Update figures and tangle matlab code
This commit is contained in:
503
index.html
503
index.html
@@ -3,7 +3,7 @@
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2020-11-24 mar. 13:54 -->
|
||||
<!-- 2020-11-24 mar. 18:24 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<title>Test Bench - Amplified Piezoelectric Actuator</title>
|
||||
<meta name="generator" content="Org mode" />
|
||||
@@ -30,75 +30,76 @@
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org692da4d">1. Experimental Setup</a></li>
|
||||
<li><a href="#orgf5c6cd7">2. Estimation of electrical/mechanical relationships</a>
|
||||
<li><a href="#orgb908526">1. Experimental Setup</a></li>
|
||||
<li><a href="#org58c6b68">2. Estimation of electrical/mechanical relationships</a>
|
||||
<ul>
|
||||
<li><a href="#orgd8c7c0b">2.1. Estimation from Data-sheet</a></li>
|
||||
<li><a href="#org901da31">2.2. Estimation from Piezoelectric parameters</a></li>
|
||||
<li><a href="#org616104b">2.3. Estimation from Experiment</a>
|
||||
<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="#orgcb3d952">2.3.1. From actuator voltage \(V_a\) to actuator force \(F_a\)</a></li>
|
||||
<li><a href="#org138110f">2.3.2. From stack strain \(\Delta h\) to generated voltage \(V_s\)</a></li>
|
||||
<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="#org7cf58ef">2.4. Conclusion</a></li>
|
||||
<li><a href="#org5e8b78e">2.4. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org36bf85c">3. Simscape model of the test-bench</a>
|
||||
<li><a href="#orgaded65a">3. Simscape model of the test-bench</a>
|
||||
<ul>
|
||||
<li><a href="#orgca28e7b">3.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
|
||||
<li><a href="#orga49ef8e">3.2. Simscape Model</a></li>
|
||||
<li><a href="#orgf1f3f75">3.3. Dynamics from Actuator Voltage to Vertical Mass Displacement</a></li>
|
||||
<li><a href="#orgd4a750c">3.4. Dynamics from Actuator Voltage to Force Sensor Voltage</a></li>
|
||||
<li><a href="#org1ac4d3c">3.5. Save Data for further use</a></li>
|
||||
<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="#org9de639d">4. Huddle Test</a>
|
||||
<li><a href="#org98bd921">4. Measurement of the ambient noise in the system</a>
|
||||
<ul>
|
||||
<li><a href="#orgaf93c0c">4.1. Time Domain Data</a></li>
|
||||
<li><a href="#org22ccdcf">4.2. PSD of Measurement Noise</a></li>
|
||||
<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="#org914a7da">5. Identification of the dynamics from actuator to displacement</a>
|
||||
<li><a href="#org90553e0">5. Identification of the dynamics from actuator Voltage to displacement</a>
|
||||
<ul>
|
||||
<li><a href="#orga1aace1">5.1. Load Data</a></li>
|
||||
<li><a href="#org1284757">5.2. Comparison of the PSD with Huddle Test</a></li>
|
||||
<li><a href="#org89453e3">5.3. Compute TF estimate and Coherence</a></li>
|
||||
<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="#orgfff18d2">6. Identification of the dynamics from actuator to force sensor</a>
|
||||
<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="#orgfa2171a">6.1. System Identification</a></li>
|
||||
<li><a href="#org633e9f1">6.2. Integral Force Feedback</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org1bf58e2">7. Integral Force Feedback</a>
|
||||
<ul>
|
||||
<li><a href="#org25092d8">7.1. First tests with few gains</a></li>
|
||||
<li><a href="#org67fa466">7.2. Second test with many Gains</a></li>
|
||||
<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="#org2c6bb95">1</a>:</li>
|
||||
<li>Section <a href="#orgb81d61b">3</a>:</li>
|
||||
<li>Section [[]]:</li>
|
||||
<li>Section [[]]:</li>
|
||||
<li>Section [[]]:</li>
|
||||
<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-org692da4d" class="outline-2">
|
||||
<h2 id="org692da4d"><span class="section-number-2">1</span> Experimental Setup</h2>
|
||||
<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="org2c6bb95"></a>
|
||||
<a id="orgac1a70e"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
A schematic of the test-bench is shown in Figure <a href="#org57478f1">1</a>.
|
||||
A schematic of the test-bench is shown in Figure <a href="#org42f14fa">1</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -111,31 +112,31 @@ 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="#org4bf2105">2</a> and <a href="#org809b25b">3</a>.
|
||||
Pictures of the test bench are shown in Figure <a href="#org934d0a3">2</a> and <a href="#orgf6513b0">3</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org57478f1" class="figure">
|
||||
<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="org4bf2105" class="figure">
|
||||
<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="org809b25b" class="figure">
|
||||
<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="org79ca795">
|
||||
<div class="note" id="orgb97925c">
|
||||
<p>
|
||||
Here are the equipment used in the test bench:
|
||||
</p>
|
||||
@@ -151,11 +152,14 @@ Here are the equipment used in the test bench:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf5c6cd7" class="outline-2">
|
||||
<h2 id="orgf5c6cd7"><span class="section-number-2">2</span> Estimation of electrical/mechanical relationships</h2>
|
||||
<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>
|
||||
In order to correctly model the piezoelectric actuator, we need to determine:
|
||||
<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>
|
||||
@@ -166,24 +170,23 @@ In order to correctly model the piezoelectric actuator, we need to determine:
|
||||
We estimate \(g_a\) and \(g_s\) using different approaches:
|
||||
</p>
|
||||
<ol class="org-ol">
|
||||
<li>Section <a href="#org640a15c">2.1</a>: \(g_a\) is estimated from the datasheet of the piezoelectric stack</li>
|
||||
<li>Section <a href="#orgeeaae90">2.2</a>: \(g_a\) and \(g_s\) are estimated using the piezoelectric constants</li>
|
||||
<li>Section <a href="#org5a92750">2.3</a>: \(g_a\) and \(g_s\) are estimated experimentally</li>
|
||||
<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-orgd8c7c0b" class="outline-3">
|
||||
<h3 id="orgd8c7c0b"><span class="section-number-3">2.1</span> Estimation from Data-sheet</h3>
|
||||
<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="org640a15c"></a>
|
||||
<a id="orgcfc54fa"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The stack parameters taken from the data-sheet are shown in Table <a href="#orgf2c9009">1</a>.
|
||||
The stack parameters taken from the data-sheet are shown in Table <a href="#org82c2f0a">1</a>.
|
||||
</p>
|
||||
|
||||
<table id="orgf2c9009" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<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>
|
||||
@@ -269,12 +272,12 @@ Vmax = 170; <span class="org-comment">% [V]</span>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">ka<span class="org-type">*</span>Lmax<span class="org-type">/</span>Vmax <span class="org-comment">% [N/V]</span>
|
||||
<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">
|
||||
27.647
|
||||
ga = 27.6 [N/V]
|
||||
</pre>
|
||||
|
||||
|
||||
@@ -284,11 +287,11 @@ From the parameters of the stack, it seems not possible to estimate the relation
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org901da31" class="outline-3">
|
||||
<h3 id="org901da31"><span class="section-number-3">2.2</span> Estimation from Piezoelectric parameters</h3>
|
||||
<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="orgeeaae90"></a>
|
||||
<a id="orgc2edf63"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -305,7 +308,7 @@ ka = 235e6; <span class="org-comment">% Stack stiffness [N/m]</span>
|
||||
The ratio of the developed force to applied voltage is:
|
||||
</p>
|
||||
\begin{equation}
|
||||
\label{org8300de8}
|
||||
\label{org6f1476c}
|
||||
F_a = g_a V_a, \quad g_a = d_{33} n k_a
|
||||
\end{equation}
|
||||
<p>
|
||||
@@ -337,7 +340,7 @@ ga = 5.6 [N/V]
|
||||
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{org312083f}
|
||||
\label{org43f15ff}
|
||||
V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
|
||||
\end{equation}
|
||||
<p>
|
||||
@@ -374,11 +377,11 @@ gs = 35.4 [V/um]
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org616104b" class="outline-3">
|
||||
<h3 id="org616104b"><span class="section-number-3">2.3</span> Estimation from Experiment</h3>
|
||||
<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="org5a92750"></a>
|
||||
<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.
|
||||
@@ -394,8 +397,8 @@ Similarly, it is fairly easy to experimentally obtain the gain from the stack di
|
||||
To link that to the strain of the sensor stack, the simscape model is used.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orgcb3d952" class="outline-4">
|
||||
<h4 id="orgcb3d952"><span class="section-number-4">2.3.1</span> From actuator voltage \(V_a\) to actuator force \(F_a\)</h4>
|
||||
<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.
|
||||
@@ -439,7 +442,7 @@ The gain from input voltage of the stack to the vertical displacement is determi
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgf3201ac" class="figure">
|
||||
<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>
|
||||
@@ -495,11 +498,11 @@ ga = 33.7 [N/V]
|
||||
|
||||
|
||||
<p>
|
||||
The obtained comparison between the Simscape model and the identified dynamics is shown in Figure <a href="#orgc727518">5</a>.
|
||||
The obtained comparison between the Simscape model and the identified dynamics is shown in Figure <a href="#orgc82fdbe">5</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgc727518" class="figure">
|
||||
<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>
|
||||
@@ -507,8 +510,8 @@ The obtained comparison between the Simscape model and the identified dynamics i
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org138110f" class="outline-4">
|
||||
<h4 id="org138110f"><span class="section-number-4">2.3.2</span> From stack strain \(\Delta h\) to generated voltage \(V_s\)</h4>
|
||||
<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.
|
||||
@@ -543,7 +546,7 @@ Here, an amplifier with a gain of 20 is used.
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Then, the transfer function from \(V_a\) to \(V_s\) is identified and its DC gain is estimated (Figure <a href="#org6245a69">6</a>).
|
||||
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);
|
||||
@@ -561,7 +564,7 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6245a69" class="figure">
|
||||
<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>
|
||||
@@ -617,7 +620,7 @@ gs = 3.5 [V/um]
|
||||
|
||||
|
||||
|
||||
<div id="org6b6b451" class="figure">
|
||||
<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>
|
||||
@@ -626,8 +629,8 @@ gs = 3.5 [V/um]
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7cf58ef" class="outline-3">
|
||||
<h3 id="org7cf58ef"><span class="section-number-3">2.4</span> Conclusion</h3>
|
||||
<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.
|
||||
@@ -644,11 +647,11 @@ The one using the experimental data are saved and further used.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org36bf85c" class="outline-2">
|
||||
<h2 id="org36bf85c"><span class="section-number-2">3</span> Simscape model of the test-bench</h2>
|
||||
<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="orgb81d61b"></a>
|
||||
<a id="org9d27452"></a>
|
||||
</p>
|
||||
<p>
|
||||
The idea here is to model the test-bench using Simscape.
|
||||
@@ -659,19 +662,19 @@ Whereas the suspended mass and metrology frame can be considered as rigid bodies
|
||||
</p>
|
||||
|
||||
<p>
|
||||
To model the APA, a Finite Element Model (FEM) is used (Figure <a href="#orgbdafa32">8</a>) and imported into Simscape.
|
||||
To model the APA, a Finite Element Model (FEM) is used (Figure <a href="#orgf4bc6a9">8</a>) and imported into Simscape.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgbdafa32" class="figure">
|
||||
<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-orgca28e7b" class="outline-3">
|
||||
<h3 id="orgca28e7b"><span class="section-number-3">3.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
|
||||
<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.
|
||||
@@ -997,7 +1000,6 @@ M = readmatrix(<span class="org-string">'APA95ML_M.CSV'</span>);
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
|
||||
<p>
|
||||
Then, we extract the coordinates of the interface nodes.
|
||||
</p>
|
||||
@@ -1007,7 +1009,7 @@ Then, we extract the coordinates of the interface nodes.
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The interface nodes are shown in Figure <a href="#org7dd0e69">9</a> and their coordinates are listed in Table <a href="#orgc8e88d6">4</a>.
|
||||
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">
|
||||
@@ -1041,7 +1043,7 @@ The interface nodes are shown in Figure <a href="#org7dd0e69">9</a> and their co
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
<table id="orgc8e88d6" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<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>
|
||||
@@ -1124,7 +1126,7 @@ The interface nodes are shown in Figure <a href="#org7dd0e69">9</a> and their co
|
||||
</table>
|
||||
|
||||
|
||||
<div id="org7dd0e69" class="figure">
|
||||
<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>
|
||||
@@ -1136,8 +1138,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <c
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga49ef8e" class="outline-3">
|
||||
<h3 id="orga49ef8e"><span class="section-number-3">3.2</span> Simscape Model</h3>
|
||||
<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.
|
||||
@@ -1152,17 +1154,22 @@ A <code>Relative Motion Sensor</code> block is added between the nodes 1 and 2 t
|
||||
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;
|
||||
<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-orgf1f3f75" class="outline-3">
|
||||
<h3 id="orgf1f3f75"><span class="section-number-3">3.3</span> Dynamics from Actuator Voltage to Vertical Mass Displacement</h3>
|
||||
<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="#orgd2ca2ca">10</a>.
|
||||
The identified dynamics is shown in Figure <a href="#org162e335">10</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
@@ -1174,12 +1181,12 @@ 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 = <span class="org-type">-</span>linearize(mdl, io);
|
||||
Ghm = linearize(mdl, io);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd2ca2ca" class="figure">
|
||||
<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>
|
||||
@@ -1187,11 +1194,11 @@ Ghm = <span class="org-type">-</span>linearize(mdl, io);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd4a750c" class="outline-3">
|
||||
<h3 id="orgd4a750c"><span class="section-number-3">3.4</span> Dynamics from Actuator Voltage to Force Sensor Voltage</h3>
|
||||
<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="#orgb135de4">11</a>.
|
||||
The obtained dynamics is shown in Figure <a href="#org2258d23">11</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
@@ -1208,7 +1215,7 @@ Gfm = linearize(mdl, io);
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgb135de4" class="figure">
|
||||
<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>
|
||||
@@ -1216,34 +1223,36 @@ Gfm = linearize(mdl, io);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1ac4d3c" class="outline-3">
|
||||
<h3 id="org1ac4d3c"><span class="section-number-3">3.5</span> Save Data for further use</h3>
|
||||
<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 class="org-src-container">
|
||||
<pre class="src src-matlab">save(<span class="org-string">'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-org9de639d" class="outline-2">
|
||||
<h2 id="org9de639d"><span class="section-number-2">4</span> Huddle Test</h2>
|
||||
<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="org9acdf87"></a>
|
||||
<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-orgaf93c0c" class="outline-3">
|
||||
<h3 id="orgaf93c0c"><span class="section-number-3">4.1</span> Time Domain Data</h3>
|
||||
<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="orgd048822" class="figure">
|
||||
<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>
|
||||
@@ -1251,8 +1260,8 @@ Gfm = linearize(mdl, io);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org22ccdcf" class="outline-3">
|
||||
<h3 id="org22ccdcf"><span class="section-number-3">4.2</span> PSD of Measurement Noise</h3>
|
||||
<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);
|
||||
@@ -1268,7 +1277,7 @@ win = hanning(ceil(1<span class="org-type">*</span>Fs));
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd309d3e" class="figure">
|
||||
<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>
|
||||
@@ -1277,24 +1286,30 @@ win = hanning(ceil(1<span class="org-type">*</span>Fs));
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org914a7da" class="outline-2">
|
||||
<h2 id="org914a7da"><span class="section-number-2">5</span> Identification of the dynamics from actuator to displacement</h2>
|
||||
<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="org8cacbcf"></a>
|
||||
<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>
|
||||
<ul class="org-ul">
|
||||
<li class="off"><code>[ ]</code> List of equipment</li>
|
||||
<li class="off"><code>[ ]</code> Schematic</li>
|
||||
<li class="off"><code>[ ]</code> Problem of matching between the models? (there is a factor 10)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
E505 with gain of 10.
|
||||
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 id="outline-container-orga1aace1" class="outline-3">
|
||||
<h3 id="orga1aace1"><span class="section-number-3">5.1</span> Load Data</h3>
|
||||
</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 “noise test” and the identification test are loaded.
|
||||
@@ -1322,14 +1337,13 @@ 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;
|
||||
ht.u = 10<span class="org-type">*</span>ht.u;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1284757" class="outline-3">
|
||||
<h3 id="org1284757"><span class="section-number-3">5.2</span> Comparison of the PSD with Huddle Test</h3>
|
||||
<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);
|
||||
@@ -1346,16 +1360,16 @@ win = hanning(ceil(1<span class="org-type">*</span>Fs));
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org87d1b81" class="figure">
|
||||
<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 14: </span>Comparison of the ASD for the identification test and the huddle test</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-org89453e3" class="outline-3">
|
||||
<h3 id="org89453e3"><span class="section-number-3">5.3</span> Compute TF estimate and Coherence</h3>
|
||||
<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);
|
||||
@@ -1364,10 +1378,10 @@ win = hanning(ceil(1<span class="org-type">*</span>Fs));
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org69f52ac" class="figure">
|
||||
<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 15: </span>Coherence</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Coherence</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
@@ -1379,31 +1393,23 @@ Comparison with the FEM model
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgac63fe3" class="figure">
|
||||
<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 16: </span>Comparison of the identified transfer function and the one estimated from the FE model</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-orgfff18d2" class="outline-2">
|
||||
<h2 id="orgfff18d2"><span class="section-number-2">6</span> Identification of the dynamics from actuator to force sensor</h2>
|
||||
<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="org2eda6f2"></a>
|
||||
<a id="orgdd284b9"></a>
|
||||
</p>
|
||||
<p>
|
||||
Two measurements are performed:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Speedgoat DAC => Voltage Amplifier (x20) => 1 Piezo Stack => … => 2 Stacks as Force Sensor (parallel) => Speedgoat ADC</li>
|
||||
<li>Speedgoat DAC => Voltage Amplifier (x20) => 2 Piezo Stacks (parallel) => … => 1 Stack as Force Sensor => Speedgoat ADC</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The obtained dynamics from force actuator to force sensor are compare with the FEM model.
|
||||
The same setup shown in Figure <a href="#org43a6d6f">14</a> is used.
|
||||
</p>
|
||||
<p>
|
||||
The data are loaded:
|
||||
@@ -1413,76 +1419,88 @@ The data are loaded:
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">u = detrend(u, 0);
|
||||
v = detrend(v, 0);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">u = 20<span class="org-type">*</span>u;
|
||||
</pre>
|
||||
</div>
|
||||
<p>
|
||||
Let’s use the amplifier gain to obtain the true voltage applied to the actuator stack(s)
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The parameters of the piezoelectric stacks are defined below:
|
||||
Any offset is removed.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<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>
|
||||
<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>
|
||||
From the FEM, we construct the transfer function from DAC voltage to ADC voltage.
|
||||
Here, the amplifier gain is 20.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<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);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<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>Gfm<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>Gfm<span class="org-type">*</span>d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>n);
|
||||
<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 input voltage to output voltage are computed and shown in Figure <a href="#org157bd9b">17</a>.
|
||||
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(10<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="org157bd9b" class="figure">
|
||||
<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 17: </span>Comparison of the identified dynamics from voltage output to voltage input and the FEM</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 id="outline-container-orgfa2171a" class="outline-3">
|
||||
<h3 id="orgfa2171a"><span class="section-number-3">6.1</span> System Identification</h3>
|
||||
<div class="outline-text-3" id="text-6-1">
|
||||
</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>
|
||||
@@ -1490,50 +1508,63 @@ 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);
|
||||
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="orgf552545" class="figure">
|
||||
|
||||
<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 18: </span>Identification of the IFF plant</p>
|
||||
</div>
|
||||
</div>
|
||||
<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>
|
||||
|
||||
<div id="outline-container-org633e9f1" class="outline-3">
|
||||
<h3 id="org633e9f1"><span class="section-number-3">6.2</span> Integral Force Feedback</h3>
|
||||
<div class="outline-text-3" id="text-6-2">
|
||||
<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>
|
||||
|
||||
<div id="orgca6756f" class="figure">
|
||||
<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 19: </span>Root Locus for IFF</p>
|
||||
</div>
|
||||
<p><span class="figure-number">Figure 22: </span>Root Locus for IFF</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1bf58e2" class="outline-2">
|
||||
<h2 id="org1bf58e2"><span class="section-number-2">7</span> Integral Force Feedback</h2>
|
||||
<div class="outline-text-2" id="text-7">
|
||||
|
||||
<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>
|
||||
<a id="org6db5225"></a>
|
||||
The controller is now implemented in practice, and few controller gains are tested: \(g = 0\), \(g = 10\) and \(g = 100\).
|
||||
</p>
|
||||
|
||||
<div id="org86a6667" class="figure">
|
||||
<p><img src="figs/test_bench_apa_schematic_iff.png" alt="test_bench_apa_schematic_iff.png" />
|
||||
<p>
|
||||
For each controller gain, the identification shown in Figure <a href="#org5ae9700">20</a> is performed.
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 20: </span>Schematic of the test bench using IFF</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org25092d8" class="outline-3">
|
||||
<h3 id="org25092d8"><span class="section-number-3">7.1</span> First tests with few gains</h3>
|
||||
<div class="outline-text-3" id="text-7-1">
|
||||
<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>);
|
||||
@@ -1556,26 +1587,37 @@ win = hann(ceil(10<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="org2da6c81" class="figure">
|
||||
|
||||
<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 21: </span>Coherence</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="org407e7b6" class="figure">
|
||||
<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 22: </span>Bode plot for different values of IFF gain</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-org67fa466" class="outline-3">
|
||||
<h3 id="org67fa466"><span class="section-number-3">7.2</span> Second test with many Gains</h3>
|
||||
<div class="outline-text-3" id="text-7-2">
|
||||
<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>
|
||||
@@ -1602,12 +1644,21 @@ g_iff = [0, 1, 5, 10, 50, 100];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The obtained dynamics are shown in Figure <a href="#org04c2472">25</a>.
|
||||
</p>
|
||||
|
||||
<div id="orgfa8bc3c" class="figure">
|
||||
|
||||
<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))};
|
||||
|
||||
@@ -1625,15 +1676,21 @@ f_end = 500; <span class="org-comment">% [Hz]</span>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org62b4b70" class="figure">
|
||||
<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="org49a571f" class="figure">
|
||||
|
||||
<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>
|
||||
@@ -1646,7 +1703,7 @@ f_end = 500; <span class="org-comment">% [Hz]</span>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-11-24 mar. 13:54</p>
|
||||
<p class="date">Created: 2020-11-24 mar. 18:24</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user