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dehaeze21_desig_compl_filte.bib
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dehaeze21_desig_compl_filte.bib
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@article{dehaeze21_new_method_desig_compl_filter,
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author = {Dehaeze, Thomas and Vermat, Mohit and Collette, Christophe},
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title = {A New Method of Designing Complementary Filters for Sensor
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Fusion Using the $\mathcal{H}_\infty$ Synthesis},
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journal = {Mechanical Systems and Signal Processing},
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year = 2021,
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month = {Nov},
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}
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index.html
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index.html
@ -3,7 +3,7 @@
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2021-06-21 lun. 11:42 -->
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<!-- 2021-09-01 mer. 11:36 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis</title>
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<meta name="author" content="Thomas Dehaeze" />
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@ -33,7 +33,7 @@
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<a accesskey="h" href="../index.html"> UP </a>
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|
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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</div><div id="content" class="content">
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<h1 class="title">Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis
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<br />
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<span class="subtitle">Dehaeze Thomas, Verma Mohit and Collette Christophe</span>
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@ -50,27 +50,27 @@ This method is shown to be easily applicable for the synthesis of complex comple
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</p>
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</blockquote>
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<div id="outline-container-org16737c1" class="outline-2">
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<h2 id="org16737c1">Paper (<a href="journal/journal.pdf">link</a>)</h2>
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<div class="outline-text-2" id="text-org16737c1">
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<div id="outline-container-orgafee5c3" class="outline-2">
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<h2 id="orgafee5c3">Paper (<a href="journal/dehaeze21_desig_compl_filte.pdf">link</a>)</h2>
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<div class="outline-text-2" id="text-orgafee5c3">
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<p>
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The paper has been created <a href="https://orgmode.org/">Org Mode</a> (generating <a href="https://www.latex-project.org/">LaTeX</a> code) under <a href="https://www.gnu.org/software/emacs/">Emacs</a>.
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</p>
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</div>
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</div>
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<div id="outline-container-orgb50a720" class="outline-2">
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<h2 id="orgb50a720">Matlab Scripts (<a href="matlab/index.html">link</a>)</h2>
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<div class="outline-text-2" id="text-orgb50a720">
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<div id="outline-container-orgb8fb429" class="outline-2">
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<h2 id="orgb8fb429">Matlab Scripts (<a href="matlab/dehaeze21_desig_compl_filte_matlab.html">link</a>)</h2>
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<div class="outline-text-2" id="text-orgb8fb429">
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<p>
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All the <a href="https://fr.mathworks.com/">Matlab</a> code that was used for the paper are accessible so that all the results are reproducible.
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</p>
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</div>
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</div>
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<div id="outline-container-org06a9b71" class="outline-2">
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<h2 id="org06a9b71">Tikz Figures (<a href="tikz/index.html">link</a>)</h2>
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<div class="outline-text-2" id="text-org06a9b71">
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||||
<div id="outline-container-org0ecd5c8" class="outline-2">
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<h2 id="org0ecd5c8">Tikz Figures (<a href="tikz/dehaeze21_desig_compl_filte_tikz.html">link</a>)</h2>
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<div class="outline-text-2" id="text-org0ecd5c8">
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<p>
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All the figures for the paper have been generated using <a href="https://sourceforge.net/projects/pgf/">TikZ</a>.
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</p>
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@ -20,13 +20,13 @@
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This method is shown to be easily applicable for the synthesis of complex complementary filters.
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#+end_quote
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* Paper ([[file:journal/journal.pdf][link]])
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* Paper ([[file:journal/dehaeze21_desig_compl_filte.pdf][link]])
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:PROPERTIES:
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:UNNUMBERED: t
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:END:
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The paper has been created [[https://orgmode.org/][Org Mode]] (generating [[https://www.latex-project.org/][LaTeX]] code) under [[https://www.gnu.org/software/emacs/][Emacs]].
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* Matlab Scripts ([[file:matlab/index.org][link]])
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* Matlab Scripts ([[file:matlab/dehaeze21_desig_compl_filte_matlab.org][link]])
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:PROPERTIES:
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:UNNUMBERED: t
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:END:
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\begin{tikzpicture}
|
||||
\node[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
|
||||
\node[above] at (P.north) {$P(s)$};
|
||||
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
|
||||
|
||||
\coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);
|
||||
|
||||
\node[block, left=1.4 of output1] (W1){$W_1(s)$};
|
||||
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
\node[block, below=0.3 of P] (H2) {$H_2(s)$};
|
||||
|
||||
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
|
||||
\draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west);
|
||||
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
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||||
\draw[->] (sub.east) -- (W1.west);
|
||||
\draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
|
||||
\draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
|
||||
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
||||
\end{tikzpicture}
|
||||
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matlab/matlab/1_synthesis_complementary_filters.m
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matlab/matlab/1_synthesis_complementary_filters.m
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@ -0,0 +1,123 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
%% Initialize Frequency Vector
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
|
||||
%% Add functions to path
|
||||
addpath('./src');
|
||||
|
||||
%% Weighting Function Design
|
||||
% Parameters
|
||||
n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2;
|
||||
|
||||
% Formulas
|
||||
W = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
%% Magnitude of the weighting function with parameters
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(W, freqs, 'Hz'))), 'k-');
|
||||
|
||||
plot([1e-3 1e0], [G0 G0], 'k--', 'LineWidth', 1)
|
||||
text(1e0, G0, '$\quad G_0$')
|
||||
|
||||
plot([1e1 1e3], [G1 G1], 'k--', 'LineWidth', 1)
|
||||
text(1e1,G1,'$G_{\infty}\quad$','HorizontalAlignment', 'right')
|
||||
|
||||
plot([w0/2/pi w0/2/pi], [1 2*Gc], 'k--', 'LineWidth', 1)
|
||||
text(w0/2/pi,1,'$\omega_c$','VerticalAlignment', 'top', 'HorizontalAlignment', 'center')
|
||||
|
||||
plot([w0/2/pi/2 2*w0/2/pi], [Gc Gc], 'k--', 'LineWidth', 1)
|
||||
text(w0/2/pi/2, Gc, '$G_c \quad$','HorizontalAlignment', 'right')
|
||||
|
||||
text(w0/5/pi/2, abs(evalfr(W, j*w0/5)), 'Slope: $n \quad$', 'HorizontalAlignment', 'right')
|
||||
|
||||
text(w0/2/pi, abs(evalfr(W, j*w0)), '$\bullet$', 'HorizontalAlignment', 'center')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([5e-4, 20]);
|
||||
yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
|
||||
|
||||
%% Design of the Weighting Functions
|
||||
W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'G1', 1/10, 'Gc', 0.45);
|
||||
W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'G1', 1000, 'Gc', 0.45);
|
||||
|
||||
%% Plot of the Weighting function magnitude
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]', 'FontSize', 10); ylabel('Magnitude', 'FontSize', 10);
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
xticks([0.1, 1, 10, 100, 1000]);
|
||||
ylim([8e-4, 20]);
|
||||
yticks([1e-3, 1e-2, 1e-1, 1, 1e1]);
|
||||
yticklabels({'', '$10^{-2}$', '', '$10^0$', ''});
|
||||
ax1.FontSize = 9;
|
||||
leg = legend('location', 'south', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
%% Generalized Plant
|
||||
P = [W1 -W1;
|
||||
0 W2;
|
||||
1 0];
|
||||
|
||||
%% H-Infinity Synthesis
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
%% Define H1 to be the complementary of H2
|
||||
H1 = 1 - H2;
|
||||
|
||||
%% Bode plot of the complementary filters
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude');
|
||||
ylim([8e-4, 20]);
|
||||
yticks([1e-3, 1e-2, 1e-1, 1, 1e1]);
|
||||
yticklabels({'', '$10^{-2}$', '', '$10^0$', ''})
|
||||
leg = legend('location', 'south', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
yticks([-180:90:180]);
|
||||
ylim([-180, 200])
|
||||
yticklabels({'-180', '', '0', '', '180'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
258
matlab/matlab/2_ligo_complementary_filters.m
Normal file
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matlab/matlab/2_ligo_complementary_filters.m
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@ -0,0 +1,258 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
%% Initialize Frequency Vector
|
||||
freqs = logspace(-3, 0, 1000);
|
||||
|
||||
%% Add functions to path
|
||||
addpath('./src');
|
||||
|
||||
%% Upper bounds for the complementary filters
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-4, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
%% Initialized CVX
|
||||
cvx_startup;
|
||||
cvx_solver sedumi;
|
||||
|
||||
%% Frequency vectors
|
||||
w1 = 0:4.06e-4:0.008;
|
||||
w2 = 0.008:4.06e-4:0.04;
|
||||
w3 = 0.04:8.12e-4:0.1;
|
||||
w4 = 0.1:8.12e-4:0.83;
|
||||
|
||||
%% Filter order
|
||||
n = 512;
|
||||
|
||||
%% Initialization of filter responses
|
||||
A1 = [ones(length(w1),1), cos(kron(w1'.*(2*pi),[1:n-1]))];
|
||||
A2 = [ones(length(w2),1), cos(kron(w2'.*(2*pi),[1:n-1]))];
|
||||
A3 = [ones(length(w3),1), cos(kron(w3'.*(2*pi),[1:n-1]))];
|
||||
A4 = [ones(length(w4),1), cos(kron(w4'.*(2*pi),[1:n-1]))];
|
||||
|
||||
B1 = [zeros(length(w1),1), sin(kron(w1'.*(2*pi),[1:n-1]))];
|
||||
B2 = [zeros(length(w2),1), sin(kron(w2'.*(2*pi),[1:n-1]))];
|
||||
B3 = [zeros(length(w3),1), sin(kron(w3'.*(2*pi),[1:n-1]))];
|
||||
B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))];
|
||||
|
||||
%% Convex optimization
|
||||
cvx_begin
|
||||
|
||||
variable y(n+1,1)
|
||||
|
||||
% t
|
||||
maximize(-y(1))
|
||||
|
||||
for i = 1:length(w1)
|
||||
norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
|
||||
end
|
||||
|
||||
for i = 1:length(w2)
|
||||
norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
|
||||
end
|
||||
|
||||
for i = 1:length(w3)
|
||||
norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
|
||||
end
|
||||
|
||||
for i = 1:length(w4)
|
||||
norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
|
||||
end
|
||||
|
||||
cvx_end
|
||||
|
||||
h = y(2:end);
|
||||
|
||||
%% Combine the frequency vectors to form the obtained filter
|
||||
w = [w1 w2 w3 w4];
|
||||
H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
|
||||
|
||||
%% Bode plot of the obtained complementary filters
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(w, abs(1-H), '-', 'DisplayName', '$L_1$');
|
||||
plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
|
||||
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(w, abs(H), '-', 'DisplayName', '$H_1$');
|
||||
plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
|
||||
plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
|
||||
plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
|
||||
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude');
|
||||
hold off;
|
||||
ylim([5e-3, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 16;
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(w, 180/pi*unwrap(angle(1-H)), '-');
|
||||
plot(w, 180/pi*unwrap(angle(H)), '-');
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'XScale', 'log');
|
||||
yticks([-360:180:180]); ylim([-380, 200]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([1e-3, 1]);
|
||||
|
||||
%% Design of the weight for the high pass filter
|
||||
w1 = 2*pi*0.008; x1 = 0.35;
|
||||
w2 = 2*pi*0.04; x2 = 0.5;
|
||||
w3 = 2*pi*0.05; x3 = 0.5;
|
||||
|
||||
% Slope of +3 from w1
|
||||
wH = 0.008*(s^2/w1^2 + 2*x1/w1*s + 1)*(s/w1 + 1);
|
||||
% Little bump from w2 to w3
|
||||
wH = wH*(s^2/w2^2 + 2*x2/w2*s + 1)/(s^2/w3^2 + 2*x3/w3*s + 1);
|
||||
% No Slope at high frequencies
|
||||
wH = wH/(s^2/w3^2 + 2*x3/w3*s + 1)/(s/w3 + 1);
|
||||
% Little bump between w2 and w3
|
||||
w0 = 2*pi*0.045; xi = 0.1; A = 2; n = 1;
|
||||
wH = wH*((s^2 + 2*w0*xi*A^(1/n)*s + w0^2)/(s^2 + 2*w0*xi*s + w0^2))^n;
|
||||
|
||||
wH = 1/wH;
|
||||
wH = minreal(ss(wH));
|
||||
|
||||
%% Design of the weight for the low pass filter
|
||||
n = 20; % Filter order
|
||||
Rp = 1; % Peak to peak passband ripple
|
||||
Wp = 2*pi*0.102; % Edge frequency
|
||||
|
||||
% Chebyshev Type I filter design
|
||||
[b,a] = cheby1(n, Rp, Wp, 'high', 's');
|
||||
wL = 0.04*tf(a, b);
|
||||
|
||||
wL = 1/wL;
|
||||
wL = minreal(ss(wL));
|
||||
|
||||
%% Magnitude of the designed Weights and initial specifications
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|W_L|^{-1}$');
|
||||
plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
|
||||
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|W_H|^{-1}$');
|
||||
plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
|
||||
plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
|
||||
plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
|
||||
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([5e-3, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 16;
|
||||
|
||||
%% Generalized plant for the H-infinity Synthesis
|
||||
P = [0 wL;
|
||||
wH -wH;
|
||||
1 0];
|
||||
|
||||
%% Standard H-Infinity synthesis
|
||||
[Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
%% High pass filter as the complementary of the low pass filter
|
||||
Hh = 1 - Hl;
|
||||
|
||||
%% Minimum realization of the filters
|
||||
Hh = minreal(Hh);
|
||||
Hl = minreal(Hl);
|
||||
|
||||
%% Bode plot of the obtained filters and comparison with the upper bounds
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'DisplayName', '$L_1^\prime$');
|
||||
plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
|
||||
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$H_1^\prime$');
|
||||
plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
|
||||
plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
|
||||
plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
|
||||
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([5e-3, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 16;
|
||||
|
||||
%% Comparison of the complementary filters obtained with H-infinity and with CVX
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', '$L_1(s)$ - $\mathcal{H}_\infty$');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', '$H_1(s)$ - $\mathcal{H}_\infty$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(w, abs(1-H), '--', ...
|
||||
'DisplayName', '$L_1(s)$ - FIR');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(w, abs(H), '--', ...
|
||||
'DisplayName', '$H_1(s)$ - FIR');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylim([5e-3, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 16;
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hl, freqs, 'Hz')))), '-');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hh, freqs, 'Hz')))), '-');
|
||||
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(w, 180/pi*unwrap(angle(1-H)), '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(w, 180/pi*unwrap(angle(H)), '--');
|
||||
set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
yticks([-360:180:180]); ylim([-380, 200]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
71
matlab/matlab/3_closed_loop_complementary_filters.m
Normal file
71
matlab/matlab/3_closed_loop_complementary_filters.m
Normal file
@ -0,0 +1,71 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
%% Initialize Frequency Vector
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
|
||||
%% Add functions to path
|
||||
addpath('./src');
|
||||
|
||||
%% Design of the Weighting Functions
|
||||
W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'G1', 1/10, 'Gc', 0.45);
|
||||
W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'G1', 1000, 'Gc', 0.45);
|
||||
|
||||
%% Generalized plant for "closed-loop" complementary filter synthesis
|
||||
P = [ W1 0 1;
|
||||
-W1 W2 -1];
|
||||
|
||||
%% Standard H-Infinity Synthesis
|
||||
[L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
%% Complementary filters
|
||||
H1 = inv(1 + L);
|
||||
H2 = 1 - H1;
|
||||
|
||||
%% Bode plot of the obtained Complementary filters with upper-bounds
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
|
||||
plot(freqs, abs(squeeze(freqresp(L, freqs, 'Hz'))), 'k--', 'DisplayName', '$|L|$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude');
|
||||
ylim([1e-3, 1e3]);
|
||||
yticks([1e-3, 1e-2, 1e-1, 1, 1e1, 1e2, 1e3]);
|
||||
yticklabels({'', '$10^{-2}$', '', '$10^0$', '', '$10^2$', ''});
|
||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
yticks([-180:90:180]);
|
||||
ylim([-180, 200])
|
||||
yticklabels({'-180', '', '0', '', '180'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
@ -4,21 +4,19 @@ clear; close all; clc;
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
freqs = logspace(-2, 4, 1000);
|
||||
freqs = logspace(-2, 3, 1000);
|
||||
|
||||
addpath('./src');
|
||||
|
||||
% Weights
|
||||
% First we define the weights.
|
||||
|
||||
n = 2; w0 = 2*pi*1; G0 = 1/10; G1 = 1000; Gc = 1/2;
|
||||
W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
%% Design of the Weighting Functions
|
||||
W1 = generateWF('n', 2, 'w0', 2*pi*1, 'G0', 1/10, 'G1', 1000, 'Gc', 0.5);
|
||||
W2 = 0.22*(1 + s/2/pi/1)^2/(sqrt(1e-4) + s/2/pi/1)^2*(1 + s/2/pi/10)^2/(1 + s/2/pi/1000)^2;
|
||||
W3 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'G1', 1/10, 'Gc', 0.5);
|
||||
|
||||
n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 1/2;
|
||||
W3 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
%% Inverse magnitude of the weighting functions
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
@ -37,6 +35,7 @@ leg.ItemTokenSize(1) = 18;
|
||||
% H-Infinity Synthesis
|
||||
% Then we create the generalized plant =P=.
|
||||
|
||||
%% Generalized plant for the synthesis of 3 complementary filters
|
||||
P = [W1 -W1 -W1;
|
||||
0 W2 0 ;
|
||||
0 0 W3;
|
||||
@ -46,15 +45,18 @@ P = [W1 -W1 -W1;
|
||||
|
||||
% And we do the $\mathcal{H}_\infty$ synthesis.
|
||||
|
||||
%% Standard H-Infinity Synthesis
|
||||
[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
% Obtained Complementary Filters
|
||||
% The obtained filters are:
|
||||
|
||||
%%
|
||||
H2 = tf(H(1));
|
||||
H3 = tf(H(2));
|
||||
H1 = 1 - H2 - H3;
|
||||
|
||||
%% Bode plot of the obtained complementary filters
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
@ -79,7 +81,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylim([1e-4, 20]);
|
||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg = legend('location', 'northeast', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
% Phase
|
||||
@ -94,7 +96,7 @@ plot(freqs, 180/pi*phase(squeeze(freqresp(H3, freqs, 'Hz'))));
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'XScale', 'log');
|
||||
yticks([-360:90:360]); ylim([-270, 270]);
|
||||
yticks([-180:90:180]); ylim([-220, 220]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
@ -1,410 +0,0 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
freqs = logspace(-3, 0, 1000);
|
||||
|
||||
addpath('./src');
|
||||
|
||||
% Specifications
|
||||
% The specifications for the filters are:
|
||||
% 1. From $0$ to $0.008\text{ Hz}$,the magnitude of the filter’s transfer function should be less than or equal to $8 \times 10^{-3}$
|
||||
% 2. From $0.008\text{ Hz}$ to $0.04\text{ Hz}$, it attenuates the input signal proportional to frequency cubed
|
||||
% 3. Between $0.04\text{ Hz}$ and $0.1\text{ Hz}$, the magnitude of the transfer function should be less than 3
|
||||
% 4. Above $0.1\text{ Hz}$, the maximum of the magnitude of the complement filter should be as close to zero as possible. In our system, we would like to have the magnitude of the complementary filter to be less than $0.1$. As the filters obtained in cite:hua05_low_ligo have a magnitude of $0.045$, we will set that as our requirement
|
||||
|
||||
% The specifications are translated in upper bounds of the complementary filters are shown on figure [[fig:ligo_specifications]].
|
||||
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-4, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
% FIR Filter
|
||||
% We here try to implement the FIR complementary filter synthesis as explained in cite:hua05_low_ligo.
|
||||
% For that, we use the [[http://cvxr.com/cvx/][CVX matlab Toolbox]].
|
||||
|
||||
% We setup the CVX toolbox and use the =SeDuMi= solver.
|
||||
|
||||
cvx_startup;
|
||||
cvx_solver sedumi;
|
||||
|
||||
|
||||
|
||||
% We define the frequency vectors on which we will constrain the norm of the FIR filter.
|
||||
|
||||
w1 = 0:4.06e-4:0.008;
|
||||
w2 = 0.008:4.06e-4:0.04;
|
||||
w3 = 0.04:8.12e-4:0.1;
|
||||
w4 = 0.1:8.12e-4:0.83;
|
||||
|
||||
|
||||
|
||||
% We then define the order of the FIR filter.
|
||||
|
||||
n = 512;
|
||||
|
||||
A1 = [ones(length(w1),1), cos(kron(w1'.*(2*pi),[1:n-1]))];
|
||||
A2 = [ones(length(w2),1), cos(kron(w2'.*(2*pi),[1:n-1]))];
|
||||
A3 = [ones(length(w3),1), cos(kron(w3'.*(2*pi),[1:n-1]))];
|
||||
A4 = [ones(length(w4),1), cos(kron(w4'.*(2*pi),[1:n-1]))];
|
||||
|
||||
B1 = [zeros(length(w1),1), sin(kron(w1'.*(2*pi),[1:n-1]))];
|
||||
B2 = [zeros(length(w2),1), sin(kron(w2'.*(2*pi),[1:n-1]))];
|
||||
B3 = [zeros(length(w3),1), sin(kron(w3'.*(2*pi),[1:n-1]))];
|
||||
B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))];
|
||||
|
||||
|
||||
|
||||
% We run the convex optimization.
|
||||
|
||||
cvx_begin
|
||||
|
||||
variable y(n+1,1)
|
||||
|
||||
% t
|
||||
maximize(-y(1))
|
||||
|
||||
for i = 1:length(w1)
|
||||
norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
|
||||
end
|
||||
|
||||
for i = 1:length(w2)
|
||||
norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
|
||||
end
|
||||
|
||||
for i = 1:length(w3)
|
||||
norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
|
||||
end
|
||||
|
||||
for i = 1:length(w4)
|
||||
norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
|
||||
end
|
||||
|
||||
cvx_end
|
||||
|
||||
h = y(2:end);
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% #+begin_example
|
||||
% cvx_begin
|
||||
% variable y(n+1,1)
|
||||
% % t
|
||||
% maximize(-y(1))
|
||||
% for i = 1:length(w1)
|
||||
% norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
|
||||
% end
|
||||
% for i = 1:length(w2)
|
||||
% norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
|
||||
% end
|
||||
% for i = 1:length(w3)
|
||||
% norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
|
||||
% end
|
||||
% for i = 1:length(w4)
|
||||
% norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
|
||||
% end
|
||||
% cvx_end
|
||||
|
||||
% Calling SeDuMi 1.34: 4291 variables, 1586 equality constraints
|
||||
% For improved efficiency, SeDuMi is solving the dual problem.
|
||||
% ------------------------------------------------------------
|
||||
% SeDuMi 1.34 (beta) by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
|
||||
% Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
|
||||
% eqs m = 1586, order n = 3220, dim = 4292, blocks = 1073
|
||||
% nnz(A) = 1100727 + 0, nnz(ADA) = 1364794, nnz(L) = 683190
|
||||
% it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
|
||||
% 0 : 4.11E+02 0.000
|
||||
% 1 : -2.58E+00 1.25E+02 0.000 0.3049 0.9000 0.9000 4.87 1 1 3.0E+02
|
||||
% 2 : -2.36E+00 3.90E+01 0.000 0.3118 0.9000 0.9000 1.83 1 1 6.6E+01
|
||||
% 3 : -1.69E+00 1.31E+01 0.000 0.3354 0.9000 0.9000 1.76 1 1 1.5E+01
|
||||
% 4 : -8.60E-01 7.10E+00 0.000 0.5424 0.9000 0.9000 2.48 1 1 4.8E+00
|
||||
% 5 : -4.91E-01 5.44E+00 0.000 0.7661 0.9000 0.9000 3.12 1 1 2.5E+00
|
||||
% 6 : -2.96E-01 3.88E+00 0.000 0.7140 0.9000 0.9000 2.62 1 1 1.4E+00
|
||||
% 7 : -1.98E-01 2.82E+00 0.000 0.7271 0.9000 0.9000 2.14 1 1 8.5E-01
|
||||
% 8 : -1.39E-01 2.00E+00 0.000 0.7092 0.9000 0.9000 1.78 1 1 5.4E-01
|
||||
% 9 : -9.99E-02 1.30E+00 0.000 0.6494 0.9000 0.9000 1.51 1 1 3.3E-01
|
||||
% 10 : -7.57E-02 8.03E-01 0.000 0.6175 0.9000 0.9000 1.31 1 1 2.0E-01
|
||||
% 11 : -5.99E-02 4.22E-01 0.000 0.5257 0.9000 0.9000 1.17 1 1 1.0E-01
|
||||
% 12 : -5.28E-02 2.45E-01 0.000 0.5808 0.9000 0.9000 1.08 1 1 5.9E-02
|
||||
% 13 : -4.82E-02 1.28E-01 0.000 0.5218 0.9000 0.9000 1.05 1 1 3.1E-02
|
||||
% 14 : -4.56E-02 5.65E-02 0.000 0.4417 0.9045 0.9000 1.02 1 1 1.4E-02
|
||||
% 15 : -4.43E-02 2.41E-02 0.000 0.4265 0.9004 0.9000 1.01 1 1 6.0E-03
|
||||
% 16 : -4.37E-02 8.90E-03 0.000 0.3690 0.9070 0.9000 1.00 1 1 2.3E-03
|
||||
% 17 : -4.35E-02 3.24E-03 0.000 0.3641 0.9164 0.9000 1.00 1 1 9.5E-04
|
||||
% 18 : -4.34E-02 1.55E-03 0.000 0.4788 0.9086 0.9000 1.00 1 1 4.7E-04
|
||||
% 19 : -4.34E-02 8.77E-04 0.000 0.5653 0.9169 0.9000 1.00 1 1 2.8E-04
|
||||
% 20 : -4.34E-02 5.05E-04 0.000 0.5754 0.9034 0.9000 1.00 1 1 1.6E-04
|
||||
% 21 : -4.34E-02 2.94E-04 0.000 0.5829 0.9136 0.9000 1.00 1 1 9.9E-05
|
||||
% 22 : -4.34E-02 1.63E-04 0.015 0.5548 0.9000 0.0000 1.00 1 1 6.6E-05
|
||||
% 23 : -4.33E-02 9.42E-05 0.000 0.5774 0.9053 0.9000 1.00 1 1 3.9E-05
|
||||
% 24 : -4.33E-02 6.27E-05 0.000 0.6658 0.9148 0.9000 1.00 1 1 2.6E-05
|
||||
% 25 : -4.33E-02 3.75E-05 0.000 0.5972 0.9187 0.9000 1.00 1 1 1.6E-05
|
||||
% 26 : -4.33E-02 1.89E-05 0.000 0.5041 0.9117 0.9000 1.00 1 1 8.6E-06
|
||||
% 27 : -4.33E-02 9.72E-06 0.000 0.5149 0.9050 0.9000 1.00 1 1 4.5E-06
|
||||
% 28 : -4.33E-02 2.94E-06 0.000 0.3021 0.9194 0.9000 1.00 1 1 1.5E-06
|
||||
% 29 : -4.33E-02 9.73E-07 0.000 0.3312 0.9189 0.9000 1.00 2 2 5.3E-07
|
||||
% 30 : -4.33E-02 2.82E-07 0.000 0.2895 0.9063 0.9000 1.00 2 2 1.6E-07
|
||||
% 31 : -4.33E-02 8.05E-08 0.000 0.2859 0.9049 0.9000 1.00 2 2 4.7E-08
|
||||
% 32 : -4.33E-02 1.43E-08 0.000 0.1772 0.9059 0.9000 1.00 2 2 8.8E-09
|
||||
|
||||
% iter seconds digits c*x b*y
|
||||
% 32 49.4 6.8 -4.3334083581e-02 -4.3334090214e-02
|
||||
% |Ax-b| = 3.7e-09, [Ay-c]_+ = 1.1E-10, |x|= 1.0e+00, |y|= 2.6e+00
|
||||
|
||||
% Detailed timing (sec)
|
||||
% Pre IPM Post
|
||||
% 3.902E+00 4.576E+01 1.035E-02
|
||||
% Max-norms: ||b||=1, ||c|| = 3,
|
||||
% Cholesky |add|=0, |skip| = 0, ||L.L|| = 4.26267.
|
||||
% ------------------------------------------------------------
|
||||
% Status: Solved
|
||||
% Optimal value (cvx_optval): -0.0433341
|
||||
% h = y(2:end);
|
||||
% #+end_example
|
||||
|
||||
% Finally, we compute the filter response over the frequency vector defined and the result is shown on figure [[fig:fir_filter_ligo]] which is very close to the filters obtain in cite:hua05_low_ligo.
|
||||
|
||||
|
||||
w = [w1 w2 w3 w4];
|
||||
H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(w, abs(H), 'k-');
|
||||
plot(w, abs(1-H), 'k--');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylim([5e-3, 5]);
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(w, 180/pi*unwrap(angle(H)), 'k-');
|
||||
plot(w, 180/pi*unwrap(angle(1-H)), 'k--');
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'XScale', 'log');
|
||||
yticks([-450:90:180]); ylim([-450, 200]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([1e-3, 1]);
|
||||
|
||||
% Weights
|
||||
% We design weights that will be used for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
|
||||
% These weights will determine the order of the obtained filters.
|
||||
% Here are the requirements on the filters:
|
||||
% - reasonable order
|
||||
% - to be as close as possible to the specified upper bounds
|
||||
% - stable minimum phase
|
||||
|
||||
% The bode plot of the weights is shown on figure [[fig:ligo_weights]].
|
||||
|
||||
|
||||
w1 = 2*pi*0.008; x1 = 0.35;
|
||||
w2 = 2*pi*0.04; x2 = 0.5;
|
||||
w3 = 2*pi*0.05; x3 = 0.5;
|
||||
|
||||
% Slope of +3 from w1
|
||||
wH = 0.008*(s^2/w1^2 + 2*x1/w1*s + 1)*(s/w1 + 1);
|
||||
% Little bump from w2 to w3
|
||||
wH = wH*(s^2/w2^2 + 2*x2/w2*s + 1)/(s^2/w3^2 + 2*x3/w3*s + 1);
|
||||
% No Slope at high frequencies
|
||||
wH = wH/(s^2/w3^2 + 2*x3/w3*s + 1)/(s/w3 + 1);
|
||||
% Little bump between w2 and w3
|
||||
w0 = 2*pi*0.045; xi = 0.1; A = 2; n = 1;
|
||||
wH = wH*((s^2 + 2*w0*xi*A^(1/n)*s + w0^2)/(s^2 + 2*w0*xi*s + w0^2))^n;
|
||||
|
||||
wH = 1/wH;
|
||||
wH = minreal(ss(wH));
|
||||
|
||||
n = 20; Rp = 1; Wp = 2*pi*0.102;
|
||||
[b,a] = cheby1(n, Rp, Wp, 'high', 's');
|
||||
wL = 0.04*tf(a, b);
|
||||
|
||||
wL = 1/wL;
|
||||
wL = minreal(ss(wL));
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|w_H|^{-1}$');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|w_L|^{-1}$');
|
||||
|
||||
plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec.');
|
||||
plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
|
||||
plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
|
||||
plot([0.1, 10], [0.045, 0.045], 'k--', 'HandleVisibility', 'off');
|
||||
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-3, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
% H-Infinity Synthesis
|
||||
% We define the generalized plant as shown on figure [[fig:h_infinity_robst_fusion]].
|
||||
|
||||
P = [0 wL;
|
||||
wH -wH;
|
||||
1 0];
|
||||
|
||||
|
||||
|
||||
% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
|
||||
|
||||
[Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% #+begin_example
|
||||
% [Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
% Resetting value of Gamma min based on D_11, D_12, D_21 terms
|
||||
|
||||
% Test bounds: 0.3276 < gamma <= 1.8063
|
||||
|
||||
% gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
|
||||
% 1.806 1.4e-02 -1.7e-16 3.6e-03 -4.8e-12 0.0000 p
|
||||
% 1.067 1.3e-02 -4.2e-14 3.6e-03 -1.9e-12 0.0000 p
|
||||
% 0.697 1.3e-02 -3.0e-01# 3.6e-03 -3.5e-11 0.0000 f
|
||||
% 0.882 1.3e-02 -9.5e-01# 3.6e-03 -1.2e-34 0.0000 f
|
||||
% 0.975 1.3e-02 -2.7e+00# 3.6e-03 -1.6e-12 0.0000 f
|
||||
% 1.021 1.3e-02 -8.7e+00# 3.6e-03 -4.5e-16 0.0000 f
|
||||
% 1.044 1.3e-02 -6.5e-14 3.6e-03 -3.0e-15 0.0000 p
|
||||
% 1.032 1.3e-02 -1.8e+01# 3.6e-03 0.0e+00 0.0000 f
|
||||
% 1.038 1.3e-02 -3.8e+01# 3.6e-03 0.0e+00 0.0000 f
|
||||
% 1.041 1.3e-02 -8.3e+01# 3.6e-03 -2.9e-33 0.0000 f
|
||||
% 1.042 1.3e-02 -1.9e+02# 3.6e-03 -3.4e-11 0.0000 f
|
||||
% 1.043 1.3e-02 -5.3e+02# 3.6e-03 -7.5e-13 0.0000 f
|
||||
|
||||
% Gamma value achieved: 1.0439
|
||||
% #+end_example
|
||||
|
||||
% The high pass filter is defined as $H_H = 1 - H_L$.
|
||||
|
||||
Hh = 1 - Hl;
|
||||
|
||||
Hh = minreal(Hh);
|
||||
Hl = minreal(Hl);
|
||||
|
||||
|
||||
|
||||
% The size of the filters is shown below.
|
||||
|
||||
|
||||
size(Hh), size(Hl)
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% #+begin_example
|
||||
% size(Hh), size(Hl)
|
||||
% State-space model with 1 outputs, 1 inputs, and 27 states.
|
||||
% State-space model with 1 outputs, 1 inputs, and 27 states.
|
||||
% #+end_example
|
||||
|
||||
% The bode plot of the obtained filters as shown on figure [[fig:hinf_synthesis_ligo_results]].
|
||||
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
|
||||
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$H_H$');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'DisplayName', '$H_L$');
|
||||
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-3, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
% Compare FIR and H-Infinity Filters
|
||||
% Let's now compare the FIR filters designed in cite:hua05_low_ligo and the one obtained with the $\mathcal{H}_\infty$ synthesis on figure [[fig:comp_fir_ligo_hinf]].
|
||||
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(w, abs(H), '--', ...
|
||||
'DisplayName', '$H_H(s)$ - FIR');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(w, abs(1-H), '--', ...
|
||||
'DisplayName', '$H_L(s)$ - FIR');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylim([5e-3, 10]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 16;
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hh, freqs, 'Hz')))), '-');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hl, freqs, 'Hz')))), '-');
|
||||
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(w, 180/pi*unwrap(angle(H)), '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(w, 180/pi*unwrap(angle(1-H)), '--');
|
||||
set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
yticks([-450:90:180]); ylim([-450, 200]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
@ -1,177 +0,0 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
|
||||
addpath('./src');
|
||||
|
||||
% Design of Weighting Function
|
||||
% A formula is proposed to help the design of the weighting functions:
|
||||
% \begin{equation}
|
||||
% W(s) = \left( \frac{
|
||||
% \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
|
||||
% }{
|
||||
% \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
|
||||
% }\right)^n
|
||||
% \end{equation}
|
||||
|
||||
% The parameters permits to specify:
|
||||
% - the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$
|
||||
% - the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$
|
||||
% - the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
|
||||
% - the absolute slope between high and low frequency: $n$
|
||||
|
||||
% The general shape of a weighting function generated using the formula is shown in figure [[fig:weight_formula]].
|
||||
|
||||
|
||||
n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2;
|
||||
|
||||
W = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(W, freqs, 'Hz'))), 'k-');
|
||||
|
||||
plot([1e-3 1e0], [G0 G0], 'k--', 'LineWidth', 1)
|
||||
text(1e0, G0, '$\quad G_0$')
|
||||
|
||||
plot([1e1 1e3], [G1 G1], 'k--', 'LineWidth', 1)
|
||||
text(1e1,G1,'$G_{\infty}\quad$','HorizontalAlignment', 'right')
|
||||
|
||||
plot([w0/2/pi w0/2/pi], [1 2*Gc], 'k--', 'LineWidth', 1)
|
||||
text(w0/2/pi,1,'$\omega_c$','VerticalAlignment', 'top', 'HorizontalAlignment', 'center')
|
||||
|
||||
plot([w0/2/pi/2 2*w0/2/pi], [Gc Gc], 'k--', 'LineWidth', 1)
|
||||
text(w0/2/pi/2, Gc, '$G_c \quad$','HorizontalAlignment', 'right')
|
||||
|
||||
text(w0/5/pi/2, abs(evalfr(W, j*w0/5)), 'Slope: $n \quad$', 'HorizontalAlignment', 'right')
|
||||
|
||||
text(w0/2/pi, abs(evalfr(W, j*w0)), '$\bullet$', 'HorizontalAlignment', 'center')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([5e-4, 20]);
|
||||
|
||||
|
||||
|
||||
% #+name: fig:weight_formula
|
||||
% #+caption: Gain of the Weighting Function formula
|
||||
% #+RESULTS:
|
||||
% [[file:figs/weight_formula.png]]
|
||||
|
||||
|
||||
n = 2; w0 = 2*pi*10; G0 = 1/10; G1 = 1000; Gc = 0.45;
|
||||
W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 0.45;
|
||||
W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-4, 20]);
|
||||
xticks([0.1, 1, 10, 100, 1000]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
% H-Infinity Synthesis
|
||||
% We define the generalized plant $P$ on matlab.
|
||||
|
||||
P = [W1 -W1;
|
||||
0 W2;
|
||||
1 0];
|
||||
|
||||
|
||||
|
||||
% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
|
||||
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% #+begin_example
|
||||
% [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
% Test bounds: 0.3223 <= gamma <= 1000
|
||||
|
||||
% gamma X>=0 Y>=0 rho(XY)<1 p/f
|
||||
% 1.795e+01 1.4e-07 0.0e+00 1.481e-16 p
|
||||
% 2.406e+00 1.4e-07 0.0e+00 3.604e-15 p
|
||||
% 8.806e-01 -3.1e+02 # -1.4e-16 7.370e-19 f
|
||||
% 1.456e+00 1.4e-07 0.0e+00 1.499e-18 p
|
||||
% 1.132e+00 1.4e-07 0.0e+00 8.587e-15 p
|
||||
% 9.985e-01 1.4e-07 0.0e+00 2.331e-13 p
|
||||
% 9.377e-01 -7.7e+02 # -6.6e-17 3.744e-14 f
|
||||
% 9.676e-01 -2.0e+03 # -5.7e-17 1.046e-13 f
|
||||
% 9.829e-01 -6.6e+03 # -1.1e-16 2.949e-13 f
|
||||
% 9.907e-01 1.4e-07 0.0e+00 2.374e-19 p
|
||||
% 9.868e-01 -1.6e+04 # -6.4e-17 5.331e-14 f
|
||||
% 9.887e-01 -5.1e+04 # -1.5e-17 2.703e-19 f
|
||||
% 9.897e-01 1.4e-07 0.0e+00 1.583e-11 p
|
||||
% Limiting gains...
|
||||
% 9.897e-01 1.5e-07 0.0e+00 1.183e-12 p
|
||||
% 9.897e-01 6.9e-07 0.0e+00 1.365e-12 p
|
||||
|
||||
% Best performance (actual): 0.9897
|
||||
% #+end_example
|
||||
|
||||
% We then define the high pass filter $H_1 = 1 - H_2$. The bode plot of both $H_1$ and $H_2$ is shown on figure [[fig:hinf_filters_results]].
|
||||
|
||||
|
||||
H1 = 1 - H2;
|
||||
|
||||
% Obtained Complementary Filters
|
||||
% The obtained complementary filters are shown on figure [[fig:hinf_filters_results]].
|
||||
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylim([1e-4, 20]);
|
||||
yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'XScale', 'log');
|
||||
yticks([-180:90:180]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
169
matlab/ref.bib
169
matlab/ref.bib
@ -1,17 +1,3 @@
|
||||
@article{collette15_sensor_fusion_method_high_perfor,
|
||||
author = {C. Collette and F. Matichard},
|
||||
title = {Sensor Fusion Methods for High Performance Active Vibration Isolation Systems},
|
||||
journal = {Journal of Sound and Vibration},
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
|
||||
|
||||
@phdthesis{hua05_low_ligo,
|
||||
author = {Hua, Wensheng},
|
||||
school = {stanford university},
|
||||
@ -20,152 +6,11 @@
|
||||
year = 2005,
|
||||
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|
||||
|
||||
|
||||
@inproceedings{hua04_polyp_fir_compl_filter_contr_system,
|
||||
author = {Wensheng Hua and Dan B. Debra and Corwin T. Hardham and Brian T. Lantz and Joseph A. Giaime},
|
||||
title = {Polyphase FIR Complementary Filters for Control Systems},
|
||||
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|
||||
year = {2004},
|
||||
pages = {109--114},
|
||||
@article{dehaeze21_new_method_desig_compl_filter,
|
||||
author = {Dehaeze, Thomas and Vermat, Mohit and Collette, Christophe},
|
||||
title = {A New Method of Designing Complementary Filters for Sensor
|
||||
Fusion Using the $\mathcal{H}_\infty$ Synthesis},
|
||||
journal = {Mechanical Systems and Signal Processing},
|
||||
year = 2021,
|
||||
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|
||||
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|
||||
|
||||
|
||||
@article{matichard15_seism_isolat_advan_ligo,
|
||||
author = {Matichard, F and Lantz, B and Mittleman, R and Mason, K and Kissel, J and Abbott, B and Biscans, S and McIver, J and Abbott, R and Abbott, S and others},
|
||||
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|
||||
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|
||||
volume = {32},
|
||||
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|
||||
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|
||||
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|
||||
publisher = {IOP Publishing},
|
||||
}
|
||||
|
||||
|
||||
@article{min15_compl_filter_desig_angle_estim,
|
||||
author = {Min, Hyung Gi and Jeung, Eun Tae},
|
||||
title = {Complementary Filter Design for Angle Estimation Using Mems Accelerometer and Gyroscope},
|
||||
journal = {Department of Control and Instrumentation, Changwon National University, Changwon, Korea},
|
||||
pages = {641--773},
|
||||
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|
||||
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|
||||
|
||||
|
||||
@article{corke04_inert_visual_sensin_system_small_auton_helic,
|
||||
author = {Peter Corke},
|
||||
title = {An Inertial and Visual Sensing System for a Small Autonomous Helicopter},
|
||||
journal = {Journal of Robotic Systems},
|
||||
volume = {21},
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
|
||||
|
||||
@inproceedings{jensen13_basic_uas,
|
||||
author = {Austin Jensen and Cal Coopmans and YangQuan Chen},
|
||||
title = {Basics and guidelines of complementary filters for small UAS
|
||||
navigation},
|
||||
booktitle = {2013 International Conference on Unmanned Aircraft Systems
|
||||
(ICUAS)},
|
||||
year = 2013,
|
||||
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|
||||
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|
||||
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|
||||
month = 5,
|
||||
}
|
||||
|
||||
|
||||
@inproceedings{pascoal99_navig_system_desig_using_time,
|
||||
author = {A. Pascoal and I. Kaminer and P. Oliveira},
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
}
|
||||
|
||||
|
||||
@article{zimmermann92_high_bandw_orien_measur_contr,
|
||||
author = {M. Zimmermann and W. Sulzer},
|
||||
title = {High Bandwidth Orientation Measurement and Control Based on Complementary Filtering},
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
publisher = {Elsevier},
|
||||
series = {Robot Control 1991},
|
||||
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|
||||
|
||||
|
||||
@inproceedings{baerveldt97_low_cost_low_weigh_attit,
|
||||
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|
||||
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|
||||
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|
||||
year = {1997},
|
||||
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|
||||
doi = {10.1109/ines.1997.632450},
|
||||
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|
||||
month = {-},
|
||||
}
|
||||
|
||||
|
||||
@article{shaw90_bandw_enhan_posit_measur_using_measur_accel,
|
||||
author = {F.R. Shaw and K. Srinivasan},
|
||||
title = {Bandwidth Enhancement of Position Measurements Using Measured
|
||||
Acceleration},
|
||||
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|
||||
volume = 4,
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
}
|
||||
|
||||
|
||||
@article{brown72_integ_navig_system_kalman_filter,
|
||||
author = {R. G. Brown},
|
||||
title = {Integrated Navigation Systems and Kalman Filtering: a
|
||||
Perspective},
|
||||
journal = {Navigation},
|
||||
volume = 19,
|
||||
number = 4,
|
||||
pages = {355-362},
|
||||
year = 1972,
|
||||
doi = {10.1002/j.2161-4296.1972.tb01706.x},
|
||||
url = {https://doi.org/10.1002/j.2161-4296.1972.tb01706.x},
|
||||
}
|
||||
|
||||
|
||||
@article{matichard15_seism_isolat_advan_ligo,
|
||||
author = {Matichard, F and Lantz, B and Mittleman, R and Mason, K and
|
||||
Kissel, J and Abbott, B and Biscans, S and McIver, J and
|
||||
Abbott, R and Abbott, S and others},
|
||||
title = {Seismic Isolation of Advanced Ligo: Review of Strategy,
|
||||
Instrumentation and Performance},
|
||||
journal = {Classical and Quantum Gravity},
|
||||
volume = 32,
|
||||
number = 18,
|
||||
pages = 185003,
|
||||
year = 2015,
|
||||
publisher = {IOP Publishing},
|
||||
}
|
||||
|
||||
|
||||
@article{mahony08_nonlin_compl_filter_special_orthog_group,
|
||||
author = {Robert Mahony and Tarek Hamel and Jean-Michel Pflimlin},
|
||||
title = {Nonlinear Complementary Filters on the Special Orthogonal
|
||||
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},
|
||||
}
|
||||
932
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932
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|
||||
<?xml version="1.0" encoding="utf-8"?>
|
||||
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|
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|
||||
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|
||||
<head>
|
||||
<!-- 2021-09-01 mer. 10:15 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<title>A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis - Tikz Figures</title>
|
||||
<meta name="author" content="Thomas Dehaeze" />
|
||||
<meta name="generator" content="Org Mode" />
|
||||
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|
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|
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|
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|
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|
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|
||||
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|
||||
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|
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|
||||
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|
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|
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|
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|
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<script id="MathJax-script" async
|
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src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js"></script>
|
||||
</head>
|
||||
<body>
|
||||
<div id="org-div-home-and-up">
|
||||
<a accesskey="h" href="../index.html"> UP </a>
|
||||
|
|
||||
<a accesskey="H" href="../index.html"> HOME </a>
|
||||
</div><div id="content" class="content">
|
||||
<h1 class="title">A new method of designing complementary filters for sensor fusion using the \(\mathcal{H}_\infty\) synthesis - Tikz Figures</h1>
|
||||
<div id="table-of-contents" role="doc-toc">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#orgaddbc4b">1. Sensor Fusion - Overview</a></li>
|
||||
<li><a href="#orgcde34ed">2. Sensor Model</a></li>
|
||||
<li><a href="#org881c7cf">3. Sensor Model with calibration</a></li>
|
||||
<li><a href="#org52dba0e">4. Sensor Fusion Architecture</a></li>
|
||||
<li><a href="#org51c6b92">5. Sensor Model with Uncertainty</a></li>
|
||||
<li><a href="#orgd0c3320">6. Sensor Model with Uncertainty - Simplified</a></li>
|
||||
<li><a href="#orgd991dd2">7. Sensor fusion architecture with sensor dynamics uncertainty</a></li>
|
||||
<li><a href="#orgb45d1ac">8. Uncertainty set of the super sensor dynamics</a></li>
|
||||
<li><a href="#orgd194580">9. Generalized plant used for \(\mathcal{H}_\infty\) synthesis of complementary filters</a></li>
|
||||
<li><a href="#orgebe7ec5">10. Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</a></li>
|
||||
<li><a href="#orgf9a2f39">11. LIGO Sensor Fusion Architecture</a></li>
|
||||
<li><a href="#orga2611bb">12. Closed-Loop Complementary Filters</a></li>
|
||||
<li><a href="#org5af72e4">13. Closed-Loop Fusion Architecture</a></li>
|
||||
<li><a href="#org4e79ee7">14. Feedback Loop Sensor Fusion Architecture</a></li>
|
||||
<li><a href="#orgb59c88b">15. Feedback Sensor Fusion - Generalized Plant</a></li>
|
||||
<li><a href="#orge593f6d">16. Sensor Fusion - Sequential</a></li>
|
||||
<li><a href="#org3b323e8">17. Sensor Fusion - Parallel</a></li>
|
||||
<li><a href="#org9baa5b5">18. Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</a></li>
|
||||
<li><a href="#org10ffb09">19. Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
<p>
|
||||
Configuration file is accessible <a href="config.html">here</a>.
|
||||
</p>
|
||||
|
||||
<div id="outline-container-orgaddbc4b" class="outline-2">
|
||||
<h2 id="orgaddbc4b"><span class="section-number-2">1.</span> Sensor Fusion - Overview</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
|
||||
<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right=0.3 and 0.5 of x](sensor1){Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below right=0.3 and 0.5 of x](sensor2){Sensor 2};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=5.0 of x](add){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor1.east) -- node[midway, above]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>} (H1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor2.east) -- node[midway, above]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>} (H2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.9, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(<span class="org-font-latex-math">$(H2.south-|x) + (0, -0.2)$</span>) (<span class="org-font-latex-math">$(H1.north-|add.east) + (0.2, 0.6)$</span>)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below] at (supersensor.north) {Super Sensor};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(sensor2.south west) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[align=center] at (sensors.center) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Sensors</span>}};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(H2.south west) (H1.north-|add.east)}, fill=black!20!white, draw, inner sep=6pt] (filters) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[align=center] at (<span class="org-font-latex-math">$(filters.center) + (-0.3, 0)$</span>) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Complementary</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Filters</span>}};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgfdc79fa" class="figure">
|
||||
<p><img src="figs/sensor_fusion_overview.png" alt="sensor_fusion_overview.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Schematic of a sensor fusion architecture using complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcde34ed" class="outline-2">
|
||||
<h2 id="orgcde34ed"><span class="section-number-2">2.</span> Sensor Model</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[addb](add1){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add1](G1){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(add1.west)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (add1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (G1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (sensor1.north east) {Sensor};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orga6c4cb8" class="figure">
|
||||
<p><img src="figs/sensor_model.png" alt="sensor_model.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Basic sensor model consisting of a noise input \(n_i\) and a linear time invariant transfer function \(G_i(s)\)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org881c7cf" class="outline-2">
|
||||
<h2 id="org881c7cf"><span class="section-number-2">3.</span> Sensor Model with calibration</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[addb](add1){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add1](G1){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G1](G1inv){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-1}(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(add1.west)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (add1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (G1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- (G1inv.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1inv.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (sensor1.north east) {Sensor};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd38eff7" class="figure">
|
||||
<p><img src="figs/sensor_model_calibrated.png" alt="sensor_model_calibrated.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Normalized sensors using the inverse of an estimate \(\hat{G}_i(s)\) of the sensor dynamics</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org52dba0e" class="outline-2">
|
||||
<h2 id="org52dba0e"><span class="section-number-2">4.</span> Sensor Fusion Architecture</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
|
||||
<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, above right=0.8 and 0.5 of x](add1){};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, below right=0.8 and 0.5 of x](add2){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add1](G1){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add2](G2){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G1](G1inv){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-1}(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G2](G2inv){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-2}(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G1inv](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G2inv](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=7 of x](add){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (add1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (add2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (G1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- (G1inv.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1inv.east) -- (H1.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add2.north) -- ++(0, 0.7)node[below right](n2){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add2.east) -- (G2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G2.east) -- (G2inv.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G2inv.east) -- (H2.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(G2.south-|x) (n1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=myblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=myblue!30!white, draw, inner sep=3pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (sensor1.north east) {Sensor 1};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add2.west |- G2inv.south) (n2.north -| G2inv.east)}, fill=myred!20!white, draw, inner sep=6pt] (sensor2cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor2cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add2.west |- G2.south) (n2.north -| G2.east)}, fill=myred!30!white, draw, inner sep=3pt] (sensor2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (sensor2.north east) {Sensor 2};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org63acf85" class="figure">
|
||||
<p><img src="figs/fusion_super_sensor.png" alt="fusion_super_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Sensor fusion architecture with two normalized sensors</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org51c6b92" class="outline-2">
|
||||
<h2 id="org51c6b92"><span class="section-number-2">5.</span> Sensor Model with Uncertainty</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (input) at (0,0) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right= 0.4 and 0.4 of input](W1){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of W1](delta1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addu) at (<span class="org-font-latex-math">$(delta1.east|-input) + (0.4, 0)$</span>) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.4 of addu] (addn) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of addn] (G1) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G1](G1inv){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-1}(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(input)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (addu);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (input.center) |- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (delta1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta1.east) -| (addu.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addu.east) -- (addn.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addn.east) -- (G1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addn.north) -- ++(0, 0.7)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- (G1inv.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1inv.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input.west |- G1inv.south) (delta1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input.west |- G1.south) (delta1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (sensor1.north east) {Sensor};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org937c435" class="figure">
|
||||
<p><img src="figs/sensor_model_uncertainty.png" alt="sensor_model_uncertainty.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Sensor with multiplicative input uncertainty</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd0c3320" class="outline-2">
|
||||
<h2 id="orgd0c3320"><span class="section-number-2">6.</span> Sensor Model with Uncertainty - Simplified</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (input) at (0,0) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right= 0.4 and 0.4 of input](W1){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of W1](delta1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addu) at (<span class="org-font-latex-math">$(delta1.east|-input) + (0.4, 0)$</span>) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.4 of addu] (addn) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(input)+(-0.8, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (addu);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (input.center) |- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (delta1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta1.east) -| (addu.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addu.east) -- (addn.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addn.north) -- ++(0, 0.6)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addn.east) -- ++(0.9, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input.west |- addu.south) (<span class="org-font-latex-math">$(delta1.north -| addn.east) + (0.1, 0)$</span>)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgce8343e" class="figure">
|
||||
<p><img src="figs/sensor_model_uncertainty_simplified.png" alt="sensor_model_uncertainty_simplified.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Simplified sensor model</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd991dd2" class="outline-2">
|
||||
<h2 id="orgd991dd2"><span class="section-number-2">7.</span> Sensor fusion architecture with sensor dynamics uncertainty</h2>
|
||||
<div class="outline-text-2" id="text-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
|
||||
<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[branch, above right=1.0 and 0.3 of x] (input1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[branch, below right=1.0 and 0.3 of x] (input2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right= 0.4 and 0.3 of input1](W1){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right= 0.4 and 0.3 of input2](W2){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of W1](delta1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of W2](delta2){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addu1) at (<span class="org-font-latex-math">$(delta1.east|-input1) + (0.4, 0)$</span>) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addu2) at (<span class="org-font-latex-math">$(delta2.east|-input2) + (0.4, 0)$</span>) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.4 of addu1] (addn1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.4 of addu2] (addn2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.9 of addn1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.9 of addn2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=7 of x](add){};
|
||||
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (addu1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (addu2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (input1.center) |- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (delta1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta1.east) -| (addu1.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addu1.east) -- (addn1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addn1.north) -- ++(0, 0.6)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (input2.center) |- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (delta2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta2.east) -| (addu2.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addu2.east) -- (addn2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addn2.north) -- ++(0, 0.6)node[below right](n2){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addn1.east) -- (H1.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addn2.east) -- (H2.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(addn2.south-|x) (delta1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input1.west |- addu1.south) (<span class="org-font-latex-math">$(delta1.north -| addn1.east) + (0.1, 0.0)$</span>)}, fill=myblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor 1</span>}};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input2.west |- addu2.south) (<span class="org-font-latex-math">$(delta2.north -| addn1.east) + (0.1, 0.0)$</span>)}, fill=myred!20!white, draw, inner sep=6pt] (sensor2cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor2cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor 2</span>}};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org5b69e5a" class="figure">
|
||||
<p><img src="figs/sensor_fusion_dynamic_uncertainty.png" alt="sensor_fusion_dynamic_uncertainty.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb45d1ac" class="outline-2">
|
||||
<h2 id="orgb45d1ac"><span class="section-number-2">8.</span> Uncertainty set of the super sensor dynamics</h2>
|
||||
<div class="outline-text-2" id="text-8">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
|
||||
<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[shift={(4, 0)}]
|
||||
|
||||
<span class="org-comment">% Uncertainty Circle</span>
|
||||
<span class="org-font-latex-sedate">\node</span>[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, fill=myblue!20!white] (0, 0) circle [radius=1.0];
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, fill=myred!20!white] (135:1.0) circle [radius=0.8];
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, dashed] (0, 0) circle [radius=1.0];
|
||||
|
||||
<span class="org-comment">% Center of Circle</span>
|
||||
<span class="org-font-latex-sedate">\node</span>[below] at (0, 0){<span class="org-font-latex-math">$1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(135:0.8);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l1) at (2, 1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, out=-90, in=0] (l1.south) to (r1);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l2) at (-3.2, 1.2) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, out=0, in=-180] (l2.east) to (r2);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l3) at (-2.5, -1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1| + |w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, out=90, in=-90] (l3.north) to (r3);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
|
||||
<span class="org-comment">% Real and Imaginary Axis</span>
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed] (1, 0) arc (0:28:1) node[midway, right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"> </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\phi</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\text</span></span><span class="org-font-latex-math">{max}$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgc80ec14" class="figure">
|
||||
<p><img src="figs/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Uncertainty region of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency \(\omega\) is here omitted.</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd194580" class="outline-2">
|
||||
<h2 id="orgd194580"><span class="section-number-2">9.</span> Generalized plant used for \(\mathcal{H}_\infty\) synthesis of complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-9">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.2 of P, opacity=0] (H2) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.75!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.35!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.75!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.35!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + ( 0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right]{<span class="org-font-latex-math">$w$</span>} -- (sub.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu) node[above right]{<span class="org-font-latex-math">$u$</span>} -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub.west)+(-0.6, 0)$</span>) node[branch]{} |- (outputv) node[above left]{<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org725e946" class="figure">
|
||||
<p><img src="figs/h_infinity_robust_fusion_plant.png" alt="h_infinity_robust_fusion_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Generalized plant used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgebe7ec5" class="outline-2">
|
||||
<h2 id="orgebe7ec5"><span class="section-number-2">10.</span> Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-10">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.2 of P] (H2) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.75!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.35!(P.north west) + (-0.4, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.75!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.35!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + ( 0.4, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right]{<span class="org-font-latex-math">$w$</span>} -- (sub.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub.west)+(-0.6, 0)$</span>) node[branch]{} |- (outputv) |- (H2.east);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2.west) -| (inputu) -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org47d276f" class="figure">
|
||||
<p><img src="figs/h_infinity_robust_fusion_fb.png" alt="h_infinity_robust_fusion_fb.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Generalized plant with the synthesized filter</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf9a2f39" class="outline-2">
|
||||
<h2 id="orgf9a2f39"><span class="section-number-2">11.</span> LIGO Sensor Fusion Architecture</h2>
|
||||
<div class="outline-text-2" id="text-11">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
|
||||
<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block, align=center] (position) at (0, 2.2) {Position<span class="org-font-latex-warning">\\</span>Sensor};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, align=center] (seismometer) at (0, 1.0) {Seismometer};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, align=center] (geophone) at (0,-0.6) {Geophone};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[branch, left=0.4 of seismometer] (x);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of seismometer](H1){<span class="org-font-latex-math">$L</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block](H2) at (H1|-geophone) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (add) at (4, 0){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of add](H2p) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block] (H1p) at (H2p|-position) {<span class="org-font-latex-math">$L</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addp) at (7, 1.0){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(x)+(-1.0, 0)$</span>) -- (seismometer.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (position.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (geophone.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (position.east) -- (H1p.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (seismometer.east) -- (H1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (geophone.east) -- (H2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- (H2p.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1p) -| (addp.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2p) -| (addp.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addp.east) -- ++(1.0, 0);
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x.west|-geophone.south) (position.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below] at (supersensor.north) {Super Sensor};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x.west|-seismometer.north) (add.east|-geophone.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[] at (superinertialsensor.center) {"Inertial" Super Sensor};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org81fe1cf" class="figure">
|
||||
<p><img src="figs/ligo_super_sensor_architecture.png" alt="ligo_super_sensor_architecture.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Simplified block diagram of the sensor blending strategy for the first stage at the LIGO</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga2611bb" class="outline-2">
|
||||
<h2 id="orga2611bb"><span class="section-number-2">12.</span> Closed-Loop Complementary Filters</h2>
|
||||
<div class="outline-text-2" id="text-12">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1 of addfb] (L){<span class="org-font-latex-math">$L(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=1 of L] (adddy){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addfb.west) -- ++(-1, 0) node[above right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addfb.east) -- (L.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (L.east) -- (adddy.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (adddy.east) -- ++(1.4, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(adddy.east) + (0.5, 0)$</span>) node[branch]{} -- ++(0, -0.8) coordinate(botc) -| (addfb.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (adddy.north) -- ++(0, 1) node[below right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(L.north-|addfb.west) (botc)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org57ca8ce" class="figure">
|
||||
<p><img src="figs/feedback_sensor_fusion.png" alt="feedback_sensor_fusion.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>“Closed-Loop” complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5af72e4" class="outline-2">
|
||||
<h2 id="org5af72e4"><span class="section-number-2">13.</span> Closed-Loop Fusion Architecture</h2>
|
||||
<div class="outline-text-2" id="text-13">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1 of addfb] (L){<span class="org-font-latex-math">$L(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=1 of L] (adddy){};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.2 of addfb] (sensor2){Sensor 2};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above=0.4 of sensor2] (sensor1){Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[branch, left=0.6 of sensor2] (x){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addfb.east) -- (L.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (L.east) -- (adddy.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (adddy.east) -- ++(1.4, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(adddy.east) + (0.5, 0)$</span>) node[branch]{} -- ++(0, -0.8) coordinate(botc) -| (addfb.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(x)-(0.8,0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (sensor2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor2.east)node[above right=0 and 0.25]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>} -- (addfb.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor1.east)node[above right=0 and 0.25]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>} -| (adddy.north);
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x|-sensor1.north) (botc)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(sensor1.north-|addfb.west) (botc)}, fill=black!20!white, draw, inner sep=6pt] (feedbackfilter) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(sensor2.west|-botc) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above, align=center] at (sensors.south) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensors</span>}};
|
||||
<span class="org-font-latex-sedate">\node</span>[below, align=center] at (feedbackfilter.north) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> "Closed-Loop"</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> complementary filters</span>}};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org8886809" class="figure">
|
||||
<p><img src="figs/feedback_sensor_fusion_arch.png" alt="feedback_sensor_fusion_arch.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Classical feedback architecture used for sensor fusion</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4e79ee7" class="outline-2">
|
||||
<h2 id="org4e79ee7"><span class="section-number-2">14.</span> Feedback Loop Sensor Fusion Architecture</h2>
|
||||
<div class="outline-text-2" id="text-14">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block] (W2) at (0,0) {<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, right=0.8 of W2] (addfb){};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=4.5 of W2] (adddy){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above=0.8 of adddy] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (W2.west) -- ++(-0.8, 0) node[above right]{<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (addfb.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{w}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addfb.east) -- ++(1, 0) node[above left]{<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (adddy.west) -- ++(-1, 0) node[above right]{<span class="org-font-latex-math">$u$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (adddy.east) -- ++(1.4, 0) node[above left]{<span class="org-font-latex-math">$z$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.south) -- (adddy.north) node[above right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{w}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (W1.north) -- ++(0, 0.8) node[below right]{<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(adddy.east) + (0.5, 0)$</span>) node[branch]{} -- ++(0, -0.8) -| (addfb.south);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org705140c" class="figure">
|
||||
<p><img src="figs/feedback_synthesis_architecture.png" alt="feedback_synthesis_architecture.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Feedback architecture with included weights</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb59c88b" class="outline-2">
|
||||
<h2 id="orgb59c88b"><span class="section-number-2">15.</span> Feedback Sensor Fusion - Generalized Plant</h2>
|
||||
<div class="outline-text-2" id="text-15">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={4.5cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">L(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw1) at (<span class="org-font-latex-math">$(P.south west)!0.75!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw2) at (<span class="org-font-latex-math">$(P.south west)!0.40!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.15!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputz) at (<span class="org-font-latex-math">$(P.south east)!0.75!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.40!(P.north east) + ( 0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.2 of inputw2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.2 of inputw1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=0.8 of W1] (add) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{-}{}{}, right=1.8 of W2] (sub) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw2) node[above right]{<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>} -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw1) node[above right]{<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>} -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu) node[above right]{<span class="org-font-latex-math">$u$</span>} -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (sub.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (add.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- (outputz)node[above left]{<span class="org-font-latex-math">$z$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub.east) -- (outputv)node[above left]{<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add-|sub) node[branch]{} -- (sub.north);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org32b295e" class="figure">
|
||||
<p><img src="figs/feedback_synthesis_architecture_generalized_plant.png" alt="feedback_synthesis_architecture_generalized_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Generalized plant used for the \(\mathcal{H}_\infty\) synthesis of “closed-loop” complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge593f6d" class="outline-2">
|
||||
<h2 id="orge593f6d"><span class="section-number-2">16.</span> Sensor Fusion - Sequential</h2>
|
||||
<div class="outline-text-2" id="text-16">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of x] (sensor2) {Sensor 2};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above=0.4 of sensor2] (sensor1) {Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.4 of sensor2] (sensor3) {Sensor 3};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (add) at (<span class="org-font-latex-math">$0.5*(H1.east)+0.5*(H2.east)+(0.6, 0)$</span>){};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add](H1p) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\prime</span></span><span class="org-font-latex-math">(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block] (H2p) at (H1p|-sensor3) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\prime</span></span><span class="org-font-latex-math">(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addp) at (<span class="org-font-latex-math">$0.5*(H1p.east)+0.5*(H2p.east)+(0.6, 0)$</span>){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(x)+(-0.8, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (sensor2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor3.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor1.east) -- (H1.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor2.east) -- (H2.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor3.east) -- (H2p.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- (H1p.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">{12}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1p) -| (addp.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2p) -| (addp.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addp.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x.west|-sensor3.south) (sensor1.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x.west|-sensor1.north) (add.east|-sensor2.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6fe58a4" class="figure">
|
||||
<p><img src="figs/sensor_fusion_three_sequential.png" alt="sensor_fusion_three_sequential.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Sequential fusion</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3b323e8" class="outline-2">
|
||||
<h2 id="org3b323e8"><span class="section-number-2">17.</span> Sensor Fusion - Parallel</h2>
|
||||
<div class="outline-text-2" id="text-17">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of x] (sensor2) {Sensor 2};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above=0.3 of sensor2] (sensor1) {Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of sensor2] (sensor3) {Sensor 3};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor3](H3){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.6 of H2](add){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.8, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (sensor2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor3.west);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor1.east) -- (H1.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor2.east) -- (H2.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor3.east) -- (H3.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -- (add.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H3) -| (add.south);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(H3.south-|x) (H1.north-|add.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6ec849a" class="figure">
|
||||
<p><img src="figs/sensor_fusion_three_parallel.png" alt="sensor_fusion_three_parallel.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Parallel fusion</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9baa5b5" class="outline-2">
|
||||
<h2 id="org9baa5b5"><span class="section-number-2">18.</span> Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-18">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"> <span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.8!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.4!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.8!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.55!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output3) at (<span class="org-font-latex-math">$(P.south east)!0.3!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + (0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output3] (W3){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P, opacity=0] (H) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s) \\ H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right](w){<span class="org-font-latex-math">$w$</span>} -- (sub2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub2.west)+(-0.5, 0)$</span>) node[branch]{} |- (outputv);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu|-W2) -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu|-W3) -- (W3.west);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub2.east) -- (sub1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub1.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left](z){<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3.east) -- (output3)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W2-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W3-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (outputv-|z){<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org7b3470a" class="figure">
|
||||
<p><img src="figs/comp_filter_three_hinf_gen_plant.png" alt="comp_filter_three_hinf_gen_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Generalized plant for the \(\mathcal{H}_\infty\) synthesis of three complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org10ffb09" class="outline-2">
|
||||
<h2 id="org10ffb09"><span class="section-number-2">19.</span> Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-19">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"> <span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.8!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.4!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.8!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.55!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output3) at (<span class="org-font-latex-math">$(P.south east)!0.3!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + (0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output3] (W3){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P] (H) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s) \\ H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right](w){<span class="org-font-latex-math">$w$</span>} -- (sub2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub2.west)+(-0.5, 0)$</span>) node[branch]{} |- (outputv) |- (H.east);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(H.south west)!0.7!(H.north west)$</span>) -| (inputu|-W2) -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(H.south west)!0.3!(H.north west)$</span>) -| (<span class="org-font-latex-math">$(inputu|-W3)+(0.4, 0)$</span>) -- (W3.west);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub2.east) -- (sub1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub1.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left](z){<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3.east) -- (output3)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W2-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W3-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (outputv-|z){<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org71495ac" class="figure">
|
||||
<p><img src="figs/comp_filter_three_hinf_fb.png" alt="comp_filter_three_hinf_fb.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Generalized plant with the synthesized filter for the \(\mathcal{H}_\infty\) synthesis of three complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2021-09-01 mer. 10:15</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,4 +1,4 @@
|
||||
#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis - Tikz Figures
|
||||
#+TITLE: A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis - Tikz Figures
|
||||
:DRAWER:
|
||||
#+HTML_LINK_HOME: ../index.html
|
||||
#+HTML_LINK_UP: ../index.html
|
||||
@ -59,7 +59,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_overview
|
||||
#+caption: Sensor Fusion Architecture - Overview
|
||||
#+caption: Schematic of a sensor fusion architecture using complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_overview.png]]
|
||||
|
||||
@ -82,7 +82,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model
|
||||
#+caption: Basic Sensor Model
|
||||
#+caption: Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model.png]]
|
||||
|
||||
@ -110,7 +110,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model_calibrated
|
||||
#+caption: Calibrated Sensor
|
||||
#+caption: Normalized sensors using the inverse of an estimate $\hat{G}_i(s)$ of the sensor dynamics
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model_calibrated.png]]
|
||||
|
||||
@ -164,7 +164,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:fusion_super_sensor
|
||||
#+caption: Sensor Fusion Architecture ([[./figs/fusion_super_sensor.png][png]], [[./figs/fusion_super_sensor.pdf][pdf]], [[./figs/fusion_super_sensor.tex][tex]]).
|
||||
#+caption: Sensor fusion architecture with two normalized sensors
|
||||
#+RESULTS:
|
||||
[[file:figs/fusion_super_sensor.png]]
|
||||
|
||||
@ -200,7 +200,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model_uncertainty
|
||||
#+caption: Input Uncertainty
|
||||
#+caption: Sensor with multiplicative input uncertainty
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model_uncertainty.png]]
|
||||
|
||||
@ -229,7 +229,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model_uncertainty_simplified
|
||||
#+caption: Input Uncertainty
|
||||
#+caption: Simplified sensor model
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model_uncertainty_simplified.png]]
|
||||
|
||||
@ -291,7 +291,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_dynamic_uncertainty
|
||||
#+caption: Sensor fusion architecture with sensor dynamics uncertainty ([[./figs/sensor_fusion_dynamic_uncertainty.png][png]], [[./figs/sensor_fusion_dynamic_uncertainty.pdf][pdf]], [[./figs/sensor_fusion_dynamic_uncertainty.tex][tex]]).
|
||||
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_dynamic_uncertainty.png]]
|
||||
|
||||
@ -331,21 +331,23 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\draw[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};
|
||||
|
||||
\draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
|
||||
\draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi$};
|
||||
\draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi_\text{max}$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:uncertainty_set_super_sensor
|
||||
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (solid circle), of the sensor 1 (dotted circle) and of the sensor 2 (dashed circle) ([[./figs/uncertainty_set_super_sensor.png][png]], [[./figs/uncertainty_set_super_sensor.pdf][pdf]], [[./figs/uncertainty_set_super_sensor.tex][tex]]).
|
||||
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency $\omega$ is here omitted.
|
||||
#+RESULTS:
|
||||
[[file:figs/uncertainty_set_super_sensor.png]]
|
||||
|
||||
* Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+begin_src latex :file h_infinity_robust_fusion.pdf :tangle figs/h_infinity_robust_fusion.tex :exports both
|
||||
* Generalized plant used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+begin_src latex :file h_infinity_robust_fusion_plant.pdf :tangle figs/h_infinity_robust_fusion_plant.tex :exports both
|
||||
\begin{tikzpicture}
|
||||
\node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
\node[above] at (P.north) {$P(s)$};
|
||||
|
||||
\node[block, below=0.2 of P, opacity=0] (H2) {$H_2(s)$};
|
||||
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
|
||||
|
||||
@ -357,8 +359,6 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
% \node[block, below=0.3 of P] (H2) {$H_2(s)$};
|
||||
|
||||
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
|
||||
\draw[->] (inputu) node[above right]{$u$} -- (W2.west);
|
||||
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
@ -369,10 +369,44 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:h_infinity_robust_fusion
|
||||
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters ([[./figs/h_infinity_robust_fusion.png][png]], [[./figs/h_infinity_robust_fusion.pdf][pdf]], [[./figs/h_infinity_robust_fusion.tex][tex]]).
|
||||
#+name: fig:h_infinity_robust_fusion_plant
|
||||
#+caption: Generalized plant used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/h_infinity_robust_fusion.png]]
|
||||
[[file:figs/h_infinity_robust_fusion_plant.png]]
|
||||
|
||||
* Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+begin_src latex :file h_infinity_robust_fusion_fb.pdf :tangle figs/h_infinity_robust_fusion_fb.tex :exports both
|
||||
\begin{tikzpicture}
|
||||
\node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
\node[above] at (P.north) {$P(s)$};
|
||||
|
||||
\node[block, below=0.2 of P] (H2) {$H_2(s)$};
|
||||
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.4, 0)$);
|
||||
|
||||
\coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.4, 0)$);
|
||||
|
||||
\node[block, left=1.4 of output1] (W1){$W_1(s)$};
|
||||
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
|
||||
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
\draw[->] (sub.east) -- (W1.west);
|
||||
\draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- (outputv) |- (H2.east);
|
||||
\draw[->] (H2.west) -| (inputu) -- (W2.west);
|
||||
\draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
|
||||
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:h_infinity_robust_fusion_fb
|
||||
#+caption: Generalized plant with the synthesized filter
|
||||
#+RESULTS:
|
||||
[[file:figs/h_infinity_robust_fusion_fb.png]]
|
||||
|
||||
* LIGO Sensor Fusion Architecture
|
||||
#+begin_src latex :file ligo_super_sensor_architecture.pdf :tangle figs/ligo_super_sensor_architecture.tex :exports both
|
||||
@ -419,7 +453,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:ligo_super_sensor_architecture
|
||||
#+caption: ([[./figs/ligo_super_sensor_architecture.png][png]], [[./figs/ligo_super_sensor_architecture.pdf][pdf]], [[./figs/ligo_super_sensor_architecture.tex][tex]]).
|
||||
#+caption: Simplified block diagram of the sensor blending strategy for the first stage at the LIGO
|
||||
#+RESULTS:
|
||||
[[file:figs/ligo_super_sensor_architecture.png]]
|
||||
|
||||
@ -439,13 +473,12 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(L.north-|addfb.west) (botc)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
% \node[below] at (supersensor.north) {Super Sensor};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:feedback_sensor_fusion
|
||||
#+caption: "Closed-Loop" complementary filters ([[./figs/feedback_sensor_fusion.png][png]], [[./figs/feedback_sensor_fusion.pdf][pdf]], [[./figs/feedback_sensor_fusion.tex][tex]]).
|
||||
#+caption: "Closed-Loop" complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/feedback_sensor_fusion.png]]
|
||||
|
||||
@ -475,37 +508,36 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\node[fit={(sensor2.west|-botc) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
|
||||
\node[above, align=center] at (sensors.south) {{\tiny Normalized}\\[-0.5em]{\tiny sensors}};
|
||||
\node[below, align=center] at (feedbackfilter.north) {{\tiny "Closed-Loop"}\\[-0.5em]{\tiny complementary filters}};
|
||||
% \node[above, align=center] at (supersensor.north) {Super Sensor};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:feedback_sensor_fusion_arch
|
||||
#+caption: ([[./figs/feedback_sensor_fusion_arch.png][png]], [[./figs/feedback_sensor_fusion_arch.pdf][pdf]], [[./figs/feedback_sensor_fusion_arch.tex][tex]]).
|
||||
#+caption: Classical feedback architecture used for sensor fusion
|
||||
#+RESULTS:
|
||||
[[file:figs/feedback_sensor_fusion_arch.png]]
|
||||
|
||||
* TODO Feedback Loop Sensor Fusion Architecture
|
||||
* Feedback Loop Sensor Fusion Architecture
|
||||
#+begin_src latex :file feedback_synthesis_architecture.pdf :tangle figs/feedback_synthesis_architecture.tex
|
||||
\begin{tikzpicture}
|
||||
\node[block] (W2) at (0,0) {$W_2$};
|
||||
\node[addb={+}{}{}{}{-}, right=1 of W2] (addfb){};
|
||||
\node[block] (W2) at (0,0) {$W_2(s)$};
|
||||
\node[addb={+}{}{}{}{-}, right=0.8 of W2] (addfb){};
|
||||
\node[addb={+}{}{}{}{}, right=4.5 of W2] (adddy){};
|
||||
\node[block, above=0.8 of adddy] (W1){$W_1$};
|
||||
\node[block, above=0.8 of adddy] (W1){$W_1(s)$};
|
||||
|
||||
\draw[<-] (W2.west) -- ++(-1, 0) node[above right]{$w_2$};
|
||||
\draw[<-] (W2.west) -- ++(-0.8, 0) node[above right]{$w_2$};
|
||||
\draw[->] (W2.east) -- (addfb.west) node[above left]{$\tilde{w}_2$};
|
||||
\draw[->] (addfb.east) -- ++(1, 0) node[above left]{$v$};
|
||||
\draw[<-] (adddy.west) -- ++(-1, 0) node[above right]{$u$};
|
||||
\draw[->] (adddy.east) -- ++(1.4, 0) node[above left]{$z$};
|
||||
\draw[->] (W1.south) -- (adddy.north) node[above right]{$\tilde{w}_1$};
|
||||
\draw[<-] (W1.north) -- ++(0, 1) node[below right]{$w_1$};
|
||||
\draw[<-] (W1.north) -- ++(0, 0.8) node[below right]{$w_1$};
|
||||
\draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -0.8) -| (addfb.south);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:feedback_synthesis_architecture
|
||||
#+caption: ([[./figs/feedback_synthesis_architecture.png][png]], [[./figs/feedback_synthesis_architecture.pdf][pdf]], [[./figs/feedback_synthesis_architecture.tex][tex]]).
|
||||
#+caption: Feedback architecture with included weights
|
||||
#+RESULTS:
|
||||
[[file:figs/feedback_synthesis_architecture.png]]
|
||||
|
||||
@ -539,10 +571,54 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:feedback_synthesis_architecture_generalized_plant
|
||||
#+caption: ([[./figs/feedback_synthesis_architecture_generalized_plant.png][png]], [[./figs/feedback_synthesis_architecture_generalized_plant.pdf][pdf]], [[./figs/feedback_synthesis_architecture_generalized_plant.tex][tex]]).
|
||||
#+caption: Generalized plant used for the $\mathcal{H}_\infty$ synthesis of "closed-loop" complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/feedback_synthesis_architecture_generalized_plant.png]]
|
||||
|
||||
* Sensor Fusion - Sequential
|
||||
#+begin_src latex :file sensor_fusion_three_sequential.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
|
||||
\node[block, right=0.4 of x] (sensor2) {Sensor 2};
|
||||
\node[block, above=0.4 of sensor2] (sensor1) {Sensor 1};
|
||||
\node[block, below=0.4 of sensor2] (sensor3) {Sensor 3};
|
||||
|
||||
\node[block, right=1.1 of sensor1](H1){$H_1(s)$};
|
||||
\node[block, right=1.1 of sensor2](H2){$H_2(s)$};
|
||||
\node[addb] (add) at ($0.5*(H1.east)+0.5*(H2.east)+(0.6, 0)$){};
|
||||
|
||||
\node[block, right=0.8 of add](H1p) {$H_1^\prime(s)$};
|
||||
\node[block] (H2p) at (H1p|-sensor3) {$H_2^\prime(s)$};
|
||||
|
||||
\node[addb] (addp) at ($0.5*(H1p.east)+0.5*(H2p.east)+(0.6, 0)$){};
|
||||
|
||||
\draw[->] ($(x)+(-0.8, 0)$) node[above right]{$x$} -- (sensor2.west);
|
||||
\draw[->] (x.center) |- (sensor1.west);
|
||||
\draw[->] (x.center) |- (sensor3.west);
|
||||
\draw[->] (sensor1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
|
||||
\draw[->] (sensor2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
|
||||
\draw[->] (sensor3.east) -- (H2p.west) node[above left]{$\hat{x}_3$};
|
||||
\draw[->] (H1) -| (add.north);
|
||||
\draw[->] (H2) -| (add.south);
|
||||
\draw[->] (add.east) -- (H1p.west) node[above left]{$\hat{x}_{12}$};
|
||||
\draw[->] (H1p) -| (addp.north);
|
||||
\draw[->] (H2p) -| (addp.south);
|
||||
\draw[->] (addp.east) -- ++(0.8, 0) node[above left]{$\hat{x}$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(x.west|-sensor3.south) (sensor1.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
|
||||
\node[fit={(x.west|-sensor1.north) (add.east|-sensor2.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_three_sequential
|
||||
#+caption: Sequential fusion
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_three_sequential.png]]
|
||||
|
||||
* Sensor Fusion - Parallel
|
||||
#+begin_src latex :file sensor_fusion_three_parallel.pdf
|
||||
\begin{tikzpicture}
|
||||
@ -578,57 +654,56 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_three_parallel
|
||||
#+caption: Sensor Fusion Architecture - Overview
|
||||
#+caption: Parallel fusion
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_three_parallel.png]]
|
||||
|
||||
* Sensor Fusion - Sequential
|
||||
#+begin_src latex :file sensor_fusion_three_sequential.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
* Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
#+begin_src latex :file comp_filter_three_hinf_gen_plant.pdf :tangle figs/comp_filter_three_hinf_gen_plant.tex
|
||||
\begin{tikzpicture}
|
||||
\node[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
||||
\node[above] at (P.north) {$P_3(s)$};
|
||||
|
||||
\node[block, right=0.4 of x] (sensor2) {Sensor 2};
|
||||
\node[block, above=0.4 of sensor2] (sensor1) {Sensor 1};
|
||||
\node[block, below=0.4 of sensor2] (sensor3) {Sensor 3};
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.4!(P.north west) + (-0.7, 0)$);
|
||||
|
||||
\node[block, right=1.1 of sensor1](H1){$H_1(s)$};
|
||||
\node[block, right=1.1 of sensor2](H2){$H_2(s)$};
|
||||
\node[addb] (add) at ($0.5*(H1.east)+0.5*(H2.east)+(0.6, 0)$){};
|
||||
\coordinate[] (output1) at ($(P.south east)!0.8!(P.north east) + (0.7, 0)$);
|
||||
\coordinate[] (output2) at ($(P.south east)!0.55!(P.north east) + (0.7, 0)$);
|
||||
\coordinate[] (output3) at ($(P.south east)!0.3!(P.north east) + (0.7, 0)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + (0.7, 0)$);
|
||||
|
||||
\node[block, right=0.8 of add](H1p) {$H_1^\prime(s)$};
|
||||
\node[block] (H2p) at (H1p|-sensor3) {$H_2^\prime(s)$};
|
||||
\node[block, left=1.4 of output1] (W1){$W_1(s)$};
|
||||
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
|
||||
\node[block, left=1.4 of output3] (W3){$W_3(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
||||
\node[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
||||
|
||||
\node[addb] (addp) at ($0.5*(H1p.east)+0.5*(H2p.east)+(0.6, 0)$){};
|
||||
\node[block, below=0.3 of P, opacity=0] (H) {$\begin{bmatrix}H_2(s) \\ H_3(s)\end{bmatrix}$};
|
||||
|
||||
\draw[->] ($(x)+(-0.8, 0)$) node[above right]{$x$} -- (sensor2.west);
|
||||
\draw[->] (x.center) |- (sensor1.west);
|
||||
\draw[->] (x.center) |- (sensor3.west);
|
||||
\draw[->] (sensor1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
|
||||
\draw[->] (sensor2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
|
||||
\draw[->] (sensor3.east) -- (H2p.west) node[above left]{$\hat{x}_3$};
|
||||
\draw[->] (H1) -| (add.north);
|
||||
\draw[->] (H2) -| (add.south);
|
||||
\draw[->] (add.east) -- (H1p.west) node[above left]{$\hat{x}_{12}$};
|
||||
\draw[->] (H1p) -| (addp.north);
|
||||
\draw[->] (H2p) -| (addp.south);
|
||||
\draw[->] (addp.east) -- ++(0.8, 0) node[above left]{$\hat{x}$};
|
||||
\draw[->] (inputw) node[above right](w){$w$} -- (sub2.west);
|
||||
\draw[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
||||
\draw[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
||||
\draw[->] ($(sub2.west)+(-0.5, 0)$) node[branch]{} |- (outputv);
|
||||
\draw[->] (inputu|-W2) -- (W2.west);
|
||||
\draw[->] (inputu|-W3) -- (W3.west);
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(x.west|-sensor3.south) (sensor1.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
% \node[below] at (supersensor.north) {Super Sensor};
|
||||
|
||||
\node[fit={(x.west|-sensor1.north) (add.east|-sensor2.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
|
||||
% \node[] at (superinertialsensor.center) {"Inertial" Super Sensor};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
\draw[->] (sub2.east) -- (sub1.west);
|
||||
\draw[->] (sub1.east) -- (W1.west);
|
||||
\draw[->] (W1.east) -- (output1)node[above left](z){$z_1$};
|
||||
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
||||
\draw[->] (W3.east) -- (output3)node[above left]{$z_3$};
|
||||
\node[above] at (W2-|w){$u_1$};
|
||||
\node[above] at (W3-|w){$u_2$};
|
||||
\node[above] at (outputv-|z){$v$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_three_sequential
|
||||
#+caption: Sensor Fusion Architecture - Overview
|
||||
#+name: fig:comp_filter_three_hinf_gen_plant
|
||||
#+caption: Generalized plant for the $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_three_sequential.png]]
|
||||
[[file:figs/comp_filter_three_hinf_gen_plant.png]]
|
||||
* Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
#+begin_src latex :file comp_filter_three_hinf.pdf :tangle figs/comp_filter_three_hinf.tex
|
||||
#+begin_src latex :file comp_filter_three_hinf_fb.pdf :tangle figs/comp_filter_three_hinf_fb.tex
|
||||
\begin{tikzpicture}
|
||||
\node[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
||||
\node[above] at (P.north) {$P_3(s)$};
|
||||
@ -667,7 +742,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:comp_filter_three_hinf
|
||||
#+caption: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters ([[./figs/comp_filter_three_hinf.png][png]], [[./figs/comp_filter_three_hinf.pdf][pdf]], [[./figs/comp_filter_three_hinf.tex][tex]]).
|
||||
#+name: fig:comp_filter_three_hinf_fb
|
||||
#+caption: Generalized plant with the synthesized filter for the $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/comp_filter_three_hinf.png]]
|
||||
[[file:figs/comp_filter_three_hinf_fb.png]]
|
||||
716
tikz/index.html
716
tikz/index.html
@ -1,716 +0,0 @@
|
||||
<?xml version="1.0" encoding="utf-8"?>
|
||||
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2020-11-12 jeu. 10:44 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<title>Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis - Tikz Figures</title>
|
||||
<meta name="generator" content="Org mode" />
|
||||
<meta name="author" content="Thomas Dehaeze" />
|
||||
<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
|
||||
<script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
|
||||
<script>MathJax = {
|
||||
tex: {
|
||||
tags: 'ams',
|
||||
macros: {bm: ["\\boldsymbol{#1}",1],}
|
||||
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|
||||
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|
||||
</script>
|
||||
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
|
||||
</head>
|
||||
<body>
|
||||
<div id="org-div-home-and-up">
|
||||
<a accesskey="h" href="../index.html"> UP </a>
|
||||
|
|
||||
<a accesskey="H" href="../index.html"> HOME </a>
|
||||
</div><div id="content">
|
||||
<h1 class="title">Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis - Tikz Figures</h1>
|
||||
<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org9059d4e">1. Fig 1: Sensor Fusion Architecture</a></li>
|
||||
<li><a href="#orgf8987c9">2. Fig 2: Sensor fusion architecture with sensor dynamics uncertainty</a></li>
|
||||
<li><a href="#org1801a90">3. Fig 3: Uncertainty set of the super sensor dynamics</a></li>
|
||||
<li><a href="#org355bde0">4. Fig 4: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</a></li>
|
||||
<li><a href="#org879ecb3">5. Fig 5: Magnitude of a weighting function generated using the proposed formula</a></li>
|
||||
<li><a href="#org2b1f7ae">6. Fig 6: Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</a></li>
|
||||
<li><a href="#org84a7e33">7. Fig 7: Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</a></li>
|
||||
<li><a href="#org20bfce3">8. Fig 8: Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</a></li>
|
||||
<li><a href="#orgb08c277">9. Fig 9: Specifications and weighting functions magnitude used for \(\mathcal{H}_\infty\) synthesis</a></li>
|
||||
<li><a href="#org74b4762">10. Fig 10: Comparison of the FIR filters (solid) with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed)</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
<p>
|
||||
Configuration file is accessible <a href="config.html">here</a>.
|
||||
</p>
|
||||
|
||||
<div id="outline-container-org9059d4e" class="outline-2">
|
||||
<h2 id="org9059d4e"><span class="section-number-2">1</span> Fig 1: Sensor Fusion Architecture</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right=0.5 and 0.5 of x](G1){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below right=0.5 and 0.5 of x](G2){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.8 of G1](add1){};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.8 of G2](add2){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=5 of x](add){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (G1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (G2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- (add1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G2.east) -- (add2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add1.north) -- ++(0, 0.8)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add2.north) -- ++(0, 0.8)node[below right](n2){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (H1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add2.east) -- (H2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(<span class="org-font-latex-math">$(G2.south-|x)+(-0.2, -0.3)$</span>) (<span class="org-font-latex-math">$(n1.north east-|add.east)+(0.2, 0.3)$</span>)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(<span class="org-font-latex-math">$(G1.south west)+(-0.3, -0.1)$</span>) (<span class="org-font-latex-math">$(n1.north east)+(0.0, 0.1)$</span>)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below right] at (sensor1.north west) {Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(<span class="org-font-latex-math">$(G2.south west)+(-0.3, -0.1)$</span>) (<span class="org-font-latex-math">$(n2.north east)+(0.0, 0.1)$</span>)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below right] at (sensor2.north west) {Sensor 2};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org9c10944" class="figure">
|
||||
<p><img src="figs/fusion_super_sensor.png" alt="fusion_super_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Sensor Fusion Architecture (<a href="./figs/fusion_super_sensor.png">png</a>, <a href="./figs/fusion_super_sensor.pdf">pdf</a>, <a href="./figs/fusion_super_sensor.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf8987c9" class="outline-2">
|
||||
<h2 id="orgf8987c9"><span class="section-number-2">2</span> Fig 2: Sensor fusion architecture with sensor dynamics uncertainty</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, above right=0.8 and 4 of x](add1){};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, below right=0.8 and 4 of x](add2){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above left=0.2 and 0.1 of add1](delta1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above left=0.2 and 0.1 of add2](delta2){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=0.5 of delta1](W1){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=0.5 of delta2](W2){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.5 of add1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.5 of add2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=6 of x](add){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (add1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (add2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(add1-|W1.west)+(-0.5, 0)$</span>)node[branch](S1){} |- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(add2-|W2.west)+(-0.5, 0)$</span>)node[branch](S1){} |- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (delta1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (delta2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta1.east) -| (add1.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta2.east) -| (add2.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (H1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add2.east) -- (H2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1.east) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2.east) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[block, fit={(<span class="org-font-latex-math">$(W1.north-|S1)+(-0.2, 0.2)$</span>) (<span class="org-font-latex-math">$(add1.south east)+(0.2, -0.3)$</span>)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above right] at (sensor1.south west) {Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, fit={(<span class="org-font-latex-math">$(W2.north-|S1)+(-0.2, 0.2)$</span>) (<span class="org-font-latex-math">$(add2.south east)+(0.2, -0.3)$</span>)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above right] at (sensor2.south west) {Sensor 2};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgfc65b94" class="figure">
|
||||
<p><img src="figs/sensor_fusion_dynamic_uncertainty.png" alt="sensor_fusion_dynamic_uncertainty.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Sensor fusion architecture with sensor dynamics uncertainty (<a href="./figs/sensor_fusion_dynamic_uncertainty.png">png</a>, <a href="./figs/sensor_fusion_dynamic_uncertainty.pdf">pdf</a>, <a href="./figs/sensor_fusion_dynamic_uncertainty.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1801a90" class="outline-2">
|
||||
<h2 id="org1801a90"><span class="section-number-2">3</span> Fig 3: Uncertainty set of the super sensor dynamics</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[shift={(4, 0)}]
|
||||
|
||||
<span class="org-comment">% Uncertainty Circle</span>
|
||||
<span class="org-font-latex-sedate">\node</span>[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, dotted] (0, 0) circle [radius=1.0];
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, dashed] (135:1.0) circle [radius=0.8];
|
||||
|
||||
<span class="org-comment">% Center of Circle</span>
|
||||
<span class="org-font-latex-sedate">\node</span>[below] at (0, 0){<span class="org-font-latex-math">$1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->, dashed] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->, dashed] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(90:0.8);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l1) at (2, 1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, dashed, out=-90, in=0] (l1.south) to (r1);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l2) at (-2.5, 1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, dashed, out=0, in=-180] (l2.east) to (r2);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->, dashed] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l3) at (-2.5, -1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1| + |w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, dashed, out=90, in=-90] (l3.north) to (r3);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
|
||||
<span class="org-comment">% Real and Imaginary Axis</span>
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed] (1, 0) arc (0:28:1) node[midway, right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"> </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\phi</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orga1a6d2e" class="figure">
|
||||
<p><img src="figs/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Uncertainty region of the super sensor dynamics in the complex plane (solid circle), of the sensor 1 (dotted circle) and of the sensor 2 (dashed circle) (<a href="./figs/uncertainty_set_super_sensor.png">png</a>, <a href="./figs/uncertainty_set_super_sensor.pdf">pdf</a>, <a href="./figs/uncertainty_set_super_sensor.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org355bde0" class="outline-2">
|
||||
<h2 id="org355bde0"><span class="section-number-2">4</span> Fig 4: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.75!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.35!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.75!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.35!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + ( 0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P] (H2) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right]{<span class="org-font-latex-math">$w$</span>} -- (sub.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2.west) -| (<span class="org-font-latex-math">$(inputu)+(0.35, 0)$</span>) node[above]{<span class="org-font-latex-math">$u$</span>} -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub.west)+(-0.6, 0)$</span>) node[branch]{} |- (<span class="org-font-latex-math">$(outputv)+(-0.35, 0)$</span>) node[above]{<span class="org-font-latex-math">$v$</span>} |- (H2.east);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org903a05c" class="figure">
|
||||
<p><img src="figs/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters (<a href="./figs/h_infinity_robust_fusion.png">png</a>, <a href="./figs/h_infinity_robust_fusion.pdf">pdf</a>, <a href="./figs/h_infinity_robust_fusion.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org879ecb3" class="outline-2">
|
||||
<h2 id="org879ecb3"><span class="section-number-2">5</span> Fig 5: Magnitude of a weighting function generated using the proposed formula</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">3.5cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=1.0<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=100,
|
||||
xtick={0.1,1,10, 100},
|
||||
xminorticks=true,
|
||||
ymode=log,
|
||||
ymin=0.0005,
|
||||
ymax=20,
|
||||
ytick={0.001, 0.01, 0.1, 1, 10},
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
xlabel={Frequency [Hz]},
|
||||
xminorgrids,
|
||||
yminorgrids,
|
||||
]
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=ampl, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matweight_formula.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dashed, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
1 10<span class="org-font-latex-warning">\\</span>
|
||||
100 10<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dashed, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.1 0.001<span class="org-font-latex-warning">\\</span>
|
||||
3 0.001<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.1 1<span class="org-font-latex-warning">\\</span>
|
||||
100 1<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dashed, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
10 2<span class="org-font-latex-warning">\\</span>
|
||||
10 1<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[below] at (2, 10) {<span class="org-font-latex-math">$G_</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\infty</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (2, 0.001) {<span class="org-font-latex-math">$G_0$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] at (10, 2){};
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed, line cap=round] (7, 2) -- (20, 2) node[right]{<span class="org-font-latex-math">$G_c$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed, line cap=round] (10, 2) -- (10, 1) node[below]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\omega</span></span><span class="org-font-latex-math">_c$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[right] at (3, 0.1) {<span class="org-font-latex-math">$+n$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org1aeddba" class="figure">
|
||||
<p><img src="figs/weight_formula.png" alt="weight_formula.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Magnitude of a weighting function generated using the proposed formula (<a href="./figs/weight_formula.png">png</a>, <a href="./figs/weight_formula.pdf">pdf</a>, <a href="./figs/weight_formula.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2b1f7ae" class="outline-2">
|
||||
<h2 id="org2b1f7ae"><span class="section-number-2">6</span> Fig 6: Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">6cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.5<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.47<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=1000,
|
||||
xtick={0.1, 1, 10, 100, 1000},
|
||||
xticklabels={{}},
|
||||
xminorticks=true,
|
||||
ymode=log,
|
||||
ymin=0.0005,
|
||||
ymax=20,
|
||||
ytick={0.001, 0.01, 0.1, 1, 10},
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
xminorgrids,
|
||||
yminorgrids,
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.45<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=1000,
|
||||
xtick={0.1, 1, 10, 100, 1000},
|
||||
xminorticks=true,
|
||||
xlabel={Frequency [Hz]},
|
||||
ymin=-200,
|
||||
ymax=200,
|
||||
ytick={-180, -90, 0, 90, 180},
|
||||
ylabel={Phase [deg]},
|
||||
xminorgrids,
|
||||
legend style={at={(1,1.1)}, outer sep=2pt , anchor=north east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}},
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addlegendimage</span>{color=mycolor1, dashed, line width=1.5pt}
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$W_1^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addlegendimage</span>{color=mycolor2, dashed, line width=1.5pt}
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$W_2^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_1$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_2$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd2fb3fa" class="figure">
|
||||
<p><img src="figs/hinf_synthesis_results.png" alt="hinf_synthesis_results.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis (<a href="./figs/hinf_synthesis_results.png">png</a>, <a href="./figs/hinf_synthesis_results.pdf">pdf</a>, <a href="./figs/hinf_synthesis_results.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org84a7e33" class="outline-2">
|
||||
<h2 id="org84a7e33"><span class="section-number-2">7</span> Fig 7: Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={5.0cm}{3.5cm}, fill=black!20!white, dashed] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.8!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.4!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.8!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.55!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output3) at (<span class="org-font-latex-math">$(P.south east)!0.3!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + (0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output3] (W3){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P] (H) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s) \\ H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right](w){<span class="org-font-latex-math">$w$</span>} -- (sub2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub2.west)+(-0.5, 0)$</span>) node[branch]{} |- (outputv) |- (H.east);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(H.south west)!0.7!(H.north west)$</span>) -| (inputu|-W2) -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(H.south west)!0.3!(H.north west)$</span>) -| (<span class="org-font-latex-math">$(inputu|-W3)+(0.4, 0)$</span>) -- (W3.west);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub2.east) -- (sub1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub1.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left](z){<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3.east) -- (output3)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W2-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W3-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (outputv-|z){<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgb697644" class="figure">
|
||||
<p><img src="figs/comp_filter_three_hinf.png" alt="comp_filter_three_hinf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters (<a href="./figs/comp_filter_three_hinf.png">png</a>, <a href="./figs/comp_filter_three_hinf.pdf">pdf</a>, <a href="./figs/comp_filter_three_hinf.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org20bfce3" class="outline-2">
|
||||
<h2 id="org20bfce3"><span class="section-number-2">8</span> Fig 8: Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</h2>
|
||||
<div class="outline-text-2" id="text-8">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">6cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.55<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.42<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=100,
|
||||
xticklabels={{}},
|
||||
xminorticks=true,
|
||||
ymode=log,
|
||||
ymin=0.0005,
|
||||
ymax=20,
|
||||
ytick={0.001, 0.01, 0.1, 1, 10},
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
xminorgrids,
|
||||
yminorgrids,
|
||||
legend columns=2,
|
||||
legend style={
|
||||
/tikz/column 2/.style={
|
||||
column sep=5pt,
|
||||
},
|
||||
at={(1,0)}, outer sep=2pt , anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}
|
||||
},
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">${W_1}^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_1$</span>};
|
||||
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">${W_2}^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_2$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor3, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=W3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">${W_3}^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor3, line width=1.5pt]
|
||||
table [x=freqs, y=H3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_3$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.4<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=100,
|
||||
xminorticks=true,
|
||||
xlabel={Frequency [Hz]},
|
||||
ymin=-240,
|
||||
ymax=240,
|
||||
ytick={-180, -90, 0, 90, 180},
|
||||
ylabel={Phase [deg]},
|
||||
xminorgrids,
|
||||
]
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor3, line width=1.5pt]
|
||||
table [x=freqs, y=H3p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgdaa24c8" class="figure">
|
||||
<p><img src="figs/hinf_three_synthesis_results.png" alt="hinf_three_synthesis_results.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis (<a href="./figs/hinf_three_synthesis_results.png">png</a>, <a href="./figs/hinf_three_synthesis_results.pdf">pdf</a>, <a href="./figs/hinf_three_synthesis_results.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb08c277" class="outline-2">
|
||||
<h2 id="orgb08c277"><span class="section-number-2">9</span> Fig 9: Specifications and weighting functions magnitude used for \(\mathcal{H}_\infty\) synthesis</h2>
|
||||
<div class="outline-text-2" id="text-9">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">3.2cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=1.0<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
separate axis lines,
|
||||
every outer x axis line/.append style={black},
|
||||
every x tick label/.append style={font=<span class="org-font-latex-sedate">\color</span>{black}},
|
||||
every x tick/.append style={black},
|
||||
xmode=log,
|
||||
xmin=0.001,
|
||||
xmax=1,
|
||||
xminorticks=true,
|
||||
xlabel={Frequency [Hz]},
|
||||
every outer y axis line/.append style={black},
|
||||
every y tick label/.append style={font=<span class="org-font-latex-sedate">\color</span>{black}},
|
||||
every y tick/.append style={black},
|
||||
ymode=log,
|
||||
ymin=0.005,
|
||||
ymax=20,
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
axis background/.style={fill=white},
|
||||
xmajorgrids,
|
||||
xminorgrids,
|
||||
ymajorgrids,
|
||||
yminorgrids,
|
||||
legend style={at={(0,1)}, outer sep=2pt, anchor=north west, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
|
||||
]
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=wHm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$|w_H|^{-1}$</span>}
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=wLm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$|w_L|^{-1}$</span>}
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.0005 0.008<span class="org-font-latex-warning">\\</span>
|
||||
0.008 0.008<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{Specifications}
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt, forget plot]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.008 0.008<span class="org-font-latex-warning">\\</span>
|
||||
0.04 1<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt, forget plot]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.04 3<span class="org-font-latex-warning">\\</span>
|
||||
0.1 3<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.1 0.045<span class="org-font-latex-warning">\\</span>
|
||||
2 0.045<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org2ba2505" class="figure">
|
||||
<p><img src="figs/ligo_weights.png" alt="ligo_weights.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Specifications and weighting functions magnitude used for \(\mathcal{H}_\infty\) synthesis (<a href="./figs/ligo_weights.png">png</a>, <a href="./figs/ligo_weights.pdf">pdf</a>, <a href="./figs/ligo_weights.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org74b4762" class="outline-2">
|
||||
<h2 id="org74b4762"><span class="section-number-2">10</span> Fig 10: Comparison of the FIR filters (solid) with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed)</h2>
|
||||
<div class="outline-text-2" id="text-10">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">6.8cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.60<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.32<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.001,
|
||||
xmax=1,
|
||||
xtick={0.001,0.01,0.1,1},
|
||||
xticklabels={{}},
|
||||
xminorticks=true,
|
||||
ymode=log,
|
||||
ymin=0.002,
|
||||
ymax=5,
|
||||
ytick={0.001, 0.01, 0.1, 1, 10},
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
xminorgrids,
|
||||
yminorgrids,
|
||||
legend style={at={(1,0)}, outer sep=2pt, anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_H(s)$</span> - <span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{H}_</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\infty</span></span><span class="org-font-latex-math">$</span>}
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_H(s)$</span> - FIR}
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_L(s)$</span> - <span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{H}_</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\infty</span></span><span class="org-font-latex-math">$</span>}
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_L(s)$</span> - FIR}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.3<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.001,
|
||||
xmax=1,
|
||||
xtick={0.001, 0.01, 0.1, 1},
|
||||
xminorticks=true,
|
||||
xlabel={Frequency [Hz]},
|
||||
ymin=-180,
|
||||
ymax=180,
|
||||
ytick={-180, -90, 0, 90, 180},
|
||||
ylabel={Phase [deg]},
|
||||
xminorgrids,
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6bce6a7" class="figure">
|
||||
<p><img src="figs/comp_fir_ligo_hinf.png" alt="comp_fir_ligo_hinf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Comparison of the FIR filters (solid) with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed) (<a href="./figs/comp_fir_ligo_hinf.png">png</a>, <a href="./figs/comp_fir_ligo_hinf.pdf">pdf</a>, <a href="./figs/comp_fir_ligo_hinf.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2020-11-12 jeu. 10:44</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
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Reference in New Issue
Block a user