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matlab/nass_uniaxial_model.slx
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matlab/nass_uniaxial_model.slx
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matlab/uniaxial_1_micro_station_model.m
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matlab/uniaxial_1_micro_station_model.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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%% Uniaxial Simscape model name
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mdl = 'nass_uniaxial_model';
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%% Frequency Vector [Hz]
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freqs = logspace(0, 3, 1000);
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% #+name: fig:micro_station_meas_dynamics_schematic
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% #+caption: Measurement setup - Schematic
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% #+RESULTS:
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% [[file:figs/micro_station_meas_dynamics_schematic.png]]
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% Due to the bad coherence at low frequency, the frequency response functions are only shown between 20 and 200Hz (Figure ref:fig:uniaxial_measured_frf_vertical).
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%% Load measured FRF
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load('meas_microstation_frf.mat');
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%% Measured Frequency Response Functions in the vertical direction
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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(f(f>20), abs(frf_Fhx_to_Dhx(f>20)), ...
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'DisplayName', '$D_{h}/F_{h}$');
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plot(f(f>20), abs(frf_Fgx_to_Dhx(f>20)), ...
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'DisplayName', '$D_{h}/F_{g}$');
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plot(f(f>20), abs(frf_Fgx_to_Dgx(f>20)), ...
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'DisplayName', '$D_{g}/F_{g}$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ylim([1e-10, 2e-6]);
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legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
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ax2 = nexttile;
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hold on;
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plot(f(f>20), 180/pi*unwrap(angle(frf_Fhx_to_Dhx(f>20))));
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plot(f(f>30), 180/pi*unwrap(angle(frf_Fgx_to_Dhx(f>30))));
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plot(f(f>20), 180/pi*unwrap(angle(frf_Fgx_to_Dgx(f>20))));
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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([-360, 10]);
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linkaxes([ax1,ax2],'x');
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xlim([1, 500]);
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% #+name: fig:uniaxial_model_micro_station
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% #+caption: Uniaxial model of the micro-station
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% #+RESULTS:
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% [[file:figs/uniaxial_model_micro_station.png]]
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% Masses are estimated from the CAD.
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%% Parameters - Mass
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mh = 15; % Micro Hexapod [kg]
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mt = 1200; % Ty + Ry + Rz [kg]
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mg = 2500; % Granite [kg]
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% And stiffnesses from the data-sheet of stage manufacturers.
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%% Parameters - Stiffnesses
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kh = 6.11e+07; % [N/m]
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kt = 5.19e+08; % [N/m]
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kg = 9.50e+08; % [N/m]
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% The damping coefficients are tuned to match the identified damping from the measurements.
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%% Parameters - damping
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ch = 2*0.05*sqrt(kh*mh); % [N/(m/s)]
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ct = 2*0.05*sqrt(kt*mt); % [N/(m/s)]
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cg = 2*0.08*sqrt(kg*mg); % [N/(m/s)]
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%% Save model parameters
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save('./mat/uniaxial_micro_station_parameters.mat', 'mh', 'mt', 'mg', 'ch', 'ct', 'cg', 'kh', 'kt', 'kg')
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%% Disable the Nano-Hexpod for now
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model_config = struct();
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model_config.nhexa = "none";
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model_config.controller = "open_loop";
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%% Identify the transfer function from u to taum
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/micro_station/Fg'], 1, 'openinput'); io_i = io_i + 1; % Hammer on Granite
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io(io_i) = linio([mdl, '/micro_station/Fh'], 1, 'openinput'); io_i = io_i + 1; % Hammer on Hexapod
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io(io_i) = linio([mdl, '/micro_station/xg'], 1, 'openoutput'); io_i = io_i + 1; % Absolute motion of Granite
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io(io_i) = linio([mdl, '/micro_station/xh'], 1, 'openoutput'); io_i = io_i + 1; % Absolute motion of Hexapod
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%% Perform the model extraction
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G_id = linearize(mdl, io, 0.0);
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G_id.InputName = {'Fg', 'Fh'};
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G_id.OutputName = {'Dg', 'Dh'};
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% Comparison of the model and measurements
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% The comparison between the measurements and the model is done in Figure ref:fig:uniaxial_comp_frf_meas_model.
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% As the model is simplistic, the goal is not to match exactly the measurement but to have a first approximation.
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% More accurate models will be used later on.
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%% Comparison of the measured FRF and identified ones from the uni-axial model
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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(f(f>20), abs(frf_Fhz_to_Dhz(f>20)), '-', 'color', colors(1,:), 'DisplayName', '$D_{h,z}/F_{h,z}$');
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plot(f(f>20), abs(frf_Fgz_to_Dhz(f>20)), '-', 'color', colors(2,:), 'DisplayName', '$D_{h,z}/F_{g,z}$');
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plot(f(f>20), abs(frf_Fgz_to_Dgz(f>20)), '-', 'color', colors(3,:), 'DisplayName', '$D_{g,z}/F_{g,z}$');
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plot(freqs, abs(squeeze(freqresp(G_id('Dh', 'Fh'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$D_{h,z}/F_{h,z}$ (model)');
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plot(freqs, abs(squeeze(freqresp(G_id('Dh', 'Fg'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '$D_{h,z}/F_{g,z}$ (model)');
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plot(freqs, abs(squeeze(freqresp(G_id('Dg', 'Fg'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'DisplayName', '$D_{g,z}/F_{g,z}$ (model)');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ylim([1e-10, 2e-7]);
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legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2);
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_id('Dh', 'Fh'), freqs, 'Hz')))), '--', 'color', colors(1,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_id('Dh', 'Fg'), freqs, 'Hz')))), '--', 'color', colors(2,:));
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_id('Dg', 'Fg'), freqs, 'Hz')))), '--', 'color', colors(3,:));
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plot(f(f>20), 180/pi*unwrap(angle(frf_Fhx_to_Dhx(f>20))), '-', 'color', colors(1,:));
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plot(f(f>30), 180/pi*unwrap(angle(frf_Fgx_to_Dhx(f>30))), '-', 'color', colors(2,:));
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plot(f(f>20), 180/pi*unwrap(angle(frf_Fgx_to_Dgx(f>20))), '-', '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([-360, 90]);
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linkaxes([ax1,ax2],'x');
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xlim([1, 500]);
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matlab/uniaxial_2_nano_hexapod_model.m
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matlab/uniaxial_2_nano_hexapod_model.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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%% Uniaxial Simscape model name
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mdl = 'nass_uniaxial_model';
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%% Frequency Vector [Hz]
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freqs = logspace(0, 3, 1000);
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%% Load the micro-station parameters
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load('uniaxial_micro_station_parameters.mat')
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% Nano-Hexapod Parameters
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% The parameters for the nano-hexapod and sample are:
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% - $m_s$ the sample mass that can vary from 1kg up to 50kg
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% - $m_n$ the nano-hexapod mass which is set to 15kg
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% - $k_n$ the nano-hexapod stiffness, which can vary depending on the chosen architecture/technology
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% As a first example, let's choose a nano-hexapod stiffness of $10\,N/\mu m$ and a sample mass of 10kg.
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%% Nano-Hexapod Parameters
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mn = 15; % [kg]
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kn = 1e7; % [N/m]
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cn = 2*0.01*sqrt(mn*kn); % [N/(m/s)]
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%% Sample Mass
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ms = 10; % [kg]
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% Obtained Dynamics
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% The sensitivity to disturbances (i.e. $x_f$, $f_t$ and $f_s$) are shown in Figure ref:fig:uniaxial_sensitivity_dist_first_params.
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% The /plant/ (i.e. the transfer function from actuator force $f$ to measured displacement $d$) is shown in Figure ref:fig:uniaxial_plant_first_params.
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% For further analysis, 9 configurations are considered: three nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) combined with three sample's masses ($m_s = 1\,kg$, $m_s = 25\,kg$ and $m_s = 50\,kg$).
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%% Use 1DoF Nano-Hexpod model
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model_config = struct();
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model_config.nhexa = "1dof";
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model_config.controller = "open_loop";
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%% Identify the transfer function from disturbances and force actuator to d
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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; % Force Actuator
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io(io_i) = linio([mdl, '/fs'], 1, 'openinput'); io_i = io_i + 1; % Force applied on the sample
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io(io_i) = linio([mdl, '/micro_station/xf'], 1, 'openinput'); io_i = io_i + 1; % Floor Motion
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io(io_i) = linio([mdl, '/micro_station/ft'], 1, 'openinput'); io_i = io_i + 1; % Stage disturbances
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io(io_i) = linio([mdl, '/d'] , 1, 'openoutput'); io_i = io_i + 1; % Metrology
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%% Perform the model extraction
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G_ol = linearize(mdl, io, 0.0);
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G_ol.InputName = {'f', 'fs', 'xf', 'ft'};
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G_ol.OutputName = {'d'};
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%% Sensitivity to disturbances
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figure;
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tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile();
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_ol('d', 'fs'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude $d/f_{s}$ [m/N]'); xlabel('Frequency [Hz]');
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xticks([1e0, 1e1, 1e2]);
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ax2 = nexttile();
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_ol('d', 'ft'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude $d/f_{t}$ [m/N]'); xlabel('Frequency [Hz]');
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xticks([1e0, 1e1, 1e2]);
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ax3 = nexttile();
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_ol('d', 'xf'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude $d/x_{f}$ [m/m]'); xlabel('Frequency [Hz]');
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xticks([1e0, 1e1, 1e2]);
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linkaxes([ax1,ax2,ax3],'x');
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xlim([1, 500]);
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% #+name: fig:uniaxial_sensitivity_dist_first_params
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% #+caption: Sensitivity to disturbances
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% #+RESULTS:
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% [[file:figs/uniaxial_sensitivity_dist_first_params.png]]
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%% Bode Plot of the transfer function from actuator forces to measured displacement by the metrology
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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_ol('d', 'f'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude $d/f$ [m/N]'); set(gca, 'XTickLabel',[]);
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol('d', 'f'), freqs, 'Hz'))));
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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, 500]);
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% Identification of all combination of stiffnesses / masses :noexport:
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%% Use 1DoF Nano-Hexpod model
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model_config = struct();
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model_config.nhexa = "1dof";
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model_config.controller = "open_loop";
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%% Nano-Hexapod Mass
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mn = 15; % Nano-Hexapod mass [kg]
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%% Identification of all combination of stiffnesses / masses
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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; % Actuator Force
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io(io_i) = linio([mdl, '/micro_station/xf'], 1, 'openinput'); io_i = io_i + 1; % Floor Motion
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io(io_i) = linio([mdl, '/micro_station/ft'], 1, 'openinput'); io_i = io_i + 1; % Stage vibrations
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io(io_i) = linio([mdl, '/fs'], 1, 'openinput'); io_i = io_i + 1; % Direct sample forces
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io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Sensor
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io(io_i) = linio([mdl, '/fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor
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io(io_i) = linio([mdl, '/vn'] , 1, 'openoutput'); io_i = io_i + 1; % Geophone
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io(io_i) = linio([mdl, '/d'] , 1, 'openoutput'); io_i = io_i + 1; % Metrology Output
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%% Light Sample
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ms = 1; % Sample Mass [kg]
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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*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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G_vc_light = linearize(mdl, io, 0.0);
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G_vc_light.InputName = {'f', 'xf', 'ft', 'fs'};
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G_vc_light.OutputName = {'dL', 'fm', 'vn', 'd'};
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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*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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G_md_light = linearize(mdl, io, 0.0);
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G_md_light.InputName = {'f', 'xf', 'ft', 'fs'};
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G_md_light.OutputName = {'dL', 'fm', 'vn', 'd'};
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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*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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G_pz_light = linearize(mdl, io, 0.0);
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G_pz_light.InputName = {'f', 'xf', 'ft', 'fs'};
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G_pz_light.OutputName = {'dL', 'fm', 'vn', 'd'};
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%% Mid Sample
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ms = 25; % Sample Mass [kg]
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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*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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G_vc_mid = linearize(mdl, io, 0.0);
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G_vc_mid.InputName = {'f', 'xf', 'ft', 'fs'};
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G_vc_mid.OutputName = {'dL', 'fm', 'vn', 'd'};
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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*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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G_md_mid = linearize(mdl, io, 0.0);
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G_md_mid.InputName = {'f', 'xf', 'ft', 'fs'};
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G_md_mid.OutputName = {'dL', 'fm', 'vn', 'd'};
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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*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
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||||
G_pz_mid = linearize(mdl, io, 0.0);
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G_pz_mid.InputName = {'f', 'xf', 'ft', 'fs'};
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||||
G_pz_mid.OutputName = {'dL', 'fm', 'vn', 'd'};
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%% Heavy Sample
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ms = 50; % Sample Mass [kg]
|
||||
|
||||
% Voice Coil (i.e. soft) Nano-Hexapod
|
||||
kn = 1e4; % Nano-Hexapod Stiffness [N/m]
|
||||
cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
|
||||
G_vc_heavy = linearize(mdl, io, 0.0);
|
||||
G_vc_heavy.InputName = {'f', 'xf', 'ft', 'fs'};
|
||||
G_vc_heavy.OutputName = {'dL', 'fm', 'vn', 'd'};
|
||||
|
||||
% APA (i.e. relatively stiff) Nano-Hexapod
|
||||
kn = 1e6; % Nano-Hexapod Stiffness [N/m]
|
||||
cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
|
||||
G_md_heavy = linearize(mdl, io, 0.0);
|
||||
G_md_heavy.InputName = {'f', 'xf', 'ft', 'fs'};
|
||||
G_md_heavy.OutputName = {'dL', 'fm', 'vn', 'd'};
|
||||
|
||||
% Piezoelectric (i.e. stiff) Nano-Hexapod
|
||||
kn = 1e8; % Nano-Hexapod Stiffness [N/m]
|
||||
cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
|
||||
G_pz_heavy = linearize(mdl, io, 0.0);
|
||||
G_pz_heavy.InputName = {'f', 'xf', 'ft', 'fs'};
|
||||
G_pz_heavy.OutputName = {'dL', 'fm', 'vn', 'd'};
|
||||
|
||||
%% Save All Identified Plants
|
||||
save('./mat/uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ...
|
||||
'G_vc_mid', 'G_md_mid', 'G_pz_mid', ...
|
||||
'G_vc_heavy', 'G_md_heavy', 'G_pz_heavy');
|
||||
183
matlab/uniaxial_3_disturbances.m
Normal file
183
matlab/uniaxial_3_disturbances.m
Normal file
@@ -0,0 +1,183 @@
|
||||
%% 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;
|
||||
|
||||
%% Uniaxial Simscape model name
|
||||
mdl = 'nass_uniaxial_model';
|
||||
|
||||
%% Frequency Vector [Hz]
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
%% Load the micro-station parameters
|
||||
load('uniaxial_micro_station_parameters.mat');
|
||||
|
||||
% Ground Motion
|
||||
% The geophone fixed to the floor to measure the floor motion.
|
||||
|
||||
|
||||
%% Load floor motion data
|
||||
% t: time in [s]
|
||||
% V: measured voltage genrated by the geophone and amplified by a 60dB gain voltage amplifier [V]
|
||||
load('ground_motion_measurement.mat', 't', 'V');
|
||||
|
||||
|
||||
|
||||
% The voltage generated by each geophone is amplified using a voltage amplifier (gain of 60dB) before going to the ADC.
|
||||
% The sensitivity of the geophone as well as the gain of the voltage amplifier are then taken into account to reconstruct the floor displacement.
|
||||
|
||||
%% Sensitivity of the geophone
|
||||
S0 = 88; % Sensitivity [V/(m/s)]
|
||||
f0 = 2; % Cut-off frequency [Hz]
|
||||
|
||||
S = S0*(s/2/pi/f0)/(1+s/2/pi/f0); % Geophone's transfer function [V/(m/s)]
|
||||
|
||||
%% Gain of the voltage amplifier
|
||||
G0_db = 60; % [dB]
|
||||
G0 = 10^(G0_db/20); % [abs]
|
||||
|
||||
%% Transfer function from measured voltage to displacement
|
||||
G_geo = 1/S/G0/s; % [m/V]
|
||||
|
||||
|
||||
|
||||
% The PSD $S_{V_f}$ of the measured voltage $V_f$ is computed.
|
||||
|
||||
%% Compute measured voltage PSD
|
||||
Fs = 1/(t(2)-t(1)); % Sampling Frequency [Hz]
|
||||
win = hanning(ceil(2*Fs)); % Hanning window
|
||||
|
||||
[psd_V, f] = pwelch(V, win, [], [], Fs); % [V^2/Hz]
|
||||
|
||||
|
||||
|
||||
% The PSD of the corresponding displacement can be computed as follows:
|
||||
% \begin{equation}
|
||||
% S_{x_f}(\omega) = \frac{S_{V_f}(\omega)}{|S_{\text{geo}}(j\omega)| \cdot G_{\text{amp}} \cdot \omega}
|
||||
% \end{equation}
|
||||
% with:
|
||||
% - $S_{\text{geo}}$ the sensitivity of the Geophone in $[Vs/m]$
|
||||
% - $G_{\text{amp}}$ the gain of the voltage amplifier
|
||||
% - $\omega$ is here to integrate and have the displacement instead of the velocity
|
||||
|
||||
|
||||
%% Ground Motion ASD
|
||||
psd_xf = psd_V.*abs(squeeze(freqresp(G_geo, f, 'Hz'))).^2; % [m^2/Hz]
|
||||
|
||||
|
||||
|
||||
% The amplitude spectral density $\Gamma_{x_f}$ of the measured displacement $x_f$ is shown in Figure ref:fig:asd_floor_motion_id31.
|
||||
|
||||
|
||||
%% Measured Amplitude Spectral Density of the Floor motion on ID31
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, sqrt(psd_xf), 'DisplayName', '$\Gamma_{x_{f}}$');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Ampl. Spectral Density $\left[\frac{m}{\sqrt{Hz}}\right]$')
|
||||
xlim([1, 500]);
|
||||
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
|
||||
% Stage Vibration
|
||||
% During Spindle rotation (here at 6rpm), the granite velocity and micro-hexapod's top platform velocity are measured with the geophones.
|
||||
|
||||
|
||||
%% Measured velocity of granite and hexapod during spindle rotation
|
||||
% t: time in [s]
|
||||
% vg: measured granite velocity [m/s]
|
||||
% vg: measured micro-hexapod's top platform velocity [m/s]
|
||||
load('meas_spindle_on.mat', 't', 'vg', 'vh');
|
||||
spindle_off = load('meas_spindle_off.mat', 't', 'vg', 'vh'); % No Rotation
|
||||
|
||||
|
||||
|
||||
% The Power Spectral Density of the relative velocity between the hexapod and the granite is computed.
|
||||
|
||||
%% Compute Power Spectral Density of the relative velocity between granite and hexapod during spindle rotation
|
||||
Fs = 1/(t(2)-t(1)); % Sampling Frequency [Hz]
|
||||
win = hanning(ceil(2*Fs)); % Hanning window
|
||||
|
||||
[psd_vft, f] = pwelch(vh-vg, win, [], [], Fs); % [(m/s)^2/Hz]
|
||||
[psd_off, ~] = pwelch(spindle_off.vh-spindle_off.vg, win, [], [], Fs); % [(m/s)^2/Hz]
|
||||
|
||||
|
||||
|
||||
% It is then integrated to obtain the Amplitude Spectral Density of the relative motion which is compared with a non-rotating case (Figure ref:fig:asd_vibration_spindle_rotation).
|
||||
% It is shown that the spindle rotation induces vibrations in a wide frequency spectrum.
|
||||
|
||||
|
||||
%% Measured Amplitude Spectral Density of the relative motion between the granite and the micro-hexapod's top platform during Spindle rotating
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, sqrt(psd_vft)./(2*pi*f), 'DisplayName', '6rpm');
|
||||
plot(f, sqrt(psd_off)./(2*pi*f), 'DisplayName', '0rpm');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Ampl. Spectral Density $\left[\frac{m}{\sqrt{Hz}}\right]$')
|
||||
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
xlim([1, 500]); ylim([1e-12, 1e-7])
|
||||
|
||||
|
||||
|
||||
% #+name: fig:asd_vibration_spindle_rotation
|
||||
% #+caption: Measured Amplitude Spectral Density of the relative motion between the granite and the micro-hexapod's top platform during Spindle rotating
|
||||
% #+RESULTS:
|
||||
% [[file:figs/asd_vibration_spindle_rotation.png]]
|
||||
|
||||
% In order to compute the equivalent disturbance force $f_t$ that induces such motion, the transfer function from $f_t$ to the relative motion of the hexapod's top platform and the granite is extracted from the model.
|
||||
|
||||
%% Disable the Nano-Hexpod for now
|
||||
model_config = struct();
|
||||
model_config.nhexa = "none";
|
||||
model_config.controller = "open_loop";
|
||||
|
||||
%% Identify the transfer function from u to taum
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/micro_station/ft'], 1, 'openinput'); io_i = io_i + 1; % Stage Disturbance Force
|
||||
io(io_i) = linio([mdl, '/micro_station/xg'], 1, 'openoutput'); io_i = io_i + 1; % Absolute motion of Granite
|
||||
io(io_i) = linio([mdl, '/micro_station/xh'], 1, 'openoutput'); io_i = io_i + 1; % Absolute motion of Hexapod
|
||||
|
||||
%% Perform the model extraction
|
||||
G = linearize(mdl, io, 0.0);
|
||||
G.InputName = {'ft'};
|
||||
G.OutputName = {'Dg', 'Dh'};
|
||||
|
||||
|
||||
|
||||
% The power spectral density $\Gamma_{f_{t}}$ of the disturbance force can be computed as follows:
|
||||
% \begin{equation}
|
||||
% \Gamma_{f_{t}}(\omega) = \frac{\Gamma_{v_{t}}(\omega)}{|G_{\text{model}}(j\omega)|^2}
|
||||
% \end{equation}
|
||||
% with:
|
||||
% - $\Gamma_{v_{t}}$ the measured power spectral density of the relative motion between the micro-hexapod's top platform and the granite during the spindle's rotation
|
||||
% - $G_{\text{model}}$ the transfer function (extracted from the uniaxial model) from $f_t$ to the relative motion between the micro-hexapod's top platform and the granite
|
||||
|
||||
|
||||
%% Power Spectral Density of the equivalent force ft
|
||||
psd_ft = (psd_vft./(2*pi*f).^2)./abs(squeeze(freqresp(G('Dh', 'ft') - G('Dg', 'ft'), f, 'Hz'))).^2;
|
||||
|
||||
|
||||
|
||||
% The obtained amplitude spectral density of the disturbance force $f_t$ is shown in Figure ref:fig:asd_disturbance_force.
|
||||
|
||||
%% Estimated disturbance force ft from measurement and uniaxial model
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, sqrt(psd_ft), 'DisplayName', '$\Gamma_{f_{t}}$');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Ampl. Spectral Density $\left[\frac{N}{\sqrt{Hz}}\right]$')
|
||||
xlim([1, 500]);
|
||||
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
|
||||
%% Save PSD of disturbances
|
||||
save('./mat/uniaxial_disturbance_psd.mat', 'f', 'psd_ft', 'psd_xf');
|
||||
150
matlab/uniaxial_4_dynamic_noise_budget.m
Normal file
150
matlab/uniaxial_4_dynamic_noise_budget.m
Normal file
@@ -0,0 +1,150 @@
|
||||
%% 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;
|
||||
|
||||
%% Frequency Vector [Hz]
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
%% Load the PSD of disturbances
|
||||
load('uniaxial_disturbance_psd.mat', 'f', 'psd_ft', 'psd_xf');
|
||||
|
||||
%% Load Plants Dynamics
|
||||
load('uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ...
|
||||
'G_vc_mid', 'G_md_mid', 'G_pz_mid', ...
|
||||
'G_vc_heavy', 'G_md_heavy', 'G_pz_heavy');
|
||||
|
||||
% Sensitivity to disturbances
|
||||
% From the Uni-axial model, the transfer function from the disturbances ($f_s$, $x_f$ and $f_t$) to the displacement $d$ are computed.
|
||||
|
||||
% This is done for *two extreme sample masses* $m_s = 1\,\text{kg}$ and $m_s = 50\,\text{kg}$ and *three nano-hexapod stiffnesses*:
|
||||
% - $k_n = 0.01\,N/\mu m$ that could represent a voice coil actuator with soft flexible guiding
|
||||
% - $k_n = 1\,N/\mu m$ that could represent a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator
|
||||
% - $k_n = 100\,N/\mu m$ that could represent a stiff piezoelectric stack actuator
|
||||
|
||||
% The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the light sample (same conclusions can be drawn with the heavy one).
|
||||
|
||||
% #+begin_important
|
||||
% From Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses, following can be concluded for the *soft nano-hexapod*:
|
||||
% - It is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to the lower stiffness
|
||||
% - Between the suspension mode of the nano-hexapod (here at 5Hz) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out.
|
||||
% - Above the suspension mode of the nano-hexapod, the sample's motion is unaffected by the floor motion, and therefore the sensitivity to floor motion is almost $1$.
|
||||
% #+end_important
|
||||
|
||||
|
||||
%% Sensitivity to disturbances for three different nano-hexpod stiffnesses
|
||||
figure;
|
||||
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_light('d', 'fs'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_light('d', 'fs'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_light('d', 'fs'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude $d/f_{s}$ [m/N]'); xlabel('Frequency [Hz]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
|
||||
ax2 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_light('d', 'ft'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_light('d', 'ft'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_light('d', 'ft'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude $d/f_{t}$ [m/N]'); xlabel('Frequency [Hz]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
|
||||
ax3 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_light('d', 'xf'), freqs, 'Hz'))), 'DisplayName', '$k_n = 0.01\,N/\mu m$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_light('d', 'xf'), freqs, 'Hz'))), 'DisplayName', '$k_n = 1 \,N/\mu m$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_light('d', 'xf'), freqs, 'Hz'))), 'DisplayName', '$k_n = 100 \,N/\mu m$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude $d/x_{f}$ [m/m]'); xlabel('Frequency [Hz]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
|
||||
linkaxes([ax1,ax2,ax3],'x');
|
||||
xlim([1, 500]);
|
||||
|
||||
% Open-Loop Dynamic Noise Budgeting
|
||||
% Now, the power spectral density of the disturbances is taken into account to estimate the residual motion $d$ in each case.
|
||||
|
||||
% The Cumulative Amplitude Spectrum of the relative motion $d$ due to both the floor motion $x_f$ and the stage vibrations $f_t$ are shown in Figure ref:fig:uniaxial_cas_d_disturbances_stiffnesses for the three nano-hexapod stiffnesses.
|
||||
|
||||
% It is shown that the effect of the floor motion is much less than the stage vibrations, except for the soft nano-hexapod below 5Hz.
|
||||
|
||||
|
||||
%% Cumulative Amplitude Spectrum of the relative motion d, due to both the floor motion and the stage vibrations
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_vc_light('d', 'ft'), f, 'Hz'))).^2)))), '-', 'color', colors(1,:), 'DisplayName', '$f_t$');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_md_light('d', 'ft'), f, 'Hz'))).^2)))), '-', 'color', colors(2,:), 'DisplayName', '$f_t$');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_pz_light('d', 'ft'), f, 'Hz'))).^2)))), '-', 'color', colors(3,:), 'DisplayName', '$f_t$');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_xf.*abs(squeeze(freqresp(G_vc_light('d', 'xf'), f, 'Hz'))).^2)))), '--', 'color', colors(1,:), 'DisplayName', '$x_f$ ($k_n = 0.01\,N/\mu m$)');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_xf.*abs(squeeze(freqresp(G_md_light('d', 'xf'), f, 'Hz'))).^2)))), '--', 'color', colors(2,:), 'DisplayName', '$x_f$ ($k_n = 1 \,N/\mu m$)');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_xf.*abs(squeeze(freqresp(G_pz_light('d', 'xf'), f, 'Hz'))).^2)))), '--', 'color', colors(3,:), 'DisplayName', '$x_f$ ($k_n = 100 \,N/\mu m$)');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Cumulative Ampl. Spectrum [m]'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
|
||||
|
||||
xlim([1, 500]);
|
||||
ylim([1e-12, 1e-6])
|
||||
|
||||
|
||||
|
||||
% #+name: fig:uniaxial_cas_d_disturbances_stiffnesses
|
||||
% #+caption: Cumulative Amplitude Spectrum of the relative motion d, due to both the floor motion and the stage vibrations (light sample)
|
||||
% #+RESULTS:
|
||||
% [[file:figs/uniaxial_cas_d_disturbances_stiffnesses.png]]
|
||||
|
||||
% The total cumulative amplitude spectrum for the three nano-hexapod stiffnesses and for the two sample's masses are shown in Figure ref:fig:uniaxial_cas_d_disturbances_payload_masses.
|
||||
% The conclusion is that the sample's mass has little effect on the cumulative amplitude spectrum of the relative motion $d$.
|
||||
|
||||
|
||||
%% Cumulative Amplitude Spectrum of the relative motion d due to all disturbances, for two sample masses
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_vc_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_vc_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(1,:), 'DisplayName', '$m_s = 1\,kg$');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_md_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_md_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(2,:), 'DisplayName', '$m_s = 1\,kg$');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_pz_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_pz_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(3,:), 'DisplayName', '$m_s = 1\,kg$');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_vc_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_vc_heavy('d', 'xf'), f, 'Hz'))).^2)))), '--', ...
|
||||
'color', colors(1,:), 'DisplayName', '$m_s = 50\,kg$ ($k_n = 0.01\,N/\mu m$)');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_md_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_md_heavy('d', 'xf'), f, 'Hz'))).^2)))), '--', ...
|
||||
'color', colors(2,:), 'DisplayName', '$m_s = 50\,kg$ ($k_n = 1\,N/\mu m$)');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_pz_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_pz_heavy('d', 'xf'), f, 'Hz'))).^2)))), '--', ...
|
||||
'color', colors(3,:), 'DisplayName', '$m_s = 50\,kg$ ($k_n = 100\,N/\mu m$)');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Cumulative Ampl. Spectrum [m]'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
|
||||
|
||||
xlim([1, 500]);
|
||||
ylim([1e-11, 3e-6])
|
||||
908
matlab/uniaxial_5_active_damping.m
Normal file
908
matlab/uniaxial_5_active_damping.m
Normal file
@@ -0,0 +1,908 @@
|
||||
%% 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;
|
||||
|
||||
%% Frequency Vector [Hz]
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
%% Load the PSD of disturbances
|
||||
load('uniaxial_disturbance_psd.mat', 'f', 'psd_ft', 'psd_xf');
|
||||
|
||||
%% Load Plants Dynamics
|
||||
load('uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ...
|
||||
'G_vc_mid', 'G_md_mid', 'G_pz_mid', ...
|
||||
'G_vc_heavy', 'G_md_heavy', 'G_pz_heavy');
|
||||
|
||||
% Plant Dynamics for Active Damping
|
||||
% The plant dynamics for all three active damping techniques are shown in Figure ref:fig:uniaxial_plant_active_damping_techniques.
|
||||
% All have *alternating poles and zeros* meaning that the phase is bounded to $\pm 90\,\text{deg}$ which makes the controller very robust.
|
||||
|
||||
% When the nano-hexapod's suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the transfer functions from IFF and DVF.
|
||||
|
||||
% For the stiff nano-hexapod, the micro-station dynamics can be seen on the transfer functions in Figure ref:fig:uniaxial_plant_active_damping_techniques.
|
||||
% Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used.
|
||||
|
||||
|
||||
%% Damped plants for three considered payload masses - Comparison of active damping techniques
|
||||
figure;
|
||||
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_light('fm', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$m_s = 1\,kg$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_mid( 'fm', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'DisplayName', '$m_s = 25\,kg$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_heavy('fm', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$m_s = 50\,kg$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_light('fm', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid( 'fm', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_heavy('fm', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_light('fm', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_mid( 'fm', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_heavy('fm', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('IFF: $f_m/f$');
|
||||
ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
ldg.ItemTokenSize = [20, 1];
|
||||
|
||||
ax2 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_light('dL', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_mid( 'dL', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_heavy('dL', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_light('dL', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid( 'dL', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_heavy('dL', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_light('dL', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_mid( 'dL', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_heavy('dL', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('RDC: $d\mathcal{L}/f$');
|
||||
|
||||
ax3 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_light('vn', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$k_n = 0.01\,N/\mu m$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_mid( 'vn', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_heavy('vn', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_light('vn', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$k_n = 1\,N/\mu m$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid( 'vn', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_heavy('vn', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_light('vn', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$k_n = 100\,N/\mu m$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_mid( 'vn', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_heavy('vn', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/s/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('DVF: $v_n/f$');
|
||||
ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
ldg.ItemTokenSize = [20, 1];
|
||||
|
||||
ax1b = nexttile();
|
||||
hold on;
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_light('fm', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid( 'fm', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_heavy('fm', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('fm', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid( 'fm', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_heavy('fm', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('fm', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid( 'fm', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_heavy('fm', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
yticks(-360:90:360);
|
||||
ylim([-200, 20]);
|
||||
|
||||
ax2b = nexttile();
|
||||
hold on;
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_light('dL', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid( 'dL', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_heavy('dL', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('dL', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid( 'dL', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_heavy('dL', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('dL', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid( 'dL', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_heavy('dL', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
yticks(-360:90:360);
|
||||
ylim([-200, 20]);
|
||||
|
||||
ax3b = nexttile();
|
||||
hold on;
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_light('vn', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid( 'vn', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_heavy('vn', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('vn', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid( 'vn', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_heavy('vn', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('vn', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid( 'vn', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_heavy('vn', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
yticks(-360:90:360);
|
||||
ylim([-110, 110]);
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax1b,ax2b,ax3b],'x');
|
||||
xlim([1, 1000]);
|
||||
|
||||
% Achievable Damping - Root Locus
|
||||
% The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques.
|
||||
% They are shown in Figure ref:fig:uniaxial_root_locus_damping_techniques.
|
||||
|
||||
% All three active damping approach can lead to *critical damping* of the nano-hexapod suspension mode.
|
||||
|
||||
% There is even a little bit of authority on micro-station modes with IFF and DVF applied on the stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques, right) and with RDC for a soft nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode).
|
||||
% This can be explained by the fact that above the suspension mode of the soft nano-hexapod, the relative motion sensor acts as an inertial sensor for the micro-station top platform. Therefore, it is like DVF was applied (the nano-hexapod acts as a geophone!).
|
||||
|
||||
|
||||
%% Active Damping Robustness to change of sample's mass - Root Locus for all three damping techniques with 3 different sample's masses
|
||||
figure;
|
||||
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
%% Soft Nano-Hexapod
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
% IFF
|
||||
plot(real(pole(G_vc_light('fm', 'f'))), imag(pole(G_vc_light('fm', 'f'))), 'x', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_vc_light('fm', 'f'))), imag(zero(G_vc_light('fm', 'f'))), 'o', 'color', colors(1,:), ...
|
||||
'DisplayName', 'IFF');
|
||||
for g = logspace(0, 2, 400)
|
||||
clpoles = pole(feedback(G_vc_light('fm', 'f'), g/s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
% RDC
|
||||
plot(real(pole(G_vc_light('dL', 'f'))), imag(pole(G_vc_light('dL', 'f'))), 'x', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_vc_light('dL', 'f'))), imag(zero(G_vc_light('dL', 'f'))), 'o', 'color', colors(2,:), ...
|
||||
'DisplayName', 'RDC');
|
||||
for g = logspace(1,3,400)
|
||||
clpoles = pole(feedback(G_vc_light('dL', 'f'), -g*s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
% DVF
|
||||
plot(real(pole(G_vc_light('vn', 'f'))), imag(pole(G_vc_light('vn', 'f'))), 'x', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_vc_light('vn', 'f'))), imag(zero(G_vc_light('vn', 'f'))), 'o', 'color', colors(3,:), ...
|
||||
'DisplayName', 'DVF');
|
||||
for g = logspace(1,3,400)
|
||||
clpoles = pole(feedback(G_vc_light('vn', 'f'), -g, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
title('$k_n = 0.01\,N/\mu m$')
|
||||
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
ldg.ItemTokenSize = [10, 1];
|
||||
xlim([-30, 0]); ylim([0, 30]);
|
||||
|
||||
%% Medium-Stiff Nano-Hexapod
|
||||
ax2 = nexttile();
|
||||
hold on;
|
||||
% IFF
|
||||
plot(real(pole(G_md_light('fm', 'f'))), imag(pole(G_md_light('fm', 'f'))), 'x', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_md_light('fm', 'f'))), imag(zero(G_md_light('fm', 'f'))), 'o', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(0,3,400)
|
||||
clpoles = pole(feedback(G_md_light('fm', 'f'), g/s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
% RDC
|
||||
plot(real(pole(G_md_light('dL', 'f'))), imag(pole(G_md_light('dL', 'f'))), 'x', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_md_light('dL', 'f'))), imag(zero(G_md_light('dL', 'f'))), 'o', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,4,400)
|
||||
clpoles = pole(feedback(G_md_light('dL', 'f'), -g*s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
% DVF
|
||||
plot(real(pole(G_md_light('vn', 'f'))), imag(pole(G_md_light('vn', 'f'))), 'x', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_md_light('vn', 'f'))), imag(zero(G_md_light('vn', 'f'))), 'o', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,4,400)
|
||||
clpoles = pole(feedback(G_md_light('vn', 'f'), -g, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
title('$k_n = 1\,N/\mu m$')
|
||||
xlim([-300, 0]); ylim([0, 300]);
|
||||
|
||||
%% Stiff Nano-Hexapod
|
||||
ax3 = nexttile();
|
||||
hold on;
|
||||
% IFF
|
||||
plot(real(pole(G_pz_light('fm', 'f'))), imag(pole(G_pz_light('fm', 'f'))), 'x', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_pz_light('fm', 'f'))), imag(zero(G_pz_light('fm', 'f'))), 'o', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,5,400)
|
||||
clpoles = pole(feedback(G_pz_light('fm', 'f'), g/s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
% RDC
|
||||
plot(real(pole(G_pz_light('dL', 'f'))), imag(pole(G_pz_light('dL', 'f'))), 'x', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_pz_light('dL', 'f'))), imag(zero(G_pz_light('dL', 'f'))), 'o', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(3,6,400)
|
||||
clpoles = pole(feedback(G_pz_light('dL', 'f'), -g*s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
% DVF
|
||||
plot(real(pole(G_pz_light('vn', 'f'))), imag(pole(G_pz_light('vn', 'f'))), 'x', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_pz_light('vn', 'f'))), imag(zero(G_pz_light('vn', 'f'))), 'o', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(3,6,400)
|
||||
clpoles = pole(feedback(G_pz_light('vn', 'f'), -g, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
title('$k_n = 100\,N/\mu m$')
|
||||
xlim([-4000, 0]); ylim([0, 4000]);
|
||||
|
||||
|
||||
|
||||
% #+name: fig:uniaxial_root_locus_damping_techniques
|
||||
% #+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses.
|
||||
% #+RESULTS:
|
||||
% [[file:figs/uniaxial_root_locus_damping_techniques.png]]
|
||||
|
||||
|
||||
%% Root Locus for the three damping techniques
|
||||
figure;
|
||||
|
||||
hold on;
|
||||
% IFF
|
||||
plot(real(pole(G_md_mid('fm', 'f'))), imag(pole(G_md_mid('fm', 'f'))), 'x', 'color', colors(1,:), ...
|
||||
'DisplayName', 'IFF');
|
||||
plot(real(zero(G_md_mid('fm', 'f'))), imag(zero(G_md_mid('fm', 'f'))), 'o', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(1,4,500)
|
||||
clpoles = pole(feedback(G_md_mid('fm', 'f'), g/s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
% RDC
|
||||
plot(real(pole(G_md_mid('dL', 'f'))), imag(pole(G_md_mid('dL', 'f'))), 'x', 'color', colors(2,:), ...
|
||||
'DisplayName', 'RDC');
|
||||
plot(real(zero(G_md_mid('dL', 'f'))), imag(zero(G_md_mid('dL', 'f'))), 'o', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,5,500)
|
||||
clpoles = pole(feedback(G_md_mid('dL', 'f'), -g*s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
% DVF
|
||||
plot(real(pole(G_md_mid('vn', 'f'))), imag(pole(G_md_mid('vn', 'f'))), 'x', 'color', colors(3,:), ...
|
||||
'DisplayName', 'DVF');
|
||||
plot(real(zero(G_md_mid('vn', 'f'))), imag(zero(G_md_mid('vn', 'f'))), 'o', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,5,500)
|
||||
clpoles = pole(feedback(G_md_mid('vn', 'f'), -tf(g), +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
xlim([-2100, 0]); ylim([0, 2100]);
|
||||
axis square;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
ldg.ItemTokenSize = [10, 1];
|
||||
|
||||
% Active Damping Controller Optimization and Damped plants :noexport:
|
||||
|
||||
%% Design of Active Damping controllers to have reasonable damping
|
||||
% IFF
|
||||
K_iff_vc = 20/(s + 2*pi*0.01);
|
||||
K_iff_vc.InputName = {'fm'};
|
||||
K_iff_vc.OutputName = {'f'};
|
||||
K_iff_md = 200/(s + 2*pi*0.01);
|
||||
K_iff_md.InputName = {'fm'};
|
||||
K_iff_md.OutputName = {'f'};
|
||||
K_iff_pz = 4000/(s + 2*pi*0.01);
|
||||
K_iff_pz.InputName = {'fm'};
|
||||
K_iff_pz.OutputName = {'f'};
|
||||
|
||||
% RDC
|
||||
K_rdc_vc = -1e3*s;
|
||||
K_rdc_vc.InputName = {'dL'};
|
||||
K_rdc_vc.OutputName = {'f'};
|
||||
K_rdc_md = -1e4*s;
|
||||
K_rdc_md.InputName = {'dL'};
|
||||
K_rdc_md.OutputName = {'f'};
|
||||
K_rdc_pz = -1e5*s;
|
||||
K_rdc_pz.InputName = {'dL'};
|
||||
K_rdc_pz.OutputName = {'f'};
|
||||
|
||||
% DVF
|
||||
K_dvf_vc = -tf(1e3);
|
||||
K_dvf_vc.InputName = {'vn'};
|
||||
K_dvf_vc.OutputName = {'f'};
|
||||
K_dvf_md = -tf(8e3);
|
||||
K_dvf_md.InputName = {'vn'};
|
||||
K_dvf_md.OutputName = {'f'};
|
||||
K_dvf_pz = -tf(2e5);
|
||||
K_dvf_pz.InputName = {'vn'};
|
||||
K_dvf_pz.OutputName = {'f'};
|
||||
|
||||
%% Save Active Damping Controller
|
||||
save('./mat/uniaxial_active_damping_controllers.mat', 'K_iff_vc', 'K_iff_md', 'K_iff_pz', ...
|
||||
'K_rdc_vc', 'K_rdc_md', 'K_rdc_pz', ...
|
||||
'K_dvf_vc', 'K_dvf_md', 'K_dvf_pz');
|
||||
|
||||
%% Compute Damped Plants
|
||||
% IFF
|
||||
G_iff_vc_light = feedback(G_vc_light, K_iff_vc, 'name', +1);
|
||||
G_iff_vc_mid = feedback(G_vc_mid , K_iff_vc, 'name', +1);
|
||||
G_iff_vc_heavy = feedback(G_vc_heavy, K_iff_vc, 'name', +1);
|
||||
G_iff_md_light = feedback(G_md_light, K_iff_md, 'name', +1);
|
||||
G_iff_md_mid = feedback(G_md_mid , K_iff_md, 'name', +1);
|
||||
G_iff_md_heavy = feedback(G_md_heavy, K_iff_md, 'name', +1);
|
||||
G_iff_pz_light = feedback(G_pz_light, K_iff_pz, 'name', +1);
|
||||
G_iff_pz_mid = feedback(G_pz_mid , K_iff_pz, 'name', +1);
|
||||
G_iff_pz_heavy = feedback(G_pz_heavy, K_iff_pz, 'name', +1);
|
||||
|
||||
% RDC
|
||||
G_rdc_vc_light = feedback(G_vc_light, K_rdc_vc, 'name', +1);
|
||||
G_rdc_vc_mid = feedback(G_vc_mid , K_rdc_vc, 'name', +1);
|
||||
G_rdc_vc_heavy = feedback(G_vc_heavy, K_rdc_vc, 'name', +1);
|
||||
G_rdc_md_light = feedback(G_md_light, K_rdc_md, 'name', +1);
|
||||
G_rdc_md_mid = feedback(G_md_mid , K_rdc_md, 'name', +1);
|
||||
G_rdc_md_heavy = feedback(G_md_heavy, K_rdc_md, 'name', +1);
|
||||
G_rdc_pz_light = feedback(G_pz_light, K_rdc_pz, 'name', +1);
|
||||
G_rdc_pz_mid = feedback(G_pz_mid , K_rdc_pz, 'name', +1);
|
||||
G_rdc_pz_heavy = feedback(G_pz_heavy, K_rdc_pz, 'name', +1);
|
||||
|
||||
% DVF
|
||||
G_dvf_vc_light = feedback(G_vc_light, K_dvf_vc, 'name', +1);
|
||||
G_dvf_vc_mid = feedback(G_vc_mid , K_dvf_vc, 'name', +1);
|
||||
G_dvf_vc_heavy = feedback(G_vc_heavy, K_dvf_vc, 'name', +1);
|
||||
G_dvf_md_light = feedback(G_md_light, K_dvf_md, 'name', +1);
|
||||
G_dvf_md_mid = feedback(G_md_mid , K_dvf_md, 'name', +1);
|
||||
G_dvf_md_heavy = feedback(G_md_heavy, K_dvf_md, 'name', +1);
|
||||
G_dvf_pz_light = feedback(G_pz_light, K_dvf_pz, 'name', +1);
|
||||
G_dvf_pz_mid = feedback(G_pz_mid , K_dvf_pz, 'name', +1);
|
||||
G_dvf_pz_heavy = feedback(G_pz_heavy, K_dvf_pz, 'name', +1);
|
||||
|
||||
%% Verify Stability
|
||||
% IFF
|
||||
isstable(G_iff_vc_light) && isstable(G_iff_vc_mid) && isstable(G_iff_vc_heavy) && ...
|
||||
isstable(G_iff_md_light) && isstable(G_iff_md_mid) && isstable(G_iff_md_heavy) && ...
|
||||
isstable(G_iff_pz_light) && isstable(G_iff_pz_mid) && isstable(G_iff_pz_heavy)
|
||||
|
||||
% RDC
|
||||
isstable(G_rdc_vc_light) && isstable(G_rdc_vc_mid) && isstable(G_rdc_vc_heavy) && ...
|
||||
isstable(G_rdc_md_light) && isstable(G_rdc_md_mid) && isstable(G_rdc_md_heavy) && ...
|
||||
isstable(G_rdc_pz_light) && isstable(G_rdc_pz_mid) && isstable(G_rdc_pz_heavy)
|
||||
|
||||
% DVF
|
||||
isstable(G_dvf_vc_light) && isstable(G_dvf_vc_mid) && isstable(G_dvf_vc_heavy) && ...
|
||||
isstable(G_dvf_md_light) && isstable(G_dvf_md_mid) && isstable(G_dvf_md_heavy) && ...
|
||||
isstable(G_dvf_pz_light) && isstable(G_dvf_pz_mid) && isstable(G_dvf_pz_heavy)
|
||||
|
||||
%% Save Active Damping Controller
|
||||
save('./mat/uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_iff_pz_light', ...
|
||||
'G_rdc_vc_light', 'G_rdc_md_light', 'G_rdc_pz_light', ...
|
||||
'G_dvf_vc_light', 'G_dvf_md_light', 'G_dvf_pz_light', ...
|
||||
'G_iff_vc_mid', 'G_iff_md_mid', 'G_iff_pz_mid', ...
|
||||
'G_rdc_vc_mid', 'G_rdc_md_mid', 'G_rdc_pz_mid', ...
|
||||
'G_dvf_vc_mid', 'G_dvf_md_mid', 'G_dvf_pz_mid', ...
|
||||
'G_iff_vc_heavy', 'G_iff_md_heavy', 'G_iff_pz_heavy', ...
|
||||
'G_rdc_vc_heavy', 'G_rdc_md_heavy', 'G_rdc_pz_heavy', ...
|
||||
'G_dvf_vc_heavy', 'G_dvf_md_heavy', 'G_dvf_pz_heavy');
|
||||
|
||||
% Change of sensitivity to disturbances
|
||||
% The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure ref:fig:uniaxial_sensitivity_dist_active_damping.
|
||||
% The comparison is done with the nano-hexapod having a stiffness $k_n = 1\,N/\mu m$.
|
||||
|
||||
% #+begin_important
|
||||
% Conclusions from Figure ref:fig:uniaxial_sensitivity_dist_active_damping are:
|
||||
% - IFF degrades the sensitivity to direct forces on the sample (i.e. the compliance) below the resonance of the nano-hexapod
|
||||
% - RDC degrades the sensitivity to stage vibrations around the nano-hexapod's resonance as compared to the other two methods
|
||||
% - both IFF and DVF degrades the sensitivity to floor motion below the resonance of the nano-hexapod
|
||||
% #+end_important
|
||||
|
||||
|
||||
%% Change of sensitivity to disturbance with all three active damping strategies
|
||||
figure;
|
||||
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'fs'), freqs, 'Hz'))), 'k-');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'fs'), freqs, 'Hz'))), 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid('d', 'fs'), freqs, 'Hz'))), 'color', colors(2,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid('d', 'fs'), freqs, 'Hz'))), 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude $d/f_{s}$ [m/N]'); xlabel('Frequency [Hz]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
|
||||
ax2 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'ft'), freqs, 'Hz'))), 'k-');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'ft'), freqs, 'Hz'))), 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid('d', 'ft'), freqs, 'Hz'))), 'color', colors(2,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid('d', 'ft'), freqs, 'Hz'))), 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude $d/f_{t}$ [m/N]'); xlabel('Frequency [Hz]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
|
||||
ax3 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'xf'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'xf'), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', 'IFF');
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid('d', 'xf'), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', 'RDC');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid('d', 'xf'), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', 'DVF');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude $d/x_{f}$ [m/m]'); xlabel('Frequency [Hz]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
|
||||
linkaxes([ax1,ax2,ax3],'x');
|
||||
xlim([1, 500]);
|
||||
|
||||
% Noise Budgeting after Active Damping
|
||||
% Cumulative Amplitude Spectrum of the distance $d$ with all three active damping techniques are compared in Figure ref:fig:uniaxial_cas_active_damping.
|
||||
% All three active damping methods are giving similar results (except the RDC which is a little bit worse for the stiff nano-hexapod).
|
||||
|
||||
% Compare to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
|
||||
|
||||
|
||||
%% Cumulative Amplitude Spectrum of the distance d with all three active damping techniques
|
||||
figure;
|
||||
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', 'black', 'DisplayName', 'OL');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(1,:), 'DisplayName', 'IFF');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_rdc_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_rdc_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(2,:), 'DisplayName', 'RDC');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_dvf_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_dvf_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(3,:), 'DisplayName', 'DVF');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('CAS of $d$ [m]'); xlabel('Frequency [Hz]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
title('$k_n = 0.01\,N/\mu m$')
|
||||
|
||||
ax2 = nexttile();
|
||||
hold on;
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', 'black', 'DisplayName', 'OL');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(1,:), 'DisplayName', 'IFF');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_rdc_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_rdc_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(2,:), 'DisplayName', 'RDC');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_dvf_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_dvf_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(3,:), 'DisplayName', 'DVF');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
title('$k_n = 1\,N/\mu m$')
|
||||
|
||||
ax3 = nexttile();
|
||||
hold on;
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', 'black', 'DisplayName', 'OL');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(1,:), 'DisplayName', 'IFF');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_rdc_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_rdc_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(2,:), 'DisplayName', 'RDC');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_dvf_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_dvf_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', colors(3,:), 'DisplayName', 'DVF');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
title('$k_n = 100\,N/\mu m$')
|
||||
legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
|
||||
linkaxes([ax1,ax2,ax3], 'xy')
|
||||
xlim([1, 500]);
|
||||
ylim([2e-10, 3e-6])
|
||||
|
||||
% Obtained Damped Plant
|
||||
% The transfer functions from the plant input $f$ to the relative displacement $d$ while the active damping is implemented are shown in Figure ref:fig:uniaxial_damped_plant_three_active_damping_techniques.
|
||||
% All three active damping techniques yield similar damped plants.
|
||||
|
||||
|
||||
%% Obtained damped transfer function from f to d for the three damping techniques
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'f'), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', 'IFF');
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid('d', 'f'), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', 'RDC');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid('d', 'f'), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', 'DVF');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude $d/f$ [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
|
||||
ax2 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz')))), 'k-');
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_mid('d', 'f'), freqs, 'Hz')))), 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_md_mid('d', 'f'), freqs, 'Hz')))), 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_md_mid('d', 'f'), freqs, 'Hz')))), 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
yticks(-360:90:360);
|
||||
ylim([-270, 90]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([1, 500]);
|
||||
|
||||
|
||||
|
||||
% #+name: fig:uniaxial_damped_plant_three_active_damping_techniques
|
||||
% #+caption: Obtained damped transfer function from f to d for the three damping techniques
|
||||
% #+RESULTS:
|
||||
% [[file:figs/uniaxial_damped_plant_three_active_damping_techniques.png]]
|
||||
|
||||
% The damped plants are shown in Figure ref:fig:uniaxial_damped_plant_change_sample_mass for all three techniques, with the three considered nano-hexapod stiffnesses and sample's masses.
|
||||
|
||||
%% Damped plant - Robustness to change of sample's mass
|
||||
figure;
|
||||
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$k_n = 0.01\,N/\mu m$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_vc_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$k_n = 1\,N/\mu m$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$k_n = 100\,N/\mu m$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_pz_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('IFF');
|
||||
ylim([5e-10, 1e-3]);
|
||||
ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
ldg.ItemTokenSize = [20, 1];
|
||||
|
||||
ax2 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_vc_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_vc_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_vc_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_md_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_md_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_md_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_pz_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_pz_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_rdc_pz_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
|
||||
title('RDC');
|
||||
ylim([5e-10, 1e-3]);
|
||||
|
||||
ax3 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_vc_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$m_s = 1\,kg$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_vc_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(1,:), 'DisplayName', '$m_s = 25\,kg$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_vc_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$m_s = 50\,kg$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_md_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_md_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_md_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_pz_light('d', 'f'), freqs, 'Hz'))), '-', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_pz_mid( 'd', 'f'), freqs, 'Hz'))), '-.', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dvf_pz_heavy('d', 'f'), freqs, 'Hz'))), '--', 'color', colors(3,:), 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
|
||||
title('DVF');
|
||||
ylim([5e-10, 1e-3]);
|
||||
ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
ldg.ItemTokenSize = [20, 1];
|
||||
|
||||
ax1b = nexttile();
|
||||
hold on;
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
yticks(-360:90:360);
|
||||
ylim([-200, 20]);
|
||||
|
||||
ax2b = nexttile();
|
||||
hold on;
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_vc_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_vc_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_vc_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_md_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_md_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_md_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_pz_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_pz_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_rdc_pz_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
yticks(-360:90:360);
|
||||
ylim([-200, 20]);
|
||||
|
||||
ax3b = nexttile();
|
||||
hold on;
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_mid('d', 'f'), freqs, 'Hz')))), '-', 'color', [0, 0, 0, 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_vc_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_vc_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_vc_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(1,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_md_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_md_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_md_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(2,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_pz_light('d', 'f'), freqs, 'Hz')))), '-', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_pz_mid( 'd', 'f'), freqs, 'Hz')))), '-.', 'color', colors(3,:));
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_pz_heavy('d', 'f'), freqs, 'Hz')))), '--', 'color', colors(3,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
yticks(-360:90:360);
|
||||
ylim([-200, 20]);
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax1b,ax2b,ax3b],'x');
|
||||
xlim([1, 1e3]);
|
||||
|
||||
% Robustness to change of payload's mass
|
||||
% The Root Locus for the three damping techniques are shown in Figure ref:fig:uniaxial_active_damping_robustness_mass_root_locus for three sample's mass (1kg, 25kg and 50kg).
|
||||
% The closed-loop poles are shown by the squares for a specific gain.
|
||||
|
||||
% We can see that having heavier samples yields larger damping for IFF and smaller damping for RDC and DVF.
|
||||
|
||||
|
||||
%% Active Damping Robustness to change of sample's mass - Root Locus for all three damping techniques with 3 different sample's masses
|
||||
figure;
|
||||
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
%% Integral Force Feedback
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
% Light Sample
|
||||
plot(real(pole(G_md_light('fm', 'f'))), imag(pole(G_md_light('fm', 'f'))), 'x', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_md_light('fm', 'f'))), imag(zero(G_md_light('fm', 'f'))), 'o', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(1,4,100)
|
||||
clpoles = pole(feedback(G_md_light('fm', 'f'), g/s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
clpoles = pole(feedback(G_md_light('fm', 'f'), K_iff_md, +1));
|
||||
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(1,:), ...
|
||||
'DisplayName', '$m_s = 1\,kg$');
|
||||
|
||||
% Mid Sample
|
||||
plot(real(pole(G_md_mid('fm', 'f'))), imag(pole(G_md_mid('fm', 'f'))), 'x', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_md_mid('fm', 'f'))), imag(zero(G_md_mid('fm', 'f'))), 'o', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(1,4,100)
|
||||
clpoles = pole(feedback(G_md_mid('fm', 'f'), g/s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
clpoles = pole(feedback(G_md_mid('fm', 'f'), K_iff_md, +1));
|
||||
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(2,:), ...
|
||||
'DisplayName', '$m_s = 25\,kg$');
|
||||
|
||||
% Heavy Sample
|
||||
plot(real(pole(G_md_heavy('fm', 'f'))), imag(pole(G_md_heavy('fm', 'f'))), 'x', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
plot(real(zero(G_md_heavy('fm', 'f'))), imag(zero(G_md_heavy('fm', 'f'))), 'o', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(1,4,100)
|
||||
clpoles = pole(feedback(G_md_heavy('fm', 'f'), g/s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
clpoles = pole(feedback(G_md_heavy('fm', 'f'), K_iff_md, +1));
|
||||
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(3,:), ...
|
||||
'DisplayName', '$m_s = 50\,kg$');
|
||||
hold off;
|
||||
axis square;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
title('IFF')
|
||||
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
ldg.ItemTokenSize = [10, 1];
|
||||
|
||||
%% Relative Damping Control
|
||||
ax2 = nexttile();
|
||||
hold on;
|
||||
% Light Sample
|
||||
plot(real(pole(G_md_light('dL', 'f'))), imag(pole(G_md_light('dL', 'f'))), 'x', 'color', colors(1,:), ...
|
||||
'DisplayName', '$m_s = 1\,kg$');
|
||||
plot(real(zero(G_md_light('dL', 'f'))), imag(zero(G_md_light('dL', 'f'))), 'o', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,5,100)
|
||||
clpoles = pole(feedback(G_md_light('dL', 'f'), -g*s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
clpoles = pole(feedback(G_md_light('dL', 'f'), K_rdc_md, +1));
|
||||
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
|
||||
% Mid Sample
|
||||
plot(real(pole(G_md_mid('dL', 'f'))), imag(pole(G_md_mid('dL', 'f'))), 'x', 'color', colors(2,:), ...
|
||||
'DisplayName', '$m_s = 25\,kg$');
|
||||
plot(real(zero(G_md_mid('dL', 'f'))), imag(zero(G_md_mid('dL', 'f'))), 'o', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,5,100)
|
||||
clpoles = pole(feedback(G_md_mid('dL', 'f'), -g*s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
clpoles = pole(feedback(G_md_mid('dL', 'f'), K_rdc_md, +1));
|
||||
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
|
||||
% Heavy Sample
|
||||
plot(real(pole(G_md_heavy('dL', 'f'))), imag(pole(G_md_heavy('dL', 'f'))), 'x', 'color', colors(3,:), ...
|
||||
'DisplayName', '$m_s = 50\,kg$');
|
||||
plot(real(zero(G_md_heavy('dL', 'f'))), imag(zero(G_md_heavy('dL', 'f'))), 'o', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,5,100)
|
||||
clpoles = pole(feedback(G_md_heavy('dL', 'f'), -g*s, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
clpoles = pole(feedback(G_md_heavy('dL', 'f'), K_rdc_md, +1));
|
||||
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
hold off;
|
||||
axis square;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
title('RDC');
|
||||
|
||||
%% Direct Velocity Feedback
|
||||
ax3 = nexttile();
|
||||
hold on;
|
||||
% Light Sample
|
||||
plot(real(pole(G_md_light('vn', 'f'))), imag(pole(G_md_light('vn', 'f'))), 'x', 'color', colors(1,:), ...
|
||||
'DisplayName', '$m_s = 1\,kg$');
|
||||
plot(real(zero(G_md_light('vn', 'f'))), imag(zero(G_md_light('vn', 'f'))), 'o', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,5,100)
|
||||
clpoles = pole(feedback(G_md_light('vn', 'f'), -g, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
clpoles = pole(feedback(G_md_light('vn', 'f'), K_dvf_md, +1));
|
||||
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
|
||||
% Mid Sample
|
||||
plot(real(pole(G_md_mid('vn', 'f'))), imag(pole(G_md_mid('vn', 'f'))), 'x', 'color', colors(2,:), ...
|
||||
'DisplayName', '$m_s = 25\,kg$');
|
||||
plot(real(zero(G_md_mid('vn', 'f'))), imag(zero(G_md_mid('vn', 'f'))), 'o', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,5,100)
|
||||
clpoles = pole(feedback(G_md_mid('vn', 'f'), -g, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
clpoles = pole(feedback(G_md_mid('vn', 'f'), K_dvf_md, +1));
|
||||
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(2,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
|
||||
% Heavy Sample
|
||||
plot(real(pole(G_md_heavy('vn', 'f'))), imag(pole(G_md_heavy('vn', 'f'))), 'x', 'color', colors(3,:), ...
|
||||
'DisplayName', '$m_s = 50\,kg$');
|
||||
plot(real(zero(G_md_heavy('vn', 'f'))), imag(zero(G_md_heavy('vn', 'f'))), 'o', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = logspace(2,5,100)
|
||||
clpoles = pole(feedback(G_md_heavy('vn', 'f'), -g, +1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
clpoles = pole(feedback(G_md_heavy('vn', 'f'), K_dvf_md, +1));
|
||||
plot(real(clpoles), imag(clpoles), 'square', 'color', colors(3,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
hold off;
|
||||
axis square;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
title('DVF');
|
||||
|
||||
linkaxes([ax1,ax2,ax3],'xy');
|
||||
xlim([-300, 0]); ylim([0, 300]);
|
||||
495
matlab/uniaxial_6_hac_lac.m
Normal file
495
matlab/uniaxial_6_hac_lac.m
Normal file
@@ -0,0 +1,495 @@
|
||||
%% 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;
|
||||
|
||||
%% Frequency Vector [Hz]
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
%% Load the PSD of disturbances
|
||||
load('uniaxial_disturbance_psd.mat', 'f', 'psd_ft', 'psd_xf');
|
||||
|
||||
%% Load Plants Dynamics
|
||||
load('uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ...
|
||||
'G_vc_mid', 'G_md_mid', 'G_pz_mid', ...
|
||||
'G_vc_heavy', 'G_md_heavy', 'G_pz_heavy');
|
||||
|
||||
%% Load Damped Plants
|
||||
load('uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_iff_pz_light', ...
|
||||
'G_rdc_vc_light', 'G_rdc_md_light', 'G_rdc_pz_light', ...
|
||||
'G_dvf_vc_light', 'G_dvf_md_light', 'G_dvf_pz_light', ...
|
||||
'G_iff_vc_mid', 'G_iff_md_mid', 'G_iff_pz_mid', ...
|
||||
'G_rdc_vc_mid', 'G_rdc_md_mid', 'G_rdc_pz_mid', ...
|
||||
'G_dvf_vc_mid', 'G_dvf_md_mid', 'G_dvf_pz_mid', ...
|
||||
'G_iff_vc_heavy', 'G_iff_md_heavy', 'G_iff_pz_heavy', ...
|
||||
'G_rdc_vc_heavy', 'G_rdc_md_heavy', 'G_rdc_pz_heavy', ...
|
||||
'G_dvf_vc_heavy', 'G_dvf_md_heavy', 'G_dvf_pz_heavy');
|
||||
|
||||
% Damped Plant Dynamics
|
||||
% As was shown in Section ref:sec:uniaxial_active_damping, all three proposed active damping techniques yield similar damping plants.
|
||||
% Therefore, /Integral Force Feedback/ will be used in this section to study the HAC-LAC performances.
|
||||
|
||||
% The obtained damped plants for the three nano-hexapod stiffnesses are shown in Figure ref:fig:uniaxial_hac_iff_damped_plants_masses.
|
||||
|
||||
|
||||
%% Damped plant - Robustness to change of sample's mass
|
||||
figure;
|
||||
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz'))), '-', 'DisplayName', '$m_s = 1\,kg$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_vc_mid( 'd', 'f'), freqs, 'Hz'))), '-', 'DisplayName', '$m_s = 25\,kg$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz'))), '-', 'DisplayName', '$m_s = 50\,kg$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('$k_n = 0.01\,N/\mu m$');
|
||||
ylim([5e-10, 1e-3]);
|
||||
ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
ldg.ItemTokenSize = [20, 1];
|
||||
|
||||
ax2 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid( 'd', 'f'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
|
||||
title('$k_n = 1\,N/\mu m$');
|
||||
ylim([5e-10, 1e-3]);
|
||||
|
||||
ax3 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_pz_mid( 'd', 'f'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
|
||||
title('$k_n = 100\,N/\mu m$');
|
||||
ylim([5e-10, 1e-3]);
|
||||
|
||||
ax1b = nexttile();
|
||||
hold on;
|
||||
plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz')))));
|
||||
plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_vc_mid( 'd', 'f'), freqs, 'Hz')))));
|
||||
plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz')))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
yticks(-360:90:360);
|
||||
ylim([-200, 20]);
|
||||
|
||||
ax2b = nexttile();
|
||||
hold on;
|
||||
plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz')))));
|
||||
plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_md_mid( 'd', 'f'), freqs, 'Hz')))));
|
||||
plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz')))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
yticks(-360:90:360);
|
||||
ylim([-200, 20]);
|
||||
|
||||
ax3b = nexttile();
|
||||
hold on;
|
||||
plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz')))));
|
||||
plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_pz_mid( 'd', 'f'), freqs, 'Hz')))));
|
||||
plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz')))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
yticks(-360:90:360);
|
||||
ylim([-200, 20]);
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax1b,ax2b,ax3b],'x');
|
||||
xlim([1, 1e3]);
|
||||
|
||||
% Position Feedback Controller
|
||||
% The objective now is to design a position feedback controller for each of the three nano-hexapods that are robust to the change of sample's mass.
|
||||
|
||||
% The required feedback bandwidth was approximately determined un Section ref:sec:uniaxial_noise_budgeting:
|
||||
% - $\approx 10\,\text{Hz}$ for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$).
|
||||
% Near this frequency, the plants are equivalent to a mass line.
|
||||
% The gain of the mass line can vary up to a fact $\approx 5$ (suspended mass from $16\,kg$ up to $65\,kg$).
|
||||
% This mean that the designed controller will need to have large gain margins to be robust to the change of sample's mass.
|
||||
% - $\approx 50\,\text{Hz}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$).
|
||||
% Similarly to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line.
|
||||
% It will be probably easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode.
|
||||
% - $\approx 100\,\text{Hz}$ for the stiff nano-hexapod ($k_n = 100\,N/\mu m$).
|
||||
% Contrary to the two first nano-hexapod stiffnesses, here the plants have more complex dynamics near the wanted crossover frequency.
|
||||
% The micro-station is not stiff enough to have a clear stiffness line at this frequency.
|
||||
% Therefore, there are both a change of phase and gain depending on the sample's mass.
|
||||
% This makes the robust design of the controller a little bit more complicated.
|
||||
|
||||
|
||||
% Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure ref:fig:uniaxial_nyquist_hac).
|
||||
% These high authority controllers are generally composed of a two integrators at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics.
|
||||
% The loop gains for the three nano-hexapod are shown in Figure ref:fig:uniaxial_loop_gain_hac.
|
||||
% We can see that:
|
||||
% - for the soft and moderately stiff nano-hexapod, the crossover frequency varies a lot with the sample mass.
|
||||
% This is due to the fact that the crossover frequency corresponds to the mass line of the plant.
|
||||
% - for the stiff nano-hexapod, the obtained crossover frequency is not at high as what was estimated necessary.
|
||||
% The crossover frequency in that case is close to the stiffness line of the plant, which makes the robust design of the controller easier.
|
||||
|
||||
% Note that these controller were quickly tuned by hand and not designed using any optimization methods.
|
||||
% The goal is just to have a first estimation of the attainable performances.
|
||||
|
||||
|
||||
%% High Authority Controller - Soft Nano-Hexapod
|
||||
% Lead to increase phase margin
|
||||
a = 5; % Amount of phase lead / width of the phase lead / high frequency gain
|
||||
wc = 2*pi*20; % Frequency with the maximum phase lead [rad/s]
|
||||
H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
|
||||
|
||||
% Low Pass filter to increase robustness
|
||||
H_lpf = 1/(1 + s/2/pi/200);
|
||||
|
||||
% Added integrator at low frequency
|
||||
H_int = (s + 2*pi*5)/(s + 2*pi*0.01);
|
||||
|
||||
% High Authority Controller
|
||||
K_hac_vc = 4e5 * ... % Gain
|
||||
H_lead * ... % Lead
|
||||
H_int * ... % Integrator
|
||||
H_lpf; % LPF
|
||||
K_hac_vc.InputName = {'d'};
|
||||
K_hac_vc.OutputName = {'f'};
|
||||
|
||||
%% High Authority Controller - Mid Stiffness Nano-Hexapod
|
||||
% Integrator as low frequency
|
||||
H_int = (s + 2*pi*10)/(s + 2*pi*0.01) * (s + 2*pi*20)/(s + 2*pi*0.01);
|
||||
|
||||
% Lead to increase phase margin
|
||||
a = 4; % Amount of phase lead / width of the phase lead / high frequency gain
|
||||
wc = 2*pi*70; % Frequency with the maximum phase lead [rad/s]
|
||||
|
||||
H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
|
||||
|
||||
% Low Pass filter to increase robustness
|
||||
H_lpf = 1/(1 + s/2/pi/300);
|
||||
|
||||
% High Authority Controller
|
||||
K_hac_md = 3e6 * ... % Gain
|
||||
H_lead * ... % Lead
|
||||
H_lpf * ... % Low Pass Filter
|
||||
H_int; % Integrator
|
||||
K_hac_md.InputName = {'d'};
|
||||
K_hac_md.OutputName = {'f'};
|
||||
|
||||
%% High Authority Controller - Stiff Nano-Hexapod
|
||||
% Integrator as low frequency
|
||||
H_int = 1/(s + 2*pi*0.01) * 1/(s + 2*pi*0.01);
|
||||
|
||||
% Lead to increase phase margin
|
||||
a = 5; % Amount of phase lead / width of the phase lead / high frequency gain
|
||||
wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s]
|
||||
H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
|
||||
|
||||
% Low Pass filter to increase robustness
|
||||
H_lpf = 1/(1 + s/2/pi/500);
|
||||
|
||||
% High Authority Controller
|
||||
K_hac_pz = 6e12 * ... % Gain
|
||||
H_lead^2 * ... % Lead
|
||||
H_lpf * ... % Low Pass Filter
|
||||
H_int; % Integrator
|
||||
K_hac_pz.InputName = {'d'};
|
||||
K_hac_pz.OutputName = {'f'};
|
||||
|
||||
%% Compute Loop gain for Nyquist Plot
|
||||
L_vc_light = squeeze(freqresp(K_hac_vc*G_iff_vc_light('d', 'f'), freqs, 'Hz'));
|
||||
L_vc_mid = squeeze(freqresp(K_hac_vc*G_iff_vc_mid( 'd', 'f'), freqs, 'Hz'));
|
||||
L_vc_heavy = squeeze(freqresp(K_hac_vc*G_iff_vc_heavy('d', 'f'), freqs, 'Hz'));
|
||||
|
||||
L_md_light = squeeze(freqresp(K_hac_md*G_iff_md_light('d', 'f'), freqs, 'Hz'));
|
||||
L_md_mid = squeeze(freqresp(K_hac_md*G_iff_md_mid( 'd', 'f'), freqs, 'Hz'));
|
||||
L_md_heavy = squeeze(freqresp(K_hac_md*G_iff_md_heavy('d', 'f'), freqs, 'Hz'));
|
||||
|
||||
L_pz_light = squeeze(freqresp(K_hac_pz*G_iff_pz_light('d', 'f'), freqs, 'Hz'));
|
||||
L_pz_mid = squeeze(freqresp(K_hac_pz*G_iff_pz_mid( 'd', 'f'), freqs, 'Hz'));
|
||||
L_pz_heavy = squeeze(freqresp(K_hac_pz*G_iff_pz_heavy('d', 'f'), freqs, 'Hz'));
|
||||
|
||||
%% Nyquist Plot - Hight Authority Controller for all three nano-hexapod stiffnesses and all sample masses
|
||||
figure;
|
||||
hold on;
|
||||
plot(real(L_vc_light), +imag(L_vc_light), '-', 'color', [colors(1,:), 0.5], 'DisplayName', '$k_n = 0.01\,N/\mu m$')
|
||||
plot(real(L_vc_light), -imag(L_vc_light), '-', 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_vc_mid ), +imag(L_vc_mid ), '-', 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_vc_mid ), -imag(L_vc_mid ), '-', 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_vc_heavy), +imag(L_vc_heavy), '-', 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_vc_heavy), -imag(L_vc_heavy), '-', 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off')
|
||||
|
||||
plot(real(L_md_light), +imag(L_md_light), '-', 'color', [colors(2,:), 0.5], 'DisplayName', '$k_n = 1\,N/\mu m$')
|
||||
plot(real(L_md_light), -imag(L_md_light), '-', 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_md_mid ), +imag(L_md_mid ), '-', 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_md_mid ), -imag(L_md_mid ), '-', 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_md_heavy), +imag(L_md_heavy), '-', 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_md_heavy), -imag(L_md_heavy), '-', 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off')
|
||||
|
||||
plot(real(L_pz_light), +imag(L_pz_light), '-', 'color', [colors(3,:), 0.5], 'DisplayName', '$k_n = 100\,N/\mu m$')
|
||||
plot(real(L_pz_light), -imag(L_pz_light), '-', 'color', [colors(3,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_pz_mid ), +imag(L_pz_mid ), '-', 'color', [colors(3,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_pz_mid ), -imag(L_pz_mid ), '-', 'color', [colors(3,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_pz_heavy), +imag(L_pz_heavy), '-', 'color', [colors(3,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(real(L_pz_heavy), -imag(L_pz_heavy), '-', 'color', [colors(3,:), 0.5], 'HandleVisibility', 'off')
|
||||
plot(-1, 0, 'kx', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Real'); ylabel('Imag');
|
||||
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
xlim([-3.8, 0.2]); ylim([-2, 2]);
|
||||
axis square;
|
||||
|
||||
|
||||
|
||||
% #+name: fig:uniaxial_nyquist_hac
|
||||
% #+caption: Nyquist Plot - Hight Authority Controller for all three nano-hexapod stiffnesses (soft one in blue, moderately stiff in red and very stiff in yellow) and all sample masses (corresponding to the three curves of each color)
|
||||
% #+RESULTS:
|
||||
% [[file:figs/uniaxial_nyquist_hac.png]]
|
||||
|
||||
|
||||
%% Loop Gain - High Authority Controller for all three nano-hexapod stiffnesses and all sample masses
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(L_vc_light), 'color', [colors(1,:), 0.5], 'DisplayName', '$k_n = 0.01\,N/\mu m$');
|
||||
plot(freqs, abs(L_vc_mid), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(L_vc_heavy), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(L_md_light), 'color', [colors(2,:), 0.5], 'DisplayName', '$k_n = 1\,N/\mu m$');
|
||||
plot(freqs, abs(L_md_mid), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(L_md_heavy), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(L_pz_light), 'color', [colors(3,:), 0.5], 'DisplayName', '$k_n = 100\,N/\mu m$');
|
||||
plot(freqs, abs(L_pz_mid), 'color', [colors(3,:), 0.5], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(L_pz_heavy), 'color', [colors(3,:), 0.5], 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
|
||||
ylim([1e-3, 1e3]);
|
||||
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*unwrap(angle(L_vc_light)), 'color', [colors(1,:), 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(L_vc_mid )), 'color', [colors(1,:), 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(L_vc_heavy)), 'color', [colors(1,:), 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(L_md_light)), 'color', [colors(2,:), 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(L_md_mid )), 'color', [colors(2,:), 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(L_md_heavy)), 'color', [colors(2,:), 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(L_pz_light)), 'color', [colors(3,:), 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(L_pz_mid )), 'color', [colors(3,:), 0.5]);
|
||||
plot(freqs, 180/pi*unwrap(angle(L_pz_heavy)), 'color', [colors(3,:), 0.5]);
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
yticks(-360:90:360);
|
||||
ylim([-270, 0]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([1, 500]);
|
||||
|
||||
% Closed-Loop Noise Budgeting
|
||||
% The high authority position feedback controllers are then implemented and the closed-loop sensitivity to disturbances are computed.
|
||||
% These are compared with the open-loop and damped plants cases in Figure ref:fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff nano-hexapod with 25kg sample's mass).
|
||||
% As expected, the sensitivity to disturbances is decreased in the controller bandwidth and slightly increase outside this bandwidth.
|
||||
|
||||
|
||||
%% Compute Closed Loop Systems
|
||||
G_hac_iff_vc_light = feedback(G_iff_vc_light, K_hac_vc, 'name', -1);
|
||||
G_hac_iff_vc_mid = feedback(G_iff_vc_mid , K_hac_vc, 'name', -1);
|
||||
G_hac_iff_vc_heavy = feedback(G_iff_vc_heavy, K_hac_vc, 'name', -1);
|
||||
|
||||
G_hac_iff_md_light = feedback(G_iff_md_light, K_hac_md, 'name', -1);
|
||||
G_hac_iff_md_mid = feedback(G_iff_md_mid , K_hac_md, 'name', -1);
|
||||
G_hac_iff_md_heavy = feedback(G_iff_md_heavy, K_hac_md, 'name', -1);
|
||||
|
||||
G_hac_iff_pz_light = feedback(G_iff_pz_light, K_hac_pz, 'name', -1);
|
||||
G_hac_iff_pz_mid = feedback(G_iff_pz_mid , K_hac_pz, 'name', -1);
|
||||
G_hac_iff_pz_heavy = feedback(G_iff_pz_heavy, K_hac_pz, 'name', -1);
|
||||
|
||||
%% Verify Stability
|
||||
isstable(G_hac_iff_vc_light) && isstable(G_hac_iff_vc_mid) && isstable(G_hac_iff_vc_heavy)
|
||||
|
||||
isstable(G_hac_iff_md_light) && isstable(G_hac_iff_md_mid) && isstable(G_hac_iff_md_heavy)
|
||||
|
||||
isstable(G_hac_iff_pz_light) && isstable(G_hac_iff_pz_mid) && isstable(G_hac_iff_pz_heavy)
|
||||
|
||||
%% Change of sensitivity to disturbances with LAC and with HAC-LAC
|
||||
figure;
|
||||
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid( 'd', 'fs'), freqs, 'Hz'))), 'k-');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'fs'), freqs, 'Hz'))), 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_hac_iff_md_mid('d', 'fs'), freqs, 'Hz'))), 'color', colors(2,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
ylabel('Amplitude $d/f_{s}$ [m/N]'); xlabel('Frequency [Hz]');
|
||||
|
||||
ax2 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid( 'd', 'ft'), freqs, 'Hz'))), 'k-');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'ft'), freqs, 'Hz'))), 'color', colors(1,:));
|
||||
plot(freqs, abs(squeeze(freqresp(G_hac_iff_md_mid('d', 'ft'), freqs, 'Hz'))), 'color', colors(2,:));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
ylabel('Amplitude $d/f_{t}$ [m/N]'); xlabel('Frequency [Hz]');
|
||||
|
||||
ax3 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_md_mid( 'd', 'xf'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff_md_mid('d', 'xf'), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', 'IFF');
|
||||
plot(freqs, abs(squeeze(freqresp(G_hac_iff_md_mid('d', 'xf'), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', 'HAC-IFF');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
ylabel('Amplitude $d/x_{f}$ [m/m]'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
|
||||
linkaxes([ax1,ax2,ax3],'x');
|
||||
xlim([1, 500]);
|
||||
|
||||
|
||||
|
||||
% #+name: fig:uniaxial_sensitivity_dist_hac_lac
|
||||
% #+caption: Change of sensitivity to disturbances with LAC and with HAC-LAC
|
||||
% #+RESULTS:
|
||||
% [[file:figs/uniaxial_sensitivity_dist_hac_lac.png]]
|
||||
|
||||
% The cumulative amplitude spectrum of the motion $d$ is computed for all nano-hexapod configurations, all sample masses and in the open-loop (OL), damped (IFF) and position controlled (HAC-IFF) cases.
|
||||
% The results are shown in Figure ref:fig:uniaxial_cas_hac_lac.
|
||||
% Obtained root mean square values of the distance $d$ are better for the soft nano-hexapod ($\approx 25\,nm$ to $\approx 35\,nm$ depending on the sample's mass) than for the stiffer nano-hexapod (from $\approx 30\,nm$ to $\approx 70\,nm$).
|
||||
|
||||
|
||||
%% Cumulative Amplitude Spectrum for all three nano-hexapod stiffnesses - Comparison of OL, IFF and HAC-LAC cases
|
||||
figure;
|
||||
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [0,0,0,0.5], 'DisplayName', 'OL');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_vc_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_vc_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [0,0,0,0.5], 'HandleVisibility', 'off');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_vc_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_vc_heavy('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [0,0,0,0.5], 'HandleVisibility', 'off');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(1,:), 0.5], 'DisplayName', 'IFF');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_vc_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_vc_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_vc_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_vc_heavy('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(2,:), 0.5], 'DisplayName', 'HAC-IFF');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_heavy('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
ylabel('CAS of $d$ [m]'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
title('$k_n = 0.01\,N/\mu m$');
|
||||
|
||||
ax2 = nexttile();
|
||||
hold on;
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [0,0,0,0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_md_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_md_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [0,0,0,0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_md_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_md_heavy('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [0,0,0,0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(1,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_md_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_md_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(1,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_md_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_md_heavy('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(1,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_md_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_hac_iff_md_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(2,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_md_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_hac_iff_md_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(2,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_md_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_hac_iff_md_heavy('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(2,:), 0.5]);
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
||||
title('$k_n = 1\,N/\mu m$');
|
||||
|
||||
ax3 = nexttile();
|
||||
hold on;
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [0,0,0,0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_pz_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_pz_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [0,0,0,0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_pz_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_pz_heavy('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [0,0,0,0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(1,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_pz_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_pz_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(1,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_iff_pz_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_iff_pz_heavy('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(1,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_mid('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_mid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(2,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(2,:), 0.5]);
|
||||
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_heavy('d', 'ft'), f, 'Hz'))).^2 + ...
|
||||
psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_heavy('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
|
||||
'color', [colors(2,:), 0.5]);
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xticks([1e0, 1e1, 1e2]);
|
||||
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
||||
title('$k_n = 100\,N/\mu m$');
|
||||
|
||||
linkaxes([ax1,ax2,ax3],'xy');
|
||||
xlim([1, 500]);
|
||||
ylim([2e-10, 3e-6])
|
||||
Reference in New Issue
Block a user