%% Clear Workspace and Close figures clear; close all; clc; %% Intialize Laplace variable s = zpk('s'); %% Path for functions, data and scripts addpath('./mat/'); % Path for data %% Colors for the figures colors = colororder; %% Frequency Vector [Hz] freqs = logspace(0, 3, 1000); %% Load the PSD of disturbances load('uniaxial_disturbance_psd.mat', 'f', 'psd_ft', 'psd_xf'); %% Load Plants Dynamics load('uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ... 'G_vc_mid', 'G_md_mid', 'G_pz_mid', ... 'G_vc_heavy', 'G_md_heavy', 'G_pz_heavy'); % Plant Dynamics for Active Damping % The plant dynamics for all three active damping techniques are shown in Figure ref:fig:uniaxial_plant_active_damping_techniques. % All have *alternating poles and zeros* meaning that the phase is bounded to $\pm 90\,\text{deg}$ which makes the controller very robust. % 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. % 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. % Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used. %% Damped plants for three considered payload masses - Comparison of active damping techniques figure; tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_vc_light('fm', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$m_s = 1\,kg$'); plot(freqs, abs(squeeze(freqresp(G_vc_mid( 'fm', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'DisplayName', '$m_s = 25\,kg$'); plot(freqs, abs(squeeze(freqresp(G_vc_heavy('fm', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$m_s = 50\,kg$'); plot(freqs, abs(squeeze(freqresp(G_md_light('fm', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_md_mid( 'fm', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_md_heavy('fm', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_pz_light('fm', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_pz_mid( 'fm', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_pz_heavy('fm', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); title('IFF: $f_m/f$'); ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [20, 1]; ax2 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_vc_light('dL', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:)); plot(freqs, abs(squeeze(freqresp(G_vc_mid( 'dL', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:)); plot(freqs, abs(squeeze(freqresp(G_vc_heavy('dL', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:)); plot(freqs, abs(squeeze(freqresp(G_md_light('dL', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:)); plot(freqs, abs(squeeze(freqresp(G_md_mid( 'dL', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:)); plot(freqs, abs(squeeze(freqresp(G_md_heavy('dL', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:)); plot(freqs, abs(squeeze(freqresp(G_pz_light('dL', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:)); plot(freqs, abs(squeeze(freqresp(G_pz_mid( 'dL', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:)); plot(freqs, abs(squeeze(freqresp(G_pz_heavy('dL', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:)); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); title('RDC: $d\mathcal{L}/f$'); ax3 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_vc_light('vn', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$k_n = 0.01\,N/\mu m$'); plot(freqs, abs(squeeze(freqresp(G_vc_mid( 'vn', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_vc_heavy('vn', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_md_light('vn', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$k_n = 1\,N/\mu m$'); plot(freqs, abs(squeeze(freqresp(G_md_mid( 'vn', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_md_heavy('vn', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_pz_light('vn', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$k_n = 100\,N/\mu m$'); plot(freqs, abs(squeeze(freqresp(G_pz_mid( 'vn', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off'); plot(freqs, abs(squeeze(freqresp(G_pz_heavy('vn', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/s/N]'); set(gca, 'XTickLabel',[]); title('DVF: $v_n/f$'); ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [20, 1]; ax1b = nexttile(); hold on; plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_light('fm', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid( 'fm', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_heavy('fm', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('fm', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid( 'fm', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_heavy('fm', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('fm', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid( 'fm', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_heavy('fm', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:)); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); 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_light('dL', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid( 'dL', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_heavy('dL', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('dL', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid( 'dL', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_heavy('dL', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('dL', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid( 'dL', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_heavy('dL', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:)); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); 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_light('vn', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid( 'vn', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_heavy('vn', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('vn', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid( 'vn', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_heavy('vn', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('vn', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid( 'vn', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:)); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_heavy('vn', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:)); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); xticks([1e0, 1e1, 1e2]); yticks(-360:90:360); ylim([-110, 110]); linkaxes([ax1,ax2,ax3,ax1b,ax2b,ax3b],'x'); xlim([1, 1000]); % Achievable Damping - Root Locus % The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques. % They are shown in Figure ref:fig:uniaxial_root_locus_damping_techniques. % All three active damping approach can lead to *critical damping* of the nano-hexapod suspension mode. % 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). % 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!). %% 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'); %% Soft Nano-Hexapod ax1 = nexttile(); hold on; % IFF plot(real(pole(G_vc_light('fm', 'f'))), imag(pole(G_vc_light('fm', 'f'))), 'x', 'color', colors(1,:), ... 'HandleVisibility', 'off'); plot(real(zero(G_vc_light('fm', 'f'))), imag(zero(G_vc_light('fm', 'f'))), 'o', 'color', colors(1,:), ... 'DisplayName', 'IFF'); for g = logspace(0, 2, 400) clpoles = pole(feedback(G_vc_light('fm', 'f'), g/s, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ... 'HandleVisibility', 'off'); end % RDC plot(real(pole(G_vc_light('dL', 'f'))), imag(pole(G_vc_light('dL', 'f'))), 'x', 'color', colors(2,:), ... 'HandleVisibility', 'off'); plot(real(zero(G_vc_light('dL', 'f'))), imag(zero(G_vc_light('dL', 'f'))), 'o', 'color', colors(2,:), ... 'DisplayName', 'RDC'); for g = logspace(1,3,400) clpoles = pole(feedback(G_vc_light('dL', 'f'), -g*s, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ... 'HandleVisibility', 'off'); end % DVF plot(real(pole(G_vc_light('vn', 'f'))), imag(pole(G_vc_light('vn', 'f'))), 'x', 'color', colors(3,:), ... 'HandleVisibility', 'off'); plot(real(zero(G_vc_light('vn', 'f'))), imag(zero(G_vc_light('vn', 'f'))), 'o', 'color', colors(3,:), ... 'DisplayName', 'DVF'); for g = logspace(1,3,400) clpoles = pole(feedback(G_vc_light('vn', 'f'), -g, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ... 'HandleVisibility', 'off'); end hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); title('$k_n = 0.01\,N/\mu m$') ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [10, 1]; xlim([-30, 0]); ylim([0, 30]); %% Medium-Stiff Nano-Hexapod ax2 = nexttile(); hold on; % IFF 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(0,3,400) clpoles = pole(feedback(G_md_light('fm', 'f'), g/s, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ... 'HandleVisibility', 'off'); end % RDC plot(real(pole(G_md_light('dL', 'f'))), imag(pole(G_md_light('dL', 'f'))), 'x', 'color', colors(2,:), ... 'HandleVisibility', 'off'); plot(real(zero(G_md_light('dL', 'f'))), imag(zero(G_md_light('dL', 'f'))), 'o', 'color', colors(2,:), ... 'HandleVisibility', 'off'); for g = logspace(2,4,400) clpoles = pole(feedback(G_md_light('dL', 'f'), -g*s, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ... 'HandleVisibility', 'off'); end % DVF plot(real(pole(G_md_light('vn', 'f'))), imag(pole(G_md_light('vn', 'f'))), 'x', 'color', colors(3,:), ... 'HandleVisibility', 'off'); plot(real(zero(G_md_light('vn', 'f'))), imag(zero(G_md_light('vn', 'f'))), 'o', 'color', colors(3,:), ... 'HandleVisibility', 'off'); for g = logspace(2,4,400) clpoles = pole(feedback(G_md_light('vn', 'f'), -g, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ... 'HandleVisibility', 'off'); end hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); title('$k_n = 1\,N/\mu m$') xlim([-300, 0]); ylim([0, 300]); %% Stiff Nano-Hexapod ax3 = nexttile(); hold on; % IFF plot(real(pole(G_pz_light('fm', 'f'))), imag(pole(G_pz_light('fm', 'f'))), 'x', 'color', colors(1,:), ... 'HandleVisibility', 'off'); plot(real(zero(G_pz_light('fm', 'f'))), imag(zero(G_pz_light('fm', 'f'))), 'o', 'color', colors(1,:), ... 'HandleVisibility', 'off'); for g = logspace(2,5,400) clpoles = pole(feedback(G_pz_light('fm', 'f'), g/s, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ... 'HandleVisibility', 'off'); end % RDC plot(real(pole(G_pz_light('dL', 'f'))), imag(pole(G_pz_light('dL', 'f'))), 'x', 'color', colors(2,:), ... 'HandleVisibility', 'off'); plot(real(zero(G_pz_light('dL', 'f'))), imag(zero(G_pz_light('dL', 'f'))), 'o', 'color', colors(2,:), ... 'HandleVisibility', 'off'); for g = logspace(3,6,400) clpoles = pole(feedback(G_pz_light('dL', 'f'), -g*s, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ... 'HandleVisibility', 'off'); end % DVF plot(real(pole(G_pz_light('vn', 'f'))), imag(pole(G_pz_light('vn', 'f'))), 'x', 'color', colors(3,:), ... 'HandleVisibility', 'off'); plot(real(zero(G_pz_light('vn', 'f'))), imag(zero(G_pz_light('vn', 'f'))), 'o', 'color', colors(3,:), ... 'HandleVisibility', 'off'); for g = logspace(3,6,400) clpoles = pole(feedback(G_pz_light('vn', 'f'), -g, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ... 'HandleVisibility', 'off'); end hold off; axis square; 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]);