Lot of works on simscape, IFF and DVF
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matlab/figs-tikz
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matlab/figs-tikz
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@@ -232,11 +30,599 @@
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Matlab Computation</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org2941be5">1. System Description and Analysis</a>
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<ul>
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<li><a href="#org4222b45">1.1. System description</a></li>
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<li><a href="#org80f8141">1.2. Equations</a></li>
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<li><a href="#orgbb90994">1.3. Numerical Values</a></li>
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<li><a href="#org3d04be0">1.4. Campbell Diagram</a></li>
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<li><a href="#org8144360">1.5. Simscape Model</a></li>
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<li><a href="#orga371e09">1.6. Comparison with the model</a></li>
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</ul>
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</li>
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<li><a href="#org912df6c">2. Integral Force Feedback</a>
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<ul>
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<li><a href="#orge201c80">2.1. Numerical Values</a></li>
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<li><a href="#orgbc26950">2.2. Equations</a></li>
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<li><a href="#org92d4b18">2.3. Poles and Zeros</a></li>
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<li><a href="#orgbb768e3">2.4. Simscape Model</a></li>
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<li><a href="#orgdf8d677">2.5. IFF Plant</a></li>
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<li><a href="#org3f264f4">2.6. Loop Gain</a></li>
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<li><a href="#org7c25fbe">2.7. Root Locus</a></li>
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<li><a href="#org82271cf">2.8. Modified IFF</a>
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<ul>
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<li><a href="#org60bb927">2.8.1. Control Law</a></li>
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<li><a href="#org6850379">2.8.2. Loop Gain</a></li>
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<li><a href="#org7314eea">2.8.3. Root Locus</a></li>
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<li><a href="#orgb4641d2">2.8.4. Optimal Gain</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orgbdc5c77">3. Direct Velocity Feedback</a>
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<ul>
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<li><a href="#org3f7448e">3.1. Equations</a></li>
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<li><a href="#org84263dc">3.2. Numerical Values</a></li>
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<li><a href="#orgbdad8de">3.3. Simscape Model</a></li>
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<li><a href="#orge253615">3.4. DVF Plant</a></li>
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<li><a href="#orgd61ece4">3.5. Loop Gain</a></li>
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<li><a href="#orgce2b090">3.6. Root Locus</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<p>
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Notations:
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</p>
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<ul class="org-ul">
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<li>\(k\): Actuator’s Stiffness [N/m]</li>
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<li>\(c\): Actuator’s Damping [N/(m/s)]</li>
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<li>\(m\): Payload’s mass [kg]</li>
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<li>\(\omega_0 = \sqrt{\frac{k}{m}}\): Resonance of the (non-rotating) mass-spring system [rad/s]</li>
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<li>\(\Omega = \dot{\theta}\): rotation speed [rad/s]</li>
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</ul>
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<tr>
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<th scope="col" class="org-left"> </th>
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<th scope="col" class="org-left">Mathematical Notation</th>
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<th scope="col" class="org-left">Matlab</th>
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<th scope="col" class="org-left">Unit</th>
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</tr>
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<tbody>
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<tr>
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<td class="org-left">Actuator Stiffness</td>
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<td class="org-left">\(k\)</td>
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<td class="org-left"><code>k</code></td>
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<td class="org-left">N/m</td>
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</tr>
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<tr>
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<td class="org-left">Actuator Damping</td>
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<td class="org-left">\(c\)</td>
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<td class="org-left"><code>c</code></td>
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<td class="org-left">N/(m/s)</td>
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</tr>
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</tbody>
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</table>
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<div id="outline-container-org2941be5" class="outline-2">
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<h2 id="org2941be5"><span class="section-number-2">1</span> System Description and Analysis</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org72741bf"></a>
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</p>
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</div>
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<div id="outline-container-org4222b45" class="outline-3">
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<h3 id="org4222b45"><span class="section-number-3">1.1</span> System description</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#org3ee83a5">1</a>).
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</p>
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<p>
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||||
<a href="figs/rotating_xy_platform.pdf">figs/rotating_xy_platform.pdf</a>
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</p>
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<p>
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The control inputs are the forces applied by the actuators of the translation stage (\(F_u\) and \(F_v\)).
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As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \(u\) and \(v\).
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</p>
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<p>
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The measurement is either the \(x-y\) displacement of the object located on top of the translation stage or the \(u-v\) displacement of the sample with respect to a fixed reference frame.
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</p>
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</div>
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</div>
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<div id="outline-container-org80f8141" class="outline-3">
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<h3 id="org80f8141"><span class="section-number-3">1.2</span> Equations</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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Based on the Figure <a href="#org3ee83a5">1</a>.
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</p>
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\begin{equation}
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\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
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\frac{\frac{1}{k}}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
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||||
\begin{bmatrix}
|
||||
\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} & 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \\
|
||||
-2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} & \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\end{equation}
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbb90994" class="outline-3">
|
||||
<h3 id="orgbb90994"><span class="section-number-3">1.3</span> Numerical Values</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">k = 2;
|
||||
m = 1;
|
||||
c = 0.05;
|
||||
xi = c/(2*sqrt(k*m));
|
||||
|
||||
w0 = sqrt(k/m);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3d04be0" class="outline-3">
|
||||
<h3 id="org3d04be0"><span class="section-number-3">1.4</span> Campbell Diagram</h3>
|
||||
<div class="outline-text-3" id="text-1-4">
|
||||
<p>
|
||||
Compute the poles
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">wrs = linspace(0, 2, 51); % [rad/s]
|
||||
|
||||
polesvc = zeros(4, length(wrs));
|
||||
|
||||
for i = 1:length(wrs)
|
||||
wr = wrs(i);
|
||||
polei = pole(1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wrs(i)^2)/(w0^2))^2 + (2*wrs(i)*s/(w0^2))^2));
|
||||
[~, i_sort] = sort(imag(polei));
|
||||
polesvc(:, i) = polei(i_sort);
|
||||
end
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8144360" class="outline-3">
|
||||
<h3 id="org8144360"><span class="section-number-3">1.5</span> Simscape Model</h3>
|
||||
<div class="outline-text-3" id="text-1-5">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">open('rotating_frame.slx');
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">k = 2;
|
||||
m = 1;
|
||||
c = 0.05;
|
||||
xi = c/(2*sqrt(k*m));
|
||||
|
||||
w0 = sqrt(k/m);
|
||||
wr = 0.1;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Fu'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Fv'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Tuv'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Tuv'], 2, 'openoutput'); io_i = io_i + 1;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">G = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G.InputName = {'Fu', 'Fv'};
|
||||
G.OutputName = {'du', 'dv'};
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga371e09" class="outline-3">
|
||||
<h3 id="orga371e09"><span class="section-number-3">1.6</span> Comparison with the model</h3>
|
||||
<div class="outline-text-3" id="text-1-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ga = (1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2))^2) * ...
|
||||
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2), 2*wr*s/(w0^2) ; ...
|
||||
-2*wr*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2)];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">figure; bode(G, 'k-', Ga, 'r--')
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org912df6c" class="outline-2">
|
||||
<h2 id="org912df6c"><span class="section-number-2">2</span> Integral Force Feedback</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
</div>
|
||||
<div id="outline-container-orge201c80" class="outline-3">
|
||||
<h3 id="orge201c80"><span class="section-number-3">2.1</span> Numerical Values</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">k = 1;
|
||||
m = 1;
|
||||
xi = 0.01;
|
||||
c = 2*xi*sqrt(k*m);
|
||||
w0 = sqrt(k/m);
|
||||
wr = 0.1*w0;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbc26950" class="outline-3">
|
||||
<h3 id="orgbc26950"><span class="section-number-3">2.2</span> Equations</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
The sensed forces are equal to:
|
||||
</p>
|
||||
\begin{equation}
|
||||
\begin{bmatrix} F_{um} \\ F_{vm} \end{bmatrix} =
|
||||
\begin{bmatrix}
|
||||
1 & 0 \\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k)
|
||||
\begin{bmatrix} d_u \\ d_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
<div class="important">
|
||||
\begin{equation}
|
||||
\begin{bmatrix} F_{um} \\ F_{vm} \end{bmatrix} =
|
||||
\frac{1}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
|
||||
\begin{bmatrix}
|
||||
(\frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2}) (\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2}) + (2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0})^2 & - (2 \xi \frac{s}{\omega_0} + 1) 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \\
|
||||
(2 \xi \frac{s}{\omega_0} + 1) 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} & (\frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2}) (\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2}) + (2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0})^2 \\
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Giffa = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - wr^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2)) + (2*wr*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*wr*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*wr*s/(w0^2), (s^2/w0^2 - wr^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))+ (2*wr*s/(w0^2))^2];
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org92d4b18" class="outline-3">
|
||||
<h3 id="org92d4b18"><span class="section-number-3">2.3</span> Poles and Zeros</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">syms wr w0 xi positive
|
||||
assumealso(w0 > wr)
|
||||
syms x
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">z = (x^2/w0^2 - wr^2/w0^2)*((x^2)/(w0^2) + 1 - (wr^2)/(w0^2)) + (2*wr*x/(w0^2))^2 == 0
|
||||
p = ((x^2)/(w0^2) + 1 - (wr^2)/(w0^2))^2 + (2*wr*x/(w0^2))^2 == 0
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">solve(p, x)
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">solve(z, x)
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The zeros are the roots of:
|
||||
</p>
|
||||
\begin{equation}
|
||||
\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 = 0
|
||||
\end{equation}
|
||||
|
||||
<p>
|
||||
Poles (without damping)
|
||||
</p>
|
||||
\begin{equation}
|
||||
\left(\begin{array}{c} -w_{0}\,1{}\mathrm{i}-\mathrm{wr}\,1{}\mathrm{i}\\ -w_{0}\,1{}\mathrm{i}+\mathrm{wr}\,1{}\mathrm{i}\\ w_{0}\,1{}\mathrm{i}-\mathrm{wr}\,1{}\mathrm{i}\\ w_{0}\,1{}\mathrm{i}+\mathrm{wr}\,1{}\mathrm{i} \end{array}\right)
|
||||
\end{equation}
|
||||
|
||||
<p>
|
||||
Zeros (without damping)
|
||||
</p>
|
||||
\begin{equation}
|
||||
\left(\begin{array}{c} -\sqrt{-\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{wr}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{wr}}^2}\\ -\sqrt{\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{wr}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{wr}}^2}\\ \sqrt{-\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{wr}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{wr}}^2}\\ \sqrt{\frac{w_{0}\,\sqrt{{w_{0}}^2+8\,{\mathrm{wr}}^2}}{2}-\frac{{w_{0}}^2}{2}-{\mathrm{wr}}^2} \end{array}\right)
|
||||
\end{equation}
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbb768e3" class="outline-3">
|
||||
<h3 id="orgbb768e3"><span class="section-number-3">2.4</span> Simscape Model</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Fu'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Fv'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Tuv'], 5, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Tuv'], 6, 'openoutput'); io_i = io_i + 1;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Giff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
Giff.InputName = {'Fu', 'Fv'};
|
||||
Giff.OutputName = {'Fmu', 'Fmv'};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">figure; bode(Giff, Giffa)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdf8d677" class="outline-3">
|
||||
<h3 id="orgdf8d677"><span class="section-number-3">2.5</span> IFF Plant</h3>
|
||||
</div>
|
||||
<div id="outline-container-org3f264f4" class="outline-3">
|
||||
<h3 id="org3f264f4"><span class="section-number-3">2.6</span> Loop Gain</h3>
|
||||
<div class="outline-text-3" id="text-2-6">
|
||||
<p>
|
||||
Let’s take \(\Omega = \frac{\omega_0}{10}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">ws = 0.1*w0;
|
||||
G_iff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2))^2 + (2*ws*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - ws^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2)) + (2*ws*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*ws*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*ws*s/(w0^2), (s^2/w0^2 - ws^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2))+ (2*ws*s/(w0^2))^2];
|
||||
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7c25fbe" class="outline-3">
|
||||
<h3 id="org7c25fbe"><span class="section-number-3">2.7</span> Root Locus</h3>
|
||||
</div>
|
||||
<div id="outline-container-org82271cf" class="outline-3">
|
||||
<h3 id="org82271cf"><span class="section-number-3">2.8</span> Modified IFF</h3>
|
||||
<div class="outline-text-3" id="text-2-8">
|
||||
</div>
|
||||
<div id="outline-container-org60bb927" class="outline-4">
|
||||
<h4 id="org60bb927"><span class="section-number-4">2.8.1</span> Control Law</h4>
|
||||
<div class="outline-text-4" id="text-2-8-1">
|
||||
<p>
|
||||
Let’s take the integral feedback controller as a low pass filter (pseudo integrator):
|
||||
</p>
|
||||
\begin{equation}
|
||||
K_{\text{IFF}} = \begin{bmatrix}
|
||||
g\frac{\omega_i}{\omega_i + s} & 0 \\
|
||||
0 & g\frac{\omega_i}{\omega_i + s}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">xi = 0.005;
|
||||
w0 = 1;
|
||||
ws = 0.1*w0;
|
||||
|
||||
G_iff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2))^2 + (2*ws*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - ws^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2)) + (2*ws*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*ws*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*ws*s/(w0^2), (s^2/w0^2 - ws^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (ws^2)/(w0^2))+ (2*ws*s/(w0^2))^2];
|
||||
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">g = 100;
|
||||
wi = ws;
|
||||
|
||||
K_iff = (g/(1+s/wi))*eye(2);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6850379" class="outline-4">
|
||||
<h4 id="org6850379"><span class="section-number-4">2.8.2</span> Loop Gain</h4>
|
||||
</div>
|
||||
<div id="outline-container-org7314eea" class="outline-4">
|
||||
<h4 id="org7314eea"><span class="section-number-4">2.8.3</span> Root Locus</h4>
|
||||
</div>
|
||||
<div id="outline-container-orgb4641d2" class="outline-4">
|
||||
<h4 id="orgb4641d2"><span class="section-number-4">2.8.4</span> Optimal Gain</h4>
|
||||
<div class="outline-text-4" id="text-2-8-4">
|
||||
<p>
|
||||
The DC gain for Giff is (for \(\Omega < \omega_0\)):
|
||||
</p>
|
||||
\begin{equation}
|
||||
G_{\text{IFF}}(\omega = 0) = \frac{1}{1 - \frac{{\omega_0}^2}{\Omega^2}} \begin{bmatrix}
|
||||
1 & 0 \\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
<p>
|
||||
The maximum gain where is system is still stable is
|
||||
</p>
|
||||
\begin{equation}
|
||||
g_\text{max} = \frac{{\omega_0}^2}{\Omega^2} - 1
|
||||
\end{equation}
|
||||
|
||||
|
||||
<p>
|
||||
Let’s find the gain that maximize the simultaneous damping of the two modes.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">K_opt = (opt_gain/(1+s/wi))*eye(2);
|
||||
|
||||
G_cl = feedback(G_iff, K_opt);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbdc5c77" class="outline-2">
|
||||
<h2 id="orgbdc5c77"><span class="section-number-2">3</span> Direct Velocity Feedback</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
</div>
|
||||
<div id="outline-container-org3f7448e" class="outline-3">
|
||||
<h3 id="org3f7448e"><span class="section-number-3">3.1</span> Equations</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
The sensed relative velocity are equal to:
|
||||
</p>
|
||||
|
||||
<div class="important">
|
||||
\begin{equation}
|
||||
\begin{bmatrix} \dot{d}_u \\ \dot{d}_v \end{bmatrix} =
|
||||
\frac{s \frac{1}{k}}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
|
||||
\begin{bmatrix}
|
||||
\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} & 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \\
|
||||
-2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} & \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org84263dc" class="outline-3">
|
||||
<h3 id="org84263dc"><span class="section-number-3">3.2</span> Numerical Values</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">k = 1;
|
||||
m = 1;
|
||||
c = 0.05;
|
||||
xi = c/(2*sqrt(k*m));
|
||||
|
||||
w0 = sqrt(k/m);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ga = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2))^2) * ...
|
||||
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2), 2*wr*s/(w0^2) ; ...
|
||||
-2*wr*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2)];
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbdad8de" class="outline-3">
|
||||
<h3 id="orgbdad8de"><span class="section-number-3">3.3</span> Simscape Model</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Fu'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Fv'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Tuv'], 3, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Tuv'], 4, 'openoutput'); io_i = io_i + 1;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">G = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G.InputName = {'Fu', 'Fv'};
|
||||
G.OutputName = {'Vu', 'Vv'};
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge253615" class="outline-3">
|
||||
<h3 id="orge253615"><span class="section-number-3">3.4</span> DVF Plant</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">figure; bode(G, Ga)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd61ece4" class="outline-3">
|
||||
<h3 id="orgd61ece4"><span class="section-number-3">3.5</span> Loop Gain</h3>
|
||||
<div class="outline-text-3" id="text-3-5">
|
||||
<p>
|
||||
Let’s take \(\Omega = \frac{\omega_0}{10}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">ws = 0.1*w0;
|
||||
G_dvf = (s/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2))^2 + (2*wr*s/(w0^2))^2) * ...
|
||||
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2), 2*wr*s/(w0^2) ; ...
|
||||
-2*wr*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (wr^2)/(w0^2)];
|
||||
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgce2b090" class="outline-3">
|
||||
<h3 id="orgce2b090"><span class="section-number-3">3.6</span> Root Locus</h3>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-03-16 lun. 10:52</p>
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2020-06-10 mer. 10:17</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
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matlab/index.org
1545
matlab/index.org
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||||
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matlab/slprj/sim/varcache/rotating_frame/varInfo.mat
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matlab/slprj/sim/varcache/rotating_frame/varInfo.mat
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Reference in New Issue
Block a user