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Update journal paper

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Thomas Dehaeze 2021-06-21 11:40:36 +02:00
parent eccd0eeba9
commit ad04f8a16d
27 changed files with 839 additions and 288 deletions
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Makefile Normal file
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.PHONY: all paper help html publish watch clean cp-figs
SHELL := /bin/bash
PAPERDIR=paper
MATLABDIR=matlab
TIKZDIR=tikz
PAPERNAME=paper
all: paper html publish
paper: cp-figs tangle tex pdf clean
help:
@echo "Usage: make <command>"
@echo " all - Cp-figs tex pdf html publish"
@echo " paper - Compile the org file to a pdf"
@echo " html - Export all the org files to html"
@echo " publish - Commit everything and push to repository"
@echo " tex - Export to paper in org format to tex"
@echo " tangle - Tangle everything that is in the org paper and tikz file"
@echo " pdf - Compile the tex file to pdf"
@echo " watch - Watch the tex file for changes and compile"
@echo " clean - Clean the paper directory"
@echo " cp-figs - Copy all the necessary figures from tikz and matlab folder to paper folder"
html:
for f in *.org; do emacsclient -e "(progn (find-file \"$$f\") (org-html-export-to-html))"; done
for f in $(TIKZDIR)/*.org; do emacsclient -e "(progn (find-file \"$$f\") (org-html-export-to-html))"; done
for f in $(MATLABDIR)/*.org; do emacsclient -e "(progn (find-file \"$$f\") (org-html-export-to-html))"; done
publish:
git add . && git commit -m "Update - $$(date +%F)" && git push origin master
tex: $(PAPERDIR)/$(PAPERNAME).org
emacsclient -e '(progn (find-file "$(PAPERDIR)/$(PAPERNAME).org") (org-latex-export-to-latex))'
tangle: $(PAPERDIR)/$(PAPERNAME).org
emacsclient -e '(progn (find-file "$(PAPERDIR)/$(PAPERNAME).org") (org-babel-tangle))'
emacsclient -e '(progn (find-file "$(TIKZDIR)/index.org") (org-babel-tangle))'
pdf: $(PAPERDIR)/$(PAPERNAME).tex
latexmk -cd -quiet -bibtex $(PREVIEW_CONTINUOUSLY) -f -pdf -pdflatex="xelatex -synctex=1 -interaction nonstopmode" -use-make $(PAPERDIR)/$(PAPERNAME).tex
# Set the PREVIEW_CONTINUOUSLY variable to -pvc to switch latexmk into the preview continuously mode
watch: PREVIEW_CONTINUOUSLY=-pvc
watch: pdf
watch-org: $(PAPERDIR)/$(PAPERNAME).org
echo $(PAPERDIR)/$(PAPERNAME).org | entr -s 'make tangle tex pdf'
clean:
latexmk -cd -c -bibtex $(PAPERDIR)/$(PAPERNAME).tex
cp-figs:
bash scripts/cp-figs.sh

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journal/figs/comp_filter_three_hinf.pdf
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@ -101,7 +101,7 @@ Sensor fusion \sep{} Optimal filters \sep{} $\mathcal{H}_\infty$ synthesis \sep{
# The applications of sensor fusion are numerous
- UAV: cite:pascoal99_navig_system_desig_using_time, cite:jensen13_basic_uas
- Gravitational wave observer: cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system
- 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
*** Kalman Filtering or Complementary filters :ignore:
@ -123,7 +123,7 @@ Sensor fusion \sep{} Optimal filters \sep{} $\mathcal{H}_\infty$ synthesis \sep{
- first order: cite:corke04_inert_visual_sensin_system_small_auton_helic
- second order: cite:baerveldt97_low_cost_low_weigh_attit, cite:stoten01_fusion_kinet_data_using_compos_filter, cite:jensen13_basic_uas
- 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
- cite:pascoal99_navig_system_desig_using_time use LMI to generate complementary filters
- cite:pascoal99_navig_system_desig_using_time use LMI to generate complementary filters (convex optimization techniques), specific for navigation systems
- cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system: FIR + convex optimization
- Similar to feedback system:
- cite:plummer06_optim_compl_filter_their_applic_motion_measur use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
@ -225,10 +225,10 @@ When two filters are complementary, usually one is a low pass filter while the o
The complementary filters are designed in such a way that their magnitude is close to one in the bandwidth of the sensor they are combined with.
This enables to measure the physical quantity over larger bandwidth.
There are two different categories of complementary filters --- frequency domain complementary filters and state space complementary filters.
Earliest application of the the frequency domain complementary filters was seen in Anderson and Fritze cite:anderson53_instr_approac_system_steer_comput.
Earliest application of the frequency domain complementary filters was seen in Anderson and Fritze cite:anderson53_instr_approac_system_steer_comput.
A simple RC circuit was used to physically realize the complementary filters.
Frequency domain complementary filters were also used in cite:shaw90_bandw_enhan_posit_measur_using_measur_accel, zimmermann92_high_bandw_orien_measur_contr, baerveldt97, roberts03_low.
State space complementary filter finds application in tracking orientation of the flexible links in a robot cite:bachmann03_desig_marg_dof, salcudean91_global_conver_angul_veloc_obser, mahony08_nonlin_compl_filter_special_orthog_group and are particularly useful for multi-input multi-output systems.
State space complementary filter finds application in tracking orientation of the flexible links in a robot cite:bachmann03_desig_marg_dof,salcudean91_global_conver_angul_veloc_obser,mahony08_nonlin_compl_filter_special_orthog_group and are particularly useful for multi-input multi-output systems.
Pascoal et al. cite:pascoal00_navig_system_desig_using_time presented complementary filters which can adapt with time for navigation system capable of estimating position and velocity using GPS and SONAR sensors.
The noise characteristics of the super sensor are governed by the norms of the complementary filters.
@ -247,11 +247,11 @@ Such a method would prove to be very useful as the noise of the "supper sensor"
This paper presents such a framework based on the $\mathcal{H}_\infty$ norm minimization.
The proposed method is quite general and can be easily extended to a case where more than two complementary filters needs to be designed.
The organization of this paper is as follows.
Section [[*Complementary filters requirements][2]] presents the design requirements of ideal complementary filters.
Section 2 presents the design requirements of ideal complementary filters.
It also demonstrates how the noise and robustness characteristics of the "super sensor" can be transformed into upper bounds on the norm of the complementary filters.
The framework for the design of complementary filters is detailed in Section [[*Design formulation using $\mathcal{H}_\infty$ synthesis][3]].
This is followed by the application of the design method to complementary filter design for the active vibration isolation at LIGO in Section [[*Application: Complementary Filter Design for Active Vibration Isolation of LIGO][4]].
Finally, concluding remarks are presented in Section [[*Concluding remarks][5]].
The framework for the design of complementary filters is detailed in Section 3.
This is followed by the application of the design method to complementary filter design for the active vibration isolation at LIGO in Section 4.
Finally, concluding remarks are presented in Section 5.
* Sensor Fusion and Complementary Filters Requirements
<<sec:requirements>>
@ -617,6 +617,8 @@ 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.
# Example where clearly manual tuning of the complementary filters is not an option
** Complementary Filters Specifications
<<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):
@ -660,22 +662,31 @@ They are found to be very close to each other and this shows the effectiveness o
[[file:figs/comp_fir_ligo_hinf.pdf]]
* Discussion
** Alternative configuration
- Feedback architecture : Similar to mixed sensitivity (add schematic of feedback loop with weights)
- 2 inputs / 1 output
<<sec:discussion>>
** Introduction :ignore:
Explain differences
** "Closed-Loop" complementary filters
<<sec:closed_loop_complementary_filters>>
*** Introduction to using feedback architecture for CF :ignore:
It is possible to use the fundamental properties of a feedback architecture to generate complementary filters.
It has been proposed by:
- 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
- Maybe also cite cite:mahony05_compl_filter_desig_special_orthog
Consider the feedback architecture of Figure ref:fig:feedback_sensor_fusion, with two inputs $\hat{x}_1$ and $\hat{x}_2$, and one output $\hat{x}$.
#+name: fig:feedback_sensor_fusion
#+caption: Classical feedback architecture for sensor fusion
#+caption: "Closed-Loop" complementary filters
#+attr_latex: :scale 1
[[file:figs/feedback_sensor_fusion.pdf]]
The output $\hat{x}$ is described by eqref:eq:closed_loop_complementary_filters.
#+name: eq:closed_loop_complementary_filters
\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
\hat{x} = \underbrace{\frac{1}{1 + L(s)}}_{S(s)} \hat{x}_1 + \underbrace{\frac{L(s)}{1 + L(s)}}_{T(s)} \hat{x}_2
\end{equation}
with the famous relationship
@ -683,6 +694,56 @@ with the famous relationship
T(s) + S(s) = 1
\end{equation}
Provided that the closed-loop system is stable, this indeed forms two complementary filters.
*** Sensor Fusion with "closed-loop" complementary filters :ignore:
Therefore, two filters can be merged as shown in Figure ref:fig:feedback_sensor_fusion_arch.
#+name: fig:feedback_sensor_fusion_arch
#+caption: Classical feedback architecture for sensor fusion
#+attr_latex: :scale 1
[[file:figs/feedback_sensor_fusion_arch.pdf]]
One of the main advantage of this configuration is that standard tools of the linear control theory can be applied.
*** Mixed Sensitivity Synthesis :ignore:
If one want to shape both the transfer functions $\frac{\hat{x}}{\hat{x}_1}(s) = S(s)$ and $\frac{\hat{x}}{\hat{x}_2}(s) = T(s)$, this corresponds to the $\mathcal{H}_\infty$ mixed-sensitivity synthesis.
The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can be perform by applying the $\mathcal{H}_\infty$ synthesis to the generalized plant $P_L(s)$ shown in Figure ref:fig:feedback_synthesis_architecture_generalized_plant and described by eqref:eq:generalized_plant_mixed_sensitivity where $W_1(s)$ and $W_2(s)$ are weighting functions used to respectively shape $S(s)$ and $T(s)$.
#+name: eq:generalized_plant_mixed_sensitivity
\begin{equation}
\begin{bmatrix} z \\ v \end{bmatrix} = P_L(s) \begin{bmatrix} w_1 \\ w_2 \\ u \end{bmatrix}; \quad P_L(s) = \begin{bmatrix}
\phantom{+}W_1(s) & 0 & \phantom{+}1 \\
-W_1(s) & W_2(s) & -1
\end{bmatrix}
\end{equation}
This is equivalent as to find a filter $L(s)$ such that eqref:eq:comp_filters_feedback_obj is verified.
#+name: eq:comp_filters_feedback_obj
\begin{equation}
\left\|\begin{matrix} \frac{1}{1 + L(s)} W_1(s) \\ \frac{L(s)}{1 + L(s)} W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
The sensor fusion can be implemented as shown in Figure ref:fig:feedback_sensor_fusion_arch using the feedback architecture or more classically as shown in Figure ref:fig:sensor_fusion_overview using eqref:eq:comp_filters_feedback.
#+name: eq:comp_filters_feedback
\begin{equation}
H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)}
\end{equation}
The two being equivalent considering only the inputs/outputs relationships.
#+name: fig:feedback_synthesis_architecture_generalized_plant
#+caption: Generalized plant for the $\mathcal{H}_\infty$ mixed-sensitivity synthesis
#+attr_latex: :scale 1
[[file:figs/feedback_synthesis_architecture_generalized_plant.pdf]]
*** Example and equivalence with our synthesis method :ignore:
Example: same weights as in ref:tab:weights_params.
Therefore, complementary filter design is very similar to mixed-sensitivity synthesis.
They are actually equivalent by taking
@ -691,19 +752,9 @@ 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
<<sec:add_features_in_filters>>
3 methods:
Link to literature about doing that with mixed sensitivity
@ -713,13 +764,19 @@ Link to literature about doing that with mixed sensitivity
*** Why it is used sometimes :ignore:
Some applications may require to merge more than two sensors.
In such a case, it is necessary to design as many complementary filters as the number of sensors used.
# Example of LIGO
# In truth two options: sequential fusion or fusion at once
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
*** Sequential vs Parallel :ignore:
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).
In the first case, typical sensor fusion synthesis techniques can be used.
However, when a parallel architecture is used, a new synthesis method for a set of more than two complementary filters is required.
Such synthesis method is presented in this section. \par
*************** TODO Say possible advantages of parallel architecture
*************** END
#+begin_export latex
\begin{figure}[htbp]
@ -740,7 +797,7 @@ In such a case, it is necessary to design as many complementary filters as the n
#+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.
The synthesis objective is to compute a set of $n$ stable transfer functions $[H_1(s),\ H_2(s),\ \dots,\ H_n(s)]$ such that eqref:eq:hinf_problem_gen is satisfied.
#+name: eq:hinf_problem_gen
\begin{subequations}
\begin{align}
@ -748,14 +805,16 @@ The synthesis problem is then to compute $n$ stable transfer functions $H_i(s)$
& \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:hinf_cond_perf_gen}
\end{align}
\end{subequations}
where $[W_1(s),\ W_2(s),\ \dots,\ W_n(s)]$ are weighting transfer functions that are chosen to specify the maximum wanted norms of the complementary filters during the synthesis.
Such synthesis objective is very close to the one described in Section ref:sec:synthesis_objective, and indeed the proposed synthesis architecture is also very similar. \par
*** H-Infinity Architecture :ignore:
The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. ref:fig:comp_filter_three_hinf.
Consider the generalized plant $P_3(s)$ shown in Figure ref:fig:comp_filter_three_hinf which is also described by eqref:eq:generalized_plant_three_filters.
The $\mathcal{H}_\infty$ synthesis objective applied on $P(s)$ is to design two stable filters $H_2(s)$ and $H_3(s)$ such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2, \ z_3]$ is less than one eqref:eq:hinf_syn_obj_three.
#+name: eq:hinf_syn_obj_three
#+name: eq:generalized_plant_three_filters
\begin{equation}
\left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
\begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix}
\end{equation}
#+name: fig:comp_filter_three_hinf
@ -763,18 +822,41 @@ The $\mathcal{H}_\infty$ synthesis objective applied on $P(s)$ is to design two
#+attr_latex: :scale 1
[[file:figs/comp_filter_three_hinf.pdf]]
By choosing $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\infty$ synthesis solves the design problem eqref:eq:hinf_problem_gen. \par
Applying the $\mathcal{H}_\infty$ synthesis on the generalized plant $P_3(s)$ is equivalent as to find two stable filters $[H_2(s),\ H_3(s)]$ (shown in Figure ref:fig:comp_filter_three_hinf) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2, \ z_3]$ is less than one eqref:eq:hinf_syn_obj_three.
#+name: eq:hinf_syn_obj_three
\begin{equation}
\left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
\end{equation}
By defining $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\infty$ synthesis solves the design problem eqref:eq:hinf_problem_gen with $n=3$. \par
*** Example of generated complementary filters :ignore:
An example is given to validate the method where three sensors are used in different frequency bands (up to $\SI{1}{Hz}$, from $1$ to $\SI{10}{Hz}$ and above $\SI{10}{Hz}$ respectively).
Three weighting functions are designed using eqref:eq:weight_formula and shown by dashed curves in Fig. ref:fig:three_complementary_filters_results.
The bode plots of the obtained complementary filters are shown in Fig. ref:fig:three_complementary_filters_results.
The bode plots of the obtained complementary filters are shown in Fig. ref:fig:three_complementary_filters_results. \par
#+name: fig:three_complementary_filters_results
#+caption: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/three_complementary_filters_results.pdf]]
*** Generalization :ignore:
Such synthesis method can be generalized to a set of $n$ complementary filters, even though there might not be any practical application for $n>3$.
#+name: eq:generalized_plant_n_filters
\begin{equation}
\begin{bmatrix} z_1 \\ \vdots \\ z_n \\ v \end{bmatrix} = P_n(s) \begin{bmatrix} w \\ u_1 \\ \vdots \\ u_{n-1} \end{bmatrix}; \quad
P_n(s) = \begin{bmatrix}
W_1 & -W_1 & \dots & \dots & -W_1 \\
0 & W_2 & 0 & \dots & 0 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots & & \ddots & \ddots & 0 \\
0 & \dots & \dots & 0 & W_n \\
1 & 0 & \dots & \dots & 0
\end{bmatrix}
\end{equation}
* Conclusion
<<sec:conclusion>>
This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor.

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@ -1,4 +1,4 @@
% Created 2021-05-21 ven. 11:56
% Created 2021-06-18 ven. 17:00
% 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:orgf465050}
\label{sec:org3356a46}
\label{sec:introduction}
\begin{itemize}
\item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion
@ -70,7 +70,7 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
\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}
\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
@ -86,7 +86,7 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
\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}
\end{itemize}
\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters
\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters (convex optimization techniques), specific for navigation systems
\item \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}: FIR + convex optimization
\item Similar to feedback system:
\begin{itemize}
@ -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:orgf888f1b}
\label{sec:org32c05cb}
\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:orgabe574c}
\label{sec:orgcfc6167}
\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:org4484191}
\label{sec:orga2c7e39}
\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:orgd1347c0}
\label{sec:org5397108}
\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:orgaa981c0}
\label{sec:org6cbe7ea}
\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:orgf912b72}
\label{sec:org3fcce50}
\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:org6a0910c}
\label{sec:org006154f}
\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:org45cf644}
\label{sec:orgd8cba14}
\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:orgb99cb9e}
\label{sec:org7aa4ffb}
\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:orgbe95f55}
\label{sec:orgb562cf2}
\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:org93403ee}
\label{sec:org60805ba}
\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:orgd0da28c}
\label{sec:orgfdd63d0}
\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:org3890dcd}
\label{sec:org916b9d5}
\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:orgd62d211}
\label{sec:orgab74bf1}
\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 \(27\).
@ -538,29 +538,33 @@ They are found to be very close to each other and this shows the effectiveness o
\end{figure}
\section{Discussion}
\label{sec:orga70d7fb}
\subsection{Alternative configuration}
\label{sec:orgccb904f}
\begin{itemize}
\item Feedback architecture : Similar to mixed sensitivity (add schematic of feedback loop with weights)
\item 2 inputs / 1 output
\end{itemize}
Explain differences
\label{sec:org5bc126e}
\label{sec:discussion}
\subsection{``Closed-Loop'' complementary filters}
\label{sec:org8731218}
\label{sec:closed_loop_complementary_filters}
It is possible to use the fundamental properties of a feedback architecture to generate complementary filters.
It has been proposed by:
\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
\item Maybe also cite \cite{mahony05_compl_filter_desig_special_orthog}
\end{itemize}
Consider the feedback architecture of Figure \ref{fig:feedback_sensor_fusion}, with two inputs \(\hat{x}_1\) and \(\hat{x}_2\), and one output \(\hat{x}\).
\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}
\caption{\label{fig:feedback_sensor_fusion}``Closed-Loop'' complementary filters}
\end{figure}
The output \(\hat{x}\) is described by \eqref{eq:closed_loop_complementary_filters}.
\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
\label{eq:closed_loop_complementary_filters}
\hat{x} = \underbrace{\frac{1}{1 + L(s)}}_{S(s)} \hat{x}_1 + \underbrace{\frac{L(s)}{1 + L(s)}}_{T(s)} \hat{x}_2
\end{equation}
with the famous relationship
@ -568,6 +572,51 @@ with the famous relationship
T(s) + S(s) = 1
\end{equation}
Provided that the closed-loop system is stable, this indeed forms two complementary filters.
Therefore, two filters can be merged as shown in Figure \ref{fig:feedback_sensor_fusion_arch}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/feedback_sensor_fusion_arch.pdf}
\caption{\label{fig:feedback_sensor_fusion_arch}Classical feedback architecture for sensor fusion}
\end{figure}
One of the main advantage of this configuration is that standard tools of the linear control theory can be applied.
If one want to shape both the transfer functions \(\frac{\hat{x}}{\hat{x}_1}(s) = S(s)\) and \(\frac{\hat{x}}{\hat{x}_2}(s) = T(s)\), this corresponds to the \(\mathcal{H}_\infty\) mixed-sensitivity synthesis.
The \(\mathcal{H}_\infty\) mixed-sensitivity synthesis can be perform by applying the \(\mathcal{H}_\infty\) synthesis to the generalized plant \(P_L(s)\) shown in Figure \ref{fig:feedback_synthesis_architecture_generalized_plant} and described by \eqref{eq:generalized_plant_mixed_sensitivity} where \(W_1(s)\) and \(W_2(s)\) are weighting functions used to respectively shape \(S(s)\) and \(T(s)\).
\begin{equation}
\label{eq:generalized_plant_mixed_sensitivity}
\begin{bmatrix} z \\ v \end{bmatrix} = P_L(s) \begin{bmatrix} w_1 \\ w_2 \\ u \end{bmatrix}; \quad P_L(s) = \begin{bmatrix}
\phantom{+}W_1(s) & 0 & \phantom{+}1 \\
-W_1(s) & W_2(s) & -1
\end{bmatrix}
\end{equation}
This is equivalent as to find a filter \(L(s)\) such that \eqref{eq:comp_filters_feedback_obj} is verified.
\begin{equation}
\label{eq:comp_filters_feedback_obj}
\left\|\begin{matrix} \frac{1}{1 + L(s)} W_1(s) \\ \frac{L(s)}{1 + L(s)} W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
The sensor fusion can be implemented as shown in Figure \ref{fig:feedback_sensor_fusion_arch} using the feedback architecture or more classically as shown in Figure \ref{fig:sensor_fusion_overview} using \eqref{eq:comp_filters_feedback}.
\begin{equation}
\label{eq:comp_filters_feedback}
H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)}
\end{equation}
The two being equivalent considering only the inputs/outputs relationships.
\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 the \(\mathcal{H}_\infty\) mixed-sensitivity synthesis}
\end{figure}
Example: same weights as in \ref{tab:weights_params}.
Therefore, complementary filter design is very similar to mixed-sensitivity synthesis.
They are actually equivalent by taking
@ -576,30 +625,36 @@ 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:org402c2aa}
\label{sec:orgdea775a}
\label{sec:add_features_in_filters}
3 methods:
Link to literature about doing that with mixed sensitivity
\subsection{Synthesis of Three Complementary Filters}
\label{sec:orgf9a165b}
\label{sec:org6446998}
\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.
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
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}).
In the first case, typical sensor fusion synthesis techniques can be used.
However, when a parallel architecture is used, a new synthesis method for a set of more than two complementary filters is required.
Such synthesis method is presented in this section. \par
\begin{center}
\fbox{
\begin{minipage}[c]{.6\textwidth}
Say possible advantages of parallel architecture
\end{minipage}
}
\end{center}
\begin{figure}[htbp]
\begin{subfigure}[b]{0.59\linewidth}
\centering
@ -615,7 +670,7 @@ In such a case, it is necessary to design as many complementary filters as the n
\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.
The synthesis objective is to compute a set of \(n\) stable transfer functions \([H_1(s),\ H_2(s),\ \dots,\ H_n(s)]\) such that \eqref{eq:hinf_problem_gen} is satisfied.
\begin{subequations}
\label{eq:hinf_problem_gen}
\begin{align}
@ -623,12 +678,14 @@ The synthesis problem is then to compute \(n\) stable transfer functions \(H_i(s
& \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:hinf_cond_perf_gen}
\end{align}
\end{subequations}
The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. \ref{fig:comp_filter_three_hinf}.
where \([W_1(s),\ W_2(s),\ \dots,\ W_n(s)]\) are weighting transfer functions that are chosen to specify the maximum wanted norms of the complementary filters during the synthesis.
Such synthesis objective is very close to the one described in Section \ref{sec:synthesis_objective}, and indeed the proposed synthesis architecture is also very similar. \par
Consider the generalized plant \(P_3(s)\) shown in Figure \ref{fig:comp_filter_three_hinf} which is also described by \eqref{eq:generalized_plant_three_filters}.
The \(\mathcal{H}_\infty\) synthesis objective applied on \(P(s)\) is to design two stable filters \(H_2(s)\) and \(H_3(s)\) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2, \ z_3]\) is less than one \eqref{eq:hinf_syn_obj_three}.
\begin{equation}
\label{eq:hinf_syn_obj_three}
\left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
\label{eq:generalized_plant_three_filters}
\begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix}
\end{equation}
\begin{figure}[htbp]
@ -637,19 +694,40 @@ The \(\mathcal{H}_\infty\) synthesis objective applied on \(P(s)\) is to design
\caption{\label{fig:comp_filter_three_hinf}Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters}
\end{figure}
By choosing \(H_1(s) \triangleq 1 - H_2(s) - H_3(s)\), the proposed \(\mathcal{H}_\infty\) synthesis solves the design problem \eqref{eq:hinf_problem_gen}. \par
Applying the \(\mathcal{H}_\infty\) synthesis on the generalized plant \(P_3(s)\) is equivalent as to find two stable filters \([H_2(s),\ H_3(s)]\) (shown in Figure \ref{fig:comp_filter_three_hinf}) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2, \ z_3]\) is less than one \eqref{eq:hinf_syn_obj_three}.
\begin{equation}
\label{eq:hinf_syn_obj_three}
\left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
\end{equation}
By defining \(H_1(s) \triangleq 1 - H_2(s) - H_3(s)\), the proposed \(\mathcal{H}_\infty\) synthesis solves the design problem \eqref{eq:hinf_problem_gen} with \(n=3\). \par
An example is given to validate the method where three sensors are used in different frequency bands (up to \(\SI{1}{Hz}\), from \(1\) to \(\SI{10}{Hz}\) and above \(\SI{10}{Hz}\) respectively).
Three weighting functions are designed using \eqref{eq:weight_formula} and shown by dashed curves in Fig. \ref{fig:three_complementary_filters_results}.
The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:three_complementary_filters_results}.
The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:three_complementary_filters_results}. \par
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/three_complementary_filters_results.pdf}
\caption{\label{fig:three_complementary_filters_results}Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
\end{figure}
Such synthesis method can be generalized to a set of \(n\) complementary filters, even though there might not be any practical application for \(n>3\).
\begin{equation}
\label{eq:generalized_plant_n_filters}
\begin{bmatrix} z_1 \\ \vdots \\ z_n \\ v \end{bmatrix} = P_n(s) \begin{bmatrix} w \\ u_1 \\ \vdots \\ u_{n-1} \end{bmatrix}; \quad
P_n(s) = \begin{bmatrix}
W_1 & -W_1 & \dots & \dots & -W_1 \\
0 & W_2 & 0 & \dots & 0 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots & & \ddots & \ddots & 0 \\
0 & \dots & \dots & 0 & W_n \\
1 & 0 & \dots & \dots & 0
\end{bmatrix}
\end{equation}
\section{Conclusion}
\label{sec:org75ed4d0}
\label{sec:orgcba6c13}
\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.
@ -657,7 +735,7 @@ 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:org0b419b1}
\label{sec:orgf175dee}
This research benefited from a FRIA grant from the French Community of Belgium.
\bibliographystyle{elsarticle-num}

27
journal/ref.bib
View File

@ -260,20 +260,6 @@
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