
\pgfplotsset{#1}%
}{%
% draws the legend:
\csname pgfplots@createlegend\endcsname
\endgroup
}%
% makes \addlegendimage available (typically only available within an
% axis environment):
\def\addlegendimage{\csname pgfplots@addlegendimage\endcsname}
% definition to insert numbers
% \pgfkeys{/pgfplots/number in legend/.style={%
% /pgfplots/legend image code/.code={%
% \node at (0.125,-0.0225){#1}; % <= changed x value
% },%
% },
% }
\pgfplotsset{
every legend to name picture/.style={west}
}
#+end_src
* Mechanical
** Spring
#+begin_src latex
\tikzset{%
spring/.style={%
thick,
decoration={
zigzag,
pre length = #1cm,
post length = #1cm,
segment length = 6
},
decorate
},
spring/.default={0.2}
}
#+end_src
** Coil
#+begin_src latex
\tikzset{%
coil/.style n args={2}{%
thick,
decoration={
coil,
pre length = #1cm,
post length = #2cm,
segment length = 4
},
decorate
},
coil/.default={0.3}{0.3}
}
#+end_src
** Damper
#+begin_src latex
\tikzset{%
damper/.style n args={2}{%
thick,
decoration={markings, mark connection node=dmp, mark=at position 0.5 with {
\node (dmp) [thick,
inner sep = 0pt,
transform shape,
rotate =-90,
minimum width = #1pt,
minimum height = #2pt,
draw=none] {};
\draw [thick] ($(dmp.north east)+(0.6*#2pt,0)$) -- (dmp.south east) -- (dmp.south west) -- ($(dmp.north west)+(0.6*#2pt,0)$);
\draw [thick] ($(dmp.north)+(0,-0.3*#1pt)$) -- ($(dmp.north)+(0,0.3*#1pt)$);
}
},
decorate
},
damper/.default={12}{3}
}
#+end_src
** Actuator
#+begin_src latex
\tikzset{%
actuator/.style n args={2}{%
thick,
draw=none,
decoration={
markings,
mark connection node=my node,
mark=at position .5 with {
\node [draw, inner sep=0pt, minimum width=#1cm, minimum height=#2cm,
transform shape, fill=white] (my node) {};
},
mark=at position .0 with {
\draw[<-] (0, 0) -- (my node);
},
mark=at position 1.0 with {
\draw[<-] (0, 0) -- (my node);
}
},
decorate
},
actuator/.default={0.5}{0.2}
}
#+end_src
** Ground
#+begin_src latex
\tikzset{%
ground/.style n args={2}{%
fill,
pattern = north east lines,
draw = none,
anchor = north,
minimum width = #1cm,
minimum height = #2cm,
append after command={
(\tikzlastnode.north west) edge (\tikzlastnode.north east)
}
},
ground/.default={2.5}{0.3}
}
#+end_src
** Force Sensor
#+begin_src latex
\tikzset{%
forcesensor/.style n args={2}{%
rectangle,
outer sep=0pt,
inner sep=0pt,
draw=black,
fill=white!60!black,
anchor=south,
minimum width =#1cm,
minimum height=#2cm,
append after command={
[every edge/.append style={
thick,
black,
}]
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
}
},
forcesensor/.default={2.0}{0.5}
}
#+end_src
** Inertial Sensor
#+begin_src latex
\tikzset{%
inertialsensor/.style={%
rectangle,
outer sep=0pt,
inner sep=0pt,
draw=black,
fill=white!60!black,
anchor=south east,
minimum size=#1cm,
append after command={
[every edge/.append style={
thick,
black,
}]
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
}
},
inertialsensor/.default={0.3}
}
#+end_src
** Cross
#+begin_src latex
\tikzstyle{cross}=[path picture={
\draw[black]
(path picture bounding box.south east) -- (path picture bounding box.north west) (path picture bounding box.south west) -- (path picture bounding box.north east);
}]
#+end_src
** Piezoelectric actuator
#+begin_src latex
\tikzset{%
piezo/.style n args={3}{%
draw,
rectangle,
minimum width = #1cm,
minimum height = #2cm,
fill=blue!10!white,
anchor=center,
append after command={
[every edge/.append style={
thick,
black,
}]
\foreach \i in {1,...,#3}{
(${\i/(1+#3)}*(\tikzlastnode.north west)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south west)+0.1*(#1,0)$) edge (${\i/(1+#3)}*(\tikzlastnode.north east)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south east)-0.1*(#1,0)$)
}
}
},
piezo/.default={2}{4}{10}
}
#+end_src
** Voice coil
#+begin_src latex
\def\voicecoil#1#2#3{
% ======================
% Parameters
% ======================
\def\voicecoilw{#1} % Total Width
\def\voicecoilh{#2} % Total Height
\def\magnetw{\voicecoilw} % Width of the magnet
\def\magneth{\voicecoilh/1.4} % Height of the magnet
\def\magnetwb{0.15*\magnetw} % Width of the borders of the magnet
\def\magnetmw{0.15*\magnetw} % Width of the middle part of the magnet
\def\magnetwg{0.5*\magnetw} % Width of the gap of the magnet
\def\magnethl{\magnetwb} % Height of the low part of the magnet
\def\magnetmh{0.15*\magneth} % Height of the middle part of the magnet
\def\magnethg{0.2*\magneth} % Height of the gap of the magnet
% ======================
\begin{scope}[shift={(0.5*\voicecoilw, 0.5*\voicecoilh)}, rotate=#3, shift={(0, -0.5*\voicecoilh)}]
% ======================
% Magnet
% ======================
\draw[fill=white] (0, 0) -| ++(0.5*\magnetw, \magneth) -| ++(-0.5*\magnetw+0.5*\magnetwg, -\magnethg) -| (0.5*\magnetw-\magnetwb, \magnethl) -| (-0.5*\magnetw+\magnetwb, \magneth-\magnethg) -| (-0.5*\magnetwg, \magneth) -| (-0.5*\magnetw, 0) -- (cycle);
\begin{scope}[shift={(0, \magnethl)}]
\draw[fill=red] (-0.5*\magnetmw, 0) rectangle (0.5*\magnetmw, \magnetmh);
\draw[fill=blue] (-0.5*\magnetmw, \magnetmh) rectangle (0.5*\magnetmw, 2*\magnetmh);
% Top conductive Magnet
\draw[fill=white] (-0.5*\magnetmw, 2*\magnetmh) -| (0.5*\magnetmw, -\magnethl+\magneth-\magnethg) -| ++(0.1, \magnethg) -| ++(-0.2-\magnetmw, -\magnethg) -| (-0.5*\magnetmw, \magnetmh);
\end{scope}
% ======================
% ======================
% Coil
% ======================
\pgfmathsetmacro{\coilwidth}{0.5*0.5*\magnetmw+0.5*0.1+0.25*\magnetwg}%
\draw[] ( \coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
\draw[] (-\coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
% Point on the coil
\foreach \x in {0,1,...,9}
{
\node[circle,inner sep=0.6pt,fill] at ( \coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
\node[circle,inner sep=0.6pt,fill] at (-\coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
}
\draw[fill=white] (-0.5*\magnetw, 1.2*\magneth) rectangle ++(\magnetw, \magnethg);
% ======================
% ======================
% Coordinates
% ======================
% Force
\coordinate[] (vc_force) at (0, \magneth-0.5*\magnethg);
% Coil
\coordinate[] (vc_coil) at (0, \voicecoilh);
% Magnet
\coordinate[] (vc_magnet) at (0, 0);
% Coil Wires
\coordinate[] (vc_wire_one) at ( \coilwidth, 1.2*\magneth);
\coordinate[] (vc_wire_two) at (-\coilwidth, 1.2*\magneth);
% ======================
\end{scope}
}
#+end_src
* Optics
#+begin_src latex
\tikzset{%
->-/.style={
decoration={
markings,
mark = at position #1 with {\arrow{>}
}
},
postaction={decorate}
}
}
\tikzset{%
-<-/.style={
decoration={
markings,
mark = at position #1 with {\arrow{<}
}
},
postaction={decorate}
}
}
#+end_src
* Misc
#+begin_src latex
\tikzset{%
labelc/.style= {%
draw,
fill=white,
shape=circle,
inner sep=2pt,
outer sep=6pt,
}
}
#+end_src
* More Defaults specific to this paper
#+begin_src latex
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
#+end_src
tikz/config.tex
+513
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@@ -0,0 +1,513 @@
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[french, english]{babel} % Last language is main language
\usepackage{lmodern} % Latin Modern Font
\usepackage{gensymb} % Generic symbols for both text and math mode
\usepackage{standalone} % Used to generate standalone Tikz
\usepackage{amsmath} % Main math Package
\usepackage{mathtools} % Extension package to amsmath
\usepackage{amsthm} % Typesetting theorems (AMS style)
\usepackage{amsfonts} % More fonts from the AMS
\usepackage{textcomp} % provide many text symbols
\usepackage{steinmetz} % For phase symbol
\usepackage{xstring} % Utils to manipulate strings
\usepackage{etoolbox} % Add basic if/then
\usepackage{esvect} % Beautyfull vectors
\usepackage{graphicx} % Enhanced support for graphics
\usepackage{grffile} % Used by matlab2tikz
\usepackage{microtype} % typographic tuning
\usepackage{setspace} % for line spacing, e.g. \onehalfspacing
\usepackage{tabularx} % table features
\usepackage{enumitem} % for simple list modifications
\usepackage{booktabs} % better table support
\usepackage{stackengine} %
\usepackage[load-configurations=abbreviations]{siunitx} % SI units
\sisetup{
locale = US,
detect-all,
range-phrase=--,
range-units=single
}
\usepackage{tikz} % Tikz
\usepackage{tikzscale} % Used to scale Tikz graphics
\usepackage{adjustbox} % Used to proper positioning of tikz pictures
\usepackage{circuitikz} % Draw electronic circuits
\usepackage{pgfpages} % Needed to use notes
\usepackage{pgfplots} % Used to plot functions
\usetikzlibrary{arrows} % Arrow tip library
\usetikzlibrary{arrows.meta} % Add some arrows
\usetikzlibrary{calc} % The library allows advanced Coordinate Calculations
\usetikzlibrary{intersections} % calculate intersections of paths
\usetikzlibrary{matrix} %
\usetikzlibrary{patterns} %
\usetikzlibrary{shapes} % Defines circle and rectangle
\usetikzlibrary{shapes.geometric} % Use for the shape diamond and isosceles triangle
\usetikzlibrary{snakes} % snake=coil and snake=zigzag using segment amplitude=10pt
\usetikzlibrary{positioning} % Additional options for placing nodes
\usetikzlibrary{3d} % Plot 3D shapes
\usetikzlibrary{spy} % Creating a magnified area
\usetikzlibrary{decorations.text} % Used to make text follows a curve
\usetikzlibrary{decorations.pathmorphing} % deformation of a path
\usetikzlibrary{decorations.markings} % Used for spring and damper
\usetikzlibrary{babel} % A tiny library that make the interaction with the babel package easier
\usetikzlibrary{plotmarks} % This library defines a number of plot marks
\usetikzlibrary{fit} % Used to make rectangle as nodes by specifying two points
\usetikzlibrary{backgrounds} % Used to put things under others
\usepgfplotslibrary{patchplots}
\usepgfplotslibrary{groupplots}
\pgfplotsset{compat=newest}
\pgfplotsset{plot coordinates/math parser=false}
\newlength{\fheight}
\newlength{\fwidth}
\setlength{\fwidth}{85mm}
\setlength{\fheight}{112mm}
\tikzset{>=Stealth}
% Setup default Linewidth
\tikzset{every path/.style={line width=1pt}}
\usepackage{xcolor}% Color extension
\definecolor{mycolor1}{RGB}{79,115,193}
\definecolor{mycolor2}{RGB}{213,91,53}
\definecolor{mycolor3}{RGB}{152,126,49}
\tikzset{%
block/.style n args={2}{%
draw,
fill=white,
minimum width = #1,
minimum height = #2,
},
block/.default={1.2cm}{1.0cm}
}
\tikzstyle{branch}=[fill,shape=circle,minimum size=4pt,inner sep=0pt]
\tikzstyle{->top}=[-{Stealth[color=black, scale=0.8]}, draw=white, double=black, double distance=1pt, line width=1pt]
\tikzstyle{<-top}=[{stealth[color=black, scale=0.8]}-, draw=white, double=black, double distance=1pt, line width=1pt]
\tikzstyle{handwriten}=[decorate,decoration={random steps,amplitude=0.1pt,segment length=0.8pt}]
\tikzset{%
DAC/.style={%
draw,
signal,
}
}
\tikzset{%
ADC/.style={%
draw,
signal,
signal to = west,
}
}
\tikzset{%
gain right/.style={%
draw,
regular polygon,
regular polygon sides = 3,
inner sep = 2pt,
shape border rotate=-90
},
gain left/.style={%
draw,
regular polygon,
regular polygon sides = 3,
inner sep = 2pt,
shape border rotate=90
},
gain top/.style={%
draw,
regular polygon,
regular polygon sides = 3,
inner sep = 2pt,
shape border rotate=0
},
gain bottom/.style={%
draw,
regular polygon,
regular polygon sides = 3,
inner sep = 2pt,
shape border rotate=180
},
}
\tikzset{% Add block with Circled operations
addc/.style n args={5}{%
draw,
fill=white,
circle,
outer sep = 0pt,
inner sep = 0pt,
minimum size = 2em,
execute at begin node={\LARGE $#1$},
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
\ifx#2\empty\else
node[draw, circle, outer sep=6pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
\fi
\ifx#3\empty\else
node[draw, circle, outer sep=6pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
\fi
\ifx#4\empty\else
node[draw, circle, outer sep=6pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
\fi
\ifx#5\empty\else
node[draw, circle, outer sep=6pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
\fi
}
},
addc/.default={+}{}{}{}{},
}
\tikzset{% Add Block
addb/.style n args={5}{%
draw,
fill=white,
circle,
outer sep = 0pt,
inner sep = 0pt,
minimum size = 2em,
execute at begin node={\LARGE $#1$},
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
\ifx#2\empty\else
node[outer sep=2pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
\fi
\ifx#3\empty\else
node[outer sep=2pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
\fi
\ifx#4\empty\else
node[outer sep=2pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
\fi
\ifx#5\empty\else
node[outer sep=2pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
\fi
}
},
addb/.default={+}{}{}{}{},
}
\pgfplotsset{
every axis plot/.append style={line join=round},
every axis plot/.append style={line cap=round},
}
\pgfplotsset{grid style={black}}
\pgfplotsset{major grid style={black!30!white}}
\pgfplotsset{minor grid style={black!10!white}}
\pgfplotsset{xmajorgrids}
\pgfplotsset{ymajorgrids}
\pgfplotsset{separate axis lines=false} % draw axis as rectangle and not as 4 lines
\pgfplotsset{every outer x axis line/.append style={black}}
\pgfplotsset{every outer y axis line/.append style={black}}
\pgfplotsset{axis background/.style={fill=white}}
\pgfplotsset{axis x line*=bottom} % solid line on the bottom with thin on the top
\pgfplotsset{axis y line*=left} % solid line on the left with thin on the right
\pgfplotsset{every y tick label/.append style={font=\color{black}}}
\pgfplotsset{every y tick/.append style={black}}
\pgfplotsset{every x tick label/.append style={font=\color{black}}}
\pgfplotsset{every x tick/.append style={black}}
\pgfplotsset{scale only axis=true}
\pgfplotsset{ylabel absolute}
% https://tex.stackexchange.com/questions/54794/using-a-pgfplots-style-legend-in-a-plain-old-tikzpicture#54834
% argument #1: any options
\newenvironment{customlegend}[1][]{%
\begingroup
% inits/clears the lists (which might be populated from previous
% axes):
\csname pgfplots@init@cleared@structures\endcsname
\pgfplotsset{#1}%
}{%
% draws the legend:
\csname pgfplots@createlegend\endcsname
\endgroup
}%
% makes \addlegendimage available (typically only available within an
% axis environment):
\def\addlegendimage{\csname pgfplots@addlegendimage\endcsname}
% definition to insert numbers
% \pgfkeys{/pgfplots/number in legend/.style={%
% /pgfplots/legend image code/.code={%
% \node at (0.125,-0.0225){#1}; % <= changed x value
% },%
% },
% }
\pgfplotsset{
every legend to name picture/.style={west}
}
\tikzset{%
spring/.style={%
thick,
decoration={
zigzag,
pre length = #1cm,
post length = #1cm,
segment length = 6
},
decorate
},
spring/.default={0.2}
}
\tikzset{%
coil/.style n args={2}{%
thick,
decoration={
coil,
pre length = #1cm,
post length = #2cm,
segment length = 4
},
decorate
},
coil/.default={0.3}{0.3}
}
\tikzset{%
damper/.style n args={2}{%
thick,
decoration={markings, mark connection node=dmp, mark=at position 0.5 with {
\node (dmp) [thick,
inner sep = 0pt,
transform shape,
rotate =-90,
minimum width = #1pt,
minimum height = #2pt,
draw=none] {};
\draw [thick] ($(dmp.north east)+(0.6*#2pt,0)$) -- (dmp.south east) -- (dmp.south west) -- ($(dmp.north west)+(0.6*#2pt,0)$);
\draw [thick] ($(dmp.north)+(0,-0.3*#1pt)$) -- ($(dmp.north)+(0,0.3*#1pt)$);
}
},
decorate
},
damper/.default={12}{3}
}
\tikzset{%
actuator/.style n args={2}{%
thick,
draw=none,
decoration={
markings,
mark connection node=my node,
mark=at position .5 with {
\node [draw, inner sep=0pt, minimum width=#1cm, minimum height=#2cm,
transform shape, fill=white] (my node) {};
},
mark=at position .0 with {
\draw[<-] (0, 0) -- (my node);
},
mark=at position 1.0 with {
\draw[<-] (0, 0) -- (my node);
}
},
decorate
},
actuator/.default={0.5}{0.2}
}
\tikzset{%
ground/.style n args={2}{%
fill,
pattern = north east lines,
draw = none,
anchor = north,
minimum width = #1cm,
minimum height = #2cm,
append after command={
(\tikzlastnode.north west) edge (\tikzlastnode.north east)
}
},
ground/.default={2.5}{0.3}
}
\tikzset{%
forcesensor/.style n args={2}{%
rectangle,
outer sep=0pt,
inner sep=0pt,
draw=black,
fill=white!60!black,
anchor=south,
minimum width =#1cm,
minimum height=#2cm,
append after command={
[every edge/.append style={
thick,
black,
}]
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
}
},
forcesensor/.default={2.0}{0.5}
}
\tikzset{%
inertialsensor/.style={%
rectangle,
outer sep=0pt,
inner sep=0pt,
draw=black,
fill=white!60!black,
anchor=south east,
minimum size=#1cm,
append after command={
[every edge/.append style={
thick,
black,
}]
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
}
},
inertialsensor/.default={0.3}
}
\tikzstyle{cross}=[path picture={
\draw[black]
(path picture bounding box.south east) -- (path picture bounding box.north west) (path picture bounding box.south west) -- (path picture bounding box.north east);
}]
\tikzset{%
piezo/.style n args={3}{%
draw,
rectangle,
minimum width = #1cm,
minimum height = #2cm,
fill=blue!10!white,
anchor=center,
append after command={
[every edge/.append style={
thick,
black,
}]
\foreach \i in {1,...,#3}{
(${\i/(1+#3)}*(\tikzlastnode.north west)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south west)+0.1*(#1,0)$) edge (${\i/(1+#3)}*(\tikzlastnode.north east)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south east)-0.1*(#1,0)$)
}
}
},
piezo/.default={2}{4}{10}
}
\def\voicecoil#1#2#3{
% ======================
% Parameters
% ======================
\def\voicecoilw{#1} % Total Width
\def\voicecoilh{#2} % Total Height
\def\magnetw{\voicecoilw} % Width of the magnet
\def\magneth{\voicecoilh/1.4} % Height of the magnet
\def\magnetwb{0.15*\magnetw} % Width of the borders of the magnet
\def\magnetmw{0.15*\magnetw} % Width of the middle part of the magnet
\def\magnetwg{0.5*\magnetw} % Width of the gap of the magnet
\def\magnethl{\magnetwb} % Height of the low part of the magnet
\def\magnetmh{0.15*\magneth} % Height of the middle part of the magnet
\def\magnethg{0.2*\magneth} % Height of the gap of the magnet
% ======================
\begin{scope}[shift={(0.5*\voicecoilw, 0.5*\voicecoilh)}, rotate=#3, shift={(0, -0.5*\voicecoilh)}]
% ======================
% Magnet
% ======================
\draw[fill=white] (0, 0) -| ++(0.5*\magnetw, \magneth) -| ++(-0.5*\magnetw+0.5*\magnetwg, -\magnethg) -| (0.5*\magnetw-\magnetwb, \magnethl) -| (-0.5*\magnetw+\magnetwb, \magneth-\magnethg) -| (-0.5*\magnetwg, \magneth) -| (-0.5*\magnetw, 0) -- (cycle);
\begin{scope}[shift={(0, \magnethl)}]
\draw[fill=red] (-0.5*\magnetmw, 0) rectangle (0.5*\magnetmw, \magnetmh);
\draw[fill=blue] (-0.5*\magnetmw, \magnetmh) rectangle (0.5*\magnetmw, 2*\magnetmh);
% Top conductive Magnet
\draw[fill=white] (-0.5*\magnetmw, 2*\magnetmh) -| (0.5*\magnetmw, -\magnethl+\magneth-\magnethg) -| ++(0.1, \magnethg) -| ++(-0.2-\magnetmw, -\magnethg) -| (-0.5*\magnetmw, \magnetmh);
\end{scope}
% ======================
% ======================
% Coil
% ======================
\pgfmathsetmacro{\coilwidth}{0.5*0.5*\magnetmw+0.5*0.1+0.25*\magnetwg}%
\draw[] ( \coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
\draw[] (-\coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
% Point on the coil
\foreach \x in {0,1,...,9}
{
\node[circle,inner sep=0.6pt,fill] at ( \coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
\node[circle,inner sep=0.6pt,fill] at (-\coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
}
\draw[fill=white] (-0.5*\magnetw, 1.2*\magneth) rectangle ++(\magnetw, \magnethg);
% ======================
% ======================
% Coordinates
% ======================
% Force
\coordinate[] (vc_force) at (0, \magneth-0.5*\magnethg);
% Coil
\coordinate[] (vc_coil) at (0, \voicecoilh);
% Magnet
\coordinate[] (vc_magnet) at (0, 0);
% Coil Wires
\coordinate[] (vc_wire_one) at ( \coilwidth, 1.2*\magneth);
\coordinate[] (vc_wire_two) at (-\coilwidth, 1.2*\magneth);
% ======================
\end{scope}
}
\tikzset{%
->-/.style={
decoration={
markings,
mark = at position #1 with {\arrow{>}
}
},
postaction={decorate}
}
}
\tikzset{%
-<-/.style={
decoration={
markings,
mark = at position #1 with {\arrow{<}
}
},
postaction={decorate}
}
}
\tikzset{%
labelc/.style= {%
draw,
fill=white,
shape=circle,
inner sep=2pt,
outer sep=6pt,
}
}
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
tikz/figs/comp_filter_three_hinf.pdf
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tikz/figs/uncertainty_set_super_sensor.pdf
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tikz/figs/uncertainty_set_super_sensor.tex
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\begin{tikzpicture}
\begin{scope}[shift={(4, 0)}]
% Uncertainty Circle
\node[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
\path[draw, dotted] (0, 0) circle [radius=1.0];
\path[draw, dashed] (135:1.0) circle [radius=0.8];
% Center of Circle
\node[below] at (0, 0){$1$};
\draw[<->, dashed] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
\draw[<->, dashed] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(90:0.8);
\node[] (l1) at (2, 1.5) {$|w_1 H_1|$};
\draw[->, dashed, out=-90, in=0] (l1.south) to (r1);
\node[] (l2) at (0, 2.0) {$|w_2 H_2|$};
\draw[->, dashed, out=-90, in=0] (l2.south) to (r2);
\draw[<->, dashed] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
\node[] (l3) at (-2.5, -1.5) {$|w_1 H_1| + |w_2 H_2|$};
\draw[->, dashed, out=90, in=-90] (l3.north) to (r3);
% % Uncertainty Regions
% \node[below left] at (45:1.8) {$\Pi_{\text{ss}}$};
% \node[above left] at (-45:1.0) {$\Pi_{1}$};
% \node[below] at ($(135:1.0)+(0, 0.8)$) {$\Pi_{2}$};
\end{scope}
% Real and Imaginary Axis
\draw[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
\draw[->] (0, -1.8) -- (0, 2.0) node[below left]{Im};
\draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
\draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi$};
\end{tikzpicture}
tikz/figs/weight_formula.pdf
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<h1 class="title">Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis - Tikz Figures</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orge7053aa">1. Fig 1: Sensor Fusion Architecture</a></li>
<li><a href="#org39b5d0c">2. Fig 2: Sensor fusion architecture with sensor dynamics uncertainty</a></li>
<li><a href="#orgfc9382d">3. Fig 3: Uncertainty set of the super sensor dynamics</a></li>
<li><a href="#orgb356b70">4. Fig 4: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</a></li>
<li><a href="#orgdeaeaf4">5. Fig 5: Magnitude of a weighting function generated using the proposed formula</a></li>
<li><a href="#org8ce545c">6. Fig 6: Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</a></li>
<li><a href="#orgac21d24">7. Fig 7: Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</a></li>
<li><a href="#org9688457">8. Fig 8: Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</a></li>
<li><a href="#org8d6867c">9. Fig 9: Specifications and weighting functions magnitude used for \(\mathcal{H}_\infty\) synthesis</a></li>
<li><a href="#orgb6ae2f5">10. Fig 10: Comparison of the FIR filters (solid) with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed)</a></li>
</ul>
</div>
</div>
<p>
Configuration file is accessible <a href="config.html">here</a>.
</p>
<div id="outline-container-orge7053aa" class="outline-2">
<h2 id="orge7053aa"><span class="section-number-2">1</span> Fig 1: Sensor Fusion Architecture</h2>
<div class="outline-text-2" id="text-1">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
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</pre>
</div>
<div id="orgc3d769f" class="figure">
<p><img src="figs/fusion_super_sensor.png" alt="fusion_super_sensor.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Sensor Fusion Architecture (<a href="./figs/fusion_super_sensor.png">png</a>, <a href="./figs/fusion_super_sensor.pdf">pdf</a>, <a href="./figs/fusion_super_sensor.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org39b5d0c" class="outline-2">
<h2 id="org39b5d0c"><span class="section-number-2">2</span> Fig 2: Sensor fusion architecture with sensor dynamics uncertainty</h2>
<div class="outline-text-2" id="text-2">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
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</pre>
</div>
<div id="orgde2b8c0" class="figure">
<p><img src="figs/sensor_fusion_dynamic_uncertainty.png" alt="sensor_fusion_dynamic_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Sensor fusion architecture with sensor dynamics uncertainty (<a href="./figs/sensor_fusion_dynamic_uncertainty.png">png</a>, <a href="./figs/sensor_fusion_dynamic_uncertainty.pdf">pdf</a>, <a href="./figs/sensor_fusion_dynamic_uncertainty.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgfc9382d" class="outline-2">
<h2 id="orgfc9382d"><span class="section-number-2">3</span> Fig 3: Uncertainty set of the super sensor dynamics</h2>
<div class="outline-text-2" id="text-3">
<div class="org-src-container">
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<span class="org-font-latex-sedate">\draw</span>[-&gt;, dashed, out=-90, in=0] (l2.south) to (r2);
<span class="org-font-latex-sedate">\draw</span>[&lt;-&gt;, dashed] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
<span class="org-font-latex-sedate">\node</span>[] (l3) at (-2.5, -1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_1</span></span><span class="org-font-latex-math"> H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_1</span></span><span class="org-font-latex-math">| + |w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_2</span></span><span class="org-font-latex-math"> H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_2</span></span><span class="org-font-latex-math">|$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;, dashed, out=90, in=-90] (l3.north) to (r3);
<span class="org-comment">% % Uncertainty Regions</span>
<span class="org-comment">% </span><span class="org-font-latex-sedate"><span class="org-comment">\node</span></span><span class="org-comment">[below left] at (45:1.8) {$</span><span class="org-font-latex-sedate"><span class="org-comment">\Pi</span></span><span class="org-comment">_{</span><span class="org-font-latex-sedate"><span class="org-comment">\text</span></span><span class="org-comment">{ss}}$};</span>
<span class="org-comment">% </span><span class="org-font-latex-sedate"><span class="org-comment">\node</span></span><span class="org-comment">[above left] at (-45:1.0) {$</span><span class="org-font-latex-sedate"><span class="org-comment">\Pi</span></span><span class="org-comment">_{1}$};</span>
<span class="org-comment">% </span><span class="org-font-latex-sedate"><span class="org-comment">\node</span></span><span class="org-comment">[below] at ($(135:1.0)+(0, 0.8)$) {$</span><span class="org-font-latex-sedate"><span class="org-comment">\Pi</span></span><span class="org-comment">_{2}$};</span>
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
<span class="org-comment">% Real and Imaginary Axis</span>
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0, -1.8) -- (0, 2.0) node[below left]{Im};
<span class="org-font-latex-sedate">\draw</span>[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
<span class="org-font-latex-sedate">\draw</span>[dashed] (1, 0) arc (0:28:1) node[midway, right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"> </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\phi</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orgf3182a4" class="figure">
<p><img src="figs/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Uncertainty region of the super sensor dynamics in the complex plane (solid circle), of the sensor 1 (dotted circle) and of the sensor 2 (dashed circle) (<a href="./figs/uncertainty_set_super_sensor.png">png</a>, <a href="./figs/uncertainty_set_super_sensor.pdf">pdf</a>, <a href="./figs/uncertainty_set_super_sensor.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgb356b70" class="outline-2">
<h2 id="orgb356b70"><span class="section-number-2">4</span> Fig 4: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</h2>
<div class="outline-text-2" id="text-4">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate">\node</span>[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.75!(P.north west) + (-0.7, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.35!(P.north west) + (-0.7, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.75!(P.north east) + ( 0.7, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.35!(P.north east) + ( 0.7, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + ( 0.7, 0)$</span>);
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_1</span></span><span class="org-font-latex-math">(s)$</span>};
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_2</span></span><span class="org-font-latex-math">(s)$</span>};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub) {};
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P] (H2) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_2</span></span><span class="org-font-latex-math">(s)$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (inputw) node[above right]{<span class="org-font-latex-math">$w$</span>} -- (sub.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (H2.west) -| (<span class="org-font-latex-math">$(inputu)+(0.35, 0)$</span>) node[above]{<span class="org-font-latex-math">$u$</span>} -- (W2.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (inputu-|sub) node[branch]{} -- (sub.south);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (sub.east) -- (W1.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(sub.west)+(-0.6, 0)$</span>) node[branch]{} |- (<span class="org-font-latex-math">$(outputv)+(-0.35, 0)$</span>) node[above]{<span class="org-font-latex-math">$v$</span>} |- (H2.east);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (W1.east) -- (output1)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_1</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_2</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="org3cb8226" class="figure">
<p><img src="figs/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters (<a href="./figs/h_infinity_robust_fusion.png">png</a>, <a href="./figs/h_infinity_robust_fusion.pdf">pdf</a>, <a href="./figs/h_infinity_robust_fusion.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgdeaeaf4" class="outline-2">
<h2 id="orgdeaeaf4"><span class="section-number-2">5</span> Fig 5: Magnitude of a weighting function generated using the proposed formula</h2>
<div class="outline-text-2" id="text-5">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">3.5cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
height=1.0<span class="org-font-latex-sedate">\fheight</span>,
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=100,
xtick={0.1,1,10, 100},
xminorticks=true,
ymode=log,
ymin=0.0005,
ymax=20,
ytick={0.001, 0.01, 0.1, 1, 10},
yminorticks=true,
ylabel={Magnitude},
xlabel={Frequency [Hz]},
xminorgrids,
yminorgrids,
]
<span class="org-font-latex-sedate">\addplot</span> [color=black, line width=1.5pt, forget plot]
table [x=freqs, y=ampl, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matweight_formula.csv};
<span class="org-font-latex-sedate">\addplot</span> [color=black, dashed, line width=1.5pt]
table[row sep=crcr]{<span class="org-comment">%</span>
1 10<span class="org-font-latex-warning">\\</span>
100 10<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addplot</span> [color=black, dashed, line width=1.5pt]
table[row sep=crcr]{<span class="org-comment">%</span>
0.1 0.001<span class="org-font-latex-warning">\\</span>
3 0.001<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addplot</span> [color=black, line width=1.5pt]
table[row sep=crcr]{<span class="org-comment">%</span>
0.1 1<span class="org-font-latex-warning">\\</span>
100 1<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addplot</span> [color=black, dashed, line width=1.5pt]
table[row sep=crcr]{<span class="org-comment">%</span>
10 2<span class="org-font-latex-warning">\\</span>
10 1<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\node</span>[below] at (2, 10) {<span class="org-font-latex-math">$G_\infty$</span>};
<span class="org-font-latex-sedate">\node</span>[above] at (2, 0.001) {<span class="org-font-latex-math">$G_0$</span>};
<span class="org-font-latex-sedate">\node</span>[branch] at (10, 2){};
<span class="org-font-latex-sedate">\draw</span>[dashed, line cap=round] (7, 2) -- (20, 2) node[right]{<span class="org-font-latex-math">$G_c$</span>};
<span class="org-font-latex-sedate">\draw</span>[dashed, line cap=round] (10, 2) -- (10, 1) node[below]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\omega</span></span><span class="org-font-latex-math">_c$</span>};
<span class="org-font-latex-sedate">\node</span>[right] at (3, 0.1) {<span class="org-font-latex-math">$+n$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orga3a1174" class="figure">
<p><img src="figs/weight_formula.png" alt="weight_formula.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Magnitude of a weighting function generated using the proposed formula (<a href="./figs/weight_formula.png">png</a>, <a href="./figs/weight_formula.pdf">pdf</a>, <a href="./figs/weight_formula.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org8ce545c" class="outline-2">
<h2 id="org8ce545c"><span class="section-number-2">6</span> Fig 6: Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</h2>
<div class="outline-text-2" id="text-6">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">6cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
height=0.5<span class="org-font-latex-sedate">\fheight</span>,
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.5<span class="org-font-latex-sedate">\fheight</span>)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=1000,
xtick={0.1, 1, 10, 100, 1000},
xticklabels={{}},
xminorticks=true,
ymode=log,
ymin=0.0005,
ymax=20,
ytick={0.001, 0.01, 0.1, 1, 10},
yminorticks=true,
ylabel={Magnitude},
xminorgrids,
yminorgrids,
]
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt, forget plot]
table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt, forget plot]
table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt, forget plot]
table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt, forget plot]
table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
height=0.45<span class="org-font-latex-sedate">\fheight</span>,
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=1000,
xtick={0.1, 1, 10, 100, 1000},
xminorticks=true,
xlabel={Frequency [Hz]},
ymin=-200,
ymax=200,
ytick={-180, -90, 0, 90, 180},
ylabel={Phase [deg]},
xminorgrids,
legend style={at={(1,1.1)}, outer sep=2pt , anchor=north east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}},
]
<span class="org-font-latex-sedate">\addlegendimage</span>{color=mycolor1, dashed, line width=1.5pt}
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$W_1^{-1}$</span>};
<span class="org-font-latex-sedate">\addlegendimage</span>{color=mycolor2, dashed, line width=1.5pt}
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$W_2^{-1}$</span>};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_1$</span>};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_2$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="org49f0f1e" class="figure">
<p><img src="figs/hinf_synthesis_results.png" alt="hinf_synthesis_results.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis (<a href="./figs/hinf_synthesis_results.png">png</a>, <a href="./figs/hinf_synthesis_results.pdf">pdf</a>, <a href="./figs/hinf_synthesis_results.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgac21d24" class="outline-2">
<h2 id="orgac21d24"><span class="section-number-2">7</span> Fig 7: Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</h2>
<div class="outline-text-2" id="text-7">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate">\node</span>[block={5.0cm}{3.5cm}, fill=black!20!white, dashed] (P) {};
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.8!(P.north west) + (-0.7, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.4!(P.north west) + (-0.7, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.8!(P.north east) + (0.7, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.55!(P.north east) + (0.7, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (output3) at (<span class="org-font-latex-math">$(P.south east)!0.3!(P.north east) + (0.7, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + (0.7, 0)$</span>);
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_1</span></span><span class="org-font-latex-math">(s)$</span>};
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_2</span></span><span class="org-font-latex-math">(s)$</span>};
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output3] (W3){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_3</span></span><span class="org-font-latex-math">(s)$</span>};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P] (H) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_2</span></span><span class="org-font-latex-math">(s) \\ H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_3</span></span><span class="org-font-latex-math">(s)</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (inputw) node[above right](w){<span class="org-font-latex-math">$w$</span>} -- (sub2.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (W3-|sub1)node[branch]{} -- (sub1.south);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (W2-|sub2)node[branch]{} -- (sub2.south);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(sub2.west)+(-0.5, 0)$</span>) node[branch]{} |- (outputv) |- (H.east);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(H.south west)!0.7!(H.north west)$</span>) -| (inputu|-W2) -- (W2.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(H.south west)!0.3!(H.north west)$</span>) -| (<span class="org-font-latex-math">$(inputu|-W3)+(0.4, 0)$</span>) -- (W3.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (sub2.east) -- (sub1.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (sub1.east) -- (W1.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (W1.east) -- (output1)node[above left](z){<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_1</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_2</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (W3.east) -- (output3)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_3</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[above] at (W2-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_1</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[above] at (W3-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_2</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[above] at (outputv-|z){<span class="org-font-latex-math">$v$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orgd2b8673" class="figure">
<p><img src="figs/comp_filter_three_hinf.png" alt="comp_filter_three_hinf.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters (<a href="./figs/comp_filter_three_hinf.png">png</a>, <a href="./figs/comp_filter_three_hinf.pdf">pdf</a>, <a href="./figs/comp_filter_three_hinf.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org9688457" class="outline-2">
<h2 id="org9688457"><span class="section-number-2">8</span> Fig 8: Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</h2>
<div class="outline-text-2" id="text-8">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">6cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
height=0.55<span class="org-font-latex-sedate">\fheight</span>,
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.45<span class="org-font-latex-sedate">\fheight</span>)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=100,
xticklabels={{}},
xminorticks=true,
ymode=log,
ymin=0.0005,
ymax=20,
ytick={0.001, 0.01, 0.1, 1, 10},
yminorticks=true,
ylabel={Magnitude},
xminorgrids,
yminorgrids,
legend columns=2,
legend style={
/tikz/column 2/.style={
column sep=5pt,
},
at={(1,0)}, outer sep=2pt , anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}
},
]
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt]
table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">${W_1}^{-1}$</span>};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_1$</span>};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt]
table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">${W_2}^{-1}$</span>};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_2$</span>};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor3, dashed, line width=1.5pt]
table [x=freqs, y=W3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">${W_3}^{-1}$</span>};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor3, line width=1.5pt]
table [x=freqs, y=H3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_3$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
height=0.4<span class="org-font-latex-sedate">\fheight</span>,
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=100,
xminorticks=true,
xlabel={Frequency [Hz]},
ymin=-240,
ymax=240,
ytick={-180, -90, 0, 90, 180},
ylabel={Phase [deg]},
xminorgrids,
]
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor3, line width=1.5pt]
table [x=freqs, y=H3p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="org044b17c" class="figure">
<p><img src="figs/hinf_three_synthesis_results.png" alt="hinf_three_synthesis_results.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis (<a href="./figs/hinf_three_synthesis_results.png">png</a>, <a href="./figs/hinf_three_synthesis_results.pdf">pdf</a>, <a href="./figs/hinf_three_synthesis_results.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org8d6867c" class="outline-2">
<h2 id="org8d6867c"><span class="section-number-2">9</span> Fig 9: Specifications and weighting functions magnitude used for \(\mathcal{H}_\infty\) synthesis</h2>
<div class="outline-text-2" id="text-9">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">4cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
height=1.0<span class="org-font-latex-sedate">\fheight</span>,
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
scale only axis,
separate axis lines,
every outer x axis line/.append style={black},
every x tick label/.append style={font=<span class="org-font-latex-sedate">\color</span>{black}},
every x tick/.append style={black},
xmode=log,
xmin=0.001,
xmax=1,
xminorticks=true,
xlabel={Frequency [Hz]},
every outer y axis line/.append style={black},
every y tick label/.append style={font=<span class="org-font-latex-sedate">\color</span>{black}},
every y tick/.append style={black},
ymode=log,
ymin=0.005,
ymax=20,
yminorticks=true,
ylabel={Magnitude},
axis background/.style={fill=white},
xmajorgrids,
xminorgrids,
ymajorgrids,
yminorgrids,
legend style={at={(0,1)}, outer sep=2pt, anchor=north west, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
]
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
table [x=freqs, y=wHm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$|w_H|^{-1}$</span>}
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
table [x=freqs, y=wLm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$|w_L|^{-1}$</span>}
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt]
table[row sep=crcr]{<span class="org-comment">%</span>
0.0005 0.008<span class="org-font-latex-warning">\\</span>
0.008 0.008<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addlegendentry</span>{Specifications}
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt, forget plot]
table[row sep=crcr]{<span class="org-comment">%</span>
0.008 0.008<span class="org-font-latex-warning">\\</span>
0.04 1<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt, forget plot]
table[row sep=crcr]{<span class="org-comment">%</span>
0.04 3<span class="org-font-latex-warning">\\</span>
0.1 3<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt]
table[row sep=crcr]{<span class="org-comment">%</span>
0.1 0.045<span class="org-font-latex-warning">\\</span>
2 0.045<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orgb9647cd" class="figure">
<p><img src="figs/ligo_weights.png" alt="ligo_weights.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Specifications and weighting functions magnitude used for \(\mathcal{H}_\infty\) synthesis (<a href="./figs/ligo_weights.png">png</a>, <a href="./figs/ligo_weights.pdf">pdf</a>, <a href="./figs/ligo_weights.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgb6ae2f5" class="outline-2">
<h2 id="orgb6ae2f5"><span class="section-number-2">10</span> Fig 10: Comparison of the FIR filters (solid) with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed)</h2>
<div class="outline-text-2" id="text-10">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">7cm</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
height=0.60<span class="org-font-latex-sedate">\fheight</span>,
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.35<span class="org-font-latex-sedate">\fheight</span>)},
scale only axis,
xmode=log,
xmin=0.001,
xmax=1,
xtick={0.001,0.01,0.1,1},
xticklabels={{}},
xminorticks=true,
ymode=log,
ymin=0.002,
ymax=5,
ytick={0.001, 0.01, 0.1, 1, 10},
yminorticks=true,
ylabel={Magnitude},
xminorgrids,
yminorgrids,
legend style={at={(1,0)}, outer sep=2pt, anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
]
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_H(s)$</span> - <span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{H}_\infty$</span>}
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt]
table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_H(s)$</span> - FIR}
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_L(s)$</span> - <span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{H}_\infty$</span>}
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt]
table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_L(s)$</span> - FIR}
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
height=0.3<span class="org-font-latex-sedate">\fheight</span>,
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
scale only axis,
xmode=log,
xmin=0.001,
xmax=1,
xtick={0.001, 0.01, 0.1, 1},
xminorticks=true,
xlabel={Frequency [Hz]},
ymin=-180,
ymax=180,
ytick={-180, -90, 0, 90, 180},
ylabel={Phase [deg]},
xminorgrids,
]
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt, forget plot]
table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt, forget plot]
table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt, forget plot]
table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt, forget plot]
table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orgc99b299" class="figure">
<p><img src="figs/comp_fir_ligo_hinf.png" alt="comp_fir_ligo_hinf.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Comparison of the FIR filters (solid) with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed) (<a href="./figs/comp_fir_ligo_hinf.png">png</a>, <a href="./figs/comp_fir_ligo_hinf.pdf">pdf</a>, <a href="./figs/comp_fir_ligo_hinf.tex">tex</a>).</p>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-09-26 jeu. 17:46</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>
tikz/index.org
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#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis - Tikz Figures
:DRAWER:
#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP: ../index.html
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
#+HTML_HEAD: <script src="../js/jquery.min.js"></script>
#+HTML_HEAD: <script src="../js/bootstrap.min.js"></script>
#+HTML_HEAD: <script src="../js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script src="../js/readtheorg.js"></script>
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/tikz/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports both
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :noweb yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
Configuration file is accessible [[file:config.org][here]].
* Fig 1: Sensor Fusion Architecture
#+begin_src latex :file fusion_super_sensor.pdf :tangle figs/fusion_super_sensor.tex
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[block, above right=0.5 and 0.5 of x](G1){$G_1(s)$};
\node[block, below right=0.5 and 0.5 of x](G2){$G_2(s)$};
\node[addb, right=0.8 of G1](add1){};
\node[addb, right=0.8 of G2](add2){};
\node[block, right=0.8 of add1](H1){$H_1(s)$};
\node[block, right=0.8 of add2](H2){$H_2(s)$};
\node[addb, right=5 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[->] (x.center) |- (G1.west);
\draw[->] (x.center) |- (G2.west);
\draw[->] (G1.east) -- (add1.west);
\draw[->] (G2.east) -- (add2.west);
\draw[<-] (add1.north) -- ++(0, 0.8)node[below right](n1){$n_1$};
\draw[<-] (add2.north) -- ++(0, 0.8)node[below right](n2){$n_2$};
\draw[->] (add1.east) -- (H1.west);
\draw[->] (add2.east) -- (H2.west);
\draw[->] (H1) -| (add.north);
\draw[->] (H2) -| (add.south);
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer]
\node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor};
\node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
\node[below right] at (sensor1.north west) {Sensor 1};
\node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
\node[below right] at (sensor2.north west) {Sensor 2};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:fusion_super_sensor
#+caption: Sensor Fusion Architecture ([[./figs/fusion_super_sensor.png][png]], [[./figs/fusion_super_sensor.pdf][pdf]], [[./figs/fusion_super_sensor.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_super_sensor.png]]
* Fig 2: Sensor fusion architecture with sensor dynamics uncertainty
#+begin_src latex :file sensor_fusion_dynamic_uncertainty.pdf :tangle figs/sensor_fusion_dynamic_uncertainty.tex
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[addb, above right=0.8 and 4 of x](add1){};
\node[addb, below right=0.8 and 4 of x](add2){};
\node[block, above left=0.2 and 0.1 of add1](delta1){$\Delta_1(s)$};
\node[block, above left=0.2 and 0.1 of add2](delta2){$\Delta_2(s)$};
\node[block, left=0.5 of delta1](W1){$w_1(s)$};
\node[block, left=0.5 of delta2](W2){$w_2(s)$};
\node[block, right=0.5 of add1](H1){$H_1(s)$};
\node[block, right=0.5 of add2](H2){$H_2(s)$};
\node[addb, right=6 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[->] (x.center) |- (add1.west);
\draw[->] (x.center) |- (add2.west);
\draw[->] ($(add1-|W1.west)+(-0.5, 0)$)node[branch](S1){} |- (W1.west);
\draw[->] ($(add2-|W2.west)+(-0.5, 0)$)node[branch](S1){} |- (W2.west);
\draw[->] (W1.east) -- (delta1.west);
\draw[->] (W2.east) -- (delta2.west);
\draw[->] (delta1.east) -| (add1.north);
\draw[->] (delta2.east) -| (add2.north);
\draw[->] (add1.east) -- (H1.west);
\draw[->] (add2.east) -- (H2.west);
\draw[->] (H1.east) -| (add.north);
\draw[->] (H2.east) -| (add.south);
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer]
\node[block, fit={($(W1.north-|S1)+(-0.2, 0.2)$) ($(add1.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
\node[above right] at (sensor1.south west) {Sensor 1};
\node[block, fit={($(W2.north-|S1)+(-0.2, 0.2)$) ($(add2.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
\node[above right] at (sensor2.south west) {Sensor 2};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty ([[./figs/sensor_fusion_dynamic_uncertainty.png][png]], [[./figs/sensor_fusion_dynamic_uncertainty.pdf][pdf]], [[./figs/sensor_fusion_dynamic_uncertainty.tex][tex]]).
#+RESULTS:
[[file:figs/sensor_fusion_dynamic_uncertainty.png]]
* Fig 3: Uncertainty set of the super sensor dynamics
#+begin_src latex :file uncertainty_set_super_sensor.pdf :tangle figs/uncertainty_set_super_sensor.tex :exports both
\begin{tikzpicture}
\begin{scope}[shift={(4, 0)}]
% Uncertainty Circle
\node[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
\path[draw, dotted] (0, 0) circle [radius=1.0];
\path[draw, dashed] (135:1.0) circle [radius=0.8];
% Center of Circle
\node[below] at (0, 0){$1$};
\draw[<->, dashed] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
\draw[<->, dashed] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(90:0.8);
\node[] (l1) at (2, 1.5) {$|w_1 H_1|$};
\draw[->, dashed, out=-90, in=0] (l1.south) to (r1);
\node[] (l2) at (-2.5, 1.5) {$|w_2 H_2|$};
\draw[->, dashed, out=0, in=-180] (l2.east) to (r2);
\draw[<->, dashed] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
\node[] (l3) at (-2.5, -1.5) {$|w_1 H_1| + |w_2 H_2|$};
\draw[->, dashed, out=90, in=-90] (l3.north) to (r3);
\end{scope}
% Real and Imaginary Axis
\draw[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
\draw[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};
\draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
\draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi$};
\end{tikzpicture}
#+end_src
#+name: fig:uncertainty_set_super_sensor
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (solid circle), of the sensor 1 (dotted circle) and of the sensor 2 (dashed circle) ([[./figs/uncertainty_set_super_sensor.png][png]], [[./figs/uncertainty_set_super_sensor.pdf][pdf]], [[./figs/uncertainty_set_super_sensor.tex][tex]]).
#+RESULTS:
[[file:figs/uncertainty_set_super_sensor.png]]
* Fig 4: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
#+begin_src latex :file h_infinity_robust_fusion.pdf :tangle figs/h_infinity_robust_fusion.tex :exports both
\begin{tikzpicture}
\node[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
\node[above] at (P.north) {$P(s)$};
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
\coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
\coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);
\node[block, left=1.4 of output1] (W1){$W_1(s)$};
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
\node[addb={+}{}{}{}{-}, left=of W1] (sub) {};
\node[block, below=0.3 of P] (H2) {$H_2(s)$};
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
\draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west);
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
\draw[->] (sub.east) -- (W1.west);
\draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
\draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
\end{tikzpicture}
#+end_src
#+name: fig:h_infinity_robust_fusion
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters ([[./figs/h_infinity_robust_fusion.png][png]], [[./figs/h_infinity_robust_fusion.pdf][pdf]], [[./figs/h_infinity_robust_fusion.tex][tex]]).
#+RESULTS:
[[file:figs/h_infinity_robust_fusion.png]]
* Fig 5: Magnitude of a weighting function generated using the proposed formula
#+begin_src matlab :exports none :results none
s = zpk('s');
freqs = logspace(-1, 2, 500);
n = 2;
w0 = 2*pi*10;
G0 = 1e-3;
G1 = 10;
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;
T = table(freqs', ...
abs(squeeze(freqresp(W, freqs, 'Hz'))), ...
'VariableNames', {'freqs', 'ampl'});
writetable(T, '../matlab/mat/weight_formula.csv');
#+end_src
#+begin_src latex :file weight_formula.pdf :tangle figs/weight_formula.tex :exports both
\setlength\fwidth{6.5cm}
\setlength\fheight{3.5cm}
\begin{tikzpicture}
\begin{axis}[%
width=1.0\fwidth,
height=1.0\fheight,
at={(0.0\fwidth, 0.0\fheight)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=100,
xtick={0.1,1,10, 100},
xminorticks=true,
ymode=log,
ymin=0.0005,
ymax=20,
ytick={0.001, 0.01, 0.1, 1, 10},
yminorticks=true,
ylabel={Magnitude},
xlabel={Frequency [Hz]},
xminorgrids,
yminorgrids,
]
\addplot [color=black, line width=1.5pt, forget plot]
table [x=freqs, y=ampl, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matweight_formula.csv};
\addplot [color=black, dashed, line width=1.5pt]
table[row sep=crcr]{%
1 10\\
100 10\\
};
\addplot [color=black, dashed, line width=1.5pt]
table[row sep=crcr]{%
0.1 0.001\\
3 0.001\\
};
\addplot [color=black, line width=1.5pt]
table[row sep=crcr]{%
0.1 1\\
100 1\\
};
\addplot [color=black, dashed, line width=1.5pt]
table[row sep=crcr]{%
10 2\\
10 1\\
};
\node[below] at (2, 10) {$G_\infty$};
\node[above] at (2, 0.001) {$G_0$};
\node[branch] at (10, 2){};
\draw[dashed, line cap=round] (7, 2) -- (20, 2) node[right]{$G_c$};
\draw[dashed, line cap=round] (10, 2) -- (10, 1) node[below]{$\omega_c$};
\node[right] at (3, 0.1) {$+n$};
\end{axis}
\end{tikzpicture}
#+end_src
#+name: fig:weight_formula
#+caption: Magnitude of a weighting function generated using the proposed formula ([[./figs/weight_formula.png][png]], [[./figs/weight_formula.pdf][pdf]], [[./figs/weight_formula.tex][tex]]).
#+RESULTS:
[[file:figs/weight_formula.png]]
* Fig 6: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
#+begin_src latex :file hinf_synthesis_results.pdf :tangle figs/hinf_synthesis_results.tex :exports both
\setlength\fwidth{6.5cm}
\setlength\fheight{6cm}
\begin{tikzpicture}
\begin{axis}[%
width=1.0\fwidth,
height=0.5\fheight,
at={(0.0\fwidth, 0.47\fheight)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=1000,
xtick={0.1, 1, 10, 100, 1000},
xticklabels={{}},
xminorticks=true,
ymode=log,
ymin=0.0005,
ymax=20,
ytick={0.001, 0.01, 0.1, 1, 10},
yminorticks=true,
ylabel={Magnitude},
xminorgrids,
yminorgrids,
]
\addplot [color=mycolor1, line width=1.5pt, forget plot]
table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
\addplot [color=mycolor2, line width=1.5pt, forget plot]
table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
\addplot [color=mycolor1, dashed, line width=1.5pt, forget plot]
table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};
\addplot [color=mycolor2, dashed, line width=1.5pt, forget plot]
table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};
\end{axis}
\begin{axis}[%
width=1.0\fwidth,
height=0.45\fheight,
at={(0.0\fwidth, 0.0\fheight)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=1000,
xtick={0.1, 1, 10, 100, 1000},
xminorticks=true,
xlabel={Frequency [Hz]},
ymin=-200,
ymax=200,
ytick={-180, -90, 0, 90, 180},
ylabel={Phase [deg]},
xminorgrids,
legend style={at={(1,1.1)}, outer sep=2pt , anchor=north east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}},
]
\addlegendimage{color=mycolor1, dashed, line width=1.5pt}
\addlegendentry{$W_1^{-1}$};
\addlegendimage{color=mycolor2, dashed, line width=1.5pt}
\addlegendentry{$W_2^{-1}$};
\addplot [color=mycolor1, line width=1.5pt]
table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
\addlegendentry{$H_1$};
\addplot [color=mycolor2, line width=1.5pt]
table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
\addlegendentry{$H_2$};
\end{axis}
\end{tikzpicture}
#+end_src
#+name: fig:hinf_synthesis_results
#+caption: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis ([[./figs/hinf_synthesis_results.png][png]], [[./figs/hinf_synthesis_results.pdf][pdf]], [[./figs/hinf_synthesis_results.tex][tex]]).
#+RESULTS:
[[file:figs/hinf_synthesis_results.png]]
* Fig 7: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
#+begin_src latex :file comp_filter_three_hinf.pdf :tangle figs/comp_filter_three_hinf.tex
\begin{tikzpicture}
\node[block={5.0cm}{3.5cm}, fill=black!20!white, dashed] (P) {};
\node[above] at (P.north) {$P(s)$};
\coordinate[] (inputw) at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.4!(P.north west) + (-0.7, 0)$);
\coordinate[] (output1) at ($(P.south east)!0.8!(P.north east) + (0.7, 0)$);
\coordinate[] (output2) at ($(P.south east)!0.55!(P.north east) + (0.7, 0)$);
\coordinate[] (output3) at ($(P.south east)!0.3!(P.north east) + (0.7, 0)$);
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + (0.7, 0)$);
\node[block, left=1.4 of output1] (W1){$W_1(s)$};
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
\node[block, left=1.4 of output3] (W3){$W_3(s)$};
\node[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
\node[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
\node[block, below=0.3 of P] (H) {$\begin{bmatrix}H_2(s) \\ H_3(s)\end{bmatrix}$};
\draw[->] (inputw) node[above right](w){$w$} -- (sub2.west);
\draw[->] (W3-|sub1)node[branch]{} -- (sub1.south);
\draw[->] (W2-|sub2)node[branch]{} -- (sub2.south);
\draw[->] ($(sub2.west)+(-0.5, 0)$) node[branch]{} |- (outputv) |- (H.east);
\draw[->] ($(H.south west)!0.7!(H.north west)$) -| (inputu|-W2) -- (W2.west);
\draw[->] ($(H.south west)!0.3!(H.north west)$) -| ($(inputu|-W3)+(0.4, 0)$) -- (W3.west);
\draw[->] (sub2.east) -- (sub1.west);
\draw[->] (sub1.east) -- (W1.west);
\draw[->] (W1.east) -- (output1)node[above left](z){$z_1$};
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
\draw[->] (W3.east) -- (output3)node[above left]{$z_3$};
\node[above] at (W2-|w){$u_1$};
\node[above] at (W3-|w){$u_2$};
\node[above] at (outputv-|z){$v$};
\end{tikzpicture}
#+end_src
#+name: fig:comp_filter_three_hinf
#+caption: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters ([[./figs/comp_filter_three_hinf.png][png]], [[./figs/comp_filter_three_hinf.pdf][pdf]], [[./figs/comp_filter_three_hinf.tex][tex]]).
#+RESULTS:
[[file:figs/comp_filter_three_hinf.png]]
* Fig 8: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
#+begin_src latex :file hinf_three_synthesis_results.pdf :tangle figs/hinf_three_synthesis_results.tex :exports both
\setlength\fwidth{6.5cm}
\setlength\fheight{6cm}
\begin{tikzpicture}
\begin{axis}[%
width=1.0\fwidth,
height=0.55\fheight,
at={(0.0\fwidth, 0.42\fheight)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=100,
xticklabels={{}},
xminorticks=true,
ymode=log,
ymin=0.0005,
ymax=20,
ytick={0.001, 0.01, 0.1, 1, 10},
yminorticks=true,
ylabel={Magnitude},
xminorgrids,
yminorgrids,
legend columns=2,
legend style={
/tikz/column 2/.style={
column sep=5pt,
},
at={(1,0)}, outer sep=2pt , anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}
},
]
\addplot [color=mycolor1, dashed, line width=1.5pt]
table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
\addlegendentry{${W_1}^{-1}$};
\addplot [color=mycolor1, line width=1.5pt]
table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
\addlegendentry{$H_1$};
\addplot [color=mycolor2, dashed, line width=1.5pt]
table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
\addlegendentry{${W_2}^{-1}$};
\addplot [color=mycolor2, line width=1.5pt]
table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
\addlegendentry{$H_2$};
\addplot [color=mycolor3, dashed, line width=1.5pt]
table [x=freqs, y=W3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
\addlegendentry{${W_3}^{-1}$};
\addplot [color=mycolor3, line width=1.5pt]
table [x=freqs, y=H3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
\addlegendentry{$H_3$};
\end{axis}
\begin{axis}[%
width=1.0\fwidth,
height=0.4\fheight,
at={(0.0\fwidth, 0.0\fheight)},
scale only axis,
xmode=log,
xmin=0.1,
xmax=100,
xminorticks=true,
xlabel={Frequency [Hz]},
ymin=-240,
ymax=240,
ytick={-180, -90, 0, 90, 180},
ylabel={Phase [deg]},
xminorgrids,
]
\addplot [color=mycolor1, line width=1.5pt]
table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
\addplot [color=mycolor2, line width=1.5pt]
table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
\addplot [color=mycolor3, line width=1.5pt]
table [x=freqs, y=H3p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
\end{axis}
\end{tikzpicture}
#+end_src
#+name: fig:hinf_three_synthesis_results
#+caption: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis ([[./figs/hinf_three_synthesis_results.png][png]], [[./figs/hinf_three_synthesis_results.pdf][pdf]], [[./figs/hinf_three_synthesis_results.tex][tex]]).
#+RESULTS:
[[file:figs/hinf_three_synthesis_results.png]]
* Fig 9: Specifications and weighting functions magnitude used for $\mathcal{H}_\infty$ synthesis
#+begin_src latex :file ligo_weights.pdf :tangle figs/ligo_weights.tex :exports both
\setlength\fwidth{6.5cm}
\setlength\fheight{3.2cm}
\begin{tikzpicture}
\begin{axis}[%
width=1.0\fwidth,
height=1.0\fheight,
at={(0.0\fwidth, 0.0\fheight)},
scale only axis,
separate axis lines,
every outer x axis line/.append style={black},
every x tick label/.append style={font=\color{black}},
every x tick/.append style={black},
xmode=log,
xmin=0.001,
xmax=1,
xminorticks=true,
xlabel={Frequency [Hz]},
every outer y axis line/.append style={black},
every y tick label/.append style={font=\color{black}},
every y tick/.append style={black},
ymode=log,
ymin=0.005,
ymax=20,
yminorticks=true,
ylabel={Magnitude},
axis background/.style={fill=white},
xmajorgrids,
xminorgrids,
ymajorgrids,
yminorgrids,
legend style={at={(0,1)}, outer sep=2pt, anchor=north west, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
]
\addplot [color=mycolor1, line width=1.5pt]
table [x=freqs, y=wHm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
\addlegendentry{$|w_H|^{-1}$}
\addplot [color=mycolor2, line width=1.5pt]
table [x=freqs, y=wLm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
\addlegendentry{$|w_L|^{-1}$}
\addplot [color=black, dotted, line width=1.5pt]
table[row sep=crcr]{%
0.0005 0.008\\
0.008 0.008\\
};
\addlegendentry{Specifications}
\addplot [color=black, dotted, line width=1.5pt, forget plot]
table[row sep=crcr]{%
0.008 0.008\\
0.04 1\\
};
\addplot [color=black, dotted, line width=1.5pt, forget plot]
table[row sep=crcr]{%
0.04 3\\
0.1 3\\
};
\addplot [color=black, dotted, line width=1.5pt]
table[row sep=crcr]{%
0.1 0.045\\
2 0.045\\
};
\end{axis}
\end{tikzpicture}
#+end_src
#+name: fig:ligo_weights
#+caption: Specifications and weighting functions magnitude used for $\mathcal{H}_\infty$ synthesis ([[./figs/ligo_weights.png][png]], [[./figs/ligo_weights.pdf][pdf]], [[./figs/ligo_weights.tex][tex]]).
#+RESULTS:
[[file:figs/ligo_weights.png]]
* Fig 10: Comparison of the FIR filters (solid) with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed)
#+begin_src latex :file comp_fir_ligo_hinf.pdf :tangle figs/comp_fir_ligo_hinf.tex :exports both
\setlength\fwidth{6.5cm}
\setlength\fheight{6.8cm}
\begin{tikzpicture}
\begin{axis}[%
width=1.0\fwidth,
height=0.60\fheight,
at={(0.0\fwidth, 0.32\fheight)},
scale only axis,
xmode=log,
xmin=0.001,
xmax=1,
xtick={0.001,0.01,0.1,1},
xticklabels={{}},
xminorticks=true,
ymode=log,
ymin=0.002,
ymax=5,
ytick={0.001, 0.01, 0.1, 1, 10},
yminorticks=true,
ylabel={Magnitude},
xminorgrids,
yminorgrids,
legend style={at={(1,0)}, outer sep=2pt, anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
]
\addplot [color=mycolor1, line width=1.5pt]
table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
\addlegendentry{$H_H(s)$ - $\mathcal{H}_\infty$}
\addplot [color=mycolor1, dashed, line width=1.5pt]
table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
\addlegendentry{$H_H(s)$ - FIR}
\addplot [color=mycolor2, line width=1.5pt]
table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
\addlegendentry{$H_L(s)$ - $\mathcal{H}_\infty$}
\addplot [color=mycolor2, dashed, line width=1.5pt]
table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
\addlegendentry{$H_L(s)$ - FIR}
\end{axis}
\begin{axis}[%
width=1.0\fwidth,
height=0.3\fheight,
at={(0.0\fwidth, 0.0\fheight)},
scale only axis,
xmode=log,
xmin=0.001,
xmax=1,
xtick={0.001, 0.01, 0.1, 1},
xminorticks=true,
xlabel={Frequency [Hz]},
ymin=-180,
ymax=180,
ytick={-180, -90, 0, 90, 180},
ylabel={Phase [deg]},
xminorgrids,
]
\addplot [color=mycolor1, line width=1.5pt, forget plot]
table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
\addplot [color=mycolor1, dashed, line width=1.5pt, forget plot]
table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
\addplot [color=mycolor2, line width=1.5pt, forget plot]
table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
\addplot [color=mycolor2, dashed, line width=1.5pt, forget plot]
table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
\end{axis}
\end{tikzpicture}
#+end_src
#+name: fig:comp_fir_ligo_hinf
#+caption: Comparison of the FIR filters (solid) with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed) ([[./figs/comp_fir_ligo_hinf.png][png]], [[./figs/comp_fir_ligo_hinf.pdf][pdf]], [[./figs/comp_fir_ligo_hinf.tex][tex]]).
#+RESULTS:
[[file:figs/comp_fir_ligo_hinf.png]]