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Thomas Dehaeze 2025-03-28 16:41:26 +01:00
parent b5e07eeedc
commit 8d97690c5c
11 changed files with 401 additions and 1712 deletions
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107
matlab/rotating_1_system_description.m
View File

@ -14,25 +14,6 @@ colors = colororder;
%% Simscape model name
mdl = 'rotating_model';
% System Poles: Campbell Diagram
% The poles of $\mathbf{G}_d$ are the complex solutions $p$ of equation eqref:eq:poles.
% #+name: eq:poles
% \begin{equation}
% \left( \frac{p^2}{{\omega_0}^2} + 2 \xi \frac{p}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{p}{\omega_0} \right)^2 = 0
% \end{equation}
% Supposing small damping ($\xi \ll 1$), two pairs of complex conjugate poles are obtained as shown in equation eqref:eq:pole_values.
% #+name: eq:pole_values
% \begin{subequations}
% \begin{align}
% p_{+} &= - \xi \omega_0 \left( 1 + \frac{\Omega}{\omega_0} \right) \pm j \omega_0 \left( 1 + \frac{\Omega}{\omega_0} \right) \\
% p_{-} &= - \xi \omega_0 \left( 1 - \frac{\Omega}{\omega_0} \right) \pm j \omega_0 \left( 1 - \frac{\Omega}{\omega_0} \right)
% \end{align}
% \end{subequations}
%% Model parameters for first analysis
kn = 1; % Actuator Stiffness [N/m]
mn = 1; % Payload Mass [kg]
@ -56,19 +37,8 @@ end
clear pole_G;
% The real and complex parts of these two pairs of complex conjugate poles are represented in Figure ref:fig:rotating_campbell_diagram as a function of the rotational speed $\Omega$.
% As the rotational speed increases, $p_{+}$ goes to higher frequencies and $p_{-}$ goes to lower frequencies.
% The system becomes unstable for $\Omega > \omega_0$ as the real part of $p_{-}$ is positive.
% Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forces exceeds the spring stiffness $k$.
%% Campbell diagram - Real and Imaginary parts of the poles as a function of the rotating velocity
figure;
tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(Wzs, real(p_ws(1, :)), '-', 'color', colors(1,:), 'DisplayName', '$p_{+}$')
plot(Wzs, real(p_ws(4, :)), '-', 'color', colors(1,:), 'HandleVisibility', 'off')
@ -86,7 +56,7 @@ ylim([-3*xin, 3*xin]);
yticks([-3*xin, -2*xin, -xin, 0, xin, 2*xin, 3*xin])
yticklabels({'', '', '$-\xi\omega_0$', '$0$', ''})
ax2 = nexttile();
figure
hold on;
plot(Wzs, imag(p_ws(1, :)), '-', 'color', colors(1,:))
plot(Wzs, imag(p_ws(4, :)), '-', 'color', colors(1,:))
@ -102,22 +72,21 @@ ylim([-3*w0n, 3*w0n]);
yticks([-3*w0n, -2*w0n, -w0n, 0, w0n, 2*w0n, 3*w0n])
yticklabels({'', '', '$-\omega_0$', '$0$', '$\omega_0$', '', ''})
% Identify Generic Dynamics :noexport:
%% Sample
%% Identify the dynamics for several rotating velocities
% Sample
ms = 0.5; % Sample mass [kg]
%% Tuv Stage
% Tuv Stage
kn = 1; % Stiffness [N/m]
mn = 0.5; % Tuv mass [kg]
cn = 0.01*2*sqrt(kn*(mn+ms)); % Damping [N/(m/s)]
%% General Configuration
% General Configuration
model_config = struct();
model_config.controller = "open_loop"; % Default: Open-Loop
model_config.Tuv_type = "normal"; % Default: 2DoF stage
%% Input/Output definition
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/controller'], 1, 'openinput'); io_i = io_i + 1; % [Fu, Fv]
io(io_i) = linio([mdl, '/fd'], 1, 'openinput'); io_i = io_i + 1; % [Fdu, Fdv]
@ -126,7 +95,7 @@ io(io_i) = linio([mdl, '/translation_stage'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/translation_stage'], 2, 'openoutput'); io_i = io_i + 1; % [Du, Dv]
io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy]
%% Tested rotating velocities [rad/s]
% Tested rotating velocities [rad/s]
Wzs = [0, 0.1, 0.2, 0.7, 1.2]; % Vector of rotation speeds [rad/s]
Gs = {zeros(2, 2, length(Wzs))}; % Direct terms
@ -144,23 +113,14 @@ for i = 1:length(Wzs)
Gs{:,:,i} = G;
end
%% Save All Identified Plants
% Save All Identified Plants
save('./mat/rotating_generic_plants.mat', 'Gs', 'Wzs');
% System Dynamics: Effect of rotation
% The system dynamics from actuator forces $[F_u, F_v]$ to the relative motion $[d_u, d_v]$ is identified for several rotating velocities.
% Looking at the transfer function matrix $\mathbf{G}_d$ in equation eqref:eq:Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite.
% The bode plot of these two terms are shown in Figure ref:fig:rotating_direct_coupling_bode_plot for several rotational speeds $\Omega$.
% These plots confirm the expected behavior: the frequency of the two pairs of complex conjugate poles are further separated as $\Omega$ increases.
% For $\Omega > \omega_0$, the low frequency pair of complex conjugate poles $p_{-}$ becomes unstable.
%% Bode plot of the direct and coupling terms for several rotating velocities
freqs = logspace(-1, 1, 1000);
figure;
tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
@ -171,26 +131,11 @@ end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
title('Direct terms: $d_u/F_u$, $d_v/F_v$');
ylim([1e-2, 1e2]);
yticks([1e-2,1e-1,1,1e1,1e2])
yticklabels({'$0.01/k$', '$0.1/k$', '$1/k$', '$10/k$', '$100/k$'})
ax2 = nexttile([2, 1]);
hold on;
for i = 1:length(Wzs)
plot(freqs, abs(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ...
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0$', Wzs(i)))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
title('Coupling Terms: $d_u/F_v$, $d_v/F_u$');
ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [10, 1];
ylim([1e-2, 1e2]);
ax3 = nexttile;
ax2 = nexttile;
hold on;
for i = 1:length(Wzs)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:))
@ -203,18 +148,40 @@ ylim([-180 180]);
xticks([1e-1,1,1e1])
xticklabels({'$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
ax4 = nexttile;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2, 1]);
hold on;
for i = 1:length(Wzs)
plot(freqs, abs(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ...
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0$', Wzs(i)))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [10, 1];
ylim([1e-2, 1e2]);
yticks([1e-2,1e-1,1,1e1,1e2])
yticklabels({'$0.01/k$', '$0.1/k$', '$1/k$', '$10/k$', '$100/k$'})
ax2 = nexttile;
hold on;
for i = 1:length(Wzs)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); set(gca, 'YTickLabel',[]);
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
xticks([1e-1,1,1e1])
xticklabels({'$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
linkaxes([ax1,ax2,ax3,ax4],'x');
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
linkaxes([ax1,ax2],'y');

88
matlab/rotating_2_iff_pure_int.m
View File

@ -17,107 +17,43 @@ mdl = 'rotating_model';
%% Load "Generic" system dynamics
load('rotating_generic_plants.mat', 'Gs', 'Wzs');
% Effect of the rotation speed on the IFF plant dynamics
% The transfer functions from actuator forces $[F_u,\ F_v]$ to the measured force sensors $[f_u,\ f_v]$ are identified for several rotating velocities and shown in Figure ref:fig:rotating_iff_bode_plot_effect_rot.
% As was expected from the derived equations of motion:
% - when $0 < \Omega < \omega_0$: the low frequency gain is no longer zero and two (non-minimum phase) real zero appears at low frequency.
% The low frequency gain increases with $\Omega$.
% A pair of (minimum phase) complex conjugate zeros appears between the two complex conjugate poles that are split further apart as $\Omega$ increases.
% - when $\omega_0 < \Omega$: the low frequency pole becomes unstable.
%% Bode plot of the direct and coupling term for Integral Force Feedback - Effect of rotation
freqs = logspace(-2, 1, 1000);
Wz_i = [1,3,4];
figure;
tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
for i = 1:length(Wzs)
plot(freqs, abs(squeeze(freqresp(Gs{i}('fu', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:))
for i = 1:length(Wz_i)
plot(freqs, abs(squeeze(freqresp(Gs{Wz_i(i)}('fu', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ...
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Wzs(Wz_i(i))),'MarkerSize',8);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
title('Direct terms: $f_u/F_u$, $f_v/F_v$');
ylim([1e-3, 1e2]);
leg = legend('location', 'northwest', 'FontSize', 8);
ax2 = nexttile([2, 1]);
ax2 = nexttile;
hold on;
for i = 1:length(Wzs)
plot(freqs, abs(squeeze(freqresp(Gs{i}('fv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ...
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0$', Wzs(i)))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
title('Coupling Terms: $f_u/F_v$, $f_v/F_u$');
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [10, 1];
ylim([1e-3, 1e2]);
ax3 = nexttile;
hold on;
for i = 1:length(Wzs)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('fu', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:))
for i = 1:length(Wz_i)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i(i)}('fu', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
ylim([0 180]);
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
ax4 = nexttile;
hold on;
for i = 1:length(Wzs)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('fv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); set(gca, 'YTickLabel',[]);
yticks(-180:90:180);
ylim([-180 180]);
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
linkaxes([ax1,ax2,ax3,ax4],'x');
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
linkaxes([ax1,ax2],'y');
% #+name: fig:rotating_iff_diagram
% #+caption: Control diagram for decentralized Integral Force Feedback
% #+RESULTS:
% [[file:figs/rotating_iff_diagram.png]]
% The decentralized IFF controller $\bm{K}_F$ corresponds to a diagonal controller with integrators:
% #+name: eq:Kf_pure_int
% \begin{equation}
% \begin{aligned}
% \mathbf{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\
% K_{F}(s) &= g \cdot \frac{1}{s}
% \end{aligned}
% \end{equation}
% In order to see how the IFF controller affects the poles of the closed loop system, a Root Locus plot (Figure ref:fig:rotating_root_locus_iff_pure_int) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain $g$ varies from $0$ to $\infty$ for the two controllers $K_{F}$ simultaneously.
% As explained in cite:preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr, the closed-loop poles start at the open-loop poles (shown by $\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};$) for $g = 0$ and coincide with the transmission zeros (shown by $\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];$) as $g \to \infty$.
% #+begin_important
% Whereas collocated IFF is usually associated with unconditional stability cite:preumont91_activ, this property is lost due to gyroscopic effects as soon as the rotation velocity in non-null.
% This can be seen in the Root Locus plot (Figure ref:fig:rotating_root_locus_iff_pure_int) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability.
% #+end_important
% Physically, this can be explained like so: at low frequency, the loop gain is very large due to the pure integrator in $K_{F}$ and the finite gain of the plant (Figure ref:fig:rotating_iff_bode_plot_effect_rot).
% The control system is thus canceling the spring forces which makes the suspended platform no able to hold the payload against centrifugal forces, hence the instability.
%% Root Locus for the Decentralized Integral Force Feedback controller
Kiff = 1/s*eye(2);

298
matlab/rotating_3_iff_hpf.m
View File

@ -17,18 +17,6 @@ mdl = 'rotating_model';
%% Load "Generic" system dynamics
load('rotating_generic_plants.mat', 'Gs', 'Wzs');
% Modified Integral Force Feedback Controller
% The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used:
% \begin{equation}
% K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix}
% 1 & 0 \\
% 0 & 1
% \end{bmatrix}
% \end{equation}
% where $\omega_i$ characterize down to which frequency the signal is integrated.
% Let's arbitrary choose the following control parameters:
%% Modified IFF - parameters
g = 2; % Controller gain
wi = 0.1; % HPF Cut-Off frequency [rad/s]
@ -36,13 +24,7 @@ wi = 0.1; % HPF Cut-Off frequency [rad/s]
Kiff = (g/s)*eye(2); % Pure IFF
Kiff_hpf = (g/(wi+s))*eye(2); % IFF with added HPF
% The loop gains ($K_F(s)$ times the direct dynamics $f_u/F_u$) with and without the added HPF are shown in Figure ref:fig:rotating_iff_modified_loop_gain.
% The effect of the added HPF limits the low frequency gain to finite values as expected.
%% Loop gain for the IFF with pure integrator and modified IFF with added high pass filter
%% Loop gain for the IFF with pure integrator and modified IFF with added high-pass filter
freqs = logspace(-2, 1, 1000);
Wz_i = 2;
@ -62,138 +44,32 @@ set(gca, 'XTickLabel',[]); ylabel('Loop Gain');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff(1,1), freqs, 'rad/s'))), 'DisplayName', 'IFF')
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff_hpf(1,1), freqs, 'rad/s'))), 'DisplayName', 'IFF + HPF')
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff_hpf(1,1), freqs, 'rad/s'))), 'DisplayName', 'IFF,HPF')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
ylim([-90 180]);
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
xticklabels({'', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
leg = legend('location', 'southwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 20;
leg.ItemTokenSize(1) = 15;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:rotating_iff_modified_loop_gain
% #+caption: Loop gain for the IFF with pure integrator and modified IFF with added high pass filter ($\Omega = 0.1\omega_0$)
% #+RESULTS:
% [[file:figs/rotating_iff_modified_loop_gain.png]]
% The Root Locus plots for the decentralized IFF with and without the HPF are displayed in Figure ref:fig:rotating_iff_root_locus_hpf.
% With the added HPF, the poles of the closed loop system are shown to be *stable up to some value of the gain* $g_\text{max}$ given by equation eqref:eq:gmax_iff_hpf.
% #+name: eq:gmax_iff_hpf
% \begin{equation}
% \boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)}
% \end{equation}
% It is interesting to note that $g_{\text{max}}$ also corresponds to the controller gain at which the low frequency loop gain (Figure ref:fig:rotating_iff_modified_loop_gain) reaches one.
%% High-Pass Filter Cut-Off Frequency
wis = [0.01, 0.05, 0.1]; % [rad/s]
%% Root Locus for the initial IFF and the modified IFF
gains = logspace(-2, 4, 200);
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([1,2]);
hold on;
for g = gains
clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), 'HandleVisibility', 'off','MarkerSize',4);
end
for g = gains
clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), 'HandleVisibility', 'off','MarkerSize',4);
end
% Pure Integrator
plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
'x', 'color', colors(1,:), 'DisplayName', 'IFF','MarkerSize',8);
plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
'o', 'color', colors(1,:), 'HandleVisibility', 'off','MarkerSize',8);
% Modified IFF
plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
'x', 'color', colors(2,:), 'DisplayName', 'IFF + HPF','MarkerSize',8);
plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
'o', 'color', colors(2,:), 'HandleVisibility', 'off','MarkerSize',8);
hold off;
axis square;
leg = legend('location', 'northwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 8;
xlabel('Real Part'); ylabel('Imaginary Part');
xlim([-2.25, 0.25]); ylim([-1.25, 1.25]);
xticks([-2, -1, 0])
xticklabels({'$-2\omega_0$', '$-\omega_0$', '$0$'})
yticks([-1, 0, 1])
yticklabels({'$-\omega_0$', '$0$', '$\omega_0$'})
ax2 = nexttile();
hold on;
for g = gains
clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4);
end
for g = gains
clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4);
end
% Pure Integrator
plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
'x', 'color', colors(1,:),'MarkerSize',8);
plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
'o', 'color', colors(1,:),'MarkerSize',8);
% Modified IFF
plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
'x', 'color', colors(2,:),'MarkerSize',8);
plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
'o', 'color', colors(2,:),'MarkerSize',8);
hold off;
axis square;
xlabel('Real Part'); ylabel('Imaginary Part');
title('Zoom near the origin');
xlim([-0.15, 0.1]); ylim([-0.15, 0.15]);
xticks([-0.1, 0, 0.1])
xticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'})
yticks([-0.1, 0, 0.1])
yticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'})
% Optimal IFF with HPF parameters $\omega_i$ and $g$
% Two parameters can be tuned for the modified controller in equation eqref:eq:iff_lhf: the gain $g$ and the pole's location $\omega_i$.
% The optimal values of $\omega_i$ and $g$ are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
% In order to visualize how $\omega_i$ does affect the attainable damping, the Root Locus plots for several $\omega_i$ are displayed in Figure ref:fig:rotating_root_locus_iff_modified_effect_wi.
% It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes, the control gain $g$ may be limited to small values due to equation eqref:eq:gmax_iff_hpf.
%% High Pass Filter Cut-Off Frequency
wis = [0.01, 0.1, 0.5, 1]*Wzs(2); % [rad/s]
%% Root Locus for the initial IFF and the modified IFF
gains = logspace(-2, 4, 200);
figure;
tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
for wi_i = 1:length(wis)
wi = wis(wi_i);
Kiff = (1/(wi+s))*eye(2);
L(wi_i) = plot(nan, nan, '.', 'color', colors(wi_i,:), 'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wi./Wzs(2)));
L(wi_i) = plot(nan, nan, '.', 'color', colors(wi_i,:), 'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wi));
for g = gains
clpoles = pole(feedback(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(wi_i,:),'MarkerSize',4);
@ -206,66 +82,20 @@ for wi_i = 1:length(wis)
'o', 'color', colors(wi_i,:),'MarkerSize',8);
end
hold off;
axis square;
axis equal;
xlim([-2.3, 0.1]); ylim([-1.2, 1.2]);
xticks([-2:1:2]); yticks([-2:1:2]);
leg = legend(L, 'location', 'northwest', 'FontSize', 8);
leg = legend(L, 'location', 'southwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 8;
xlabel('Real Part'); ylabel('Imaginary Part');
clear L
xlim([-2.25, 0.25]); ylim([-1.25, 1.25]);
xticks([-2, -1, 0])
xticklabels({'$-2\omega_0$', '$-\omega_0$', '$0$'})
xlim([-1.25, 0.25]); ylim([-1.25, 1.25]);
xticks([-1, 0])
xticklabels({'$-\omega_0$', '$0$'})
yticks([-1, 0, 1])
yticklabels({'$-\omega_0$', '$0$', '$\omega_0$'})
ax2 = nexttile();
hold on;
for wi_i = 1:length(wis)
wi = wis(wi_i);
Kiff = (1/(wi+s))*eye(2);
L(wi_i) = plot(nan, nan, '.', 'color', colors(wi_i,:));
for g = gains
clpoles = pole(feedback(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(wi_i,:),'MarkerSize',4);
end
plot(real(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
imag(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
'x', 'color', colors(wi_i,:),'MarkerSize',8);
plot(real(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
imag(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
'o', 'color', colors(wi_i,:),'MarkerSize',8);
end
hold off;
axis square;
xlim([-2.3, 0.1]); ylim([-1.2, 1.2]);
xticks([-2:1:2]); yticks([-2:1:2]);
xlabel('Real Part'); ylabel('Imaginary Part');
title('Zoom near the origin');
clear L
xlim([-0.15, 0.1]); ylim([-0.15, 0.15]);
xticks([-0.1, 0, 0.1])
xticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'})
yticks([-0.1, 0, 0.1])
yticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'})
% #+name: fig:rotating_root_locus_iff_modified_effect_wi
% #+caption: Root Locus for several high pass filter cut-off frequency
% #+RESULTS:
% [[file:figs/rotating_root_locus_iff_modified_effect_wi.png]]
% In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of $\omega_i/\omega_0$.
% The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also displayed and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure ref:fig:rotating_iff_hpf_optimal_gain).
% Three regions can be observed:
% - $\omega_i/\omega_0 < 0.02$: the added damping is limited by the maximum allowed control gain $g_{\text{max}}$
% - $0.02 < \omega_i/\omega_0 < 0.2$: the attainable damping ratio is maximized and is reached for $g \approx 2$
% - $0.2 < \omega_i/\omega_0$: the added damping decreases as $\omega_i/\omega_0$ increases.
ytickangle(90)
%% Compute the optimal control gain
wis = logspace(-2, 1, 100); % [rad/s]
@ -296,6 +126,7 @@ yyaxis right
hold on;
plot(wis, opt_gain, '-', 'DisplayName', '$g_{opt}$');
plot(wis, wis*((1/Wzs(2))^2 - 1), '--', 'DisplayName', '$g_{max}$');
hold off;
set(gca, 'YScale', 'lin');
ylim([0,10]);
ylabel('Controller gain $g$');
@ -304,79 +135,6 @@ xlabel('$\omega_i/\omega_0$');
set(gca, 'XScale', 'log');
legend('location', 'northeast', 'FontSize', 8);
% Obtained Damped Plant
% Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compute the damped plant.
% The undamped and damped plants are compared in Figure ref:fig:rotating_iff_hpf_damped_plant in blue and red respectively.
% A well damped plant is indeed obtained.
% However, the magnitude of the coupling term ($d_v/F_u$) is larger then IFF is applied.
%% Compute damped plant
wi = 0.1; % [rad/s]
g = 2; % Gain
Kiff_hpf = (g/(wi+s))*eye(2); % IFF with added HPF
Kiff_hpf.InputName = {'fu', 'fv'};
Kiff_hpf.OutputName = {'Fu', 'Fv'};
G_iff_hpf = feedback(Gs{2}, Kiff_hpf, 'name');
isstable(G_iff_hpf) % Verify stability
%% Damped plant with IFF and added HPF - Transfer function from $F_u$ to $d_u$
freqs = logspace(-2, 1, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gs{2}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros(1,3)], ...
'DisplayName', '$d_u/F_u$, OL')
plot(freqs, abs(squeeze(freqresp(G_iff_hpf('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:)], ...
'DisplayName', '$d_u/F_u$, IFF')
plot(freqs, abs(squeeze(freqresp(Gs{2}('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros(1,3), 0.5], ...
'DisplayName', '$d_v/F_u$, OL')
plot(freqs, abs(squeeze(freqresp(G_iff_hpf('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:), 0.5], ...
'DisplayName', '$d_v/F_u$, IFF')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 20;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{2}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', zeros(1,3))
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_hpf('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(1,:))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:rotating_iff_hpf_damped_plant
% #+caption: Damped plant with IFF and added HPF - Transfer function from $F_u$ to $d_u$, $\omega_i = 0.1 \cdot \omega_0$, $\Omega = 0.1 \cdot \omega_0$
% #+RESULTS:
% [[file:figs/rotating_iff_hpf_damped_plant.png]]
% In order to study how $\omega_i$ affects the coupling of the damped plant, the closed-loop plant is identified for several $\omega_i$.
% The direct and coupling terms of the plants are shown in Figure ref:fig:rotating_iff_hpf_damped_plant_effect_wi_coupling (left) and the ratio between the two (i.e. the coupling ratio) is shown in Figure ref:fig:rotating_iff_hpf_damped_plant_effect_wi_coupling (right).
% The coupling ratio is decreasing as $\omega_i$ increases.
% There is therefore a *trade-off between achievable damping and coupling ratio* for the choice of $\omega_i$.
% The same trade-off can be seen between achievable damping and loss of compliance at low frequency (see Figure ref:fig:rotating_iff_hpf_effect_wi_compliance).
%% Compute damped plant
wis = [0.03, 0.1, 0.5]; % [rad/s]
g = 2; % Gain
@ -395,7 +153,7 @@ end
freqs = logspace(-2, 1, 1000);
figure;
tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
@ -412,20 +170,6 @@ set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 20;
ax3 = nexttile([3,1]);
hold on;
for i = 1:length(wis)
plot(freqs, abs(squeeze(freqresp(Gs_iff_hpf{i}('Dv', 'Fu')/Gs_iff_hpf{i}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(i,:)], ...
'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wis(i)))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [rad/s]'); ylabel('Coupling Ratio');
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 20;
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
ax2 = nexttile;
hold on;
for i = 1:length(wis)
@ -435,21 +179,13 @@ hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
ylim([-180 0]);
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:rotating_iff_hpf_damped_plant_effect_wi_coupling
% #+caption: Effect of $\omega_i$ on the damped plant coupling
% #+RESULTS:
% [[file:figs/rotating_iff_hpf_damped_plant_effect_wi_coupling.png]]
%% Effect of $\omega_i$ on the obtained compliance
freqs = logspace(-2, 1, 1000);

180
matlab/rotating_4_iff_kp.m
View File

@ -17,8 +17,6 @@ mdl = 'rotating_model';
%% Load "Generic" system dynamics
load('rotating_generic_plants.mat', 'Gs', 'Wzs');
% Identify plant with parallel stiffnesses :noexport:
%% Tuv Stage
mn = 0.5; % Tuv mass [kg]
@ -38,13 +36,6 @@ io(io_i) = linio([mdl, '/translation_stage'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/translation_stage'], 2, 'openoutput'); io_i = io_i + 1; % [Du, Dv]
io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy]
% Effect of the parallel stiffness on the IFF plant
% The IFF plant (transfer function from $[F_u, F_v]$ to $[f_u, f_v]$) is identified in three different cases:
% - without parallel stiffness $k_p = 0$
% - with a small parallel stiffness $k_p < m \Omega^2$
% - with a large parallel stiffness $k_p > m \Omega^2$
Wz = 0.1; % The rotation speed [rad/s]
%% No parallel Stiffness
@ -77,12 +68,6 @@ G_high_kp = linearize(mdl, io, 0);
G_high_kp.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy'};
G_high_kp.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
% The Bode plots of the obtained dynamics are shown in Figure ref:fig:rotating_iff_effect_kp.
% One can see that for $k_p > m \Omega^2$, the two real zeros with $k_p < m \Omega^2$ are transformed into two complex conjugate zeros and the systems shows alternating complex conjugate poles and zeros.
%% Effect of the parallel stiffness on the IFF plant
freqs = logspace(-2, 1, 1000);
@ -101,7 +86,7 @@ plot(freqs, abs(squeeze(freqresp(G_high_kp('fu', 'Fu'), freqs, 'rad/s'))), '-',
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
ylim([1e-5, 5e1]);
ylim([1e-4, 5e1]);
legend('location', 'southeast', 'FontSize', 8);
% Phase
@ -113,7 +98,7 @@ plot(freqs, 180/pi*angle(squeeze(freqresp(G_high_kp('fu', 'Fu'), freqs, 'rad/s')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
ylim([0 180]);
hold off;
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
@ -121,24 +106,10 @@ xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:rotating_iff_effect_kp
% #+caption: Effect of the parallel stiffness on the IFF plant: Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
% #+RESULTS:
% [[file:figs/rotating_iff_effect_kp.png]]
% Figure ref:fig:rotating_iff_kp_root_locus shows the Root Locus plots for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$ when $K_F$ is a pure integrator as in Eq. eqref:eq:Kf_pure_int.
% It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system are bounded on the (stable) left half-plane, and hence the *unconditional stability of IFF is recovered*.
%% Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring
gains = logspace(-2, 2, 200);
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([1,2]);
hold on;
plot(real(pole(G_no_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), imag(pole(G_no_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), 'x', 'color', colors(1,:), ...
'DisplayName', '$k_p = 0$','MarkerSize',8);
@ -181,48 +152,6 @@ xlabel('Real Part'); ylabel('Imaginary Part');
leg = legend('location', 'northwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 8;
ax2 = nexttile();
hold on;
plot(real(pole(G_no_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), imag(pole(G_no_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), 'x', 'color', colors(1,:), 'MarkerSize',8);
plot(real(tzero(G_no_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), imag(tzero(G_no_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), 'o', 'color', colors(1,:), 'MarkerSize',8);
for g = gains
cl_poles = pole(feedback(G_no_kp({'fu','fv'},{'Fu','Fv'}), (g/s)*eye(2)));
plot(real(cl_poles), imag(cl_poles), '.', 'color', colors(1,:), 'MarkerSize', 4);
end
plot(real(pole(G_low_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), imag(pole(G_low_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), 'x', 'color', colors(2,:), 'MarkerSize',8);
plot(real(tzero(G_low_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), imag(tzero(G_low_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), 'o', 'color', colors(2,:), 'MarkerSize',8);
for g = gains
cl_poles = pole(feedback(G_low_kp({'fu','fv'},{'Fu','Fv'}), (g/s)*eye(2)));
plot(real(cl_poles), imag(cl_poles), '.', 'color', colors(2,:), 'MarkerSize', 4);
end
plot(real(pole(G_high_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), imag(pole(G_high_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), 'x', 'color', colors(3,:), 'MarkerSize',8);
plot(real(tzero(G_high_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), imag(tzero(G_high_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), 'o', 'color', colors(3,:), 'MarkerSize',8);
for g = gains
cl_poles = pole(feedback(G_high_kp({'fu','fv'},{'Fu','Fv'}), (g/s)*eye(2)));
plot(real(cl_poles), imag(cl_poles), '.', 'color', colors(3,:), 'MarkerSize',4);
end
hold off;
axis equal;
xlim([-0.04, 0.04]); ylim([-0.08, 0.08]);
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
xlabel('Real Part'); ylabel('Imaginary Part');
title('Zoom on controller pole')
% Effect of $k_p$ on the attainable damping
% Even though the parallel stiffness $k_p$ has no impact on the open-loop poles (as the overall stiffness $k$ is kept constant), it has a large impact on the transmission zeros.
% Moreover, as the attainable damping is generally proportional to the distance between poles and zeros cite:preumont18_vibrat_contr_activ_struc_fourt_edition, the parallel stiffness $k_p$ is foreseen to have some impact on the attainable damping.
% To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure ref:fig:rotating_iff_kp_root_locus_effect_kp.
% The frequencies of the transmission zeros of the system are increasing with an increase of the parallel stiffness $k_p$ (thus getting closer to the poles) and the associated attainable damping is reduced.
% #+begin_important
% Therefore, even though the parallel stiffness $k_p$ should be larger than $m \Omega^2$ for stability reasons, it should not be taken too large as this would limit the attainable damping.
% #+end_important
%% Tested parallel stiffnesses
kps = [2, 20, 40]*(mn + ms)*Wz^2;
@ -265,16 +194,6 @@ xlabel('Real Part'); ylabel('Imaginary Part');
leg = legend('location', 'northwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 12;
% #+name: fig:rotating_iff_kp_root_locus_effect_kp
% #+caption: Root Locus: Effect of the parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$
% #+RESULTS:
% [[file:figs/rotating_iff_kp_root_locus_effect_kp.png]]
% This is confirmed by the Figure ref:fig:rotating_iff_kp_optimal_gain where the attainable closed-loop damping ratio $\xi_{\text{cl}}$ and the associated optimal control gain $g_\text{opt}$ are computed as a function of the parallel stiffness.
%% Computes the optimal parameters and attainable simultaneous damping
alphas = logspace(-2, 0, 100);
alphas(end) = []; % Remove last point
@ -325,14 +244,6 @@ xlim([0.01, 1]);
xticks([0.01, 0.1, 1])
xticklabels({'$m\Omega^2$', '$10m\Omega^2$', '$100m\Omega^2$'})
% Damped plant
% Let's choose a parallel stiffness equal to $k_p = 2 m \Omega^2$ and compute the damped plant.
% The damped and undamped transfer functions from $F_u$ to $d_u$ are compared in Figure ref:fig:rotating_iff_kp_damped_plant.
% Even though the two resonances are well damped, the IFF changes the low frequency behavior of the plant which is usually not wanted.
% This is due to the fact that "pure" integrators are used, and that the low frequency loop gains becomes large below some frequency.
%% Identify dynamics with parallel stiffness = 2mW^2
Wz = 0.1; % [rad/s]
kp = 2*(mn + ms)*Wz^2; % Parallel Stiffness [N/m]
@ -353,65 +264,6 @@ Kiff_kp.OutputName = {'Fu', 'Fv'};
%% Compute the damped plant
G_cl_iff_kp = feedback(G, Kiff_kp, 'name');
%% Damped plant with IFF - Transfer function from $F_u$ to $d_u$
freqs = logspace(-3, 1, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', zeros(1,3), ...
'DisplayName', '$d_u/F_u$ - OL')
plot(freqs, abs(squeeze(freqresp(G_cl_iff_kp('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(1,:), ...
'DisplayName', '$d_u/F_u$ - IFF + $k_p$')
plot(freqs, abs(squeeze(freqresp(G('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros(1,3), 0.5], ...
'DisplayName', '$d_v/F_u$ - OL')
plot(freqs, abs(squeeze(freqresp(G_cl_iff_kp('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:), 0.5], ...
'DisplayName', '$d_v/F_u$ - IFF + $k_p$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
ldg.ItemTokenSize(1) = 20;
ylim([1e-6, 1e2]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', zeros(1,3))
plot(freqs, 180/pi*angle(squeeze(freqresp(G_cl_iff_kp('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(1,:))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
xticks([1e-3,1e-2,1e-1,1,1e1])
xticklabels({'$0.001 \omega_0$', '$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:rotating_iff_kp_damped_plant
% #+caption: Damped plant with IFF - Transfer function from $F_u$ to $d_u$
% #+RESULTS:
% [[file:figs/rotating_iff_kp_damped_plant.png]]
% In order to lower the low frequency gain, an high pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane):
% \begin{equation}
% K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix}
% 1 & 0 \\
% 0 & 1
% \end{bmatrix}
% \end{equation}
% Let's see how the high pass filter impacts the attainable damping.
% The controller gain $g$ is kept constant while $\omega_i$ is changed, and the minimum damping ratio of the damped plant is computed.
% The obtained damping ratio as a function of $\omega_i/\omega_0$ (where $\omega_0$ is the resonance of the system without rotation) is shown in Figure ref:fig:rotating_iff_kp_added_hpf_effect_damping.
w0 = sqrt((kn+kp)/(mn+ms)); % Resonance frequency [rad/s]
wis = w0*logspace(-2, 0, 100); % LPF cut-off [rad/s]
@ -427,7 +279,7 @@ for wi_i = 1:length(wis)
opt_xi(wi_i) = min(xi);
end
%% Effect of the high pass filter cut-off frequency on the obtained damping
%% Effect of the high-pass filter cut-off frequency on the obtained damping
figure;
plot(wis, opt_xi, '-');
set(gca, 'XScale', 'log');
@ -436,19 +288,7 @@ ylim([0,1]);
ylabel('Damping Ratio $\xi$');
xlabel('$\omega_i/\omega_0$');
% #+name: fig:rotating_iff_kp_added_hpf_effect_damping
% #+caption: Effect of the high pass filter cut-off frequency on the obtained damping
% #+RESULTS:
% [[file:figs/rotating_iff_kp_added_hpf_effect_damping.png]]
% Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compute the damped plant again.
% The Bode plots of the undamped, damped with "pure" IFF, and with added high pass filters are shown in Figure ref:fig:rotating_iff_kp_added_hpf_damped_plant.
% The added high pass filter gives almost the same damping properties while giving acceptable low frequency behavior.
%% Compute the damped plant with added High Pass Filter
%% Compute the damped plant with added High-Pass Filter
Kiff_kp_hpf = (2.2/(s + 0.1*w0))*eye(2);
Kiff_kp_hpf.InputName = {'fu', 'fv'};
Kiff_kp_hpf.OutputName = {'Fu', 'Fv'};
@ -471,16 +311,16 @@ plot(freqs, abs(squeeze(freqresp(G_cl_iff_kp( 'Du', 'Fu'), freqs, 'rad/s'))),
plot(freqs, abs(squeeze(freqresp(G_cl_iff_hpf_kp('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(2,:), ...
'DisplayName', '$d_u/F_u$ - IFF + $k_p$ + HPF')
plot(freqs, abs(squeeze(freqresp(G( 'Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros( 1,3), 0.5], ...
'DisplayName', '$d_v/F_u$ - OL')
'HandleVisibility', 'off')
plot(freqs, abs(squeeze(freqresp(G_cl_iff_kp( 'Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:), 0.5], ...
'DisplayName', '$d_v/F_u$ - IFF + $k_p$')
'HandleVisibility', 'off')
plot(freqs, abs(squeeze(freqresp(G_cl_iff_hpf_kp('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(2,:), 0.5], ...
'DisplayName', '$d_v/F_u$ - IFF + $k_p$ + HPF')
'HandleVisibility', 'off')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
ldg.ItemTokenSize(1) = 20;
ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize(1) = 10;
ax2 = nexttile;
hold on;
@ -491,7 +331,7 @@ hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
ylim([-180 90]);
xticks([1e-3,1e-2,1e-1,1,1e1])
xticklabels({'$0.001 \omega_0$', '$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})

90
matlab/rotating_5_rdc.m
View File

@ -17,84 +17,6 @@ mdl = 'rotating_model';
%% Load "Generic" system dynamics
load('rotating_generic_plants.mat', 'Gs', 'Wzs');
% Decentralized Relative Damping Control
% The transfer functions from $[F_u,\ F_v]$ to $[d_u,\ d_v]$ is identified and shown in Figure ref:fig:rotating_rdc_plant_effect_rot for several rotating velocities.
%% Bode plot of the direct and coupling term for the "relative damping control" plant - Effect of rotation
freqs = logspace(-2, 1, 1000);
figure;
tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
for i = 1:length(Wzs)
plot(freqs, abs(squeeze(freqresp(Gs{i}('du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
title('Direct terms: $d_u/F_u$, $d_v/F_v$');
ylim([1e-3, 1e2]);
ax2 = nexttile([2, 1]);
hold on;
for i = 1:length(Wzs)
plot(freqs, abs(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ...
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0$', Wzs(i)))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
title('Coupling Terms: $d_u/F_v$, $d_v/F_u$');
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [10, 1];
ylim([1e-3, 1e2]);
ax3 = nexttile;
hold on;
for i = 1:length(Wzs)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
ax4 = nexttile;
hold on;
for i = 1:length(Wzs)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); set(gca, 'YTickLabel',[]);
yticks(-180:90:180);
ylim([-180 180]);
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
linkaxes([ax1,ax2,ax3,ax4],'x');
xlim([freqs(1), freqs(end)]);
linkaxes([ax1,ax2],'y');
% #+name: fig:rotating_rdc_plant_effect_rot
% #+caption: Bode plot of the direct and coupling term for the "relative damping control" plant - Effect of rotation
% #+RESULTS:
% [[file:figs/rotating_rdc_plant_effect_rot.png]]
% In order to see if large damping can be added with Relative Damping Control, the root locus is computed (Figure ref:fig:rotating_rdc_root_locus).
% The closed-loop system is unconditionally stable and the poles can be damped as much as wanted.
%% Root Locus for Relative Damping Control
Krdc = s*eye(2); % Relative damping controller
@ -126,14 +48,6 @@ xlabel('Real Part'); ylabel('Imaginary Part');
leg = legend('location', 'northwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 8;
% Damped Plant
% Let's select a reasonable "Relative Damping Control" gain, and compute the closed-loop damped system.
% The open-loop and damped plants are compared in Figure ref:fig:rotating_rdc_damped_plant.
% The rotating aspect does not add any complexity for the use of Relative Damping Control.
% It does not increase the low frequency coupling as compared to Integral Force Feedback.
i = 2;
%% Relative Damping Controller
@ -166,7 +80,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
ldg.ItemTokenSize(1) = 20;
ylim([1e-6, 1e2]);
ylim([1e-4, 1e2]);
ax2 = nexttile;
hold on;
@ -176,7 +90,7 @@ hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
ylim([-180 0]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);

60
matlab/rotating_6_act_damp_comparison.m
View File

@ -14,11 +14,6 @@ colors = colororder;
%% Simscape model name
mdl = 'rotating_model';
%% Load "Generic" system dynamics
load('rotating_generic_plants.mat', 'Gs', 'Wzs');
% Identify plants :noexport:
%% The rotating speed is set to $\Omega = 0.1 \omega_0$.
Wz = 0.1; % [rad/s]
@ -40,7 +35,6 @@ io(io_i) = linio([mdl, '/translation_stage'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy]
%% Identifying plant with parallel stiffness
model_config.Tuv_type = "parallel_k";
% Parallel stiffness
@ -57,7 +51,6 @@ G_kp.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'};
G_kp.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
%% Identifying plant with no parallel stiffness
model_config.Tuv_type = "normal";
% Tuv Stage
@ -84,18 +77,6 @@ Krdc = 2*s*eye(2);
Krdc.InputName = {'Du', 'Dv'};
Krdc.OutputName = {'Fu', 'Fv'};
% Root Locus
% Figure ref:fig:rotating_comp_techniques_root_locus shows the Root Locus plots for the two proposed IFF modifications as well as for relative damping control.
% While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not.
% This means that the closed-loop behavior of both systems will differ when large control gains are used.
% One can observe that the closed loop poles corresponding to the system with added springs (in red) are bounded to the left half plane implying unconditional stability.
% This is not the case for the system where the controller is augmented with an HPF (in blue).
% It is interesting to note that the maximum added damping is very similar for both techniques.
%% Comparison of active damping techniques for rotating platform - Root Locus
gains = logspace(-2, 2, 500);
@ -139,22 +120,13 @@ xlabel('Real Part'); ylabel('Imaginary Part');
leg = legend('location', 'northwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 12;
% Obtained Damped Plant
% The actively damped plants are computed for the three techniques and compared in Figure ref:fig:rotating_comp_techniques_dampled_plants.
% #+begin_important
% It is shown that while the diagonal (direct) terms of the damped plants are similar for the three active damping techniques, of off-diagonal (coupling) terms are not.
% Integral Force Feedback strategy is adding some coupling at low frequency which may negatively impact the positioning performances.
% #+end_important
%% Compute Damped plants
G_cl_iff = feedback(G, Kiff, 'name');
G_cl_iff_kp = feedback(G_kp, Kiff_kp, 'name');
G_cl_rdc = feedback(G, Krdc, 'name');
%% Comparison of the damped plants obtained with the three active damping techniques
freqs = logspace(-3, 2, 1000);
freqs = logspace(-2, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
@ -195,37 +167,18 @@ hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
ylim([-180 15]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
xticks([1e-2,1e-1,1,1e1,1e2])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$', '$100 \omega_0$'})
% Transmissibility And Compliance
% The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance.
% The transmissibility is here defined as the transfer function from a displacement of the rotating stage along $\vec{i}_x$ to the displacement of the payload along the same direction.
% It is used to characterize how much vibration is transmitted through the suspended platform to the payload.
% The compliance describes the displacement response of the payload to external forces applied to it.
% This is a useful metric when disturbances are directly applied to the payload.
% It is here defined as the transfer function from external forces applied on the payload along $\vec{i}_x$ to the displacement of the payload along the same direction.
% Very similar results are obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure ref:fig:rotating_comp_techniques_transmissibility_compliance).
% #+begin_important
% Using IFF degrades the compliance at low frequency while using relative damping control degrades the transmissibility at high frequency.
% This is very well known characteristics of these common active damping techniques that holds when applied to rotating platforms.
% #+end_important
%% Comparison of the obtained transmissibilty and compliance for the three tested active damping techniques
%% Comparison of the obtained transmissibility and compliance for the three tested active damping techniques
freqs = logspace(-2, 2, 1000);
% transmissibility
figure;
tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
% Transmissibility
ax1 = nexttile();
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Dx', 'Dfx'), freqs, 'rad/s'))), '-', 'color', zeros(1,3), ...
'DisplayName', 'OL')
@ -241,6 +194,7 @@ xlabel('Frequency [rad/s]'); ylabel('Transmissibility [m/m]');
xlim([freqs(1), freqs(end)]);
% Compliance
figure;
ax1 = nexttile();
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Dx', 'Fdx'), freqs, 'rad/s'))), '-', 'color', zeros(1,3), ...

969
matlab/rotating_7_nano_hexapod.m
View File

File diff suppressed because it is too large Load Diff

315
matlab/rotating_8_nass.m
View File

@ -20,21 +20,16 @@ load('uniaxial_micro_station_parameters.mat')
%% Load controllers
load('nass_controllers.mat');
% System dynamics
%% Nano-Hexapod
%% System parameters
mn = 15; % Nano-Hexapod mass [kg]
%% Light Sample
ms = 1; % Sample Mass [kg]
%% General Configuration
% General Configuration
model_config = struct();
model_config.controller = "open_loop"; % Default: Open-Loop
model_config.Tuv_type = "normal"; % Default: 2DoF stage
%% Input/Output definition
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces [Fu, Fv]
io(io_i) = linio([mdl, '/fd'], 1, 'openinput'); io_i = io_i + 1; % Direct Forces on Sample [Fdx, Fdy]
@ -44,13 +39,8 @@ io(io_i) = linio([mdl, '/nano_hexapod'], 1, 'openoutput'); io_i = io_i + 1; % [F
io(io_i) = linio([mdl, '/nano_hexapod'], 2, 'openoutput'); io_i = io_i + 1; % [Du, Dv]
io(io_i) = linio([mdl, '/ext_metrology'],1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy]
% The dynamics of the undamped and damped plants are identified.
% The active damping parameters used are the optimal ones previously identified (i.e. for the rotating nano-hexapod fixed on a rigid platform).
%% Voice Coil (i.e. soft) Nano-Hexapod
%% Identify plant without parallel stiffness
% Voice Coil (i.e. soft) Nano-Hexapod
kn = 1e4; % Nano-Hexapod Stiffness [N/m]
cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
@ -64,7 +54,7 @@ G_vc_fast = linearize(mdl, io, 0.0);
G_vc_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy', 'Ftx', 'Fty'};
G_vc_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
%% APA (i.e. relatively stiff) Nano-Hexapod
% APA (i.e. relatively stiff) Nano-Hexapod
kn = 1e6; % Nano-Hexapod Stiffness [N/m]
cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
@ -78,7 +68,7 @@ G_md_fast = linearize(mdl, io, 0.0);
G_md_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy', 'Ftx', 'Fty'};
G_md_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
%% Piezoelectric (i.e. stiff) Nano-Hexapod
% Piezoelectric (i.e. stiff) Nano-Hexapod
kn = 1e8; % Nano-Hexapod Stiffness [N/m]
cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
@ -146,7 +136,8 @@ G_pz_kp_norot = linearize(mdl, io, 0);
G_pz_kp_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy', 'Ftx', 'Fty'};
G_pz_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
%% Closed Loop Plants - IFF with HPF
%% Compute dampepd plants
% Closed-Loop Plants - IFF with HPF
G_vc_norot_iff_hpf = feedback(G_vc_norot, Kiff_hpf_vc, 'name');
G_vc_fast_iff_hpf = feedback(G_vc_fast, Kiff_hpf_vc, 'name');
@ -156,7 +147,7 @@ G_md_fast_iff_hpf = feedback(G_md_fast, Kiff_hpf_md, 'name');
G_pz_norot_iff_hpf = feedback(G_pz_norot, Kiff_hpf_pz, 'name');
G_pz_fast_iff_hpf = feedback(G_pz_fast, Kiff_hpf_pz, 'name');
%% Closed Loop Plants - IFF with Parallel Stiffness
% Closed-Loop Plants - IFF with Parallel Stiffness
G_vc_norot_iff_kp = feedback(G_vc_kp_norot, Kiff_kp_vc, 'name');
G_vc_fast_iff_kp = feedback(G_vc_kp_fast, Kiff_kp_vc, 'name');
@ -166,7 +157,7 @@ G_md_fast_iff_kp = feedback(G_md_kp_fast, Kiff_kp_md, 'name');
G_pz_norot_iff_kp = feedback(G_pz_kp_norot, Kiff_kp_pz, 'name');
G_pz_fast_iff_kp = feedback(G_pz_kp_fast, Kiff_kp_pz, 'name');
%% Closed Loop Plants - RDC
% Closed-Loop Plants - RDC
G_vc_norot_rdc = feedback(G_vc_norot, Krdc_vc, 'name');
G_vc_fast_rdc = feedback(G_vc_fast, Krdc_vc, 'name');
@ -176,28 +167,11 @@ G_md_fast_rdc = feedback(G_md_fast, Krdc_md, 'name');
G_pz_norot_rdc = feedback(G_pz_norot, Krdc_pz, 'name');
G_pz_fast_rdc = feedback(G_pz_fast, Krdc_pz, 'name');
% The undamped and damped plants are shown in Figure ref:fig:rotating_nass_plant_comp_stiffness.
% Three nano-hexapod velocities are shown (from left to right): $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$.
% The direct terms are shown by the solid curves while the coupling terms are shown by the shaded ones.
% #+begin_important
% It can be observed on Figure ref:fig:rotating_nass_plant_comp_stiffness that:
% - Coupling (ratio between the off-diagonal and direct terms) is larger for the soft nano-hexapod
% - Damping added by the three proposed techniques is quite high and the obtained plant is rather easy to control
% - There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz)
% - The two proposed IFF modification yields similar results
% #+end_important
%% Bode plot of the transfer function from nano-hexapod actuator to measured motion by the external metrology
freqs_vc = logspace(-1, 2, 1000);
freqs_md = logspace(0, 3, 1000);
freqs_pz = logspace(0, 3, 1000);
figure;
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
@ -220,10 +194,30 @@ plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_rdc('Dy', 'Fu'), freqs_vc, 'Hz')))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-12, 1e-2])
title('$k_n = 0.01\,N/\mu m$');
ylim([1e-8, 1e-2])
ax2 = nexttile([2,1]);
ax2 = nexttile;
hold on;
plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', zeros(1,3));
plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_iff_hpf('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(1,:));
plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_iff_kp('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(2,:));
plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_rdc('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(3,:));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([ -200, 20]);
linkaxes([ax,ax2],'x');
xlim([freqs_vc(1), freqs_vc(end)]);
xticks([1e-1, 1e0, 1e1]);
freqs_md = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs_md, abs(squeeze(freqresp(G_md_fast('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -243,11 +237,31 @@ plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc('Dy', 'Fu'), freqs_md, 'Hz')))
'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
ylim([1e-12, 1e-2])
title('$k_n = 1\,N/\mu m$');
ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-10, 1e-4])
ax3 = nexttile([2,1]);
ax2 = nexttile;
hold on;
plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', zeros(1,3));
plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_iff_hpf('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(1,:));
plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_iff_kp('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(2,:));
plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_rdc('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(3,:));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([ -200, 20]);
linkaxes([ax1,ax2],'x');
xlim([freqs_md(1), freqs_md(end)]);
xticks([1e0, 1e1, 1e2]);
freqs_pz = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -267,39 +281,12 @@ plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_rdc('Dy', 'Fu'), freqs_pz, 'Hz')))
'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
ylim([1e-12, 1e-2])
ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-12, 1e-6])
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [20, 1];
title('$k_n = 100\,N/\mu m$');
ax1b = nexttile;
hold on;
plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', zeros(1,3));
plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_iff_hpf('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(1,:));
plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_iff_kp('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(2,:));
plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_rdc('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(3,:));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-270, 90]);
ax2b = nexttile;
hold on;
plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', zeros(1,3));
plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_iff_hpf('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(1,:));
plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_iff_kp('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(2,:));
plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_rdc('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(3,:));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
hold off;
yticks(-360:90:360);
ylim([-270, 90]);
ax3b = nexttile;
ax2 = nexttile;
hold on;
plot(freqs_pz, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_fast('Dx', 'Fu'), freqs_pz, 'Hz')))), 'color', zeros(1,3));
plot(freqs_pz, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_fast_iff_hpf('Dx', 'Fu'), freqs_pz, 'Hz')))), 'color', colors(1,:));
@ -307,111 +294,19 @@ plot(freqs_pz, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_fast_iff_kp('Dx', 'Fu')
plot(freqs_pz, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_fast_rdc('Dx', 'Fu'), freqs_pz, 'Hz')))), 'color', colors(3,:));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-270, 90]);
ylim([ -200, 20]);
linkaxes([ax1,ax1b],'x');
xlim([freqs_vc(1), freqs_vc(end)]);
linkaxes([ax2,ax2b],'x');
xlim([freqs_md(1), freqs_md(end)]);
linkaxes([ax3,ax3b],'x');
linkaxes([ax1,ax2],'x');
xlim([freqs_pz(1), freqs_pz(end)]);
% #+name: fig:rotating_nass_plant_comp_stiffness
% #+caption: Bode plot of the transfer function from nano-hexapod actuator to measured motion by the external metrology
% #+RESULTS:
% [[file:figs/rotating_nass_plant_comp_stiffness.png]]
% To confirm that the coupling is smaller when the stiffness of the nano-hexapod is increase, the /coupling ratio/ for the three nano-hexapod stiffnesses are shown in Figure ref:fig:rotating_nass_plant_coupling_comp.
%% Coupling ratio for the proposed active damping techniques evaluated for the three nano-hexapod stiffnesses
freqs_vc = logspace(-1, 2, 1000);
freqs_md = logspace(0, 3, 1000);
freqs_pz = logspace(0, 3, 1000);
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast('Dy', 'Fu')/G_vc_fast('Dx', 'Fu'), freqs_vc, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_iff_hpf('Dy', 'Fu')/G_vc_fast_iff_hpf('Dx', 'Fu'), freqs_vc, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', 'IFF + $k_p$');
plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_iff_kp('Dy', 'Fu')/G_vc_fast_iff_kp('Dx', 'Fu'), freqs_vc, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', 'IFF + HPF');
plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_rdc('Dy', 'Fu')/G_vc_fast_rdc('Dx', 'Fu'), freqs_vc, 'Hz'))), 'color', colors(3,:), ...
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Coupling Ratio');
title('$k_n = 0.01\,N/\mu m$');
ax2 = nexttile();
hold on;
plot(freqs_md, abs(squeeze(freqresp(G_md_fast('Dy', 'Fu')/G_md_fast('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
plot(freqs_md, abs(squeeze(freqresp(G_md_fast_iff_hpf('Dy', 'Fu')/G_md_fast_iff_hpf('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', 'IFF + $k_p$');
plot(freqs_md, abs(squeeze(freqresp(G_md_fast_iff_kp('Dy', 'Fu')/G_md_fast_iff_kp('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', 'IFF + HPF');
plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc('Dy', 'Fu')/G_md_fast_rdc('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', colors(3,:), ...
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
title('$k_n = 1\,N/\mu m$');
ax3 = nexttile();
hold on;
plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast('Dy', 'Fu')/G_pz_fast('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_iff_hpf('Dy', 'Fu')/G_pz_fast_iff_hpf('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', 'IFF + $k_p$');
plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_iff_kp('Dy', 'Fu')/G_pz_fast_iff_kp('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', 'IFF + HPF');
plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_rdc('Dy', 'Fu')/G_pz_fast_rdc('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', colors(3,:), ...
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
title('$k_n = 100\,N/\mu m$');
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [20, 1];
linkaxes([ax1,ax2,ax3], 'y')
% Effect of disturbances
% The effect of three disturbances are considered:
% - Floor motion (Figure ref:fig:rotating_nass_effect_floor_motion)
% - Micro-Station vibrations (Figure ref:fig:rotating_nass_effect_stage_vibration)
% - Direct force applied on the payload (Figure ref:fig:rotating_nass_effect_direct_forces)
% #+begin_important
% Conclusions are similar than with the uniaxial (non-rotating) model:
% - Regarding the effect of floor motion and forces applied on the payload:
% - The stiffer, the better (magnitudes are lower for the right curves, Figures ref:fig:rotating_nass_effect_floor_motion and ref:fig:rotating_nass_effect_direct_forces)
% - Integral Force Feedback degrades the performances at low frequency compared to relative damping control
% - Regarding the effect of micro-station vibrations:
% - Having a soft nano-hexapod allows to filter these vibrations between the suspensions modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduce this filtering (Figure ref:fig:rotating_nass_effect_stage_vibration, left).
% #+end_important
xticks([1e0, 1e1, 1e2]);
%% Effect of Floor motion on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
freqs = logspace(-1, 3, 1000);
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(freqs, abs(squeeze(freqresp(G_vc_fast('Dx', 'Dfx'), freqs, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -423,12 +318,12 @@ plot(freqs, abs(squeeze(freqresp(G_vc_fast_rdc('Dx', 'Dfx'), freqs, 'Hz'))), 'co
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/D_{f,x}$ [m/N]');
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/x_{f,x}$ [m/N]');
xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
xtickangle(0)
title('$k_n = 0.01\,N/\mu m$');
ylim([1e-4, 1e2]);
ax2 = nexttile();
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_md_fast('Dx', 'Dfx'), freqs, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -440,12 +335,12 @@ plot(freqs, abs(squeeze(freqresp(G_md_fast_rdc('Dx', 'Dfx'), freqs, 'Hz'))), 'co
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/x_{f,x}$ [m/N]');
xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
xtickangle(0)
title('$k_n = 1\,N/\mu m$');
ylim([1e-4, 1e2]);
ax3 = nexttile();
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_pz_fast('Dx', 'Dfx'), freqs, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -457,28 +352,15 @@ plot(freqs, abs(squeeze(freqresp(G_pz_fast_rdc('Dx', 'Dfx'), freqs, 'Hz'))), 'co
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/x_{f,x}$ [m/N]');
xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
xtickangle(0)
title('$k_n = 100\,N/\mu m$');
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [20, 1];
linkaxes([ax1,ax2,ax3], 'y')
% #+name: fig:rotating_nass_effect_floor_motion
% #+caption: Effect of Floor motion on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
% #+RESULTS:
% [[file:figs/rotating_nass_effect_floor_motion.png]]
ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [15, 1];
ylim([1e-4, 1e2]);
%% Effect of micro-station vibrations on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(freqs, abs(squeeze(freqresp(G_vc_fast('Dx', 'Ftx'), freqs, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -493,9 +375,9 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{t,x}$ [m/N]');
xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
xtickangle(0)
title('$k_n = 0.01\,N/\mu m$');
ylim([1e-12, 2e-7]);
ax2 = nexttile();
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_md_fast('Dx', 'Ftx'), freqs, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -507,12 +389,12 @@ plot(freqs, abs(squeeze(freqresp(G_md_fast_rdc('Dx', 'Ftx'), freqs, 'Hz'))), 'co
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{t,x}$ [m/N]');
xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
xtickangle(0)
title('$k_n = 1\,N/\mu m$');
ylim([1e-12, 2e-7]);
ax3 = nexttile();
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_pz_fast('Dx', 'Ftx'), freqs, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -524,28 +406,15 @@ plot(freqs, abs(squeeze(freqresp(G_pz_fast_rdc('Dx', 'Ftx'), freqs, 'Hz'))), 'co
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{t,x}$ [m/N]');
xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
xtickangle(0)
title('$k_n = 100\,N/\mu m$');
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [20, 1];
linkaxes([ax1,ax2,ax3], 'y')
% #+name: fig:rotating_nass_effect_stage_vibration
% #+caption: Effect of micro-station vibrations on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
% #+RESULTS:
% [[file:figs/rotating_nass_effect_stage_vibration.png]]
ylim([1e-12, 2e-7]);
%% Effect of sample forces on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(freqs, abs(squeeze(freqresp(G_vc_fast('Dx', 'Fdx'), freqs, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -560,9 +429,9 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{s,x}$ [m/N]');
xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
xtickangle(0)
title('$k_n = 0.01\,N/\mu m$');
ylim([1e-8, 1e-2])
ax2 = nexttile();
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_md_fast('Dx', 'Fdx'), freqs, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -574,12 +443,12 @@ plot(freqs, abs(squeeze(freqresp(G_md_fast_rdc('Dx', 'Fdx'), freqs, 'Hz'))), 'co
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{s,x}$ [m/N]');
xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
xtickangle(0)
title('$k_n = 1\,N/\mu m$');
ylim([1e-8, 1e-2])
ax3 = nexttile();
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_pz_fast('Dx', 'Fdx'), freqs, 'Hz'))), 'color', zeros(1,3), ...
'DisplayName', 'OL');
@ -591,11 +460,11 @@ plot(freqs, abs(squeeze(freqresp(G_pz_fast_rdc('Dx', 'Fdx'), freqs, 'Hz'))), 'co
'DisplayName', 'RDC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{s,x}$ [m/N]');
xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
xtickangle(0)
title('$k_n = 100\,N/\mu m$');
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [20, 1];
linkaxes([ax1,ax2,ax3], 'y')
ylim([1e-8, 1e-2])

2
matlab/src/computeSimultaneousDamping.m
View File

@ -1,5 +1,3 @@
% =computeSimultaneousDamping=
function [xi_min] = computeSimultaneousDamping(g, G, K)
[~, xi] = damp(minreal(feedback(G, g*K), [], false));
xi_min = 1/min(xi);

2
matlab/src/rootLocusPolesSorted.m
View File

@ -1,5 +1,3 @@
% =rootLocusPolesSorted=
function [poles] = rootLocusPolesSorted(G, K, gains, args)
% rootLocusPolesSorted -
%

2
nass-rotating-3dof-model.org
View File

@ -22,7 +22,7 @@
#+BIND: org-latex-bib-compiler "biber"
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :comments no
#+PROPERTY: header-args:matlab+ :exports none
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :eval no-export