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Removed DVF from the study (replace with passive)

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Thomas Dehaeze
2020-06-29 10:22:52 +02:00
parent 66714c6974
commit 75e3dacd71
19 changed files with 6197 additions and 6607 deletions
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@@ -29,7 +29,6 @@
- Section [[sec:iff_pure_int]]
- Section [[sec:iff_pseudo_int]]
- Section [[sec:iff_parallel_stiffness]]
- Section [[sec:dvf]]
- Section [[sec:comparison]]
- Section [[sec:notations]]
@@ -215,7 +214,6 @@ Define the rotating speed for the Simscape Model.
#+begin_src matlab :exports none
Kiff = tf(zeros(2));
Kdvf = tf(zeros(2));
kp = 0; % Parallel Stiffness [N/m]
cp = 0; % Parallel Damping [N/(m/s)]
@@ -492,7 +490,6 @@ The rotation speed is set to $\Omega = 0.1 \omega_0$.
#+begin_src matlab :exports none
Kiff = tf(zeros(2));
Kdvf = tf(zeros(2));
#+end_src
#+begin_src matlab
@@ -1294,7 +1291,6 @@ The same transfer function from $[F_u, F_v]$ to $[f_u, f_v]$ is written down fro
#+begin_src matlab :exports none
Kiff = tf(zeros(2));
Kdvf = tf(zeros(2));
#+end_src
#+begin_src matlab
@@ -1897,280 +1893,6 @@ Let's take $k_p = 5 m \Omega^2$ and find the optimal IFF control gain $g$ such t
exportFig('figs-inkscape/root_locus_opt_gain_iff_kp.pdf', 'width', 'wide', 'height', 'tall', 'png', false, 'pdf', false, 'svg', true);
#+end_src
* Direct Velocity Feedback
:PROPERTIES:
:header-args:matlab+: :tangle matlab/s5_dvf.m
:header-args:matlab+: :comments org :mkdirp yes
:END:
<<sec:dvf>>
** Introduction :ignore:
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no
addpath('./matlab/');
addpath('./src/');
#+end_src
** Schematic
#+name: fig:system_dvf
#+caption: Figure caption
[[file:figs-tikz/system_dvf.png]]
** Equations
The sensed relative velocity are equal to:
#+begin_important
\begin{equation}
\begin{bmatrix} \dot{d}_u \\ \dot{d}_v \end{bmatrix} =
\bm{G}_v
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
\begin{equation}
\begin{bmatrix} \dot{d}_u \\ \dot{d}_v \end{bmatrix} =
\frac{s}{k} \frac{1}{G_{vp}}
\begin{bmatrix}
G_{vz} & G_{vc} \\
-G_{vc} & G_{vz}
\end{bmatrix}
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
With:
\begin{align}
G_{vp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{vz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
G_{vc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}
\end{align}
#+end_important
** Plant Parameters
Let's define initial values for the model.
#+begin_src matlab
k = 1; % Actuator Stiffness [N/m]
c = 0.05; % Actuator Damping [N/(m/s)]
m = 1; % Payload mass [kg]
#+end_src
#+begin_src matlab
xi = c/(2*sqrt(k*m));
w0 = sqrt(k/m); % [rad/s]
#+end_src
#+begin_src matlab :exports none
kp = 0; % [N/m]
cp = 0; % [N/(m/s)]
#+end_src
** Comparison of the Analytical Model and the Simscape Model
The rotating speed is set to $\Omega = 0.1 \omega_0$.
#+begin_src matlab
W = 0.1*w0;
#+end_src
#+begin_src matlab :exports none
Kiff = tf(zeros(2));
Kdvf = tf(zeros(2));
#+end_src
#+begin_src matlab
open('rotating_frame.slx');
#+end_src
And the transfer function from $[F_u, F_v]$ to $[v_u, v_v]$ is identified using the Simscape model.
#+begin_src matlab
%% Name of the Simulink File
mdl = 'rotating_frame';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
#+end_src
#+begin_src matlab
Gdvf = linearize(mdl, io, 0);
%% Input/Output definition
Gdvf.InputName = {'Fu', 'Fv'};
Gdvf.OutputName = {'Vu', 'Vv'};
#+end_src
The same transfer function from $[F_u, F_v]$ to $[v_u, v_v]$ is written down from the analytical model.
#+begin_src matlab
Gdvf_th = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
Gdvf_th.InputName = {'Fu', 'Fv'};
Gdvf_th.OutputName = {'vu', 'vv'};
#+end_src
The two are compared in Figure [[fig:plant_iff_comp_simscape_analytical]] and found to perfectly match.
#+begin_src matlab :exports none
freqs = logspace(-1, 1, 1000);
figure;
ax1 = subplot(2, 2, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs))), '-')
plot(freqs, abs(squeeze(freqresp(Gdvf_th(1,1), freqs))), '--')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [$\frac{m/s}{N}$]');
title('$v_u/F_u$, $v_v/F_v$');
ax3 = subplot(2, 2, 3);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(1,1), freqs))), '-')
plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf_th(1,1), freqs))), '--')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
hold off;
ax2 = subplot(2, 2, 2);
hold on;
plot(freqs, abs(squeeze(freqresp(Gdvf(1,2), freqs))), '-')
plot(freqs, abs(squeeze(freqresp(Gdvf_th(1,2), freqs))), '--')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [$\frac{m/s}{N}$]');
title('$v_u/F_v$, $v_v/F_u$');
ax4 = subplot(2, 2, 4);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(1,2), freqs))), '-')
plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf_th(1,2), freqs))), '--')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
hold off;
linkaxes([ax1,ax2,ax3,ax4],'x');
xlim([freqs(1), freqs(end)]);
linkaxes([ax1,ax2],'y');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/plant_dvf_comp_simscape_analytical.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:plant_dvf_comp_simscape_analytical
#+caption: Comparison of the transfer functions from $[F_u, F_v]$ to $[v_u, v_v]$ between the Simscape model and the analytical one
#+RESULTS:
[[file:figs/plant_dvf_comp_simscape_analytical.png]]
** Root Locus
The Decentralized Direct Velocity Feedback controller consist of a pure gain on the diagonal:
\begin{equation}
K_{\text{DVF}}(s) = g \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\end{equation}
The corresponding Root Locus plots for the following rotating speeds are shown in Figure [[fig:root_locus_dvf]].
#+begin_src matlab
Ws = [0, 0.2, 0.7, 1.1]*w0; % Rotating Speeds [rad/s]
#+end_src
It is shown that for rotating speed $\Omega < \omega_0$, the closed loop system is unconditionally stable and arbitrary damping can be added to the poles.
#+begin_src matlab :exports none
gains = logspace(-2, 1, 100);
figure;
hold on;
for W_i = 1:length(Ws)
W = Ws(W_i);
Gdvf = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
set(gca,'ColorOrderIndex',W_i);
plot(real(pole(Gdvf)), imag(pole(Gdvf)), 'x', ...
'DisplayName', sprintf('$\\Omega = %.2f \\omega_0 $', W/w0));
set(gca,'ColorOrderIndex',W_i);
plot(real(tzero(Gdvf)), imag(tzero(Gdvf)), 'o', ...
'HandleVisibility', 'off');
for g = gains
set(gca,'ColorOrderIndex',W_i);
cl_poles = pole(feedback(Gdvf, g*eye(2)));
plot(real(cl_poles), imag(cl_poles), '.', ...
'HandleVisibility', 'off');
end
end
hold off;
axis square;
xlim([-2, 0.5]); ylim([0, 2.5]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/root_locus_dvf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:root_locus_dvf
#+caption: Root Locus for the Decentralized Direct Velocity Feedback controller. Several rotating speed are shown.
#+RESULTS:
[[file:figs/root_locus_dvf.png]]
#+begin_src matlab :exports none :tangle no
gains = logspace(-2, 1, 1000);
figure;
hold on;
for W_i = 1:length(Ws)
W = Ws(W_i);
Gdvf = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
set(gca,'ColorOrderIndex',W_i);
plot(real(pole(Gdvf)), imag(pole(Gdvf)), 'x', ...
'DisplayName', sprintf('$\\Omega = %.2f \\omega_0 $', W/w0));
set(gca,'ColorOrderIndex',W_i);
plot(real(tzero(Gdvf)), imag(tzero(Gdvf)), 'o', ...
'HandleVisibility', 'off');
poles_dvf = rootLocusPolesSorted(Gdvf, eye(2), gains, 'd_max', 1e-4);
for p_i = 1:size(poles_dvf, 2)
set(gca,'ColorOrderIndex', W_i);
plot(real(poles_dvf(:, p_i)), imag(poles_dvf(:, p_i)), '-', ...
'HandleVisibility', 'off');
end
end
hold off;
axis square;
xlim([-2, 0.5]); ylim([0, 2.5]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src
#+begin_src matlab :tangle no :exports none :results none
exportFig('figs-inkscape/root_locus_dvf.pdf', 'width', 'wide', 'height', 'tall', 'png', false, 'pdf', false, 'svg', true);
#+end_src
* Comparison
:PROPERTIES:
:header-args:matlab+: :tangle matlab/s6_act_damp_comparison.m
@@ -2241,13 +1963,6 @@ IFF With parallel Stiffness
k = k + kp;
#+end_src
DVF
#+begin_src matlab
Gdvf = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
#+end_src
#+begin_src matlab :exports none
figure;
@@ -2281,20 +1996,6 @@ DVF
plot(real(cl_poles), imag(cl_poles), '.', ...
'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',3);
plot(real(pole(Gdvf)), imag(pole(Gdvf)), 'x', ...
'DisplayName', 'DVF');
set(gca,'ColorOrderIndex',3);
plot(real(tzero(Gdvf)), imag(tzero(Gdvf)), 'o', ...
'HandleVisibility', 'off');
for g = gains
Kdvf = g*eye(2);
cl_poles = pole(feedback(Gdvf, Kdvf));
set(gca,'ColorOrderIndex',3);
plot(real(cl_poles), imag(cl_poles), '.', ...
'HandleVisibility', 'off');
end
hold off;
axis square;
xlim([-1.2, 0.05]); ylim([0, 1.25]);
@@ -2308,7 +2009,7 @@ DVF
#+end_src
#+name: fig:comp_root_locus
#+caption: Root Locus plot - Comparison of IFF with additional high pass filter, IFF with additional parallel stiffness and DVF
#+caption: Root Locus plot - Comparison of IFF with additional high pass filter, IFF with additional parallel stiffness
#+RESULTS:
[[file:figs/comp_root_locus.png]]
@@ -2317,7 +2018,6 @@ DVF
poles_iff_hpf = rootLocusPolesSorted(Giff, 1/(s + wi)*eye(2), gains, 'd_max', 1e-4);
poles_iff_kp = rootLocusPolesSorted(Giff_kp, 1/s*eye(2), gains, 'd_max', 1e-4);
poles_dvf = rootLocusPolesSorted(Gdvf, eye(2), gains, 'd_max', 1e-4);
figure;
@@ -2345,18 +2045,6 @@ DVF
plot(real(poles_iff_kp(:, p_i)), imag(poles_iff_kp(:, p_i)), '-', ...
'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',3);
plot(real(pole(Gdvf)), imag(pole(Gdvf)), 'x', ...
'DisplayName', 'DVF');
set(gca,'ColorOrderIndex',3);
plot(real(tzero(Gdvf)), imag(tzero(Gdvf)), 'o', ...
'HandleVisibility', 'off');
for p_i = 1:size(poles_dvf, 2)
set(gca,'ColorOrderIndex',3);
plot(real(poles_dvf(:, p_i)), imag(poles_dvf(:, p_i)), '-', ...
'HandleVisibility', 'off');
end
hold off;
axis square;
xlim([-1.2, 0.05]); ylim([0, 1.25]);
@@ -2408,28 +2096,10 @@ In order to compare to three considered Active Damping techniques, gains that yi
end
#+end_src
#+begin_src matlab :exports none
%% Direct Velocity Feedback
gains = logspace(0, 2, 100);
opt_zeta_dvf = 0;
opt_gain_dvf = 0;
for g = gains
Kdvf = g*eye(2);
[w, zeta] = damp(minreal(feedback(Gdvf, Kdvf)));
if min(zeta) > opt_zeta_dvf && all(zeta > 0) && min(zeta) < 0.85
opt_zeta_dvf = min(zeta);
opt_gain_dvf = g;
end
end
#+end_src
The obtained damping ratio and control are shown below.
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([opt_zeta_iff, opt_zeta_kp, opt_zeta_dvf; opt_gain_iff, opt_gain_kp, opt_gain_dvf]', {'Modified IFF', 'IFF with $k_p$', 'DVF'}, {'Obtained $\xi$', 'Control Gain'}, ' %.2f ');
data2orgtable([opt_zeta_iff, opt_zeta_kp; opt_gain_iff, opt_gain_kp]', {'Modified IFF', 'IFF with $k_p$'}, {'Obtained $\xi$', 'Control Gain'}, ' %.2f ');
#+end_src
#+RESULTS:
@@ -2437,24 +2107,19 @@ The obtained damping ratio and control are shown below.
|----------------+----------------+--------------|
| Modified IFF | 0.83 | 2.0 |
| IFF with $k_p$ | 0.83 | 2.01 |
| DVF | 0.85 | 1.67 |
** Transmissibility
** Passive Damping - Critical Damping
#+begin_src matlab
c_opt = 2*sqrt(k*m);
#+end_src
** Transmissibility And Compliance
<<sec:comp_transmissibilty>>
#+begin_src matlab
open('rotating_frame.slx');
#+end_src
*** Open Loop :ignore:
#+begin_src matlab :exports none
Kdvf = tf(zeros(2));
Kiff = tf(zeros(2));
kp = 0;
cp = 0;
#+end_src
#+begin_src matlab
%% Name of the Simulink File
mdl = 'rotating_frame';
@@ -2462,23 +2127,57 @@ The obtained damping ratio and control are shown below.
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/dw'], 1, 'input'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/fd'], 1, 'input'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Meas'], 1, 'output'); io_i = io_i + 1;
#+end_src
*** Open Loop :ignore:
#+begin_src matlab :exports none
Kiff = tf(zeros(2));
kp = 0;
cp = 0;
#+end_src
#+begin_src matlab
Tol = linearize(mdl, io, 0);
G_ol = linearize(mdl, io, 0);
%% Input/Output definition
Tol.InputName = {'Dwx', 'Dwy'};
Tol.OutputName = {'Dx', 'Dy'};
G_ol.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
G_ol.OutputName = {'Dx', 'Dy'};
#+end_src
*** Passive Damping
#+begin_src matlab
kp = 0;
cp = 0;
#+end_src
#+begin_src matlab
c_old = c;
c = c_opt;
#+end_src
#+begin_src matlab :exports none
Kiff = tf(zeros(2));
#+end_src
#+begin_src matlab
G_pas = linearize(mdl, io, 0);
%% Input/Output definition
G_pas.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
G_pas.OutputName = {'Dx', 'Dy'};
#+end_src
#+begin_src matlab
c = c_old;
#+end_src
*** Pseudo Integrator IFF :ignore:
#+begin_src matlab :exports none
kp = 0;
cp = 0;
Kdvf = tf(zeros(2));
#+end_src
#+begin_src matlab
@@ -2486,11 +2185,11 @@ The obtained damping ratio and control are shown below.
#+end_src
#+begin_src matlab
Tiff = linearize(mdl, io, 0);
G_iff = linearize(mdl, io, 0);
%% Input/Output definition
Tiff.InputName = {'Dwx', 'Dwy'};
Tiff.OutputName = {'Dx', 'Dy'};
G_iff.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
G_iff.OutputName = {'Dx', 'Dy'};
#+end_src
*** IFF With parallel Stiffness :ignore:
@@ -2503,36 +2202,12 @@ The obtained damping ratio and control are shown below.
Kiff = opt_gain_kp/s*tf(eye(2));
#+end_src
#+begin_src matlab :exports none
Kdvf = tf(zeros(2));
#+end_src
#+begin_src matlab
Tiff_kp = linearize(mdl, io, 0);
G_kp = linearize(mdl, io, 0);
%% Input/Output definition
Tiff_kp.InputName = {'Dwx', 'Dwy'};
Tiff_kp.OutputName = {'Dx', 'Dy'};
#+end_src
*** DVF :ignore:
#+begin_src matlab :exports none
kp = 0;
cp = 0;
Kiff = tf(zeros(2));
#+end_src
#+begin_src matlab
Kdvf = opt_gain_kp*tf(eye(2));
#+end_src
#+begin_src matlab
Tdvf = linearize(mdl, io, 0);
%% Input/Output definition
Tdvf.InputName = {'Dwx', 'Dwy'};
Tdvf.OutputName = {'Dx', 'Dy'};
G_kp.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
G_kp.OutputName = {'Dx', 'Dy'};
#+end_src
*** Transmissibility :ignore:
@@ -2541,13 +2216,13 @@ The obtained damping ratio and control are shown below.
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Tiff(1,1), freqs))), ...
plot(freqs, abs(squeeze(freqresp(G_iff({'Dx'}, {'Dwx'}), freqs))), ...
'DisplayName', 'IFF + HPF')
plot(freqs, abs(squeeze(freqresp(Tiff_kp(1,1), freqs))), ...
plot(freqs, abs(squeeze(freqresp(G_kp( {'Dx'}, {'Dwx'}), freqs))), ...
'DisplayName', 'IFF + $k_p$')
plot(freqs, abs(squeeze(freqresp(Tdvf(1,1), freqs))), ...
'DisplayName', 'DVF')
plot(freqs, abs(squeeze(freqresp(Tol(1,1), freqs))), 'k-', ...
plot(freqs, abs(squeeze(freqresp(G_pas({'Dx'}, {'Dwx'}), freqs))), ...
'DisplayName', 'Passive')
plot(freqs, abs(squeeze(freqresp(G_ol( {'Dx'}, {'Dwx'}), freqs))), 'k-', ...
'DisplayName', 'Open-Loop')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -2568,110 +2243,19 @@ The obtained damping ratio and control are shown below.
exportFig('figs-inkscape/comp_transmissibility.pdf', 'width', 'half', 'height', 'tall', 'png', false, 'pdf', false, 'svg', true);
#+end_src
** Compliance
<<sec:comp_compliance>>
*** Open Loop :ignore:
#+begin_src matlab :exports none
Kdvf = tf(zeros(2));
Kiff = tf(zeros(2));
kp = 0;
cp = 0;
#+end_src
#+begin_src matlab
%% Name of the Simulink File
mdl = 'rotating_frame';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/fd'], 1, 'input'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Meas'], 1, 'output'); io_i = io_i + 1;
#+end_src
#+begin_src matlab
Col = linearize(mdl, io, 0);
%% Input/Output definition
Col.InputName = {'Fdx', 'Fdy'};
Col.OutputName = {'Dx', 'Dy'};
#+end_src
*** Pseudo Integrator IFF :ignore:
#+begin_src matlab :exports none
kp = 0;
cp = 0;
Kdvf = tf(zeros(2));
#+end_src
#+begin_src matlab
Kiff = opt_gain_iff/(wi + s)*tf(eye(2));
#+end_src
#+begin_src matlab
Ciff = linearize(mdl, io, 0);
%% Input/Output definition
Ciff.InputName = {'Fdx', 'Fdy'};
Ciff.OutputName = {'Dx', 'Dy'};
#+end_src
*** IFF With parallel Stiffness :ignore:
#+begin_src matlab
kp = 5*m*W^2;
cp = 0.01;
#+end_src
#+begin_src matlab
Kiff = opt_gain_kp/s*tf(eye(2));
#+end_src
#+begin_src matlab :exports none
Kdvf = tf(zeros(2));
#+end_src
#+begin_src matlab
Ciff_kp = linearize(mdl, io, 0);
%% Input/Output definition
Ciff_kp.InputName = {'Fdx', 'Fdy'};
Ciff_kp.OutputName = {'Dx', 'Dy'};
#+end_src
*** DVF :ignore:
#+begin_src matlab :exports none
kp = 0;
cp = 0;
Kiff = tf(zeros(2));
#+end_src
#+begin_src matlab
Kdvf = opt_gain_kp*tf(eye(2));
#+end_src
#+begin_src matlab
Cdvf = linearize(mdl, io, 0);
%% Input/Output definition
Cdvf.InputName = {'Fdx', 'Fdy'};
Cdvf.OutputName = {'Dx', 'Dy'};
#+end_src
*** Compliance :ignore:
#+begin_src matlab :exports none
freqs = logspace(-2, 1, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Ciff(1,1), freqs))), ...
plot(freqs, abs(squeeze(freqresp(G_iff({'Dx'}, {'Fdx'}), freqs))), ...
'DisplayName', 'IFF + HPF')
plot(freqs, abs(squeeze(freqresp(Ciff_kp(1,1), freqs))), ...
plot(freqs, abs(squeeze(freqresp(G_kp( {'Dx'}, {'Fdx'}), freqs))), ...
'DisplayName', 'IFF + $k_p$')
plot(freqs, abs(squeeze(freqresp(Cdvf(1,1), freqs))), ...
'DisplayName', 'DVF')
plot(freqs, abs(squeeze(freqresp(Col(1,1), freqs))), 'k-', ...
plot(freqs, abs(squeeze(freqresp(G_pas({'Dx'}, {'Fdx'}), freqs))), ...
'DisplayName', 'Passive')
plot(freqs, abs(squeeze(freqresp(G_ol( {'Dx'}, {'Fdx'}), freqs))), 'k-', ...
'DisplayName', 'Open-Loop')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');

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matlab/matlab/rotating_frame.slx
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