Identification of Piezo parameters

index.org

@@ -67,18 +67,442 @@ Here are the equipment used in the test bench:
 - Low Noise Voltage Preamplifier from Ametek ([[file:doc/model_5113.pdf][doc]])
 #+end_note
 
+* Estimation of electrical/mechanical relationships
+** Introduction                                                      :ignore:
+In order to correctly model the piezoelectric actuator with Simscape, we need to determine:
+1. $g_a$: the ratio of the generated force $F_a$ to the supply voltage $V_a$ across the piezoelectric stack
+2. $g_s$: the ratio of the generated voltage $V_s$ across the piezoelectric stack when subject to a strain $\Delta h$
+
+We estimate $g_a$ and $g_s$ using different approaches:
+1. Section [[sec:estimation_datasheet]]: $g_a$ is estimated from the datasheet of the piezoelectric stack
+2. Section [[sec:estimation_piezo_params]]: $g_a$ and $g_s$ are estimated using the piezoelectric constants
+3. Section [[sec:estimation_identification]]: $g_a$ and $g_s$ are estimated experimentally
+
+** 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/mat/');
+  addpath('./matlab/');
+#+end_src
+
+#+begin_src matlab :eval no
+  addpath('./mat/');
+#+end_src
+
+** 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}
+
+#+begin_src matlab
+  ka = 235e6; % [N/m]
+  Lmax = 20e-6; % [m]
+  Vmax = 170; % [V]
+#+end_src
+
+#+begin_src matlab :results value replace
+  ka*Lmax/Vmax % [N/V]
+#+end_src
+
+#+RESULTS:
+: 27.647
+
+From the parameters of the stack, it seems not possible to estimate the relation between the strain and the generated voltage.
+
+** 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:
+#+begin_src matlab
+  d33 = 300e-12; % Strain constant [m/V]
+  n   = 80;      % Number of layers per stack
+  ka  = 235e6;   % Stack stiffness [N/m]
+#+end_src
+
+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:
+#+begin_src matlab
+  ga = d33*n*ka; % [N/V]
+#+end_src
+
+#+begin_src matlab :results value replace :exports results :tangle no
+  sprintf('ga = %.1f [N/V]', ga)
+#+end_src
+
+#+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:
+#+begin_src matlab
+  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]
+#+end_src
+
+#+begin_src matlab
+  gs = d33/(eT*sD*n); % [V/m]
+#+end_src
+
+#+begin_src matlab :results value replace :exports results :tangle no
+  sprintf('gs = %.1f [V/um]', 1e-6*gs)
+#+end_src
+
+#+RESULTS:
+: gs = 35.4 [V/um]
+
+** Estimation from Experiment
+<<sec:estimation_identification>>
+*** Introduction                                                    :ignore:
+The idea here is to obtain the parameters $g_a$ and $g_s$ from the comparison of an experimental identification and the identification using Simscape.
+
+Using the experimental identification, we can easily obtain the gain from the applied voltage to the generated displacement, but not to the generated force.
+However, from the Simscape model, we can easily have the link from the generated force to the displacement, them we can computed $g_a$.
+
+Similarly, it is fairly easy to experimentally obtain the gain from the stack displacement to the generated voltage across the stack.
+To link that to the strain of the sensor stack, the simscape model is used.
+
+*** From actuator voltage $V_a$ to actuator force $F_a$
+The data from the identification test is loaded.
+#+begin_src matlab
+  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]
+#+end_src
+
+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.
+#+begin_src matlab
+  um = 10*um; % Stack Actuator Input Voltage [V]
+#+end_src
+
+Then, the transfer function from the stack voltage to the vertical displacement is computed.
+#+begin_src matlab
+  Ts = t(end)/(length(t)-1);
+  Fs = 1/Ts;
+
+  win = hanning(ceil(1*Fs));
+
+  [tf_est, f] = tfestimate(um, y, win, [], [], 1/Ts);
+#+end_src
+
+The gain from input voltage of the stack to the vertical displacement is determined:
+#+begin_src matlab
+  g_d_Va = 4e-7; % [m/V]
+#+end_src
+
+#+begin_src matlab :exports none
+  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');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/gain_Va_to_d.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+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.
+#+begin_src matlab :exports none
+  % 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]
+#+end_src
+
+#+begin_src matlab
+  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);
+#+end_src
+
+The DC gain the the identified dynamics
+#+begin_src matlab
+  g_d_Fa = abs(dcgain(Gd)); % [m/N]
+#+end_src
+
+#+begin_src matlab :results value replace :exports results :tangle no
+  sprintf('G_d_Fa = %.2g [m/N]', g_d_Fa)
+#+end_src
+
+#+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}
+
+#+begin_src matlab
+  ga = g_d_Va/g_d_Fa;
+#+end_src
+
+#+begin_src matlab :results value replace :exports results :tangle no
+  sprintf('ga = %.1f [N/V]', ga)
+#+end_src
+
+#+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]].
+
+#+begin_src matlab :exports none
+  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');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/compare_Gd_id_simscape.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:compare_Gd_id_simscape
+#+caption: Comparison of the identified transfer function between actuator voltage $V_a$ and vertical mass displacement $d$
+#+RESULTS:
+[[file:figs/compare_Gd_id_simscape.png]]
+
+*** 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.
+#+begin_src matlab
+  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]
+#+end_src
+
+Here, an amplifier with a gain of 20 is used.
+#+begin_src matlab
+  u = 20*u; % Input Voltage of the Amplifier [V]
+#+end_src
+
+Then, the transfer function from $V_a$ to $V_s$ is identified and its DC gain is estimated (Figure [[fig:gain_Va_to_Vs]]).
+#+begin_src matlab
+  Ts = t(end)/(length(t)-1);
+  Fs = 1/Ts;
+
+  win = hann(ceil(10/Ts));
+
+  [tf_est, f] = tfestimate(u, v, win, [], [], 1/Ts);
+#+end_src
+
+#+begin_src matlab
+  g_Vs_Va = 0.022; % [V/V]
+#+end_src
+
+#+begin_src matlab :exports none
+  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');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/gain_Va_to_Vs.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+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.
+#+begin_src matlab
+  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);
+#+end_src
+
+The gain from the actuator stack voltage to the sensor stack strain is estimated below.
+#+begin_src matlab
+  G_dh_Va = abs(dcgain(Gf));
+#+end_src
+
+#+begin_src matlab :results value replace :exports results :tangle no
+  sprintf('G_dh_Va = %.2g [m/V]', G_dh_Va);
+#+end_src
+
+#+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}
+
+#+begin_src matlab
+  gs = g_Vs_Va/G_dh_Va; % [V/m]
+#+end_src
+
+#+begin_src matlab :results value replace :exports results :tangle no
+  sprintf('gs = %.1f [V/um]', 1e-6*gs)
+#+end_src
+
+#+RESULTS:
+: gs = 3.5 [V/um]
+
+#+begin_src matlab :exports none
+  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');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/compare_Gf_id_simscape.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:compare_Gf_id_simscape
+#+caption: Comparison of the identified transfer function between actuator voltage $V_a$ and sensor stack voltage
+#+RESULTS:
+[[file:figs/compare_Gf_id_simscape.png]]
+
+** Conclusion
+The obtained parameters $g_a$ and $g_s$ are not consistent between the different methods.
+
+The one using the experimental data are saved and further used.
+#+begin_src matlab :tangle no
+  save('./matlab/mat/apa95ml_params.mat', 'ga', 'gs');
+#+end_src
+
+#+begin_src matlab :exports none :eval no
+  save('./mat/apa95ml_params.mat', 'ga', 'gs');
+#+end_src
+
 * Simscape model of the test-bench
 :PROPERTIES:
 :header-args:matlab+: :tangle matlab/simscape_model.m
 :header-args:matlab+: :comments org :mkdirp yes
 :END:
 <<sec:simscape_model>>
+
 ** Introduction                                                      :ignore:
 The idea here is to model the test-bench using Simscape.
 
 Whereas the suspended mass and metrology frame can be considered as rigid bodies in the frequency range of interest, the Amplified Piezoelectric Actuator (APA) is flexible.
 
-To model the APA, a Finite Element Model (FEM) is used and imported into Simscape.
+To model the APA, a Finite Element Model (FEM) is used (Figure [[fig:APA95ML_FEM]]) and imported into Simscape.
+
+#+name: fig:APA95ML_FEM
+#+caption: Finite Element Model of the APA95ML
+[[file:figs/APA95ML_FEM.png]]
 
 ** Matlab Init                                             :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -101,8 +525,8 @@ To model the APA, a Finite Element Model (FEM) is used and imported into Simscap
 ** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
 We first extract the stiffness and mass matrices.
 #+begin_src matlab
-  K = extractMatrix('APA95ML_K.txt');
-  M = extractMatrix('APA95ML_M.txt');
+  K = readmatrix('APA95ML_K.CSV');
+  M = readmatrix('APA95ML_M.CSV');
 #+end_src
 
 #+begin_src matlab :exports results :results value table replace :tangle no
@@ -111,16 +535,16 @@ We first extract the stiffness and mass matrices.
 
 #+caption: First 10x10 elements of the Stiffness matrix
 #+RESULTS:
-| 300000000.0 |    -30000.0 |     8000.0 |    -200.0 |   -30.0 | -60000.0 |  20000000.0 |     -4000.0 |      500.0 |        8 |
-|    -30000.0 | 100000000.0 |      400.0 |      30.0 |   200.0 |       -1 |      4000.0 |  -8000000.0 |      800.0 |        7 |
-|      8000.0 |       400.0 | 50000000.0 | -800000.0 |  -300.0 |    -40.0 |       300.0 |       100.0 |  5000000.0 |  40000.0 |
-|      -200.0 |        30.0 |  -800000.0 |   20000.0 |       5 |        1 |       -10.0 |          -2 |   -40000.0 |   -300.0 |
-|       -30.0 |       200.0 |     -300.0 |         5 | 40000.0 |      0.3 |          -4 |       -10.0 |       40.0 |      0.4 |
-|    -60000.0 |          -1 |      -40.0 |         1 |     0.3 |   3000.0 |      7000.0 |         0.8 |         -1 |   0.0003 |
-|  20000000.0 |      4000.0 |      300.0 |     -10.0 |      -4 |   7000.0 | 300000000.0 |     20000.0 |     3000.0 |     80.0 |
-|     -4000.0 |  -8000000.0 |      100.0 |        -2 |   -10.0 |      0.8 |     20000.0 | 100000000.0 |    -4000.0 |   -100.0 |
-|       500.0 |       800.0 |  5000000.0 |  -40000.0 |    40.0 |       -1 |      3000.0 |     -4000.0 | 50000000.0 | 800000.0 |
-|           8 |           7 |    40000.0 |    -300.0 |     0.4 |   0.0003 |        80.0 |      -100.0 |   800000.0 |  20000.0 |
+| 300000000.0 |    -1000.0 |    -30000.0 |    -40.0 | 70000.0 |   300.0 |  20000000.0 |      -30.0 |     -5000.0 |         5 |
+|     -1000.0 | 50000000.0 |     -7000.0 | 800000.0 |   -20.0 |   300.0 |      3000.0 |  5000000.0 |       400.0 |  -40000.0 |
+|    -30000.0 |    -7000.0 | 100000000.0 |   -200.0 |   -60.0 |    70.0 |      3000.0 |     3000.0 |  -8000000.0 |     -30.0 |
+|       -40.0 |   800000.0 |      -200.0 |  20000.0 |    -0.4 |       4 |        30.0 |    40000.0 |           7 |    -300.0 |
+|     70000.0 |      -20.0 |       -60.0 |     -0.4 |  3000.0 |       1 |     -6000.0 |       10.0 |           8 |      -0.1 |
+|       300.0 |      300.0 |        70.0 |        4 |       1 | 40000.0 |       -10.0 |      -10.0 |        30.0 |       0.1 |
+|  20000000.0 |     3000.0 |      3000.0 |     30.0 | -6000.0 |   -10.0 | 300000000.0 |     2000.0 |      9000.0 |    -100.0 |
+|       -30.0 |  5000000.0 |      3000.0 |  40000.0 |    10.0 |   -10.0 |      2000.0 | 50000000.0 |     -3000.0 | -800000.0 |
+|     -5000.0 |      400.0 |  -8000000.0 |        7 |       8 |    30.0 |      9000.0 |    -3000.0 | 100000000.0 |     100.0 |
+|           5 |   -40000.0 |       -30.0 |   -300.0 |    -0.1 |     0.1 |      -100.0 |  -800000.0 |       100.0 |   20000.0 |
 
 
 #+begin_src matlab :exports results :results value table replace :tangle no
@@ -129,16 +553,16 @@ We first extract the stiffness and mass matrices.
 
 #+caption: First 10x10 elements of the Mass matrix
 #+RESULTS:
-|   0.03 |  2e-06 |  -2e-07 |  1e-08 |  2e-08 | 0.0002 | -0.001 |  2e-07 |  -8e-08 | -9e-10 |
-|  2e-06 |   0.02 |  -5e-07 |  7e-09 |  3e-08 |  2e-08 | -3e-07 | 0.0003 |  -1e-08 |  1e-10 |
-| -2e-07 | -5e-07 |    0.02 | -9e-05 |  4e-09 | -1e-08 |  2e-07 | -2e-08 | -0.0006 | -5e-06 |
-|  1e-08 |  7e-09 |  -9e-05 |  1e-06 |  6e-11 |  4e-10 | -1e-09 |  3e-11 |   5e-06 |  3e-08 |
-|  2e-08 |  3e-08 |   4e-09 |  6e-11 |  1e-06 |  2e-10 | -2e-09 |  2e-10 |  -7e-09 | -4e-11 |
-| 0.0002 |  2e-08 |  -1e-08 |  4e-10 |  2e-10 |  2e-06 | -2e-06 | -1e-09 |  -7e-10 | -9e-12 |
-| -0.001 | -3e-07 |   2e-07 | -1e-09 | -2e-09 | -2e-06 |   0.03 | -2e-06 |  -1e-07 | -5e-09 |
-|  2e-07 | 0.0003 |  -2e-08 |  3e-11 |  2e-10 | -1e-09 | -2e-06 |   0.02 |  -8e-07 | -1e-08 |
-| -8e-08 | -1e-08 | -0.0006 |  5e-06 | -7e-09 | -7e-10 | -1e-07 | -8e-07 |    0.02 |  9e-05 |
-| -9e-10 |  1e-10 |  -5e-06 |  3e-08 | -4e-11 | -9e-12 | -5e-09 | -1e-08 |   9e-05 |  1e-06 |
+|    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.
@@ -146,100 +570,47 @@ Then, we extract the coordinates of the interface nodes.
   [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
 #+end_src
 
+The interface nodes are shown in Figure [[fig:APA95ML_nodes_1]] and their coordinates are listed in Table [[tab:apa95ml_nodes_coordinates]].
+
 #+begin_src matlab :exports results :results value table replace :tangle no
   data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
 #+end_src
 
 #+RESULTS:
-| Total number of Nodes     | 168959 |
-| Number of interface Nodes |     13 |
-| Number of Modes           |     30 |
-| Size of M and K matrices  |    108 |
+| Total number of Nodes     |  7 |
+| Number of interface Nodes |  7 |
+| Number of Modes           |  6 |
+| Size of M and K matrices  | 48 |
 
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
   data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
 #+end_src
 
+#+name: tab:apa95ml_nodes_coordinates
 #+caption: Coordinates of the interface nodes
 #+RESULTS:
 | Node i | Node Number |  x [m] | y [m] |     z [m] |
-|--------+-------------+--------+-------+-------|
-|    1.0 |    168947.0 |    0.0 |  0.03 |   0.0 |
-|    2.0 |    168949.0 |    0.0 | -0.03 |   0.0 |
-|    3.0 |    168950.0 | -0.035 |   0.0 |   0.0 |
-|    4.0 |    168951.0 | -0.028 |   0.0 |   0.0 |
-|    5.0 |    168952.0 | -0.021 |   0.0 |   0.0 |
-|    6.0 |    168953.0 | -0.014 |   0.0 |   0.0 |
-|    7.0 |    168954.0 | -0.007 |   0.0 |   0.0 |
-|    8.0 |    168955.0 |    0.0 |   0.0 |   0.0 |
-|    9.0 |    168956.0 |  0.007 |   0.0 |   0.0 |
-|   10.0 |    168957.0 |  0.014 |   0.0 |   0.0 |
-|   11.0 |    168958.0 |  0.021 |   0.0 |   0.0 |
-|   12.0 |    168959.0 |  0.035 |   0.0 |   0.0 |
-|   13.0 |    168960.0 |  0.028 |   0.0 |   0.0 |
+|--------+-------------+--------+-------+-----------|
+|    1.0 |     40467.0 |    0.0 |   0.0 |  0.029997 |
+|    2.0 |     40469.0 |    0.0 |   0.0 | -0.029997 |
+|    3.0 |     40470.0 | -0.035 |   0.0 |       0.0 |
+|    4.0 |     40475.0 | -0.015 |   0.0 |       0.0 |
+|    5.0 |     40476.0 | -0.005 |   0.0 |       0.0 |
+|    6.0 |     40477.0 |  0.015 |   0.0 |       0.0 |
+|    7.0 |     40478.0 |  0.035 |   0.0 |       0.0 |
+
+#+name: fig:APA95ML_nodes_1
+#+caption: Interface Nodes chosen for the APA95ML
+[[file:figs/APA95ML_nodes_1.png]]
 
 Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
 
-** Piezoelectric parameters
-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:
-#+begin_src matlab
-  d33 = 300e-12; % Strain constant [m/V]
-  n   = 80;      % Number of layers per stack
-  eT  = 1.6e-8;  % Permittivity under constant stress [F/m]
-  sD  = 1e-11;   % Compliance under constant electric displacement [m2/N]
-  ka  = 235e6;   % Stack stiffness [N/m]
-  C   = 5e-6;    % Stack capactiance [F]
-#+end_src
-
-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:
-#+begin_src matlab :results replace value
-  d33*n*ka % [N/V]
-#+end_src
-
-#+RESULTS:
-: 5.64
-
-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:
-#+begin_src matlab :results replace value
-  1e-6*d33/(eT*sD*n) % [V/um]
-#+end_src
-
-#+RESULTS:
-: 23.438
-
 ** 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.
 
-- [ ] Add schematic of the model with interface nodes
-
 One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
 #+begin_src matlab
   m = 5;
@@ -354,236 +725,6 @@ The obtained dynamics is shown in Figure [[fig:dynamics_force_force_sensor_comp_
   save('mat/fem_simscape_models.mat', 'Ghm', 'Gfm')
 #+end_src
 
-* TODO Estimation of piezoelectric parameters
-** 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/mat/');
-  addpath('./matlab/');
-#+end_src
-
-#+begin_src matlab :eval no
-  addpath('./mat/');
-#+end_src
-
-** From actuator voltage to vertical displacement
-The data from the "noise test" and the identification test are loaded.
-#+begin_src matlab
-  load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y');
-#+end_src
-
-Any offset value is removed:
-#+begin_src matlab
-  um = detrend(um, 0); % Amplifier Input Voltage [V]
-  y  = detrend(y , 0); % Mass displacement [m]
-#+end_src
-
-Now we add a factor 10 to take into account the gain of the voltage amplifier.
-#+begin_src matlab
-  um = 10*um; % Stack Actuator Input Voltage [V]
-#+end_src
-
-#+begin_src matlab
-  Ts = t(end)/(length(t)-1);
-  Fs = 1/Ts;
-
-  win = hanning(ceil(1*Fs));
-#+end_src
-
-#+begin_src matlab
-  [tf_est, f] = tfestimate(um, y, win, [], [], 1/Ts);
-#+end_src
-
-The gain from input voltage of the stack to the vertical displacement is determined:
-#+begin_src matlab
-  gD = 4e-7; % [m/V]
-#+end_src
-
-#+begin_src matlab :exports none
-  freqs = logspace(0, 4, 1000);
-
-  figure;
-  hold on;
-  plot(f, abs(tf_est))
-  plot([f(2) f(end)], [gD gD])
-  hold off;
-  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
-  ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel', []);
-  xlim([10, 5e3]);
-#+end_src
-
-#+begin_src matlab
-  K = extractMatrix('APA95ML_K.txt');
-  M = extractMatrix('APA95ML_M.txt');
-  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
-#+end_src
-
-Define parameters just for the simulation. Should not change any results
-#+begin_src matlab
-  d33 = 1; % Strain constant [m/V]
-  n   = 1; % Number of layers per stack
-  eT  = 1; % Permittivity under constant stress [F/m]
-  sD  = 1; % Compliance under constant electric displacement [m2/N]
-  ka  = 1; % Stack stiffness [N/m]
-  C   = 1; % Stack capactiance [F]
-#+end_src
-
-#+begin_src matlab
-  m = 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);
-#+end_src
-
-#+begin_src matlab :results value replace
-  gF = abs(dcgain(Gd)); % [m/N]
-  ans = gF
-#+end_src
-
-#+RESULTS:
-: 6.1695e-09
-
-\[ g_a = g_D/g_F \]
-in [N/V]
-#+begin_src matlab :results value replace
-  ga = gD/gF
-  ans = ga
-#+end_src
-
-#+RESULTS:
-: 64.835
-
-#+begin_src matlab :results value replace
-  na = 2; ns = 1; d33 = 300e-12; n = 80; ka = 235e6; gL = 1.7;
-  na/(na+ns)*gL*d33*n*ka
-#+end_src
-
-#+RESULTS:
-: 6.392
-
-- [ ] Why is there a factor 10 with the "theoretical estimation?"
-
-#+begin_src matlab :exports none
-  freqs = logspace(1, 4, 1000);
-
-  figure;
-  hold on;
-  plot(f, abs(tf_est))
-  plot(f, ga*abs(squeeze(freqresp(Gd, f, 'Hz'))))
-  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
-  ylabel('Amplitude'); xlabel('Frequency [Hz]');
-  hold off;
-#+end_src
-
-** From actuator voltage to sensor Voltage
-#+begin_src matlab
-  load('apa95ml_5kg_2a_1s.mat', 't', 'u', 'v');
-#+end_src
-
-#+begin_src matlab
-  u = detrend(u, 0); % Input Voltage of the Amplifier [V]
-  v = detrend(v, 0); % Voltage accross the stack sensor [V]
-#+end_src
-
-#+begin_src matlab
-  u = 20*u; % Input Voltage of the Amplifier [V]
-#+end_src
-
-#+begin_src matlab
-  Ts = t(end)/(length(t)-1);
-  Fs = 1/Ts;
-
-  win = hann(ceil(10/Ts));
-
-  [tf_est, f] = tfestimate(u, v, win, [], [], 1/Ts);
-#+end_src
-
-#+begin_src matlab
-  gV = 0.022; % [V/V]
-#+end_src
-
-#+begin_src matlab :exports none
-  freqs = logspace(1, 4, 1000);
-
-  figure;
-  hold on;
-  plot(f, abs(tf_est))
-  plot([f(2); f(end)], [gV; gV])
-  hold off;
-  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
-  ylabel('Amplitude'); xlabel('Frequency [Hz]');
-  ylim([1e-3, 1e1]);
-#+end_src
-
-#+begin_src matlab
-  m = 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, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Stack displacement [m]
-
-  Gf = linearize(mdl, io);
-#+end_src
-
-$g_F$ in [m/N]
-#+begin_src matlab :results value replace
-  gF = abs(dcgain(Gf));
-  ans = gF
-#+end_src
-
-#+RESULTS:
-: 2.1546e-10
-
-Finally, we compute the gain from strain of the sensor stack to its generated voltage:
-\[ g_S = \frac{g_V}{g_F g_a} \]
-Gs in [V/m]
-#+begin_src matlab :results value replace
-  gS = gV/gF/ga;
-  ans = gS
-#+end_src
-
-#+RESULTS:
-: 15749000.0
-
-#+begin_src matlab :results value replace
-  d33 = 300e-12; eT = 5.3e-9; sD = 2e-11; n = 80;
-  d33/(eT*sD*n)
-#+end_src
-
-#+RESULTS:
-: 35377000.0
-
-#+begin_src matlab :exports none
-  freqs = logspace(1, 4, 1000);
-
-  figure;
-  hold on;
-  plot(f, abs(tf_est))
-  plot(f, ga*gS*abs(squeeze(freqresp(Gf, f, 'Hz'))))
-  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
-  ylabel('Amplitude'); xlabel('Frequency [Hz]');
-  hold off;
-  ylim([1e-3, 1e1]);
-#+end_src
-
 * Huddle Test
 :PROPERTIES:
 :header-args:matlab+: :tangle matlab/huddle_test.m