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123
matlab/matlab/1_synthesis_complementary_filters.m
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123
matlab/matlab/1_synthesis_complementary_filters.m
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Initialize Frequency Vector
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freqs = logspace(-1, 3, 1000);
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%% Add functions to path
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addpath('./src');
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%% Weighting Function Design
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% Parameters
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n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2;
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% Formulas
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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;
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%% Magnitude of the weighting function with parameters
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(W, freqs, 'Hz'))), 'k-');
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plot([1e-3 1e0], [G0 G0], 'k--', 'LineWidth', 1)
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text(1e0, G0, '$\quad G_0$')
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plot([1e1 1e3], [G1 G1], 'k--', 'LineWidth', 1)
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text(1e1,G1,'$G_{\infty}\quad$','HorizontalAlignment', 'right')
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plot([w0/2/pi w0/2/pi], [1 2*Gc], 'k--', 'LineWidth', 1)
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text(w0/2/pi,1,'$\omega_c$','VerticalAlignment', 'top', 'HorizontalAlignment', 'center')
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plot([w0/2/pi/2 2*w0/2/pi], [Gc Gc], 'k--', 'LineWidth', 1)
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text(w0/2/pi/2, Gc, '$G_c \quad$','HorizontalAlignment', 'right')
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text(w0/5/pi/2, abs(evalfr(W, j*w0/5)), 'Slope: $n \quad$', 'HorizontalAlignment', 'right')
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text(w0/2/pi, abs(evalfr(W, j*w0)), '$\bullet$', 'HorizontalAlignment', 'center')
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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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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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%% Design of the Weighting Functions
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W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'G1', 1/10, 'Gc', 0.45);
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W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'G1', 1000, 'Gc', 0.45);
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%% Plot of the Weighting function magnitude
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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]', '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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xticks([0.1, 1, 10, 100, 1000]);
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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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%% Generalized Plant
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P = [W1 -W1;
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0 W2;
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1 0];
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%% H-Infinity Synthesis
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[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
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%% Define H1 to be the complementary of H2
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H1 = 1 - H2;
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%% Bode plot of the complementary filters
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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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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([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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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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258
matlab/matlab/2_ligo_complementary_filters.m
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258
matlab/matlab/2_ligo_complementary_filters.m
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Initialize Frequency Vector
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freqs = logspace(-3, 0, 1000);
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%% Add functions to path
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addpath('./src');
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%% Upper bounds for the complementary filters
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figure;
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hold on;
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set(gca,'ColorOrderIndex',1)
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plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
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set(gca,'ColorOrderIndex',1)
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plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',1)
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plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',2)
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plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
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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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hold off;
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xlim([freqs(1), freqs(end)]);
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ylim([1e-4, 10]);
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leg = legend('location', 'southeast', 'FontSize', 8);
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leg.ItemTokenSize(1) = 18;
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%% Initialized CVX
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cvx_startup;
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cvx_solver sedumi;
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%% Frequency vectors
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w1 = 0:4.06e-4:0.008;
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w2 = 0.008:4.06e-4:0.04;
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w3 = 0.04:8.12e-4:0.1;
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w4 = 0.1:8.12e-4:0.83;
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%% Filter order
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n = 512;
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%% Initialization of filter responses
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A1 = [ones(length(w1),1), cos(kron(w1'.*(2*pi),[1:n-1]))];
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A2 = [ones(length(w2),1), cos(kron(w2'.*(2*pi),[1:n-1]))];
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A3 = [ones(length(w3),1), cos(kron(w3'.*(2*pi),[1:n-1]))];
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A4 = [ones(length(w4),1), cos(kron(w4'.*(2*pi),[1:n-1]))];
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B1 = [zeros(length(w1),1), sin(kron(w1'.*(2*pi),[1:n-1]))];
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B2 = [zeros(length(w2),1), sin(kron(w2'.*(2*pi),[1:n-1]))];
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B3 = [zeros(length(w3),1), sin(kron(w3'.*(2*pi),[1:n-1]))];
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B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))];
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%% Convex optimization
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cvx_begin
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variable y(n+1,1)
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% t
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maximize(-y(1))
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for i = 1:length(w1)
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norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
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end
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for i = 1:length(w2)
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norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
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end
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for i = 1:length(w3)
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norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
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end
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for i = 1:length(w4)
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norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
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end
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cvx_end
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h = y(2:end);
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%% Combine the frequency vectors to form the obtained filter
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w = [w1 w2 w3 w4];
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H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
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%% Bode plot of the obtained complementary filters
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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(w, abs(1-H), '-', 'DisplayName', '$L_1$');
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plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
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set(gca,'ColorOrderIndex',2)
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plot(w, abs(H), '-', 'DisplayName', '$H_1$');
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plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
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plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
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plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', '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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hold off;
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ylim([5e-3, 10]);
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leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
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leg.ItemTokenSize(1) = 16;
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% Phase
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ax2 = nexttile;
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hold on;
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plot(w, 180/pi*unwrap(angle(1-H)), '-');
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plot(w, 180/pi*unwrap(angle(H)), '-');
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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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yticks([-360:180:180]); ylim([-380, 200]);
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linkaxes([ax1,ax2],'x');
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xlim([1e-3, 1]);
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%% Design of the weight for the high pass filter
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w1 = 2*pi*0.008; x1 = 0.35;
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w2 = 2*pi*0.04; x2 = 0.5;
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w3 = 2*pi*0.05; x3 = 0.5;
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% Slope of +3 from w1
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wH = 0.008*(s^2/w1^2 + 2*x1/w1*s + 1)*(s/w1 + 1);
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% Little bump from w2 to w3
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wH = wH*(s^2/w2^2 + 2*x2/w2*s + 1)/(s^2/w3^2 + 2*x3/w3*s + 1);
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% No Slope at high frequencies
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wH = wH/(s^2/w3^2 + 2*x3/w3*s + 1)/(s/w3 + 1);
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% Little bump between w2 and w3
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w0 = 2*pi*0.045; xi = 0.1; A = 2; n = 1;
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wH = wH*((s^2 + 2*w0*xi*A^(1/n)*s + w0^2)/(s^2 + 2*w0*xi*s + w0^2))^n;
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wH = 1/wH;
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wH = minreal(ss(wH));
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%% Design of the weight for the low pass filter
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n = 20; % Filter order
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Rp = 1; % Peak to peak passband ripple
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Wp = 2*pi*0.102; % Edge frequency
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% Chebyshev Type I filter design
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[b,a] = cheby1(n, Rp, Wp, 'high', 's');
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wL = 0.04*tf(a, b);
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wL = 1/wL;
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wL = minreal(ss(wL));
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%% Magnitude of the designed Weights and initial specifications
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figure;
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hold on;
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|W_L|^{-1}$');
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plot([0.1, 10], [0.045, 0.045], 'k:', 'DisplayName', 'Spec. on $L_1$');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|W_H|^{-1}$');
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plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec. on $H_1$');
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plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
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plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
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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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hold off;
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xlim([freqs(1), freqs(end)]);
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ylim([5e-3, 10]);
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leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
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leg.ItemTokenSize(1) = 16;
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%% Generalized plant for the H-infinity Synthesis
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P = [0 wL;
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wH -wH;
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1 0];
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%% Standard H-Infinity synthesis
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[Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
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%% High pass filter as the complementary of the low pass filter
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Hh = 1 - Hl;
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%% Minimum realization of the filters
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Hh = minreal(Hh);
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Hl = minreal(Hl);
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%% Bode plot of the obtained filters and comparison with the upper bounds
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figure;
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hold on;
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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
|
%% Intialize Laplace variable
|
||||||
s = zpk('s');
|
s = zpk('s');
|
||||||
|
|
||||||
freqs = logspace(-2, 4, 1000);
|
freqs = logspace(-2, 3, 1000);
|
||||||
|
|
||||||
addpath('./src');
|
addpath('./src');
|
||||||
|
|
||||||
% Weights
|
% Weights
|
||||||
% First we define the weights.
|
% First we define the weights.
|
||||||
|
|
||||||
n = 2; w0 = 2*pi*1; G0 = 1/10; G1 = 1000; Gc = 1/2;
|
%% Design of the Weighting Functions
|
||||||
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;
|
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;
|
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;
|
%% Inverse magnitude of the weighting functions
|
||||||
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;
|
|
||||||
|
|
||||||
figure;
|
figure;
|
||||||
hold on;
|
hold on;
|
||||||
set(gca,'ColorOrderIndex',1)
|
set(gca,'ColorOrderIndex',1)
|
||||||
@@ -37,6 +35,7 @@ leg.ItemTokenSize(1) = 18;
|
|||||||
% H-Infinity Synthesis
|
% H-Infinity Synthesis
|
||||||
% Then we create the generalized plant =P=.
|
% Then we create the generalized plant =P=.
|
||||||
|
|
||||||
|
%% Generalized plant for the synthesis of 3 complementary filters
|
||||||
P = [W1 -W1 -W1;
|
P = [W1 -W1 -W1;
|
||||||
0 W2 0 ;
|
0 W2 0 ;
|
||||||
0 0 W3;
|
0 0 W3;
|
||||||
@@ -46,15 +45,18 @@ P = [W1 -W1 -W1;
|
|||||||
|
|
||||||
% And we do the $\mathcal{H}_\infty$ synthesis.
|
% 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');
|
[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||||
|
|
||||||
% Obtained Complementary Filters
|
% Obtained Complementary Filters
|
||||||
% The obtained filters are:
|
% The obtained filters are:
|
||||||
|
|
||||||
|
%%
|
||||||
H2 = tf(H(1));
|
H2 = tf(H(1));
|
||||||
H3 = tf(H(2));
|
H3 = tf(H(2));
|
||||||
H1 = 1 - H2 - H3;
|
H1 = 1 - H2 - H3;
|
||||||
|
|
||||||
|
%% Bode plot of the obtained complementary filters
|
||||||
figure;
|
figure;
|
||||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||||
|
|
||||||
@@ -79,7 +81,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|||||||
ylabel('Magnitude');
|
ylabel('Magnitude');
|
||||||
set(gca, 'XTickLabel',[]);
|
set(gca, 'XTickLabel',[]);
|
||||||
ylim([1e-4, 20]);
|
ylim([1e-4, 20]);
|
||||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
|
leg = legend('location', 'northeast', 'FontSize', 8);
|
||||||
leg.ItemTokenSize(1) = 18;
|
leg.ItemTokenSize(1) = 18;
|
||||||
|
|
||||||
% Phase
|
% Phase
|
||||||
@@ -94,7 +96,7 @@ plot(freqs, 180/pi*phase(squeeze(freqresp(H3, freqs, 'Hz'))));
|
|||||||
hold off;
|
hold off;
|
||||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||||
set(gca, 'XScale', 'log');
|
set(gca, 'XScale', 'log');
|
||||||
yticks([-360:90:360]); ylim([-270, 270]);
|
yticks([-180:90:180]); ylim([-220, 220]);
|
||||||
|
|
||||||
linkaxes([ax1,ax2],'x');
|
linkaxes([ax1,ax2],'x');
|
||||||
xlim([freqs(1), freqs(end)]);
|
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)]);
|
|
||||||
Reference in New Issue
Block a user