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238
matlab/dehaeze26_decoupling.m
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238
matlab/dehaeze26_decoupling.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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%% Path for functions, data and scripts
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addpath('./src/'); % Path for functions
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%% Colors for the figures
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colors = colororder;
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%% Initialize Frequency Vector
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freqs = logspace(-1, 3, 1000);
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%% Analytical Complementary Filters - Effect of alpha
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freqs_study = logspace(-2, 2, 1000);
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alphas = [0.1, 1, 10];
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w0 = 2*pi*1;
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s = tf('s');
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figure;
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hold on;
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for i = 1:length(alphas)
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alpha = alphas(i);
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Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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plot(freqs_study, abs(squeeze(freqresp(Hh2, freqs_study, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$\\alpha = %g$', alphas(i)));
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plot(freqs_study, abs(squeeze(freqresp(Hl2, freqs_study, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
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end
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Relative Frequency $\frac{\omega}{\omega_0}$'); ylabel('Magnitude');
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hold off;
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ylim([1e-3, 20]);
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leg = legend('location', 'northeast', 'FontSize', 8);
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leg.ItemTokenSize(1) = 18;
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%% Analytical Complementary Filters - Effect of w0
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freqs_study = logspace(-1, 3, 1000);
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alpha = [1];
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w0s = [2*pi*1, 2*pi*10, 2*pi*100];
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s = tf('s');
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figure;
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hold on;
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for i = 1:length(w0s)
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w0 =w0s(i);
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Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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plot(freqs_study, abs(squeeze(freqresp(Hh2, freqs_study, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$\\omega_0 = %g$ Hz', w0/2/pi));
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plot(freqs_study, abs(squeeze(freqresp(Hl2, freqs_study, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
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end
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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_study(1), freqs_study(end)]); ylim([1e-3, 20]);
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leg = legend('location', 'southeast', 'FontSize', 8);
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leg.ItemTokenSize(1) = 18;
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%% Test model
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freqs = logspace(0, 3, 1000); % Frequency Vector [Hz]
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m = 20; % mass [kg]
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k = 1e6; % stiffness [N/m]
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c = 1e2; % damping [N/(m/s)]
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% Plant dynamics
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G = 1/(m*s^2 + c*s + k);
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% Uncertainty weight
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wI = generateWF('n', 2, 'w0', 2*pi*50, 'G0', 0.1, 'Ginf', 10, 'Gc', 1);
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%% Bode plot of the plant with dynamical uncertainty
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', '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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plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-', 'DisplayName', 'G');
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plotMagUncertainty(wI, freqs, 'G', G, 'DisplayName', '$\Pi_i$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
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ylim([1e-8, 7e-5]);
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hold off;
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leg = legend('location', 'northeast', 'FontSize', 8);
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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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plotPhaseUncertainty(wI, freqs, 'G', G);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))), 'k-');
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set(gca,'xscale','log');
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yticks(-360:90:90);
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ylim([-270 45]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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%% Analytical Complementary Filters
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w0 = 2*pi*20;
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alpha = 1;
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Hh = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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Hl = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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%% Specifications
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figure;
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hold on;
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plot([1, 100], [0.01, 100], ':', 'color', colors(2,:));
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plot([300, 1000], [0.01, 0.01], ':', 'color', colors(1,:));
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plot(freqs, 1./abs(squeeze(freqresp(wI, freqs, 'Hz'))), ':', 'color', colors(1,:));
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plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'color', colors(1,:));
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plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'color', colors(2,:));
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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-3, 10]);
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xticks([0.1, 1, 10, 100, 1000]);
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%% Obtained controller
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omega = 2*pi*1000;
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K = 1/(Hh*G) * 1/((1+s/omega+(s/omega)^2));
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K = zpk(minreal(K));
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%% Bode plot of the controller K
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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% Magnitude
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ax1 = nexttile([2, 1]);
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plot(freqs, abs(squeeze(freqresp(K*Hl, freqs, 'Hz'))), 'k-');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
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ylim([8e3, 1e8])
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% Phase
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ax2 = nexttile;
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plot(freqs, 180/pi*angle(squeeze(freqresp(K*Hl, freqs, 'Hz'))), 'k-');
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set(gca,'xscale','log');
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yticks(-180:45:180);
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ylim([-180 45]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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num_delta_points = 50;
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theta = linspace(0, 2*pi, num_delta_points);
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delta_points = exp(1j * theta);
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% Get frequency responses for all components
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G_resp = squeeze(freqresp(G, freqs, 'Hz'));
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K_resp = squeeze(freqresp(K, freqs, 'Hz'));
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Hl_resp = squeeze(freqresp(Hl, freqs, 'Hz'));
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wI_resp = squeeze(freqresp(wI, freqs, 'Hz'));
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% Calculate nominal responses
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nom_L = G_resp .* K_resp .* Hl_resp;
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nom_S = 1 ./ (1 + nom_L);
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nom_T = nom_L ./ (1 + nom_L);
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% Store all the points in the complex plane that L can take
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loop_region_points = zeros(length(freqs), num_delta_points);
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% Initialize arrays to store magnitude bounds
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S_mag_min = ones(length(freqs), 1) * inf;
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S_mag_max = zeros(length(freqs), 1);
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T_mag_min = ones(length(freqs), 1) * inf;
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T_mag_max = zeros(length(freqs), 1);
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% Calculate magnitude bounds for all delta values
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for i = 1:num_delta_points
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% Perturbed loop gain
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loop_perturbed = nom_L .* (1 + wI_resp .* delta_points(i));
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loop_region_points(:,i) = loop_perturbed;
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% Perturbed sensitivity function
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S_perturbed = 1 ./ (1 + loop_perturbed);
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S_mag = abs(S_perturbed);
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% Update S magnitude bounds
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S_mag_min = min(S_mag_min, S_mag);
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S_mag_max = max(S_mag_max, S_mag);
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% Perturbed complementary sensitivity function
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T_perturbed = loop_perturbed ./ (1 + loop_perturbed);
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T_mag = abs(T_perturbed);
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% Update T magnitude bounds
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T_mag_min = min(T_mag_min, T_mag);
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T_mag_max = max(T_mag_max, T_mag);
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end
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% At frequencies where |wI| > 1, T min is zero
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T_mag_min(abs(wI_resp)>1) = 1e-10;
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%% Nyquist plot to check Robust Stability
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figure;
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hold on;
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plot(real(squeeze(freqresp(G*K*Hl, freqs, 'Hz'))), imag(squeeze(freqresp(G*K*Hl, freqs, 'Hz'))), 'k', 'DisplayName', '$L(j\omega)$ - Nominal');
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plot(alphaShape(real(loop_region_points(:)), imag(loop_region_points(:)), 0.1), 'FaceColor', [0, 0, 0], 'EdgeColor', 'none', 'FaceAlpha', 0.3, 'DisplayName', '$L(j\omega)$ - $\forall G \in \Pi_i$');
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plot(-1, 0, 'k+', 'MarkerSize', 5, 'HandleVisibility', 'off');
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hold off;
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grid on;
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axis equal
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xlim([-1.4, 0.2]); ylim([-1.2, 0.4]);
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xticks(-1.4:0.2:0.2); yticks(-1.2:0.2:0.4);
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xlabel('Real Part'); ylabel('Imaginary Part');
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leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
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leg.ItemTokenSize(1) = 18;
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%% Robust Performance
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figure;
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hold on;
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plot(freqs, abs(nom_S), 'color', colors(2,:), 'DisplayName', '$|S|$ - Nom.');
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plot(freqs, abs(nom_T), 'color', colors(1,:), 'DisplayName', '$|T|$ - Nom.');
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patch([freqs, fliplr(freqs)], [S_mag_max', fliplr(S_mag_min')], colors(2,:), 'FaceAlpha', 0.2, 'EdgeColor', 'none', 'HandleVisibility', 'off');
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patch([freqs, fliplr(freqs)], [T_mag_max', fliplr(T_mag_min')], colors(1,:), 'FaceAlpha', 0.2, 'EdgeColor', 'none', 'HandleVisibility', 'off');
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plot([1, 100], [0.01, 100], ':', 'color', colors(2,:), 'DisplayName', 'Specs.');
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plot([300, 1000], [0.01, 0.01], ':', 'color', colors(1,:), 'DisplayName', 'Specs.');
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plot(freqs, 1./abs(squeeze(freqresp(wI, freqs, 'Hz'))), ':', 'color', colors(1,:), 'HandleVisibility', 'off');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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hold off;
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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xlim([freqs(1), freqs(end)]);
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ylim([1e-4, 5]);
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xticks([0.1, 1, 10, 100, 1000]);
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leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3);
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leg.ItemTokenSize(1) = 18;
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1
matlab/figs
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1
matlab/figs
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../paper/figs
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529
matlab/index.org
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529
matlab/index.org
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#+TITLE: Closed-Loop Shaping using Complementary Filters
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:DRAWER:
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments no
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#+PROPERTY: header-args:matlab+ :exports none
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:matlab+ :tangle dehaeze26_control.m
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:END:
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* Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :noweb yes :results silent
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<<m-init-path-tangle>>
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#+end_src
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#+begin_src matlab :noweb yes :results silent
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<<m-init-other>>
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%% Initialize Frequency Vector
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freqs = logspace(-1, 3, 1000);
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#+end_src
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* Complementary filter design
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#+begin_src matlab :exports none :results none
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%% Analytical Complementary Filters - Effect of alpha
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freqs_study = logspace(-2, 2, 1000);
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alphas = [0.1, 1, 10];
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w0 = 2*pi*1;
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s = tf('s');
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figure;
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hold on;
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for i = 1:length(alphas)
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alpha = alphas(i);
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Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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plot(freqs_study, abs(squeeze(freqresp(Hh2, freqs_study, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$\\alpha = %g$', alphas(i)));
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plot(freqs_study, abs(squeeze(freqresp(Hl2, freqs_study, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
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end
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Relative Frequency $\frac{\omega}{\omega_0}$'); ylabel('Magnitude');
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hold off;
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ylim([1e-3, 20]);
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leg = legend('location', 'northeast', '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/detail_control_cf_analytical_effect_alpha.pdf', 'width', 'half', 'height', 'normal');
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#+end_src
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#+begin_src matlab :exports none :results none
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%% Analytical Complementary Filters - Effect of w0
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freqs_study = logspace(-1, 3, 1000);
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alpha = [1];
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w0s = [2*pi*1, 2*pi*10, 2*pi*100];
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s = tf('s');
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figure;
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hold on;
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for i = 1:length(w0s)
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w0 =w0s(i);
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Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
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plot(freqs_study, abs(squeeze(freqresp(Hh2, freqs_study, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$\\omega_0 = %g$ Hz', w0/2/pi));
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plot(freqs_study, abs(squeeze(freqresp(Hl2, freqs_study, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
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end
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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_study(1), freqs_study(end)]); ylim([1e-3, 20]);
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leg = legend('location', 'southeast', '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/detail_control_cf_analytical_effect_w0.pdf', 'width', 'half', 'height', 'normal');
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#+end_src
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* Numerical Example
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*** Plant
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#+begin_src matlab
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%% Test model
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freqs = logspace(0, 3, 1000); % Frequency Vector [Hz]
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m = 20; % mass [kg]
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k = 1e6; % stiffness [N/m]
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c = 1e2; % damping [N/(m/s)]
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% Plant dynamics
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G = 1/(m*s^2 + c*s + k);
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% Uncertainty weight
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wI = generateWF('n', 2, 'w0', 2*pi*50, 'G0', 0.1, 'Ginf', 10, 'Gc', 1);
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#+end_src
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#+begin_src matlab :exports none :results none
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%% Bode plot of the plant with dynamical uncertainty
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', '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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plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-', 'DisplayName', 'G');
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plotMagUncertainty(wI, freqs, 'G', G, 'DisplayName', '$\Pi_i$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
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ylim([1e-8, 7e-5]);
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hold off;
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leg = legend('location', 'northeast', 'FontSize', 8);
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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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plotPhaseUncertainty(wI, freqs, 'G', G);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))), 'k-');
|
||||
set(gca,'xscale','log');
|
||||
yticks(-360:90:90);
|
||||
ylim([-270 45]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/detail_control_cf_bode_plot_mech_sys.pdf', 'width', 'half', 'height', 450);
|
||||
#+end_src
|
||||
|
||||
*** Requirements and choice of complementary filters
|
||||
|
||||
#+begin_src matlab
|
||||
%% Analytical Complementary Filters
|
||||
w0 = 2*pi*20;
|
||||
alpha = 1;
|
||||
|
||||
Hh = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
|
||||
Hl = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results none
|
||||
%% Specifications
|
||||
figure;
|
||||
hold on;
|
||||
plot([1, 100], [0.01, 100], ':', 'color', colors(2,:));
|
||||
plot([300, 1000], [0.01, 0.01], ':', 'color', colors(1,:));
|
||||
plot(freqs, 1./abs(squeeze(freqresp(wI, freqs, 'Hz'))), ':', 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'color', colors(2,:));
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-3, 10]);
|
||||
xticks([0.1, 1, 10, 100, 1000]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
% exportFig('figs/detail_control_cf_specs_S_T.pdf', 'width', 'half', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
*** Controller analysis
|
||||
|
||||
#+begin_src matlab
|
||||
%% Obtained controller
|
||||
omega = 2*pi*1000;
|
||||
|
||||
K = 1/(Hh*G) * 1/((1+s/omega+(s/omega)^2));
|
||||
K = zpk(minreal(K));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results none
|
||||
%% Bode plot of the controller K
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
plot(freqs, abs(squeeze(freqresp(K*Hl, freqs, 'Hz'))), 'k-');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
|
||||
ylim([8e3, 1e8])
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(K*Hl, freqs, 'Hz'))), 'k-');
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:45:180);
|
||||
ylim([-180 45]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/detail_control_cf_bode_Kfb.pdf', 'width', 'half', 'height', 500);
|
||||
#+end_src
|
||||
|
||||
*** Robustness and Performance analysis
|
||||
|
||||
#+begin_src matlab
|
||||
num_delta_points = 50;
|
||||
theta = linspace(0, 2*pi, num_delta_points);
|
||||
delta_points = exp(1j * theta);
|
||||
|
||||
% Get frequency responses for all components
|
||||
G_resp = squeeze(freqresp(G, freqs, 'Hz'));
|
||||
K_resp = squeeze(freqresp(K, freqs, 'Hz'));
|
||||
Hl_resp = squeeze(freqresp(Hl, freqs, 'Hz'));
|
||||
wI_resp = squeeze(freqresp(wI, freqs, 'Hz'));
|
||||
|
||||
% Calculate nominal responses
|
||||
nom_L = G_resp .* K_resp .* Hl_resp;
|
||||
nom_S = 1 ./ (1 + nom_L);
|
||||
nom_T = nom_L ./ (1 + nom_L);
|
||||
|
||||
% Store all the points in the complex plane that L can take
|
||||
loop_region_points = zeros(length(freqs), num_delta_points);
|
||||
|
||||
% Initialize arrays to store magnitude bounds
|
||||
S_mag_min = ones(length(freqs), 1) * inf;
|
||||
S_mag_max = zeros(length(freqs), 1);
|
||||
T_mag_min = ones(length(freqs), 1) * inf;
|
||||
T_mag_max = zeros(length(freqs), 1);
|
||||
|
||||
% Calculate magnitude bounds for all delta values
|
||||
for i = 1:num_delta_points
|
||||
% Perturbed loop gain
|
||||
loop_perturbed = nom_L .* (1 + wI_resp .* delta_points(i));
|
||||
loop_region_points(:,i) = loop_perturbed;
|
||||
|
||||
% Perturbed sensitivity function
|
||||
S_perturbed = 1 ./ (1 + loop_perturbed);
|
||||
S_mag = abs(S_perturbed);
|
||||
|
||||
% Update S magnitude bounds
|
||||
S_mag_min = min(S_mag_min, S_mag);
|
||||
S_mag_max = max(S_mag_max, S_mag);
|
||||
|
||||
% Perturbed complementary sensitivity function
|
||||
T_perturbed = loop_perturbed ./ (1 + loop_perturbed);
|
||||
T_mag = abs(T_perturbed);
|
||||
|
||||
% Update T magnitude bounds
|
||||
T_mag_min = min(T_mag_min, T_mag);
|
||||
T_mag_max = max(T_mag_max, T_mag);
|
||||
end
|
||||
|
||||
% At frequencies where |wI| > 1, T min is zero
|
||||
T_mag_min(abs(wI_resp)>1) = 1e-10;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% Nyquist plot to check Robust Stability
|
||||
figure;
|
||||
hold on;
|
||||
plot(real(squeeze(freqresp(G*K*Hl, freqs, 'Hz'))), imag(squeeze(freqresp(G*K*Hl, freqs, 'Hz'))), 'k', 'DisplayName', '$L(j\omega)$ - Nominal');
|
||||
plot(alphaShape(real(loop_region_points(:)), imag(loop_region_points(:)), 0.1), 'FaceColor', [0, 0, 0], 'EdgeColor', 'none', 'FaceAlpha', 0.3, 'DisplayName', '$L(j\omega)$ - $\forall G \in \Pi_i$');
|
||||
plot(-1, 0, 'k+', 'MarkerSize', 5, 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
grid on;
|
||||
axis equal
|
||||
xlim([-1.4, 0.2]); ylim([-1.2, 0.4]);
|
||||
xticks(-1.4:0.2:0.2); yticks(-1.2:0.2:0.4);
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/detail_control_cf_nyquist_robustness', 'width', 'half', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results none
|
||||
%% Robust Performance
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(nom_S), 'color', colors(2,:), 'DisplayName', '$|S|$ - Nom.');
|
||||
plot(freqs, abs(nom_T), 'color', colors(1,:), 'DisplayName', '$|T|$ - Nom.');
|
||||
|
||||
patch([freqs, fliplr(freqs)], [S_mag_max', fliplr(S_mag_min')], colors(2,:), 'FaceAlpha', 0.2, 'EdgeColor', 'none', 'HandleVisibility', 'off');
|
||||
patch([freqs, fliplr(freqs)], [T_mag_max', fliplr(T_mag_min')], colors(1,:), 'FaceAlpha', 0.2, 'EdgeColor', 'none', 'HandleVisibility', 'off');
|
||||
|
||||
plot([1, 100], [0.01, 100], ':', 'color', colors(2,:), 'DisplayName', 'Specs.');
|
||||
plot([300, 1000], [0.01, 0.01], ':', 'color', colors(1,:), 'DisplayName', 'Specs.');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(wI, freqs, 'Hz'))), ':', 'color', colors(1,:), 'HandleVisibility', 'off');
|
||||
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-4, 5]);
|
||||
xticks([0.1, 1, 10, 100, 1000]);
|
||||
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3);
|
||||
leg.ItemTokenSize(1) = 18;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/detail_control_cf_robust_perf.pdf', 'width', 'half', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
* Matlab Functions :noexport:
|
||||
** =generateWF=: Generate Weighting Functions
|
||||
|
||||
#+begin_src matlab :tangle src/generateWF.m :comments none :mkdirp yes :eval no
|
||||
function [W] = generateWF(args)
|
||||
% generateWF -
|
||||
%
|
||||
% Syntax: [W] = generateWeight(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - n - Weight Order (integer)
|
||||
% - G0 - Low frequency Gain
|
||||
% - G1 - High frequency Gain
|
||||
% - Gc - Gain of the weight at frequency w0
|
||||
% - w0 - Frequency at which |W(j w0)| = Gc [rad/s]
|
||||
%
|
||||
% Outputs:
|
||||
% - W - Generated Weighting Function
|
||||
|
||||
%% Argument validation
|
||||
arguments
|
||||
args.n (1,1) double {mustBeInteger, mustBePositive} = 1
|
||||
args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
||||
args.Ginf (1,1) double {mustBeNumeric, mustBePositive} = 10
|
||||
args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
args.w0 (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
end
|
||||
|
||||
% Verification of correct relation between G0, Gc and Ginf
|
||||
mustBeBetween(args.G0, args.Gc, args.Ginf);
|
||||
|
||||
%% Initialize the Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
%% Create the weighting function according to formula
|
||||
W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
|
||||
(args.G0/args.Gc)^(1/args.n))/...
|
||||
((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 + ...
|
||||
(1/args.Gc)^(1/args.n)))^args.n;
|
||||
|
||||
%% Custom validation function
|
||||
function mustBeBetween(a,b,c)
|
||||
if ~((a > b && b > c) || (c > b && b > a))
|
||||
eid = 'createWeight:inputError';
|
||||
msg = 'Gc should be between G0 and Ginf.';
|
||||
throwAsCaller(MException(eid,msg))
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** =generateCF=: Generate Complementary Filters
|
||||
|
||||
#+begin_src matlab :tangle src/generateCF.m :comments none :mkdirp yes :eval no
|
||||
function [H1, H2] = generateCF(W1, W2, args)
|
||||
% generateCF -
|
||||
%
|
||||
% Syntax: [H1, H2] = generateCF(W1, W2, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - W1 - Weighting Function for H1
|
||||
% - W2 - Weighting Function for H2
|
||||
% - args:
|
||||
% - method - H-Infinity solver ('lmi' or 'ric')
|
||||
% - display - Display synthesis results ('on' or 'off')
|
||||
%
|
||||
% Outputs:
|
||||
% - H1 - Generated H1 Filter
|
||||
% - H2 - Generated H2 Filter
|
||||
|
||||
%% Argument validation
|
||||
arguments
|
||||
W1
|
||||
W2
|
||||
args.method char {mustBeMember(args.method,{'lmi', 'ric'})} = 'ric'
|
||||
args.display char {mustBeMember(args.display,{'on', 'off'})} = 'on'
|
||||
end
|
||||
|
||||
%% The generalized plant is defined
|
||||
P = [W1 -W1;
|
||||
0 W2;
|
||||
1 0];
|
||||
|
||||
%% The standard H-infinity synthesis is performed
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', args.method, 'DISPLAY', args.display);
|
||||
|
||||
%% H1 is defined as the complementary of H2
|
||||
H1 = 1 - H2;
|
||||
#+end_src
|
||||
|
||||
** =plotMagUncertainty=
|
||||
|
||||
#+begin_src matlab :tangle src/plotMagUncertainty.m :comments none :mkdirp yes :eval no
|
||||
function [p] = plotMagUncertainty(W, freqs, args)
|
||||
% plotMagUncertainty -
|
||||
%
|
||||
% Syntax: [p] = plotMagUncertainty(W, freqs, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - W - Multiplicative Uncertainty Weight
|
||||
% - freqs - Frequency Vector [Hz]
|
||||
% - args - Optional Arguments:
|
||||
% - G
|
||||
% - color_i
|
||||
% - opacity
|
||||
%
|
||||
% Outputs:
|
||||
% - p - Plot Handle
|
||||
|
||||
arguments
|
||||
W
|
||||
freqs double {mustBeNumeric, mustBeNonnegative}
|
||||
args.G = tf(1)
|
||||
args.color_i (1,1) double {mustBeInteger, mustBeNonnegative} = 0
|
||||
args.opacity (1,1) double {mustBeNumeric, mustBeNonnegative} = 0.3
|
||||
args.DisplayName char = ''
|
||||
end
|
||||
|
||||
% Get defaults colors
|
||||
colors = get(groot, 'defaultAxesColorOrder');
|
||||
|
||||
p = patch([freqs flip(freqs)], ...
|
||||
[abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*(1 + abs(squeeze(freqresp(W, freqs, 'Hz')))); ...
|
||||
flip(abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*max(1 - abs(squeeze(freqresp(W, freqs, 'Hz'))), 1e-6))], 'w', ...
|
||||
'DisplayName', args.DisplayName);
|
||||
|
||||
if args.color_i == 0
|
||||
p.FaceColor = [0; 0; 0];
|
||||
else
|
||||
p.FaceColor = colors(args.color_i, :);
|
||||
end
|
||||
p.EdgeColor = 'none';
|
||||
p.FaceAlpha = args.opacity;
|
||||
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** =plotPhaseUncertainty=
|
||||
|
||||
#+begin_src matlab :tangle src/plotPhaseUncertainty.m :comments none :mkdirp yes :eval no
|
||||
function [p] = plotPhaseUncertainty(W, freqs, args)
|
||||
% plotPhaseUncertainty -
|
||||
%
|
||||
% Syntax: [p] = plotPhaseUncertainty(W, freqs, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - W - Multiplicative Uncertainty Weight
|
||||
% - freqs - Frequency Vector [Hz]
|
||||
% - args - Optional Arguments:
|
||||
% - G
|
||||
% - color_i
|
||||
% - opacity
|
||||
%
|
||||
% Outputs:
|
||||
% - p - Plot Handle
|
||||
|
||||
arguments
|
||||
W
|
||||
freqs double {mustBeNumeric, mustBeNonnegative}
|
||||
args.G = tf(1)
|
||||
args.unwrap logical {mustBeNumericOrLogical} = false
|
||||
args.color_i (1,1) double {mustBeInteger, mustBeNonnegative} = 0
|
||||
args.opacity (1,1) double {mustBeNumeric, mustBePositive} = 0.3
|
||||
args.DisplayName char = ''
|
||||
end
|
||||
|
||||
% Get defaults colors
|
||||
colors = get(groot, 'defaultAxesColorOrder');
|
||||
|
||||
% Compute Phase Uncertainty
|
||||
Dphi = 180/pi*asin(abs(squeeze(freqresp(W, freqs, 'Hz'))));
|
||||
Dphi(abs(squeeze(freqresp(W, freqs, 'Hz'))) > 1) = 360;
|
||||
|
||||
% Compute Plant Phase
|
||||
if args.unwrap
|
||||
G_ang = 180/pi*unwrap(angle(squeeze(freqresp(args.G, freqs, 'Hz'))));
|
||||
else
|
||||
G_ang = 180/pi*angle(squeeze(freqresp(args.G, freqs, 'Hz')));
|
||||
end
|
||||
|
||||
p = patch([freqs flip(freqs)], [G_ang+Dphi; flip(G_ang-Dphi)], 'w', ...
|
||||
'DisplayName', args.DisplayName);
|
||||
|
||||
if args.color_i == 0
|
||||
p.FaceColor = [0; 0; 0];
|
||||
else
|
||||
p.FaceColor = colors(args.color_i, :);
|
||||
end
|
||||
p.EdgeColor = 'none';
|
||||
p.FaceAlpha = args.opacity;
|
||||
|
||||
end
|
||||
#+end_src
|
||||
|
||||
* Helping Functions :noexport:
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :tangle no
|
||||
:END:
|
||||
** Initialize Path
|
||||
#+NAME: m-init-path-tangle
|
||||
#+BEGIN_SRC matlab
|
||||
%% Path for functions, data and scripts
|
||||
addpath('./src/'); % Path for functions
|
||||
#+END_SRC
|
||||
|
||||
** Initialize other elements
|
||||
#+NAME: m-init-other
|
||||
#+BEGIN_SRC matlab
|
||||
%% Colors for the figures
|
||||
colors = colororder;
|
||||
#+END_SRC
|
||||
34
matlab/src/generateCF.m
Normal file
34
matlab/src/generateCF.m
Normal file
@@ -0,0 +1,34 @@
|
||||
function [H1, H2] = generateCF(W1, W2, args)
|
||||
% generateCF -
|
||||
%
|
||||
% Syntax: [H1, H2] = generateCF(W1, W2, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - W1 - Weighting Function for H1
|
||||
% - W2 - Weighting Function for H2
|
||||
% - args:
|
||||
% - method - H-Infinity solver ('lmi' or 'ric')
|
||||
% - display - Display synthesis results ('on' or 'off')
|
||||
%
|
||||
% Outputs:
|
||||
% - H1 - Generated H1 Filter
|
||||
% - H2 - Generated H2 Filter
|
||||
|
||||
%% Argument validation
|
||||
arguments
|
||||
W1
|
||||
W2
|
||||
args.method char {mustBeMember(args.method,{'lmi', 'ric'})} = 'ric'
|
||||
args.display char {mustBeMember(args.display,{'on', 'off'})} = 'on'
|
||||
end
|
||||
|
||||
%% The generalized plant is defined
|
||||
P = [W1 -W1;
|
||||
0 W2;
|
||||
1 0];
|
||||
|
||||
%% The standard H-infinity synthesis is performed
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', args.method, 'DISPLAY', args.display);
|
||||
|
||||
%% H1 is defined as the complementary of H2
|
||||
H1 = 1 - H2;
|
||||
43
matlab/src/generateWF.m
Normal file
43
matlab/src/generateWF.m
Normal file
@@ -0,0 +1,43 @@
|
||||
function [W] = generateWF(args)
|
||||
% generateWF -
|
||||
%
|
||||
% Syntax: [W] = generateWeight(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - n - Weight Order (integer)
|
||||
% - G0 - Low frequency Gain
|
||||
% - G1 - High frequency Gain
|
||||
% - Gc - Gain of the weight at frequency w0
|
||||
% - w0 - Frequency at which |W(j w0)| = Gc [rad/s]
|
||||
%
|
||||
% Outputs:
|
||||
% - W - Generated Weighting Function
|
||||
|
||||
%% Argument validation
|
||||
arguments
|
||||
args.n (1,1) double {mustBeInteger, mustBePositive} = 1
|
||||
args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
||||
args.Ginf (1,1) double {mustBeNumeric, mustBePositive} = 10
|
||||
args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
args.w0 (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
end
|
||||
|
||||
% Verification of correct relation between G0, Gc and Ginf
|
||||
mustBeBetween(args.G0, args.Gc, args.Ginf);
|
||||
|
||||
%% Initialize the Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
%% Create the weighting function according to formula
|
||||
W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
|
||||
(args.G0/args.Gc)^(1/args.n))/...
|
||||
((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 + ...
|
||||
(1/args.Gc)^(1/args.n)))^args.n;
|
||||
|
||||
%% Custom validation function
|
||||
function mustBeBetween(a,b,c)
|
||||
if ~((a > b && b > c) || (c > b && b > a))
|
||||
eid = 'createWeight:inputError';
|
||||
msg = 'Gc should be between G0 and Ginf.';
|
||||
throwAsCaller(MException(eid,msg))
|
||||
end
|
||||
42
matlab/src/plotMagUncertainty.m
Normal file
42
matlab/src/plotMagUncertainty.m
Normal file
@@ -0,0 +1,42 @@
|
||||
function [p] = plotMagUncertainty(W, freqs, args)
|
||||
% plotMagUncertainty -
|
||||
%
|
||||
% Syntax: [p] = plotMagUncertainty(W, freqs, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - W - Multiplicative Uncertainty Weight
|
||||
% - freqs - Frequency Vector [Hz]
|
||||
% - args - Optional Arguments:
|
||||
% - G
|
||||
% - color_i
|
||||
% - opacity
|
||||
%
|
||||
% Outputs:
|
||||
% - p - Plot Handle
|
||||
|
||||
arguments
|
||||
W
|
||||
freqs double {mustBeNumeric, mustBeNonnegative}
|
||||
args.G = tf(1)
|
||||
args.color_i (1,1) double {mustBeInteger, mustBeNonnegative} = 0
|
||||
args.opacity (1,1) double {mustBeNumeric, mustBeNonnegative} = 0.3
|
||||
args.DisplayName char = ''
|
||||
end
|
||||
|
||||
% Get defaults colors
|
||||
colors = get(groot, 'defaultAxesColorOrder');
|
||||
|
||||
p = patch([freqs flip(freqs)], ...
|
||||
[abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*(1 + abs(squeeze(freqresp(W, freqs, 'Hz')))); ...
|
||||
flip(abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*max(1 - abs(squeeze(freqresp(W, freqs, 'Hz'))), 1e-6))], 'w', ...
|
||||
'DisplayName', args.DisplayName);
|
||||
|
||||
if args.color_i == 0
|
||||
p.FaceColor = [0; 0; 0];
|
||||
else
|
||||
p.FaceColor = colors(args.color_i, :);
|
||||
end
|
||||
p.EdgeColor = 'none';
|
||||
p.FaceAlpha = args.opacity;
|
||||
|
||||
end
|
||||
52
matlab/src/plotPhaseUncertainty.m
Normal file
52
matlab/src/plotPhaseUncertainty.m
Normal file
@@ -0,0 +1,52 @@
|
||||
function [p] = plotPhaseUncertainty(W, freqs, args)
|
||||
% plotPhaseUncertainty -
|
||||
%
|
||||
% Syntax: [p] = plotPhaseUncertainty(W, freqs, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - W - Multiplicative Uncertainty Weight
|
||||
% - freqs - Frequency Vector [Hz]
|
||||
% - args - Optional Arguments:
|
||||
% - G
|
||||
% - color_i
|
||||
% - opacity
|
||||
%
|
||||
% Outputs:
|
||||
% - p - Plot Handle
|
||||
|
||||
arguments
|
||||
W
|
||||
freqs double {mustBeNumeric, mustBeNonnegative}
|
||||
args.G = tf(1)
|
||||
args.unwrap logical {mustBeNumericOrLogical} = false
|
||||
args.color_i (1,1) double {mustBeInteger, mustBeNonnegative} = 0
|
||||
args.opacity (1,1) double {mustBeNumeric, mustBePositive} = 0.3
|
||||
args.DisplayName char = ''
|
||||
end
|
||||
|
||||
% Get defaults colors
|
||||
colors = get(groot, 'defaultAxesColorOrder');
|
||||
|
||||
% Compute Phase Uncertainty
|
||||
Dphi = 180/pi*asin(abs(squeeze(freqresp(W, freqs, 'Hz'))));
|
||||
Dphi(abs(squeeze(freqresp(W, freqs, 'Hz'))) > 1) = 360;
|
||||
|
||||
% Compute Plant Phase
|
||||
if args.unwrap
|
||||
G_ang = 180/pi*unwrap(angle(squeeze(freqresp(args.G, freqs, 'Hz'))));
|
||||
else
|
||||
G_ang = 180/pi*angle(squeeze(freqresp(args.G, freqs, 'Hz')));
|
||||
end
|
||||
|
||||
p = patch([freqs flip(freqs)], [G_ang+Dphi; flip(G_ang-Dphi)], 'w', ...
|
||||
'DisplayName', args.DisplayName);
|
||||
|
||||
if args.color_i == 0
|
||||
p.FaceColor = [0; 0; 0];
|
||||
else
|
||||
p.FaceColor = colors(args.color_i, :);
|
||||
end
|
||||
p.EdgeColor = 'none';
|
||||
p.FaceAlpha = args.opacity;
|
||||
|
||||
end
|
||||
Reference in New Issue
Block a user