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<title>Control Requirements</title> <title>Control Requirements</title>
@ -248,52 +247,57 @@
<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="#org4418235">1. Goal</a></li> <li><a href="#org0341df1">1. Simplify Model for the Nano-Hexapod</a>
<li><a href="#org0341df1">2. Simplify Model for the Nano-Hexapod</a>
<ul> <ul>
<li><a href="#org136c9af">2.1. Model of the nano-hexapod</a></li> <li><a href="#org136c9af">1.1. Model of the nano-hexapod</a></li>
<li><a href="#org2fbecfd">2.2. How to include Ground Motion in the model?</a></li> <li><a href="#org2fbecfd">1.2. How to include Ground Motion in the model?</a></li>
<li><a href="#org8c1e462">1.3. Motion of the micro-station</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org8c1e462">3. Motion of the micro-station</a></li> <li><a href="#org92b1239">2. Control with the Stiff Nano-Hexapod</a>
<li><a href="#org6182074">4. Values and Plant</a>
<ul> <ul>
<li><a href="#org19b83b7">4.1. Definition of the values</a></li> <li><a href="#org19b83b7">2.1. Definition of the values</a></li>
<li><a href="#org0e9811a">2.2. Control using \(d\)</a>
<ul>
<li><a href="#org02a7ab1">2.2.1. Control Architecture</a></li>
<li><a href="#org5a120e1">2.2.2. Analytical Analysis</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org0e9811a">5. Control using \(d\)</a> <li><a href="#orga741e48">2.3. Control using \(F_m\)</a>
<ul> <ul>
<li><a href="#org6a29cc6">5.1. Control Architecture</a></li> <li><a href="#org9828aed">2.3.1. Control Architecture</a></li>
<li><a href="#org5a120e1">5.2. Analytical Analysis</a></li> <li><a href="#orgdd5134e">2.3.2. Pure Integrator</a></li>
<li><a href="#org5011ab0">2.3.3. Low pass filter</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orga741e48">6. Control using \(F_m\)</a> <li><a href="#org4fce174">2.4. Comparison</a></li>
<li><a href="#org5e0585d">2.5. Control using \(x\)</a>
<ul> <ul>
<li><a href="#org02a7ab1">6.1. Control Architecture</a></li> <li><a href="#orgfab8395">2.5.1. Analytical analysis</a></li>
<li><a href="#orgdd5134e">6.2. Pure Integrator</a></li> <li><a href="#org625e3c2">2.5.2. Control implementation</a></li>
<li><a href="#org5011ab0">6.3. Low pass filter</a></li> <li><a href="#org8d34d7f">2.5.3. Results</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org4fce174">7. Comparison</a></li>
<li><a href="#org5e0585d">8. Control using \(x\)</a>
<ul>
<li><a href="#orgfab8395">8.1. Analytical analysis</a></li>
<li><a href="#org625e3c2">8.2. Control implementation</a></li>
<li><a href="#org8d34d7f">8.3. Results</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgd2547fd">9. Two degree of freedom control</a></li> <li><a href="#org7c4b4fc">3. Comparison with the use of a Soft nano-hexapod</a></li>
<li><a href="#orgab31e9f">10. Soft nano-hexapod</a></li> <li><a href="#orgc0253c3">4. Estimate the level of vibration</a></li>
<li><a href="#orgcb7520f">11. Compare Soft and Stiff nano-hexapods</a></li> <li><a href="#org764c4a9">5. Requirements on the norm of closed-loop transfer functions</a>
<li><a href="#orgc0253c3">12. Estimate the level of vibration</a></li> <ul>
<li><a href="#org764c4a9">13. Requirements on the norm of closed-loop transfer functions</a></li> <li><a href="#org27379f3">5.1. Approximation of the ASD of perturbations</a></li>
<li><a href="#orgff3d823">5.2. Wanted ASD of outputs</a></li>
<li><a href="#org8c6b37c">5.3. Limiting the bandwidth</a></li>
<li><a href="#org50054f2">5.4. Generalized Weighted plant</a></li>
<li><a href="#org949ab66">5.5. Synthesis</a></li>
<li><a href="#orgfe970e4">5.6. Loop Gain</a></li>
<li><a href="#org3db77f5">5.7. Results</a></li>
<li><a href="#orgb18d7df">5.8. Requirements</a></li>
</ul>
</li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org4418235" class="outline-2">
<h2 id="org4418235"><span class="section-number-2">1</span> Goal</h2>
<div class="outline-text-2" id="text-1">
<p> <p>
The goal here is to write clear specifications for the NASS. The goal here is to write clear specifications for the NASS.
</p> </p>
@ -305,16 +309,14 @@ This can then be used for the control synthesis and for the design of the nano-h
<p> <p>
Ideal, specifications on the norm of closed loop transfer function should be written. Ideal, specifications on the norm of closed loop transfer function should be written.
</p> </p>
</div>
</div>
<div id="outline-container-org0341df1" class="outline-2"> <div id="outline-container-org0341df1" class="outline-2">
<h2 id="org0341df1"><span class="section-number-2">2</span> Simplify Model for the Nano-Hexapod</h2> <h2 id="org0341df1"><span class="section-number-2">1</span> Simplify Model for the Nano-Hexapod</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-org136c9af" class="outline-3"> <div id="outline-container-org136c9af" class="outline-3">
<h3 id="org136c9af"><span class="section-number-3">2.1</span> Model of the nano-hexapod</h3> <h3 id="org136c9af"><span class="section-number-3">1.1</span> Model of the nano-hexapod</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
Let&rsquo;s consider the simple mechanical system in Figure <a href="#orgfa3391a">1</a>. Let&rsquo;s consider the simple mechanical system in Figure <a href="#orgfa3391a">1</a>.
</p> </p>
@ -445,8 +447,8 @@ The nano-hexapod can thus be represented as in Figure <a href="#orgb2d1168">2</a
</div> </div>
<div id="outline-container-org2fbecfd" class="outline-3"> <div id="outline-container-org2fbecfd" class="outline-3">
<h3 id="org2fbecfd"><span class="section-number-3">2.2</span> How to include Ground Motion in the model?</h3> <h3 id="org2fbecfd"><span class="section-number-3">1.2</span> How to include Ground Motion in the model?</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
What we measure is not the absolute motion \(x\), but the relative motion \(x - w\) where \(w\) is the motion of the granite. What we measure is not the absolute motion \(x\), but the relative motion \(x - w\) where \(w\) is the motion of the granite.
</p> </p>
@ -454,19 +456,12 @@ What we measure is not the absolute motion \(x\), but the relative motion \(x -
<p> <p>
Also, \(w\) induces some motion \(x_\mu\) through the transmissibility of the micro-station. Also, \(w\) induces some motion \(x_\mu\) through the transmissibility of the micro-station.
</p> </p>
<div class="figure">
<p><img src="figs/nano_station_inputs_outputs_ground_motion.png" alt="nano_station_inputs_outputs_ground_motion.png" />
</p>
</div>
</div>
</div> </div>
</div> </div>
<div id="outline-container-org8c1e462" class="outline-2"> <div id="outline-container-org8c1e462" class="outline-3">
<h2 id="org8c1e462"><span class="section-number-2">3</span> Motion of the micro-station</h2> <h3 id="org8c1e462"><span class="section-number-3">1.3</span> Motion of the micro-station</h3>
<div class="outline-text-2" id="text-3"> <div class="outline-text-3" id="text-1-3">
<p> <p>
As explained, we consider \(x_\mu\) as an external input (\(F\) has no influence on \(x_\mu\)). As explained, we consider \(x_\mu\) as an external input (\(F\) has no influence on \(x_\mu\)).
</p> </p>
@ -503,14 +498,14 @@ Also, here, we suppose that the granite is not moving.
</p> </p>
<p> <p>
If we now include the motion of the granite \(w\), we obtain the block diagram shown in Figure <a href="#org974c98f">4</a>. If we now include the motion of the granite \(w\), we obtain the block diagram shown in Figure <a href="#org974c98f">3</a>.
</p> </p>
<div id="org974c98f" class="figure"> <div id="org974c98f" class="figure">
<p><img src="figs/nano_station_ground_motion.png" alt="nano_station_ground_motion.png" /> <p><img src="figs/nano_station_ground_motion.png" alt="nano_station_ground_motion.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Ground Motion \(w\) included</p> <p><span class="figure-number">Figure 3: </span>Ground Motion \(w\) included</p>
</div> </div>
<p> <p>
@ -520,14 +515,15 @@ We can approximate this transfer function by a second order low pass filter:
</p> </p>
</div> </div>
</div> </div>
</div>
<div id="outline-container-org6182074" class="outline-2"> <div id="outline-container-org92b1239" class="outline-2">
<h2 id="org6182074"><span class="section-number-2">4</span> Values and Plant</h2> <h2 id="org92b1239"><span class="section-number-2">2</span> Control with the Stiff Nano-Hexapod</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-2">
</div> </div>
<div id="outline-container-org19b83b7" class="outline-3"> <div id="outline-container-org19b83b7" class="outline-3">
<h3 id="org19b83b7"><span class="section-number-3">4.1</span> Definition of the values</h3> <h3 id="org19b83b7"><span class="section-number-3">2.1</span> Definition of the values</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
Let&rsquo;s define the mass and stiffness of the nano-hexapod. Let&rsquo;s define the mass and stiffness of the nano-hexapod.
</p> </p>
@ -585,31 +581,30 @@ Gpz = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {<span class="org-string">'Fd'</span>,
</div> </div>
</div> </div>
</div> </div>
</div>
<div id="outline-container-org0e9811a" class="outline-2"> <div id="outline-container-org0e9811a" class="outline-3">
<h2 id="org0e9811a"><span class="section-number-2">5</span> Control using \(d\)</h2> <h3 id="org0e9811a"><span class="section-number-3">2.2</span> Control using \(d\)</h3>
<div class="outline-text-2" id="text-5"> <div class="outline-text-3" id="text-2-2">
</div> </div>
<div id="outline-container-org6a29cc6" class="outline-3"> <div id="outline-container-org02a7ab1" class="outline-4">
<h3 id="org6a29cc6"><span class="section-number-3">5.1</span> Control Architecture</h3> <h4 id="org02a7ab1"><span class="section-number-4">2.2.1</span> Control Architecture</h4>
<div class="outline-text-3" id="text-5-1"> <div class="outline-text-4" id="text-2-2-1">
<p> <p>
Let&rsquo;s consider a feedback loop using \(d\) as shown in Figure <a href="#orgb50386a">5</a>. Let&rsquo;s consider a feedback loop using \(d\) as shown in Figure <a href="#orgb50386a">4</a>.
</p> </p>
<div id="orgb50386a" class="figure"> <div id="orgb50386a" class="figure">
<p><img src="figs/nano_station_control_d.png" alt="nano_station_control_d.png" /> <p><img src="figs/nano_station_control_d.png" alt="nano_station_control_d.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Feedback diagram using \(d\)</p> <p><span class="figure-number">Figure 4: </span>Feedback diagram using \(d\)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org5a120e1" class="outline-3"> <div id="outline-container-org5a120e1" class="outline-4">
<h3 id="org5a120e1"><span class="section-number-3">5.2</span> Analytical Analysis</h3> <h4 id="org5a120e1"><span class="section-number-4">2.2.2</span> Analytical Analysis</h4>
<div class="outline-text-3" id="text-5-2"> <div class="outline-text-4" id="text-2-2-2">
<p> <p>
Let&rsquo;s apply a direct velocity feedback by deriving \(d\): Let&rsquo;s apply a direct velocity feedback by deriving \(d\):
\[ F = F^\prime - g s d \] \[ F = F^\prime - g s d \]
@ -671,29 +666,29 @@ And \(\epsilon = r - x\):
</div> </div>
</div> </div>
<div id="outline-container-orga741e48" class="outline-2"> <div id="outline-container-orga741e48" class="outline-3">
<h2 id="orga741e48"><span class="section-number-2">6</span> Control using \(F_m\)</h2> <h3 id="orga741e48"><span class="section-number-3">2.3</span> Control using \(F_m\)</h3>
<div class="outline-text-2" id="text-6"> <div class="outline-text-3" id="text-2-3">
</div> </div>
<div id="outline-container-org02a7ab1" class="outline-3"> <div id="outline-container-org9828aed" class="outline-4">
<h3 id="org02a7ab1"><span class="section-number-3">6.1</span> Control Architecture</h3> <h4 id="org9828aed"><span class="section-number-4">2.3.1</span> Control Architecture</h4>
<div class="outline-text-3" id="text-6-1"> <div class="outline-text-4" id="text-2-3-1">
<p> <p>
Let&rsquo;s consider a feedback loop using \(d\) as shown in Figure <a href="#orgb50386a">5</a>. Let&rsquo;s consider a feedback loop using \(Fm\) as shown in Figure <a href="#org5012ef2">5</a>.
</p> </p>
<div id="org5012ef2" class="figure"> <div id="org5012ef2" class="figure">
<p><img src="figs/nano_station_control_Fm.png" alt="nano_station_control_Fm.png" /> <p><img src="figs/nano_station_control_Fm.png" alt="nano_station_control_Fm.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Feedback diagram using \(F_m\)</p> <p><span class="figure-number">Figure 5: </span>Feedback diagram using \(F_m\)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgdd5134e" class="outline-3"> <div id="outline-container-orgdd5134e" class="outline-4">
<h3 id="orgdd5134e"><span class="section-number-3">6.2</span> Pure Integrator</h3> <h4 id="orgdd5134e"><span class="section-number-4">2.3.2</span> Pure Integrator</h4>
<div class="outline-text-3" id="text-6-2"> <div class="outline-text-4" id="text-2-3-2">
<p> <p>
Let&rsquo;s apply integral force feedback by integration \(F_m\): Let&rsquo;s apply integral force feedback by integration \(F_m\):
\[ F = F^\prime - \frac{g}{s} F_m \] \[ F = F^\prime - \frac{g}{s} F_m \]
@ -735,9 +730,9 @@ And \(\epsilon = r - x\):
</div> </div>
</div> </div>
<div id="outline-container-org5011ab0" class="outline-3"> <div id="outline-container-org5011ab0" class="outline-4">
<h3 id="org5011ab0"><span class="section-number-3">6.3</span> Low pass filter</h3> <h4 id="org5011ab0"><span class="section-number-4">2.3.3</span> Low pass filter</h4>
<div class="outline-text-3" id="text-6-3"> <div class="outline-text-4" id="text-2-3-3">
<p> <p>
Instead of a pure integrator, let&rsquo;s use a low pass filter with a cut-off frequency above the bandwidth of the micro-station \(\omega_mu\) Instead of a pure integrator, let&rsquo;s use a low pass filter with a cut-off frequency above the bandwidth of the micro-station \(\omega_mu\)
</p> </p>
@ -753,14 +748,14 @@ Instead of a pure integrator, let&rsquo;s use a low pass filter with a cut-off f
</div> </div>
</div> </div>
<div id="outline-container-org4fce174" class="outline-2"> <div id="outline-container-org4fce174" class="outline-3">
<h2 id="org4fce174"><span class="section-number-2">7</span> Comparison</h2> <h3 id="org4fce174"><span class="section-number-3">2.4</span> Comparison</h3>
<div class="outline-text-2" id="text-7"> <div class="outline-text-3" id="text-2-4">
<div id="orgc10daac" class="figure"> <div id="orgc10daac" class="figure">
<p><img src="figs/comp_iff_dvf_simplified.png" alt="comp_iff_dvf_simplified.png" /> <p><img src="figs/comp_iff_dvf_simplified.png" alt="comp_iff_dvf_simplified.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Obtained transfer functions for DVF and IFF (<a href="./figs/comp_iff_dvf_simplified.png">png</a>, <a href="./figs/comp_iff_dvf_simplified.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 6: </span>Obtained transfer functions for DVF and IFF (<a href="./figs/comp_iff_dvf_simplified.png">png</a>, <a href="./figs/comp_iff_dvf_simplified.pdf">pdf</a>)</p>
</div> </div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -802,22 +797,22 @@ Instead of a pure integrator, let&rsquo;s use a low pass filter with a cut-off f
</div> </div>
</div> </div>
<div id="outline-container-org5e0585d" class="outline-2"> <div id="outline-container-org5e0585d" class="outline-3">
<h2 id="org5e0585d"><span class="section-number-2">8</span> Control using \(x\)</h2> <h3 id="org5e0585d"><span class="section-number-3">2.5</span> Control using \(x\)</h3>
<div class="outline-text-2" id="text-8"> <div class="outline-text-3" id="text-2-5">
</div> </div>
<div id="outline-container-orgfab8395" class="outline-3"> <div id="outline-container-orgfab8395" class="outline-4">
<h3 id="orgfab8395"><span class="section-number-3">8.1</span> Analytical analysis</h3> <h4 id="orgfab8395"><span class="section-number-4">2.5.1</span> Analytical analysis</h4>
<div class="outline-text-3" id="text-8-1"> <div class="outline-text-4" id="text-2-5-1">
<p> <p>
Let&rsquo;s first consider that only the output \(x\) is used for feedback (Figure <a href="#orgd366408">8</a>) Let&rsquo;s first consider that only the output \(x\) is used for feedback (Figure <a href="#orgd366408">7</a>)
</p> </p>
<div id="orgd366408" class="figure"> <div id="orgd366408" class="figure">
<p><img src="figs/nano_station_control_x.png" alt="nano_station_control_x.png" /> <p><img src="figs/nano_station_control_x.png" alt="nano_station_control_x.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Feedback diagram using \(x\)</p> <p><span class="figure-number">Figure 7: </span>Feedback diagram using \(x\)</p>
</div> </div>
<p> <p>
@ -904,9 +899,9 @@ Some implications on the design are summarized on table <a href="#orga5207fc">2<
</div> </div>
</div> </div>
<div id="outline-container-org625e3c2" class="outline-3"> <div id="outline-container-org625e3c2" class="outline-4">
<h3 id="org625e3c2"><span class="section-number-3">8.2</span> Control implementation</h3> <h4 id="org625e3c2"><span class="section-number-4">2.5.2</span> Control implementation</h4>
<div class="outline-text-3" id="text-8-2"> <div class="outline-text-4" id="text-2-5-2">
<p> <p>
Controller for the damped plant using DVF. Controller for the damped plant using DVF.
</p> </p>
@ -927,8 +922,7 @@ K = Hi<span class="org-type">*</span>H<span class="org-type">*</span>(1<span cla
Kpz_dvf = K<span class="org-type">/</span>abs(freqresp(K<span class="org-type">*</span>Gpz_dvf(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb)); Kpz_dvf = K<span class="org-type">/</span>abs(freqresp(K<span class="org-type">*</span>Gpz_dvf(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
Kpz_dvf.InputName = {<span class="org-string">'e'</span>}; Kpz_dvf.InputName = {<span class="org-string">'e'</span>};
Kpz_dvf.OutputName = {<span class="org-string">'F'</span>}; Kpz_dvf.OutputName = {<span class="org-string">'Fi'</span>};
</pre> </pre>
</div> </div>
@ -952,7 +946,7 @@ K = Hi<span class="org-type">*</span>H<span class="org-type">*</span>(1<span cla
Kpz_iff = K<span class="org-type">/</span>abs(freqresp(K<span class="org-type">*</span>Gpz_iff(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb)); Kpz_iff = K<span class="org-type">/</span>abs(freqresp(K<span class="org-type">*</span>Gpz_iff(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
Kpz_iff.InputName = {<span class="org-string">'e'</span>}; Kpz_iff.InputName = {<span class="org-string">'e'</span>};
Kpz_iff.OutputName = {<span class="org-string">'F'</span>}; Kpz_iff.OutputName = {<span class="org-string">'Fi'</span>};
</pre> </pre>
</div> </div>
@ -963,12 +957,12 @@ Loop gain
<div id="org0d0fb80" class="figure"> <div id="org0d0fb80" class="figure">
<p><img src="figs/simple_loop_gain_pz.png" alt="simple_loop_gain_pz.png" /> <p><img src="figs/simple_loop_gain_pz.png" alt="simple_loop_gain_pz.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Loop Gain (<a href="./figs/simple_loop_gain_pz.png">png</a>, <a href="./figs/simple_loop_gain_pz.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 8: </span>Loop Gain (<a href="./figs/simple_loop_gain_pz.png">png</a>, <a href="./figs/simple_loop_gain_pz.pdf">pdf</a>)</p>
</div> </div>
<p> <p>
Let&rsquo;s connect all the systems as shown in Figure <a href="#orgd366408">8</a>. Let&rsquo;s connect all the systems as shown in Figure <a href="#orgd366408">7</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r2 - y'</span>); <pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r2 - y'</span>);
@ -977,71 +971,71 @@ R = [tf(1); tf(1)];
R.InputName = {<span class="org-string">'r'</span>}; R.InputName = {<span class="org-string">'r'</span>};
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>}; R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
Gpz_fb_dvf = connect(Gpz_dvf, Kpz_dvf, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>}); F = [tf(1); tf(1)];
Gpz_fb_iff = connect(Gpz_iff, Kpz_iff, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>}); F.InputName = {<span class="org-string">'Fi'</span>};
F.OutputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fu'</span>};
Gpz_fb_dvf = connect(Gpz_dvf, Kpz_dvf, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
Gpz_fb_iff = connect(Gpz_iff, Kpz_iff, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
</pre> </pre>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org8d34d7f" class="outline-3"> <div id="outline-container-org8d34d7f" class="outline-4">
<h3 id="org8d34d7f"><span class="section-number-3">8.3</span> Results</h3> <h4 id="org8d34d7f"><span class="section-number-4">2.5.3</span> Results</h4>
<div class="outline-text-3" id="text-8-3"> <div class="outline-text-4" id="text-2-5-3">
<div id="org2b4e783" class="figure"> <div id="org2b4e783" class="figure">
<p><img src="figs/simple_hac_lac_results.png" alt="simple_hac_lac_results.png" /> <p><img src="figs/simple_hac_lac_results.png" alt="simple_hac_lac_results.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Obtained closed-loop transfer functions (<a href="./figs/simple_hac_lac_results.png">png</a>, <a href="./figs/simple_hac_lac_results.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 9: </span>Obtained closed-loop transfer functions (<a href="./figs/simple_hac_lac_results.png">png</a>, <a href="./figs/simple_hac_lac_results.pdf">pdf</a>)</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left">Reference Tracking</th>
<th scope="col" class="org-left">Vibration Filtering</th>
<th scope="col" class="org-left">Compliance</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">DVF</td>
<td class="org-left">Similar behavior</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">Better for \(\omega < \omega_\nu\)</td>
</tr>
<tr>
<td class="org-left">IFF</td>
<td class="org-left">Similar behavior</td>
<td class="org-left">Better for \(\omega > \omega_\nu\)</td>
<td class="org-left">&#xa0;</td>
</tr>
</tbody>
</table>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgd2547fd" class="outline-2"> <div id="outline-container-org7c4b4fc" class="outline-2">
<h2 id="orgd2547fd"><span class="section-number-2">9</span> Two degree of freedom control</h2> <h2 id="org7c4b4fc"><span class="section-number-2">3</span> Comparison with the use of a Soft nano-hexapod</h2>
<div class="outline-text-2" id="text-9"> <div class="outline-text-2" id="text-3">
<p>
Let&rsquo;s try to implement the control architecture shown in Figure <a href="#org7ce0167">11</a>.
</p>
<p>
The pre-filter \(K_r\) is added in order to improve the reference tracking performances.
</p>
<div id="org7ce0167" class="figure">
<p><img src="figs/nano_station_control_2dof_x.png" alt="nano_station_control_2dof_x.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Two degrees of freedom feedback control</p>
</div>
<p>
In order to design the pre-filter \(K_r\), the dynamics of the system should be known quite precisely (Dynamics of the nano-hexapod + \(T_\mu\)).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Krpz = inv(Gpz_fb(<span class="org-string">'y'</span>, <span class="org-string">'r'</span>));
Krpz.InputName = {<span class="org-string">'r2'</span>};
Krpz.OutputName = {<span class="org-string">'r3'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r3 - y'</span>);
R = [tf(1); tf(1)];
R.InputName = {<span class="org-string">'r'</span>};
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
Gpz_2dof = connect(Gpz_dvf, Krpz, Kpz, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>});
</pre>
</div>
</div>
</div>
<div id="outline-container-orgab31e9f" class="outline-2">
<h2 id="orgab31e9f"><span class="section-number-2">10</span> Soft nano-hexapod</h2>
<div class="outline-text-2" id="text-10">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">m = 50; <span class="org-comment">% [kg]</span> <pre class="src src-matlab">m = 50; <span class="org-comment">% [kg]</span>
k = 1e3; <span class="org-comment">% [N/m]</span> k = 1e3; <span class="org-comment">% [N/m]</span>
@ -1110,11 +1104,11 @@ H = 1<span class="org-type">/</span>h<span class="org-type">*</span>(1 <span cla
Kvc_dvf = H<span class="org-type">/</span>abs(freqresp(H<span class="org-type">*</span>Gvc_dvf(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb)); Kvc_dvf = H<span class="org-type">/</span>abs(freqresp(H<span class="org-type">*</span>Gvc_dvf(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
Kvc_dvf.InputName = {<span class="org-string">'e'</span>}; Kvc_dvf.InputName = {<span class="org-string">'e'</span>};
Kvc_dvf.OutputName = {<span class="org-string">'F'</span>}; Kvc_dvf.OutputName = {<span class="org-string">'Fi'</span>};
Kvc_iff = H<span class="org-type">/</span>abs(freqresp(H<span class="org-type">*</span>Gvc_iff(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb)); Kvc_iff = H<span class="org-type">/</span>abs(freqresp(H<span class="org-type">*</span>Gvc_iff(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
Kvc_iff.InputName = {<span class="org-string">'e'</span>}; Kvc_iff.InputName = {<span class="org-string">'e'</span>};
Kvc_iff.OutputName = {<span class="org-string">'F'</span>}; Kvc_iff.OutputName = {<span class="org-string">'Fi'</span>};
</pre> </pre>
</div> </div>
@ -1125,8 +1119,13 @@ R = [tf(1); tf(1)];
R.InputName = {<span class="org-string">'r'</span>}; R.InputName = {<span class="org-string">'r'</span>};
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>}; R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
Gvc_fb_dvf = connect(Gvc_dvf, Kvc_dvf, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>}); F = [tf(1); tf(1)];
Gvc_fb_iff = connect(Gvc_iff, Kvc_iff, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>}); F.InputName = {<span class="org-string">'Fi'</span>};
F.OutputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fu'</span>};
Gvc_fb_dvf = connect(Gvc_dvf, Kvc_dvf, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
Gvc_fb_iff = connect(Gvc_iff, Kvc_iff, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
</pre> </pre>
</div> </div>
@ -1134,19 +1133,51 @@ Gvc_fb_iff = connect(Gvc_iff, Kvc_iff, R, Sfb, {<span class="org-string">'r'</sp
<div id="org3817d8a" class="figure"> <div id="org3817d8a" class="figure">
<p><img src="figs/simple_hac_lac_results_soft.png" alt="simple_hac_lac_results_soft.png" /> <p><img src="figs/simple_hac_lac_results_soft.png" alt="simple_hac_lac_results_soft.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Obtained closed-loop transfer functions (<a href="./figs/simple_hac_lac_results_soft.png">png</a>, <a href="./figs/simple_hac_lac_results_soft.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 10: </span>Obtained closed-loop transfer functions (<a href="./figs/simple_hac_lac_results_soft.png">png</a>, <a href="./figs/simple_hac_lac_results_soft.pdf">pdf</a>)</p>
</div>
</div>
</div> </div>
<div id="outline-container-orgcb7520f" class="outline-2"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<h2 id="orgcb7520f"><span class="section-number-2">11</span> Compare Soft and Stiff nano-hexapods</h2>
<div class="outline-text-2" id="text-11">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left">Reference Tracking</th>
<th scope="col" class="org-left">Vibration Filtering</th>
<th scope="col" class="org-left">Compliance</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">DVF</td>
<td class="org-left">Similar behavior</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">Better for \(\omega < \omega_\nu\)</td>
</tr>
<tr>
<td class="org-left">IFF</td>
<td class="org-left">Similar behavior</td>
<td class="org-left">Better for \(\omega > \omega_\nu\)</td>
<td class="org-left">&#xa0;</td>
</tr>
</tbody>
</table>
<div id="org55e0fe2" class="figure"> <div id="org55e0fe2" class="figure">
<p><img src="figs/simple_comp_vc_pz.png" alt="simple_comp_vc_pz.png" /> <p><img src="figs/simple_comp_vc_pz.png" alt="simple_comp_vc_pz.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Comparison of the closed-loop transfer functions for Soft and Stiff nano-hexapod (<a href="./figs/simple_comp_vc_pz.png">png</a>, <a href="./figs/simple_comp_vc_pz.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 11: </span>Comparison of the closed-loop transfer functions for Soft and Stiff nano-hexapod (<a href="./figs/simple_comp_vc_pz.png">png</a>, <a href="./figs/simple_comp_vc_pz.pdf">pdf</a>)</p>
</div> </div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -1173,6 +1204,12 @@ Gvc_fb_iff = connect(Gvc_iff, Kvc_iff, R, Sfb, {<span class="org-string">'r'</sp
<td class="org-center">=</td> <td class="org-center">=</td>
</tr> </tr>
<tr>
<td class="org-left"><b>Ground Motion</b></td>
<td class="org-center">=</td>
<td class="org-center">=</td>
</tr>
<tr> <tr>
<td class="org-left"><b>Vibration Isolation</b></td> <td class="org-left"><b>Vibration Isolation</b></td>
<td class="org-center">+</td> <td class="org-center">+</td>
@ -1190,8 +1227,8 @@ Gvc_fb_iff = connect(Gvc_iff, Kvc_iff, R, Sfb, {<span class="org-string">'r'</sp
</div> </div>
<div id="outline-container-orgc0253c3" class="outline-2"> <div id="outline-container-orgc0253c3" class="outline-2">
<h2 id="orgc0253c3"><span class="section-number-2">12</span> Estimate the level of vibration</h2> <h2 id="orgc0253c3"><span class="section-number-2">4</span> Estimate the level of vibration</h2>
<div class="outline-text-2" id="text-12"> <div class="outline-text-2" id="text-4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">gm = load(<span class="org-string">'./mat/psd_gm.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'psd_gm'</span>); <pre class="src src-matlab">gm = load(<span class="org-string">'./mat/psd_gm.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'psd_gm'</span>);
rz = load(<span class="org-string">'./mat/pxsp_r.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'pxsp_r'</span>); rz = load(<span class="org-string">'./mat/pxsp_r.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'pxsp_r'</span>);
@ -1199,17 +1236,240 @@ tyz = load(<span class="org-string">'./mat/pxz_ty_r.mat'</span>, <span class="or
</pre> </pre>
</div> </div>
<p>
If we note the PSD \(\Gamma\):
\[ \Gamma_y = |G_{\frac{y}{w}}|^2 \Gamma_w + |G_{\frac{y}{x_\mu}}|^2 \Gamma_{x_\mu} \]
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">x_pz = sqrt(abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm)); <pre class="src src-matlab">x_pz = abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm);
x_vc = sqrt(abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm)); x_vc = abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm);
</pre>
</div>
<div id="org858053c" class="figure">
<p><img src="figs/simple_asd_motion_error.png" alt="simple_asd_motion_error.png" />
</p>
<p><span class="figure-number">Figure 12: </span>ASD of the position error due to Ground Motion and Vibration (<a href="./figs/simple_asd_motion_error.png">png</a>, <a href="./figs/simple_asd_motion_error.pdf">pdf</a>)</p>
</div>
<p>
Actuator usage
</p>
<div class="org-src-container">
<pre class="src src-matlab">F_pz = abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'Fu'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'Fu'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm);
F_vc = abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'Fu'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'Fu'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">sqrt(trapz(f, F_pz))
sqrt(trapz(f, F_vc))
</pre>
</div>
<pre class="example">
sqrt(trapz(f, F_pz))
ans =
84.8961762069446
sqrt(trapz(f, F_vc))
ans =
0.0387785981815527
</pre>
</div>
</div>
<div id="outline-container-org764c4a9" class="outline-2">
<h2 id="org764c4a9"><span class="section-number-2">5</span> Requirements on the norm of closed-loop transfer functions</h2>
<div class="outline-text-2" id="text-5">
</div>
<div id="outline-container-org27379f3" class="outline-3">
<h3 id="org27379f3"><span class="section-number-3">5.1</span> Approximation of the ASD of perturbations</h3>
<div class="outline-text-3" id="text-5-1">
<div class="org-src-container">
<pre class="src src-matlab">G_rz = 1e<span class="org-type">-</span>9<span class="org-type">*</span>1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>0.5)<span class="org-type">^</span>2<span class="org-type">*</span>(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>1)<span class="org-type">*</span>(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10)<span class="org-type">*</span>(1<span class="org-type">/</span>((1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>100)<span class="org-type">^</span>2));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">G_gm = 1e<span class="org-type">-</span>8<span class="org-type">*</span>1<span class="org-type">/</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>1)<span class="org-type">^</span>2<span class="org-type">*</span>(1<span class="org-type">/</span>((1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10)<span class="org-type">^</span>3));
</pre> </pre>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org764c4a9" class="outline-2"> <div id="outline-container-orgff3d823" class="outline-3">
<h2 id="org764c4a9"><span class="section-number-2">13</span> Requirements on the norm of closed-loop transfer functions</h2> <h3 id="orgff3d823"><span class="section-number-3">5.2</span> Wanted ASD of outputs</h3>
<div class="outline-text-2" id="text-13"> <div class="outline-text-3" id="text-5-2">
<p>
Wanted ASD of motion error
</p>
<div class="org-src-container">
<pre class="src src-matlab">y_wanted = 100e<span class="org-type">-</span>9; <span class="org-comment">% 10nm rms wanted</span>
y_bw = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100; <span class="org-comment">% bandwidth [rad/s]</span>
G_y = 2<span class="org-type">*</span>y_wanted<span class="org-type">/</span>sqrt(y_bw) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>y_bw<span class="org-type">/</span>10) <span class="org-type">/</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>y_bw);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2))
</pre>
</div>
<pre class="example">
sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, 'Hz'))).^2))
ans =
9.47118350214793e-08
</pre>
</div>
</div>
<div id="outline-container-org8c6b37c" class="outline-3">
<h3 id="org8c6b37c"><span class="section-number-3">5.3</span> Limiting the bandwidth</h3>
<div class="outline-text-3" id="text-5-3">
<div class="org-src-container">
<pre class="src src-matlab">wF = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10;
G_F = 100000<span class="org-type">*</span>(wF <span class="org-type">+</span> s)<span class="org-type">^</span>2;
</pre>
</div>
</div>
</div>
<div id="outline-container-org50054f2" class="outline-3">
<h3 id="org50054f2"><span class="section-number-3">5.4</span> Generalized Weighted plant</h3>
<div class="outline-text-3" id="text-5-4">
<p>
Let&rsquo;s create a generalized weighted plant for controller synthesis.
</p>
<p>
Let&rsquo;s start simple:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left"><b>Symbol</b></th>
<th scope="col" class="org-left"><b>Meaning</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"><b>Exogenous Inputs</b></td>
<td class="org-left">\(x_\mu\)</td>
<td class="org-left">Motion of the $&nu;$-hexapod&rsquo;s base</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Exogenous Outputs</b></td>
<td class="org-left">\(y\)</td>
<td class="org-left">Motion error of the Payload</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Sensed Outputs</b></td>
<td class="org-left">\(y\)</td>
<td class="org-left">Motion error of the Payload</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><b>Control Signals</b></td>
<td class="org-left">\(F\)</td>
<td class="org-left">Actuator Inputs</td>
</tr>
</tbody>
</table>
<p>
Add \(F\) as output.
</p>
<div class="org-src-container">
<pre class="src src-matlab">F = [tf(1); tf(1)];
F.InputName = {<span class="org-string">'Fi'</span>};
F.OutputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fu'</span>};
P_pz = connect(F, Gpz_dvf, {<span class="org-string">'dmu'</span>, <span class="org-string">'Fi'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'Fu'</span>, <span class="org-string">'y'</span>})
P_vc = connect(F, Gvc_dvf, {<span class="org-string">'dmu'</span>, <span class="org-string">'Fi'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'Fu'</span>, <span class="org-string">'y'</span>})
</pre>
</div>
<p>
Normalization.
</p>
<p>
We multiply the plant input by \(G_{rz}\) and the plant output by \(G_y^{-1}\):
</p>
<div class="org-src-container">
<pre class="src src-matlab">P_pz_norm = blkdiag(inv(G_y), inv(G_F), 1)<span class="org-type">*</span>P_pz<span class="org-type">*</span>blkdiag(G_rz, 1);
P_pz_norm.OutputName = {<span class="org-string">'z'</span>, <span class="org-string">'F'</span>, <span class="org-string">'y'</span>};
P_pz_norm.InputName = {<span class="org-string">'w'</span>, <span class="org-string">'u'</span>};
P_vc_norm = blkdiag(inv(G_y), inv(G_F), 1)<span class="org-type">*</span>P_vc<span class="org-type">*</span>blkdiag(G_rz, 1);
P_vc_norm.OutputName = {<span class="org-string">'z'</span>, <span class="org-string">'F'</span>, <span class="org-string">'y'</span>};
P_vc_norm.InputName = {<span class="org-string">'w'</span>, <span class="org-string">'u'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-org949ab66" class="outline-3">
<h3 id="org949ab66"><span class="section-number-3">5.5</span> Synthesis</h3>
<div class="outline-text-3" id="text-5-5">
<div class="org-src-container">
<pre class="src src-matlab">[Kpz_dvf,CL_vc,<span class="org-type">~</span>] = hinfsyn(minreal(P_pz_norm), 1, 1, <span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'LMI'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
Kpz_dvf.InputName = {<span class="org-string">'e'</span>};
Kpz_dvf.OutputName = {<span class="org-string">'Fi'</span>};
[Kvc_dvf,CL_pz,<span class="org-type">~</span>] = hinfsyn(minreal(P_vc_norm), 1, 1, <span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'LMI'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
Kvc_dvf.InputName = {<span class="org-string">'e'</span>};
Kvc_dvf.OutputName = {<span class="org-string">'Fi'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-orgfe970e4" class="outline-3">
<h3 id="orgfe970e4"><span class="section-number-3">5.6</span> Loop Gain</h3>
<div class="outline-text-3" id="text-5-6">
<div class="org-src-container">
<pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r2 - y'</span>);
R = [tf(1); tf(1)];
R.InputName = {<span class="org-string">'r'</span>};
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
F = [tf(1); tf(1)];
F.InputName = {<span class="org-string">'Fi'</span>};
F.OutputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fu'</span>};
Gpz_fb_dvf = connect(Gpz_dvf, <span class="org-type">-</span>Kpz_dvf, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
Gvc_fb_dvf = connect(Gvc_dvf, <span class="org-type">-</span>Kvc_dvf, R, Sfb, F, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'Fu'</span>});
</pre>
</div>
</div>
</div>
<div id="outline-container-org3db77f5" class="outline-3">
<h3 id="org3db77f5"><span class="section-number-3">5.7</span> Results</h3>
</div>
<div id="outline-container-orgb18d7df" class="outline-3">
<h3 id="orgb18d7df"><span class="section-number-3">5.8</span> Requirements</h3>
<div class="outline-text-3" id="text-5-8">
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -1243,9 +1503,10 @@ x_vc = sqrt(abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'y'</span>,
</div> </div>
</div> </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: 2020-03-06 ven. 15:10</p> <p class="date">Created: 2020-03-17 mar. 11:22</p>
</div> </div>
</body> </body>
</html> </html>

17
docs/hac_lac.html
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@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-03-13 ven. 17:39 --> <!-- 2020-03-17 mar. 11:21 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>HAC-LAC applied on the Simscape Model</title> <title>HAC-LAC applied on the Simscape Model</title>
@ -277,9 +277,10 @@ It is then compare to the wanted position of the Sample \(\bm{r}_\mathcal{X}\) i
</p> </p>
<div class="figure"> <div id="org250b69d" class="figure">
<p><img src="figs/hac_lac_control_schematic.png" alt="hac_lac_control_schematic.png" /> <p><img src="figs/hac_lac_control_schematic.png" alt="hac_lac_control_schematic.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>HAC-LAC Control Architecture used for the Control of the NASS</p>
</div> </div>
<div id="outline-container-org9a782be" class="outline-2"> <div id="outline-container-org9a782be" class="outline-2">
@ -519,6 +520,16 @@ And we simulate the system.
<div id="outline-container-org9498b7b" class="outline-2"> <div id="outline-container-org9498b7b" class="outline-2">
<h2 id="org9498b7b"><span class="section-number-2">5</span> Results</h2> <h2 id="org9498b7b"><span class="section-number-2">5</span> Results</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<p>
Let&rsquo;s load the simulation when no control is applied.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/experiment_tomography.mat'</span>, <span class="org-string">'tomo_align_dist'</span>);
t = tomo_align_dist.t;
MTr = tomo_align_dist.MTr;
</pre>
</div>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/tomo_exp_hac_lac.mat'</span>, <span class="org-string">'simout'</span>); <pre class="src src-matlab">load(<span class="org-string">'./mat/tomo_exp_hac_lac.mat'</span>, <span class="org-string">'simout'</span>);
</pre> </pre>
@ -528,7 +539,7 @@ And we simulate the system.
</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: 2020-03-13 ven. 17:39</p> <p class="date">Created: 2020-03-17 mar. 11:21</p>
</div> </div>
</body> </body>
</html> </html>

846
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2
org/control_requirements.org
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@ -160,6 +160,8 @@ The nano-hexapod can thus be represented as in Figure [[fig:nano_station_inputs_
#+name: fig:nano_station_inputs_outputs #+name: fig:nano_station_inputs_outputs
#+caption: Block representation of the nano-hexapod #+caption: Block representation of the nano-hexapod
#+RESULTS:
[[file:figs/nano_station_inputs_outputs.png]]
** How to include Ground Motion in the model? ** How to include Ground Motion in the model?
What we measure is not the absolute motion $x$, but the relative motion $x - w$ where $w$ is the motion of the granite. What we measure is not the absolute motion $x$, but the relative motion $x - w$ where $w$ is the motion of the granite.

19
org/hac_lac.org
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@ -89,9 +89,28 @@ It is then compare to the wanted position of the Sample $\bm{r}_\mathcal{X}$ in
\end{tikzpicture} \end{tikzpicture}
#+end_src #+end_src
#+name: fig:hac_lac_control_schematic
#+caption: HAC-LAC Control Architecture used for the Control of the NASS
#+RESULTS: #+RESULTS:
[[file:figs/hac_lac_control_schematic.png]] [[file:figs/hac_lac_control_schematic.png]]
* Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no
simulinkproject('../');
#+end_src
#+begin_src matlab
open('nass_model.slx')
#+end_src
* Initialization * Initialization
We initialize all the stages with the default parameters. We initialize all the stages with the default parameters.
#+begin_src matlab #+begin_src matlab