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Work on discussion section + add new figures

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Thomas Dehaeze
2021-05-21 11:57:21 +02:00
parent 1fda0d3383
commit eccd0eeba9
31 changed files with 1858 additions and 767 deletions
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@@ -313,7 +313,7 @@ The sensor dynamics estimate $\hat{G}_1(s)$ can be a simple gain or more complex
\includegraphics[scale=1]{figs/sensor_model_calibrated.pdf}
\caption{\label{fig:sensor_model_calibrated} Calibrated sensors using the inverse of an estimate $\hat{G}_1(s)$ of the sensor dynamics}
\end{subfigure}
\caption{\label{fig:figure_name}Sensor models with an without normalization}
\caption{\label{fig:sensor_models}Sensor models with an without normalization}
\centering
\end{figure}
#+end_export
@@ -666,6 +666,43 @@ They are found to be very close to each other and this shows the effectiveness o
Explain differences
- cite:plummer06_optim_compl_filter_their_applic_motion_measur use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
- cite:jensen13_basic_uas design of complementary filters with classical control theory, PID
#+name: fig:feedback_sensor_fusion
#+caption: Classical feedback architecture for sensor fusion
#+attr_latex: :scale 1
[[file:figs/feedback_sensor_fusion.pdf]]
\begin{equation}
\hat{x} = \underbrace{\frac{L(s)}{1 + L(s)}}_{T(s)} \hat{x}_1 + \underbrace{\frac{1}{1 + L(s)}}_{S(s)} \hat{x}_2
\end{equation}
with the famous relationship
\begin{equation}
T(s) + S(s) = 1
\end{equation}
Therefore, complementary filter design is very similar to mixed-sensitivity synthesis.
They are actually equivalent by taking
\begin{equation}
L = H_H^{-1} - 1
\end{equation}
(provided $H_H$ is invertible, therefore bi-proper)
\begin{equation}
P_L(s) = \begin{bmatrix}
\phantom{+}W_2(s) & 0 & \phantom{+}1 \\
-W_2(s) & W_1(s) & -1
\end{bmatrix}
\end{equation}
#+name: fig:feedback_synthesis_architecture_generalized_plant
#+caption: Generalized plant for mixed-sensitivity shaping
#+attr_latex: :scale 1
[[file:figs/feedback_synthesis_architecture_generalized_plant.pdf]]
** Imposing zero at origin / roll-off
3 methods:
@@ -678,7 +715,31 @@ Link to literature about doing that with mixed sensitivity
Some applications may require to merge more than two sensors.
In such a case, it is necessary to design as many complementary filters as the number of sensors used.
*** Mathematical Problem :ignore:
# Example of LIGO
# In truth two options: sequential fusion or fusion at once
*** Sequential vs Parallel :ignore:
#+begin_export latex
\begin{figure}[htbp]
\begin{subfigure}[b]{0.59\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_sequential.pdf}
\caption{\label{fig:sensor_fusion_three_sequential}Sequential fusion}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.39\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_parallel.pdf}
\caption{\label{fig:sensor_fusion_three_parallel}Parallel fusion}
\end{subfigure}
\caption{\label{fig:sensor_fusion_three}Sensor fusion architecture with more than two sensors}
\centering
\end{figure}
#+end_export
*** Mathematical Problem :ignore:
The synthesis problem is then to compute $n$ stable transfer functions $H_i(s)$ such that eqref:eq:hinf_problem_gen is satisfied.
#+name: eq:hinf_problem_gen
\begin{subequations}

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@@ -1,4 +1,4 @@
% Created 2021-05-20 jeu. 16:26
% Created 2021-05-21 ven. 11:56
% 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:orgc28ff6a}
\label{sec:orgf465050}
\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:org5a2a4a4}
\label{sec:orgf888f1b}
\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:orgb29eeb0}
\label{sec:orgabe574c}
\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:org749994d}
\label{sec:org4484191}
\label{sec:sensor_models}
In order to study such sensor fusion architecture, a model of the sensors is required.
@@ -165,7 +165,7 @@ The sensor dynamics estimate \(\hat{G}_1(s)\) can be a simple gain or more compl
\includegraphics[scale=1]{figs/sensor_model_calibrated.pdf}
\caption{\label{fig:sensor_model_calibrated} Calibrated sensors using the inverse of an estimate $\hat{G}_1(s)$ of the sensor dynamics}
\end{subfigure}
\caption{\label{fig:figure_name}Sensor models with an without normalization}
\caption{\label{fig:sensor_models}Sensor models with an without normalization}
\centering
\end{figure}
@@ -187,7 +187,7 @@ The super sensor output is therefore equal to:
\end{figure}
\subsection{Noise Sensor Filtering}
\label{sec:org0f07eb6}
\label{sec:orgd1347c0}
\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:orgdb0117a}
\label{sec:orgaa981c0}
\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:orgba2ddcd}
\label{sec:orgf912b72}
\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:orgef373f7}
\label{sec:org6a0910c}
\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:org9d293ca}
\label{sec:org45cf644}
\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:org552a374}
\label{sec:orgb99cb9e}
\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:orgdf78400}
\label{sec:orgbe95f55}
\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:
@@ -465,7 +465,7 @@ 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:orgd51fb42}
\label{sec:org93403ee}
\label{sec:application_ligo}
Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO \cite{matichard15_seism_isolat_advan_ligo,hua05_low_ligo}, the VIRGO \cite{lucia18_low_frequen_optim_perfor_advan,heijningen18_low} and the KAGRA \cite{akutsu21_vibrat_isolat_system_beam_split}.
@@ -488,7 +488,7 @@ After synthesis, the obtained FIR filters were found to be compliant with the re
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:org45a6d0a}
\label{sec:orgd0da28c}
\label{sec:ligo_specifications}
The specifications for the set of complementary filters (\(L_1,H_1\)) used at the LIGO are summarized below (for further details, refer to \cite{hua04_polyp_fir_compl_filter_contr_system}):
\begin{itemize}
@@ -508,7 +508,7 @@ They are physically represented in Figure \ref{fig:fir_filter_ligo} as well as t
\end{figure}
\subsection{Weighting Functions Design}
\label{sec:orgeca6c60}
\label{sec:org3890dcd}
\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.
@@ -524,7 +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:orgcaaff49}
\label{sec:orgd62d211}
\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 \(27\).
@@ -538,9 +538,9 @@ They are found to be very close to each other and this shows the effectiveness o
\end{figure}
\section{Discussion}
\label{sec:orga827f5f}
\label{sec:orga70d7fb}
\subsection{Alternative configuration}
\label{sec:org7abd40b}
\label{sec:orgccb904f}
\begin{itemize}
\item Feedback architecture : Similar to mixed sensitivity (add schematic of feedback loop with weights)
\item 2 inputs / 1 output
@@ -548,17 +548,73 @@ They are found to be very close to each other and this shows the effectiveness o
Explain differences
\begin{itemize}
\item \cite{plummer06_optim_compl_filter_their_applic_motion_measur} use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
\item \cite{jensen13_basic_uas} design of complementary filters with classical control theory, PID
\end{itemize}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/feedback_sensor_fusion.pdf}
\caption{\label{fig:feedback_sensor_fusion}Classical feedback architecture for sensor fusion}
\end{figure}
\begin{equation}
\hat{x} = \underbrace{\frac{L(s)}{1 + L(s)}}_{T(s)} \hat{x}_1 + \underbrace{\frac{1}{1 + L(s)}}_{S(s)} \hat{x}_2
\end{equation}
with the famous relationship
\begin{equation}
T(s) + S(s) = 1
\end{equation}
Therefore, complementary filter design is very similar to mixed-sensitivity synthesis.
They are actually equivalent by taking
\begin{equation}
L = H_H^{-1} - 1
\end{equation}
(provided \(H_H\) is invertible, therefore bi-proper)
\begin{equation}
P_L(s) = \begin{bmatrix}
\phantom{+}W_2(s) & 0 & \phantom{+}1 \\
-W_2(s) & W_1(s) & -1
\end{bmatrix}
\end{equation}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/feedback_synthesis_architecture_generalized_plant.pdf}
\caption{\label{fig:feedback_synthesis_architecture_generalized_plant}Generalized plant for mixed-sensitivity shaping}
\end{figure}
\subsection{Imposing zero at origin / roll-off}
\label{sec:orge105bdf}
\label{sec:org402c2aa}
3 methods:
Link to literature about doing that with mixed sensitivity
\subsection{Synthesis of Three Complementary Filters}
\label{sec:org3fd7562}
\label{sec:orgf9a165b}
\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.
\begin{figure}[htbp]
\begin{subfigure}[b]{0.59\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_sequential.pdf}
\caption{\label{fig:sensor_fusion_three_sequential}Sequential fusion}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.39\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_parallel.pdf}
\caption{\label{fig:sensor_fusion_three_parallel}Parallel fusion}
\end{subfigure}
\caption{\label{fig:sensor_fusion_three}Sensor fusion architecture with more than two sensors}
\centering
\end{figure}
The synthesis problem is then to compute \(n\) stable transfer functions \(H_i(s)\) such that \eqref{eq:hinf_problem_gen} is satisfied.
\begin{subequations}
\label{eq:hinf_problem_gen}
@@ -593,7 +649,7 @@ The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:
\end{figure}
\section{Conclusion}
\label{sec:orgc3f0120}
\label{sec:org75ed4d0}
\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.
@@ -601,7 +657,7 @@ 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:org1be2c18}
\label{sec:org0b419b1}
This research benefited from a FRIA grant from the French Community of Belgium.
\bibliographystyle{elsarticle-num}