909 lines
48 KiB
Matlab
909 lines
48 KiB
Matlab
%% 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('./mat/'); % Path for data
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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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%% Frequency Vector [Hz]
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freqs = logspace(0, 3, 1000);
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%% Load the PSD of disturbances
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load('uniaxial_disturbance_psd.mat', 'f', 'psd_ft', 'psd_xf');
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%% Load Plants Dynamics
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load('uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ...
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'G_vc_mid', 'G_md_mid', 'G_pz_mid', ...
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'G_vc_heavy', 'G_md_heavy', 'G_pz_heavy');
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% Plant Dynamics for Active Damping
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% The plant dynamics for all three active damping techniques are shown in Figure ref:fig:uniaxial_plant_active_damping_techniques.
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% All have *alternating poles and zeros* meaning that the phase is bounded to $\pm 90\,\text{deg}$ which makes the controller very robust.
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% When the nano-hexapod's suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the transfer functions from IFF and DVF.
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% For the stiff nano-hexapod, the micro-station dynamics can be seen on the transfer functions in Figure ref:fig:uniaxial_plant_active_damping_techniques.
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% Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used.
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%% Damped plants for three considered payload masses - Comparison of active damping techniques
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figure;
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tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_vc_light('fm', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$m_s = 1\,kg$');
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plot(freqs, abs(squeeze(freqresp(G_vc_mid( 'fm', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'DisplayName', '$m_s = 25\,kg$');
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plot(freqs, abs(squeeze(freqresp(G_vc_heavy('fm', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$m_s = 50\,kg$');
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plot(freqs, abs(squeeze(freqresp(G_md_light('fm', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_md_mid( 'fm', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_md_heavy('fm', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_pz_light('fm', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_pz_mid( 'fm', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_pz_heavy('fm', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
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title('IFF: $f_m/f$');
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ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
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ldg.ItemTokenSize = [20, 1];
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ax2 = nexttile([2,1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_vc_light('dL', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:));
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plot(freqs, abs(squeeze(freqresp(G_vc_mid( 'dL', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:));
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plot(freqs, abs(squeeze(freqresp(G_vc_heavy('dL', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:));
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plot(freqs, abs(squeeze(freqresp(G_md_light('dL', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:));
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plot(freqs, abs(squeeze(freqresp(G_md_mid( 'dL', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:));
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plot(freqs, abs(squeeze(freqresp(G_md_heavy('dL', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:));
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plot(freqs, abs(squeeze(freqresp(G_pz_light('dL', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:));
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plot(freqs, abs(squeeze(freqresp(G_pz_mid( 'dL', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:));
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plot(freqs, abs(squeeze(freqresp(G_pz_heavy('dL', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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title('RDC: $d\mathcal{L}/f$');
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ax3 = nexttile([2,1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_vc_light('vn', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$k_n = 0.01\,N/\mu m$');
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plot(freqs, abs(squeeze(freqresp(G_vc_mid( 'vn', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_vc_heavy('vn', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_md_light('vn', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$k_n = 1\,N/\mu m$');
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plot(freqs, abs(squeeze(freqresp(G_md_mid( 'vn', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_md_heavy('vn', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_pz_light('vn', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$k_n = 100\,N/\mu m$');
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plot(freqs, abs(squeeze(freqresp(G_pz_mid( 'vn', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_pz_heavy('vn', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/s/N]'); set(gca, 'XTickLabel',[]);
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title('DVF: $v_n/f$');
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ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
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ldg.ItemTokenSize = [20, 1];
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ax1b = nexttile();
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hold on;
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_light('fm', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid( 'fm', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_heavy('fm', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('fm', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid( 'fm', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_heavy('fm', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('fm', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid( 'fm', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_heavy('fm', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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xticks([1e0, 1e1, 1e2]);
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yticks(-360:90:360);
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ylim([-200, 20]);
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ax2b = nexttile();
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hold on;
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_light('dL', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid( 'dL', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_heavy('dL', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('dL', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid( 'dL', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_heavy('dL', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('dL', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid( 'dL', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_heavy('dL', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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xticks([1e0, 1e1, 1e2]);
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yticks(-360:90:360);
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ylim([-200, 20]);
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ax3b = nexttile();
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hold on;
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_light('vn', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid( 'vn', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_heavy('vn', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('vn', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid( 'vn', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_heavy('vn', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('vn', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid( 'vn', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_heavy('vn', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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xticks([1e0, 1e1, 1e2]);
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yticks(-360:90:360);
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ylim([-110, 110]);
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linkaxes([ax1,ax2,ax3,ax1b,ax2b,ax3b],'x');
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xlim([1, 1000]);
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% Achievable Damping - Root Locus
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% The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques.
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% They are shown in Figure ref:fig:uniaxial_root_locus_damping_techniques.
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% All three active damping approach can lead to *critical damping* of the nano-hexapod suspension mode.
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% There is even a little bit of authority on micro-station modes with IFF and DVF applied on the stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques, right) and with RDC for a soft nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode).
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% This can be explained by the fact that above the suspension mode of the soft nano-hexapod, the relative motion sensor acts as an inertial sensor for the micro-station top platform. Therefore, it is like DVF was applied (the nano-hexapod acts as a geophone!).
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%% Active Damping Robustness to change of sample's mass - Root Locus for all three damping techniques with 3 different sample's masses
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figure;
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tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
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%% Soft Nano-Hexapod
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ax1 = nexttile();
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hold on;
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% IFF
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plot(real(pole(G_vc_light('fm', 'f'))), imag(pole(G_vc_light('fm', 'f'))), 'x', 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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plot(real(zero(G_vc_light('fm', 'f'))), imag(zero(G_vc_light('fm', 'f'))), 'o', 'color', colors(1,:), ...
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'DisplayName', 'IFF');
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for g = logspace(0, 2, 400)
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clpoles = pole(feedback(G_vc_light('fm', 'f'), g/s, +1));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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end
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% RDC
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plot(real(pole(G_vc_light('dL', 'f'))), imag(pole(G_vc_light('dL', 'f'))), 'x', 'color', colors(2,:), ...
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'HandleVisibility', 'off');
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plot(real(zero(G_vc_light('dL', 'f'))), imag(zero(G_vc_light('dL', 'f'))), 'o', 'color', colors(2,:), ...
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'DisplayName', 'RDC');
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for g = logspace(1,3,400)
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clpoles = pole(feedback(G_vc_light('dL', 'f'), -g*s, +1));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
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'HandleVisibility', 'off');
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end
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% DVF
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plot(real(pole(G_vc_light('vn', 'f'))), imag(pole(G_vc_light('vn', 'f'))), 'x', 'color', colors(3,:), ...
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'HandleVisibility', 'off');
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plot(real(zero(G_vc_light('vn', 'f'))), imag(zero(G_vc_light('vn', 'f'))), 'o', 'color', colors(3,:), ...
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'DisplayName', 'DVF');
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for g = logspace(1,3,400)
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clpoles = pole(feedback(G_vc_light('vn', 'f'), -g, +1));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
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'HandleVisibility', 'off');
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end
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hold off;
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axis square;
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xlabel('Real Part'); ylabel('Imaginary Part');
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title('$k_n = 0.01\,N/\mu m$')
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ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
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ldg.ItemTokenSize = [10, 1];
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xlim([-30, 0]); ylim([0, 30]);
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%% Medium-Stiff Nano-Hexapod
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ax2 = nexttile();
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hold on;
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% IFF
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plot(real(pole(G_md_light('fm', 'f'))), imag(pole(G_md_light('fm', 'f'))), 'x', 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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plot(real(zero(G_md_light('fm', 'f'))), imag(zero(G_md_light('fm', 'f'))), 'o', 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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for g = logspace(0,3,400)
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clpoles = pole(feedback(G_md_light('fm', 'f'), g/s, +1));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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end
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% RDC
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plot(real(pole(G_md_light('dL', 'f'))), imag(pole(G_md_light('dL', 'f'))), 'x', 'color', colors(2,:), ...
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'HandleVisibility', 'off');
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plot(real(zero(G_md_light('dL', 'f'))), imag(zero(G_md_light('dL', 'f'))), 'o', 'color', colors(2,:), ...
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'HandleVisibility', 'off');
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for g = logspace(2,4,400)
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clpoles = pole(feedback(G_md_light('dL', 'f'), -g*s, +1));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
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'HandleVisibility', 'off');
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end
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% DVF
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plot(real(pole(G_md_light('vn', 'f'))), imag(pole(G_md_light('vn', 'f'))), 'x', 'color', colors(3,:), ...
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'HandleVisibility', 'off');
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plot(real(zero(G_md_light('vn', 'f'))), imag(zero(G_md_light('vn', 'f'))), 'o', 'color', colors(3,:), ...
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'HandleVisibility', 'off');
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for g = logspace(2,4,400)
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clpoles = pole(feedback(G_md_light('vn', 'f'), -g, +1));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
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'HandleVisibility', 'off');
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end
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hold off;
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axis square;
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xlabel('Real Part'); ylabel('Imaginary Part');
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title('$k_n = 1\,N/\mu m$')
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xlim([-300, 0]); ylim([0, 300]);
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%% Stiff Nano-Hexapod
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ax3 = nexttile();
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hold on;
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% IFF
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plot(real(pole(G_pz_light('fm', 'f'))), imag(pole(G_pz_light('fm', 'f'))), 'x', 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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plot(real(zero(G_pz_light('fm', 'f'))), imag(zero(G_pz_light('fm', 'f'))), 'o', 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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for g = logspace(2,5,400)
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clpoles = pole(feedback(G_pz_light('fm', 'f'), g/s, +1));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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end
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% RDC
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plot(real(pole(G_pz_light('dL', 'f'))), imag(pole(G_pz_light('dL', 'f'))), 'x', 'color', colors(2,:), ...
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'HandleVisibility', 'off');
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plot(real(zero(G_pz_light('dL', 'f'))), imag(zero(G_pz_light('dL', 'f'))), 'o', 'color', colors(2,:), ...
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'HandleVisibility', 'off');
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for g = logspace(3,6,400)
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clpoles = pole(feedback(G_pz_light('dL', 'f'), -g*s, +1));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
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'HandleVisibility', 'off');
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end
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% DVF
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plot(real(pole(G_pz_light('vn', 'f'))), imag(pole(G_pz_light('vn', 'f'))), 'x', 'color', colors(3,:), ...
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'HandleVisibility', 'off');
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plot(real(zero(G_pz_light('vn', 'f'))), imag(zero(G_pz_light('vn', 'f'))), 'o', 'color', colors(3,:), ...
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'HandleVisibility', 'off');
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for g = logspace(3,6,400)
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clpoles = pole(feedback(G_pz_light('vn', 'f'), -g, +1));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
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'HandleVisibility', 'off');
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end
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hold off;
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axis square;
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xlabel('Real Part'); ylabel('Imaginary Part');
|
|
title('$k_n = 100\,N/\mu m$')
|
|
xlim([-4000, 0]); ylim([0, 4000]);
|
|
|
|
|
|
|
|
% #+name: fig:uniaxial_root_locus_damping_techniques
|
|
% #+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses.
|
|
% #+RESULTS:
|
|
% [[file:figs/uniaxial_root_locus_damping_techniques.png]]
|
|
|
|
|
|
%% Root Locus for the three damping techniques
|
|
figure;
|
|
|
|
hold on;
|
|
% IFF
|
|
plot(real(pole(G_md_mid('fm', 'f'))), imag(pole(G_md_mid('fm', 'f'))), 'x', 'color', colors(1,:), ...
|
|
'DisplayName', 'IFF');
|
|
plot(real(zero(G_md_mid('fm', 'f'))), imag(zero(G_md_mid('fm', 'f'))), 'o', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(1,4,500)
|
|
clpoles = pole(feedback(G_md_mid('fm', 'f'), g/s, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
|
|
% RDC
|
|
plot(real(pole(G_md_mid('dL', 'f'))), imag(pole(G_md_mid('dL', 'f'))), 'x', 'color', colors(2,:), ...
|
|
'DisplayName', 'RDC');
|
|
plot(real(zero(G_md_mid('dL', 'f'))), imag(zero(G_md_mid('dL', 'f'))), 'o', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(2,5,500)
|
|
clpoles = pole(feedback(G_md_mid('dL', 'f'), -g*s, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
|
|
% DVF
|
|
plot(real(pole(G_md_mid('vn', 'f'))), imag(pole(G_md_mid('vn', 'f'))), 'x', 'color', colors(3,:), ...
|
|
'DisplayName', 'DVF');
|
|
plot(real(zero(G_md_mid('vn', 'f'))), imag(zero(G_md_mid('vn', 'f'))), 'o', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(2,5,500)
|
|
clpoles = pole(feedback(G_md_mid('vn', 'f'), -tf(g), +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
hold off;
|
|
xlim([-2100, 0]); ylim([0, 2100]);
|
|
axis square;
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
|
ldg.ItemTokenSize = [10, 1];
|
|
|
|
% Active Damping Controller Optimization and Damped plants :noexport:
|
|
|
|
%% Design of Active Damping controllers to have reasonable damping
|
|
% IFF
|
|
K_iff_vc = 20/(s + 2*pi*0.01);
|
|
K_iff_vc.InputName = {'fm'};
|
|
K_iff_vc.OutputName = {'f'};
|
|
K_iff_md = 200/(s + 2*pi*0.01);
|
|
K_iff_md.InputName = {'fm'};
|
|
K_iff_md.OutputName = {'f'};
|
|
K_iff_pz = 4000/(s + 2*pi*0.01);
|
|
K_iff_pz.InputName = {'fm'};
|
|
K_iff_pz.OutputName = {'f'};
|
|
|
|
% RDC
|
|
K_rdc_vc = -1e3*s;
|
|
K_rdc_vc.InputName = {'dL'};
|
|
K_rdc_vc.OutputName = {'f'};
|
|
K_rdc_md = -1e4*s;
|
|
K_rdc_md.InputName = {'dL'};
|
|
K_rdc_md.OutputName = {'f'};
|
|
K_rdc_pz = -1e5*s;
|
|
K_rdc_pz.InputName = {'dL'};
|
|
K_rdc_pz.OutputName = {'f'};
|
|
|
|
% DVF
|
|
K_dvf_vc = -tf(1e3);
|
|
K_dvf_vc.InputName = {'vn'};
|
|
K_dvf_vc.OutputName = {'f'};
|
|
K_dvf_md = -tf(8e3);
|
|
K_dvf_md.InputName = {'vn'};
|
|
K_dvf_md.OutputName = {'f'};
|
|
K_dvf_pz = -tf(2e5);
|
|
K_dvf_pz.InputName = {'vn'};
|
|
K_dvf_pz.OutputName = {'f'};
|
|
|
|
%% Save Active Damping Controller
|
|
save('./mat/uniaxial_active_damping_controllers.mat', 'K_iff_vc', 'K_iff_md', 'K_iff_pz', ...
|
|
'K_rdc_vc', 'K_rdc_md', 'K_rdc_pz', ...
|
|
'K_dvf_vc', 'K_dvf_md', 'K_dvf_pz');
|
|
|
|
%% Compute Damped Plants
|
|
% IFF
|
|
G_iff_vc_light = feedback(G_vc_light, K_iff_vc, 'name', +1);
|
|
G_iff_vc_mid = feedback(G_vc_mid , K_iff_vc, 'name', +1);
|
|
G_iff_vc_heavy = feedback(G_vc_heavy, K_iff_vc, 'name', +1);
|
|
G_iff_md_light = feedback(G_md_light, K_iff_md, 'name', +1);
|
|
G_iff_md_mid = feedback(G_md_mid , K_iff_md, 'name', +1);
|
|
G_iff_md_heavy = feedback(G_md_heavy, K_iff_md, 'name', +1);
|
|
G_iff_pz_light = feedback(G_pz_light, K_iff_pz, 'name', +1);
|
|
G_iff_pz_mid = feedback(G_pz_mid , K_iff_pz, 'name', +1);
|
|
G_iff_pz_heavy = feedback(G_pz_heavy, K_iff_pz, 'name', +1);
|
|
|
|
% RDC
|
|
G_rdc_vc_light = feedback(G_vc_light, K_rdc_vc, 'name', +1);
|
|
G_rdc_vc_mid = feedback(G_vc_mid , K_rdc_vc, 'name', +1);
|
|
G_rdc_vc_heavy = feedback(G_vc_heavy, K_rdc_vc, 'name', +1);
|
|
G_rdc_md_light = feedback(G_md_light, K_rdc_md, 'name', +1);
|
|
G_rdc_md_mid = feedback(G_md_mid , K_rdc_md, 'name', +1);
|
|
G_rdc_md_heavy = feedback(G_md_heavy, K_rdc_md, 'name', +1);
|
|
G_rdc_pz_light = feedback(G_pz_light, K_rdc_pz, 'name', +1);
|
|
G_rdc_pz_mid = feedback(G_pz_mid , K_rdc_pz, 'name', +1);
|
|
G_rdc_pz_heavy = feedback(G_pz_heavy, K_rdc_pz, 'name', +1);
|
|
|
|
% DVF
|
|
G_dvf_vc_light = feedback(G_vc_light, K_dvf_vc, 'name', +1);
|
|
G_dvf_vc_mid = feedback(G_vc_mid , K_dvf_vc, 'name', +1);
|
|
G_dvf_vc_heavy = feedback(G_vc_heavy, K_dvf_vc, 'name', +1);
|
|
G_dvf_md_light = feedback(G_md_light, K_dvf_md, 'name', +1);
|
|
G_dvf_md_mid = feedback(G_md_mid , K_dvf_md, 'name', +1);
|
|
G_dvf_md_heavy = feedback(G_md_heavy, K_dvf_md, 'name', +1);
|
|
G_dvf_pz_light = feedback(G_pz_light, K_dvf_pz, 'name', +1);
|
|
G_dvf_pz_mid = feedback(G_pz_mid , K_dvf_pz, 'name', +1);
|
|
G_dvf_pz_heavy = feedback(G_pz_heavy, K_dvf_pz, 'name', +1);
|
|
|
|
%% Verify Stability
|
|
% IFF
|
|
isstable(G_iff_vc_light) && isstable(G_iff_vc_mid) && isstable(G_iff_vc_heavy) && ...
|
|
isstable(G_iff_md_light) && isstable(G_iff_md_mid) && isstable(G_iff_md_heavy) && ...
|
|
isstable(G_iff_pz_light) && isstable(G_iff_pz_mid) && isstable(G_iff_pz_heavy)
|
|
|
|
% RDC
|
|
isstable(G_rdc_vc_light) && isstable(G_rdc_vc_mid) && isstable(G_rdc_vc_heavy) && ...
|
|
isstable(G_rdc_md_light) && isstable(G_rdc_md_mid) && isstable(G_rdc_md_heavy) && ...
|
|
isstable(G_rdc_pz_light) && isstable(G_rdc_pz_mid) && isstable(G_rdc_pz_heavy)
|
|
|
|
% DVF
|
|
isstable(G_dvf_vc_light) && isstable(G_dvf_vc_mid) && isstable(G_dvf_vc_heavy) && ...
|
|
isstable(G_dvf_md_light) && isstable(G_dvf_md_mid) && isstable(G_dvf_md_heavy) && ...
|
|
isstable(G_dvf_pz_light) && isstable(G_dvf_pz_mid) && isstable(G_dvf_pz_heavy)
|
|
|
|
%% Save Active Damping Controller
|
|
save('./mat/uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_iff_pz_light', ...
|
|
'G_rdc_vc_light', 'G_rdc_md_light', 'G_rdc_pz_light', ...
|
|
'G_dvf_vc_light', 'G_dvf_md_light', 'G_dvf_pz_light', ...
|
|
'G_iff_vc_mid', 'G_iff_md_mid', 'G_iff_pz_mid', ...
|
|
'G_rdc_vc_mid', 'G_rdc_md_mid', 'G_rdc_pz_mid', ...
|
|
'G_dvf_vc_mid', 'G_dvf_md_mid', 'G_dvf_pz_mid', ...
|
|
'G_iff_vc_heavy', 'G_iff_md_heavy', 'G_iff_pz_heavy', ...
|
|
'G_rdc_vc_heavy', 'G_rdc_md_heavy', 'G_rdc_pz_heavy', ...
|
|
'G_dvf_vc_heavy', 'G_dvf_md_heavy', 'G_dvf_pz_heavy');
|
|
|
|
% Change of sensitivity to disturbances
|
|
% The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure ref:fig:uniaxial_sensitivity_dist_active_damping.
|
|
% The comparison is done with the nano-hexapod having a stiffness $k_n = 1\,N/\mu m$.
|
|
|
|
% #+begin_important
|
|
% Conclusions from Figure ref:fig:uniaxial_sensitivity_dist_active_damping are:
|
|
% - IFF degrades the sensitivity to direct forces on the sample (i.e. the compliance) below the resonance of the nano-hexapod
|
|
% - RDC degrades the sensitivity to stage vibrations around the nano-hexapod's resonance as compared to the other two methods
|
|
% - both IFF and DVF degrades the sensitivity to floor motion below the resonance of the nano-hexapod
|
|
% #+end_important
|
|
|
|
|
|
%% Change of sensitivity to disturbance with all three active damping strategies
|
|
figure;
|
|
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile();
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'fs'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'fs'), freqs, 'Hz'))), 'color', colors(1,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid('d', 'fs'), freqs, 'Hz'))), 'color', colors(2,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid('d', 'fs'), freqs, 'Hz'))), 'color', colors(3,:));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d/f_{s}$ [m/N]'); xlabel('Frequency [Hz]');
|
|
xticks([1e0, 1e1, 1e2]);
|
|
|
|
ax2 = nexttile();
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'ft'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'ft'), freqs, 'Hz'))), 'color', colors(1,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid('d', 'ft'), freqs, 'Hz'))), 'color', colors(2,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid('d', 'ft'), freqs, 'Hz'))), 'color', colors(3,:));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d/f_{t}$ [m/N]'); xlabel('Frequency [Hz]');
|
|
xticks([1e0, 1e1, 1e2]);
|
|
|
|
ax3 = nexttile();
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'xf'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'xf'), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid('d', 'xf'), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', 'RDC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid('d', 'xf'), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d/x_{f}$ [m/m]'); xlabel('Frequency [Hz]');
|
|
xticks([1e0, 1e1, 1e2]);
|
|
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
|
|
|
linkaxes([ax1,ax2,ax3],'x');
|
|
xlim([1, 500]);
|
|
|
|
% Noise Budgeting after Active Damping
|
|
% Cumulative Amplitude Spectrum of the distance $d$ with all three active damping techniques are compared in Figure ref:fig:uniaxial_cas_active_damping.
|
|
% All three active damping methods are giving similar results (except the RDC which is a little bit worse for the stiff nano-hexapod).
|
|
|
|
% Compare to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
|
|
|
|
|
|
%% Cumulative Amplitude Spectrum of the distance d with all three active damping techniques
|
|
figure;
|
|
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile();
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', 'black', 'DisplayName', 'OL');
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_iff_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', colors(1,:), 'DisplayName', 'IFF');
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_rdc_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_rdc_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', colors(2,:), 'DisplayName', 'RDC');
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_dvf_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_dvf_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', colors(3,:), 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('CAS of $d$ [m]'); xlabel('Frequency [Hz]');
|
|
xticks([1e0, 1e1, 1e2]);
|
|
title('$k_n = 0.01\,N/\mu m$')
|
|
|
|
ax2 = nexttile();
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', 'black', 'DisplayName', 'OL');
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_iff_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', colors(1,:), 'DisplayName', 'IFF');
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_rdc_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_rdc_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', colors(2,:), 'DisplayName', 'RDC');
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_dvf_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_dvf_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', colors(3,:), 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
|
xticks([1e0, 1e1, 1e2]);
|
|
title('$k_n = 1\,N/\mu m$')
|
|
|
|
ax3 = nexttile();
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', 'black', 'DisplayName', 'OL');
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_iff_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', colors(1,:), 'DisplayName', 'IFF');
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_rdc_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_rdc_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', colors(2,:), 'DisplayName', 'RDC');
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_dvf_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
|
psd_xf.*abs(squeeze(freqresp(G_dvf_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
|
'color', colors(3,:), 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
|
xticks([1e0, 1e1, 1e2]);
|
|
title('$k_n = 100\,N/\mu m$')
|
|
legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
|
|
|
|
linkaxes([ax1,ax2,ax3], 'xy')
|
|
xlim([1, 500]);
|
|
ylim([2e-10, 3e-6])
|
|
|
|
% Obtained Damped Plant
|
|
% The transfer functions from the plant input $f$ to the relative displacement $d$ while the active damping is implemented are shown in Figure ref:fig:uniaxial_damped_plant_three_active_damping_techniques.
|
|
% All three active damping techniques yield similar damped plants.
|
|
|
|
|
|
%% Obtained damped transfer function from f to d for the three damping techniques
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'f'), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid('d', 'f'), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', 'RDC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid('d', 'f'), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d/f$ [m/N]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
|
|
|
ax2 = nexttile();
|
|
hold on;
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz')))), 'k-');
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_mid('d', 'f'), freqs, 'Hz')))), 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_md_mid('d', 'f'), freqs, 'Hz')))), 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_md_mid('d', 'f'), freqs, 'Hz')))), 'color', colors(3,:));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
yticks(-360:90:360);
|
|
ylim([-270, 90]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([1, 500]);
|
|
|
|
|
|
|
|
% #+name: fig:uniaxial_damped_plant_three_active_damping_techniques
|
|
% #+caption: Obtained damped transfer function from f to d for the three damping techniques
|
|
% #+RESULTS:
|
|
% [[file:figs/uniaxial_damped_plant_three_active_damping_techniques.png]]
|
|
|
|
% The damped plants are shown in Figure ref:fig:uniaxial_damped_plant_change_sample_mass for all three techniques, with the three considered nano-hexapod stiffnesses and sample's masses.
|
|
|
|
%% Damped plant - Robustness to change of sample's mass
|
|
figure;
|
|
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$k_n = 0.01\,N/\mu m$');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_vc_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$k_n = 1\,N/\mu m$');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$k_n = 100\,N/\mu m$');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_pz_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
title('IFF');
|
|
ylim([5e-10, 1e-3]);
|
|
ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
|
ldg.ItemTokenSize = [20, 1];
|
|
|
|
ax2 = nexttile([2,1]);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, abs(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_vc_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_vc_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_vc_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_md_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_md_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_pz_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_pz_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:));
|
|
plot(freqs, abs(squeeze(freqresp(G_rdc_pz_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
|
|
title('RDC');
|
|
ylim([5e-10, 1e-3]);
|
|
|
|
ax3 = nexttile([2,1]);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_vc_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$m_s = 1\,kg$');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_vc_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'DisplayName', '$m_s = 25\,kg$');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_vc_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$m_s = 50\,kg$');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_md_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_md_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_pz_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_pz_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf_pz_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
|
|
title('DVF');
|
|
ylim([5e-10, 1e-3]);
|
|
ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
|
ldg.ItemTokenSize = [20, 1];
|
|
|
|
ax1b = nexttile();
|
|
hold on;
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
xticks([1e0, 1e1, 1e2]);
|
|
yticks(-360:90:360);
|
|
ylim([-200, 20]);
|
|
|
|
ax2b = nexttile();
|
|
hold on;
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_vc_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_vc_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_vc_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_md_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_md_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_md_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_pz_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_pz_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_pz_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
|
xticks([1e0, 1e1, 1e2]);
|
|
yticks(-360:90:360);
|
|
ylim([-200, 20]);
|
|
|
|
ax3b = nexttile();
|
|
hold on;
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_vc_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_vc_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_vc_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_md_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_md_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_md_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_pz_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_pz_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_pz_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
|
xticks([1e0, 1e1, 1e2]);
|
|
yticks(-360:90:360);
|
|
ylim([-200, 20]);
|
|
|
|
linkaxes([ax1,ax2,ax3,ax1b,ax2b,ax3b],'x');
|
|
xlim([1, 1e3]);
|
|
|
|
% Robustness to change of payload's mass
|
|
% The Root Locus for the three damping techniques are shown in Figure ref:fig:uniaxial_active_damping_robustness_mass_root_locus for three sample's mass (1kg, 25kg and 50kg).
|
|
% The closed-loop poles are shown by the squares for a specific gain.
|
|
|
|
% We can see that having heavier samples yields larger damping for IFF and smaller damping for RDC and DVF.
|
|
|
|
|
|
%% Active Damping Robustness to change of sample's mass - Root Locus for all three damping techniques with 3 different sample's masses
|
|
figure;
|
|
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
%% Integral Force Feedback
|
|
ax1 = nexttile();
|
|
hold on;
|
|
% Light Sample
|
|
plot(real(pole(G_md_light('fm', 'f'))), imag(pole(G_md_light('fm', 'f'))), 'x', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
plot(real(zero(G_md_light('fm', 'f'))), imag(zero(G_md_light('fm', 'f'))), 'o', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(1,4,100)
|
|
clpoles = pole(feedback(G_md_light('fm', 'f'), g/s, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
clpoles = pole(feedback(G_md_light('fm', 'f'), K_iff_md, +1));
|
|
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(1,:), ...
|
|
'DisplayName', '$m_s = 1\,kg$');
|
|
|
|
% Mid Sample
|
|
plot(real(pole(G_md_mid('fm', 'f'))), imag(pole(G_md_mid('fm', 'f'))), 'x', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
plot(real(zero(G_md_mid('fm', 'f'))), imag(zero(G_md_mid('fm', 'f'))), 'o', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(1,4,100)
|
|
clpoles = pole(feedback(G_md_mid('fm', 'f'), g/s, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
clpoles = pole(feedback(G_md_mid('fm', 'f'), K_iff_md, +1));
|
|
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(2,:), ...
|
|
'DisplayName', '$m_s = 25\,kg$');
|
|
|
|
% Heavy Sample
|
|
plot(real(pole(G_md_heavy('fm', 'f'))), imag(pole(G_md_heavy('fm', 'f'))), 'x', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
plot(real(zero(G_md_heavy('fm', 'f'))), imag(zero(G_md_heavy('fm', 'f'))), 'o', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(1,4,100)
|
|
clpoles = pole(feedback(G_md_heavy('fm', 'f'), g/s, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
clpoles = pole(feedback(G_md_heavy('fm', 'f'), K_iff_md, +1));
|
|
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(3,:), ...
|
|
'DisplayName', '$m_s = 50\,kg$');
|
|
hold off;
|
|
axis square;
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
title('IFF')
|
|
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
|
ldg.ItemTokenSize = [10, 1];
|
|
|
|
%% Relative Damping Control
|
|
ax2 = nexttile();
|
|
hold on;
|
|
% Light Sample
|
|
plot(real(pole(G_md_light('dL', 'f'))), imag(pole(G_md_light('dL', 'f'))), 'x', 'color', colors(1,:), ...
|
|
'DisplayName', '$m_s = 1\,kg$');
|
|
plot(real(zero(G_md_light('dL', 'f'))), imag(zero(G_md_light('dL', 'f'))), 'o', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(2,5,100)
|
|
clpoles = pole(feedback(G_md_light('dL', 'f'), -g*s, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
clpoles = pole(feedback(G_md_light('dL', 'f'), K_rdc_md, +1));
|
|
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
|
|
% Mid Sample
|
|
plot(real(pole(G_md_mid('dL', 'f'))), imag(pole(G_md_mid('dL', 'f'))), 'x', 'color', colors(2,:), ...
|
|
'DisplayName', '$m_s = 25\,kg$');
|
|
plot(real(zero(G_md_mid('dL', 'f'))), imag(zero(G_md_mid('dL', 'f'))), 'o', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(2,5,100)
|
|
clpoles = pole(feedback(G_md_mid('dL', 'f'), -g*s, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
clpoles = pole(feedback(G_md_mid('dL', 'f'), K_rdc_md, +1));
|
|
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
|
|
% Heavy Sample
|
|
plot(real(pole(G_md_heavy('dL', 'f'))), imag(pole(G_md_heavy('dL', 'f'))), 'x', 'color', colors(3,:), ...
|
|
'DisplayName', '$m_s = 50\,kg$');
|
|
plot(real(zero(G_md_heavy('dL', 'f'))), imag(zero(G_md_heavy('dL', 'f'))), 'o', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(2,5,100)
|
|
clpoles = pole(feedback(G_md_heavy('dL', 'f'), -g*s, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
clpoles = pole(feedback(G_md_heavy('dL', 'f'), K_rdc_md, +1));
|
|
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
hold off;
|
|
axis square;
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
title('RDC');
|
|
|
|
%% Direct Velocity Feedback
|
|
ax3 = nexttile();
|
|
hold on;
|
|
% Light Sample
|
|
plot(real(pole(G_md_light('vn', 'f'))), imag(pole(G_md_light('vn', 'f'))), 'x', 'color', colors(1,:), ...
|
|
'DisplayName', '$m_s = 1\,kg$');
|
|
plot(real(zero(G_md_light('vn', 'f'))), imag(zero(G_md_light('vn', 'f'))), 'o', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(2,5,100)
|
|
clpoles = pole(feedback(G_md_light('vn', 'f'), -g, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
clpoles = pole(feedback(G_md_light('vn', 'f'), K_dvf_md, +1));
|
|
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
|
|
% Mid Sample
|
|
plot(real(pole(G_md_mid('vn', 'f'))), imag(pole(G_md_mid('vn', 'f'))), 'x', 'color', colors(2,:), ...
|
|
'DisplayName', '$m_s = 25\,kg$');
|
|
plot(real(zero(G_md_mid('vn', 'f'))), imag(zero(G_md_mid('vn', 'f'))), 'o', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(2,5,100)
|
|
clpoles = pole(feedback(G_md_mid('vn', 'f'), -g, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
clpoles = pole(feedback(G_md_mid('vn', 'f'), K_dvf_md, +1));
|
|
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(2,:), ...
|
|
'HandleVisibility', 'off');
|
|
|
|
% Heavy Sample
|
|
plot(real(pole(G_md_heavy('vn', 'f'))), imag(pole(G_md_heavy('vn', 'f'))), 'x', 'color', colors(3,:), ...
|
|
'DisplayName', '$m_s = 50\,kg$');
|
|
plot(real(zero(G_md_heavy('vn', 'f'))), imag(zero(G_md_heavy('vn', 'f'))), 'o', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
for g = logspace(2,5,100)
|
|
clpoles = pole(feedback(G_md_heavy('vn', 'f'), -g, +1));
|
|
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
clpoles = pole(feedback(G_md_heavy('vn', 'f'), K_dvf_md, +1));
|
|
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(3,:), ...
|
|
'HandleVisibility', 'off');
|
|
hold off;
|
|
axis square;
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
title('DVF');
|
|
|
|
linkaxes([ax1,ax2,ax3],'xy');
|
|
xlim([-300, 0]); ylim([0, 300]);
|