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diff --git a/journal/figs/three_complementary_filters_results.png b/journal/figs/three_complementary_filters_results.png
index 7ffe7cb..06dcaea 100644
Binary files a/journal/figs/three_complementary_filters_results.png and b/journal/figs/three_complementary_filters_results.png differ
diff --git a/journal/journal.org b/journal/journal.org
index cd57fc2..e060256 100644
--- a/journal/journal.org
+++ b/journal/journal.org
@@ -596,20 +596,42 @@ A more complex real life example is taken up in the next section.
* Application: Design of Complementary Filters used in the Active Vibration Isolation System at the 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.
+Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO cite:matichard15_seism_isolat_advan_ligo,hua05_low_ligo, the VIRGO cite:lucia18_low_frequen_optim_perfor_advan,heijningen18_low and the KAGRA cite:akutsu21_vibrat_isolat_system_beam_split.
+
+In the first isolation stage at the LIGO, two sets of complementary filters are used and included in a feedback loop cite:hua04_low_ligo.
+A set of complementary filters ($L_2,H_2$) is first used to fuse a seismometer and a geophone.
+Then, another set of complementary filters ($L_1,H_1$) is used to merge the output of the first "inertial super sensor" with a position sensor.
+A simplified block diagram of the sensor fusion architecture is shown in Figure ref:fig:ligo_super_sensor_architecture.
+
+#+name: fig:ligo_super_sensor_architecture
+#+caption: Simplified block diagram of the sensor blending strategy for the first stage at the LIGO cite:hua04_low_ligo
+#+attr_latex: :scale 1
+[[file:figs/ligo_super_sensor_architecture.pdf]]
+
+The fusion of the position sensor at low frequency with the "inertial super sensor" at high frequency using the complementary filters ($L_1,H_1$) is done for several reasons, first of which is to give the super sensor a DC sensibility and therefore allow the feedback loop to have authority at zero frequency.
+The requirements on those filters are very tight and thus their design is complex and should be expressed as an optimization problem.
+
+The approach used in cite:hua05_low_ligo is to use FIR complementary filters and to write the synthesis as a convex optimization problem.
+After synthesis, the obtained FIR filters were found to be compliant with the requirements.
+However they are of very high order so their implementation is quite complex.
+In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated on the same set of requirements.
** Complementary Filters 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 $0.045$
+The specifications for the set of complementary filters ($L_1,H_1$) used at the LIGO are summarized below (for further details, refer to cite:hua04_polyp_fir_compl_filter_contr_system):
+- From $0$ to $\SI{0.008}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less or equal to $8 \times 10^{-4}$
+- Between $\SI{0.008}{Hz}$ to $\SI{0.04}{Hz}$, the filter $L_1(s)$ should attenuate the input signal proportional to frequency cubed
+- Between $\SI{0.04}{Hz}$ to $\SI{0.1}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less than $3$
+- Above $\SI{0.1}{Hz}$, the magnitude $|H_1(j\omega)|$ should be less than $0.045$
+
+These specifications are therefore upper bounds on the complementary filters' magnitudes.
+They are physically represented in Figure ref:fig:fir_filter_ligo as well as the obtained magnitude of the FIR filters in cite:hua05_low_ligo.
+
+#+name: fig:fir_filter_ligo
+#+caption: Specifications and Bode plot of the obtained FIR filters in cite:hua05_low_ligo
+#+attr_latex: :scale 1
+[[file:figs/fir_filter_ligo.pdf]]
** Weighting Functions Design
<>
diff --git a/journal/journal.pdf b/journal/journal.pdf
index 7d23240..1eda9ff 100644
Binary files a/journal/journal.pdf and b/journal/journal.pdf differ
diff --git a/journal/journal.tex b/journal/journal.tex
index 7316f21..13ae9df 100644
--- a/journal/journal.tex
+++ b/journal/journal.tex
@@ -1,4 +1,4 @@
-% Created 2021-05-20 jeu. 11:13
+% Created 2021-05-20 jeu. 16:26
% Intended LaTeX compiler: pdflatex
\documentclass[preprint, sort&compress]{elsarticle}
\usepackage[utf8]{inputenc}
@@ -58,7 +58,7 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
\end{frontmatter}
\section{Introduction}
-\label{sec:org810a8ae}
+\label{sec:orgc28ff6a}
\label{sec:introduction}
\begin{itemize}
\item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion
@@ -105,13 +105,13 @@ Most of the requirements => shape of the complementary filters
=> propose a way to shape complementary filters.
\section{Sensor Fusion and Complementary Filters Requirements}
-\label{sec:orge97289b}
+\label{sec:org5a2a4a4}
\label{sec:requirements}
Complementary filters provides a framework for fusing signals from different sensors.
As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
These requirements are discussed in this section.
\subsection{Sensor Fusion Architecture}
-\label{sec:org0a17c4e}
+\label{sec:orgb29eeb0}
\label{sec:sensor_fusion}
A general sensor fusion architecture using complementary filters is shown in Figure \ref{fig:sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\).
@@ -138,7 +138,7 @@ Therefore, a pair of strict complementary filter needs to satisfy the following
It will soon become clear why the complementary property is important.
\subsection{Sensor Models and Sensor Normalization}
-\label{sec:org52d9122}
+\label{sec:org749994d}
\label{sec:sensor_models}
In order to study such sensor fusion architecture, a model of the sensors is required.
@@ -187,7 +187,7 @@ The super sensor output is therefore equal to:
\end{figure}
\subsection{Noise Sensor Filtering}
-\label{sec:orgf08876b}
+\label{sec:org0f07eb6}
\label{sec:noise_filtering}
In this section, it is supposed that all the sensors are perfectly calibrated, such that:
@@ -227,7 +227,7 @@ In such case, to lower the noise of the super sensor, the value of the norm \(|H
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
\subsection{Sensor Fusion Robustness}
-\label{sec:org3269d9e}
+\label{sec:orgdb0117a}
\label{sec:fusion_robustness}
In practical systems the sensor normalization is not perfect and condition \eqref{eq:perfect_dynamics} is not verified.
@@ -289,14 +289,14 @@ As it is generally desired to limit the maximum phase added by the super sensor,
Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain.
\section{Complementary Filters Shaping}
-\label{sec:orga11797e}
+\label{sec:orgba2ddcd}
\label{sec:hinf_method}
As shown in Section \ref{sec:requirements}, the noise and robustness of the ``super sensor'' are determined by the complementary filters norms.
Therefore, a complementary filters synthesis method that allows to shape their norms would be of great use.
In this section, such synthesis is proposed by expressing this problem as a \(\mathcal{H}_\infty\) norm optimization.
\subsection{Synthesis Objective}
-\label{sec:orgc196ed6}
+\label{sec:orgef373f7}
\label{sec:synthesis_objective}
The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:comp_filter}.
@@ -313,7 +313,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
where \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are chosen to specify the maximum wanted norms of the complementary filters during the synthesis.
\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
-\label{sec:orgf037f4b}
+\label{sec:org9d293ca}
\label{sec:hinf_synthesis}
In this section, it is shown that the synthesis objective can be easily expressed as a standard \(\mathcal{H}_\infty\) optimal control problem and therefore solved using convenient tools readily available.
@@ -354,7 +354,7 @@ Therefore, applying the \(\mathcal{H}_\infty\) synthesis on the standard plant \
The above optimization problem can be efficiently solved in Matlab \cite{matlab20} using the Robust Control Toolbox.
\subsection{Weighting Functions Design}
-\label{sec:org80ebc41}
+\label{sec:org552a374}
\label{sec:hinf_weighting_func}
Weighting functions are used during the synthesis to specify what is the maximum allowed norms of the complementary filters.
@@ -404,7 +404,7 @@ The typical shape of a weighting function generated using \eqref{eq:weight_formu
\end{figure}
\subsection{Validation of the proposed synthesis method}
-\label{sec:orge831635}
+\label{sec:orgdf78400}
\label{sec:hinf_example}
The proposed methodology for the design of complementary filters is now applied on a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
@@ -465,27 +465,50 @@ This simple example illustrates the fact that the proposed methodology for compl
A more complex real life example is taken up in the next section.
\section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
-\label{sec:org5405518}
+\label{sec:orgd51fb42}
\label{sec:application_ligo}
-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.
+Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO \cite{matichard15_seism_isolat_advan_ligo,hua05_low_ligo}, the VIRGO \cite{lucia18_low_frequen_optim_perfor_advan,heijningen18_low} and the KAGRA \cite{akutsu21_vibrat_isolat_system_beam_split}.
-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}.
+In the first isolation stage at the LIGO, two sets of complementary filters are used and included in a feedback loop \cite{hua04_low_ligo}.
+A set of complementary filters (\(L_2,H_2\)) is first used to fuse a seismometer and a geophone.
+Then, another set of complementary filters (\(L_1,H_1\)) is used to merge the output of the first ``inertial super sensor'' with a position sensor.
+A simplified block diagram of the sensor fusion architecture is shown in Figure \ref{fig:ligo_super_sensor_architecture}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,scale=1]{figs/ligo_super_sensor_architecture.pdf}
+\caption{\label{fig:ligo_super_sensor_architecture}Simplified block diagram of the sensor blending strategy for the first stage at the LIGO \cite{hua04_low_ligo}}
+\end{figure}
+
+The fusion of the position sensor at low frequency with the ``inertial super sensor'' at high frequency using the complementary filters (\(L_1,H_1\)) is done for several reasons, first of which is to give the super sensor a DC sensibility and therefore allow the feedback loop to have authority at zero frequency.
+The requirements on those filters are very tight and thus their design is complex and should be expressed as an optimization problem.
+
+The approach used in \cite{hua05_low_ligo} is to use FIR complementary filters and to write the synthesis as a convex optimization problem.
+After synthesis, the obtained FIR filters were found to be compliant with the requirements.
+However they are of very high order so their implementation is quite complex.
+In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated on the same set of requirements.
\subsection{Complementary Filters Specifications}
-\label{sec:orgd3d9b91}
+\label{sec:org45a6d0a}
\label{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}:
+The specifications for the set of complementary filters (\(L_1,H_1\)) used at the LIGO are summarized below (for further details, refer to \cite{hua04_polyp_fir_compl_filter_contr_system}):
\begin{itemize}
-\item From \(0\) to \(\SI{0.008}{Hz}\), the magnitude of the filter's transfer function should be less or equal to \(8 \times 10^{-4}\)
-\item Between \(\SI{0.008}{Hz}\) to \(\SI{0.04}{Hz}\), the filter should attenuate the input signal proportional to frequency cubed
-\item Between \(\SI{0.04}{Hz}\) to \(\SI{0.1}{Hz}\), the magnitude of the transfer function should be less than \(3\)
-\item Above \(\SI{0.1}{Hz}\), the magnitude of the complementary filter should be less than \(0.045\)
+\item From \(0\) to \(\SI{0.008}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less or equal to \(8 \times 10^{-4}\)
+\item Between \(\SI{0.008}{Hz}\) to \(\SI{0.04}{Hz}\), the filter \(L_1(s)\) should attenuate the input signal proportional to frequency cubed
+\item Between \(\SI{0.04}{Hz}\) to \(\SI{0.1}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less than \(3\)
+\item Above \(\SI{0.1}{Hz}\), the magnitude \(|H_1(j\omega)|\) should be less than \(0.045\)
\end{itemize}
+These specifications are therefore upper bounds on the complementary filters' magnitudes.
+They are physically represented in Figure \ref{fig:fir_filter_ligo} as well as the obtained magnitude of the FIR filters in \cite{hua05_low_ligo}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,scale=1]{figs/fir_filter_ligo.pdf}
+\caption{\label{fig:fir_filter_ligo}Specifications and Bode plot of the obtained FIR filters in \cite{hua05_low_ligo}}
+\end{figure}
+
\subsection{Weighting Functions Design}
-\label{sec:orgcda22a3}
+\label{sec:orgeca6c60}
\label{sec:ligo_weights}
The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem.
However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters.
@@ -501,7 +524,7 @@ The magnitudes of the weighting functions are shown in Fig. \ref{fig:ligo_weight
\end{figure}
\subsection{\(\mathcal{H}_\infty\) Synthesis}
-\label{sec:org5fc1ebf}
+\label{sec:orgcaaff49}
\label{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 \(27\).
@@ -515,9 +538,9 @@ They are found to be very close to each other and this shows the effectiveness o
\end{figure}
\section{Discussion}
-\label{sec:org2363233}
+\label{sec:orga827f5f}
\subsection{Alternative configuration}
-\label{sec:org164637d}
+\label{sec:org7abd40b}
\begin{itemize}
\item Feedback architecture : Similar to mixed sensitivity (add schematic of feedback loop with weights)
\item 2 inputs / 1 output
@@ -526,13 +549,13 @@ They are found to be very close to each other and this shows the effectiveness o
Explain differences
\subsection{Imposing zero at origin / roll-off}
-\label{sec:orgf116368}
+\label{sec:orge105bdf}
3 methods:
Link to literature about doing that with mixed sensitivity
\subsection{Synthesis of Three Complementary Filters}
-\label{sec:orgbdd0317}
+\label{sec:org3fd7562}
\label{sec:hinf_three_comp_filters}
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.
@@ -570,7 +593,7 @@ The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:
\end{figure}
\section{Conclusion}
-\label{sec:org150800b}
+\label{sec:orgc3f0120}
\label{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.
@@ -578,7 +601,7 @@ Therefore, a synthesis method that permits the shaping of the complementary filt
Future work will aim at further developing this synthesis method for the robust and optimal synthesis of complementary filters used in sensor fusion.
\section*{Acknowledgment}
-\label{sec:org6c6a099}
+\label{sec:org1be2c18}
This research benefited from a FRIA grant from the French Community of Belgium.
\bibliographystyle{elsarticle-num}
diff --git a/journal/ref.bib b/journal/ref.bib
index ab551cf..2f2471a 100644
--- a/journal/ref.bib
+++ b/journal/ref.bib
@@ -93,19 +93,6 @@
series = {Robot Control 1991},
}
-@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,
- doi = {10.1002/j.2161-4296.1972.tb01706.x},
- url = {https://doi.org/10.1002/j.2161-4296.1972.tb01706.x},
-}
-
@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
@@ -471,7 +458,7 @@
number = 3,
pages = {321--325},
year = 1975,
- doi = 10.1109/TAES.1975.308081,
+ doi = {10.1109/TAES.1975.308081},
url = {https://doi.org/10.1109/TAES.1975.308081},
keywords = {sensor fusion, complementary filters},
publisher = {IEEE},
@@ -488,8 +475,8 @@
}
@inproceedings{becker15_compl_filter_desig_three_frequen_bands,
- author = {Becker, Thiago; Fabro, Joao Alberto; Oliveira, Andre
- Schneider de; Reis, Luis Paulo},
+ author = {Becker, Thiago and Fabro, Joao Alberto and Oliveira, Andre
+ Schneider de and Reis, Luis Paulo},
title = {Complementary Filter Design with Three Frequency Bands:
Robot Attitude Estimation},
booktitle = {International Conference on Autonomous Robot Systems and
@@ -538,7 +525,7 @@
number = 5,
pages = {483--497},
year = 2001,
- doi = 10.1177/095965180121500505,
+ doi = {10.1177/095965180121500505},
url = {https://doi.org/10.1177/095965180121500505},
keywords = {sensor fusion, complementary filters},
publisher = {SAGE Publications Sage UK: London, England},
@@ -554,3 +541,48 @@
url = {https://doi.org/10.1109/tct.1957.1086345},
page = {--},
}
+
+@inproceedings{hua04_low_ligo,
+ author = {Hua, Wensheng and Adhikari, R and DeBra, Daniel B and
+ Giaime, Joseph A and Hammond, Giles Dominic and Hardham, C and
+ Hennessy, Mike and How, Jonathan P and Lantz, Brian T and
+ Macinnis, M and others},
+ title = {Low-frequency active vibration isolation for advanced LIGO},
+ booktitle = {Gravitational Wave and Particle Astrophysics Detectors},
+ year = 2004,
+ volume = 5500,
+ pages = {194--205},
+ organization = {International Society for Optics and Photonics},
+}
+
+@article{akutsu21_vibrat_isolat_system_beam_split,
+ author = {Akutsu, T and Ando, M and Arai, K and Arai, Y and Araki, S
+ and Araya, A and Aritomi, N and Asada, H and Aso, Y and Bae, S
+ and others},
+ title = {Vibration Isolation Systems for the Beam Splitter and
+ Signal Recycling Mirrors of the Kagra Gravitational Wave
+ Detector},
+ journal = {Classical and Quantum Gravity},
+ volume = 38,
+ number = 6,
+ pages = 065011,
+ year = 2021,
+ publisher = {IOP Publishing},
+}
+
+@phdthesis{heijningen18_low,
+ author = {van Heijningen, JV},
+ school = {Vrije Universiteit},
+ title = {Low-frequency performance improvement of seismic
+ attenuation systems and vibration sensors for next generation
+ gravitational wave detectors},
+ year = 2018,
+}
+
+@phdthesis{lucia18_low_frequen_optim_perfor_advan,
+ author = {Trozzo Lucia},
+ school = {University of Siena},
+ title = {Low Frequency Optimization and Performance of Advanced
+ Virgo Seismic Isolation System},
+ year = 2018,
+}
diff --git a/matlab/index.org b/matlab/index.org
index 5190a0a..ebcceab 100644
--- a/matlab/index.org
+++ b/matlab/index.org
@@ -292,7 +292,7 @@ xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/hinf_filters_results.pdf', 'width', 'wide', 'height', 'tall');
+exportFig('figs/hinf_filters_results.pdf', 'width', 'wide', 'height', 600);
#+end_src
#+name: fig:hinf_filters_results
@@ -492,14 +492,14 @@ plot(freqs, 180/pi*phase(squeeze(freqresp(H3, freqs, 'Hz'))));
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
-yticks([-360:90:360]); ylim([-270, 270]);
+yticks([-180:90:180]); ylim([-220, 220]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/three_complementary_filters_results.pdf', 'width', 'wide', 'height', 'tall');
+exportFig('figs/three_complementary_filters_results.pdf', 'width', 'wide', 'height', 600);
#+end_src
#+name: fig:three_complementary_filters_results
@@ -739,30 +739,39 @@ tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
-plot(w, abs(H), 'k-');
-plot(w, abs(1-H), 'k--');
-hold off;
+set(gca,'ColorOrderIndex',1)
+plot(w, abs(1-H), '-', 'DisplayName', '$L_1$');
+plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
+
+set(gca,'ColorOrderIndex',2)
+plot(w, abs(H), '-', 'DisplayName', '$H_1$');
+plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
+plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
+plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
+
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-ylabel('Magnitude');
-set(gca, 'XTickLabel',[]);
-ylim([5e-3, 5]);
+set(gca, 'XTickLabel',[]); ylabel('Magnitude');
+hold off;
+ylim([5e-3, 10]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 16;
% Phase
ax2 = nexttile;
hold on;
-plot(w, 180/pi*unwrap(angle(H)), 'k-');
-plot(w, 180/pi*unwrap(angle(1-H)), 'k--');
+plot(w, 180/pi*unwrap(angle(1-H)), '-');
+plot(w, 180/pi*unwrap(angle(H)), '-');
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
-yticks([-450:90:180]); ylim([-450, 200]);
+yticks([-360:180:180]); ylim([-380, 200]);
linkaxes([ax1,ax2],'x');
xlim([1e-3, 1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/fir_filter_ligo.pdf', 'width', 'wide', 'height', 'tall');
+exportFig('figs/fir_filter_ligo.pdf', 'width', 'wide', 'height', 600);
#+end_src
#+name: fig:fir_filter_ligo
@@ -812,22 +821,22 @@ wL = minreal(ss(wL));
figure;
hold on;
set(gca,'ColorOrderIndex',1);
-plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|w_H|^{-1}$');
-set(gca,'ColorOrderIndex',2);
-plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|w_L|^{-1}$');
+plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|W_L|^{-1}$');
+plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
-plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec.');
-plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
-plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
-plot([0.1, 10], [0.045, 0.045], 'k--', 'HandleVisibility', 'off');
+set(gca,'ColorOrderIndex',2);
+plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|W_H|^{-1}$');
+plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
+plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
+plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
-ylim([1e-3, 10]);
-leg = legend('location', 'southeast', 'FontSize', 8);
-leg.ItemTokenSize(1) = 18;
+ylim([5e-3, 10]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 16;
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
@@ -905,27 +914,22 @@ The bode plot of the obtained filters as shown on figure [[fig:hinf_synthesis_li
figure;
hold on;
set(gca,'ColorOrderIndex',1);
-plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
-set(gca,'ColorOrderIndex',1);
-plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
-set(gca,'ColorOrderIndex',1);
-plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
+plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'DisplayName', '$L_1^\prime$');
+plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
set(gca,'ColorOrderIndex',2);
-plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
-
-set(gca,'ColorOrderIndex',1);
-plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$H_H$');
-set(gca,'ColorOrderIndex',2);
-plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'DisplayName', '$H_L$');
+plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$H_1^\prime$');
+plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
+plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
+plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
-ylim([1e-3, 10]);
-leg = legend('location', 'southeast', 'FontSize', 8);
-leg.ItemTokenSize(1) = 18;
+ylim([5e-3, 10]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 16;
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
@@ -948,18 +952,18 @@ tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2, 1]);
hold on;
set(gca,'ColorOrderIndex',1);
-plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', ...
- 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$');
-set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', ...
- 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$');
+ 'DisplayName', '$L_1(s)$ - $\mathcal{H}_\infty$');
+set(gca,'ColorOrderIndex',2);
+plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', ...
+ 'DisplayName', '$H_1(s)$ - $\mathcal{H}_\infty$');
set(gca,'ColorOrderIndex',1);
-plot(w, abs(H), '--', ...
- 'DisplayName', '$H_H(s)$ - FIR');
-set(gca,'ColorOrderIndex',2);
plot(w, abs(1-H), '--', ...
- 'DisplayName', '$H_L(s)$ - FIR');
+ 'DisplayName', '$L_1(s)$ - FIR');
+set(gca,'ColorOrderIndex',2);
+plot(w, abs(H), '--', ...
+ 'DisplayName', '$H_1(s)$ - FIR');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude');
@@ -972,25 +976,25 @@ leg.ItemTokenSize(1) = 16;
ax2 = nexttile;
hold on;
set(gca,'ColorOrderIndex',1);
-plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hh, freqs, 'Hz')))), '-');
-set(gca,'ColorOrderIndex',2);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hl, freqs, 'Hz')))), '-');
+set(gca,'ColorOrderIndex',2);
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hh, freqs, 'Hz')))), '-');
set(gca,'ColorOrderIndex',1);
-plot(w, 180/pi*unwrap(angle(H)), '--');
-set(gca,'ColorOrderIndex',2);
plot(w, 180/pi*unwrap(angle(1-H)), '--');
+set(gca,'ColorOrderIndex',2);
+plot(w, 180/pi*unwrap(angle(H)), '--');
set(gca, 'XScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
-yticks([-450:90:180]); ylim([-450, 200]);
+yticks([-360:180:180]); ylim([-380, 200]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/comp_fir_ligo_hinf.pdf', 'width', 'wide', 'height', 'tall');
+exportFig('figs/comp_fir_ligo_hinf.pdf', 'width', 'wide', 'height', 600);
#+end_src
#+name: fig:comp_fir_ligo_hinf
diff --git a/matlab/matlab/comp_filters_ligo.m b/matlab/matlab/comp_filters_ligo.m
index fc1c8f9..77b8f17 100644
--- a/matlab/matlab/comp_filters_ligo.m
+++ b/matlab/matlab/comp_filters_ligo.m
@@ -6,6 +6,8 @@ s = zpk('s');
freqs = logspace(-3, 0, 1000);
+addpath('./src');
+
% Specifications
% The specifications for the filters are:
% 1. From $0$ to $0.008\text{ Hz}$,the magnitude of the filter’s transfer function should be less than or equal to $8 \times 10^{-3}$
@@ -31,7 +33,8 @@ xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([1e-4, 10]);
-legend('location', 'northeast');
+leg = legend('location', 'southeast', 'FontSize', 8);
+leg.ItemTokenSize(1) = 18;
% FIR Filter
% We here try to implement the FIR complementary filter synthesis as explained in cite:hua05_low_ligo.
@@ -184,8 +187,10 @@ w = [w1 w2 w3 w4];
H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
-ax1 = subplot(2,1,1);
+% Magnitude
+ax1 = nexttile([2, 1]);
hold on;
plot(w, abs(H), 'k-');
plot(w, abs(1-H), 'k--');
@@ -195,18 +200,18 @@ ylabel('Magnitude');
set(gca, 'XTickLabel',[]);
ylim([5e-3, 5]);
-ax2 = subplot(2,1,2);
+% Phase
+ax2 = nexttile;
hold on;
-plot(w, 180/pi*angle(H), 'k-');
-plot(w, 180/pi*angle(1-H), 'k--');
+plot(w, 180/pi*unwrap(angle(H)), 'k-');
+plot(w, 180/pi*unwrap(angle(1-H)), 'k--');
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
-yticks([-540:90:360]);
+yticks([-450:90:180]); ylim([-450, 200]);
linkaxes([ax1,ax2],'x');
xlim([1e-3, 1]);
-xticks([0.01, 0.1, 1, 10, 100, 1000]);
% Weights
% We design weights that will be used for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
@@ -260,7 +265,8 @@ xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([1e-3, 10]);
-legend('location', 'southeast');
+leg = legend('location', 'southeast', 'FontSize', 8);
+leg.ItemTokenSize(1) = 18;
% H-Infinity Synthesis
% We define the generalized plant as shown on figure [[fig:h_infinity_robst_fusion]].
@@ -349,47 +355,56 @@ xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([1e-3, 10]);
-legend('location', 'southeast');
+leg = legend('location', 'southeast', 'FontSize', 8);
+leg.ItemTokenSize(1) = 18;
% Compare FIR and H-Infinity Filters
% Let's now compare the FIR filters designed in cite:hua05_low_ligo and the one obtained with the $\mathcal{H}_\infty$ synthesis on figure [[fig:comp_fir_ligo_hinf]].
figure;
-ax1 = subplot(2,1,1);
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+% Magnitude
+ax1 = nexttile([2, 1]);
hold on;
set(gca,'ColorOrderIndex',1);
-plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-');
+plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', ...
+ 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$');
set(gca,'ColorOrderIndex',2);
-plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-');
+plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', ...
+ 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$');
set(gca,'ColorOrderIndex',1);
-plot(w, abs(H), '--');
+plot(w, abs(H), '--', ...
+ 'DisplayName', '$H_H(s)$ - FIR');
set(gca,'ColorOrderIndex',2);
-plot(w, abs(1-H), '--');
+plot(w, abs(1-H), '--', ...
+ 'DisplayName', '$H_L(s)$ - FIR');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude');
set(gca, 'XTickLabel',[]);
-ylim([1e-3, 10]);
+ylim([5e-3, 10]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 16;
-ax2 = subplot(2,1,2);
+% Phase
+ax2 = nexttile;
hold on;
set(gca,'ColorOrderIndex',1);
-plot(freqs, 180/pi*angle(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$\mathcal{H}_\infty$ filters');
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hh, freqs, 'Hz')))), '-');
set(gca,'ColorOrderIndex',2);
-plot(freqs, 180/pi*angle(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'HandleVisibility', 'off');
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hl, freqs, 'Hz')))), '-');
set(gca,'ColorOrderIndex',1);
-plot(w, 180/pi*angle(H), '--', 'DisplayName', 'FIR filters');
+plot(w, 180/pi*unwrap(angle(H)), '--');
set(gca,'ColorOrderIndex',2);
-plot(w, 180/pi*angle(1-H), '--', 'HandleVisibility', 'off');
+plot(w, 180/pi*unwrap(angle(1-H)), '--');
set(gca, 'XScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
-yticks([-540:90:360]);
-legend('location', 'northeast');
+yticks([-450:90:180]); ylim([-450, 200]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
-xticks([0.001, 0.01, 0.1, 1]);
diff --git a/matlab/matlab/h_inf_synthesis_complementary_filters.m b/matlab/matlab/h_inf_synthesis_complementary_filters.m
index 2eff502..7dd9ca0 100644
--- a/matlab/matlab/h_inf_synthesis_complementary_filters.m
+++ b/matlab/matlab/h_inf_synthesis_complementary_filters.m
@@ -6,6 +6,8 @@ s = zpk('s');
freqs = logspace(-1, 3, 1000);
+addpath('./src');
+
% Design of Weighting Function
% A formula is proposed to help the design of the weighting functions:
% \begin{equation}
@@ -24,15 +26,48 @@ freqs = logspace(-1, 3, 1000);
% The general shape of a weighting function generated using the formula is shown in figure [[fig:weight_formula]].
+
+n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2;
+
+W = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
+
+figure;
+hold on;
+plot(freqs, abs(squeeze(freqresp(W, freqs, 'Hz'))), 'k-');
+
+plot([1e-3 1e0], [G0 G0], 'k--', 'LineWidth', 1)
+text(1e0, G0, '$\quad G_0$')
+
+plot([1e1 1e3], [G1 G1], 'k--', 'LineWidth', 1)
+text(1e1,G1,'$G_{\infty}\quad$','HorizontalAlignment', 'right')
+
+plot([w0/2/pi w0/2/pi], [1 2*Gc], 'k--', 'LineWidth', 1)
+text(w0/2/pi,1,'$\omega_c$','VerticalAlignment', 'top', 'HorizontalAlignment', 'center')
+
+plot([w0/2/pi/2 2*w0/2/pi], [Gc Gc], 'k--', 'LineWidth', 1)
+text(w0/2/pi/2, Gc, '$G_c \quad$','HorizontalAlignment', 'right')
+
+text(w0/5/pi/2, abs(evalfr(W, j*w0/5)), 'Slope: $n \quad$', 'HorizontalAlignment', 'right')
+
+text(w0/2/pi, abs(evalfr(W, j*w0)), '$\bullet$', 'HorizontalAlignment', 'center')
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+hold off;
+xlim([freqs(1), freqs(end)]);
+ylim([5e-4, 20]);
+
+
+
% #+name: fig:weight_formula
-% #+caption: Amplitude of the proposed formula for the weighting functions
-% [[file:figs-tikz/weight_formula.png]]
+% #+caption: Gain of the Weighting Function formula
+% #+RESULTS:
+% [[file:figs/weight_formula.png]]
-n = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; Gc = 1/2;
+n = 2; w0 = 2*pi*10; G0 = 1/10; G1 = 1000; Gc = 0.45;
W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
-n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 1/2;
+n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 0.45;
W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
figure;
@@ -45,9 +80,10 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
-ylim([5e-4, 20]);
+ylim([1e-4, 20]);
xticks([0.1, 1, 10, 100, 1000]);
-legend('location', 'northeast');
+leg = legend('location', 'southeast', 'FontSize', 8);
+leg.ItemTokenSize(1) = 18;
% H-Infinity Synthesis
% We define the generalized plant $P$ on matlab.
@@ -67,35 +103,28 @@ P = [W1 -W1;
% #+RESULTS:
% #+begin_example
% [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-% Resetting value of Gamma min based on D_11, D_12, D_21 terms
-% Test bounds: 0.1000 < gamma <= 1050.0000
+% Test bounds: 0.3223 <= gamma <= 1000
-% gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
-% 1.050e+03 2.8e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p
-% 525.050 2.8e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p
-% 262.575 2.8e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p
-% 131.337 2.8e+01 2.4e-07 4.1e+00 -1.0e-13 0.0000 p
-% 65.719 2.8e+01 2.4e-07 4.1e+00 -9.5e-14 0.0000 p
-% 32.909 2.8e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p
-% 16.505 2.8e+01 2.4e-07 4.1e+00 -1.0e-13 0.0000 p
-% 8.302 2.8e+01 2.4e-07 4.1e+00 -7.2e-14 0.0000 p
-% 4.201 2.8e+01 2.4e-07 4.1e+00 -2.5e-25 0.0000 p
-% 2.151 2.7e+01 2.4e-07 4.1e+00 -3.8e-14 0.0000 p
-% 1.125 2.6e+01 2.4e-07 4.1e+00 -5.4e-24 0.0000 p
-% 0.613 2.3e+01 -3.7e+01# 4.1e+00 0.0e+00 0.0000 f
-% 0.869 2.6e+01 -3.7e+02# 4.1e+00 0.0e+00 0.0000 f
-% 0.997 2.6e+01 -1.1e+04# 4.1e+00 0.0e+00 0.0000 f
-% 1.061 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p
-% 1.029 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p
-% 1.013 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p
-% 1.005 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p
-% 1.001 2.6e+01 -3.1e+04# 4.1e+00 -3.8e-14 0.0000 f
-% 1.003 2.6e+01 -2.8e+05# 4.1e+00 0.0e+00 0.0000 f
-% 1.004 2.6e+01 2.4e-07 4.1e+00 -5.8e-24 0.0000 p
-% 1.004 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p
+% gamma X>=0 Y>=0 rho(XY)<1 p/f
+% 1.795e+01 1.4e-07 0.0e+00 1.481e-16 p
+% 2.406e+00 1.4e-07 0.0e+00 3.604e-15 p
+% 8.806e-01 -3.1e+02 # -1.4e-16 7.370e-19 f
+% 1.456e+00 1.4e-07 0.0e+00 1.499e-18 p
+% 1.132e+00 1.4e-07 0.0e+00 8.587e-15 p
+% 9.985e-01 1.4e-07 0.0e+00 2.331e-13 p
+% 9.377e-01 -7.7e+02 # -6.6e-17 3.744e-14 f
+% 9.676e-01 -2.0e+03 # -5.7e-17 1.046e-13 f
+% 9.829e-01 -6.6e+03 # -1.1e-16 2.949e-13 f
+% 9.907e-01 1.4e-07 0.0e+00 2.374e-19 p
+% 9.868e-01 -1.6e+04 # -6.4e-17 5.331e-14 f
+% 9.887e-01 -5.1e+04 # -1.5e-17 2.703e-19 f
+% 9.897e-01 1.4e-07 0.0e+00 1.583e-11 p
+% Limiting gains...
+% 9.897e-01 1.5e-07 0.0e+00 1.183e-12 p
+% 9.897e-01 6.9e-07 0.0e+00 1.365e-12 p
-% Gamma value achieved: 1.0036
+% Best performance (actual): 0.9897
% #+end_example
% We then define the high pass filter $H_1 = 1 - H_2$. The bode plot of both $H_1$ and $H_2$ is shown on figure [[fig:hinf_filters_results]].
@@ -108,8 +137,10 @@ H1 = 1 - H2;
figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
-ax1 = subplot(2,1,1);
+% Magnitude
+ax1 = nexttile([2, 1]);
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
@@ -125,10 +156,13 @@ hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude');
set(gca, 'XTickLabel',[]);
-ylim([5e-4, 20]);
-legend('location', 'northeast');
+ylim([1e-4, 20]);
+yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 18;
-ax2 = subplot(2,1,2);
+% Phase
+ax2 = nexttile;
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
@@ -137,8 +171,7 @@ plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
-yticks([-360:90:360]);
+yticks([-180:90:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
-xticks([0.1, 1, 10, 100, 1000]);
diff --git a/matlab/matlab/three_comp_filters.m b/matlab/matlab/three_comp_filters.m
index 6161383..aba0008 100644
--- a/matlab/matlab/three_comp_filters.m
+++ b/matlab/matlab/three_comp_filters.m
@@ -6,6 +6,8 @@ s = zpk('s');
freqs = logspace(-2, 4, 1000);
+addpath('./src');
+
% Weights
% First we define the weights.
@@ -28,9 +30,9 @@ plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), '--', 'DisplayName', '$|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
-xlim([freqs(1), freqs(end)]);
-xticks([0.01, 0.1, 1, 10, 100, 1000]);
-legend('location', 'northeast');
+xlim([freqs(1), freqs(end)]); ylim([2e-4, 1.3e1])
+leg = legend('location', 'northeast', 'FontSize', 8);
+leg.ItemTokenSize(1) = 18;
% H-Infinity Synthesis
% Then we create the generalized plant =P=.
@@ -54,8 +56,10 @@ H3 = tf(H(2));
H1 = 1 - H2 - H3;
figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
-ax1 = subplot(2,1,1);
+% Magnitude
+ax1 = nexttile([2, 1]);
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
@@ -74,10 +78,12 @@ hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude');
set(gca, 'XTickLabel',[]);
-ylim([5e-4, 20]);
-legend('location', 'northeast');
+ylim([1e-4, 20]);
+leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 18;
-ax2 = subplot(2,1,2);
+% Phase
+ax2 = nexttile;
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))));
@@ -88,8 +94,7 @@ plot(freqs, 180/pi*phase(squeeze(freqresp(H3, freqs, 'Hz'))));
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
-yticks([-360:90:360]);
+yticks([-360:90:360]); ylim([-270, 270]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
-xticks([0.1, 1, 10, 100, 1000]);
diff --git a/mohit/paper.org b/mohit/paper.org
index 90b5a6c..596fd02 100644
--- a/mohit/paper.org
+++ b/mohit/paper.org
@@ -418,7 +418,7 @@ Filter Design for Active Vibration Isolation of LIGO][4]]).
functions and designed complementary filters
[[file:figs/hinf_filters_results.pdf]]
-** Extension to set of three complementary filters
+** TODO Extension to set of three complementary filters
In certain applications, more than two sensors are used to measure the
same quantity and can be merged together to form a "super sensor". In
such case, a set of three (or more) complementary filters is required.
diff --git a/tikz/index.org b/tikz/index.org
index 29679b2..18e9a32 100644
--- a/tikz/index.org
+++ b/tikz/index.org
@@ -374,6 +374,55 @@ Configuration file is accessible [[file:config.org][here]].
#+RESULTS:
[[file:figs/h_infinity_robust_fusion.png]]
+* Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
+#+begin_src latex :file ligo_super_sensor_architecture.pdf :tangle figs/ligo_super_sensor_architecture.tex :exports both
+\definecolor{myblue}{rgb}{0, 0.447, 0.741}
+\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
+
+\begin{tikzpicture}
+ \node[block, align=center] (position) at (0, 2.2) {Position\\Sensor};
+ \node[block, align=center] (seismometer) at (0, 1.0) {Seismometer};
+ \node[block, align=center] (geophone) at (0,-0.6) {Geophone};
+
+ \node[branch, left=0.4 of seismometer] (x);
+
+ \node[block, right=1.1 of seismometer](H1){$L_2(s)$};
+ \node[block](H2) at (H1|-geophone) {$H_2(s)$};
+
+ \node[addb] (add) at (4, 0){};
+ \node[block, right=1.1 of add](H2p) {$H_1(s)$};
+ \node[block] (H1p) at (H2p|-position) {$L_1(s)$};
+
+ \node[addb] (addp) at (7, 1.0){};
+
+ \draw[->] ($(x)+(-1.0, 0)$) -- (seismometer.west);
+ \draw[->] (x.center) |- (position.west);
+ \draw[->] (x.center) |- (geophone.west);
+ \draw[->] (position.east) -- (H1p.west);
+ \draw[->] (seismometer.east) -- (H1.west);
+ \draw[->] (geophone.east) -- (H2.west);
+ \draw[->] (H1) -| (add.north);
+ \draw[->] (H2) -| (add.south);
+ \draw[->] (add.east) -- (H2p.west);
+ \draw[->] (H1p) -| (addp.north);
+ \draw[->] (H2p) -| (addp.south);
+ \draw[->] (addp.east) -- ++(1.0, 0);
+
+ \begin{scope}[on background layer]
+ \node[fit={(x.west|-geophone.south) (position.north-|addp.east)}, fill=black!20!white, draw, inner sep=6pt] (supersensor) {};
+ \node[below] at (supersensor.north) {Super Sensor};
+
+ \node[fit={(x.west|-seismometer.north) (add.east|-geophone.south)}, fill=black!10!white, draw, inner sep=3pt] (superinertialsensor) {};
+ \node[] at (superinertialsensor.center) {"Inertial" Super Sensor};
+ \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:ligo_super_sensor_architecture
+#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters ([[./figs/ligo_super_sensor_architecture.png][png]], [[./figs/ligo_super_sensor_architecture.pdf][pdf]], [[./figs/ligo_super_sensor_architecture.tex][tex]]).
+#+RESULTS:
+[[file:figs/ligo_super_sensor_architecture.png]]
+
* 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}