Rework Matlab figure size + Start section 4
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@ -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
|
||||
<<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.
|
||||
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
|
||||
<<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 $0.045$
|
||||
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
|
||||
<<sec:ligo_weights>>
|
||||
|
||||
Binary file not shown.
@ -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}
|
||||
|
||||
@ -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,
|
||||
}
|
||||
|
||||
102
matlab/index.org
102
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.');
|
||||
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');
|
||||
plot([0.1, 10], [0.045, 0.045], 'k--', 'HandleVisibility', 'off');
|
||||
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-3, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
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
|
||||
|
||||
@ -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]);
|
||||
|
||||
@ -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]);
|
||||
|
||||
@ -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]);
|
||||
|
||||
@ -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.
|
||||
|
||||
@ -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}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user