Update journal paper
56
Makefile
Normal file
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.PHONY: all paper help html publish watch clean cp-figs
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SHELL := /bin/bash
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PAPERDIR=paper
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MATLABDIR=matlab
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TIKZDIR=tikz
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PAPERNAME=paper
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all: paper html publish
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paper: cp-figs tangle tex pdf clean
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help:
|
||||
@echo "Usage: make <command>"
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||||
@echo " all - Cp-figs tex pdf html publish"
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||||
@echo " paper - Compile the org file to a pdf"
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@echo " html - Export all the org files to html"
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@echo " publish - Commit everything and push to repository"
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@echo " tex - Export to paper in org format to tex"
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@echo " tangle - Tangle everything that is in the org paper and tikz file"
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@echo " pdf - Compile the tex file to pdf"
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@echo " watch - Watch the tex file for changes and compile"
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@echo " clean - Clean the paper directory"
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@echo " cp-figs - Copy all the necessary figures from tikz and matlab folder to paper folder"
|
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|
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html:
|
||||
for f in *.org; do emacsclient -e "(progn (find-file \"$$f\") (org-html-export-to-html))"; done
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for f in $(TIKZDIR)/*.org; do emacsclient -e "(progn (find-file \"$$f\") (org-html-export-to-html))"; done
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for f in $(MATLABDIR)/*.org; do emacsclient -e "(progn (find-file \"$$f\") (org-html-export-to-html))"; done
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publish:
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git add . && git commit -m "Update - $$(date +%F)" && git push origin master
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tex: $(PAPERDIR)/$(PAPERNAME).org
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||||
emacsclient -e '(progn (find-file "$(PAPERDIR)/$(PAPERNAME).org") (org-latex-export-to-latex))'
|
||||
|
||||
tangle: $(PAPERDIR)/$(PAPERNAME).org
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||||
emacsclient -e '(progn (find-file "$(PAPERDIR)/$(PAPERNAME).org") (org-babel-tangle))'
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emacsclient -e '(progn (find-file "$(TIKZDIR)/index.org") (org-babel-tangle))'
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pdf: $(PAPERDIR)/$(PAPERNAME).tex
|
||||
latexmk -cd -quiet -bibtex $(PREVIEW_CONTINUOUSLY) -f -pdf -pdflatex="xelatex -synctex=1 -interaction nonstopmode" -use-make $(PAPERDIR)/$(PAPERNAME).tex
|
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|
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# Set the PREVIEW_CONTINUOUSLY variable to -pvc to switch latexmk into the preview continuously mode
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watch: PREVIEW_CONTINUOUSLY=-pvc
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||||
watch: pdf
|
||||
|
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watch-org: $(PAPERDIR)/$(PAPERNAME).org
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||||
echo $(PAPERDIR)/$(PAPERNAME).org | entr -s 'make tangle tex pdf'
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||||
|
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clean:
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latexmk -cd -c -bibtex $(PAPERDIR)/$(PAPERNAME).tex
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cp-figs:
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bash scripts/cp-figs.sh
|
Before Width: | Height: | Size: 19 KiB After Width: | Height: | Size: 19 KiB |
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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
|
||||
@ -711,15 +762,21 @@ Link to literature about doing that with mixed sensitivity
|
||||
** Synthesis of Three Complementary Filters
|
||||
<<sec:hinf_three_comp_filters>>
|
||||
|
||||
*** Why it is used sometimes :ignore:
|
||||
*** 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.
|
||||
|
||||
*** 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.
|
||||
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
|
||||
|
||||
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
|
||||
*** H-Infinity Architecture :ignore:
|
||||
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.
|
||||
|
||||
#+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.
|
||||
|
||||
*** Example of generated complementary filters :ignore:
|
||||
#+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.
|
||||
|
@ -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}
|
||||
|
@ -260,20 +260,6 @@
|
||||
publisher = {IEEE},
|
||||
}
|
||||
|
||||
@article{mahony08_nonlin_compl_filter_special_orthog_group,
|
||||
author = {Mahony, Robert and Hamel, Tarek and Pflimlin, Jean-Michel},
|
||||
title = {Nonlinear Complementary Filters on the Special Orthogonal
|
||||
Group},
|
||||
journal = {IEEE Transactions on automatic control},
|
||||
volume = 53,
|
||||
number = 5,
|
||||
pages = {1203--1218},
|
||||
year = 2008,
|
||||
doi = {10.1109/TAC.2008.923738},
|
||||
url = {https://doi.org/10.1109/TAC.2008.923738},
|
||||
publisher = {IEEE},
|
||||
}
|
||||
|
||||
@article{pascoal00_navig_system_desig_using_time,
|
||||
author = {Pascoal, Antonio and Kaminer, Isaac and Oliveira, Paulo},
|
||||
title = {Navigation System Design Using Time-Varying Complementary
|
||||
@ -586,3 +572,16 @@
|
||||
Virgo Seismic Isolation System},
|
||||
year = 2018,
|
||||
}
|
||||
|
||||
@inproceedings{mahony05_compl_filter_desig_special_orthog,
|
||||
author = {R. Mahony and T. Hamel and J.-M. Pflimlin},
|
||||
title = {Complementary Filter Design on the Special Orthogonal Group
|
||||
SO(3)},
|
||||
booktitle = {Proceedings of the 44th IEEE Conference on Decision and
|
||||
Control},
|
||||
year = 2005,
|
||||
pages = {nil},
|
||||
doi = {10.1109/cdc.2005.1582367},
|
||||
url = {https://doi.org/10.1109/cdc.2005.1582367},
|
||||
month = {-},
|
||||
}
|
||||
|