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<h1 class="title">Test Bench - Amplified Piezoelectric Actuator</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgb908526">1. Experimental Setup</a></li>
<li><a href="#org58c6b68">2. Estimation of electrical/mechanical relationships</a>
<ul>
<li><a href="#org5a0dde5">2.1. Estimation from Data-sheet</a></li>
<li><a href="#org20f0327">2.2. Estimation from Piezoelectric parameters</a></li>
<li><a href="#org29014d6">2.3. Estimation from Experiment</a>
<ul>
<li><a href="#org3f63fed">2.3.1. From actuator voltage \(V_a\) to actuator force \(F_a\)</a></li>
<li><a href="#org0877a6d">2.3.2. From stack strain \(\Delta h\) to generated voltage \(V_s\)</a></li>
</ul>
</li>
<li><a href="#org5e8b78e">2.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgaded65a">3. Simscape model of the test-bench</a>
<ul>
<li><a href="#orgbd18d02">3.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org6df0376">3.2. Simscape Model</a></li>
<li><a href="#org1d92e82">3.3. Dynamics from Actuator Voltage to Vertical Mass Displacement</a></li>
<li><a href="#org18d1e8d">3.4. Dynamics from Actuator Voltage to Force Sensor Voltage</a></li>
<li><a href="#org74335e3">3.5. Save Data for further use</a></li>
</ul>
</li>
<li><a href="#org98bd921">4. Measurement of the ambient noise in the system</a>
<ul>
<li><a href="#orge831f73">4.1. Time Domain Data</a></li>
<li><a href="#org38d16b4">4.2. PSD of Measurement Noise</a></li>
</ul>
</li>
<li><a href="#org90553e0">5. Identification of the dynamics from actuator Voltage to displacement</a>
<ul>
<li><a href="#orgbdf7a7b">5.1. Load Data</a></li>
<li><a href="#org5513479">5.2. Comparison of the PSD with Huddle Test</a></li>
<li><a href="#org4d4bebc">5.3. Compute TF estimate and Coherence</a></li>
</ul>
</li>
<li><a href="#org12553c5">6. Identification of the dynamics from actuator Voltage to force sensor Voltage</a></li>
<li><a href="#org6f69286">7. Integral Force Feedback</a>
<ul>
<li><a href="#orgbc62cba">7.1. IFF Plant</a></li>
<li><a href="#org4d62e16">7.2. First tests with few gains</a></li>
<li><a href="#org4304a4e">7.3. Second test with many Gains</a></li>
</ul>
</li>
</ul>
</div>
</div>
<p>
This document is divided in the following sections:
</p>
<ul class="org-ul">
<li>Section <a href="#orgac1a70e">1</a>: the experimental setup is described</li>
<li>Section <a href="#org0db4f21">2</a>: the parameters which are important for the Simscape model of the piezoelectric stack actuator and sensors are estimated</li>
<li>Section <a href="#org9d27452">3</a>: the Simscape model of the test bench is presented</li>
<li>Section <a href="#orga049d7f">4</a>: as usual, a first measurement of the noise/disturbances present in the system is performed</li>
<li>Section <a href="#org72688d3">5</a>: the transfer function from the actuator voltage to the displacement of the mass is identified and compared with the model</li>
<li>Section <a href="#orgdd284b9">6</a>: the tranfer function from the actuator voltage to the sensor stack voltage is identified and compare with the model</li>
<li>Section <a href="#org13dae17">7</a>: the Integral Force Feedback control architecture is applied on the system using the force sensor stack in order to add damping to the suspension resonance</li>
</ul>
<div id="outline-container-orgb908526" class="outline-2">
<h2 id="orgb908526"><span class="section-number-2">1</span> Experimental Setup</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="orgac1a70e"></a>
</p>
<p>
A schematic of the test-bench is shown in Figure <a href="#org42f14fa">1</a>.
</p>
<p>
A mass can be vertically moved using the amplified piezoelectric actuator (APA95ML).
The displacement of the mass (relative to the mechanical frame) is measured by the interferometer.
</p>
<p>
The APA95ML has three stacks that can be used as actuator or as sensors.
</p>
<p>
Pictures of the test bench are shown in Figure <a href="#org934d0a3">2</a> and <a href="#orgf6513b0">3</a>.
</p>
<div id="org42f14fa" class="figure">
<p><img src="figs/test_bench_apa_schematic.png" alt="test_bench_apa_schematic.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the Setup</p>
</div>
<div id="org934d0a3" class="figure">
<p><img src="figs/setup_picture.png" alt="setup_picture.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Picture of the Setup</p>
</div>
<div id="orgf6513b0" class="figure">
<p><img src="figs/setup_zoom.png" alt="setup_zoom.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Zoom on the APA</p>
</div>
<div class="note" id="orgb97925c">
<p>
Here are the equipment used in the test bench:
</p>
<ul class="org-ul">
<li>Attocube interferometer (<a href="doc/IDS3010.pdf">doc</a>)</li>
<li>Cedrat Amplified Piezoelectric Actuator APA95ML (<a href="doc/APA95ML.pdf">doc</a>)</li>
<li>Voltage Amplifier LA75B (<a href="doc/LA75B.pdf">doc</a>)</li>
<li>Speedgoat IO131 with 16bits ADC and DAC (<a href="doc/IO130 IO131 OEM Datasheet.pdf">doc</a>)</li>
<li>Low Noise Voltage Preamplifier from Ametek (<a href="doc/model_5113.pdf">doc</a>)</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-org58c6b68" class="outline-2">
<h2 id="org58c6b68"><span class="section-number-2">2</span> Estimation of electrical/mechanical relationships</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org0db4f21"></a>
</p>
<p>
In order to correctly model the piezoelectric actuator with Simscape, we need to determine:
</p>
<ol class="org-ol">
<li>\(g_a\): the ratio of the generated force \(F_a\) to the supply voltage \(V_a\) across the piezoelectric stack</li>
<li>\(g_s\): the ratio of the generated voltage \(V_s\) across the piezoelectric stack when subject to a strain \(\Delta h\)</li>
</ol>
<p>
We estimate \(g_a\) and \(g_s\) using different approaches:
</p>
<ol class="org-ol">
<li>Section <a href="#orgcfc54fa">2.1</a>: \(g_a\) is estimated from the datasheet of the piezoelectric stack</li>
<li>Section <a href="#org0db4f21">2</a>: \(g_a\) and \(g_s\) are estimated using the piezoelectric constants</li>
<li>Section <a href="#orgcdcd11f">2.3</a>: \(g_a\) and \(g_s\) are estimated experimentally</li>
</ol>
</div>
<div id="outline-container-org5a0dde5" class="outline-3">
<h3 id="org5a0dde5"><span class="section-number-3">2.1</span> Estimation from Data-sheet</h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="orgcfc54fa"></a>
</p>
<p>
The stack parameters taken from the data-sheet are shown in Table <a href="#org82c2f0a">1</a>.
</p>
<table id="org82c2f0a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Stack Parameters</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Parameter</th>
<th scope="col" class="org-left">Unit</th>
<th scope="col" class="org-right">Value</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Nominal Stroke</td>
<td class="org-left">\(\mu m\)</td>
<td class="org-right">20</td>
</tr>
<tr>
<td class="org-left">Blocked force</td>
<td class="org-left">\(N\)</td>
<td class="org-right">4700</td>
</tr>
<tr>
<td class="org-left">Stiffness</td>
<td class="org-left">\(N/\mu m\)</td>
<td class="org-right">235</td>
</tr>
<tr>
<td class="org-left">Voltage Range</td>
<td class="org-left">\(V\)</td>
<td class="org-right">-20..150</td>
</tr>
<tr>
<td class="org-left">Capacitance</td>
<td class="org-left">\(\mu F\)</td>
<td class="org-right">4.4</td>
</tr>
<tr>
<td class="org-left">Length</td>
<td class="org-left">\(mm\)</td>
<td class="org-right">20</td>
</tr>
<tr>
<td class="org-left">Stack Area</td>
<td class="org-left">\(mm^2\)</td>
<td class="org-right">10x10</td>
</tr>
</tbody>
</table>
<p>
Let&rsquo;s compute the generated force
</p>
<p>
The stroke is \(L_{\max} = 20\mu m\) for a voltage range of \(V_{\max} = 170 V\).
Furthermore, the stiffness is \(k_a = 235 \cdot 10^6 N/m\).
</p>
<p>
The relation between the applied voltage and the generated force can be estimated as follows:
</p>
\begin{equation}
g_a = k_a \frac{L_{\max}}{V_{\max}}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab">ka = 235e6; <span class="org-comment">% [N/m]</span>
Lmax = 20e<span class="org-type">-</span>6; <span class="org-comment">% [m]</span>
Vmax = 170; <span class="org-comment">% [V]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">ga = ka<span class="org-type">*</span>Lmax<span class="org-type">/</span>Vmax; <span class="org-comment">% [N/V]</span>
</pre>
</div>
<pre class="example">
ga = 27.6 [N/V]
</pre>
<p>
From the parameters of the stack, it seems not possible to estimate the relation between the strain and the generated voltage.
</p>
</div>
</div>
<div id="outline-container-org20f0327" class="outline-3">
<h3 id="org20f0327"><span class="section-number-3">2.2</span> Estimation from Piezoelectric parameters</h3>
<div class="outline-text-3" id="text-2-2">
<p>
<a id="orgc2edf63"></a>
</p>
<p>
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:
</p>
<div class="org-src-container">
<pre class="src src-matlab">d33 = 300e<span class="org-type">-</span>12; <span class="org-comment">% Strain constant [m/V]</span>
n = 80; <span class="org-comment">% Number of layers per stack</span>
ka = 235e6; <span class="org-comment">% Stack stiffness [N/m]</span>
</pre>
</div>
<p>
The ratio of the developed force to applied voltage is:
</p>
\begin{equation}
\label{org6f1476c}
F_a = g_a V_a, \quad g_a = d_{33} n k_a
\end{equation}
<p>
where:
</p>
<ul class="org-ul">
<li>\(F_a\): developed force in [N]</li>
<li>\(n\): number of layers of the actuator stack</li>
<li>\(d_{33}\): strain constant in [m/V]</li>
<li>\(k_a\): actuator stack stiffness in [N/m]</li>
<li>\(V_a\): applied voltage in [V]</li>
</ul>
<p>
If we take the numerical values, we obtain:
</p>
<div class="org-src-container">
<pre class="src src-matlab">ga = d33<span class="org-type">*</span>n<span class="org-type">*</span>ka; <span class="org-comment">% [N/V]</span>
</pre>
</div>
<pre class="example">
ga = 5.6 [N/V]
</pre>
<p>
From (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>) (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
</p>
\begin{equation}
\label{org43f15ff}
V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
\end{equation}
<p>
where:
</p>
<ul class="org-ul">
<li>\(V_s\): measured voltage in [V]</li>
<li>\(d_{33}\): strain constant in [m/V]</li>
<li>\(\epsilon^T\): permittivity under constant stress in [F/m]</li>
<li>\(s^D\): elastic compliance under constant electric displacement in [m^2/N]</li>
<li>\(n\): number of layers of the sensor stack</li>
<li>\(\Delta h\): relative displacement in [m]</li>
</ul>
<p>
If we take the numerical values, we obtain:
</p>
<div class="org-src-container">
<pre class="src src-matlab">d33 = 300e<span class="org-type">-</span>12; <span class="org-comment">% Strain constant [m/V]</span>
n = 80; <span class="org-comment">% Number of layers per stack</span>
eT = 5.3e<span class="org-type">-</span>9; <span class="org-comment">% Permittivity under constant stress [F/m]</span>
sD = 2e<span class="org-type">-</span>11; <span class="org-comment">% Compliance under constant electric displacement [m2/N]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">gs = d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>n); <span class="org-comment">% [V/m]</span>
</pre>
</div>
<pre class="example">
gs = 35.4 [V/um]
</pre>
</div>
</div>
<div id="outline-container-org29014d6" class="outline-3">
<h3 id="org29014d6"><span class="section-number-3">2.3</span> Estimation from Experiment</h3>
<div class="outline-text-3" id="text-2-3">
<p>
<a id="orgcdcd11f"></a>
</p>
<p>
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.
</p>
<p>
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 \(g_a\).
</p>
<p>
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.
</p>
</div>
<div id="outline-container-org3f63fed" class="outline-4">
<h4 id="org3f63fed"><span class="section-number-4">2.3.1</span> From actuator voltage \(V_a\) to actuator force \(F_a\)</h4>
<div class="outline-text-4" id="text-2-3-1">
<p>
The data from the identification test is loaded.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'apa95ml_5kg_Amp_E505.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'um'</span>, <span class="org-string">'y'</span>);
<span class="org-comment">% Any offset value is removed:</span>
um = detrend(um, 0); <span class="org-comment">% Amplifier Input Voltage [V]</span>
y = detrend(y , 0); <span class="org-comment">% Mass displacement [m]</span>
</pre>
</div>
<p>
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.
</p>
<div class="org-src-container">
<pre class="src src-matlab">um = 10<span class="org-type">*</span>um; <span class="org-comment">% Stack Actuator Input Voltage [V]</span>
</pre>
</div>
<p>
Then, the transfer function from the stack voltage to the vertical displacement is computed.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
Fs = 1<span class="org-type">/</span>Ts;
win = hanning(ceil(1<span class="org-type">*</span>Fs));
[tf_est, f] = tfestimate(um, y, win, [], [], 1<span class="org-type">/</span>Ts);
</pre>
</div>
<p>
The gain from input voltage of the stack to the vertical displacement is determined:
</p>
<div class="org-src-container">
<pre class="src src-matlab">g_d_Va = 4e<span class="org-type">-</span>7; <span class="org-comment">% [m/V]</span>
</pre>
</div>
<div id="org85d884f" class="figure">
<p><img src="figs/gain_Va_to_d.png" alt="gain_Va_to_d.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Transfer function from actuator stack voltage \(V_a\) to vertical displacement of the mass \(d\)</p>
</div>
<p>
Then, the transfer function from forces applied by the stack actuator to the vertical displacement of the mass is identified from the Simscape model.
</p>
<div class="org-src-container">
<pre class="src src-matlab">m = 5.5;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'piezo_amplified_3d'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Fa'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force [N]</span>
io(io_i) = linio([mdl, <span class="org-string">'/y'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Vertical Displacement [m]</span>
Gd = linearize(mdl, io);
</pre>
</div>
<p>
The DC gain the the identified dynamics
</p>
<div class="org-src-container">
<pre class="src src-matlab">g_d_Fa = abs(dcgain(Gd)); <span class="org-comment">% [m/N]</span>
</pre>
</div>
<pre class="example">
G_d_Fa = 1.2e-08 [m/N]
</pre>
<p>
And finally, the gain \(g_a\) from the the actuator voltage \(V_a\) to the generated force \(F_a\) can be computed:
</p>
\begin{equation}
g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \frac{d}{V_a}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab">ga = g_d_Va<span class="org-type">/</span>g_d_Fa;
</pre>
</div>
<pre class="example">
ga = 33.7 [N/V]
</pre>
<p>
The obtained comparison between the Simscape model and the identified dynamics is shown in Figure <a href="#orgc82fdbe">5</a>.
</p>
<div id="orgc82fdbe" class="figure">
<p><img src="figs/compare_Gd_id_simscape.png" alt="compare_Gd_id_simscape.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Comparison of the identified transfer function between actuator voltage \(V_a\) and vertical mass displacement \(d\)</p>
</div>
</div>
</div>
<div id="outline-container-org0877a6d" class="outline-4">
<h4 id="org0877a6d"><span class="section-number-4">2.3.2</span> From stack strain \(\Delta h\) to generated voltage \(V_s\)</h4>
<div class="outline-text-4" id="text-2-3-2">
<p>
Now, the gain from the stack strain \(\Delta h\) to the generated voltage \(V_s\) is estimated.
</p>
<p>
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 \(\Delta h\).
</p>
<p>
Finally, using these two values, we can compute the gain \(g_s\) from the stack strain \(\Delta h\) to the generated Voltage \(V_s\).
</p>
<p>
Identification data is loaded.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'apa95ml_5kg_2a_1s.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
u = detrend(u, 0); <span class="org-comment">% Input Voltage of the Amplifier [V]</span>
v = detrend(v, 0); <span class="org-comment">% Voltage accross the stack sensor [V]</span>
</pre>
</div>
<p>
Here, an amplifier with a gain of 20 is used.
</p>
<div class="org-src-container">
<pre class="src src-matlab">u = 20<span class="org-type">*</span>u; <span class="org-comment">% Input Voltage of the Amplifier [V]</span>
</pre>
</div>
<p>
Then, the transfer function from \(V_a\) to \(V_s\) is identified and its DC gain is estimated (Figure <a href="#orgf3891ce">6</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
Fs = 1<span class="org-type">/</span>Ts;
win = hann(ceil(10<span class="org-type">/</span>Ts));
[tf_est, f] = tfestimate(u, v, win, [], [], 1<span class="org-type">/</span>Ts);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">g_Vs_Va = 0.022; <span class="org-comment">% [V/V]</span>
</pre>
</div>
<div id="orgf3891ce" class="figure">
<p><img src="figs/gain_Va_to_Vs.png" alt="gain_Va_to_Vs.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Transfer function from actuator stack voltage \(V_a\) to sensor stack voltage \(V_s\)</p>
</div>
<p>
Now the transfer function from the actuator stack voltage to the sensor stack strain is estimated using the Simscape model.
</p>
<div class="org-src-container">
<pre class="src src-matlab">m = 5.5;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'piezo_amplified_3d'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Va'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Voltage [V]</span>
io(io_i) = linio([mdl, <span class="org-string">'/dL'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Sensor Stack displacement [m]</span>
Gf = linearize(mdl, io);
</pre>
</div>
<p>
The gain from the actuator stack voltage to the sensor stack strain is estimated below.
</p>
<div class="org-src-container">
<pre class="src src-matlab">G_dh_Va = abs(dcgain(Gf));
</pre>
</div>
<pre class="example">
G_dh_Va = 6.2e-09 [m/V]
</pre>
<p>
And finally, the gain \(g_s\) from the sensor stack strain to the generated voltage can be estimated:
</p>
\begin{equation}
g_s = \frac{V_s}{\Delta h} = \frac{V_s}{V_a} \frac{V_a}{\Delta h}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab">gs = g_Vs_Va<span class="org-type">/</span>G_dh_Va; <span class="org-comment">% [V/m]</span>
</pre>
</div>
<pre class="example">
gs = 3.5 [V/um]
</pre>
<div id="org5abdbfd" class="figure">
<p><img src="figs/compare_Gf_id_simscape.png" alt="compare_Gf_id_simscape.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Comparison of the identified transfer function between actuator voltage \(V_a\) and sensor stack voltage</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org5e8b78e" class="outline-3">
<h3 id="org5e8b78e"><span class="section-number-3">2.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-4">
<p>
The obtained parameters \(g_a\) and \(g_s\) are not consistent between the different methods.
</p>
<p>
The one using the experimental data are saved and further used.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./matlab/mat/apa95ml_params.mat'</span>, <span class="org-string">'ga'</span>, <span class="org-string">'gs'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgaded65a" class="outline-2">
<h2 id="orgaded65a"><span class="section-number-2">3</span> Simscape model of the test-bench</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org9d27452"></a>
</p>
<p>
The idea here is to model the test-bench using Simscape.
</p>
<p>
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.
</p>
<p>
To model the APA, a Finite Element Model (FEM) is used (Figure <a href="#orgf4bc6a9">8</a>) and imported into Simscape.
</p>
<div id="orgf4bc6a9" class="figure">
<p><img src="figs/APA95ML_FEM.png" alt="APA95ML_FEM.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Finite Element Model of the APA95ML</p>
</div>
</div>
<div id="outline-container-orgbd18d02" class="outline-3">
<h3 id="orgbd18d02"><span class="section-number-3">3.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div class="outline-text-3" id="text-3-1">
<p>
We first extract the stiffness and mass matrices.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = readmatrix(<span class="org-string">'APA95ML_K.CSV'</span>);
M = readmatrix(<span class="org-string">'APA95ML_M.CSV'</span>);
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> First 10x10 elements of the Stiffness matrix</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">300000000.0</td>
<td class="org-right">-1000.0</td>
<td class="org-right">-30000.0</td>
<td class="org-right">-40.0</td>
<td class="org-right">70000.0</td>
<td class="org-right">300.0</td>
<td class="org-right">20000000.0</td>
<td class="org-right">-30.0</td>
<td class="org-right">-5000.0</td>
<td class="org-right">5</td>
</tr>
<tr>
<td class="org-right">-1000.0</td>
<td class="org-right">50000000.0</td>
<td class="org-right">-7000.0</td>
<td class="org-right">800000.0</td>
<td class="org-right">-20.0</td>
<td class="org-right">300.0</td>
<td class="org-right">3000.0</td>
<td class="org-right">5000000.0</td>
<td class="org-right">400.0</td>
<td class="org-right">-40000.0</td>
</tr>
<tr>
<td class="org-right">-30000.0</td>
<td class="org-right">-7000.0</td>
<td class="org-right">100000000.0</td>
<td class="org-right">-200.0</td>
<td class="org-right">-60.0</td>
<td class="org-right">70.0</td>
<td class="org-right">3000.0</td>
<td class="org-right">3000.0</td>
<td class="org-right">-8000000.0</td>
<td class="org-right">-30.0</td>
</tr>
<tr>
<td class="org-right">-40.0</td>
<td class="org-right">800000.0</td>
<td class="org-right">-200.0</td>
<td class="org-right">20000.0</td>
<td class="org-right">-0.4</td>
<td class="org-right">4</td>
<td class="org-right">30.0</td>
<td class="org-right">40000.0</td>
<td class="org-right">7</td>
<td class="org-right">-300.0</td>
</tr>
<tr>
<td class="org-right">70000.0</td>
<td class="org-right">-20.0</td>
<td class="org-right">-60.0</td>
<td class="org-right">-0.4</td>
<td class="org-right">3000.0</td>
<td class="org-right">1</td>
<td class="org-right">-6000.0</td>
<td class="org-right">10.0</td>
<td class="org-right">8</td>
<td class="org-right">-0.1</td>
</tr>
<tr>
<td class="org-right">300.0</td>
<td class="org-right">300.0</td>
<td class="org-right">70.0</td>
<td class="org-right">4</td>
<td class="org-right">1</td>
<td class="org-right">40000.0</td>
<td class="org-right">-10.0</td>
<td class="org-right">-10.0</td>
<td class="org-right">30.0</td>
<td class="org-right">0.1</td>
</tr>
<tr>
<td class="org-right">20000000.0</td>
<td class="org-right">3000.0</td>
<td class="org-right">3000.0</td>
<td class="org-right">30.0</td>
<td class="org-right">-6000.0</td>
<td class="org-right">-10.0</td>
<td class="org-right">300000000.0</td>
<td class="org-right">2000.0</td>
<td class="org-right">9000.0</td>
<td class="org-right">-100.0</td>
</tr>
<tr>
<td class="org-right">-30.0</td>
<td class="org-right">5000000.0</td>
<td class="org-right">3000.0</td>
<td class="org-right">40000.0</td>
<td class="org-right">10.0</td>
<td class="org-right">-10.0</td>
<td class="org-right">2000.0</td>
<td class="org-right">50000000.0</td>
<td class="org-right">-3000.0</td>
<td class="org-right">-800000.0</td>
</tr>
<tr>
<td class="org-right">-5000.0</td>
<td class="org-right">400.0</td>
<td class="org-right">-8000000.0</td>
<td class="org-right">7</td>
<td class="org-right">8</td>
<td class="org-right">30.0</td>
<td class="org-right">9000.0</td>
<td class="org-right">-3000.0</td>
<td class="org-right">100000000.0</td>
<td class="org-right">100.0</td>
</tr>
<tr>
<td class="org-right">5</td>
<td class="org-right">-40000.0</td>
<td class="org-right">-30.0</td>
<td class="org-right">-300.0</td>
<td class="org-right">-0.1</td>
<td class="org-right">0.1</td>
<td class="org-right">-100.0</td>
<td class="org-right">-800000.0</td>
<td class="org-right">100.0</td>
<td class="org-right">20000.0</td>
</tr>
</tbody>
</table>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> First 10x10 elements of the Mass matrix</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">0.03</td>
<td class="org-right">7e-08</td>
<td class="org-right">2e-06</td>
<td class="org-right">-3e-09</td>
<td class="org-right">-0.0002</td>
<td class="org-right">-6e-08</td>
<td class="org-right">-0.001</td>
<td class="org-right">8e-07</td>
<td class="org-right">6e-07</td>
<td class="org-right">-8e-09</td>
</tr>
<tr>
<td class="org-right">7e-08</td>
<td class="org-right">0.02</td>
<td class="org-right">-1e-06</td>
<td class="org-right">9e-05</td>
<td class="org-right">-3e-09</td>
<td class="org-right">-4e-09</td>
<td class="org-right">-1e-06</td>
<td class="org-right">-0.0006</td>
<td class="org-right">-4e-08</td>
<td class="org-right">5e-06</td>
</tr>
<tr>
<td class="org-right">2e-06</td>
<td class="org-right">-1e-06</td>
<td class="org-right">0.02</td>
<td class="org-right">-3e-08</td>
<td class="org-right">-4e-08</td>
<td class="org-right">1e-08</td>
<td class="org-right">1e-07</td>
<td class="org-right">-2e-07</td>
<td class="org-right">0.0003</td>
<td class="org-right">1e-09</td>
</tr>
<tr>
<td class="org-right">-3e-09</td>
<td class="org-right">9e-05</td>
<td class="org-right">-3e-08</td>
<td class="org-right">1e-06</td>
<td class="org-right">-3e-11</td>
<td class="org-right">-3e-13</td>
<td class="org-right">-7e-09</td>
<td class="org-right">-5e-06</td>
<td class="org-right">-3e-10</td>
<td class="org-right">3e-08</td>
</tr>
<tr>
<td class="org-right">-0.0002</td>
<td class="org-right">-3e-09</td>
<td class="org-right">-4e-08</td>
<td class="org-right">-3e-11</td>
<td class="org-right">2e-06</td>
<td class="org-right">6e-10</td>
<td class="org-right">2e-06</td>
<td class="org-right">-7e-09</td>
<td class="org-right">-2e-09</td>
<td class="org-right">7e-11</td>
</tr>
<tr>
<td class="org-right">-6e-08</td>
<td class="org-right">-4e-09</td>
<td class="org-right">1e-08</td>
<td class="org-right">-3e-13</td>
<td class="org-right">6e-10</td>
<td class="org-right">1e-06</td>
<td class="org-right">1e-08</td>
<td class="org-right">3e-09</td>
<td class="org-right">-2e-09</td>
<td class="org-right">2e-13</td>
</tr>
<tr>
<td class="org-right">-0.001</td>
<td class="org-right">-1e-06</td>
<td class="org-right">1e-07</td>
<td class="org-right">-7e-09</td>
<td class="org-right">2e-06</td>
<td class="org-right">1e-08</td>
<td class="org-right">0.03</td>
<td class="org-right">4e-08</td>
<td class="org-right">-2e-06</td>
<td class="org-right">8e-09</td>
</tr>
<tr>
<td class="org-right">8e-07</td>
<td class="org-right">-0.0006</td>
<td class="org-right">-2e-07</td>
<td class="org-right">-5e-06</td>
<td class="org-right">-7e-09</td>
<td class="org-right">3e-09</td>
<td class="org-right">4e-08</td>
<td class="org-right">0.02</td>
<td class="org-right">-9e-07</td>
<td class="org-right">-9e-05</td>
</tr>
<tr>
<td class="org-right">6e-07</td>
<td class="org-right">-4e-08</td>
<td class="org-right">0.0003</td>
<td class="org-right">-3e-10</td>
<td class="org-right">-2e-09</td>
<td class="org-right">-2e-09</td>
<td class="org-right">-2e-06</td>
<td class="org-right">-9e-07</td>
<td class="org-right">0.02</td>
<td class="org-right">2e-08</td>
</tr>
<tr>
<td class="org-right">-8e-09</td>
<td class="org-right">5e-06</td>
<td class="org-right">1e-09</td>
<td class="org-right">3e-08</td>
<td class="org-right">7e-11</td>
<td class="org-right">2e-13</td>
<td class="org-right">8e-09</td>
<td class="org-right">-9e-05</td>
<td class="org-right">2e-08</td>
<td class="org-right">1e-06</td>
</tr>
</tbody>
</table>
<p>
Then, we extract the coordinates of the interface nodes.
</p>
<div class="org-src-container">
<pre class="src src-matlab">[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes(<span class="org-string">'APA95ML_out_nodes_3D.txt'</span>);
</pre>
</div>
<p>
The interface nodes are shown in Figure <a href="#org25039e2">9</a> and their coordinates are listed in Table <a href="#orga84d9cf">4</a>.
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-left">Total number of Nodes</td>
<td class="org-right">7</td>
</tr>
<tr>
<td class="org-left">Number of interface Nodes</td>
<td class="org-right">7</td>
</tr>
<tr>
<td class="org-left">Number of Modes</td>
<td class="org-right">6</td>
</tr>
<tr>
<td class="org-left">Size of M and K matrices</td>
<td class="org-right">48</td>
</tr>
</tbody>
</table>
<table id="orga84d9cf" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Coordinates of the interface nodes</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">Node i</th>
<th scope="col" class="org-right">Node Number</th>
<th scope="col" class="org-right">x [m]</th>
<th scope="col" class="org-right">y [m]</th>
<th scope="col" class="org-right">z [m]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">40467.0</td>
<td class="org-right">0.0</td>
<td class="org-right">0.0</td>
<td class="org-right">0.029997</td>
</tr>
<tr>
<td class="org-right">2.0</td>
<td class="org-right">40469.0</td>
<td class="org-right">0.0</td>
<td class="org-right">0.0</td>
<td class="org-right">-0.029997</td>
</tr>
<tr>
<td class="org-right">3.0</td>
<td class="org-right">40470.0</td>
<td class="org-right">-0.035</td>
<td class="org-right">0.0</td>
<td class="org-right">0.0</td>
</tr>
<tr>
<td class="org-right">4.0</td>
<td class="org-right">40475.0</td>
<td class="org-right">-0.015</td>
<td class="org-right">0.0</td>
<td class="org-right">0.0</td>
</tr>
<tr>
<td class="org-right">5.0</td>
<td class="org-right">40476.0</td>
<td class="org-right">-0.005</td>
<td class="org-right">0.0</td>
<td class="org-right">0.0</td>
</tr>
<tr>
<td class="org-right">6.0</td>
<td class="org-right">40477.0</td>
<td class="org-right">0.015</td>
<td class="org-right">0.0</td>
<td class="org-right">0.0</td>
</tr>
<tr>
<td class="org-right">7.0</td>
<td class="org-right">40478.0</td>
<td class="org-right">0.035</td>
<td class="org-right">0.0</td>
<td class="org-right">0.0</td>
</tr>
</tbody>
</table>
<div id="org25039e2" class="figure">
<p><img src="figs/APA95ML_nodes_1.png" alt="APA95ML_nodes_1.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Interface Nodes chosen for the APA95ML</p>
</div>
<p>
Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <code>Reduced Order Flexible Solid</code> simscape block.
</p>
</div>
</div>
<div id="outline-container-org6df0376" class="outline-3">
<h3 id="org6df0376"><span class="section-number-3">3.2</span> Simscape Model</h3>
<div class="outline-text-3" id="text-3-2">
<p>
The flexible element is imported using the <code>Reduced Order Flexible Solid</code> Simscape block.
</p>
<p>
To model the actuator, an <code>Internal Force</code> block is added between the nodes 3 and 12.
A <code>Relative Motion Sensor</code> block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
</p>
<p>
One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
</p>
<div class="org-src-container">
<pre class="src src-matlab">m = 5.5;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'apa95ml_params.mat'</span>, <span class="org-string">'ga'</span>, <span class="org-string">'gs'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org1d92e82" class="outline-3">
<h3 id="org1d92e82"><span class="section-number-3">3.3</span> Dynamics from Actuator Voltage to Vertical Mass Displacement</h3>
<div class="outline-text-3" id="text-3-3">
<p>
The identified dynamics is shown in Figure <a href="#org162e335">10</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'piezo_amplified_3d'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Va'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Voltage [V]</span>
io(io_i) = linio([mdl, <span class="org-string">'/y'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Vertical Displacement [m]</span>
Ghm = linearize(mdl, io);
</pre>
</div>
<div id="org162e335" class="figure">
<p><img src="figs/dynamics_act_disp_comp_mass.png" alt="dynamics_act_disp_comp_mass.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Dynamics from \(F\) to \(d\) without a payload and with a 5kg payload</p>
</div>
</div>
</div>
<div id="outline-container-org18d1e8d" class="outline-3">
<h3 id="org18d1e8d"><span class="section-number-3">3.4</span> Dynamics from Actuator Voltage to Force Sensor Voltage</h3>
<div class="outline-text-3" id="text-3-4">
<p>
The obtained dynamics is shown in Figure <a href="#org2258d23">11</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'piezo_amplified_3d'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Va'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Voltage Actuator [V]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Vs'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Sensor Voltage [V]</span>
Gfm = linearize(mdl, io);
</pre>
</div>
<div id="org2258d23" class="figure">
<p><img src="figs/dynamics_force_force_sensor_comp_mass.png" alt="dynamics_force_force_sensor_comp_mass.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Dynamics from \(F\) to \(F_m\) for \(m=0\) and \(m = 10kg\)</p>
</div>
</div>
</div>
<div id="outline-container-org74335e3" class="outline-3">
<h3 id="org74335e3"><span class="section-number-3">3.5</span> Save Data for further use</h3>
<div class="outline-text-3" id="text-3-5">
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'matlab/mat/fem_simscape_models.mat'</span>, <span class="org-string">'Ghm'</span>, <span class="org-string">'Gfm'</span>)
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org98bd921" class="outline-2">
<h2 id="org98bd921"><span class="section-number-2">4</span> Measurement of the ambient noise in the system</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="orga049d7f"></a>
</p>
<p>
This first measurement consist of measuring the displacement of the mass using the interferometer when no voltage is applied to the actuator.
</p>
<p>
This can help determining the actuator voltage necessary to generate a motion way above the measured noise and disturbances, and thus obtain a good identification.
</p>
</div>
<div id="outline-container-orge831f73" class="outline-3">
<h3 id="orge831f73"><span class="section-number-3">4.1</span> Time Domain Data</h3>
<div class="outline-text-3" id="text-4-1">
<div id="org21f1e60" class="figure">
<p><img src="figs/huddle_test_time_domain.png" alt="huddle_test_time_domain.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Measurement of the Mass displacement during Huddle Test</p>
</div>
</div>
</div>
<div id="outline-container-org38d16b4" class="outline-3">
<h3 id="org38d16b4"><span class="section-number-3">4.2</span> PSD of Measurement Noise</h3>
<div class="outline-text-3" id="text-4-2">
<div class="org-src-container">
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
Fs = 1<span class="org-type">/</span>Ts;
win = hanning(ceil(1<span class="org-type">*</span>Fs));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">[pxx, f] = pwelch(y(1000<span class="org-type">:</span>end), win, [], [], Fs);
</pre>
</div>
<div id="org5264502" class="figure">
<p><img src="figs/huddle_test_pdf.png" alt="huddle_test_pdf.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Amplitude Spectral Density of the Displacement during Huddle Test</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org90553e0" class="outline-2">
<h2 id="org90553e0"><span class="section-number-2">5</span> Identification of the dynamics from actuator Voltage to displacement</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="org72688d3"></a>
</p>
<p>
The setup used for the identification of the dynamics from \(V_a\) to \(d\) is shown in Figure <a href="#org43a6d6f">14</a>.
</p>
<p>
A Voltage amplifier with a gain of \(10\) is used.
Two stacks are used as actuators while one stack is used as a force sensor.
</p>
<div id="org43a6d6f" class="figure">
<p><img src="figs/test_bench_apa_identification.png" alt="test_bench_apa_identification.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Test Bench used for the identification of the dynaimcs from \(V_a\) to \(d\)</p>
</div>
</div>
<div id="outline-container-orgbdf7a7b" class="outline-3">
<h3 id="orgbdf7a7b"><span class="section-number-3">5.1</span> Load Data</h3>
<div class="outline-text-3" id="text-5-1">
<p>
The data from the &ldquo;noise test&rdquo; and the identification test are loaded.
</p>
<div class="org-src-container">
<pre class="src src-matlab">ht = load(<span class="org-string">'huddle_test.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'u'</span>, <span class="org-string">'y'</span>);
load(<span class="org-string">'apa95ml_5kg_Amp_E505.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'um'</span>, <span class="org-string">'y'</span>);
</pre>
</div>
<p>
Any offset value is removed:
</p>
<div class="org-src-container">
<pre class="src src-matlab">um = detrend(um, 0); <span class="org-comment">% Input Voltage [V]</span>
y = detrend(y , 0); <span class="org-comment">% Mass displacement [m]</span>
ht.u = detrend(ht.u, 0);
ht.y = detrend(ht.y, 0);
</pre>
</div>
<p>
Now we add a factor 10 to take into account the gain of the voltage amplifier.
</p>
<div class="org-src-container">
<pre class="src src-matlab">um = 10<span class="org-type">*</span>um;
</pre>
</div>
</div>
</div>
<div id="outline-container-org5513479" class="outline-3">
<h3 id="org5513479"><span class="section-number-3">5.2</span> Comparison of the PSD with Huddle Test</h3>
<div class="outline-text-3" id="text-5-2">
<div class="org-src-container">
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
Fs = 1<span class="org-type">/</span>Ts;
win = hanning(ceil(1<span class="org-type">*</span>Fs));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">[pxx, f] = pwelch(y, win, [], [], Fs);
[pht, <span class="org-type">~</span>] = pwelch(ht.y, win, [], [], Fs);
</pre>
</div>
<div id="orgf5a2201" class="figure">
<p><img src="figs/apa95ml_5kg_PI_pdf_comp_huddle.png" alt="apa95ml_5kg_PI_pdf_comp_huddle.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Comparison of the ASD for the identification test and the huddle test</p>
</div>
</div>
</div>
<div id="outline-container-org4d4bebc" class="outline-3">
<h3 id="org4d4bebc"><span class="section-number-3">5.3</span> Compute TF estimate and Coherence</h3>
<div class="outline-text-3" id="text-5-3">
<div class="org-src-container">
<pre class="src src-matlab">[tf_est, f] = tfestimate(um, <span class="org-type">-</span>y, win, [], [], 1<span class="org-type">/</span>Ts);
[co_est, <span class="org-type">~</span>] = mscohere( um, <span class="org-type">-</span>y, win, [], [], 1<span class="org-type">/</span>Ts);
</pre>
</div>
<div id="org23ac087" class="figure">
<p><img src="figs/apa95ml_5kg_PI_coh.png" alt="apa95ml_5kg_PI_coh.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Coherence</p>
</div>
<p>
Comparison with the FEM model
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/fem_simscape_models.mat'</span>, <span class="org-string">'Ghm'</span>);
</pre>
</div>
<div id="orged73e47" class="figure">
<p><img src="figs/apa95ml_5kg_pi_comp_fem.png" alt="apa95ml_5kg_pi_comp_fem.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Comparison of the identified transfer function and the one estimated from the FE model</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org12553c5" class="outline-2">
<h2 id="org12553c5"><span class="section-number-2">6</span> Identification of the dynamics from actuator Voltage to force sensor Voltage</h2>
<div class="outline-text-2" id="text-6">
<p>
<a id="orgdd284b9"></a>
</p>
<p>
The same setup shown in Figure <a href="#org43a6d6f">14</a> is used.
</p>
<p>
The data are loaded:
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'apa95ml_5kg_2a_1s.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
</pre>
</div>
<p>
Any offset is removed.
</p>
<div class="org-src-container">
<pre class="src src-matlab">u = detrend(u, 0); <span class="org-comment">% Speedgoat DAC output Voltage [V]</span>
v = detrend(v, 0); <span class="org-comment">% Speedgoat ADC input Voltage (sensor stack) [V]</span>
</pre>
</div>
<p>
Here, the amplifier gain is 20.
</p>
<div class="org-src-container">
<pre class="src src-matlab">u = 20<span class="org-type">*</span>u; <span class="org-comment">% Actuator Stack Voltage [V]</span>
</pre>
</div>
<p>
The transfer function from the actuator voltage \(V_a\) to the force sensor stack voltage \(V_s\) is computed.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
Fs = 1<span class="org-type">/</span>Ts;
win = hann(ceil(5<span class="org-type">/</span>Ts));
[tf_est, f] = tfestimate(u, v, win, [], [], 1<span class="org-type">/</span>Ts);
[coh, <span class="org-type">~</span>] = mscohere( u, v, win, [], [], 1<span class="org-type">/</span>Ts);
</pre>
</div>
<p>
The coherence is shown in Figure <a href="#org0c44552">18</a>.
</p>
<div id="org0c44552" class="figure">
<p><img src="figs/apa95ml_5kg_cedrat_coh.png" alt="apa95ml_5kg_cedrat_coh.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Coherence</p>
</div>
<p>
The Simscape model is loaded and compared with the identified dynamics in Figure <a href="#org5ad15cc">19</a>.
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.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/fem_simscape_models.mat'</span>, <span class="org-string">'Gfm'</span>);
</pre>
</div>
<div id="org5ad15cc" class="figure">
<p><img src="figs/bode_plot_force_sensor_voltage_comp_fem.png" alt="bode_plot_force_sensor_voltage_comp_fem.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Comparison of the identified dynamics from voltage output to voltage input and the FEM</p>
</div>
</div>
</div>
<div id="outline-container-org6f69286" class="outline-2">
<h2 id="org6f69286"><span class="section-number-2">7</span> Integral Force Feedback</h2>
<div class="outline-text-2" id="text-7">
<p>
<a id="org13dae17"></a>
</p>
<div id="org5ae9700" class="figure">
<p><img src="figs/test_bench_apa_schematic_iff.png" alt="test_bench_apa_schematic_iff.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Schematic of the test bench using IFF</p>
</div>
</div>
<div id="outline-container-orgbc62cba" class="outline-3">
<h3 id="orgbc62cba"><span class="section-number-3">7.1</span> IFF Plant</h3>
<div class="outline-text-3" id="text-7-1">
<p>
From the identified plant, a model of the transfer function from the actuator stack voltage to the force sensor generated voltage is developed.
</p>
<div class="org-src-container">
<pre class="src src-matlab">w_z = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>111; <span class="org-comment">% Zeros frequency [rad/s]</span>
w_p = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>255; <span class="org-comment">% Pole frequency [rad/s]</span>
xi_z = 0.05;
xi_p = 0.015;
G_inf = 0.1;
Gi = G_inf<span class="org-type">*</span>(s<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xi_z<span class="org-type">*</span>w_z<span class="org-type">*</span>s <span class="org-type">+</span> w_z<span class="org-type">^</span>2)<span class="org-type">/</span>(s<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xi_p<span class="org-type">*</span>w_p<span class="org-type">*</span>s <span class="org-type">+</span> w_p<span class="org-type">^</span>2);
</pre>
</div>
<p>
Its bode plot is shown in Figure <a href="#org255cfc1">21</a>.
</p>
<div id="org255cfc1" class="figure">
<p><img src="figs/iff_plant_identification_apa95ml.png" alt="iff_plant_identification_apa95ml.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Bode plot of the IFF plant</p>
</div>
<p>
The controller used in the Integral Force Feedback Architecture is:
</p>
\begin{equation}
K_{\text{IFF}}(s) = \frac{g}{s + 2\cdot 2\pi} \cdot \frac{s}{s + 0.5 \cdot 2\pi}
\end{equation}
<p>
where \(g\) is a gain that can be tuned.
</p>
<p>
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.
</p>
<p>
In the frequency band of interest, this controller should mostly act as a pure integrator.
</p>
<p>
The Root Locus corresponding to this controller is shown in Figure <a href="#org70704b4">22</a>.
</p>
<div id="org70704b4" class="figure">
<p><img src="figs/root_locus_iff_apa95ml_identification.png" alt="root_locus_iff_apa95ml_identification.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Root Locus for IFF</p>
</div>
</div>
</div>
<div id="outline-container-org4d62e16" class="outline-3">
<h3 id="org4d62e16"><span class="section-number-3">7.2</span> First tests with few gains</h3>
<div class="outline-text-3" id="text-7-2">
<p>
The controller is now implemented in practice, and few controller gains are tested: \(g = 0\), \(g = 10\) and \(g = 100\).
</p>
<p>
For each controller gain, the identification shown in Figure <a href="#org5ae9700">20</a> is performed.
</p>
<div class="org-src-container">
<pre class="src src-matlab">iff_g10 = load(<span class="org-string">'apa95ml_iff_g10_res.mat'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>, <span class="org-string">'y'</span>, <span class="org-string">'v'</span>);
iff_g100 = load(<span class="org-string">'apa95ml_iff_g100_res.mat'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>, <span class="org-string">'y'</span>, <span class="org-string">'v'</span>);
iff_of = load(<span class="org-string">'apa95ml_iff_off_res.mat'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>, <span class="org-string">'y'</span>, <span class="org-string">'v'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4;
win = hann(ceil(10<span class="org-type">/</span>Ts));
[tf_iff_g10, f] = tfestimate(iff_g10.u, iff_g10.y, win, [], [], 1<span class="org-type">/</span>Ts);
[co_iff_g10, <span class="org-type">~</span>] = mscohere(iff_g10.u, iff_g10.y, win, [], [], 1<span class="org-type">/</span>Ts);
[tf_iff_g100, <span class="org-type">~</span>] = tfestimate(iff_g100.u, iff_g100.y, win, [], [], 1<span class="org-type">/</span>Ts);
[co_iff_g100, <span class="org-type">~</span>] = mscohere(iff_g100.u, iff_g100.y, win, [], [], 1<span class="org-type">/</span>Ts);
[tf_iff_of, <span class="org-type">~</span>] = tfestimate(iff_of.u, iff_of.y, win, [], [], 1<span class="org-type">/</span>Ts);
[co_iff_of, <span class="org-type">~</span>] = mscohere(iff_of.u, iff_of.y, win, [], [], 1<span class="org-type">/</span>Ts);
</pre>
</div>
<p>
The coherence between the excitation signal and the mass displacement as measured by the interferometer is shown in Figure <a href="#orga102681">23</a>.
</p>
<div id="orga102681" class="figure">
<p><img src="figs/iff_first_test_coherence.png" alt="iff_first_test_coherence.png" />
</p>
<p><span class="figure-number">Figure 23: </span>Coherence</p>
</div>
<p>
The obtained transfer functions are shown in Figure <a href="#org64a82bf">24</a>.
It is clear that the IFF architecture can actively damp the main resonance of the system.
</p>
<div id="org64a82bf" class="figure">
<p><img src="figs/iff_first_test_bode_plot.png" alt="iff_first_test_bode_plot.png" />
</p>
<p><span class="figure-number">Figure 24: </span>Bode plot for different values of IFF gain</p>
</div>
</div>
</div>
<div id="outline-container-org4304a4e" class="outline-3">
<h3 id="org4304a4e"><span class="section-number-3">7.3</span> Second test with many Gains</h3>
<div class="outline-text-3" id="text-7-3">
<p>
Then, the same identification test is performed for many more gains.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'apa95ml_iff_test.mat'</span>, <span class="org-string">'results'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4;
win = hann(ceil(10<span class="org-type">/</span>Ts));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">tf_iff = {zeros(1, length(results))};
co_iff = {zeros(1, length(results))};
g_iff = [0, 1, 5, 10, 50, 100];
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span>=<span class="org-constant">1:length(results)</span>
[tf_est, f] = tfestimate(results{<span class="org-constant">i</span>}.u, results{<span class="org-constant">i</span>}.y, win, [], [], 1<span class="org-type">/</span>Ts);
[co_est, <span class="org-type">~</span>] = mscohere(results{<span class="org-constant">i</span>}.u, results{<span class="org-constant">i</span>}.y, win, [], [], 1<span class="org-type">/</span>Ts);
tf_iff(<span class="org-constant">i</span>) = {tf_est};
co_iff(<span class="org-constant">i</span>) = {co_est};
<span class="org-keyword">end</span>
</pre>
</div>
<p>
The obtained dynamics are shown in Figure <a href="#org04c2472">25</a>.
</p>
<div id="org04c2472" class="figure">
<p><img src="figs/iff_results_bode_plots.png" alt="iff_results_bode_plots.png" />
</p>
<p><span class="figure-number">Figure 25: </span>Identified dynamics from excitation voltage to the mass displacement</p>
</div>
<p>
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 <a href="#org21dc325">26</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">G_id = {zeros(1,length(results))};
f_start = 70; <span class="org-comment">% [Hz]</span>
f_end = 500; <span class="org-comment">% [Hz]</span>
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(results)</span>
tf_id = tf_iff{<span class="org-constant">i</span>}(sum(f<span class="org-type">&lt;</span>f_start)<span class="org-type">:</span>length(f)<span class="org-type">-</span>sum(f<span class="org-type">&gt;</span>f_end));
f_id = f(sum(f<span class="org-type">&lt;</span>f_start)<span class="org-type">:</span>length(f)<span class="org-type">-</span>sum(f<span class="org-type">&gt;</span>f_end));
gfr = idfrd(tf_id, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>f_id, Ts);
G_id(<span class="org-constant">i</span>) = {procest(gfr,<span class="org-string">'P2UDZ'</span>)};
<span class="org-keyword">end</span>
</pre>
</div>
<div id="org21dc325" class="figure">
<p><img src="figs/iff_results_bode_plots_identification.png" alt="iff_results_bode_plots_identification.png" />
</p>
<p><span class="figure-number">Figure 26: </span>Comparison of the measured dynamic and the identified 2nd order resonant systems that best fits the data</p>
</div>
<p>
Finally, we can represent the position of the poles of the 2nd order systems on the Root Locus plot (Figure <a href="#orgd96aad3">27</a>).
</p>
<div id="orgd96aad3" class="figure">
<p><img src="figs/iff_results_root_locus.png" alt="iff_results_root_locus.png" />
</p>
<p><span class="figure-number">Figure 27: </span>Root Locus plot of the identified IFF plant as well as the identified poles of the damped system</p>
</div>
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
<div class="csl-bib-body">
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Fleming, Andrew J., and Kam K. Leang. 2014. <i>Design, Modeling and Control of Nanopositioning Systems</i>. Advances in Industrial Control. Springer International Publishing. <a href="https://doi.org/10.1007/978-3-319-06617-2">https://doi.org/10.1007/978-3-319-06617-2</a>.</div>
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<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-24 mar. 18:24</p>
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