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B3-control/detail_control_3_close_loop_shaping.m
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238
B3-control/detail_control_3_close_loop_shaping.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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