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A2-nass-rotating-3dof-model/rotating_7_nano_hexapod.m
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682
A2-nass-rotating-3dof-model/rotating_7_nano_hexapod.m
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Path for functions, data and scripts
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addpath('./mat/'); % Path for data
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addpath('./src/'); % Path for Functions
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%% Colors for the figures
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colors = colororder;
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%% Simscape model name
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mdl = 'rotating_model';
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%% Nano-Hexapod
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mn = 15; % Nano-Hexapod mass [kg]
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%% Light Sample
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ms = 1; % Sample Mass [kg]
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%% General Configuration
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model_config = struct();
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model_config.controller = "open_loop"; % Default: Open-Loop
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model_config.Tuv_type = "normal"; % Default: 2DoF stage
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/controller'], 1, 'openinput'); io_i = io_i + 1; % [Fu, Fv]
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io(io_i) = linio([mdl, '/fd'], 1, 'openinput'); io_i = io_i + 1; % [Fdx, Fdy]
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io(io_i) = linio([mdl, '/xf'], 1, 'openinput'); io_i = io_i + 1; % [Dfx, Dfy]
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io(io_i) = linio([mdl, '/translation_stage'], 1, 'openoutput'); io_i = io_i + 1; % [Fmu, Fmv]
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io(io_i) = linio([mdl, '/translation_stage'], 2, 'openoutput'); io_i = io_i + 1; % [Du, Dv]
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io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy]
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%% Voice Coil (i.e. soft) Nano-Hexapod
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kn = 1e4; % Nano-Hexapod Stiffness [N/m]
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cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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Wz = 0; % Rotating Velocity [rad/s]
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G_vc_norot = linearize(mdl, io, 0.0);
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G_vc_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'};
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G_vc_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
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Wz = 2*pi; % Rotating Velocity [rad/s]
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G_vc_fast = linearize(mdl, io, 0.0);
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G_vc_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'};
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G_vc_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
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%% APA (i.e. relatively stiff) Nano-Hexapod
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kn = 1e6; % Nano-Hexapod Stiffness [N/m]
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cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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Wz = 0; % Rotating Velocity [rad/s]
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G_md_norot = linearize(mdl, io, 0.0);
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G_md_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'};
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G_md_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
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Wz = 2*pi; % Rotating Velocity [rad/s]
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G_md_fast = linearize(mdl, io, 0.0);
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G_md_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'};
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G_md_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
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%% Piezoelectric (i.e. stiff) Nano-Hexapod
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kn = 1e8; % Nano-Hexapod Stiffness [N/m]
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cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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Wz = 0; % Rotating Velocity [rad/s]
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G_pz_norot = linearize(mdl, io, 0.0);
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G_pz_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'};
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G_pz_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
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Wz = 2*pi; % Rotating Velocity [rad/s]
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G_pz_fast = linearize(mdl, io, 0.0);
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G_pz_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'};
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G_pz_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
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%% Compute negative spring in [N/m]
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Kneg_light = (15+1)*(2*pi)^2;
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%% Effect of rotation on the nano-hexapod dynamics
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freqs = logspace(0, 1, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_vc_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(1,:), ...
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'DisplayName', '$\Omega = 0\,$rpm, $D_u/F_u$');
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plot(freqs, abs(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(1,:), ...
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'DisplayName', '$\Omega = 60\,$rpm, $D_u/F_u$');
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plot(freqs, abs(squeeze(freqresp(G_vc_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , 'color', [colors(1,:), 0.5], ...
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'DisplayName', '$\Omega = 60\,$rpm, $D_v/F_u$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
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ylim([1e-6, 1e-2])
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_vc_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(1,:));
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(1,:));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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yticks(-360:90:360);
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ylim([-180, 0]);
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linkaxes([ax1,ax2],'x');
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% xlim([1, 1e3]);
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%% Effect of rotation on the nano-hexapod dynamics
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freqs = logspace(1, 2, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_md_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(2,:), ...
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'DisplayName', '$\Omega = 0\,$rpm, $D_u/F_u$');
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plot(freqs, abs(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(2,:), ...
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'DisplayName', '$\Omega = 60\,$rpm, $D_u/F_u$');
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plot(freqs, abs(squeeze(freqresp(G_md_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , 'color', [colors(2,:), 0.5], ...
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'DisplayName', '$\Omega = 60\,$rpm, $D_v/F_u$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
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ylim([1e-8, 1e-4])
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_md_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(2,:));
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(2,:));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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yticks(-360:90:360);
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ylim([-180, 0]);
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linkaxes([ax1,ax2],'x');
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%% Effect of rotation on the nano-hexapod dynamics
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freqs = logspace(2, 3, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_pz_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(3,:), ...
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'DisplayName', '$\Omega = 0\,$rpm, $D_u/F_u$');
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plot(freqs, abs(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(3,:), ...
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'DisplayName', '$\Omega = 60\,$rpm, $D_u/F_u$');
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plot(freqs, abs(squeeze(freqresp(G_pz_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , 'color', [colors(3,:), 0.5], ...
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'DisplayName', '$\Omega = 60\,$rpm, $D_v/F_u$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
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ylim([1e-10, 1e-6])
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(3,:));
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(3,:));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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yticks(-360:90:360);
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ylim([-180, 0]);
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linkaxes([ax1,ax2],'x');
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%% Compute the optimal control gain
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wis = logspace(-2, 3, 200); % [rad/s]
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opt_iff_hpf_xi_vc = zeros(1, length(wis)); % Optimal simultaneous damping
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opt_iff_hpf_gain_vc = zeros(1, length(wis)); % Corresponding optimal gain
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opt_iff_hpf_xi_md = zeros(1, length(wis)); % Optimal simultaneous damping
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opt_iff_hpf_gain_md = zeros(1, length(wis)); % Corresponding optimal gain
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opt_iff_hpf_xi_pz = zeros(1, length(wis)); % Optimal simultaneous damping
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opt_iff_hpf_gain_pz = zeros(1, length(wis)); % Corresponding optimal gain
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for wi_i = 1:length(wis)
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wi = wis(wi_i);
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Kiff = 1/(s + wi)*eye(2);
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fun = @(g)computeSimultaneousDamping(g, G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff);
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[g_opt, xi_opt] = fminsearch(fun, 0.1);
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opt_iff_hpf_xi_vc(wi_i) = 1/xi_opt;
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opt_iff_hpf_gain_vc(wi_i) = g_opt;
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fun = @(g)computeSimultaneousDamping(g, G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff);
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[g_opt, xi_opt] = fminsearch(fun, 0.1);
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opt_iff_hpf_xi_md(wi_i) = 1/xi_opt;
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opt_iff_hpf_gain_md(wi_i) = g_opt;
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fun = @(g)computeSimultaneousDamping(g, G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff);
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[g_opt, xi_opt] = fminsearch(fun, 0.1);
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opt_iff_hpf_xi_pz(wi_i) = 1/xi_opt;
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opt_iff_hpf_gain_pz(wi_i) = g_opt;
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end
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%% Find optimal parameters with at least a gain margin of 2
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i_iff_hpf_vc = find(opt_iff_hpf_gain_vc < 0.5*(wis*((sqrt(1e4/16)/(2*pi))^2 - 1)));
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i_iff_hpf_vc = i_iff_hpf_vc(1);
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i_iff_hpf_md = find(opt_iff_hpf_xi_md > 0.95*max(opt_iff_hpf_xi_md));
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i_iff_hpf_md = i_iff_hpf_md(end)+1;
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i_iff_hpf_pz = find(opt_iff_hpf_xi_pz > 0.95*max(opt_iff_hpf_xi_pz));
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i_iff_hpf_pz = i_iff_hpf_pz(end)+1;
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%% Optimal modified IFF parameters that yields maximum simultaneous damping
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figure;
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yyaxis left
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hold on;
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plot(wis, opt_iff_hpf_xi_vc, '-', 'DisplayName', '$\xi_{cl}$');
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plot(wis(i_iff_hpf_vc), opt_iff_hpf_xi_vc(i_iff_hpf_vc), '.', 'MarkerSize', 15, 'HandleVisibility', 'off');
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hold off;
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set(gca, 'YScale', 'lin');
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ylim([0,1]);
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ylabel('Damping Ratio $\xi$');
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yyaxis right
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hold on;
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plot(wis, opt_iff_hpf_gain_vc, '-', 'DisplayName', '$g_{opt}$');
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plot(wis, wis*((sqrt(1e4/16)/(2*pi))^2 - 1), '--', 'DisplayName', '$g_{max}$');
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plot(wis(i_iff_hpf_vc), opt_iff_hpf_gain_vc(i_iff_hpf_vc), '.', 'MarkerSize', 15, 'HandleVisibility', 'off');
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set(gca, 'YScale', 'lin');
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ylim([0,200]);
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xlabel('$\omega_i$ [rad/s]');
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set(gca, 'YTickLabel',[]);
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ylabel('Controller gain $g$');
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set(gca, 'XScale', 'log');
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xticks([1e-2,1,1e2])
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legend('location', 'northwest', 'FontSize', 8);
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figure;
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yyaxis left
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hold on;
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plot(wis, opt_iff_hpf_xi_md, '-');
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plot(wis(i_iff_hpf_md), opt_iff_hpf_xi_md(i_iff_hpf_md), '.', 'MarkerSize', 15, 'HandleVisibility', 'off');
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hold off;
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set(gca, 'YScale', 'lin');
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ylim([0,1]);
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ylabel('Damping Ratio $\xi$');
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yyaxis right
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hold on;
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plot(wis, opt_iff_hpf_gain_md, '-');
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plot(wis(i_iff_hpf_md), opt_iff_hpf_gain_md(i_iff_hpf_md), '.', 'MarkerSize', 15, 'HandleVisibility', 'off');
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plot(wis, wis*((sqrt(1e6/16)/(2*pi))^2 - 1), '--');
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set(gca, 'YScale', 'lin');
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ylim([0,1000]);
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xlabel('$\omega_i$ [rad/s]');
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ylabel('Controller gain $g$');
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set(gca, 'YTickLabel',[]);
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set(gca, 'XScale', 'log');
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xticks([1e-2,1,1e2])
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figure;
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yyaxis left
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hold on;
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plot(wis, opt_iff_hpf_xi_pz, '-');
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plot(wis(i_iff_hpf_pz), opt_iff_hpf_xi_pz(i_iff_hpf_pz), '.', 'MarkerSize', 15, 'HandleVisibility', 'off');
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hold off;
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set(gca, 'YScale', 'lin');
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ylim([0,1]);
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ylabel('Damping Ratio $\xi$');
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yyaxis right
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hold on;
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plot(wis, opt_iff_hpf_gain_pz, '-');
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plot(wis(i_iff_hpf_pz), opt_iff_hpf_gain_pz(i_iff_hpf_pz), '.', 'MarkerSize', 15, 'HandleVisibility', 'off');
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plot(wis, wis*((sqrt(1e8/16)/(2*pi))^2 - 1), '--');
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set(gca, 'YScale', 'lin');
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ylim([0,10000]);
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xlabel('$\omega_i$ [rad/s]');
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set(gca, 'YTickLabel',[]);
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ylabel('Controller gain $g$');
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set(gca, 'XScale', 'log');
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xticks([1e-2,1,1e2])
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%% Maximum rotating velocity
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Wz = 2*pi; % [rad/s]
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%% Minimum parallel stiffness
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kp_min = (mn + ms) * Wz^2; % [N/m]
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%% Parameters for simulation
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mn = 15; % Nano-Hexapod mass [kg]
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ms = 1; % Sample Mass [kg]
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%% IFF Controller
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Kiff_vc = 1/(s + 0.1*sqrt(1e4/(mn+ms)))*eye(2); % IFF
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Kiff_md = 1/(s + 0.1*sqrt(1e6/(mn+ms)))*eye(2); % IFF
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Kiff_pz = 1/(s + 0.1*sqrt(1e8/(mn+ms)))*eye(2); % IFF
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%% General Configuration
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model_config = struct();
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model_config.controller = "open_loop"; % Default: Open-Loop
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model_config.Tuv_type = "parallel_k"; % Default: 2DoF stage
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%% Computes the optimal parameters and attainable simultaneous damping - Voice Coil nano-hexapod
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kps_vc = logspace(log10(kp_min), log10(1e4), 100); % Tested parallel stiffnesses [N/m]
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kps_vc(end) = [];
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opt_iff_kp_xi_vc = zeros(1, length(kps_vc)); % Optimal simultaneous damping
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opt_iff_kp_gain_vc = zeros(1, length(kps_vc)); % Corresponding optimal gain
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for kp_i = 1:length(kps_vc)
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% Voice Coil Nano-Hexapod
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kp = kps_vc(kp_i);
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cp = 2*0.001*sqrt((ms + mn)*kp);
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kn = 1e4 - kp; % Nano-Hexapod Stiffness [N/m]
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cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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% Identify dynamics
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Giff_vc = linearize(mdl, io, 0);
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Giff_vc.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'};
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Giff_vc.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
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fun = @(g)computeSimultaneousDamping(g, Giff_vc({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_vc);
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[g_opt, xi_opt] = fminsearch(fun, 0.1);
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opt_iff_kp_xi_vc(kp_i) = 1/xi_opt;
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opt_iff_kp_gain_vc(kp_i) = g_opt;
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end
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%% Computes the optimal parameters and attainable simultaneous damping - APA nano-hexapod
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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
|
||||
|
||||
%% 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));
|
||||
|
||||
%% Identify plants with choosen Parallel stiffnesses
|
||||
model_config.Tuv_type = "parallel_k"; % Default: 2DoF stage
|
||||
|
||||
% Voice Coil
|
||||
kp = 1e3;
|
||||
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'};
|
||||
|
||||
% APA
|
||||
kp = 1e4;
|
||||
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'};
|
||||
|
||||
% Piezo
|
||||
kp = 1e6;
|
||||
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'};
|
||||
|
||||
%% 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, '-', ...
|
||||
'color', colors(1,:), 'DisplayName', '$k_n = 0.01\,N/\mu m$');
|
||||
plot(kps_vc(i_kp_vc), opt_iff_kp_xi_vc(i_kp_vc), '.', ...
|
||||
'color', colors(1,:), 'MarkerSize', 15, 'HandleVisibility', 'off');
|
||||
plot(kps_md, opt_iff_kp_xi_md, '-', ...
|
||||
'color', colors(2,:), 'DisplayName', '$k_n = 1\,N/\mu m$');
|
||||
plot(kps_md(i_kp_md), opt_iff_kp_xi_md(i_kp_md), '.', ...
|
||||
'color', colors(2,:), 'MarkerSize', 15, 'HandleVisibility', 'off');
|
||||
plot(kps_pz, opt_iff_kp_xi_pz, '-', ...
|
||||
'color', colors(3,:), 'DisplayName', '$k_n = 100\,N/\mu m$');
|
||||
plot(kps_pz(i_kp_pz), opt_iff_kp_xi_pz(i_kp_pz), '.', ...
|
||||
'color', colors(3,:), 'MarkerSize', 15, 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
xlabel('$k_p [N/m]$');
|
||||
ylabel('Damping Ratio $\xi$');
|
||||
set(gca, 'XScale', 'log');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,1]);
|
||||
yticks([0:0.2:1])
|
||||
legend('location', 'southeast', 'FontSize', 8);
|
||||
xlim([kps_pz(1), kps_pz(end)])
|
||||
|
||||
%% 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 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;
|
||||
|
||||
%% Optimal IFF gain and associated simultaneous damping as a function of the parallel stiffness
|
||||
figure;
|
||||
hold on;
|
||||
plot(rdc_gains, rdc_xi_vc, '-', ...
|
||||
'color', colors(1,:), 'DisplayName', '$k_n = 0.01\,N/\mu m$');
|
||||
plot(rdc_gains(i_rdc_vc), rdc_xi_vc(i_rdc_vc), '.', ...
|
||||
'color', colors(1,:), 'MarkerSize', 15, 'HandleVisibility', 'off');
|
||||
plot(rdc_gains, rdc_xi_md, '-', ...
|
||||
'color', colors(2,:), 'DisplayName', '$k_n = 1\,N/\mu m$');
|
||||
plot(rdc_gains(i_rdc_md), rdc_xi_md(i_rdc_md), '.', ...
|
||||
'color', colors(2,:), 'MarkerSize', 15, 'HandleVisibility', 'off');
|
||||
plot(rdc_gains, rdc_xi_pz, '-', ...
|
||||
'color', colors(3,:), 'DisplayName', '$k_n = 100\,N/\mu m$');
|
||||
plot(rdc_gains(i_rdc_pz), rdc_xi_pz(i_rdc_pz), '.', ...
|
||||
'color', colors(3,:), 'MarkerSize', 15, 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
xlabel('Relative Damping Controller gain $g$');
|
||||
ylabel('Damping Ratio $\xi$');
|
||||
set(gca, 'XScale', 'log');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,1]);
|
||||
yticks([0:0.2:1])
|
||||
xlim([rdc_gains(1), rdc_gains(end)])
|
||||
legend('location', 'southeast', 'FontSize', 8);
|
||||
|
||||
%% 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
|
||||
freqs_vc = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, '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',[]);
|
||||
ylim([1e-8, 1e-2])
|
||||
|
||||
ax2 = 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, 0]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs_vc(1), freqs_vc(end)]);
|
||||
|
||||
freqs_md = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = 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');
|
||||
ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
ylim([1e-10, 1e-4])
|
||||
|
||||
ax2 = 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]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
yticks(-360:90:360);
|
||||
ylim([-180, 0]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs_md(1), freqs_md(end)]);
|
||||
|
||||
freqs_pz = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = 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');
|
||||
ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
ylim([1e-12, 1e-6])
|
||||
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
ldg.ItemTokenSize = [20, 1];
|
||||
|
||||
ax2 = 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]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
yticks(-360:90:360);
|
||||
ylim([-180, 0]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs_pz(1), freqs_pz(end)]);
|
||||
Reference in New Issue
Block a user