Save diagonal Controller

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Thomas Dehaeze 2019-09-17 16:21:42 +02:00
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<title>Cercalo Test Bench</title> <title>Cercalo Test Bench</title>
@ -276,92 +276,93 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org3519192">1. Introduction</a> <li><a href="#orga1af2bd">1. Introduction</a>
<ul> <ul>
<li><a href="#org872df99">1.1. Block Diagram</a></li> <li><a href="#orgba30a6f">1.1. Block Diagram</a></li>
<li><a href="#orgeb18df8">1.2. Cercalo</a></li> <li><a href="#org7551d72">1.2. Cercalo</a></li>
<li><a href="#orgabd5452">1.3. Optical Setup</a></li> <li><a href="#org346972d">1.3. Optical Setup</a></li>
<li><a href="#orga857b69">1.4. Newport</a></li> <li><a href="#org592a330">1.4. Newport</a></li>
<li><a href="#org65c50b5">1.5. 4 quadrant Diode</a></li> <li><a href="#org7cbae0f">1.5. 4 quadrant Diode</a></li>
<li><a href="#orge0ab4f9">1.6. ADC/DAC</a></li> <li><a href="#orgdd83994">1.6. ADC/DAC</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org29ba44d">2. Identification of the system dynamics</a> <li><a href="#org0c91379">2. Identification of the system dynamics</a>
<ul> <ul>
<li><a href="#orgf19c09b">2.1. Calibration of the 4 Quadrant Diode</a> <li><a href="#orga71352f">2.1. Calibration of the 4 Quadrant Diode</a>
<ul> <ul>
<li><a href="#org488413f">2.1.1. Input / Output data</a></li> <li><a href="#org38399c1">2.1.1. Input / Output data</a></li>
<li><a href="#orgbbc3ad2">2.1.2. Linear Regression to obtain the gain of the 4QD</a></li> <li><a href="#org88480ae">2.1.2. Linear Regression to obtain the gain of the 4QD</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org7dd4e6c">2.2. Identification of the Cercalo Impedance, Current Amplifier and Voltage Amplifier dynamics</a> <li><a href="#org45084e8">2.2. Identification of the Cercalo Impedance, Current Amplifier and Voltage Amplifier dynamics</a>
<ul> <ul>
<li><a href="#org6078ef0">2.2.1. Electrical Schematic</a></li> <li><a href="#orgd91f44a">2.2.1. Electrical Schematic</a></li>
<li><a href="#orgaa575f2">2.2.2. Theoretical Transfer Functions</a></li> <li><a href="#orge075768">2.2.2. Theoretical Transfer Functions</a></li>
<li><a href="#orgc24d0bd">2.2.3. Identified Transfer Functions</a></li> <li><a href="#org31831c5">2.2.3. Identified Transfer Functions</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org8b914fd">2.3. Identification of the Cercalo Dynamics</a> <li><a href="#orge7aa024">2.3. Identification of the Cercalo Dynamics</a>
<ul> <ul>
<li><a href="#org45f478a">2.3.1. Input / Output data</a></li> <li><a href="#orgcf382ac">2.3.1. Input / Output data</a></li>
<li><a href="#orga76fba2">2.3.2. Coherence</a></li> <li><a href="#orgc61c149">2.3.2. Coherence</a></li>
<li><a href="#org131ac39">2.3.3. Estimation of the Frequency Response Function Matrix</a></li> <li><a href="#orge1f3965">2.3.3. Estimation of the Frequency Response Function Matrix</a></li>
<li><a href="#orgf0a3476">2.3.4. Time Delay</a></li> <li><a href="#orgef1a3c2">2.3.4. Time Delay</a></li>
<li><a href="#org0174055">2.3.5. Extraction of a transfer function matrix</a></li> <li><a href="#org36d1dfc">2.3.5. Extraction of a transfer function matrix</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org29f709e">2.4. Identification of the Newport Dynamics</a> <li><a href="#org87a7d68">2.4. Identification of the Newport Dynamics</a>
<ul> <ul>
<li><a href="#org18a03ed">2.4.1. Input / Output data</a></li> <li><a href="#org83a4f28">2.4.1. Input / Output data</a></li>
<li><a href="#org978a0c9">2.4.2. Coherence</a></li> <li><a href="#orge19fa44">2.4.2. Coherence</a></li>
<li><a href="#orgc0c5d02">2.4.3. Estimation of the Frequency Response Function Matrix</a></li> <li><a href="#orgc36f0eb">2.4.3. Estimation of the Frequency Response Function Matrix</a></li>
<li><a href="#orgdfcee7e">2.4.4. Time Delay</a></li> <li><a href="#org830e6db">2.4.4. Time Delay</a></li>
<li><a href="#org314a223">2.4.5. Extraction of a transfer function matrix</a></li> <li><a href="#orgab7b96b">2.4.5. Extraction of a transfer function matrix</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org8fe71fd">2.5. Full System</a></li> <li><a href="#orgee9b6d9">2.5. Full System</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org07fb936">3. Active Damping</a> <li><a href="#orgbb8218f">3. Active Damping</a>
<ul> <ul>
<li><a href="#org07da045">3.1. Load Plant</a></li> <li><a href="#org584508e">3.1. Load Plant</a></li>
<li><a href="#orge1669f2">3.2. Test</a></li> <li><a href="#org0df875d">3.2. Test</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgf4f85b9">4. <span class="todo TODO">TODO</span> Huddle Test</a></li> <li><a href="#org4ac14f0">4. <span class="todo TODO">TODO</span> Huddle Test</a></li>
<li><a href="#org03828c5">5. Plant Scaling</a></li> <li><a href="#orgb1866ab">5. Plant Scaling</a></li>
<li><a href="#org8254d1a">6. Plant Analysis</a> <li><a href="#orgdc654c4">6. Plant Analysis</a>
<ul> <ul>
<li><a href="#org592969b">6.1. Load Plant</a></li> <li><a href="#org59cd7d2">6.1. Load Plant</a></li>
<li><a href="#org966d85b">6.2. RGA-Number</a></li> <li><a href="#org62e9187">6.2. RGA-Number</a></li>
<li><a href="#org7063733">6.3. Rotation Matrix</a></li> <li><a href="#org4bc71a6">6.3. Rotation Matrix</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org759298e">7. Control Objective</a></li> <li><a href="#org29f5209">7. Control Objective</a></li>
<li><a href="#org96df63d">8. Decentralized Control</a> <li><a href="#org2b9b5f8">8. Decentralized Control</a>
<ul> <ul>
<li><a href="#orgde6febc">8.1. Load Plant</a></li> <li><a href="#orgff694d6">8.1. Load Plant</a></li>
<li><a href="#orgf815918">8.2. Diagonal Controller</a></li> <li><a href="#orgb485008">8.2. Diagonal Controller</a></li>
<li><a href="#org37b475d">8.3. Save the Controller</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orge641704">9. Measurement of the non-repeatability</a></li> <li><a href="#org58d3080">9. Measurement of the non-repeatability</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org3519192" class="outline-2"> <div id="outline-container-orga1af2bd" class="outline-2">
<h2 id="org3519192"><span class="section-number-2">1</span> Introduction</h2> <h2 id="orga1af2bd"><span class="section-number-2">1</span> Introduction</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-org872df99" class="outline-3"> <div id="outline-container-orgba30a6f" class="outline-3">
<h3 id="org872df99"><span class="section-number-3">1.1</span> Block Diagram</h3> <h3 id="orgba30a6f"><span class="section-number-3">1.1</span> Block Diagram</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
The block diagram of the setup to be controlled is shown in Fig. <a href="#org3b04014">1</a>. The block diagram of the setup to be controlled is shown in Fig. <a href="#orgea07581">1</a>.
</p> </p>
<div id="org3b04014" class="figure"> <div id="orgea07581" class="figure">
<p><img src="figs/cercalo_diagram_simplify.png" alt="cercalo_diagram_simplify.png" /> <p><img src="figs/cercalo_diagram_simplify.png" alt="cercalo_diagram_simplify.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Block Diagram of the Experimental Setup</p> <p><span class="figure-number">Figure 1: </span>Block Diagram of the Experimental Setup</p>
@ -391,10 +392,10 @@ The transfer functions in the system are:
</ul> </ul>
<p> <p>
The block diagram with each transfer function is shown in Fig. <a href="#org0ea7908">2</a>. The block diagram with each transfer function is shown in Fig. <a href="#orgb2e86b9">2</a>.
</p> </p>
<div id="org0ea7908" class="figure"> <div id="orgb2e86b9" class="figure">
<p><img src="figs/cercalo_diagram.png" alt="cercalo_diagram.png" /> <p><img src="figs/cercalo_diagram.png" alt="cercalo_diagram.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Block Diagram of the Experimental Setup with detailed dynamics</p> <p><span class="figure-number">Figure 2: </span>Block Diagram of the Experimental Setup with detailed dynamics</p>
@ -402,14 +403,14 @@ The block diagram with each transfer function is shown in Fig. <a href="#org0ea7
</div> </div>
</div> </div>
<div id="outline-container-orgeb18df8" class="outline-3"> <div id="outline-container-org7551d72" class="outline-3">
<h3 id="orgeb18df8"><span class="section-number-3">1.2</span> Cercalo</h3> <h3 id="org7551d72"><span class="section-number-3">1.2</span> Cercalo</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
From the Cercalo documentation, we have the parameters shown on table <a href="#org6430d98">1</a>. From the Cercalo documentation, we have the parameters shown on table <a href="#orgf1714bf">1</a>.
</p> </p>
<table id="org6430d98" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgf1714bf" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Cercalo Parameters</caption> <caption class="t-above"><span class="table-number">Table 1:</span> Cercalo Parameters</caption>
<colgroup> <colgroup>
@ -467,11 +468,11 @@ The Inductance and DC resistance of the two axis of the Cercalo have been measur
</ul> </ul>
<p> <p>
Let's first consider the <b>horizontal direction</b> and we try to model the Cercalo by a spring/mass/damper system (Fig. <a href="#org1c65c6f">3</a>). Let's first consider the <b>horizontal direction</b> and we try to model the Cercalo by a spring/mass/damper system (Fig. <a href="#org898f123">3</a>).
</p> </p>
<div id="org1c65c6f" class="figure"> <div id="org898f123" class="figure">
<p><img src="figs/mech_cercalo.png" alt="mech_cercalo.png" /> <p><img src="figs/mech_cercalo.png" alt="mech_cercalo.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>1 degree-of-freedom model of the Cercalo</p> <p><span class="figure-number">Figure 3: </span>1 degree-of-freedom model of the Cercalo</p>
@ -510,7 +511,7 @@ The current \(I\) is also proportional to the voltage at the output of the buffe
<p> <p>
Let's try to determine the equivalent mass and spring values. Let's try to determine the equivalent mass and spring values.
From table <a href="#org6430d98">1</a>, for the horizontal direction: From table <a href="#orgf1714bf">1</a>, for the horizontal direction:
\[ \left| \frac{x}{I} \right|(0) = \left| \alpha \frac{x}{F} \right|(0) = 28.4\ \frac{mA}{deg} = 1.63\ \frac{A}{rad} \] \[ \left| \frac{x}{I} \right|(0) = \left| \alpha \frac{x}{F} \right|(0) = 28.4\ \frac{mA}{deg} = 1.63\ \frac{A}{rad} \]
</p> </p>
@ -571,18 +572,18 @@ This will be done using the Newport.
</div> </div>
</div> </div>
<div id="outline-container-orgabd5452" class="outline-3"> <div id="outline-container-org346972d" class="outline-3">
<h3 id="orgabd5452"><span class="section-number-3">1.3</span> Optical Setup</h3> <h3 id="org346972d"><span class="section-number-3">1.3</span> Optical Setup</h3>
</div> </div>
<div id="outline-container-orga857b69" class="outline-3"> <div id="outline-container-org592a330" class="outline-3">
<h3 id="orga857b69"><span class="section-number-3">1.4</span> Newport</h3> <h3 id="org592a330"><span class="section-number-3">1.4</span> Newport</h3>
<div class="outline-text-3" id="text-1-4"> <div class="outline-text-3" id="text-1-4">
<p> <p>
Parameters of the Newport are shown in Fig. <a href="#org882caed">4</a>. Parameters of the Newport are shown in Fig. <a href="#org6050b00">4</a>.
</p> </p>
<p> <p>
It's dynamics for small angle excitation is shown in Fig. <a href="#orgbe2df74">5</a>. It's dynamics for small angle excitation is shown in Fig. <a href="#org57aa82b">5</a>.
</p> </p>
<p> <p>
@ -594,14 +595,14 @@ And we have:
\end{align*} \end{align*}
<div id="org882caed" class="figure"> <div id="org6050b00" class="figure">
<p><img src="figs/newport_doc.png" alt="newport_doc.png" /> <p><img src="figs/newport_doc.png" alt="newport_doc.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Documentation of the Newport</p> <p><span class="figure-number">Figure 4: </span>Documentation of the Newport</p>
</div> </div>
<div id="orgbe2df74" class="figure"> <div id="org57aa82b" class="figure">
<p><img src="figs/newport_gain.png" alt="newport_gain.png" /> <p><img src="figs/newport_gain.png" alt="newport_gain.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Transfer function of the Newport</p> <p><span class="figure-number">Figure 5: </span>Transfer function of the Newport</p>
@ -609,25 +610,25 @@ And we have:
</div> </div>
</div> </div>
<div id="outline-container-org65c50b5" class="outline-3"> <div id="outline-container-org7cbae0f" class="outline-3">
<h3 id="org65c50b5"><span class="section-number-3">1.5</span> 4 quadrant Diode</h3> <h3 id="org7cbae0f"><span class="section-number-3">1.5</span> 4 quadrant Diode</h3>
<div class="outline-text-3" id="text-1-5"> <div class="outline-text-3" id="text-1-5">
<p> <p>
The front view of the 4 quadrant photo-diode is shown in Fig. <a href="#org0b9b9ab">6</a>. The front view of the 4 quadrant photo-diode is shown in Fig. <a href="#org74e3a9d">6</a>.
</p> </p>
<div id="org0b9b9ab" class="figure"> <div id="org74e3a9d" class="figure">
<p><img src="figs/4qd_naming.png" alt="4qd_naming.png" /> <p><img src="figs/4qd_naming.png" alt="4qd_naming.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Front view of the 4QD</p> <p><span class="figure-number">Figure 6: </span>Front view of the 4QD</p>
</div> </div>
<p> <p>
Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig. <a href="#org9c0de93">7</a>. Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig. <a href="#org47ae36a">7</a>.
</p> </p>
<div id="org9c0de93" class="figure"> <div id="org47ae36a" class="figure">
<p><img src="figs/4qd_amplifier.png" alt="4qd_amplifier.png" /> <p><img src="figs/4qd_amplifier.png" alt="4qd_amplifier.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Wiring of the amplifier. The amplifier is located on the bottom right of the board</p> <p><span class="figure-number">Figure 7: </span>Wiring of the amplifier. The amplifier is located on the bottom right of the board</p>
@ -635,8 +636,8 @@ Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig
</div> </div>
</div> </div>
<div id="outline-container-orge0ab4f9" class="outline-3"> <div id="outline-container-orgdd83994" class="outline-3">
<h3 id="orge0ab4f9"><span class="section-number-3">1.6</span> ADC/DAC</h3> <h3 id="orgdd83994"><span class="section-number-3">1.6</span> ADC/DAC</h3>
<div class="outline-text-3" id="text-1-6"> <div class="outline-text-3" id="text-1-6">
<p> <p>
Let's compute the theoretical noise of the ADC/DAC. Let's compute the theoretical noise of the ADC/DAC.
@ -656,14 +657,14 @@ with \(\Delta V\) the total range of the ADC, \(n\) its number of bits, \(q\) th
</div> </div>
</div> </div>
<div id="outline-container-org29ba44d" class="outline-2"> <div id="outline-container-org0c91379" class="outline-2">
<h2 id="org29ba44d"><span class="section-number-2">2</span> Identification of the system dynamics</h2> <h2 id="org0c91379"><span class="section-number-2">2</span> Identification of the system dynamics</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="org15c1ca3"></a> <a id="org4716847"></a>
</p> </p>
<p> <p>
In this section, we seek to identify all the blocks as shown in Fig. <a href="#org3b04014">1</a>. In this section, we seek to identify all the blocks as shown in Fig. <a href="#orgea07581">1</a>.
</p> </p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -760,8 +761,8 @@ All the files (data and Matlab scripts) are accessible <a href="data/cercalo_ide
</div> </div>
</div> </div>
<div id="outline-container-orgf19c09b" class="outline-3"> <div id="outline-container-orga71352f" class="outline-3">
<h3 id="orgf19c09b"><span class="section-number-3">2.1</span> Calibration of the 4 Quadrant Diode</h3> <h3 id="orga71352f"><span class="section-number-3">2.1</span> Calibration of the 4 Quadrant Diode</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
Prior to any dynamic identification, we would like to be able to determine the meaning of the 4 quadrant diode measurement. Prior to any dynamic identification, we would like to be able to determine the meaning of the 4 quadrant diode measurement.
@ -776,8 +777,8 @@ We then should be able to obtain the "gain" of the 4QD in [V/rad].
</p> </p>
</div> </div>
<div id="outline-container-org488413f" class="outline-4"> <div id="outline-container-org38399c1" class="outline-4">
<h4 id="org488413f"><span class="section-number-4">2.1.1</span> Input / Output data</h4> <h4 id="org38399c1"><span class="section-number-4">2.1.1</span> Input / Output data</h4>
<div class="outline-text-4" id="text-2-1-1"> <div class="outline-text-4" id="text-2-1-1">
<p> <p>
The identification data is loaded The identification data is loaded
@ -811,7 +812,7 @@ uv.t = uv.t <span class="org-type">-</span> uv.t<span class="org-rainbow-delimit
</div> </div>
<div id="orgc9d2415" class="figure"> <div id="org0bf6949" class="figure">
<p><img src="figs/calib_4qd_h.png" alt="calib_4qd_h.png" /> <p><img src="figs/calib_4qd_h.png" alt="calib_4qd_h.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Identification signals when exciting the horizontal direction (<a href="./figs/calib_4qd_h.png">png</a>, <a href="./figs/calib_4qd_h.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 8: </span>Identification signals when exciting the horizontal direction (<a href="./figs/calib_4qd_h.png">png</a>, <a href="./figs/calib_4qd_h.pdf">pdf</a>)</p>
@ -819,7 +820,7 @@ uv.t = uv.t <span class="org-type">-</span> uv.t<span class="org-rainbow-delimit
<div id="org116dcc2" class="figure"> <div id="org4b43d2d" class="figure">
<p><img src="figs/calib_4qd_v.png" alt="calib_4qd_v.png" /> <p><img src="figs/calib_4qd_v.png" alt="calib_4qd_v.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Identification signals when exciting in the vertical direction (<a href="./figs/calib_4qd_v.png">png</a>, <a href="./figs/calib_4qd_v.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 9: </span>Identification signals when exciting in the vertical direction (<a href="./figs/calib_4qd_v.png">png</a>, <a href="./figs/calib_4qd_v.pdf">pdf</a>)</p>
@ -827,8 +828,8 @@ uv.t = uv.t <span class="org-type">-</span> uv.t<span class="org-rainbow-delimit
</div> </div>
</div> </div>
<div id="outline-container-orgbbc3ad2" class="outline-4"> <div id="outline-container-org88480ae" class="outline-4">
<h4 id="orgbbc3ad2"><span class="section-number-4">2.1.2</span> Linear Regression to obtain the gain of the 4QD</h4> <h4 id="org88480ae"><span class="section-number-4">2.1.2</span> Linear Regression to obtain the gain of the 4QD</h4>
<div class="outline-text-4" id="text-2-1-2"> <div class="outline-text-4" id="text-2-1-2">
<p> <p>
We plot the angle of mirror We plot the angle of mirror
@ -858,7 +859,7 @@ where:
</ul> </ul>
<p> <p>
The linear regression is shown in Fig. <a href="#org3b7b1cd">10</a>. The linear regression is shown in Fig. <a href="#org2eca186">10</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -868,17 +869,17 @@ bv = <span class="org-rainbow-delimiters-depth-1">[</span>ones<span class="org-r
</div> </div>
<div id="org3b7b1cd" class="figure"> <div id="org2eca186" class="figure">
<p><img src="figs/4qd_linear_reg.png" alt="4qd_linear_reg.png" /> <p><img src="figs/4qd_linear_reg.png" alt="4qd_linear_reg.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Linear Regression (<a href="./figs/4qd_linear_reg.png">png</a>, <a href="./figs/4qd_linear_reg.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 10: </span>Linear Regression (<a href="./figs/4qd_linear_reg.png">png</a>, <a href="./figs/4qd_linear_reg.pdf">pdf</a>)</p>
</div> </div>
<p> <p>
Thus, we obtain the "gain of the 4 quadrant photo-diode as shown on table <a href="#org27870f3">2</a>. Thus, we obtain the "gain of the 4 quadrant photo-diode as shown on table <a href="#org6e6cf68">2</a>.
</p> </p>
<table id="org27870f3" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org6e6cf68" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Identified Gain of the 4 quadrant diode</caption> <caption class="t-above"><span class="table-number">Table 2:</span> Identified Gain of the 4 quadrant diode</caption>
<colgroup> <colgroup>
@ -922,11 +923,11 @@ We obtain:
</div> </div>
</div> </div>
<div id="outline-container-org7dd4e6c" class="outline-3"> <div id="outline-container-org45084e8" class="outline-3">
<h3 id="org7dd4e6c"><span class="section-number-3">2.2</span> Identification of the Cercalo Impedance, Current Amplifier and Voltage Amplifier dynamics</h3> <h3 id="org45084e8"><span class="section-number-3">2.2</span> Identification of the Cercalo Impedance, Current Amplifier and Voltage Amplifier dynamics</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
We wish here to determine \(G_i\) and \(G_a\) shown in Fig. <a href="#org3b04014">1</a>. We wish here to determine \(G_i\) and \(G_a\) shown in Fig. <a href="#orgea07581">1</a>.
</p> </p>
<p> <p>
@ -934,15 +935,15 @@ We ignore the electro-mechanical coupling.
</p> </p>
</div> </div>
<div id="outline-container-org6078ef0" class="outline-4"> <div id="outline-container-orgd91f44a" class="outline-4">
<h4 id="org6078ef0"><span class="section-number-4">2.2.1</span> Electrical Schematic</h4> <h4 id="orgd91f44a"><span class="section-number-4">2.2.1</span> Electrical Schematic</h4>
<div class="outline-text-4" id="text-2-2-1"> <div class="outline-text-4" id="text-2-2-1">
<p> <p>
The schematic of the electrical circuit used to drive the Cercalo is shown in Fig. <a href="#orge5d70b8">11</a>. The schematic of the electrical circuit used to drive the Cercalo is shown in Fig. <a href="#org7e35a0b">11</a>.
</p> </p>
<div id="orge5d70b8" class="figure"> <div id="org7e35a0b" class="figure">
<p><img src="figs/cercalo_amplifier.png" alt="cercalo_amplifier.png" /> <p><img src="figs/cercalo_amplifier.png" alt="cercalo_amplifier.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>Current Amplifier Schematic</p> <p><span class="figure-number">Figure 11: </span>Current Amplifier Schematic</p>
@ -1029,8 +1030,8 @@ with
</div> </div>
</div> </div>
<div id="outline-container-orgaa575f2" class="outline-4"> <div id="outline-container-orge075768" class="outline-4">
<h4 id="orgaa575f2"><span class="section-number-4">2.2.2</span> Theoretical Transfer Functions</h4> <h4 id="orge075768"><span class="section-number-4">2.2.2</span> Theoretical Transfer Functions</h4>
<div class="outline-text-4" id="text-2-2-2"> <div class="outline-text-4" id="text-2-2-2">
<p> <p>
The values of the components in the current amplifier have been measured. The values of the components in the current amplifier have been measured.
@ -1060,7 +1061,7 @@ Ga = blkdiag<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
</div> </div>
<div id="org3761ab1" class="figure"> <div id="orga49918f" class="figure">
<p><img src="figs/current_amplifier_tf.png" alt="current_amplifier_tf.png" /> <p><img src="figs/current_amplifier_tf.png" alt="current_amplifier_tf.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Transfer function for the current amplifier (<a href="./figs/current_amplifier_tf.png">png</a>, <a href="./figs/current_amplifier_tf.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 12: </span>Transfer function for the current amplifier (<a href="./figs/current_amplifier_tf.png">png</a>, <a href="./figs/current_amplifier_tf.pdf">pdf</a>)</p>
@ -1082,8 +1083,8 @@ Zc = tf<span class="org-rainbow-delimiters-depth-1">(</span>blkdiag<span class="
</div> </div>
</div> </div>
<div id="outline-container-orgc24d0bd" class="outline-4"> <div id="outline-container-org31831c5" class="outline-4">
<h4 id="orgc24d0bd"><span class="section-number-4">2.2.3</span> Identified Transfer Functions</h4> <h4 id="org31831c5"><span class="section-number-4">2.2.3</span> Identified Transfer Functions</h4>
<div class="outline-text-4" id="text-2-2-3"> <div class="outline-text-4" id="text-2-2-3">
<p> <p>
Noise is generated using the DAC (\([U_{c,h}\ U_{c,v}]\)) and we measure the output of the voltage amplifier \([V_{c,h}, V_{c,v}]\). Noise is generated using the DAC (\([U_{c,h}\ U_{c,v}]\)) and we measure the output of the voltage amplifier \([V_{c,h}, V_{c,v}]\).
@ -1110,7 +1111,7 @@ We remove the first seconds where the Cercalo is turned on.
</div> </div>
<div id="org96f6b44" class="figure"> <div id="orgd1b9369" class="figure">
<p><img src="figs/current_amplifier_comp_theory_id.png" alt="current_amplifier_comp_theory_id.png" /> <p><img src="figs/current_amplifier_comp_theory_id.png" alt="current_amplifier_comp_theory_id.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Identified and Theoretical Transfer Function \(G_a G_i\) (<a href="./figs/current_amplifier_comp_theory_id.png">png</a>, <a href="./figs/current_amplifier_comp_theory_id.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 13: </span>Identified and Theoretical Transfer Function \(G_a G_i\) (<a href="./figs/current_amplifier_comp_theory_id.png">png</a>, <a href="./figs/current_amplifier_comp_theory_id.pdf">pdf</a>)</p>
@ -1128,7 +1129,7 @@ Gi = tf<span class="org-rainbow-delimiters-depth-1">(</span>blkdiag<span class="
</div> </div>
<div id="org4ac2439" class="figure"> <div id="org1ba887a" class="figure">
<p><img src="figs/current_amplifier_comp_theory_id_bis.png" alt="current_amplifier_comp_theory_id_bis.png" /> <p><img src="figs/current_amplifier_comp_theory_id_bis.png" alt="current_amplifier_comp_theory_id_bis.png" />
</p> </p>
<p><span class="figure-number">Figure 14: </span>Identified and Theoretical Transfer Function \(G_a G_i\) (<a href="./figs/current_amplifier_comp_theory_id_bis.png">png</a>, <a href="./figs/current_amplifier_comp_theory_id_bis.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 14: </span>Identified and Theoretical Transfer Function \(G_a G_i\) (<a href="./figs/current_amplifier_comp_theory_id_bis.png">png</a>, <a href="./figs/current_amplifier_comp_theory_id_bis.pdf">pdf</a>)</p>
@ -1206,11 +1207,11 @@ Continuous-time zero/pole/gain model.
</div> </div>
</div> </div>
<div id="outline-container-org8b914fd" class="outline-3"> <div id="outline-container-orge7aa024" class="outline-3">
<h3 id="org8b914fd"><span class="section-number-3">2.3</span> Identification of the Cercalo Dynamics</h3> <h3 id="orge7aa024"><span class="section-number-3">2.3</span> Identification of the Cercalo Dynamics</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
We now wish to identify the dynamics of the Cercalo identified by \(G_c\) on the block diagram in Fig. <a href="#org3b04014">1</a>. We now wish to identify the dynamics of the Cercalo identified by \(G_c\) on the block diagram in Fig. <a href="#orgea07581">1</a>.
</p> </p>
<p> <p>
@ -1222,8 +1223,8 @@ The transfer function obtained will be \(G_c G_i\), and because we have already
</p> </p>
</div> </div>
<div id="outline-container-org45f478a" class="outline-4"> <div id="outline-container-orgcf382ac" class="outline-4">
<h4 id="org45f478a"><span class="section-number-4">2.3.1</span> Input / Output data</h4> <h4 id="orgcf382ac"><span class="section-number-4">2.3.1</span> Input / Output data</h4>
<div class="outline-text-4" id="text-2-3-1"> <div class="outline-text-4" id="text-2-3-1">
<p> <p>
The identification data is loaded The identification data is loaded
@ -1258,7 +1259,7 @@ uv.t = uv.t <span class="org-type">-</span> uv.t<span class="org-rainbow-delimit
</div> </div>
<div id="org11b6434" class="figure"> <div id="org88dabc9" class="figure">
<p><img src="figs/identification_uh.png" alt="identification_uh.png" /> <p><img src="figs/identification_uh.png" alt="identification_uh.png" />
</p> </p>
<p><span class="figure-number">Figure 15: </span>Identification signals when exciting the horizontal direction (<a href="./figs/identification_uh.png">png</a>, <a href="./figs/identification_uh.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 15: </span>Identification signals when exciting the horizontal direction (<a href="./figs/identification_uh.png">png</a>, <a href="./figs/identification_uh.pdf">pdf</a>)</p>
@ -1266,7 +1267,7 @@ uv.t = uv.t <span class="org-type">-</span> uv.t<span class="org-rainbow-delimit
<div id="org12f9542" class="figure"> <div id="orgaf83542" class="figure">
<p><img src="figs/identification_uv.png" alt="identification_uv.png" /> <p><img src="figs/identification_uv.png" alt="identification_uv.png" />
</p> </p>
<p><span class="figure-number">Figure 16: </span>Identification signals when exciting in the vertical direction (<a href="./figs/identification_uv.png">png</a>, <a href="./figs/identification_uv.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 16: </span>Identification signals when exciting in the vertical direction (<a href="./figs/identification_uv.png">png</a>, <a href="./figs/identification_uv.pdf">pdf</a>)</p>
@ -1274,8 +1275,8 @@ uv.t = uv.t <span class="org-type">-</span> uv.t<span class="org-rainbow-delimit
</div> </div>
</div> </div>
<div id="outline-container-orga76fba2" class="outline-4"> <div id="outline-container-orgc61c149" class="outline-4">
<h4 id="orga76fba2"><span class="section-number-4">2.3.2</span> Coherence</h4> <h4 id="orgc61c149"><span class="section-number-4">2.3.2</span> Coherence</h4>
<div class="outline-text-4" id="text-2-3-2"> <div class="outline-text-4" id="text-2-3-2">
<p> <p>
The window used for the spectral analysis is an <code>hanning</code> windows with temporal size equal to 1 second. The window used for the spectral analysis is an <code>hanning</code> windows with temporal size equal to 1 second.
@ -1294,7 +1295,7 @@ The window used for the spectral analysis is an <code>hanning</code> windows wit
</div> </div>
<div id="org7963278" class="figure"> <div id="org05a7b44" class="figure">
<p><img src="figs/coh_cercalo.png" alt="coh_cercalo.png" /> <p><img src="figs/coh_cercalo.png" alt="coh_cercalo.png" />
</p> </p>
<p><span class="figure-number">Figure 17: </span>Coherence (<a href="./figs/coh_cercalo.png">png</a>, <a href="./figs/coh_cercalo.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 17: </span>Coherence (<a href="./figs/coh_cercalo.png">png</a>, <a href="./figs/coh_cercalo.pdf">pdf</a>)</p>
@ -1302,8 +1303,8 @@ The window used for the spectral analysis is an <code>hanning</code> windows wit
</div> </div>
</div> </div>
<div id="outline-container-org131ac39" class="outline-4"> <div id="outline-container-orge1f3965" class="outline-4">
<h4 id="org131ac39"><span class="section-number-4">2.3.3</span> Estimation of the Frequency Response Function Matrix</h4> <h4 id="orge1f3965"><span class="section-number-4">2.3.3</span> Estimation of the Frequency Response Function Matrix</h4>
<div class="outline-text-4" id="text-2-3-3"> <div class="outline-text-4" id="text-2-3-3">
<p> <p>
We compute an estimate of the transfer functions. We compute an estimate of the transfer functions.
@ -1317,14 +1318,14 @@ We compute an estimate of the transfer functions.
</div> </div>
<div id="org7bbc71b" class="figure"> <div id="org0dad434" class="figure">
<p><img src="figs/frf_cercalo_gain.png" alt="frf_cercalo_gain.png" /> <p><img src="figs/frf_cercalo_gain.png" alt="frf_cercalo_gain.png" />
</p> </p>
<p><span class="figure-number">Figure 18: </span>Frequency Response Matrix (<a href="./figs/frf_cercalo_gain.png">png</a>, <a href="./figs/frf_cercalo_gain.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 18: </span>Frequency Response Matrix (<a href="./figs/frf_cercalo_gain.png">png</a>, <a href="./figs/frf_cercalo_gain.pdf">pdf</a>)</p>
</div> </div>
<div id="orge616ff8" class="figure"> <div id="org6c451f4" class="figure">
<p><img src="figs/frf_cercalo_phase.png" alt="frf_cercalo_phase.png" /> <p><img src="figs/frf_cercalo_phase.png" alt="frf_cercalo_phase.png" />
</p> </p>
<p><span class="figure-number">Figure 19: </span>Frequency Response Matrix<sub>Phase</sub> (<a href="./figs/frf_cercalo_phase.png">png</a>, <a href="./figs/frf_cercalo_phase.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 19: </span>Frequency Response Matrix<sub>Phase</sub> (<a href="./figs/frf_cercalo_phase.png">png</a>, <a href="./figs/frf_cercalo_phase.pdf">pdf</a>)</p>
@ -1332,8 +1333,8 @@ We compute an estimate of the transfer functions.
</div> </div>
</div> </div>
<div id="outline-container-orgf0a3476" class="outline-4"> <div id="outline-container-orgef1a3c2" class="outline-4">
<h4 id="orgf0a3476"><span class="section-number-4">2.3.4</span> Time Delay</h4> <h4 id="orgef1a3c2"><span class="section-number-4">2.3.4</span> Time Delay</h4>
<div class="outline-text-4" id="text-2-3-4"> <div class="outline-text-4" id="text-2-3-4">
<p> <p>
Now, we would like to remove the time delay included in the FRF prior to the model extraction. Now, we would like to remove the time delay included in the FRF prior to the model extraction.
@ -1364,8 +1365,8 @@ tf_Ucv_Vpv = tf_Ucv_Vpv<span class="org-type">./</span>G_delay_resp;
</div> </div>
</div> </div>
<div id="outline-container-org0174055" class="outline-4"> <div id="outline-container-org36d1dfc" class="outline-4">
<h4 id="org0174055"><span class="section-number-4">2.3.5</span> Extraction of a transfer function matrix</h4> <h4 id="org36d1dfc"><span class="section-number-4">2.3.5</span> Extraction of a transfer function matrix</h4>
<div class="outline-text-4" id="text-2-3-5"> <div class="outline-text-4" id="text-2-3-5">
<p> <p>
First we define the initial guess for the resonance frequencies and the weights associated. First we define the initial guess for the resonance frequencies and the weights associated.
@ -1415,11 +1416,11 @@ weight_Ucv_Vpv<span class="org-rainbow-delimiters-depth-1">(</span>f<span class=
</div> </div>
<p> <p>
The weights are shown in Fig. <a href="#org38a4998">20</a>. The weights are shown in Fig. <a href="#org0274c3d">20</a>.
</p> </p>
<div id="org38a4998" class="figure"> <div id="org0274c3d" class="figure">
<p><img src="figs/weights_cercalo.png" alt="weights_cercalo.png" /> <p><img src="figs/weights_cercalo.png" alt="weights_cercalo.png" />
</p> </p>
<p><span class="figure-number">Figure 20: </span>Weights amplitude (<a href="./figs/weights_cercalo.png">png</a>, <a href="./figs/weights_cercalo.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 20: </span>Weights amplitude (<a href="./figs/weights_cercalo.png">png</a>, <a href="./figs/weights_cercalo.pdf">pdf</a>)</p>
@ -1471,7 +1472,7 @@ An we run the <code>vectfit3</code> algorithm.
</div> </div>
<div id="org6e49a66" class="figure"> <div id="orga5de183" class="figure">
<p><img src="figs/identification_matrix_fit.png" alt="identification_matrix_fit.png" /> <p><img src="figs/identification_matrix_fit.png" alt="identification_matrix_fit.png" />
</p> </p>
<p><span class="figure-number">Figure 21: </span>Transfer Function Extraction of the FRF matrix (<a href="./figs/identification_matrix_fit.png">png</a>, <a href="./figs/identification_matrix_fit.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 21: </span>Transfer Function Extraction of the FRF matrix (<a href="./figs/identification_matrix_fit.png">png</a>, <a href="./figs/identification_matrix_fit.pdf">pdf</a>)</p>
@ -1479,7 +1480,7 @@ An we run the <code>vectfit3</code> algorithm.
<div id="orgfafe5c8" class="figure"> <div id="org3e1b13e" class="figure">
<p><img src="figs/identification_matrix_fit_phase.png" alt="identification_matrix_fit_phase.png" /> <p><img src="figs/identification_matrix_fit_phase.png" alt="identification_matrix_fit_phase.png" />
</p> </p>
<p><span class="figure-number">Figure 22: </span>Transfer Function Extraction of the FRF matrix (<a href="./figs/identification_matrix_fit_phase.png">png</a>, <a href="./figs/identification_matrix_fit_phase.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 22: </span>Transfer Function Extraction of the FRF matrix (<a href="./figs/identification_matrix_fit_phase.png">png</a>, <a href="./figs/identification_matrix_fit_phase.pdf">pdf</a>)</p>
@ -1502,8 +1503,8 @@ Gc = <span class="org-rainbow-delimiters-depth-1">[</span>G_Uch_Vph, G_Ucv_Vph;
</div> </div>
</div> </div>
<div id="outline-container-org29f709e" class="outline-3"> <div id="outline-container-org87a7d68" class="outline-3">
<h3 id="org29f709e"><span class="section-number-3">2.4</span> Identification of the Newport Dynamics</h3> <h3 id="org87a7d68"><span class="section-number-3">2.4</span> Identification of the Newport Dynamics</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<p> <p>
We here identify the transfer function from a reference sent to the Newport \([U_{n,h},\ U_{n,v}]\) to the measurement made by the 4QD \([V_{p,h},\ V_{p,v}]\). We here identify the transfer function from a reference sent to the Newport \([U_{n,h},\ U_{n,v}]\) to the measurement made by the 4QD \([V_{p,h},\ V_{p,v}]\).
@ -1514,8 +1515,8 @@ To do so, we inject noise to the Newport \([U_{n,h},\ U_{n,v}]\) and we record t
</p> </p>
</div> </div>
<div id="outline-container-org18a03ed" class="outline-4"> <div id="outline-container-org83a4f28" class="outline-4">
<h4 id="org18a03ed"><span class="section-number-4">2.4.1</span> Input / Output data</h4> <h4 id="org83a4f28"><span class="section-number-4">2.4.1</span> Input / Output data</h4>
<div class="outline-text-4" id="text-2-4-1"> <div class="outline-text-4" id="text-2-4-1">
<p> <p>
The identification data is loaded The identification data is loaded
@ -1550,14 +1551,14 @@ uv.t = uv.t <span class="org-type">-</span> uv.t<span class="org-rainbow-delimit
</div> </div>
<div id="org906edc3" class="figure"> <div id="orgfaa7160" class="figure">
<p><img src="figs/identification_unh.png" alt="identification_unh.png" /> <p><img src="figs/identification_unh.png" alt="identification_unh.png" />
</p> </p>
<p><span class="figure-number">Figure 23: </span>Identification signals when exciting the horizontal direction (<a href="./figs/identification_unh.png">png</a>, <a href="./figs/identification_unh.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 23: </span>Identification signals when exciting the horizontal direction (<a href="./figs/identification_unh.png">png</a>, <a href="./figs/identification_unh.pdf">pdf</a>)</p>
</div> </div>
<div id="org32607b6" class="figure"> <div id="org9ec6c46" class="figure">
<p><img src="figs/identification_unv.png" alt="identification_unv.png" /> <p><img src="figs/identification_unv.png" alt="identification_unv.png" />
</p> </p>
<p><span class="figure-number">Figure 24: </span>Identification signals when exciting in the vertical direction (<a href="./figs/identification_unv.png">png</a>, <a href="./figs/identification_unv.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 24: </span>Identification signals when exciting in the vertical direction (<a href="./figs/identification_unv.png">png</a>, <a href="./figs/identification_unv.pdf">pdf</a>)</p>
@ -1565,8 +1566,8 @@ uv.t = uv.t <span class="org-type">-</span> uv.t<span class="org-rainbow-delimit
</div> </div>
</div> </div>
<div id="outline-container-org978a0c9" class="outline-4"> <div id="outline-container-orge19fa44" class="outline-4">
<h4 id="org978a0c9"><span class="section-number-4">2.4.2</span> Coherence</h4> <h4 id="orge19fa44"><span class="section-number-4">2.4.2</span> Coherence</h4>
<div class="outline-text-4" id="text-2-4-2"> <div class="outline-text-4" id="text-2-4-2">
<p> <p>
The window used for the spectral analysis is an <code>hanning</code> windows with temporal size equal to 1 second. The window used for the spectral analysis is an <code>hanning</code> windows with temporal size equal to 1 second.
@ -1585,7 +1586,7 @@ The window used for the spectral analysis is an <code>hanning</code> windows wit
</div> </div>
<div id="org69a538e" class="figure"> <div id="org288353f" class="figure">
<p><img src="figs/id_newport_coherence.png" alt="id_newport_coherence.png" /> <p><img src="figs/id_newport_coherence.png" alt="id_newport_coherence.png" />
</p> </p>
<p><span class="figure-number">Figure 25: </span>Coherence (<a href="./figs/id_newport_coherence.png">png</a>, <a href="./figs/id_newport_coherence.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 25: </span>Coherence (<a href="./figs/id_newport_coherence.png">png</a>, <a href="./figs/id_newport_coherence.pdf">pdf</a>)</p>
@ -1593,8 +1594,8 @@ The window used for the spectral analysis is an <code>hanning</code> windows wit
</div> </div>
</div> </div>
<div id="outline-container-orgc0c5d02" class="outline-4"> <div id="outline-container-orgc36f0eb" class="outline-4">
<h4 id="orgc0c5d02"><span class="section-number-4">2.4.3</span> Estimation of the Frequency Response Function Matrix</h4> <h4 id="orgc36f0eb"><span class="section-number-4">2.4.3</span> Estimation of the Frequency Response Function Matrix</h4>
<div class="outline-text-4" id="text-2-4-3"> <div class="outline-text-4" id="text-2-4-3">
<p> <p>
We compute an estimate of the transfer functions. We compute an estimate of the transfer functions.
@ -1608,14 +1609,14 @@ We compute an estimate of the transfer functions.
</div> </div>
<div id="org2cdb3e3" class="figure"> <div id="orga7d56d3" class="figure">
<p><img src="figs/frf_newport_gain.png" alt="frf_newport_gain.png" /> <p><img src="figs/frf_newport_gain.png" alt="frf_newport_gain.png" />
</p> </p>
<p><span class="figure-number">Figure 26: </span>Frequency Response Matrix (<a href="./figs/frf_newport_gain.png">png</a>, <a href="./figs/frf_newport_gain.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 26: </span>Frequency Response Matrix (<a href="./figs/frf_newport_gain.png">png</a>, <a href="./figs/frf_newport_gain.pdf">pdf</a>)</p>
</div> </div>
<div id="org02a1305" class="figure"> <div id="org8bc5549" class="figure">
<p><img src="figs/frf_newport_phase.png" alt="frf_newport_phase.png" /> <p><img src="figs/frf_newport_phase.png" alt="frf_newport_phase.png" />
</p> </p>
<p><span class="figure-number">Figure 27: </span>Frequency Response Matrix Phase (<a href="./figs/frf_newport_phase.png">png</a>, <a href="./figs/frf_newport_phase.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 27: </span>Frequency Response Matrix Phase (<a href="./figs/frf_newport_phase.png">png</a>, <a href="./figs/frf_newport_phase.pdf">pdf</a>)</p>
@ -1623,8 +1624,8 @@ We compute an estimate of the transfer functions.
</div> </div>
</div> </div>
<div id="outline-container-orgdfcee7e" class="outline-4"> <div id="outline-container-org830e6db" class="outline-4">
<h4 id="orgdfcee7e"><span class="section-number-4">2.4.4</span> Time Delay</h4> <h4 id="org830e6db"><span class="section-number-4">2.4.4</span> Time Delay</h4>
<div class="outline-text-4" id="text-2-4-4"> <div class="outline-text-4" id="text-2-4-4">
<p> <p>
Now, we would like to remove the time delay included in the FRF prior to the model extraction. Now, we would like to remove the time delay included in the FRF prior to the model extraction.
@ -1646,7 +1647,7 @@ G_delay_resp = squeeze<span class="org-rainbow-delimiters-depth-1">(</span>freqr
We then remove the time delay from the frequency response function. We then remove the time delay from the frequency response function.
</p> </p>
<div id="org28407ed" class="figure"> <div id="org90cd7da" class="figure">
<p><img src="figs/time_delay_newport.png" alt="time_delay_newport.png" /> <p><img src="figs/time_delay_newport.png" alt="time_delay_newport.png" />
</p> </p>
<p><span class="figure-number">Figure 28: </span>Phase change due to time-delay in the Newport dynamics (<a href="./figs/time_delay_newport.png">png</a>, <a href="./figs/time_delay_newport.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 28: </span>Phase change due to time-delay in the Newport dynamics (<a href="./figs/time_delay_newport.png">png</a>, <a href="./figs/time_delay_newport.pdf">pdf</a>)</p>
@ -1654,11 +1655,11 @@ We then remove the time delay from the frequency response function.
</div> </div>
</div> </div>
<div id="outline-container-org314a223" class="outline-4"> <div id="outline-container-orgab7b96b" class="outline-4">
<h4 id="org314a223"><span class="section-number-4">2.4.5</span> Extraction of a transfer function matrix</h4> <h4 id="orgab7b96b"><span class="section-number-4">2.4.5</span> Extraction of a transfer function matrix</h4>
<div class="outline-text-4" id="text-2-4-5"> <div class="outline-text-4" id="text-2-4-5">
<p> <p>
From Fig. <a href="#org2cdb3e3">26</a>, it seems reasonable to model the Newport dynamics as diagonal and constant. From Fig. <a href="#orga7d56d3">26</a>, it seems reasonable to model the Newport dynamics as diagonal and constant.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -1669,8 +1670,8 @@ From Fig. <a href="#org2cdb3e3">26</a>, it seems reasonable to model the Newport
</div> </div>
</div> </div>
<div id="outline-container-org8fe71fd" class="outline-3"> <div id="outline-container-orgee9b6d9" class="outline-3">
<h3 id="org8fe71fd"><span class="section-number-3">2.5</span> Full System</h3> <h3 id="orgee9b6d9"><span class="section-number-3">2.5</span> Full System</h3>
<div class="outline-text-3" id="text-2-5"> <div class="outline-text-3" id="text-2-5">
<p> <p>
We now have identified: We now have identified:
@ -1732,12 +1733,12 @@ The file <code>mat/plant.mat</code> is accessible <a href="./mat/plant.mat">here
</div> </div>
</div> </div>
<div id="outline-container-org07fb936" class="outline-2"> <div id="outline-container-orgbb8218f" class="outline-2">
<h2 id="org07fb936"><span class="section-number-2">3</span> Active Damping</h2> <h2 id="orgbb8218f"><span class="section-number-2">3</span> Active Damping</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
</div> </div>
<div id="outline-container-org07da045" class="outline-3"> <div id="outline-container-org584508e" class="outline-3">
<h3 id="org07da045"><span class="section-number-3">3.1</span> Load Plant</h3> <h3 id="org584508e"><span class="section-number-3">3.1</span> Load Plant</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -1746,8 +1747,8 @@ The file <code>mat/plant.mat</code> is accessible <a href="./mat/plant.mat">here
</div> </div>
</div> </div>
<div id="outline-container-orge1669f2" class="outline-3"> <div id="outline-container-org0df875d" class="outline-3">
<h3 id="orge1669f2"><span class="section-number-3">3.2</span> Test</h3> <h3 id="org0df875d"><span class="section-number-3">3.2</span> Test</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">bode<span class="org-rainbow-delimiters-depth-1">(</span>sys<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">{</span><span class="org-string">'Vch', 'Vcv'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">}</span></span><span class="org-string">, </span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">{</span></span><span class="org-string">'Uch', 'Ucv'</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">bode<span class="org-rainbow-delimiters-depth-1">(</span>sys<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">{</span><span class="org-string">'Vch', 'Vcv'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">}</span></span><span class="org-string">, </span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">{</span></span><span class="org-string">'Uch', 'Ucv'</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -1775,8 +1776,8 @@ sys_cl = connect<span class="org-rainbow-delimiters-depth-1">(</span>sys, Kppf,
</div> </div>
</div> </div>
<div id="outline-container-orgf4f85b9" class="outline-2"> <div id="outline-container-org4ac14f0" class="outline-2">
<h2 id="orgf4f85b9"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> Huddle Test</h2> <h2 id="org4ac14f0"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> Huddle Test</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
We load the data taken during the Huddle Test. We load the data taken during the Huddle Test.
@ -1861,8 +1862,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-org03828c5" class="outline-2"> <div id="outline-container-orgb1866ab" class="outline-2">
<h2 id="org03828c5"><span class="section-number-2">5</span> Plant Scaling</h2> <h2 id="orgb1866ab"><span class="section-number-2">5</span> Plant Scaling</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<ul class="org-ul"> <ul class="org-ul">
<li>measured noise</li> <li>measured noise</li>
@ -1873,12 +1874,12 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-org8254d1a" class="outline-2"> <div id="outline-container-orgdc654c4" class="outline-2">
<h2 id="org8254d1a"><span class="section-number-2">6</span> Plant Analysis</h2> <h2 id="orgdc654c4"><span class="section-number-2">6</span> Plant Analysis</h2>
<div class="outline-text-2" id="text-6"> <div class="outline-text-2" id="text-6">
</div> </div>
<div id="outline-container-org592969b" class="outline-3"> <div id="outline-container-org59cd7d2" class="outline-3">
<h3 id="org592969b"><span class="section-number-3">6.1</span> Load Plant</h3> <h3 id="org59cd7d2"><span class="section-number-3">6.1</span> Load Plant</h3>
<div class="outline-text-3" id="text-6-1"> <div class="outline-text-3" id="text-6-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/plant.mat', 'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/plant.mat', 'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -1887,8 +1888,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-org966d85b" class="outline-3"> <div id="outline-container-org62e9187" class="outline-3">
<h3 id="org966d85b"><span class="section-number-3">6.2</span> RGA-Number</h3> <h3 id="org62e9187"><span class="section-number-3">6.2</span> RGA-Number</h3>
<div class="outline-text-3" id="text-6-2"> <div class="outline-text-3" id="text-6-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">freqs = logspace<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">4</span>, <span class="org-highlight-numbers-number">1000</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">freqs = logspace<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">4</span>, <span class="org-highlight-numbers-number">1000</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -1927,8 +1928,8 @@ V = zeros<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-h
</div> </div>
</div> </div>
<div id="outline-container-org7063733" class="outline-3"> <div id="outline-container-org4bc71a6" class="outline-3">
<h3 id="org7063733"><span class="section-number-3">6.3</span> Rotation Matrix</h3> <h3 id="org4bc71a6"><span class="section-number-3">6.3</span> Rotation Matrix</h3>
<div class="outline-text-3" id="text-6-3"> <div class="outline-text-3" id="text-6-3">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">G0 = freqresp<span class="org-rainbow-delimiters-depth-1">(</span>G, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">G0 = freqresp<span class="org-rainbow-delimiters-depth-1">(</span>G, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -1938,8 +1939,8 @@ V = zeros<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-h
</div> </div>
</div> </div>
<div id="outline-container-org759298e" class="outline-2"> <div id="outline-container-org29f5209" class="outline-2">
<h2 id="org759298e"><span class="section-number-2">7</span> Control Objective</h2> <h2 id="org29f5209"><span class="section-number-2">7</span> Control Objective</h2>
<div class="outline-text-2" id="text-7"> <div class="outline-text-2" id="text-7">
<p> <p>
The maximum expected stroke is \(y_\text{max} = 3mm \approx 5e^{-2} rad\) at \(1Hz\). The maximum expected stroke is \(y_\text{max} = 3mm \approx 5e^{-2} rad\) at \(1Hz\).
@ -1961,11 +1962,11 @@ In terms of loop gain, this is equivalent to:
</div> </div>
</div> </div>
<div id="outline-container-org96df63d" class="outline-2"> <div id="outline-container-org2b9b5f8" class="outline-2">
<h2 id="org96df63d"><span class="section-number-2">8</span> Decentralized Control</h2> <h2 id="org2b9b5f8"><span class="section-number-2">8</span> Decentralized Control</h2>
<div class="outline-text-2" id="text-8"> <div class="outline-text-2" id="text-8">
<p> <p>
<a id="org4e742a7"></a> <a id="orgc668c78"></a>
</p> </p>
<p> <p>
In this section, we try to implement a simple decentralized controller. In this section, we try to implement a simple decentralized controller.
@ -1978,8 +1979,8 @@ All the files (data and Matlab scripts) are accessible <a href="data/decentraliz
</div> </div>
</div> </div>
<div id="outline-container-orgde6febc" class="outline-3"> <div id="outline-container-orgff694d6" class="outline-3">
<h3 id="orgde6febc"><span class="section-number-3">8.1</span> Load Plant</h3> <h3 id="orgff694d6"><span class="section-number-3">8.1</span> Load Plant</h3>
<div class="outline-text-3" id="text-8-1"> <div class="outline-text-3" id="text-8-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -1988,12 +1989,12 @@ All the files (data and Matlab scripts) are accessible <a href="data/decentraliz
</div> </div>
</div> </div>
<div id="outline-container-orgf815918" class="outline-3"> <div id="outline-container-orgb485008" class="outline-3">
<h3 id="orgf815918"><span class="section-number-3">8.2</span> Diagonal Controller</h3> <h3 id="orgb485008"><span class="section-number-3">8.2</span> Diagonal Controller</h3>
<div class="outline-text-3" id="text-8-2"> <div class="outline-text-3" id="text-8-2">
<p> <p>
Using <code>SISOTOOL</code>, a diagonal controller is designed. Using <code>SISOTOOL</code>, a diagonal controller is designed.
The two SISO loop gains are shown in Fig. <a href="#org8ca046f">29</a>. The two SISO loop gains are shown in Fig. <a href="#org6e368fe">29</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Kh = <span class="org-type">-</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">25598</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">+</span><span class="org-highlight-numbers-number">112</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span> <span class="org-type">+</span> <span class="org-highlight-numbers-number">15</span>.<span class="org-highlight-numbers-number">93</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-highlight-numbers-number">6</span>.<span class="org-highlight-numbers-number">686e06</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">(</span>s<span class="org-type">+</span><span class="org-highlight-numbers-number">352</span>.<span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">+</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2000</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">Kh = <span class="org-type">-</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">25598</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">+</span><span class="org-highlight-numbers-number">112</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span> <span class="org-type">+</span> <span class="org-highlight-numbers-number">15</span>.<span class="org-highlight-numbers-number">93</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-highlight-numbers-number">6</span>.<span class="org-highlight-numbers-number">686e06</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">(</span>s<span class="org-type">+</span><span class="org-highlight-numbers-number">352</span>.<span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">+</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2000</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -2006,14 +2007,14 @@ K.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class=
</div> </div>
<div id="org8ca046f" class="figure"> <div id="org6e368fe" class="figure">
<p><img src="figs/diag_contr_loop_gain.png" alt="diag_contr_loop_gain.png" /> <p><img src="figs/diag_contr_loop_gain.png" alt="diag_contr_loop_gain.png" />
</p> </p>
<p><span class="figure-number">Figure 29: </span>Loop Gain using the Decentralized Diagonal Controller (<a href="./figs/diag_contr_loop_gain.png">png</a>, <a href="./figs/diag_contr_loop_gain.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 29: </span>Loop Gain using the Decentralized Diagonal Controller (<a href="./figs/diag_contr_loop_gain.png">png</a>, <a href="./figs/diag_contr_loop_gain.pdf">pdf</a>)</p>
</div> </div>
<p> <p>
We then close the loop and we look at the transfer function from the Newport rotation signal to the beam angle (Fig. <a href="#org1a57ffe">30</a>). We then close the loop and we look at the transfer function from the Newport rotation signal to the beam angle (Fig. <a href="#orgf836e78">30</a>).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">inputs = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Uch', 'Ucv', 'Unh', 'Unv'</span><span class="org-rainbow-delimiters-depth-1">}</span>; <pre class="src src-matlab">inputs = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Uch', 'Ucv', 'Unh', 'Unv'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
@ -2024,22 +2025,40 @@ sys_cl = connect<span class="org-rainbow-delimiters-depth-1">(</span>sys, <span
</div> </div>
<div id="org1a57ffe" class="figure"> <div id="orgf836e78" class="figure">
<p><img src="figs/diag_contr_effect_newport.png" alt="diag_contr_effect_newport.png" /> <p><img src="figs/diag_contr_effect_newport.png" alt="diag_contr_effect_newport.png" />
</p> </p>
<p><span class="figure-number">Figure 30: </span>Effect of the Newport rotation on the beam position when the loop is closed using the Decentralized Diagonal Controller (<a href="./figs/diag_contr_effect_newport.png">png</a>, <a href="./figs/diag_contr_effect_newport.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 30: </span>Effect of the Newport rotation on the beam position when the loop is closed using the Decentralized Diagonal Controller (<a href="./figs/diag_contr_effect_newport.png">png</a>, <a href="./figs/diag_contr_effect_newport.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org37b475d" class="outline-3">
<h3 id="org37b475d"><span class="section-number-3">8.3</span> Save the Controller</h3>
<div class="outline-text-3" id="text-8-3">
<div class="org-src-container">
<pre class="src src-matlab">Kd = c2d<span class="org-rainbow-delimiters-depth-1">(</span>K, <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">4</span>, <span class="org-string">'tustin'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div> </div>
<div id="outline-container-orge641704" class="outline-2"> <p>
<h2 id="orge641704"><span class="section-number-2">9</span> Measurement of the non-repeatability</h2> The diagonal controller is accessible <a href="./mat/K_diag.mat">here</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/K_diag.mat', 'K', 'Kd'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org58d3080" class="outline-2">
<h2 id="org58d3080"><span class="section-number-2">9</span> Measurement of the non-repeatability</h2>
</div> </div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-09-17 mar. 15:53</p> <p class="date">Created: 2019-09-17 mar. 16:21</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

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@ -2147,4 +2147,14 @@ We then close the loop and we look at the transfer function from the Newport rot
#+CAPTION: Effect of the Newport rotation on the beam position when the loop is closed using the Decentralized Diagonal Controller ([[./figs/diag_contr_effect_newport.png][png]], [[./figs/diag_contr_effect_newport.pdf][pdf]]) #+CAPTION: Effect of the Newport rotation on the beam position when the loop is closed using the Decentralized Diagonal Controller ([[./figs/diag_contr_effect_newport.png][png]], [[./figs/diag_contr_effect_newport.pdf][pdf]])
[[file:figs/diag_contr_effect_newport.png]] [[file:figs/diag_contr_effect_newport.png]]
** Save the Controller
#+begin_src matlab
Kd = c2d(K, 1e-4, 'tustin');
#+end_src
The diagonal controller is accessible [[./mat/K_diag.mat][here]].
#+begin_src matlab
save('mat/K_diag.mat', 'K', 'Kd');
#+end_src
* Measurement of the non-repeatability * Measurement of the non-repeatability

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