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# Windows default autosave extension
*.asv
*rtw/
test_at**.m
data/
# OSX / *nix default autosave extension
*.m~
# Compiled MEX binaries (all platforms)
*.mex*
# Packaged app and toolbox files
*.mlappinstall
*.slxc
*.mldatx
*.mltbx
*.xml
piezoapa_slrt_rtw/
# Generated helpsearch folders
helpsearch*/
# Simulink code generation folders
slprj/
sccprj/
# Matlab code generation folders
codegen/
# Simulink autosave extension
*.autosave
# Simulink cache files
*.slxc
# Octave session info
octave-workspace

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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
addpath('./mat/');
% Load Data :ignore:
% The data are loaded:
load('apa95ml_5kg_2a_1s.mat', 't', 'u', 'v');
% Any offset is removed.
u = detrend(u, 0); % Speedgoat DAC output Voltage [V]
v = detrend(v, 0); % Speedgoat ADC input Voltage (sensor stack) [V]
% Here, the amplifier gain is 20.
u = 20*u; % Actuator Stack Voltage [V]
% Compute TF estimate and Coherence :ignore:
% The transfer function from the actuator voltage $V_a$ to the force sensor stack voltage $V_s$ is computed.
Ts = t(end)/(length(t)-1);
Fs = 1/Ts;
win = hann(ceil(5/Ts));
[tf_est, f] = tfestimate(u, v, win, [], [], 1/Ts);
[coh, ~] = mscohere( u, v, win, [], [], 1/Ts);
% The coherence is shown in Figure [[fig:apa95ml_5kg_cedrat_coh]].
figure;
hold on;
plot(f, coh, 'k-')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Coherence'); xlabel('Frequency [Hz]');
hold off;
xlim([10, 5e3]);
% #+name: fig:apa95ml_5kg_cedrat_coh
% #+caption: Coherence
% #+RESULTS:
% [[file:figs/apa95ml_5kg_cedrat_coh.png]]
% The Simscape model is loaded and compared with the identified dynamics in Figure [[fig:bode_plot_force_sensor_voltage_comp_fem]].
% The non-minimum phase zero is just a side effect of the not so great identification.
% Taking longer measurements would results in a minimum phase zero.
load('mat/fem_simscape_models.mat', 'Gfm');
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(tf_est))
plot(freqs, abs(squeeze(freqresp(Gfm, freqs, 'Hz'))))
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-3, 1e1]);
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(tf_est), ...
'DisplayName', 'Identification')
plot(freqs, 180/pi*angle(squeeze(freqresp(Gfm, freqs, 'Hz'))), ...
'DisplayName', 'FEM')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Phase'); xlabel('Frequency [Hz]');
hold off;
ylim([-180, 180]);
yticks(-180:90:180);
legend('location', 'northeast')
linkaxes([ax1,ax2], 'x');
xlim([10, 5e3]);

34
matlab/huddle_test.m Normal file
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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
addpath('./mat/');
% Load Data :noexport:
load('huddle_test.mat', 't', 'y');
y = y - mean(y(1000:end));
% Time Domain Data
figure;
plot(t(1000:end), y(1000:end))
ylabel('Output Displacement [m]'); xlabel('Time [s]');
% PSD of Measurement Noise
Ts = t(end)/(length(t)-1);
Fs = 1/Ts;
win = hanning(ceil(1*Fs));
[pxx, f] = pwelch(y(1000:end), win, [], [], Fs);
figure;
plot(f, sqrt(pxx));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
xlim([1, Fs/2]);

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@ -0,0 +1,307 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
addpath('./mat/');
% IFF Plant
% From the identified plant, a model of the transfer function from the actuator stack voltage to the force sensor generated voltage is developed.
w_z = 2*pi*111; % Zeros frequency [rad/s]
w_p = 2*pi*255; % Pole frequency [rad/s]
xi_z = 0.05;
xi_p = 0.015;
G_inf = 0.1;
Gi = G_inf*(s^2 + 2*xi_z*w_z*s + w_z^2)/(s^2 + 2*xi_p*w_p*s + w_p^2);
% Its bode plot is shown in Figure [[fig:iff_plant_identification_apa95ml]].
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gi, freqs, 'Hz'))), 'k-')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel', []);
hold off;
ylim([1e-3, 1e1]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gi, freqs, 'Hz'))), 'k-')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Phase'); xlabel('Frequency [Hz]');
hold off;
ylim([-180, 180]);
yticks(-180:90:180);
linkaxes([ax1,ax2], 'x');
xlim([10, 5e3]);
% #+name: fig:iff_plant_identification_apa95ml
% #+caption: Bode plot of the IFF plant
% #+RESULTS:
% [[file:figs/iff_plant_identification_apa95ml.png]]
% The controller used in the Integral Force Feedback Architecture is:
% \begin{equation}
% K_{\text{IFF}}(s) = \frac{g}{s + 2\cdot 2\pi} \cdot \frac{s}{s + 0.5 \cdot 2\pi}
% \end{equation}
% where $g$ is a gain that can be tuned.
% Above 2 Hz the controller is basically an integrator, whereas an high pass filter is added at 0.5Hz to further reduce the low frequency gain.
% In the frequency band of interest, this controller should mostly act as a pure integrator.
% The Root Locus corresponding to this controller is shown in Figure [[fig:root_locus_iff_apa95ml_identification]].
gains = logspace(0, 5, 1000);
figure;
hold on;
plot(real(pole(Gi)), imag(pole(Gi)), 'kx');
plot(real(tzero(Gi)), imag(tzero(Gi)), 'ko');
for i = 1:length(gains)
cl_poles = pole(feedback(Gi, (gains(i)/(s + 2*2*pi)*s/(s + 0.5*2*pi))));
plot(real(cl_poles), imag(cl_poles), 'k.');
end
ylim([0, 1800]);
xlim([-1600,200]);
xlabel('Real Part')
ylabel('Imaginary Part')
axis square
% First tests with few gains
% The controller is now implemented in practice, and few controller gains are tested: $g = 0$, $g = 10$ and $g = 100$.
% For each controller gain, the identification shown in Figure [[fig:test_bench_apa_schematic_iff]] is performed.
iff_g10 = load('apa95ml_iff_g10_res.mat', 'u', 't', 'y', 'v');
iff_g100 = load('apa95ml_iff_g100_res.mat', 'u', 't', 'y', 'v');
iff_of = load('apa95ml_iff_off_res.mat', 'u', 't', 'y', 'v');
Ts = 1e-4;
win = hann(ceil(10/Ts));
[tf_iff_g10, f] = tfestimate(iff_g10.u, iff_g10.y, win, [], [], 1/Ts);
[co_iff_g10, ~] = mscohere( iff_g10.u, iff_g10.y, win, [], [], 1/Ts);
[tf_iff_g100, ~] = tfestimate(iff_g100.u, iff_g100.y, win, [], [], 1/Ts);
[co_iff_g100, ~] = mscohere( iff_g100.u, iff_g100.y, win, [], [], 1/Ts);
[tf_iff_of, ~] = tfestimate(iff_of.u, iff_of.y, win, [], [], 1/Ts);
[co_iff_of, ~] = mscohere( iff_of.u, iff_of.y, win, [], [], 1/Ts);
% The coherence between the excitation signal and the mass displacement as measured by the interferometer is shown in Figure [[fig:iff_first_test_coherence]].
figure;
hold on;
plot(f, co_iff_of, '-', 'DisplayName', 'g=0')
plot(f, co_iff_g10, '-', 'DisplayName', 'g=10')
plot(f, co_iff_g100, '-', 'DisplayName', 'g=100')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Coherence'); xlabel('Frequency [Hz]');
hold off;
legend();
xlim([60, 600])
% #+name: fig:iff_first_test_coherence
% #+caption: Coherence
% #+RESULTS:
% [[file:figs/iff_first_test_coherence.png]]
% The obtained transfer functions are shown in Figure [[fig:iff_first_test_bode_plot]].
% It is clear that the IFF architecture can actively damp the main resonance of the system.
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2, 1]);
hold on;
plot(f, abs(tf_iff_of), '-', 'DisplayName', 'g=0')
plot(f, abs(tf_iff_g10), '-', 'DisplayName', 'g=10')
plot(f, abs(tf_iff_g100), '-', 'DisplayName', 'g=100')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
hold off;
legend();
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(-tf_iff_of), '-')
plot(f, 180/pi*angle(-tf_iff_g10), '-')
plot(f, 180/pi*angle(-tf_iff_g100), '-')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Phase'); xlabel('Frequency [Hz]');
hold off;
yticks(-360:90:360);
linkaxes([ax1,ax2], 'x');
xlim([60, 600]);
% Second test with many Gains
% Then, the same identification test is performed for many more gains.
load('apa95ml_iff_test.mat', 'results');
Ts = 1e-4;
win = hann(ceil(10/Ts));
tf_iff = {zeros(1, length(results))};
co_iff = {zeros(1, length(results))};
g_iff = [0, 1, 5, 10, 50, 100];
for i=1:length(results)
[tf_est, f] = tfestimate(results{i}.u, results{i}.y, win, [], [], 1/Ts);
[co_est, ~] = mscohere(results{i}.u, results{i}.y, win, [], [], 1/Ts);
tf_iff(i) = {tf_est};
co_iff(i) = {co_est};
end
% The obtained dynamics are shown in Figure [[fig:iff_results_bode_plots]].
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2, 1]);
hold on;
for i = 1:length(results)
plot(f, abs(tf_iff{i}), '-', 'DisplayName', sprintf('g = %0.f', g_iff(i)))
end
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
hold off;
legend();
ax2 = nexttile;
hold on;
for i = 1:length(results)
plot(f, 180/pi*angle(-tf_iff{i}), '-')
end
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Phase'); xlabel('Frequency [Hz]');
yticks(-360:90:360);
axis padded 'auto x'
linkaxes([ax1,ax2], 'x');
xlim([60, 600]);
% #+name: fig:iff_results_bode_plots
% #+caption: Identified dynamics from excitation voltage to the mass displacement
% #+RESULTS:
% [[file:figs/iff_results_bode_plots.png]]
% For each gain, the parameters of a second order resonant system that best fits the data are estimated and are compared with the data in Figure [[fig:iff_results_bode_plots_identification]].
G_id = {zeros(1,length(results))};
f_start = 70; % [Hz]
f_end = 500; % [Hz]
for i = 1:length(results)
tf_id = tf_iff{i}(sum(f<f_start):length(f)-sum(f>f_end));
f_id = f(sum(f<f_start):length(f)-sum(f>f_end));
gfr = idfrd(tf_id, 2*pi*f_id, Ts);
G_id(i) = {procest(gfr,'P2UDZ')};
end
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2, 1]);
hold on;
for i = 1:length(results)
set(gca,'ColorOrderIndex',i)
plot(f, abs(tf_iff{i}), '-', 'DisplayName', sprintf('g = %0.f', g_iff(i)))
set(gca,'ColorOrderIndex',i)
plot(f, abs(squeeze(freqresp(G_id{i}, f, 'Hz'))), '--', 'HandleVisibility', 'off')
end
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
hold off;
legend();
ax2 = nexttile;
hold on;
for i = 1:length(results)
set(gca,'ColorOrderIndex',i)
plot(f, 180/pi*angle(tf_iff{i}), '-')
set(gca,'ColorOrderIndex',i)
plot(f, 180/pi*angle(squeeze(freqresp(G_id{i}, f, 'Hz'))), '--')
end
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Phase'); xlabel('Frequency [Hz]');
hold off;
yticks(-360:90:360);
axis padded 'auto x'
linkaxes([ax1,ax2], 'x');
xlim([60, 600]);
% #+name: fig:iff_results_bode_plots_identification
% #+caption: Comparison of the measured dynamic and the identified 2nd order resonant systems that best fits the data
% #+RESULTS:
% [[file:figs/iff_results_bode_plots_identification.png]]
% Finally, we can represent the position of the poles of the 2nd order systems on the Root Locus plot (Figure [[fig:iff_results_root_locus]]).
w_z = 2*pi*111; % Zeros frequency [rad/s]
w_p = 2*pi*255; % Pole frequency [rad/s]
xi_z = 0.05;
xi_p = 0.015;
G_inf = 2;
Gi = G_inf*(s^2 - 2*xi_z*w_z*s + w_z^2)/(s^2 + 2*xi_p*w_p*s + w_p^2);
gains = logspace(0, 5, 1000);
figure;
hold on;
plot(real(pole(Gi)), imag(pole(Gi)), 'kx', 'HandleVisibility', 'off');
plot(real(tzero(Gi)), imag(tzero(Gi)), 'ko', 'HandleVisibility', 'off');
for i = 1:length(results)
set(gca,'ColorOrderIndex',i)
plot(real(pole(G_id{i})), imag(pole(G_id{i})), 'o', 'DisplayName', sprintf('g = %0.f', g_iff(i)));
end
for i = 1:length(gains)
cl_poles = pole(feedback(Gi, (gains(i)/(s + 2*pi*2))));
plot(real(cl_poles), imag(cl_poles), 'k.', 'HandleVisibility', 'off');
end
ylim([0, 1800]);
xlim([-1600,200]);
xlabel('Real Part')
ylabel('Imaginary Part')
axis square
legend('location', 'northwest');

48
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@ -0,0 +1,75 @@
LIST ALL SELECTED NODES. DSYS= 0
*** ANSYS - ENGINEERING ANALYSIS SYSTEM RELEASE 2020 R2 20.2 ***
DISTRIBUTED ANSYS Academic Research
00208316 VERSION=WINDOWS x64 15:24:46 NOV 18, 2020 CP= 2.031
Piezo-APA95ML-3DElm--Modal (A5)
NODE X Y Z THXY THYZ THZX
40467 0.0000 0.0000 0.29997E-001 0.00 0.00 0.00
40469 0.0000 0.0000 -0.29997E-001 0.00 0.00 0.00
40470 -0.35000E-001 0.0000 0.0000 0.00 0.00 0.00
40475 -0.15000E-001 0.0000 0.0000 0.00 0.00 0.00
40476 -0.50000E-002 0.0000 0.0000 0.00 0.00 0.00
40477 0.15000E-001 0.0000 0.0000 0.00 0.00 0.00
40478 0.35000E-001 0.0000 0.0000 0.00 0.00 0.00
LIST MASTERS ON ALL SELECTED NODES.
CURRENT DOF SET= UX UY UZ ROTX ROTY ROTZ
*** ANSYS - ENGINEERING ANALYSIS SYSTEM RELEASE 2020 R2 20.2 ***
DISTRIBUTED ANSYS Academic Research
00208316 VERSION=WINDOWS x64 15:24:46 NOV 18, 2020 CP= 2.156
Piezo-APA95ML-3DElm--Modal (A5)
NODE LABEL SUPPORT
40467 UX
40467 UY
40467 UZ
40467 ROTX
40467 ROTY
40467 ROTZ
40469 UX
40469 UY
40469 UZ
40469 ROTX
40469 ROTY
40469 ROTZ
40470 UX
40470 UY
40470 UZ
40470 ROTX
40470 ROTY
40470 ROTZ
40475 UX
40475 UY
40475 UZ
40475 ROTX
40475 ROTY
40475 ROTZ
40476 UX
40476 UY
40476 UZ
40476 ROTX
40476 ROTY
40476 ROTZ
40477 UX
40477 UY
40477 UZ
40477 ROTX
40477 ROTY
40477 ROTZ
40478 UX
40478 UY
40478 UZ
40478 ROTX
40478 ROTY
40478 ROTZ

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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
addpath('./mat/');
% Load Data
% The data from the "noise test" and the identification test are loaded.
ht = load('huddle_test.mat', 't', 'y');
load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y');
% Any offset value is removed:
um = detrend(um, 0); % Generated DAC Voltage [V]
y = detrend(y , 0); % Mass displacement [m]
ht.y = detrend(ht.y, 0);
% Now we add a factor 10 to take into account the gain of the voltage amplifier.
Va = 10*um;
% Comparison of the PSD with Huddle Test
Ts = t(end)/(length(t)-1);
Fs = 1/Ts;
win = hanning(ceil(1*Fs));
[pxx, f] = pwelch(y, win, [], [], Fs);
[pht, ~] = pwelch(ht.y, win, [], [], Fs);
figure;
hold on;
plot(f, sqrt(pxx), 'DisplayName', '5kg');
plot(f, sqrt(pht), 'DisplayName', 'Huddle Test');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
legend('location', 'northeast');
xlim([1, Fs/2]);
% Compute TF estimate and Coherence
[tf_est, f] = tfestimate(Va, -y, win, [], [], 1/Ts);
[co_est, ~] = mscohere( Va, -y, win, [], [], 1/Ts);
figure;
hold on;
plot(f, co_est, 'k-')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Coherence'); xlabel('Frequency [Hz]');
hold off;
xlim([10, 5e3]);
% #+name: fig:apa95ml_5kg_PI_coh
% #+caption: Coherence
% #+RESULTS:
% [[file:figs/apa95ml_5kg_PI_coh.png]]
% Comparison with the FEM model
load('mat/fem_simscape_models.mat', 'Ghm');
freqs = logspace(0, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(tf_est), 'DisplayName', 'Identification')
plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), 'DisplayName', 'FEM')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel', []);
legend('location', 'northeast')
hold off;
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(tf_est))
plot(freqs, 180/pi*angle(squeeze(freqresp(Ghm, freqs, 'Hz'))))
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Phase'); xlabel('Frequency [Hz]');
hold off;
ylim([-180, 180]);
yticks(-180:90:180);
linkaxes([ax1,ax2], 'x');
xlim([10, 5e3]);

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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
addpath('./mat/');
% Estimation from Data-sheet
% <<sec:estimation_datasheet>>
% The stack parameters taken from the data-sheet are shown in Table [[tab:stack_parameters]].
% #+name: tab:stack_parameters
% #+caption: Stack Parameters
% | Parameter | Unit | Value |
% |----------------+-----------+----------|
% | Nominal Stroke | $\mu m$ | 20 |
% | Blocked force | $N$ | 4700 |
% | Stiffness | $N/\mu m$ | 235 |
% | Voltage Range | $V$ | -20..150 |
% | Capacitance | $\mu F$ | 4.4 |
% | Length | $mm$ | 20 |
% | Stack Area | $mm^2$ | 10x10 |
% Let's compute the generated force
% The stroke is $L_{\max} = 20\mu m$ for a voltage range of $V_{\max} = 170 V$.
% Furthermore, the stiffness is $k_a = 235 \cdot 10^6 N/m$.
% The relation between the applied voltage and the generated force can be estimated as follows:
% \begin{equation}
% g_a = k_a \frac{L_{\max}}{V_{\max}}
% \end{equation}
ka = 235e6; % [N/m]
Lmax = 20e-6; % [m]
Vmax = 170; % [V]
ga = ka*Lmax/Vmax; % [N/V]
% Estimation from Piezoelectric parameters
% <<sec:estimation_piezo_params>>
% In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
d33 = 300e-12; % Strain constant [m/V]
n = 80; % Number of layers per stack
ka = 235e6; % Stack stiffness [N/m]
% The ratio of the developed force to applied voltage is:
% #+name: eq:piezo_voltage_to_force
% \begin{equation}
% F_a = g_a V_a, \quad g_a = d_{33} n k_a
% \end{equation}
% where:
% - $F_a$: developed force in [N]
% - $n$: number of layers of the actuator stack
% - $d_{33}$: strain constant in [m/V]
% - $k_a$: actuator stack stiffness in [N/m]
% - $V_a$: applied voltage in [V]
% If we take the numerical values, we obtain:
ga = d33*n*ka; % [N/V]
% #+RESULTS:
% : ga = 5.6 [N/V]
% From cite:fleming14_desig_model_contr_nanop_system (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
% #+name: eq:piezo_strain_to_voltage
% \begin{equation}
% V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
% \end{equation}
% where:
% - $V_s$: measured voltage in [V]
% - $d_{33}$: strain constant in [m/V]
% - $\epsilon^T$: permittivity under constant stress in [F/m]
% - $s^D$: elastic compliance under constant electric displacement in [m^2/N]
% - $n$: number of layers of the sensor stack
% - $\Delta h$: relative displacement in [m]
% If we take the numerical values, we obtain:
d33 = 300e-12; % Strain constant [m/V]
n = 80; % Number of layers per stack
eT = 5.3e-9; % Permittivity under constant stress [F/m]
sD = 2e-11; % Compliance under constant electric displacement [m2/N]
gs = d33/(eT*sD*n); % [V/m]
% From actuator voltage $V_a$ to actuator force $F_a$
% The data from the identification test is loaded.
load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y');
% Any offset value is removed:
um = detrend(um, 0); % Amplifier Input Voltage [V]
y = detrend(y , 0); % Mass displacement [m]
% Now we add a factor 10 to take into account the gain of the voltage amplifier and thus obtain the voltage across the piezoelectric stack.
um = 10*um; % Stack Actuator Input Voltage [V]
% Then, the transfer function from the stack voltage to the vertical displacement is computed.
Ts = t(end)/(length(t)-1);
Fs = 1/Ts;
win = hanning(ceil(1*Fs));
[tf_est, f] = tfestimate(um, y, win, [], [], 1/Ts);
% The gain from input voltage of the stack to the vertical displacement is determined:
g_d_Va = 4e-7; % [m/V]
freqs = logspace(0, 4, 1000);
figure;
hold on;
plot(f, abs(tf_est), 'DisplayName', 'Identification')
plot([f(2) f(end)], [g_d_Va g_d_Va], '--', 'DisplayName', sprintf('$d/V_a \\approx %.0g\\ m/V$', g_d_Va))
hold off;
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude $d/V_a$ [m/V]');
xlim([10, 5e3]); ylim([1e-8, 1e-5]);
legend('location', 'southwest');
% #+name: fig:gain_Va_to_d
% #+caption: Transfer function from actuator stack voltage $V_a$ to vertical displacement of the mass $d$
% #+RESULTS:
% [[file:figs/gain_Va_to_d.png]]
% Then, the transfer function from forces applied by the stack actuator to the vertical displacement of the mass is identified from the Simscape model.
% Load Parameters
K = readmatrix('APA95ML_K.CSV');
M = readmatrix('APA95ML_M.CSV');
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
% Define parameters just for the simulation
ga = 1; % [N/V]
gs = 1; % [V/m]
m = 5.5;
%% Name of the Simulink File
mdl = 'piezo_amplified_3d';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
Gd = linearize(mdl, io);
% The DC gain the the identified dynamics
g_d_Fa = abs(dcgain(Gd)); % [m/N]
% #+RESULTS:
% : G_d_Fa = 1.2e-08 [m/N]
% And finally, the gain $g_a$ from the the actuator voltage $V_a$ to the generated force $F_a$ can be computed:
% \begin{equation}
% g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \frac{d}{V_a}
% \end{equation}
ga = g_d_Va/g_d_Fa;
% #+RESULTS:
% : ga = 33.7 [N/V]
% The obtained comparison between the Simscape model and the identified dynamics is shown in Figure [[fig:compare_Gd_id_simscape]].
freqs = logspace(1, 4, 1000);
figure;
hold on;
plot(f, abs(tf_est), 'DisplayName', 'Identification')
plot(f, ga*abs(squeeze(freqresp(Gd, f, 'Hz'))), 'DisplayName', 'Simscape Model')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude $d/V_a$ [m/V]'); xlabel('Frequency [Hz]');
hold off;
xlim([1, 5e3]); ylim([1e-9, 1e-5]);
legend('location', 'southwest');
% From stack strain $\Delta h$ to generated voltage $V_s$
% Now, the gain from the stack strain $\Delta h$ to the generated voltage $V_s$ is estimated.
% We can determine the gain from actuator voltage $V_a$ to sensor voltage $V_s$ thanks to the identification.
% Using the simscape model, we can have the transfer function from the actuator voltage $V_a$ (using the previously estimated gain $g_a$) to the sensor stack strain $\Delta h$.
% Finally, using these two values, we can compute the gain $g_s$ from the stack strain $\Delta h$ to the generated Voltage $V_s$.
% Identification data is loaded.
load('apa95ml_5kg_2a_1s.mat', 't', 'u', 'v');
u = detrend(u, 0); % Input Voltage of the Amplifier [V]
v = detrend(v, 0); % Voltage accross the stack sensor [V]
% Here, an amplifier with a gain of 20 is used.
u = 20*u; % Input Voltage of the Amplifier [V]
% Then, the transfer function from $V_a$ to $V_s$ is identified and its DC gain is estimated (Figure [[fig:gain_Va_to_Vs]]).
Ts = t(end)/(length(t)-1);
Fs = 1/Ts;
win = hann(ceil(10/Ts));
[tf_est, f] = tfestimate(u, v, win, [], [], 1/Ts);
g_Vs_Va = 0.022; % [V/V]
freqs = logspace(1, 4, 1000);
figure;
hold on;
plot(f, abs(tf_est), 'DisplayName', 'Identification')
plot([f(2); f(end)], [g_Vs_Va; g_Vs_Va], '--', 'DisplayName', sprintf('$V_s/V_a \\approx %.0g\\ m/V$', g_Vs_Va))
hold off;
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); xlabel('Frequency [Hz]');
xlim([5e-1 5e3]); ylim([1e-3, 1e1]);
legend('location', 'northwest');
% #+name: fig:gain_Va_to_Vs
% #+caption: Transfer function from actuator stack voltage $V_a$ to sensor stack voltage $V_s$
% #+RESULTS:
% [[file:figs/gain_Va_to_Vs.png]]
% Now the transfer function from the actuator stack voltage to the sensor stack strain is estimated using the Simscape model.
m = 5.5;
%% Name of the Simulink File
mdl = 'piezo_amplified_3d';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage [V]
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Stack displacement [m]
Gf = linearize(mdl, io);
% The gain from the actuator stack voltage to the sensor stack strain is estimated below.
G_dh_Va = abs(dcgain(Gf));
% #+RESULTS:
% : G_dh_Va = 6.2e-09 [m/V]
% And finally, the gain $g_s$ from the sensor stack strain to the generated voltage can be estimated:
% \begin{equation}
% g_s = \frac{V_s}{\Delta h} = \frac{V_s}{V_a} \frac{V_a}{\Delta h}
% \end{equation}
gs = g_Vs_Va/G_dh_Va; % [V/m]
% #+RESULTS:
% : gs = 3.5 [V/um]
freqs = logspace(1, 4, 1000);
figure;
hold on;
plot(f, abs(tf_est), 'DisplayName', 'Identification')
plot(f, gs*abs(squeeze(freqresp(Gf, f, 'Hz'))), 'DisplayName', 'Simscape Model')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); xlabel('Frequency [Hz]');
hold off;
xlim([1, 5e3]); ylim([1e-3, 1e1]);
legend('location', 'southwest');
save('./mat/apa95ml_params.mat', 'ga', 'gs');

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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
addpath('./mat/');
% Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
% We first extract the stiffness and mass matrices.
K = readmatrix('APA95ML_K.CSV');
M = readmatrix('APA95ML_M.CSV');
% #+caption: First 10x10 elements of the Mass matrix
% #+RESULTS:
% | 0.03 | 7e-08 | 2e-06 | -3e-09 | -0.0002 | -6e-08 | -0.001 | 8e-07 | 6e-07 | -8e-09 |
% | 7e-08 | 0.02 | -1e-06 | 9e-05 | -3e-09 | -4e-09 | -1e-06 | -0.0006 | -4e-08 | 5e-06 |
% | 2e-06 | -1e-06 | 0.02 | -3e-08 | -4e-08 | 1e-08 | 1e-07 | -2e-07 | 0.0003 | 1e-09 |
% | -3e-09 | 9e-05 | -3e-08 | 1e-06 | -3e-11 | -3e-13 | -7e-09 | -5e-06 | -3e-10 | 3e-08 |
% | -0.0002 | -3e-09 | -4e-08 | -3e-11 | 2e-06 | 6e-10 | 2e-06 | -7e-09 | -2e-09 | 7e-11 |
% | -6e-08 | -4e-09 | 1e-08 | -3e-13 | 6e-10 | 1e-06 | 1e-08 | 3e-09 | -2e-09 | 2e-13 |
% | -0.001 | -1e-06 | 1e-07 | -7e-09 | 2e-06 | 1e-08 | 0.03 | 4e-08 | -2e-06 | 8e-09 |
% | 8e-07 | -0.0006 | -2e-07 | -5e-06 | -7e-09 | 3e-09 | 4e-08 | 0.02 | -9e-07 | -9e-05 |
% | 6e-07 | -4e-08 | 0.0003 | -3e-10 | -2e-09 | -2e-09 | -2e-06 | -9e-07 | 0.02 | 2e-08 |
% | -8e-09 | 5e-06 | 1e-09 | 3e-08 | 7e-11 | 2e-13 | 8e-09 | -9e-05 | 2e-08 | 1e-06 |
% Then, we extract the coordinates of the interface nodes.
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
% Simscape Model
% The flexible element is imported using the =Reduced Order Flexible Solid= Simscape block.
% To model the actuator, an =Internal Force= block is added between the nodes 3 and 12.
% A =Relative Motion Sensor= block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
% One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
m = 5.5;
load('apa95ml_params.mat', 'ga', 'gs');
% Dynamics from Actuator Voltage to Vertical Mass Displacement
% The identified dynamics is shown in Figure [[fig:dynamics_act_disp_comp_mass]].
%% Name of the Simulink File
mdl = 'piezo_amplified_3d';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage [V]
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
Ghm = linearize(mdl, io);
freqs = logspace(0, 4, 5000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Ghm, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% Dynamics from Actuator Voltage to Force Sensor Voltage
% The obtained dynamics is shown in Figure [[fig:dynamics_force_force_sensor_comp_mass]].
%% Name of the Simulink File
mdl = 'piezo_amplified_3d';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Voltage Actuator [V]
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage [V]
Gfm = linearize(mdl, io);
freqs = logspace(1, 5, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gfm, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gfm, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-360, 180]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
save('mat/fem_simscape_models.mat', 'Ghm', 'Gfm')