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% Created 2021-06-18 ven. 17:00 % Created 2021-08-27 ven. 11:24
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[preprint, sort&compress]{elsarticle} \documentclass[preprint, sort&compress]{elsarticle}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -58,31 +58,56 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
\end{frontmatter} \end{frontmatter}
\section{Introduction} \section{Introduction}
\label{sec:org3356a46} \label{sec:org5737795}
\label{sec:introduction} \label{sec:introduction}
\begin{itemize} \begin{itemize}
\item \cite{anderson53_instr_approac_system_steer_comput} earliest application of complementary filters (A simple RC circuit was used to physically realize the complementary filters)
\item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion \item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion
\end{itemize} \end{itemize}
\begin{itemize} Sensor fusion can have many advantages.
\item Increase the bandwidth: \cite{zimmermann92_high_bandw_orien_measur_contr}
\item Increased robustness: \cite{collette15_sensor_fusion_method_high_perfor}
\item Decrease the noise:
\end{itemize}
\begin{itemize}
\item UAV: \cite{pascoal99_navig_system_desig_using_time}, \cite{jensen13_basic_uas}
\item Gravitational wave observer: \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,lucia18_low_frequen_optim_perfor_advan,heijningen18_low,akutsu21_vibrat_isolat_system_beam_split}
\end{itemize}
\begin{itemize}
\item \cite{brown72_integ_navig_system_kalman_filter} alternate form of complementary filters => Kalman filtering
\item \cite{higgins75_compar_compl_kalman_filter} Compare Kalman Filtering with sensor fusion using complementary filters
\item \cite{robert12_introd_random_signal_applied_kalman} advantage of complementary filters over Kalman filtering
\end{itemize}
\begin{itemize}
\item Analog complementary filters: \cite{yong16_high_speed_vertic_posit_stage}, \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop}
In some situations, it is used to increase the bandwidth of the sensor \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim}.
For instance by increasing the high frequency bandwidth of a position sensor using an accelerometer.
Decrease the noise: \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur}
\cite[chapter 8]{robert12_introd_random_signal_applied_kalman}
Increased robustness (sensor measuring different quantities): \cite{collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage}
\par
The applications of sensor fusion are numerous.
It is widely used for attitude estimation of unmanned aerial vehicle
\cite{baerveldt97_low_cost_low_weigh_attit,pascoal99_navig_system_desig_using_time,corke04_inert_visual_sensin_system_small_auton_helic,batista10_optim_posit_veloc_navig_filter_auton_vehic,jensen13_basic_uas,min15_compl_filter_desig_angle_estim}
Motion control
\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr}
Tjepkema et al. \cite{tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip} used sensor fusion to isolate precision equipment from the ground motion.
Gravitational wave observer \cite{heijningen18_low}:
LIGO \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}
VIRGO \cite{lucia18_low_frequen_optim_perfor_advan}
There are mainly two ways to perform sensor fusion: using complementary filters or using Kalman filtering \cite{brown72_integ_navig_system_kalman_filter}.
Kalman filtering \cite{odry18_kalman_filter_mobil_robot_attit_estim}
Relations between CF and Kalman: \cite{becker15_compl_filter_desig_three_frequen_bands}
Advantages of complementary filtering over Kalman filtering for sensor fusion:
\begin{itemize}
\item Less computation \cite{higgins75_compar_compl_kalman_filter}
\item For Kalman filtering, we are forced to make assumption about the probabilistic character of the sensor noises \cite{robert12_introd_random_signal_applied_kalman}
\item More intuitive frequency domain technique
\end{itemize}
In some cases, complementary filters are implemented in an analog way such as in \cite{yong16_high_speed_vertic_posit_stage,moore19_capac_instr_sensor_fusion_high_bandw_nanop}, but most of the time it is implemented numerically which allows much more complex
Multiple design methods have been used for complementary filters
\begin{itemize}
\item Analytical methods: \item Analytical methods:
\begin{itemize} \begin{itemize}
\item first order: \cite{corke04_inert_visual_sensin_system_small_auton_helic} \item first order: \cite{corke04_inert_visual_sensin_system_small_auton_helic,yong16_high_speed_vertic_posit_stage}
\item second order: \cite{baerveldt97_low_cost_low_weigh_attit}, \cite{stoten01_fusion_kinet_data_using_compos_filter}, \cite{jensen13_basic_uas} \item second order: \cite{baerveldt97_low_cost_low_weigh_attit}, \cite{stoten01_fusion_kinet_data_using_compos_filter}, \cite{jensen13_basic_uas}
\item higher order: \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}, \cite{zimmermann92_high_bandw_orien_measur_contr}, \cite{collette15_sensor_fusion_method_high_perfor}, \cite{matichard15_seism_isolat_advan_ligo} \item higher order: \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}, \cite{zimmermann92_high_bandw_orien_measur_contr}, \cite{collette15_sensor_fusion_method_high_perfor}, \cite{matichard15_seism_isolat_advan_ligo}
\end{itemize} \end{itemize}
@ -97,21 +122,28 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
\item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands} \item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands}
\end{itemize} \end{itemize}
\begin{itemize} \begin{itemize}
\item Robustness problems: \cite{zimmermann92_high_bandw_orien_measur_contr} change of phase near the merging frequency \item Robustness problems: \cite{zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur} change of phase near the merging frequency
\item Trial and error \item Trial and error
\item Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available. \item Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available.
\end{itemize} \end{itemize}
Most of the requirements => shape of the complementary filters Most of the requirements => shape of the complementary filters
=> propose a way to shape complementary filters. => propose a way to shape complementary filters.
Section \ref{sec:requirements}
Section \ref{sec:hinf_method}
Section \ref{sec:application_ligo}
Section \ref{sec:discussion}
\section{Sensor Fusion and Complementary Filters Requirements} \section{Sensor Fusion and Complementary Filters Requirements}
\label{sec:org32c05cb} \label{sec:orgbd86d49}
\label{sec:requirements} \label{sec:requirements}
Complementary filters provides a framework for fusing signals from different sensors. 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. 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. These requirements are discussed in this section.
\subsection{Sensor Fusion Architecture} \subsection{Sensor Fusion Architecture}
\label{sec:orgcfc6167} \label{sec:org56b9e47}
\label{sec:sensor_fusion} \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 \(x\). 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 +170,7 @@ Therefore, a pair of strict complementary filter needs to satisfy the following
It will soon become clear why the complementary property is important. It will soon become clear why the complementary property is important.
\subsection{Sensor Models and Sensor Normalization} \subsection{Sensor Models and Sensor Normalization}
\label{sec:orga2c7e39} \label{sec:org684f136}
\label{sec:sensor_models} \label{sec:sensor_models}
In order to study such sensor fusion architecture, a model of the sensors is required. In order to study such sensor fusion architecture, a model of the sensors is required.
@ -187,7 +219,7 @@ The super sensor output is therefore equal to:
\end{figure} \end{figure}
\subsection{Noise Sensor Filtering} \subsection{Noise Sensor Filtering}
\label{sec:org5397108} \label{sec:org99631b9}
\label{sec:noise_filtering} \label{sec:noise_filtering}
In this section, it is supposed that all the sensors are perfectly calibrated, such that: In this section, it is supposed that all the sensors are perfectly calibrated, such that:
@ -220,14 +252,14 @@ As shown in \eqref{eq:noise_filtering_psd}, the Power Spectral Density (PSD) of
\end{equation} \end{equation}
If the two sensors have identical noise characteristics (\(\Phi_{n_1}(\omega) = \Phi_{n_2}(\omega)\)), a simple averaging (\(H_1(s) = H_2(s) = 0.5\)) is what would minimize the super sensor noise. If the two sensors have identical noise characteristics (\(\Phi_{n_1}(\omega) = \Phi_{n_2}(\omega)\)), a simple averaging (\(H_1(s) = H_2(s) = 0.5\)) is what would minimize the super sensor noise.
This the simplest form of sensor fusion with complementary filters. This is the simplest form of sensor fusion with complementary filters.
However, the two sensors have usually high noise levels over distinct frequency regions. However, the two sensors have usually high noise levels over distinct frequency regions.
In such case, to lower the noise of the super sensor, the value of the norm \(|H_1|\) has to be lowered when \(\Phi_{n_1}\) is larger than \(\Phi_{n_2}\) and that of \(|H_2|\) lowered when \(\Phi_{n_2}\) is larger than \(\Phi_{n_1}\). In such case, to lower the noise of the super sensor, the value of the norm \(|H_1|\) has to be lowered when \(\Phi_{n_1}\) is larger than \(\Phi_{n_2}\) and that of \(|H_2|\) lowered when \(\Phi_{n_2}\) is larger than \(\Phi_{n_1}\).
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise. 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} \subsection{Sensor Fusion Robustness}
\label{sec:org6cbe7ea} \label{sec:org3f9e403}
\label{sec:fusion_robustness} \label{sec:fusion_robustness}
In practical systems the sensor normalization is not perfect and condition \eqref{eq:perfect_dynamics} is not verified. In practical systems the sensor normalization is not perfect and condition \eqref{eq:perfect_dynamics} is not verified.
@ -289,14 +321,14 @@ 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. 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} \section{Complementary Filters Shaping}
\label{sec:org3fcce50} \label{sec:org82bc276}
\label{sec:hinf_method} \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. 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. 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. In this section, such synthesis is proposed by expressing this problem as a \(\mathcal{H}_\infty\) norm optimization.
\subsection{Synthesis Objective} \subsection{Synthesis Objective}
\label{sec:org006154f} \label{sec:orgceb5825}
\label{sec:synthesis_objective} \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}. 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 +345,7 @@ 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. 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} \subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
\label{sec:orgd8cba14} \label{sec:org79feac5}
\label{sec:hinf_synthesis} \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. 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 +386,7 @@ 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. The above optimization problem can be efficiently solved in Matlab \cite{matlab20} using the Robust Control Toolbox.
\subsection{Weighting Functions Design} \subsection{Weighting Functions Design}
\label{sec:org7aa4ffb} \label{sec:orgd27beed}
\label{sec:hinf_weighting_func} \label{sec:hinf_weighting_func}
Weighting functions are used during the synthesis to specify what is the maximum allowed norms of the complementary filters. Weighting functions are used during the synthesis to specify what is the maximum allowed norms of the complementary filters.
@ -404,7 +436,7 @@ The typical shape of a weighting function generated using \eqref{eq:weight_formu
\end{figure} \end{figure}
\subsection{Validation of the proposed synthesis method} \subsection{Validation of the proposed synthesis method}
\label{sec:orgb562cf2} \label{sec:orgc8f3eb3}
\label{sec:hinf_example} \label{sec:hinf_example}
The proposed methodology for the design of complementary filters is now applied on a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that: The proposed methodology for the design of complementary filters is now applied on a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
@ -465,9 +497,9 @@ 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. 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} \section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
\label{sec:org60805ba} \label{sec:org8cb3b2e}
\label{sec:application_ligo} \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}. Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO \cite{matichard15_seism_isolat_advan_ligo,hua05_low_ligo}, the VIRGO \cite{lucia18_low_frequen_optim_perfor_advan,heijningen18_low} and the KAGRA \cite[Chap. 5]{sekiguchi16_study_low_frequen_vibrat_isolat_system}.
In the first isolation stage at the LIGO, two sets of complementary filters are used and included in a feedback loop \cite{hua04_low_ligo}. In the first isolation stage at the LIGO, two sets of complementary filters are used 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. A set of complementary filters (\(L_2,H_2\)) is first used to fuse a seismometer and a geophone.
@ -488,7 +520,7 @@ 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. 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. In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated on the same set of requirements.
\subsection{Complementary Filters Specifications} \subsection{Complementary Filters Specifications}
\label{sec:orgfdd63d0} \label{sec:orgb603be6}
\label{sec:ligo_specifications} \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}): 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} \begin{itemize}
@ -508,7 +540,7 @@ They are physically represented in Figure \ref{fig:fir_filter_ligo} as well as t
\end{figure} \end{figure}
\subsection{Weighting Functions Design} \subsection{Weighting Functions Design}
\label{sec:org916b9d5} \label{sec:orgd94a6e5}
\label{sec:ligo_weights} \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. 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. 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 +556,7 @@ The magnitudes of the weighting functions are shown in Fig. \ref{fig:ligo_weight
\end{figure} \end{figure}
\subsection{\(\mathcal{H}_\infty\) Synthesis} \subsection{\(\mathcal{H}_\infty\) Synthesis}
\label{sec:orgab74bf1} \label{sec:org1f03af8}
\label{sec:ligo_results} \label{sec:ligo_results}
\(\mathcal{H}_\infty\) synthesis is performed using the architecture shown in Fig. \ref{eq:generalized_plant}. \(\mathcal{H}_\infty\) synthesis is performed using the architecture shown in Fig. \ref{eq:generalized_plant}.
The complementary filters obtained are of order \(27\). The complementary filters obtained are of order \(27\).
@ -538,10 +570,10 @@ They are found to be very close to each other and this shows the effectiveness o
\end{figure} \end{figure}
\section{Discussion} \section{Discussion}
\label{sec:org5bc126e} \label{sec:org013b9e6}
\label{sec:discussion} \label{sec:discussion}
\subsection{``Closed-Loop'' complementary filters} \subsection{``Closed-Loop'' complementary filters}
\label{sec:org8731218} \label{sec:orga1ea439}
\label{sec:closed_loop_complementary_filters} \label{sec:closed_loop_complementary_filters}
It is possible to use the fundamental properties of a feedback architecture to generate complementary filters. It is possible to use the fundamental properties of a feedback architecture to generate complementary filters.
@ -626,7 +658,7 @@ L = H_H^{-1} - 1
(provided \(H_H\) is invertible, therefore bi-proper) (provided \(H_H\) is invertible, therefore bi-proper)
\subsection{Imposing zero at origin / roll-off} \subsection{Imposing zero at origin / roll-off}
\label{sec:orgdea775a} \label{sec:org293cf77}
\label{sec:add_features_in_filters} \label{sec:add_features_in_filters}
3 methods: 3 methods:
@ -634,10 +666,14 @@ L = H_H^{-1} - 1
Link to literature about doing that with mixed sensitivity Link to literature about doing that with mixed sensitivity
\subsection{Synthesis of Three Complementary Filters} \subsection{Synthesis of Three Complementary Filters}
\label{sec:org6446998} \label{sec:orgd44eb72}
\label{sec:hinf_three_comp_filters} \label{sec:hinf_three_comp_filters}
Some applications may require to merge more than two sensors. Some applications may require to merge more than two sensors.
For instance at the LIGO, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor (Figure \ref{fig:ligo_super_sensor_architecture}). \par For instance at the LIGO, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor (Figure \ref{fig:ligo_super_sensor_architecture}). \par
\begin{itemize}
\item[{$\square$}] \cite{becker15_compl_filter_desig_three_frequen_bands}
\end{itemize}
When merging \(n>2\) sensors using complementary filters, two architectures can be used as shown in Figure \ref{fig:sensor_fusion_three}. When merging \(n>2\) sensors using complementary filters, two architectures can be used as shown in Figure \ref{fig:sensor_fusion_three}.
The fusion can either be done in a ``sequential'' way where \(n-1\) sets of two complementary filters are used (Figure \ref{fig:sensor_fusion_three_sequential}), or in a ``parallel'' way where one set of \(n\) complementary filters is used (Figure \ref{fig:sensor_fusion_three_parallel}). The fusion can either be done in a ``sequential'' way where \(n-1\) sets of two complementary filters are used (Figure \ref{fig:sensor_fusion_three_sequential}), or in a ``parallel'' way where one set of \(n\) complementary filters is used (Figure \ref{fig:sensor_fusion_three_parallel}).
@ -727,7 +763,7 @@ Such synthesis method can be generalized to a set of \(n\) complementary filters
\end{equation} \end{equation}
\section{Conclusion} \section{Conclusion}
\label{sec:orgcba6c13} \label{sec:orgc6071ad}
\label{sec:conclusion} \label{sec:conclusion}
This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor. 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. 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.
@ -735,7 +771,7 @@ 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. 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} \section*{Acknowledgment}
\label{sec:orgf175dee} \label{sec:org4efce57}
This research benefited from a FRIA grant from the French Community of Belgium. This research benefited from a FRIA grant from the French Community of Belgium.
\bibliographystyle{elsarticle-num} \bibliographystyle{elsarticle-num}