495 lines
27 KiB
Mathematica
495 lines
27 KiB
Mathematica
|
%% Clear Workspace and Close figures
|
||
|
|
clear; close all; clc;
|
||
|
|
|
||
|
|
%% Intialize Laplace variable
|
||
|
|
s = zpk('s');
|
||
|
|
|
||
|
|
%% Path for functions, data and scripts
|
||
|
|
addpath('./mat/'); % Path for data
|
||
|
|
|
||
|
|
%% Colors for the figures
|
||
|
|
colors = colororder;
|
||
|
|
|
||
|
|
%% Frequency Vector [Hz]
|
||
|
|
freqs = logspace(0, 3, 1000);
|
||
|
|
|
||
|
|
%% Load the PSD of disturbances
|
||
|
|
load('uniaxial_disturbance_psd.mat', 'f', 'psd_ft', 'psd_xf');
|
||
|
|
|
||
|
|
%% Load Plants Dynamics
|
||
|
|
load('uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ...
|
||
|
|
'G_vc_mid', 'G_md_mid', 'G_pz_mid', ...
|
||
|
|
'G_vc_heavy', 'G_md_heavy', 'G_pz_heavy');
|
||
|
|
|
||
|
|
%% 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])
|