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Finish analysis of required bw as a function of K

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Thomas Dehaeze
2020-04-08 12:12:41 +02:00
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@@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-04-07 mar. 19:33 -->
<!-- 2020-04-08 mer. 12:12 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Determination of the optimal nano-hexapod&rsquo;s stiffness for reducing the effect of disturbances</title>
@@ -257,7 +257,7 @@
<li><a href="#org78dd34d">2.3. Sensitivity to Stages vibration (Filtering)</a></li>
<li><a href="#orgd4ea2f4">2.4. Effect of Ground motion (Transmissibility).</a></li>
<li><a href="#org0448746">2.5. Direct Forces (Compliance).</a></li>
<li><a href="#orga29f90b">2.6. Conclusion</a></li>
<li><a href="#orge784867">2.6. Conclusion</a></li>
</ul>
</li>
<li><a href="#org6527e58">3. Effect of granite stiffness</a>
@@ -270,7 +270,7 @@
</li>
<li><a href="#org9215f81">3.2. Soft Granite</a></li>
<li><a href="#org8878556">3.3. Effect of the Granite transfer function</a></li>
<li><a href="#org6362e01">3.4. Conclusion</a></li>
<li><a href="#org4b4fa39">3.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
@@ -278,7 +278,7 @@
<li><a href="#org6bd588f">4.1. Noise Budgeting - Theory</a></li>
<li><a href="#orgcc86f59">4.2. Power Spectral Densities</a></li>
<li><a href="#orgef96b89">4.3. Cumulative Amplitude Spectrum</a></li>
<li><a href="#org2852fc6">4.4. Conclusion</a></li>
<li><a href="#org2b9df24">4.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org34c0f38">5. Closed Loop Budget Error</a>
@@ -287,7 +287,7 @@
<li><a href="#orgf2d36a1">5.2. Reduction thanks to feedback - Required bandwidth</a></li>
</ul>
</li>
<li><a href="#orge784867">6. Conclusion</a></li>
<li><a href="#orgbf0fb63">6. Conclusion</a></li>
</ul>
</div>
</div>
@@ -497,8 +497,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
</div>
</div>
<div id="outline-container-orga29f90b" class="outline-3">
<h3 id="orga29f90b"><span class="section-number-3">2.6</span> Conclusion</h3>
<div id="outline-container-orge784867" class="outline-3">
<h3 id="orge784867"><span class="section-number-3">2.6</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-6">
<div class="important">
<p>
@@ -678,8 +678,8 @@ From Figures <a href="#orgc4c14fb">11</a> and <a href="#org533cc4b">12</a>, we s
</div>
</div>
<div id="outline-container-org6362e01" class="outline-3">
<h3 id="org6362e01"><span class="section-number-3">3.4</span> Conclusion</h3>
<div id="outline-container-org4b4fa39" class="outline-3">
<h3 id="org4b4fa39"><span class="section-number-3">3.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-4">
<div class="important">
<p>
@@ -900,8 +900,8 @@ xlim([1, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-typ
</div>
</div>
<div id="outline-container-org2852fc6" class="outline-3">
<h3 id="org2852fc6"><span class="section-number-3">4.4</span> Conclusion</h3>
<div id="outline-container-org2b9df24" class="outline-3">
<h3 id="org2b9df24"><span class="section-number-3">4.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-4-4">
<div class="important">
<p>
@@ -919,6 +919,9 @@ From Figure <a href="#orgf6888f0">20</a>, we can see that a soft nano-hexapod \(
<p>
<a id="orgd3503fb"></a>
</p>
<p>
From the total open-loop power spectral density of the sample&rsquo;s motion error, we can estimate what is the required control bandwidth for the sample&rsquo;s motion error to be reduced down to \(10nm\).
</p>
</div>
<div id="outline-container-orgdfef0eb" class="outline-3">
<h3 id="orgdfef0eb"><span class="section-number-3">5.1</span> Approximation of the effect of feedback on the motion error</h3>
@@ -940,10 +943,13 @@ The reduction of the impact of \(d\) on \(y\) thanks to feedback is described by
\begin{equation}
\frac{y}{d} = \frac{G_d}{1 + KG}
\end{equation}
<p>
The transfer functions corresponding to \(G_d\) are those identified in Section <a href="#orgf9e4300">2</a>.
</p>
<p>
As a first approximation, we can consider that the controller is designed in such a way that the loop gain \(KG\) is a pure integrator:
As a first approximation, we can consider that the controller \(K\) is designed in such a way that the loop gain \(KG\) is a pure integrator:
\[ L_1(s) = K_1(s) G(s) = \frac{\omega_c}{s} \]
where \(\omega_c\) is the crossover frequency.
</p>
@@ -953,107 +959,131 @@ where \(\omega_c\) is the crossover frequency.
We may then consider another controller in such a way that the loop gain corresponds to a double integrator with a lead centered with the crossover frequency \(\omega_c\):
\[ L_2(s) = K_2(s) G(s) = \left( \frac{\omega_c}{s} \right)^2 \cdot \frac{1 + \frac{s}{\omega_c/2}}{1 + \frac{s}{2\omega_c}} \]
</p>
<p>
In the next section, we see how the power spectral density of \(y\) is reduced as a function of the control bandwidth \(\omega_c\).
This will help to determine what is the approximate control bandwidth required such that the rms value of \(y\) is below \(10nm\).
</p>
</div>
</div>
<div id="outline-container-orgf2d36a1" class="outline-3">
<h3 id="orgf2d36a1"><span class="section-number-3">5.2</span> Reduction thanks to feedback - Required bandwidth</h3>
<div class="outline-text-3" id="text-5-2">
<div class="org-src-container">
<pre class="src src-matlab">wc = 1<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
xic = 0.5;
<p>
Let&rsquo;s first see how does the Cumulative Amplitude Spectrum of the sample&rsquo;s motion error is modified by the control.
</p>
S = (s<span class="org-type">/</span>wc)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc);
<p>
In Figure <a href="#orgcbef465">22</a> is shown the Cumulative Amplitude Spectrum of the sample&rsquo;s motion error in Open-Loop and in Closed Loop for several control bandwidth (from 1Hz to 200Hz) and 4 different nano-hexapod stiffnesses.
The controller used in this simulation is \(K_1\). The loop gain is then a pure integrator.
</p>
bodeFig({S}, logspace(<span class="org-type">-</span>1,2,1000))
</pre>
<p>
In Figure <a href="#orgd677910">23</a> is shown the expected RMS value of the sample&rsquo;s position error as a function of the control bandwidth, both for \(K_1\) (left plot) and \(K_2\) (right plot).
As expected, it is shown that \(K_2\) performs better than \(K_1\).
This Figure tells us how much control bandwidth is required to attain a certain level of performance, and that for all the considered nano-hexapod stiffnesses.
</p>
<p>
The obtained required bandwidth (approximate upper and lower bounds) to obtained a sample&rsquo;s motion error less than 10nm rms are gathered in Table <a href="#org5ab4860">1</a>.
</p>
<div id="orgcbef465" class="figure">
<p><img src="figs/opt_stiff_cas_closed_loop.png" alt="opt_stiff_cas_closed_loop.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Cumulative Amplitude Spectrum of the sample&rsquo;s motion error in Open-Loop and in Closed Loop for several control bandwidth and 4 different nano-hexapod stiffnesses (<a href="./figs/opt_stiff_cas_closed_loop.png">png</a>, <a href="./figs/opt_stiff_cas_closed_loop.pdf">pdf</a>)</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">wc = [1, 5, 10, 20, 50, 100, 200];
S1 = {zeros(length(wc), 1)};
S2 = {zeros(length(wc), 1)};
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(wc)</span>
L = (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s; <span class="org-comment">% Simple integrator</span>
S1{<span class="org-constant">j</span>} = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> L);
L = ((2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s)<span class="org-type">^</span>2<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>wc(<span class="org-constant">j</span>)<span class="org-type">/</span>2))<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>wc(<span class="org-constant">j</span>)<span class="org-type">*</span>2));
S2{<span class="org-constant">j</span>} = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> L);
<span class="org-keyword">end</span>
</pre>
<div id="orgd677910" class="figure">
<p><img src="figs/opt_stiff_req_bandwidth_K1_K2.png" alt="opt_stiff_req_bandwidth_K1_K2.png" />
</p>
<p><span class="figure-number">Figure 23: </span>Expected RMS value of the sample&rsquo;s motion error \(E_z\) as a function of the control bandwidth when using \(K_1\) and \(K_2\) (<a href="./figs/opt_stiff_req_bandwidth_K1_K2.png">png</a>, <a href="./figs/opt_stiff_req_bandwidth_K1_K2.pdf">pdf</a>)</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">freqs = dist_f.f;
<table id="org5ab4860" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Approximate required control bandwidth such that the motion error is below \(10nm\)</caption>
<span class="org-type">figure</span>;
hold on;
<span class="org-constant">i</span> = 6;
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(wc)</span>
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>,<span class="org-string">'ColorOrderIndex'</span>,<span class="org-constant">j</span>);
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(abs(squeeze(freqresp(S1{<span class="org-constant">j</span>}, freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>))))), <span class="org-string">'-'</span>, ...
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$\\omega_c = %.0f$ [Hz]'</span>, wc(<span class="org-constant">j</span>)));
<span class="org-keyword">end</span>
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>))))), <span class="org-string">'k-'</span>, ...
<span class="org-string">'DisplayName'</span>, <span class="org-string">'Open-Loop'</span>);
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
xlim([0.5, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
</pre>
</div>
<colgroup>
<col class="org-left" />
<div class="org-src-container">
<pre class="src src-matlab">wc = logspace(0, 3, 100);
<col class="org-right" />
Dz1_rms = zeros(length(Ks), length(wc));
Dz2_rms = zeros(length(Ks), length(wc));
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(wc)</span>
L = (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s;
Dz1_rms(<span class="org-constant">i</span>, <span class="org-constant">j</span>) = sqrt(trapz(freqs, abs(squeeze(freqresp(1<span class="org-type">/</span>(1 <span class="org-type">+</span> L), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>)));
<col class="org-right" />
L = ((2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s)<span class="org-type">^</span>2<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>wc(<span class="org-constant">j</span>)<span class="org-type">/</span>2))<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>wc(<span class="org-constant">j</span>)<span class="org-type">*</span>2));
Dz2_rms(<span class="org-constant">i</span>, <span class="org-constant">j</span>) = sqrt(trapz(freqs, abs(squeeze(freqresp(1<span class="org-type">/</span>(1 <span class="org-type">+</span> L), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>)));
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
</div>
<col class="org-right" />
<div class="org-src-container">
<pre class="src src-matlab">freqs = dist_f.f;
<col class="org-right" />
<span class="org-type">figure</span>;
hold on;
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>,<span class="org-string">'ColorOrderIndex'</span>,<span class="org-constant">i</span>);
plot(wc, Dz1_rms(<span class="org-constant">i</span>, <span class="org-type">:</span>), <span class="org-string">'-'</span>, ...
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)))
<col class="org-right" />
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>,<span class="org-string">'ColorOrderIndex'</span>,<span class="org-constant">i</span>);
plot(wc, Dz2_rms(<span class="org-constant">i</span>, <span class="org-type">:</span>), <span class="org-string">'--'</span>, ...
<span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>)
<span class="org-keyword">end</span>
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Control Bandwidth [Hz]'</span>); ylabel(<span class="org-string">'$E_z\ [m, rms]$'</span>)
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
xlim([1, 500]);
</pre>
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Nano-Hexapod stiffness [N/m]</th>
<th scope="col" class="org-right">10<sup>3</sup></th>
<th scope="col" class="org-right">10<sup>4</sup></th>
<th scope="col" class="org-right">10<sup>5</sup></th>
<th scope="col" class="org-right">10<sup>6</sup></th>
<th scope="col" class="org-right">10<sup>7</sup></th>
<th scope="col" class="org-right">10<sup>8</sup></th>
<th scope="col" class="org-right">10<sup>9</sup></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Required wc with L1 [Hz]</td>
<td class="org-right">152</td>
<td class="org-right">305</td>
<td class="org-right">1000</td>
<td class="org-right">870</td>
<td class="org-right">933</td>
<td class="org-right">870</td>
<td class="org-right">870</td>
</tr>
<tr>
<td class="org-left">Required wc with L2 [Hz]</td>
<td class="org-right">57</td>
<td class="org-right">66</td>
<td class="org-right">152</td>
<td class="org-right">152</td>
<td class="org-right">248</td>
<td class="org-right">266</td>
<td class="org-right">248</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="outline-container-orgbf0fb63" class="outline-2">
<h2 id="orgbf0fb63"><span class="section-number-2">6</span> Conclusion</h2>
<div class="outline-text-2" id="text-6">
<div class="important">
<p>
From Figure <a href="#orgd677910">23</a> and Table <a href="#org5ab4860">1</a>, we can clearly see three different results depending on the nano-hexapod stiffness:
</p>
<ul class="org-ul">
<li>For a soft nano-hexapod (\(k < 10^4\ [N/m]\)), the required bandwidth is \(\omega_c \approx 50-100\ Hz\)</li>
<li>For a nano-hexapods with \(10^5 < k < 10^6\ [N/m]\)), the required bandwidth is \(\omega_c \approx 150-300\ Hz\)</li>
<li>For a stiff nano-hexapods (\(k > 10^7\ [N/m]\)), the required bandwidth is \(\omega_c \approx 250-500\ Hz\)</li>
</ul>
</div>
</div>
<div id="outline-container-orge784867" class="outline-2">
<h2 id="orge784867"><span class="section-number-2">6</span> Conclusion</h2>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-04-07 mar. 19:33</p>
<p class="date">Created: 2020-04-08 mer. 12:12</p>
</div>
</body>
</html>