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<h1 class="title">Test Bench APA95ML</h1>
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<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org90a334e">1. Huddle Test</a>
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<ul>
<li><a href="#org285da21">1.1. Time Domain Data</a></li>
<li><a href="#org5e1c721">1.2. PSD of Measurement Noise</a></li>
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</ul>
</li>
<li><a href="#orgd37467e">2. Identification of the dynamics from actuator to displacement</a>
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<ul>
<li><a href="#org9434b07">2.1. Load Data</a></li>
<li><a href="#org1eba886">2.2. Comparison of the PSD with Huddle Test</a></li>
<li><a href="#orgcfe0cb7">2.3. Compute TF estimate and Coherence</a></li>
<li><a href="#org1cdc3bc">2.4. Comparison with the FEM model</a></li>
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</ul>
</li>
<li><a href="#orgcce11f7">3. Identification of the dynamics from actuator to force sensor</a>
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<ul>
<li><a href="#org48d4bd0">3.1. System Identification</a></li>
<li><a href="#orge58d929">3.2. Integral Force Feedback</a></li>
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</ul>
</li>
<li><a href="#orgfe5bc71">4. Integral Force Feedback</a>
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<ul>
<li><a href="#orgd854491">4.1. First tests with few gains</a></li>
<li><a href="#org899fef5">4.2. Second test with many Gains</a></li>
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</ul>
</li>
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</ul>
</div>
</div>
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<div id="org392f56d" class="figure">
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<p><img src="figs/setup_picture.png" alt="setup_picture.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Picture of the Setup</p>
</div>
<div id="org6970f6d" class="figure">
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<p><img src="figs/setup_zoom.png" alt="setup_zoom.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Zoom on the APA</p>
</div>
<div id="outline-container-org90a334e" class="outline-2">
<h2 id="org90a334e"><span class="section-number-2">1</span> Huddle Test</h2>
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<div class="outline-text-2" id="text-1">
<p>
<a id="orga0e8ba1"></a>
</p>
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</div>
<div id="outline-container-org285da21" class="outline-3">
<h3 id="org285da21"><span class="section-number-3">1.1</span> Time Domain Data</h3>
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<div class="outline-text-3" id="text-1-1">
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<div id="orgde9f524" class="figure">
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<p><img src="figs/huddle_test_time_domain.png" alt="huddle_test_time_domain.png" />
</p>
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<p><span class="figure-number">Figure 3: </span>Measurement of the Mass displacement during Huddle Test</p>
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</div>
</div>
</div>
<div id="outline-container-org5e1c721" class="outline-3">
<h3 id="org5e1c721"><span class="section-number-3">1.2</span> PSD of Measurement Noise</h3>
<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<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;
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win = hanning(ceil(1<span class="org-type">*</span>Fs));
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</pre>
</div>
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<div class="org-src-container">
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<pre class="src src-matlab">[pxx, f] = pwelch(y(1000<span class="org-type">:</span>end), win, [], [], Fs);
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</pre>
</div>
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<div id="org71c1833" class="figure">
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<p><img src="figs/huddle_test_pdf.png" alt="huddle_test_pdf.png" />
</p>
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<p><span class="figure-number">Figure 4: </span>Amplitude Spectral Density of the Displacement during Huddle Test</p>
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</div>
</div>
</div>
</div>
<div id="outline-container-orgd37467e" class="outline-2">
<h2 id="orgd37467e"><span class="section-number-2">2</span> Identification of the dynamics from actuator to displacement</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org7bb414b"></a>
</p>
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</div>
<div id="outline-container-org9434b07" class="outline-3">
<h3 id="org9434b07"><span class="section-number-3">2.1</span> Load Data</h3>
<div class="outline-text-3" id="text-2-1">
<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">'u'</span>, <span class="org-string">'um'</span>, <span class="org-string">'y'</span>);
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</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">u = 10<span class="org-type">*</span>(u <span class="org-type">-</span> mean(u)); <span class="org-comment">% Input Voltage of Piezo [V]</span>
um = 10<span class="org-type">*</span>(um <span class="org-type">-</span> mean(um)); <span class="org-comment">% Monitor [V]</span>
y = y <span class="org-type">-</span> mean(y); <span class="org-comment">% Mass displacement [m]</span>
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ht.u = 10<span class="org-type">*</span>(ht.u <span class="org-type">-</span> mean(ht.u));
ht.y = ht.y <span class="org-type">-</span> mean(ht.y);
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</pre>
</div>
</div>
</div>
<div id="outline-container-org1eba886" class="outline-3">
<h3 id="org1eba886"><span class="section-number-3">2.2</span> Comparison of the PSD with Huddle Test</h3>
<div class="outline-text-3" id="text-2-2">
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<div class="org-src-container">
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<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;
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win = hanning(ceil(1<span class="org-type">*</span>Fs));
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</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">[pxx, f] = pwelch(y, win, [], [], Fs);
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[pht, <span class="org-type">~</span>] = pwelch(ht.y, win, [], [], Fs);
</pre>
</div>
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<div id="orgd9832c1" class="figure">
<p><img src="figs/apa95ml_5kg_PI_pdf_comp_huddle.png" alt="apa95ml_5kg_PI_pdf_comp_huddle.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Comparison of the ASD for the identification test and the huddle test</p>
</div>
</div>
</div>
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<div id="outline-container-orgcfe0cb7" class="outline-3">
<h3 id="orgcfe0cb7"><span class="section-number-3">2.3</span> Compute TF estimate and Coherence</h3>
<div class="outline-text-3" id="text-2-3">
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<div class="org-src-container">
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<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;
</pre>
</div>
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<div class="org-src-container">
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<pre class="src src-matlab">win = hann(ceil(1<span class="org-type">/</span>Ts));
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[tf_est, f] = tfestimate(u, <span class="org-type">-</span>y, win, [], [], 1<span class="org-type">/</span>Ts);
[tf_um , <span class="org-type">~</span>] = 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);
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</pre>
</div>
<div id="orgcd90594" class="figure">
<p><img src="figs/apa95ml_5kg_PI_coh.png" alt="apa95ml_5kg_PI_coh.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Coherence</p>
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</div>
<div id="org55f5177" class="figure">
<p><img src="figs/apa95ml_5kg_PI_tf.png" alt="apa95ml_5kg_PI_tf.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Estimation of the transfer function from input voltage to displacement</p>
</div>
</div>
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</div>
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<div id="outline-container-org1cdc3bc" class="outline-3">
<h3 id="org1cdc3bc"><span class="section-number-3">2.4</span> Comparison with the FEM model</h3>
<div class="outline-text-3" id="text-2-4">
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<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'fem_model_5kg.mat'</span>, <span class="org-string">'G'</span>);
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</pre>
</div>
<div id="org7e7ad3c" class="figure">
<p><img src="figs/apa95ml_5kg_pi_comp_fem.png" alt="apa95ml_5kg_pi_comp_fem.png" />
</p>
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<p><span class="figure-number">Figure 8: </span>Comparison of the identified transfer function and the one estimated from the FE model</p>
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</div>
</div>
</div>
</div>
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<div id="outline-container-orgcce11f7" class="outline-2">
<h2 id="orgcce11f7"><span class="section-number-2">3</span> Identification of the dynamics from actuator to force sensor</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org7d9ec69"></a>
</p>
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<p>
Two measurements are performed:
</p>
<ul class="org-ul">
<li>Speedgoat DAC =&gt; Voltage Amplifier (x20) =&gt; 1 Piezo Stack =&gt; &#x2026; =&gt; 2 Stacks as Force Sensor (parallel) =&gt; Speedgoat ADC</li>
<li>Speedgoat DAC =&gt; Voltage Amplifier (x20) =&gt; 2 Piezo Stacks (parallel) =&gt; &#x2026; =&gt; 1 Stack as Force Sensor =&gt; Speedgoat ADC</li>
</ul>
<p>
The obtained dynamics from force actuator to force sensor are compare with the FEM model.
</p>
<p>
The data are loaded:
</p>
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<div class="org-src-container">
<pre class="src src-matlab">a_ss = load(<span class="org-string">'apa95ml_5kg_1a_2s.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'u'</span>, <span class="org-string">'y'</span>, <span class="org-string">'v'</span>);
aa_s = 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">'y'</span>, <span class="org-string">'v'</span>);
load(<span class="org-string">'G_force_sensor_5kg.mat'</span>, <span class="org-string">'G'</span>);
</pre>
</div>
<p>
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Let&rsquo;s use the amplifier gain to obtain the true voltage applied to the actuator stack(s)
</p>
<p>
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The parameters of the piezoelectric stacks are defined below:
</p>
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<div class="org-src-container">
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<pre class="src src-matlab">d33 = 3e<span class="org-type">-</span>10; <span class="org-comment">% Strain constant [m/V]</span>
n = 80; <span class="org-comment">% Number of layers per stack</span>
eT = 1.6e<span class="org-type">-</span>8; <span class="org-comment">% Permittivity under constant stress [F/m]</span>
sD = 2e<span class="org-type">-</span>11; <span class="org-comment">% Elastic compliance under constant electric displacement [m2/N]</span>
ka = 235e6; <span class="org-comment">% Stack stiffness [N/m]</span>
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</pre>
</div>
<p>
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From the FEM, we construct the transfer function from DAC voltage to ADC voltage.
</p>
<div class="org-src-container">
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<pre class="src src-matlab">Gfem_aa_s = exp(<span class="org-type">-</span>s<span class="org-type">/</span>1e4)<span class="org-type">*</span>20<span class="org-type">*</span>(2<span class="org-type">*</span>d33<span class="org-type">*</span>n<span class="org-type">*</span>ka)<span class="org-type">*</span>(G(3,1)<span class="org-type">+</span>G(3,2))<span class="org-type">*</span>d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>n);
Gfem_a_ss = exp(<span class="org-type">-</span>s<span class="org-type">/</span>1e4)<span class="org-type">*</span>20<span class="org-type">*</span>( d33<span class="org-type">*</span>n<span class="org-type">*</span>ka)<span class="org-type">*</span>(G(3,1)<span class="org-type">+</span>G(2,1))<span class="org-type">*</span>d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>n);
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</pre>
</div>
<p>
The transfer function from input voltage to output voltage are computed and shown in Figure <a href="#org93e238f">9</a>.
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</p>
<div class="org-src-container">
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<pre class="src src-matlab">Ts = a_ss.t(end)<span class="org-type">/</span>(length(a_ss.t)<span class="org-type">-</span>1);
Fs = 1<span class="org-type">/</span>Ts;
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win = hann(ceil(10<span class="org-type">/</span>Ts));
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[tf_a_ss, f] = tfestimate(a_ss.u, a_ss.v, win, [], [], 1<span class="org-type">/</span>Ts);
[coh_a_ss, <span class="org-type">~</span>] = mscohere( a_ss.u, a_ss.v, win, [], [], 1<span class="org-type">/</span>Ts);
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[tf_aa_s, f] = tfestimate(aa_s.u, aa_s.v, win, [], [], 1<span class="org-type">/</span>Ts);
[coh_aa_s, <span class="org-type">~</span>] = mscohere( aa_s.u, aa_s.v, win, [], [], 1<span class="org-type">/</span>Ts);
</pre>
</div>
<div id="org93e238f" class="figure">
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<p><img src="figs/bode_plot_force_sensor_voltage_comp_fem.png" alt="bode_plot_force_sensor_voltage_comp_fem.png" />
</p>
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<p><span class="figure-number">Figure 9: </span>Comparison of the identified dynamics from voltage output to voltage input and the FEM</p>
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</div>
</div>
<div id="outline-container-org48d4bd0" class="outline-3">
<h3 id="org48d4bd0"><span class="section-number-3">3.1</span> System Identification</h3>
<div class="outline-text-3" id="text-3-1">
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<div class="org-src-container">
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<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>
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xi_z = 0.05;
xi_p = 0.015;
G_inf = 2;
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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);
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</pre>
</div>
<div id="orga3e3e7b" class="figure">
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<p><img src="figs/iff_plant_identification_apa95ml.png" alt="iff_plant_identification_apa95ml.png" />
</p>
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<p><span class="figure-number">Figure 10: </span>Identification of the IFF plant</p>
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</div>
</div>
</div>
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<div id="outline-container-orge58d929" class="outline-3">
<h3 id="orge58d929"><span class="section-number-3">3.2</span> Integral Force Feedback</h3>
<div class="outline-text-3" id="text-3-2">
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<div id="org55e0695" class="figure">
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<p><img src="figs/root_locus_iff_apa95ml_identification.png" alt="root_locus_iff_apa95ml_identification.png" />
</p>
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<p><span class="figure-number">Figure 11: </span>Root Locus for IFF</p>
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</div>
</div>
</div>
</div>
<div id="outline-container-orgfe5bc71" class="outline-2">
<h2 id="orgfe5bc71"><span class="section-number-2">4</span> Integral Force Feedback</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org9e016b1"></a>
</p>
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</div>
<div id="outline-container-orgd854491" class="outline-3">
<h3 id="orgd854491"><span class="section-number-3">4.1</span> First tests with few gains</h3>
<div class="outline-text-3" id="text-4-1">
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<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>);
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<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4;
win = hann(ceil(10<span class="org-type">/</span>Ts));
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[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);
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[tf_iff_g100, f] = 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);
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[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);
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<div id="org2c6892b" class="figure">
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<p><img src="figs/iff_first_test_coherence.png" alt="iff_first_test_coherence.png" />
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<p><span class="figure-number">Figure 12: </span>Coherence</p>
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<div id="org91fa41a" class="figure">
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<p><img src="figs/iff_first_test_bode_plot.png" alt="iff_first_test_bode_plot.png" />
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<p><span class="figure-number">Figure 13: </span>Bode plot for different values of IFF gain</p>
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<h3 id="org899fef5"><span class="section-number-3">4.2</span> Second test with many Gains</h3>
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<pre class="src src-matlab">load(<span class="org-string">'apa95ml_iff_test.mat'</span>, <span class="org-string">'results'</span>);
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<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4;
win = hann(ceil(10<span class="org-type">/</span>Ts));
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<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];
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<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);
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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>
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</pre>
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<div id="orgcf10be1" class="figure">
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<p><img src="figs/iff_results_bode_plots.png" alt="iff_results_bode_plots.png" />
</p>
</div>
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<pre class="src src-matlab">G_id = {zeros(1,length(results))};
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f_start = 70; <span class="org-comment">% [Hz]</span>
f_end = 500; <span class="org-comment">% [Hz]</span>
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<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));
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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>
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</div>
<div id="orgda781ba" class="figure">
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<p><img src="figs/iff_results_bode_plots_identification.png" alt="iff_results_bode_plots_identification.png" />
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<div id="org473937a" class="figure">
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<p><img src="figs/iff_results_root_locus.png" alt="iff_results_root_locus.png" />
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<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-12 jeu. 09:29</p>
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