%% 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 addpath('./src/'); % Path for Functions %% Colors for the figures colors = colororder; %% Simscape model name mdl = 'rotating_model'; % Identify NASS dynamics :noexport: % First, the dynamics is identified for all the considered cases. %% Nano-Hexapod mn = 15; % Nano-Hexapod mass [kg] %% Light Sample ms = 1; % Sample Mass [kg] %% General Configuration model_config = struct(); model_config.controller = "open_loop"; % Default: Open-Loop model_config.Tuv_type = "normal"; % Default: 2DoF stage %% Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/controller'], 1, 'openinput'); io_i = io_i + 1; % [Fu, Fv] io(io_i) = linio([mdl, '/fd'], 1, 'openinput'); io_i = io_i + 1; % [Fdx, Fdy] io(io_i) = linio([mdl, '/xf'], 1, 'openinput'); io_i = io_i + 1; % [Dfx, Dfy] io(io_i) = linio([mdl, '/translation_stage'], 1, 'openoutput'); io_i = io_i + 1; % [Fmu, Fmv] io(io_i) = linio([mdl, '/translation_stage'], 2, 'openoutput'); io_i = io_i + 1; % [Du, Dv] io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy] %% Voice Coil (i.e. soft) Nano-Hexapod kn = 1e4; % Nano-Hexapod Stiffness [N/m] cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] Wz = 0; % Rotating Velocity [rad/s] G_vc_norot = linearize(mdl, io, 0.0); G_vc_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_vc_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; Wz = 2*pi; % Rotating Velocity [rad/s] G_vc_fast = linearize(mdl, io, 0.0); G_vc_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_vc_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; %% APA (i.e. relatively stiff) Nano-Hexapod kn = 1e6; % Nano-Hexapod Stiffness [N/m] cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] Wz = 0; % Rotating Velocity [rad/s] G_md_norot = linearize(mdl, io, 0.0); G_md_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_md_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; Wz = 2*pi; % Rotating Velocity [rad/s] G_md_fast = linearize(mdl, io, 0.0); G_md_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_md_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; %% Piezoelectric (i.e. stiff) Nano-Hexapod kn = 1e8; % Nano-Hexapod Stiffness [N/m] cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] Wz = 0; % Rotating Velocity [rad/s] G_pz_norot = linearize(mdl, io, 0.0); G_pz_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_pz_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; Wz = 2*pi; % Rotating Velocity [rad/s] G_pz_fast = linearize(mdl, io, 0.0); G_pz_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_pz_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; % Nano-Active-Stabilization-System - Plant Dynamics % For the NASS, the maximum rotating velocity is $\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}$ for a suspended mass on top of the nano-hexapod's actuators equal to $m_n + m_s = \SI{16}{\kilo\gram}$. % The parallel stiffness corresponding to the centrifugal forces is $m \Omega^2 \approx \SI{0.6}{\newton\per\mm}$. %% Compute negative spring in [N/m] Kneg_light = (15+1)*(2*pi)^2; % The transfer functions from nano-hexapod actuator force $F_u$ to the displacement of the nano-hexapod in the same direction $d_u$ as well as in the orthogonal direction $d_v$ (coupling) are shown in Figure ref:fig:rotating_nano_hexapod_dynamics for all three considered nano-hexapod stiffnesses. % #+begin_important % It is shown that the rotation has the largest effect on the soft nano-hexapod: % - larger coupling (the ratio of the coupling term to the direct term is larger for the sort nano-hexapod) % - larger shift of poles as a function of the rotating velocity % #+end_important %% Effect of rotation on the nano-hexapod dynamics freqs = logspace(0, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_vc_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(1,:), ... 'DisplayName', '$k_n = 0.01\,N/\mu m$'); plot(freqs, abs(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(1,:), ... 'DisplayName', '$\Omega = 60\,$rpm, $D_u/F_u$'); plot(freqs, abs(squeeze(freqresp(G_vc_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , 'color', [colors(1,:), 0.5], ... 'DisplayName', '$\Omega = 60\,$rpm, $D_v/F_u$'); plot(freqs, abs(squeeze(freqresp(G_md_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(2,:), ... 'DisplayName', '$k_n = 1\,N/\mu m$'); plot(freqs, abs(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(2,:), ... 'DisplayName', '$\Omega = 60\,$rpm, $D_u/F_u$'); plot(freqs, abs(squeeze(freqresp(G_md_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , 'color', [colors(2,:), 0.5], ... 'DisplayName', '$\Omega = 60\,$rpm, $D_v/F_u$'); plot(freqs, abs(squeeze(freqresp(G_pz_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(3,:), ... 'DisplayName', '$k_n = 100\,N/\mu m$'); plot(freqs, abs(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(3,:), ... 'DisplayName', '$\Omega = 60\,$rpm, $D_u/F_u$'); plot(freqs, abs(squeeze(freqresp(G_pz_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , 'color', [colors(3,:), 0.5], ... 'DisplayName', '$\Omega = 60\,$rpm, $D_v/F_u$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); ylim([1e-12, 1e-2]) ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); ldg.ItemTokenSize = [20, 1]; ax2 = nexttile; hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(G_vc_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(1,:)); plot(freqs, 180/pi*angle(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(1,:)); plot(freqs, 180/pi*angle(squeeze(freqresp(G_md_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(2,:)); plot(freqs, 180/pi*angle(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(2,:)); plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(3,:)); plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(3,:)); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 0]); linkaxes([ax1,ax2],'x'); xlim([1, 1e3]); % Coupling :noexport: % Let's define the /coupling ratio/ $\mathcal{C}$ in the system as the ratio of the magnitude of the coupling term $G_c$ to the magnitude of the direct term $G_d$: % \begin{equation} % \mathcal{C}(\omega) = \frac{|G_c(j\omega)|}{|G_d(j\omega)|} % \end{equation} % This gives some information in the coupling in the system. % This is quite important as high coupling can affect the closed-loop stability. % The coupling ratio for the three nano-hexapod stiffnesses are shown in Figure ref:fig:rotating_coupling_ratio_nano_hexapod for the maximum rotating velocity $\Omega = 60\,\text{rpm}$. % #+begin_important % It is shown that the low frequency coupling is inversely proportional to the nano-hexapod stiffness (Figure ref:fig:rotating_coupling_ratio_nano_hexapod). % High nano-hexapod stiffness makes the system better decoupled and therefore easier to control. % #+end_important %% Coupling ratio of the nano-hexapod at maximum velocity figure; tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); hold on; plot(freqs, abs(squeeze(freqresp(G_vc_fast('Dv', 'Fu')/G_vc_fast('Du', 'Fu'), freqs, 'Hz'))), ... 'DisplayName', '$k_n = 0.01\,N/\mu m$'); plot(freqs, abs(squeeze(freqresp(G_md_fast('Dv', 'Fu')/G_md_fast('Du', 'Fu'), freqs, 'Hz'))), ... 'DisplayName', '$k_n = 1\,N/\mu m$'); plot(freqs, abs(squeeze(freqresp(G_pz_fast('Dv', 'Fu')/G_pz_fast('Du', 'Fu'), freqs, 'Hz'))), ... 'DisplayName', '$k_n = 100\,N/\mu m$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Coupling ratio'); xlim([1, 500]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1); % Optimal IFF with High Pass Filter % Let's apply Integral Force Feedback with an added High Pass Filter to the three nano-hexapods. % First, let's find the parameters of the IFF controller that yield best simultaneous damping. % The results are shown in Figure ref:fig:rotating_iff_hpf_nass_optimal_gain. % The added damping for the soft nano-hexapod is quite low and is limited by the maximum usable gain. %% Compute the optimal control gain wis = logspace(-2, 3, 200); % [rad/s] opt_iff_hpf_xi_vc = zeros(1, length(wis)); % Optimal simultaneous damping opt_iff_hpf_gain_vc = zeros(1, length(wis)); % Corresponding optimal gain opt_iff_hpf_xi_md = zeros(1, length(wis)); % Optimal simultaneous damping opt_iff_hpf_gain_md = zeros(1, length(wis)); % Corresponding optimal gain opt_iff_hpf_xi_pz = zeros(1, length(wis)); % Optimal simultaneous damping opt_iff_hpf_gain_pz = zeros(1, length(wis)); % Corresponding optimal gain for wi_i = 1:length(wis) wi = wis(wi_i); Kiff = 1/(s + wi)*eye(2); fun = @(g)computeSimultaneousDamping(g, G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff); [g_opt, xi_opt] = fminsearch(fun, 0.1); opt_iff_hpf_xi_vc(wi_i) = 1/xi_opt; opt_iff_hpf_gain_vc(wi_i) = g_opt; fun = @(g)computeSimultaneousDamping(g, G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff); [g_opt, xi_opt] = fminsearch(fun, 0.1); opt_iff_hpf_xi_md(wi_i) = 1/xi_opt; opt_iff_hpf_gain_md(wi_i) = g_opt; fun = @(g)computeSimultaneousDamping(g, G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff); [g_opt, xi_opt] = fminsearch(fun, 0.1); opt_iff_hpf_xi_pz(wi_i) = 1/xi_opt; opt_iff_hpf_gain_pz(wi_i) = g_opt; end %% Optimal modified IFF parameters that yields maximum simultaneous damping figure; tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile(); yyaxis left plot(wis, opt_iff_hpf_xi_vc, '-', 'DisplayName', '$\xi_{cl}$'); set(gca, 'YScale', 'lin'); ylim([0,1]); ylabel('Damping Ratio $\xi$'); yyaxis right hold on; plot(wis, opt_iff_hpf_gain_vc, '-', 'DisplayName', '$g_{opt}$'); plot(wis, wis*((sqrt(1e4/16)/(2*pi))^2 - 1), '--', 'DisplayName', '$g_{max}$'); set(gca, 'YScale', 'lin'); ylim([0,200]); xlabel('$\omega_i$ [rad/s]'); set(gca, 'YTickLabel',[]); % ylabel('Controller gain $g$'); set(gca, 'XScale', 'log'); xticks([1e-2,1,1e2]) legend('location', 'northwest', 'FontSize', 8); title('$k_n = 0.01\,N/\mu m$'); ax2 = nexttile(); yyaxis left plot(wis, opt_iff_hpf_xi_md, '-'); set(gca, 'YScale', 'lin'); ylim([0,1]); % ylabel('Damping Ratio $\xi$'); set(gca, 'YTickLabel',[]); yyaxis right hold on; plot(wis, opt_iff_hpf_gain_md, '-'); plot(wis, wis*((sqrt(1e6/16)/(2*pi))^2 - 1), '--'); set(gca, 'YScale', 'lin'); ylim([0,1000]); xlabel('$\omega_i$ [rad/s]'); % ylabel('Controller gain $g$'); set(gca, 'YTickLabel',[]); set(gca, 'XScale', 'log'); xticks([1e-2,1,1e2]) title('$k_n = 1\,N/\mu m$'); ax3 = nexttile(); yyaxis left plot(wis, opt_iff_hpf_xi_pz, '-'); set(gca, 'YScale', 'lin'); ylim([0,1]); set(gca, 'YTickLabel',[]); % ylabel('Damping Ratio $\xi$'); yyaxis right hold on; plot(wis, opt_iff_hpf_gain_pz, '-'); plot(wis, wis*((sqrt(1e8/16)/(2*pi))^2 - 1), '--'); set(gca, 'YScale', 'lin'); ylim([0,10000]); xlabel('$\omega_i$ [rad/s]'); set(gca, 'YTickLabel',[]); ylabel('Controller gain $g$'); set(gca, 'XScale', 'log'); xticks([1e-2,1,1e2]) title('$k_n = 100\,N/\mu m$'); % #+name: fig:rotating_iff_hpf_nass_optimal_gain % #+caption: Optimal high pass filter cut-off frequency $\omega_i$ that yields maximum simultaneous damping % #+RESULTS: % [[file:figs/rotating_iff_hpf_nass_optimal_gain.png]] % The IFF parameters are chosen as follow: % - for $k_n = \SI{0.01}{\N\per\mu\m}$: $\omega_i$ is chosen such that the maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. % This is done to have some control robustness. % - for $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$: the largest $\omega_i$ is chosen such that obtained damping is $\SI{95}{\percent}$ of the maximum achievable damping. % Large $\omega_i$ is chosen here to limit the loss of compliance and the increase of coupling at low frequency as was shown in Section ref:sec:rotating_iff_pseudo_int. % The obtained IFF parameters and the achievable damping are summarized in Table ref:tab:iff_hpf_opt_iff_hpf_params_nass. %% Find optimal parameters with at least a gain margin of 2 i_iff_hpf_vc = find(opt_iff_hpf_gain_vc < 0.5*(wis*((sqrt(1e4/16)/(2*pi))^2 - 1))); i_iff_hpf_vc = i_iff_hpf_vc(1); i_iff_hpf_md = find(opt_iff_hpf_xi_md > 0.95*max(opt_iff_hpf_xi_md)); i_iff_hpf_md = i_iff_hpf_md(end)+1; i_iff_hpf_pz = find(opt_iff_hpf_xi_pz > 0.95*max(opt_iff_hpf_xi_pz)); i_iff_hpf_pz = i_iff_hpf_pz(end)+1; % #+name: tab:iff_hpf_opt_iff_hpf_params_nass % #+caption: Obtained optimal parameters for the modified IFF controller % #+attr_latex: :environment tabularx :width 0.5\linewidth :align lXXX % #+attr_latex: :center t :booktabs t % #+RESULTS: % | | $\omega_i$ | $g$ | $\xi$ | % |-----------------------+------------+---------+-------| % | $k_n = 0.01\,N/\mu m$ | 7.32 | 51.13 | 0.45 | % | $k_n = 1\,N/\mu m$ | 39.17 | 426.95 | 0.93 | % | $k_n = 100\,N/\mu m$ | 499.45 | 3774.63 | 0.94 | % The Root Locus for all three nano-hexapods are shown in Figure ref:fig:rotating_root_locus_iff_hpf_nass with included optimal chosen gains. %% Optimal IFF with added HPF Kiff_hpf_vc = opt_iff_hpf_gain_vc(i_iff_hpf_vc)/(s + wis(i_iff_hpf_vc))*eye(2); Kiff_hpf_vc.InputName = {'fu', 'fv'}; Kiff_hpf_vc.OutputName = {'Fu', 'Fv'}; Kiff_hpf_md = opt_iff_hpf_gain_md(i_iff_hpf_md)/(s + wis(i_iff_hpf_md))*eye(2); Kiff_hpf_md.InputName = {'fu', 'fv'}; Kiff_hpf_md.OutputName = {'Fu', 'Fv'}; Kiff_hpf_pz = opt_iff_hpf_gain_pz(i_iff_hpf_pz)/(s + wis(i_iff_hpf_pz))*eye(2); Kiff_hpf_pz.InputName = {'fu', 'fv'}; Kiff_hpf_pz.OutputName = {'Fu', 'Fv'}; %% Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately stiff nano-hexapods gains = logspace(-2, 3, 200); figure; tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); % Voice coil Nano-Hexapod ax1 = nexttile(); hold on; for g = gains clpoles = pole(feedback(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4, ... 'HandleVisibility', 'off'); end clpoles = pole(feedback(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15, ... 'DisplayName', '$\Omega = 0$'); plot(real(pole(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... imag(pole(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... 'x', 'color', colors(1,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); plot(real(tzero(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... imag(tzero(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... 'o', 'color', colors(1,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); for g = gains clpoles = pole(feedback(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4, ... 'HandleVisibility', 'off'); end clpoles = pole(feedback(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15, ... 'DisplayName', '$\Omega = 60$ rpm'); plot(real(pole(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... imag(pole(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... 'x', 'color', colors(2,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); plot(real(tzero(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... imag(tzero(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... 'o', 'color', colors(2,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); plot([0, -1e2*opt_iff_hpf_xi_vc(i_iff_hpf_vc)], [0, 1e2*cos(asin(opt_iff_hpf_xi_vc(i_iff_hpf_vc)))], '-', ... 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_hpf_xi_vc(i_iff_hpf_vc)), 'color', [zeros(1,3), 0.5]) hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); xlim([-65, 5]); ylim([-35, 35]); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title('$k_n = 0.01\,N/\mu m$'); ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [10, 1]; % APA Nano-Hexapod ax2 = nexttile(); hold on; for g = gains clpoles = pole(feedback(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); end clpoles = pole(feedback(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); plot(real(pole(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... imag(pole(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... 'x', 'color', colors(1,:),'MarkerSize',8); plot(real(tzero(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... imag(tzero(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... 'o', 'color', colors(1,:),'MarkerSize',8); for g = gains clpoles = pole(feedback(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); end clpoles = pole(feedback(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); plot(real(pole(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... imag(pole(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... 'x', 'color', colors(2,:),'MarkerSize',8); plot(real(tzero(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... imag(tzero(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... 'o', 'color', colors(2,:),'MarkerSize',8); L = plot([0, -1e3*opt_iff_hpf_xi_md(i_iff_hpf_md)], [0, 1e3*cos(asin(opt_iff_hpf_xi_md(i_iff_hpf_md)))], '-', ... 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_hpf_xi_md(i_iff_hpf_md)), 'color', [zeros(1,3), 0.5]); leg = legend(L, 'location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 10; hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); xlim([-520, 20]); ylim([-270, 270]); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title('$k_n = 1\,N/\mu m$'); % Piezo Nano-Hexapod ax3 = nexttile(); hold on; for g = gains clpoles = pole(feedback(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); end clpoles = pole(feedback(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); plot(real(pole(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... imag(pole(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... 'x', 'color', colors(1,:),'MarkerSize',8); plot(real(tzero(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... imag(tzero(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... 'o', 'color', colors(1,:),'MarkerSize',8); for g = gains clpoles = pole(feedback(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); end clpoles = pole(feedback(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); plot(real(pole(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... imag(pole(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... 'x', 'color', colors(2,:),'MarkerSize',8); plot(real(tzero(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... imag(tzero(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... 'o', 'color', colors(2,:),'MarkerSize',8); L = plot([0, -1e4*opt_iff_hpf_xi_pz(i_iff_hpf_pz)], [0, 1e4*cos(asin(opt_iff_hpf_xi_pz(i_iff_hpf_pz)))], '-', ... 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_hpf_xi_pz(i_iff_hpf_pz)), 'color', [zeros(1,3), 0.5]); leg = legend(L, 'location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 10; hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); xlim([-5200, 20]); ylim([-2700, 2700]); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title('$k_n = 100\,N/\mu m$'); % Optimal IFF with Parallel Stiffness % For each considered nano-hexapod stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness to have unconditional stability) to $k_{p,\text{max}} = k_n$ (the total nano-hexapod stiffness). % In order to keep the overall stiffness constant, the actuator stiffness $k_a$ is decreased when $k_p$ is increased: % \begin{equation} % k_a = k_n - k_p % \end{equation} % With $k_n$ the total nano-hexapod stiffness. % An high pass filter is also added to limit the low frequency gain. % The cut-off frequency $\omega_i$ is chosen to be one tenth of the system resonance: % \begin{equation} % \omega_i = \omega_0/10 % \end{equation} %% Maximum rotating velocity Wz = 2*pi; % [rad/s] %% Minimum parallel stiffness kp_min = (mn + ms) * Wz^2; % [N/m] %% Parameters for simulation mn = 15; % Nano-Hexapod mass [kg] ms = 1; % Sample Mass [kg] %% IFF Controller Kiff_vc = 1/(s + 0.1*sqrt(1e4/(mn+ms)))*eye(2); % IFF Kiff_md = 1/(s + 0.1*sqrt(1e6/(mn+ms)))*eye(2); % IFF Kiff_pz = 1/(s + 0.1*sqrt(1e8/(mn+ms)))*eye(2); % IFF %% General Configuration model_config = struct(); model_config.controller = "open_loop"; % Default: Open-Loop model_config.Tuv_type = "parallel_k"; % Default: 2DoF stage %% Computes the optimal parameters and attainable simultaneous damping - Voice Coil nano-hexapod kps_vc = logspace(log10(kp_min), log10(1e4), 100); % Tested parallel stiffnesses [N/m] kps_vc(end) = []; opt_iff_kp_xi_vc = zeros(1, length(kps_vc)); % Optimal simultaneous damping opt_iff_kp_gain_vc = zeros(1, length(kps_vc)); % Corresponding optimal gain for kp_i = 1:length(kps_vc) % Voice Coil Nano-Hexapod kp = kps_vc(kp_i); cp = 2*0.001*sqrt((ms + mn)*kp); kn = 1e4 - kp; % Nano-Hexapod Stiffness [N/m] cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] % Identify dynamics Giff_vc = linearize(mdl, io, 0); Giff_vc.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; Giff_vc.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; fun = @(g)computeSimultaneousDamping(g, Giff_vc({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_vc); [g_opt, xi_opt] = fminsearch(fun, 0.1); opt_iff_kp_xi_vc(kp_i) = 1/xi_opt; opt_iff_kp_gain_vc(kp_i) = g_opt; end %% Computes the optimal parameters and attainable simultaneous damping - APA nano-hexapod kps_md = logspace(log10(kp_min), log10(1e6), 100); % Tested parallel stiffnesses [N/m] kps_md(end) = []; opt_iff_kp_xi_md = zeros(1, length(kps_md)); % Optimal simultaneous damping opt_iff_kp_gain_md = zeros(1, length(kps_md)); % Corresponding optimal gain for kp_i = 1:length(kps_md) % Voice Coil Nano-Hexapod kp = kps_md(kp_i); cp = 2*0.001*sqrt((ms + mn)*kp); kn = 1e6 - kp; % Nano-Hexapod Stiffness [N/m] cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] % Identify dynamics Giff_md = linearize(mdl, io, 0); Giff_md.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; Giff_md.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; fun = @(g)computeSimultaneousDamping(g, Giff_md({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_md); [g_opt, xi_opt] = fminsearch(fun, 0.1); opt_iff_kp_xi_md(kp_i) = 1/xi_opt; opt_iff_kp_gain_md(kp_i) = g_opt; end %% Computes the optimal parameters and attainable simultaneous damping - Piezo nano-hexapod kps_pz = logspace(log10(kp_min), log10(1e8), 100); % Tested parallel stiffnesses [N/m] kps_pz(end) = []; opt_iff_kp_xi_pz = zeros(1, length(kps_pz)); % Optimal simultaneous damping opt_iff_kp_gain_pz = zeros(1, length(kps_pz)); % Corresponding optimal gain for kp_i = 1:length(kps_pz) % Voice Coil Nano-Hexapod kp = kps_pz(kp_i); cp = 2*0.001*sqrt((ms + mn)*kp); kn = 1e8 - kp; % Nano-Hexapod Stiffness [N/m] cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] % Identify dynamics Giff_pz = linearize(mdl, io, 0); Giff_pz.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; Giff_pz.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; fun = @(g)computeSimultaneousDamping(g, Giff_pz({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_pz); [g_opt, xi_opt] = fminsearch(fun, 0.1); opt_iff_kp_xi_pz(kp_i) = 1/xi_opt; opt_iff_kp_gain_pz(kp_i) = g_opt; end % The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses. % The comparison for the nano-hexapod stiffnesses is done in Figure ref:fig:rotating_iff_kp_nass_optimal_gain. % It is shown that *the soft nano-hexapod cannot yield good damping*. % For the two stiff options, the achievable damping starts to significantly decrease when the parallel stiffness is one tenth of the total stiffness $k_p = k_n/10$. %% Optimal IFF gain and associated simultaneous damping as a function of the parallel stiffness figure; hold on; plot(kps_vc, opt_iff_kp_xi_vc, '-', 'DisplayName', '$k_n = 0.01\,N/\mu m$'); plot(kps_md, opt_iff_kp_xi_md, '-', 'DisplayName', '$k_n = 1\,N/\mu m$'); plot(kps_pz, opt_iff_kp_xi_pz, '-', 'DisplayName', '$k_n = 100\,N/\mu m$'); hold off; xlabel('$k_p [N/m]$'); ylabel('Damping Ratio $\xi$'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylim([0,1]); legend('location', 'southeast', 'FontSize', 8); xlim([kps_pz(1), kps_pz(end)]) % #+name: fig:rotating_iff_kp_nass_optimal_gain % #+caption: Maximum achievable simultaneous damping with IFF as a function of the parallel stiffness for all three nano-hexapod stiffnesses % #+RESULTS: % [[file:figs/rotating_iff_kp_nass_optimal_gain.png]] % Let's choose $k_p = \SI{e3}{\newton\per\m}$, $k_p = \SI{e4}{\newton\per\m}$ and $k_p = \SI{e6}{\newton\per\m}$ for the three considered nano-hexapods respectively based on Figure ref:fig:rotating_iff_kp_nass_optimal_gain. % The corresponding optimal controller gains are shown in Table ref:tab:iff_kp_opt_iff_kp_params_nass. %% Find result with wanted parallel stiffness [~, i_kp_vc] = min(abs(kps_vc - 1e3)); [~, i_kp_md] = min(abs(kps_md - 1e4)); [~, i_kp_pz] = min(abs(kps_pz - 1e6)); % #+name: tab:iff_kp_opt_iff_kp_params_nass % #+caption: Obtained optimal parameters for the modified IFF controller % #+attr_latex: :environment tabularx :width 0.4\linewidth :align lXX % #+attr_latex: :center t :booktabs t % #+RESULTS: % | | $g$ | $\xi_{\text{opt}}$ | % |-----------------------+---------+--------------------| % | $k_n = 0.01\,N/\mu m$ | 47.9 | 0.44 | % | $k_n = 1\,N/\mu m$ | 465.57 | 0.97 | % | $k_n = 100\,N/\mu m$ | 4624.25 | 1.0 | % The root locus for the three nano-hexapod with parallel stiffnesses are shown in Figure ref:fig:rotating_root_locus_iff_kp_nass. % #+begin_important % Similarly to what was found with the IFF and added High Pass Filter: % - the stiff nano-hexapod is less affected by the rotation than the soft one % - the achievable damping is much larger with the stiff nano-hexapods % #+end_important %% Optimal IFF with added parallel stiffness Kiff_kp_vc = opt_iff_kp_gain_vc(i_kp_vc)/(s + 0.1*sqrt(1e4/(ms+mn)))*eye(2); Kiff_kp_vc.InputName = {'fu', 'fv'}; Kiff_kp_vc.OutputName = {'Fu', 'Fv'}; Kiff_kp_md = opt_iff_kp_gain_md(i_kp_md)/(s + 0.1*sqrt(1e6/(ms+mn)))*eye(2); Kiff_kp_md.InputName = {'fu', 'fv'}; Kiff_kp_md.OutputName = {'Fu', 'Fv'}; Kiff_kp_pz = opt_iff_kp_gain_pz(i_kp_pz)/(s + 0.1*sqrt(1e8/(ms+mn)))*eye(2); Kiff_kp_pz.InputName = {'fu', 'fv'}; Kiff_kp_pz.OutputName = {'Fu', 'Fv'}; %% Identify plant with optimal parallel stiffness - Soft nano-hexapod kp = kps_vc(i_kp_vc); cp = 2*0.001*sqrt((ms + mn)*kp); kn = 1e4-kp; % Nano-Hexapod Stiffness [N/m] cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] % Identify dynamics Wz = 2*pi; % [rad/s] G_vc_kp_fast = linearize(mdl, io, 0); G_vc_kp_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_vc_kp_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; Wz = 0; % [rad/s] G_vc_kp_norot = linearize(mdl, io, 0); G_vc_kp_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_vc_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; %% Identify plant with optimal parallel stiffness - Stiff nano-hexapod kp = kps_md(i_kp_md); cp = 2*0.001*sqrt((ms + mn)*kp); kn = 1e6 - kp; % Nano-Hexapod Stiffness [N/m] cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] % Identify dynamics Wz = 2*pi; % [rad/s] G_md_kp_fast = linearize(mdl, io, 0); G_md_kp_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_md_kp_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; Wz = 0; % [rad/s] G_md_kp_norot = linearize(mdl, io, 0); G_md_kp_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_md_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; %% Identify plant with optimal parallel stiffness - Stiff nano-hexapod kp = kps_pz(i_kp_pz); cp = 2*0.001*sqrt((ms + mn)*kp); kn = 1e8 - kp; % Nano-Hexapod Stiffness [N/m] cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] % Identify dynamics Wz = 2*pi; % [rad/s] G_pz_kp_fast = linearize(mdl, io, 0); G_pz_kp_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_pz_kp_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; Wz = 0; % [rad/s] G_pz_kp_norot = linearize(mdl, io, 0); G_pz_kp_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_pz_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; %% Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately stiff nano-hexapods gains = logspace(-2, 2, 400); figure; tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile(); hold on; % Soft Nano-Hexapod - No Rotation for g = gains clpoles = pole(feedback(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4, ... 'HandleVisibility', 'off'); end clpoles = pole(feedback(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15, ... 'DisplayName', '$\Omega = 0$'); plot(real(pole(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... imag(pole(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... 'x', 'color', colors(1,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); plot(real(tzero(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... imag(tzero(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... 'o', 'color', colors(1,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); % Soft Nano-Hexapod - High Speed Rotation for g = gains clpoles = pole(feedback(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4, ... 'HandleVisibility', 'off'); end clpoles = pole(feedback(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15, ... 'DisplayName', '$\Omega = 60$ rpm'); plot(real(pole(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... imag(pole(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... 'x', 'color', colors(2,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); plot(real(tzero(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... imag(tzero(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... 'o', 'color', colors(2,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); plot([0, -1e2*opt_iff_kp_xi_vc(i_kp_vc)], [0, 1e2*cos(asin(opt_iff_kp_xi_vc(i_kp_vc)))], '-', ... 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_kp_xi_vc(i_kp_vc)), 'color', [zeros(1,3), 0.5]); hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); xlim([-65, 5]); ylim([-35, 35]); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title('$k_n = 0.01\,N/\mu m$'); ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [10, 1]; ax2 = nexttile(); hold on; % Stiff Nano-Hexapod - No Rotation for g = gains clpoles = pole(feedback(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); end clpoles = pole(feedback(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); plot(real(pole(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... imag(pole(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... 'x', 'color', colors(1,:),'MarkerSize',8); plot(real(tzero(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... imag(tzero(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... 'o', 'color', colors(1,:),'MarkerSize',8); % Stiff Nano-Hexapod - High Speed Rotation for g = gains clpoles = pole(feedback(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); end clpoles = pole(feedback(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); plot(real(pole(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... imag(pole(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... 'x', 'color', colors(2,:),'MarkerSize',8); plot(real(tzero(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... imag(tzero(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... 'o', 'color', colors(2,:),'MarkerSize',8); L = plot([0, -1e3*opt_iff_kp_xi_md(i_kp_md)], [0, 1e3*cos(asin(opt_iff_kp_xi_md(i_kp_md)))], '-', ... 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_kp_xi_md(i_kp_md)), 'color', [zeros(1,3), 0.5]); leg = legend(L, 'location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 10; hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); xlim([-520, 20]); ylim([-270, 270]); title('$k_n = 1\,N/\mu m$'); ax3 = nexttile(); hold on; % Stiff Nano-Hexapod - No Rotation for g = gains clpoles = pole(feedback(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); end clpoles = pole(feedback(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); plot(real(pole(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... imag(pole(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... 'x', 'color', colors(1,:),'MarkerSize',8); plot(real(tzero(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... imag(tzero(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... 'o', 'color', colors(1,:),'MarkerSize',8); % Stiff Nano-Hexapod - High Speed Rotation for g = gains clpoles = pole(feedback(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); end clpoles = pole(feedback(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); plot(real(pole(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... imag(pole(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... 'x', 'color', colors(2,:),'MarkerSize',8); plot(real(tzero(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... imag(tzero(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... 'o', 'color', colors(2,:),'MarkerSize',8); L = plot([0, -1e4*opt_iff_kp_xi_pz(i_kp_pz)], [0, 1e4*cos(asin(opt_iff_kp_xi_pz(i_kp_pz)))], '-', ... 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_kp_xi_pz(i_kp_pz)), 'color', [zeros(1,3), 0.5]); leg = legend(L, 'location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 10; hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); xlim([-5200, 200]); ylim([-2700, 2700]); title('$k_n = 100\,N/\mu m$'); % Optimal Relative Motion Control % For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is shown in Figure ref:fig:rotating_rdc_optimal_gain. %% Computes the optimal parameters and attainable simultaneous damping - Piezo nano-hexapod rdc_gains = 2*logspace(1, 5, 200); % Obtained simultaneous damping rdc_xi_vc = zeros(1, length(rdc_gains)); rdc_xi_md = zeros(1, length(rdc_gains)); rdc_xi_pz = zeros(1, length(rdc_gains)); Krdc = s*eye(2); Krdc.InputName = {'Du', 'Dv'}; Krdc.OutputName = {'Fu', 'Fv'}; for g_i = 1:length(rdc_gains) [~, xi] = damp(feedback(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), rdc_gains(g_i)*Krdc)); rdc_xi_vc(g_i) = min(xi); [~, xi] = damp(feedback(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), rdc_gains(g_i)*Krdc)); rdc_xi_md(g_i) = min(xi); [~, xi] = damp(feedback(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), rdc_gains(g_i)*Krdc)); rdc_xi_pz(g_i) = min(xi); end %% Optimal IFF gain and associated simultaneous damping as a function of the parallel stiffness figure; hold on; plot(rdc_gains, rdc_xi_vc, '-', 'DisplayName', '$k_n = 0.01\,N/\mu m$'); plot(rdc_gains, rdc_xi_md, '-', 'DisplayName', '$k_n = 1\,N/\mu m$'); plot(rdc_gains, rdc_xi_pz, '-', 'DisplayName', '$k_n = 100\,N/\mu m$'); hold off; xlabel('Relative Damping Controller gain $g$'); ylabel('Damping Ratio $\xi$'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylim([0,1]); xlim([rdc_gains(1), rdc_gains(end)]) legend('location', 'southeast', 'FontSize', 8); % #+name: fig:rotating_rdc_optimal_gain % #+caption: Achievable simultaneous damping with "Relative Damping Control" as a function of the controller gain for all three nano-hexapod stiffnesses % #+RESULTS: % [[file:figs/rotating_rdc_optimal_gain.png]] % The gain is chosen is chosen such that 99% of modal damping is obtained. % The root locus for all three nano-hexapod stiffnesses are shown in Figure ref:fig:rotating_root_locus_rdc_nass. % #+begin_important % Relative damping control is much less impacted by gyroscopic effects. % It can be easily applied on the nano-hexapod with and without rotation without much differences. % #+end_important %% Optimal RDC [~, i_rdc_vc] = min(abs(rdc_xi_vc - 0.99)); [~, i_rdc_md] = min(abs(rdc_xi_md - 0.99)); [~, i_rdc_pz] = min(abs(rdc_xi_pz - 0.99)); Krdc_vc = rdc_gains(i_rdc_vc)*Krdc; Krdc_md = rdc_gains(i_rdc_md)*Krdc; Krdc_pz = rdc_gains(i_rdc_pz)*Krdc; %% Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately strdc nano-hexapods gains = logspace(-2, 3, 200); figure; tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); % Voice coil Nano-Hexapod ax1 = nexttile(); hold on; for g = gains clpoles = pole(feedback(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4, ... 'HandleVisibility', 'off'); end clpoles = pole(feedback(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15, ... 'DisplayName', '$\Omega = 0$'); plot(real(pole(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... imag(pole(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... 'x', 'color', colors(1,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); plot(real(tzero(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... imag(tzero(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... 'o', 'color', colors(1,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); for g = gains clpoles = pole(feedback(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4, ... 'HandleVisibility', 'off'); end clpoles = pole(feedback(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_vc)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15, ... 'DisplayName', '$\Omega = 60$ rpm'); plot(real(pole(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... imag(pole(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... 'x', 'color', colors(2,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); plot(real(tzero(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... imag(tzero(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... 'o', 'color', colors(2,:),'MarkerSize',8, ... 'HandleVisibility', 'off'); plot([0, -1e2*rdc_xi_vc(i_rdc_vc)], [0, 1e2*cos(asin(rdc_xi_vc(i_rdc_vc)))], '-', ... 'DisplayName', sprintf('$\\xi = %.2f$', rdc_xi_vc(i_rdc_vc)), 'color', [zeros(1,3), 0.5]) hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); xlim([-65, 5]); ylim([-35, 35]); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title('$k_n = 0.01\,N/\mu m$'); ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [10, 1]; % APA Nano-Hexapod ax2 = nexttile(); hold on; for g = gains clpoles = pole(feedback(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); end clpoles = pole(feedback(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); plot(real(pole(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... imag(pole(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... 'x', 'color', colors(1,:),'MarkerSize',8); plot(real(tzero(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... imag(tzero(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... 'o', 'color', colors(1,:),'MarkerSize',8); for g = gains clpoles = pole(feedback(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); end clpoles = pole(feedback(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_md)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); plot(real(pole(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... imag(pole(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... 'x', 'color', colors(2,:),'MarkerSize',8); plot(real(tzero(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... imag(tzero(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... 'o', 'color', colors(2,:),'MarkerSize',8); L = plot([0, -1e3*rdc_xi_md(i_rdc_md)], [0, 1e3*cos(asin(rdc_xi_md(i_rdc_md)))], '-', ... 'DisplayName', sprintf('$\\xi = %.2f$', rdc_xi_md(i_rdc_md)), 'color', [zeros(1,3), 0.5]); leg = legend(L, 'location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 10; hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); xlim([-520, 20]); ylim([-270, 270]); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title('$k_n = 1\,N/\mu m$'); % Piezo Nano-Hexapod ax3 = nexttile(); hold on; for g = gains clpoles = pole(feedback(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); end clpoles = pole(feedback(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); plot(real(pole(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... imag(pole(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... 'x', 'color', colors(1,:),'MarkerSize',8); plot(real(tzero(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... imag(tzero(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... 'o', 'color', colors(1,:),'MarkerSize',8); for g = gains clpoles = pole(feedback(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); end clpoles = pole(feedback(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_pz)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); plot(real(pole(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... imag(pole(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... 'x', 'color', colors(2,:),'MarkerSize',8); plot(real(tzero(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... imag(tzero(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... 'o', 'color', colors(2,:),'MarkerSize',8); L = plot([0, -1e4*rdc_xi_pz(i_rdc_pz)], [0, 1e4*cos(asin(rdc_xi_pz(i_rdc_pz)))], '-', ... 'DisplayName', sprintf('$\\xi = %.2f$', rdc_xi_pz(i_rdc_pz)), 'color', [zeros(1,3), 0.5]); leg = legend(L, 'location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 10; hold off; axis square; xlabel('Real Part'); ylabel('Imaginary Part'); xlim([-5200, 20]); ylim([-2700, 2700]); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title('$k_n = 100\,N/\mu m$'); %% Closed Loop Plants - IFF with HPF G_vc_norot_iff_hpf = feedback(G_vc_norot, Kiff_hpf_vc, 'name'); G_vc_fast_iff_hpf = feedback(G_vc_fast, Kiff_hpf_vc, 'name'); G_md_norot_iff_hpf = feedback(G_md_norot, Kiff_hpf_md, 'name'); G_md_fast_iff_hpf = feedback(G_md_fast, Kiff_hpf_md, 'name'); G_pz_norot_iff_hpf = feedback(G_pz_norot, Kiff_hpf_pz, 'name'); G_pz_fast_iff_hpf = feedback(G_pz_fast, Kiff_hpf_pz, 'name'); %% Closed Loop Plants - IFF with Parallel Stiffness G_vc_norot_iff_kp = feedback(G_vc_kp_norot, Kiff_kp_vc, 'name'); G_vc_fast_iff_kp = feedback(G_vc_kp_fast, Kiff_kp_vc, 'name'); G_md_norot_iff_kp = feedback(G_md_kp_norot, Kiff_kp_md, 'name'); G_md_fast_iff_kp = feedback(G_md_kp_fast, Kiff_kp_md, 'name'); G_pz_norot_iff_kp = feedback(G_pz_kp_norot, Kiff_kp_pz, 'name'); G_pz_fast_iff_kp = feedback(G_pz_kp_fast, Kiff_kp_pz, 'name'); %% Closed Loop Plants - RDC G_vc_norot_rdc = feedback(G_vc_norot, Krdc_vc, 'name'); G_vc_fast_rdc = feedback(G_vc_fast, Krdc_vc, 'name'); G_md_norot_rdc = feedback(G_md_norot, Krdc_md, 'name'); G_md_fast_rdc = feedback(G_md_fast, Krdc_md, 'name'); G_pz_norot_rdc = feedback(G_pz_norot, Krdc_pz, 'name'); G_pz_fast_rdc = feedback(G_pz_fast, Krdc_pz, 'name'); %% Comparison of the damped plants (direct and coupling terms) for the three proposed active damping techniques (IFF with HPF, IFF with $k_p$ and RDC) applied on the three nano-hexapod stiffnesses figure; freqs_vc = logspace(-1, 2, 1000); freqs_md = logspace(0, 3, 1000); freqs_pz = logspace(0, 3, 1000); figure; tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [zeros(1,3)]); plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast( 'Dv', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [zeros(1,3), 0.5]); plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_iff_hpf( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(1,:)]); plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_iff_hpf( 'Dv', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(1,:), 0.5]); plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_iff_kp( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(2,:)]); plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_iff_kp( 'Dv', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(2,:), 0.5]); plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_rdc( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(3,:)]); plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_rdc( 'Dv', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(3,:), 0.5]); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); title('$k_n = 0.01\,N/\mu m$'); ax2 = nexttile([2,1]); hold on; plot(freqs_md, abs(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [zeros(1,3)]); plot(freqs_md, abs(squeeze(freqresp(G_md_fast( 'Dv', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [zeros(1,3), 0.5]); plot(freqs_md, abs(squeeze(freqresp(G_md_fast_iff_hpf( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(1,:)]); plot(freqs_md, abs(squeeze(freqresp(G_md_fast_iff_hpf( 'Dv', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(1,:), 0.5]); plot(freqs_md, abs(squeeze(freqresp(G_md_fast_iff_kp( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(2,:)]); plot(freqs_md, abs(squeeze(freqresp(G_md_fast_iff_kp( 'Dv', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(2,:), 0.5]); plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(3,:)]); plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc( 'Dv', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(3,:), 0.5]); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title('$k_n = 1\,N/\mu m$'); ax3 = nexttile([2,1]); hold on; plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [zeros(1,3)], ... 'DisplayName', 'OL'); plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast( 'Dv', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [zeros(1,3), 0.5], ... 'DisplayName', 'Coupling'); plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_iff_hpf( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(1,:)], ... 'DisplayName', 'IFF + HPF'); plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_iff_hpf( 'Dv', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(1,:), 0.5], ... 'HandleVisibility', 'off'); plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_iff_kp( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(2,:)], ... 'DisplayName', 'IFF + $k_p$'); plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_iff_kp( 'Dv', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(2,:), 0.5], ... 'HandleVisibility', 'off'); plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_rdc( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(3,:)], ... 'DisplayName', 'RDC'); plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_rdc( 'Dv', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(3,:), 0.5], ... 'HandleVisibility', 'off'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); ylim([1e-12, 1e-2]) ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [20, 1]; title('$k_n = 100\,N/\mu m$'); ax1b = nexttile; hold on; plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [zeros(1,3)]); plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast_iff_hpf( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(1,:)]); plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast_iff_kp( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(2,:)]); plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast_rdc( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(3,:)]); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); xlim([freqs_vc(1), freqs_vc(end)]); ax2b = nexttile; hold on; plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [zeros(1,3)]); plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast_iff_hpf( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(1,:)]); plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast_iff_kp( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(2,:)]); plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast_rdc( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(3,:)]); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); hold off; yticks(-360:90:360); ylim([-180, 180]); xlim([freqs_md(1), freqs_md(end)]); ax3b = nexttile; hold on; plot(freqs_pz, 180/pi*angle(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [zeros(1,3)]); plot(freqs_pz, 180/pi*angle(squeeze(freqresp(G_pz_fast_iff_hpf( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(1,:)]); plot(freqs_pz, 180/pi*angle(squeeze(freqresp(G_pz_fast_iff_kp( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(2,:)]); plot(freqs_pz, 180/pi*angle(squeeze(freqresp(G_pz_fast_rdc( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(3,:)]); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); hold off; yticks(-360:90:360); ylim([-180, 180]); xlim([freqs_pz(1), freqs_pz(end)]); linkaxes([ax1,ax2,ax3],'y'); linkaxes([ax1b,ax2b,ax3b],'y'); linkaxes([ax1,ax1b],'x'); linkaxes([ax2,ax2b],'x'); linkaxes([ax3,ax3b],'x');