Rename main orgmode matlab file
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@ -1,4 +1,4 @@
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#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis - Matlab Computation
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#+TITLE: A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis - Matlab Computation
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:DRAWER:
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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@ -34,23 +34,12 @@
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:END:
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* Introduction :ignore:
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In this document, the design of complementary filters is studied.
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One use of complementary filter is described below:
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#+begin_quote
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The basic idea of a complementary filter involves taking two or more sensors, filtering out unreliable frequencies for each sensor, and combining the filtered outputs to get a better estimate throughout the entire bandwidth of the system.
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To achieve this, the sensors included in the filter should complement one another by performing better over specific parts of the system bandwidth.
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#+end_quote
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This document is divided into several sections:
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- in section [[#sec:h_inf_synthesis_complementary_filters]], the $\mathcal{H}_\infty$ synthesis is used for generating two complementary filters
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- in section [[sec:three_comp_filters]], a method using the $\mathcal{H}_\infty$ synthesis is proposed to shape three of more complementary filters
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- in section [[sec:comp_filters_ligo]], the $\mathcal{H}_\infty$ synthesis is used and compared with FIR complementary filters used for LIGO
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#+begin_note
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Add the Matlab code use to obtain the results presented in the paper are accessible [[file:matlab.zip][here]] and presented below.
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#+end_note
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* H-Infinity synthesis of complementary filters
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/h_inf_synthesis_complementary_filters.m
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@ -155,6 +144,7 @@ xlabel('Frequency [Hz]'); ylabel('Magnitude');
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hold off;
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xlim([freqs(1), freqs(end)]);
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ylim([5e-4, 20]);
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yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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@ -176,23 +166,28 @@ W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G
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#+begin_src matlab :exports none
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figure;
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tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile();
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hold on;
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set(gca,'ColorOrderIndex',1)
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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xlabel('Frequency [Hz]', 'FontSize', 10); ylabel('Magnitude', 'FontSize', 10);
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hold off;
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xlim([freqs(1), freqs(end)]);
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ylim([1e-4, 20]);
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xticks([0.1, 1, 10, 100, 1000]);
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leg = legend('location', 'southeast', 'FontSize', 8);
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ylim([8e-4, 20]);
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yticks([1e-3, 1e-2, 1e-1, 1, 1e1]);
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yticklabels({'', '$10^{-2}$', '', '$10^0$', ''});
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ax1.FontSize = 9;
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leg = legend('location', 'south', 'FontSize', 8);
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leg.ItemTokenSize(1) = 18;
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/weights_W1_W2.pdf', 'width', 'wide', 'height', 'normal');
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exportFig('figs/weights_W1_W2.pdf', 'width', 'half', 'height', 350);
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#+end_src
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#+name: fig:weights_W1_W2
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@ -279,22 +274,21 @@ tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2, 1]);
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hold on;
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set(gca,'ColorOrderIndex',1)
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
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plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude');
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set(gca, 'XTickLabel',[]);
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ylim([1e-4, 20]);
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yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
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leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
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set(gca, 'XTickLabel',[]); ylabel('Magnitude');
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ylim([8e-4, 20]);
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yticks([1e-3, 1e-2, 1e-1, 1, 1e1]);
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yticklabels({'', '$10^{-2}$', '', '$10^0$', ''})
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leg = legend('location', 'south', 'FontSize', 8, 'NumColumns', 2);
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leg.ItemTokenSize(1) = 18;
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% Phase
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@ -305,16 +299,18 @@ plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
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hold off;
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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set(gca, 'XScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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yticks([-180:90:180]);
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ylim([-180, 200])
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yticklabels({'-180', '', '0', '', '180'})
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/hinf_filters_results.pdf', 'width', 'wide', 'height', 600);
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exportFig('figs/hinf_filters_results.pdf', 'width', 700, 'height', 450);
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#+end_src
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#+name: fig:hinf_filters_results
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@ -1352,7 +1348,7 @@ P = [ W1 0 1;
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#+end_src
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#+begin_src matlab :results output replace :exports both
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[L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on');
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[L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
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#+end_src
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#+begin_src matlab
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@ -1368,20 +1364,80 @@ zpk(H2)
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#+RESULTS:
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#+begin_example
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zpk(H1)
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ans =
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(s+2.115e07) (s+153.6) (s+4.613) (s^2 + 6.858s + 12.03)
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--------------------------------------------------------
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(s+2.117e07) (s^2 + 102.1s + 2732) (s^2 + 69.43s + 3271)
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(s+3.842)^3 (s+153.6) (s+1.289e05)
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-------------------------------------------------------
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(s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272)
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Continuous-time zero/pole/gain model.
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zpk(H2)
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ans =
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20455 (s+3425) (s+3318) (s^2 + 46.58s + 813.2)
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--------------------------------------------------------
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(s+2.117e07) (s^2 + 102.1s + 2732) (s^2 + 69.43s + 3271)
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125.61 (s+3358)^2 (s^2 + 46.61s + 813.8)
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-------------------------------------------------------
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(s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272)
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Continuous-time zero/pole/gain model.
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#+end_example
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#+begin_src matlab :exports none
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freqs = logspace(-1, 3, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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% Magnitude
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ax1 = nexttile([2, 1]);
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hold on;
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set(gca,'ColorOrderIndex',1)
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
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plot(freqs, abs(squeeze(freqresp(L, freqs, 'Hz'))), 'k--', 'DisplayName', '$|L|$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]); ylabel('Magnitude');
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ylim([1e-3, 1e3]);
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yticks([1e-3, 1e-2, 1e-1, 1, 1e1, 1e2, 1e3]);
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yticklabels({'', '$10^{-2}$', '', '$10^0$', '', '$10^2$', ''});
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leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3);
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leg.ItemTokenSize(1) = 18;
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% Phase
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ax2 = nexttile;
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hold on;
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set(gca,'ColorOrderIndex',1)
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plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
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hold off;
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set(gca, 'XScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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yticks([-180:90:180]);
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ylim([-180, 200])
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yticklabels({'-180', '', '0', '', '180'})
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/hinf_filters_results_mixed_sensitivity.pdf', 'width', 700 , 'height', 600);
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#+end_src
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#+name: fig:hinf_filters_results_mixed_sensitivity
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#+caption:
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#+RESULTS:
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[[file:figs/hinf_filters_results_mixed_sensitivity.png]]
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#+begin_src matlab :exports none
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freqs = logspace(-2, 4, 1000);
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@ -2592,10 +2648,14 @@ bibliographystyle:unsrt
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bibliography:ref.bib
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* Functions
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:PROPERTIES:
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:functions>>
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** =generateWF=: Generate Weighting Functions
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/src/generateWF.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:generateWF>>
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@ -2620,7 +2680,7 @@ function [W] = generateWF(args)
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% - w0 - Frequency at which |W(j w0)| = Gc [rad/s]
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%
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% Outputs:
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% - W - Generated Weight
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% - W - Generated Weighting Function
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#+end_src
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*** Optional Parameters
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@ -2628,6 +2688,7 @@ function [W] = generateWF(args)
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:UNNUMBERED: t
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:END:
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#+begin_src matlab
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%% Argument validation
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arguments
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args.n (1,1) double {mustBeInteger, mustBePositive} = 1
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args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
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@ -2635,7 +2696,19 @@ arguments
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args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
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args.w0 (1,1) double {mustBeNumeric, mustBePositive} = 1
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end
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#+end_src
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Verification that the parameters $G_0$, $G_c$ and $G_\infty$ are satisfy condition eqref:eq:cond_formula_1 or eqref:eq:cond_formula_2.
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#+name: eq:condition_params_formula
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\begin{subequations}
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\begin{align}
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G_0 < 1 < G_\infty \text{ and } G_0 < G_c < G_\infty \label{eq:cond_formula_1}\\
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G_\infty < 1 < G_0 \text{ and } G_\infty < G_c < G_0 \label{eq:cond_formula_2}
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\end{align}
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\end{subequations}
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#+begin_src matlab
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% Verification of correct relation between G0, Gc and Ginf
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mustBeBetween(args.G0, args.Gc, args.Ginf);
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#+end_src
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@ -2644,10 +2717,22 @@ mustBeBetween(args.G0, args.Gc, args.Ginf);
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:UNNUMBERED: t
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:END:
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#+begin_src matlab
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%% Initialize the Laplace variable
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s = zpk('s');
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#+end_src
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The weighting function formula use is:
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#+name: eq:weight_formula
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\begin{equation}
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W(s) = \left( \frac{
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\hfill{} \frac{1}{\omega_c} \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}}
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}{
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\left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_c} \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}}
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}\right)^n
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\end{equation}
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#+begin_src matlab
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%% Create the weighting function according to formula
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W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
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(args.G0/args.Gc)^(1/args.n))/...
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((1/args.Ginf)^(1/args.n)*(1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
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@ -2660,7 +2745,7 @@ W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(
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:UNNUMBERED: t
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:END:
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#+begin_src matlab
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% Custom validation function
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%% Custom validation function
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function mustBeBetween(a,b,c)
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if ~((a > b && b > c) || (c > b && b > a))
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eid = 'createWeight:inputError';
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@ -2672,7 +2757,6 @@ function mustBeBetween(a,b,c)
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** =generateCF=: Generate Complementary Filters
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/src/generateCF.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:generateCF>>
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@ -2687,7 +2771,7 @@ This Matlab function is accessible [[file:matlab/src/generateCF.m][here]].
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function [H1, H2] = generateCF(W1, W2, args)
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% createWeight -
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%
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% Syntax: [W] = generateCF(args)
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% Syntax: [H1, H2] = generateCF(W1, W2, args)
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%
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% Inputs:
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% - W1 - Weighting Function for H1
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@ -2706,6 +2790,7 @@ function [H1, H2] = generateCF(W1, W2, args)
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:UNNUMBERED: t
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:END:
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#+begin_src matlab
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%% Argument validation
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arguments
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W1
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W2
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@ -2719,15 +2804,18 @@ end
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:UNNUMBERED: t
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:END:
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#+begin_src matlab
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%% The generalized plant is defined
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P = [W1 -W1;
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0 W2;
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1 0];
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#+end_src
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#+begin_src matlab :results output replace :exports both
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%% The standard H-infinity synthesis is performed
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[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', args.method, 'DISPLAY', args.display);
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#+end_src
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#+begin_src matlab
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%% H1 is defined as the complementary of H2
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H1 = 1 - H2;
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#+end_src
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