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Add cambell plot. Add plot about simscape analysis

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tdehaeze committed 2019-01-24 14:07:14 +01:00
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2019-01-23 mer. 15:21 -->
<!-- 2019-01-24 jeu. 14:05 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Control in a rotating frame</title>
@@ -204,7 +204,7 @@
@licstart The following is the entire license notice for the
JavaScript code in this tag.
Copyright (C) 2012-2018 Free Software Foundation, Inc.
Copyright (C) 2012-2019 Free Software Foundation, Inc.
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
@@ -275,75 +275,86 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org9988e59">1. Introduction</a></li>
<li><a href="#orgfd01695">2. System</a>
<li><a href="#orgf151bb1">1. Introduction</a></li>
<li><a href="#org268da4c">2. System Description and Analysis</a>
<ul>
<li><a href="#orged1d203">2.1. System description</a></li>
<li><a href="#orgbcc53a3">2.2. Equations</a></li>
<li><a href="#org28e82a2">2.3. Numerical Values for the NASS</a></li>
<li><a href="#orgd1ccc62">2.4. Euler and Coriolis forces - Numerical Result</a></li>
<li><a href="#org8db03ec">2.5. Negative Spring Effect - Numerical Result</a></li>
<li><a href="#org7e45369">2.6. Limitations due to coupling</a>
<li><a href="#orge7664a5">2.1. System description</a></li>
<li><a href="#org60bfb2d">2.2. Equations</a></li>
<li><a href="#orgf397277">2.3. Numerical Values for the NASS</a></li>
<li><a href="#org7aa02cb">2.4. Euler and Coriolis forces - Numerical Result</a></li>
<li><a href="#org9672480">2.5. Negative Spring Effect - Numerical Result</a></li>
<li><a href="#org57a38eb">2.6. Limitations due to coupling</a>
<ul>
<li><a href="#orgc898fca">2.6.1. Numerical Analysis</a></li>
<li><a href="#org6274181">2.6.1. Numerical Analysis</a></li>
</ul>
</li>
<li><a href="#org87cd267">2.7. Limitations due to negative stiffness effect</a></li>
</ul>
</li>
<li><a href="#org86cc8ca">3. Control Strategies</a>
<li><a href="#org4030106">2.7. Limitations due to negative stiffness effect</a></li>
<li><a href="#org7049dc3">2.8. Effect of rotation speed on the plant</a>
<ul>
<li><a href="#orga1abb2c">3.1. Measurement in the fixed reference frame</a></li>
<li><a href="#org08a5499">3.2. Measurement in the rotating frame</a></li>
</ul>
</li>
<li><a href="#org5b0bef3">4. Multi Body Model - Simscape</a>
<ul>
<li><a href="#org13aaa95">4.1. Parameter for the Simscape simulations</a></li>
<li><a href="#orgd334995">4.2. Identification in the rotating referenced frame</a>
<ul>
<li><a href="#org5cb3ac6">4.2.1. Low rotation speed and High rotation speed</a></li>
</ul>
</li>
<li><a href="#orgb159f85">4.3. Identification in the fixed frame</a></li>
<li><a href="#org6b50e4b">4.4. Identification from actuator forces to displacement in the fixed frame</a></li>
<li><a href="#org6a8d002">4.5. Effect of the rotating Speed</a>
<ul>
<li><a href="#org4a07d2b">4.5.1. <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</a></li>
<li><a href="#org01d22ae">4.5.2. <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</a></li>
</ul>
</li>
<li><a href="#org6cdc442">4.6. Effect of the X-Y stage stiffness</a>
<ul>
<li><a href="#org74a0c06">4.6.1. <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</a></li>
<li><a href="#org755ed06">2.8.1. Voice coil actuator</a></li>
<li><a href="#org53fec97">2.8.2. Piezoelectric actuator</a></li>
<li><a href="#orgf6be1a4">2.8.3. Analysis</a></li>
<li><a href="#org2262aaa">2.8.4. Campbell diagram</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#orge84791a">5. Control Implementation</a>
<li><a href="#orga07d0dd">3. Control Strategies</a>
<ul>
<li><a href="#org86d67af">5.1. Measurement in the fixed reference frame</a></li>
<li><a href="#orgeb25ab0">3.1. Measurement in the fixed reference frame</a></li>
<li><a href="#org9456905">3.2. Measurement in the rotating frame</a></li>
</ul>
</li>
<li><a href="#orgab7ac9c">4. Multi Body Model - Simscape</a>
<ul>
<li><a href="#org123b2ae">4.1. Initialize</a></li>
<li><a href="#orgc1906bb">4.2. Parameter for the Simscape simulations</a></li>
<li><a href="#org255159f">4.3. Identification in the rotating referenced frame</a></li>
<li><a href="#org5f1926d">4.4. Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</a></li>
<li><a href="#orge5f2b9f">4.5. Plant Control</a>
<ul>
<li><a href="#orgb9cef97">4.5.1. Low rotation speed and High rotation speed</a></li>
</ul>
</li>
<li><a href="#org09ff6ca">4.6. Identification in the fixed frame</a></li>
<li><a href="#org588dae5">4.7. Identification from actuator forces to displacement in the fixed frame</a></li>
<li><a href="#org53255e3">4.8. Effect of the rotating Speed</a>
<ul>
<li><a href="#org14c5fe5">4.8.1. <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</a></li>
<li><a href="#org5347efa">4.8.2. <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</a></li>
</ul>
</li>
<li><a href="#orgd2cb6ed">4.9. Effect of the X-Y stage stiffness</a>
<ul>
<li><a href="#org177c370">4.9.1. <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org4965ab2">5. Control Implementation</a>
<ul>
<li><a href="#org9f42bc5">5.1. Measurement in the fixed reference frame</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org9988e59" class="outline-2">
<h2 id="org9988e59"><span class="section-number-2">1</span> Introduction</h2>
<div id="outline-container-orgf151bb1" class="outline-2">
<h2 id="orgf151bb1"><span class="section-number-2">1</span> Introduction</h2>
<div class="outline-text-2" id="text-1">
<p>
The objective of this note it to highlight some control problems that arises when controlling the position of an object using actuators that are rotating with respect to a fixed reference frame.
</p>
<p>
In section <a href="#org4f1fd4b">2</a>, a simple system composed of a spindle and a translation stage is defined and the equations of motion are written.
In section <a href="#org3669843">2</a>, a simple system composed of a spindle and a translation stage is defined and the equations of motion are written.
The rotation induces some coupling between the actuators and their displacement, and modifies the dynamics of the system.
This is studied using the equations, and some numerical computations are used to compare the use of voice coil and piezoelectric actuators.
</p>
<p>
Then, in section <a href="#orgdb88326">3</a>, two different control approach are compared where:
Then, in section <a href="#org3747048">3</a>, two different control approach are compared where:
</p>
<ul class="org-ul">
<li>the measurement is made in the fixed frame</li>
@@ -351,27 +362,27 @@ Then, in section <a href="#orgdb88326">3</a>, two different control approach are
</ul>
<p>
In section <a href="#org8ef210c">4</a>, the analytical study will be validated using a multi body model of the studied system.
In section <a href="#org149db50">4</a>, the analytical study will be validated using a multi body model of the studied system.
</p>
<p>
Finally, in section <a href="#orgd9942b8">5</a>, the control strategies are implemented using Simulink and Simscape and compared.
Finally, in section <a href="#org9e7daf4">5</a>, the control strategies are implemented using Simulink and Simscape and compared.
</p>
</div>
</div>
<div id="outline-container-orgfd01695" class="outline-2">
<h2 id="orgfd01695"><span class="section-number-2">2</span> System</h2>
<div id="outline-container-org268da4c" class="outline-2">
<h2 id="org268da4c"><span class="section-number-2">2</span> System Description and Analysis</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org4f1fd4b"></a>
<a id="org3669843"></a>
</p>
</div>
<div id="outline-container-orged1d203" class="outline-3">
<h3 id="orged1d203"><span class="section-number-3">2.1</span> System description</h3>
<div id="outline-container-orge7664a5" class="outline-3">
<h3 id="orge7664a5"><span class="section-number-3">2.1</span> System description</h3>
<div class="outline-text-3" id="text-2-1">
<p>
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#orgce6c963">1</a>).
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#org5ddd11b">1</a>).
</p>
<p>
@@ -384,7 +395,7 @@ The measurement is either the \(x-y\) displacement of the object located on top
</p>
<div id="orgce6c963" class="figure">
<div id="org5ddd11b" class="figure">
<p><img src="./Figures/rotating_frame_2dof.png" alt="rotating_frame_2dof.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the mecanical system</p>
@@ -418,19 +429,19 @@ Indices \(u\) and \(v\) corresponds to signals in the rotating reference frame (
</div>
</div>
<div id="outline-container-orgbcc53a3" class="outline-3">
<h3 id="orgbcc53a3"><span class="section-number-3">2.2</span> Equations</h3>
<div id="outline-container-org60bfb2d" class="outline-3">
<h3 id="org60bfb2d"><span class="section-number-3">2.2</span> Equations</h3>
<div class="outline-text-3" id="text-2-2">
<p>
<a id="org2f020df"></a>
Based on the figure <a href="#orgce6c963">1</a>, we can write the equations of motion of the system.
<a id="orgc4fe841"></a>
Based on the figure <a href="#org5ddd11b">1</a>, we can write the equations of motion of the system.
</p>
<p>
Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\):
</p>
\begin{align}
\label{org93a4d45}
\label{org7c77780}
T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
V & = \frac{1}{2} k \left( x^2 + y^2 \right)
\end{align}
@@ -439,7 +450,7 @@ V & = \frac{1}{2} k \left( x^2 + y^2 \right)
The Lagrangian is the kinetic energy minus the potential energy.
</p>
\begin{equation}
\label{org19136da}
\label{orgc4495ac}
L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right)
\end{equation}
@@ -448,7 +459,7 @@ L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \le
The partial derivatives of the Lagrangian with respect to the variables \((x, y)\) are:
</p>
\begin{align*}
\label{org4fc9f2b}
\label{org5e103d6}
\frac{\partial L}{\partial x} & = -kx \\
\frac{\partial L}{\partial y} & = -ky \\
\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\
@@ -518,11 +529,11 @@ We can then subtract and add the previous equations to obtain the following equa
</p>
<div class="important">
\begin{equation}
\label{orgf3ca0ca}
\label{orgb342505}
m \ddot{d_u} + (k - m\dot{\theta}^2) d_u = F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta}
\end{equation}
\begin{equation}
\label{org5e2eb96}
\label{org97a2349}
m \ddot{d_v} + (k - m\dot{\theta}^2) d_v = F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta}
\end{equation}
@@ -548,8 +559,8 @@ The resulting effect of those forces should then be higher when using softer act
</div>
</div>
<div id="outline-container-org28e82a2" class="outline-3">
<h3 id="org28e82a2"><span class="section-number-3">2.3</span> Numerical Values for the NASS</h3>
<div id="outline-container-orgf397277" class="outline-3">
<h3 id="orgf397277"><span class="section-number-3">2.3</span> Numerical Values for the NASS</h3>
<div class="outline-text-3" id="text-2-3">
<p>
Let's define the parameters for the NASS.
@@ -612,8 +623,8 @@ Let's define the parameters for the NASS.
</div>
</div>
<div id="outline-container-orgd1ccc62" class="outline-3">
<h3 id="orgd1ccc62"><span class="section-number-3">2.4</span> Euler and Coriolis forces - Numerical Result</h3>
<div id="outline-container-org7aa02cb" class="outline-3">
<h3 id="org7aa02cb"><span class="section-number-3">2.4</span> Euler and Coriolis forces - Numerical Result</h3>
<div class="outline-text-3" id="text-2-4">
<p>
First we will determine the value for Euler and Coriolis forces during regular experiment.
@@ -624,10 +635,10 @@ First we will determine the value for Euler and Coriolis forces during regular e
</ul>
<p>
The obtained values are displayed in table <a href="#orga16401f">1</a>.
The obtained values are displayed in table <a href="#org3e40f1c">1</a>.
</p>
<table id="orga16401f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org3e40f1c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Euler and Coriolis forces for the NASS</caption>
<colgroup>
@@ -661,22 +672,22 @@ The obtained values are displayed in table <a href="#orga16401f">1</a>.
</div>
</div>
<div id="outline-container-org8db03ec" class="outline-3">
<h3 id="org8db03ec"><span class="section-number-3">2.5</span> Negative Spring Effect - Numerical Result</h3>
<div id="outline-container-org9672480" class="outline-3">
<h3 id="org9672480"><span class="section-number-3">2.5</span> Negative Spring Effect - Numerical Result</h3>
<div class="outline-text-3" id="text-2-5">
<p>
The negative stiffness due to the rotation is equal to \(-m{\omega_0}^2\).
</p>
<p>
The values for the negative spring effect are displayed in table <a href="#org8cad235">2</a>.
The values for the negative spring effect are displayed in table <a href="#org7b1aaf6">2</a>.
</p>
<p>
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
</p>
<table id="org8cad235" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org7b1aaf6" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Negative Spring effect</caption>
<colgroup>
@@ -704,15 +715,15 @@ This is definitely negligible when using piezoelectric actuators. It may not be
</div>
</div>
<div id="outline-container-org7e45369" class="outline-3">
<h3 id="org7e45369"><span class="section-number-3">2.6</span> Limitations due to coupling</h3>
<div id="outline-container-org57a38eb" class="outline-3">
<h3 id="org57a38eb"><span class="section-number-3">2.6</span> Limitations due to coupling</h3>
<div class="outline-text-3" id="text-2-6">
<p>
To simplify, we consider a constant rotating speed \(\dot{\theta} = {\omega_0}\) and thus \(\ddot{\theta} = 0\).
</p>
<p>
From equations \eqref{orgf3ca0ca} and \eqref{org5e2eb96}, we obtain:
From equations \eqref{orgb342505} and \eqref{org97a2349}, we obtain:
</p>
\begin{align*}
(m s^2 + (k - m{\omega_0}^2)) d_u &= F_u + 2 m {\omega_0} s d_v \\
@@ -750,6 +761,7 @@ The two previous equations can be written in a matrix form:
</p>
<div class="important">
\begin{equation}
\label{orga4820eb}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
\frac{1}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}
\begin{bmatrix}
@@ -766,26 +778,26 @@ Then, coupling is negligible if \(|-m \omega^2 + (k - m{\omega_0}^2)| \gg |2 m {
</p>
</div>
<div id="outline-container-orgc898fca" class="outline-4">
<h4 id="orgc898fca"><span class="section-number-4">2.6.1</span> Numerical Analysis</h4>
<div id="outline-container-org6274181" class="outline-4">
<h4 id="org6274181"><span class="section-number-4">2.6.1</span> Numerical Analysis</h4>
<div class="outline-text-4" id="text-2-6-1">
<p>
We plot on the same graph \(\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \omega_0 \omega|}\) for the voice coil and the piezo:
</p>
<ul class="org-ul">
<li>with the light sample (figure <a href="#org2b9a0e8">2</a>).</li>
<li>with the heavy sample (figure <a href="#org24d5dc4">3</a>).</li>
<li>with the light sample (figure <a href="#orgeb8c982">2</a>).</li>
<li>with the heavy sample (figure <a href="#orga3125c6">3</a>).</li>
</ul>
<div id="org2b9a0e8" class="figure">
<div id="orgeb8c982" class="figure">
<p><img src="Figures/coupling_light.png" alt="coupling_light.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Relative Coupling for light mass and high rotation speed</p>
</div>
<div id="org24d5dc4" class="figure">
<div id="orga3125c6" class="figure">
<p><img src="Figures/coupling_heavy.png" alt="coupling_heavy.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Relative Coupling for heavy mass and low rotation speed</p>
@@ -801,17 +813,17 @@ Coupling is higher for actuators with small stiffness.
</div>
</div>
<div id="outline-container-org87cd267" class="outline-3">
<h3 id="org87cd267"><span class="section-number-3">2.7</span> Limitations due to negative stiffness effect</h3>
<div id="outline-container-org4030106" class="outline-3">
<h3 id="org4030106"><span class="section-number-3">2.7</span> Limitations due to negative stiffness effect</h3>
<div class="outline-text-3" id="text-2-7">
<p>
If \(\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}\), then the negative spring effect is negligible and \(k - m\dot{\theta}^2 \approx k\).
</p>
<p>
Let's estimate what is the maximum rotation speed for which the negative stiffness effect is still negligible (\(\omega_\text{max} = 0.1 \sqrt{\frac{k}{m}}\)). Results are shown table <a href="#org84660ee">3</a>.
Let's estimate what is the maximum rotation speed for which the negative stiffness effect is still negligible (\(\omega_\text{max} = 0.1 \sqrt{\frac{k}{m}}\)). Results are shown table <a href="#org7eddfba">3</a>.
</p>
<table id="org84660ee" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org7eddfba" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> Maximum rotation speed at which negative stiffness is negligible (\(0.1\sqrt{\frac{k}{m}}\))</caption>
<colgroup>
@@ -860,10 +872,10 @@ The system can even goes unstable when \(m \omega^2 > k\), that is when the cent
</p>
<p>
From this analysis, we can determine the lowest practical stiffness that is possible to use: \(k_\text{min} = 10 m \omega^2\) (table <a href="#orgbea17e3">4</a>)
From this analysis, we can determine the lowest practical stiffness that is possible to use: \(k_\text{min} = 10 m \omega^2\) (table <a href="#org63d2716">4</a>)
</p>
<table id="orgbea17e3" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org63d2716" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Minimum possible stiffness</caption>
<colgroup>
@@ -890,17 +902,166 @@ From this analysis, we can determine the lowest practical stiffness that is poss
</table>
</div>
</div>
</div>
<div id="outline-container-org86cc8ca" class="outline-2">
<h2 id="org86cc8ca"><span class="section-number-2">3</span> Control Strategies</h2>
<div class="outline-text-2" id="text-3">
<div id="outline-container-org7049dc3" class="outline-3">
<h3 id="org7049dc3"><span class="section-number-3">2.8</span> Effect of rotation speed on the plant</h3>
<div class="outline-text-3" id="text-2-8">
<p>
<a id="orgdb88326"></a>
As shown in equation \eqref{orga4820eb}, the plant changes with the rotation speed \(\omega_0\).
</p>
<p>
Then, we compute the bode plot of the direct term and coupling term for multiple rotating speed.
</p>
<p>
Then we compare the result between voice coil and piezoelectric actuators.
</p>
</div>
<div id="outline-container-orga1abb2c" class="outline-3">
<h3 id="orga1abb2c"><span class="section-number-3">3.1</span> Measurement in the fixed reference frame</h3>
<div id="outline-container-org755ed06" class="outline-4">
<h4 id="org755ed06"><span class="section-number-4">2.8.1</span> Voice coil actuator</h4>
<div class="outline-text-4" id="text-2-8-1">
<div id="org4eafd24" class="figure">
<p><img src="Figures/G_ws_vc.png" alt="G_ws_vc.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Voice coil</p>
</div>
<div id="org9ef750d" class="figure">
<p><img src="Figures/Gc_ws_vc.png" alt="Gc_ws_vc.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Bode plot of the coupling transfer function term (from \(F_u\) to \(D_v\)) for multiple rotation speed - Voice coil</p>
</div>
</div>
</div>
<div id="outline-container-org53fec97" class="outline-4">
<h4 id="org53fec97"><span class="section-number-4">2.8.2</span> Piezoelectric actuator</h4>
<div class="outline-text-4" id="text-2-8-2">
<div id="orgc028298" class="figure">
<p><img src="Figures/G_ws_pz.png" alt="G_ws_pz.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Piezoelectric actuator</p>
</div>
<div id="org991a168" class="figure">
<p><img src="Figures/Gc_ws_pz.png" alt="Gc_ws_pz.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Bode plot of the coupling transfer function term (from \(F_u\) to \(D_v\)) for multiple rotation speed - Piezoelectric actuator</p>
</div>
</div>
</div>
<div id="outline-container-orgf6be1a4" class="outline-4">
<h4 id="orgf6be1a4"><span class="section-number-4">2.8.3</span> Analysis</h4>
<div class="outline-text-4" id="text-2-8-3">
<p>
When the rotation speed is null, the coupling terms are equal to zero and the diagonal terms corresponds to one degree of freedom mass spring system.
</p>
<p>
When the rotation speed in not null, the resonance frequency is duplicated into two pairs of complex conjugate poles.
</p>
<p>
As the rotation speed increases, one of the two resonant frequency goes to lower frequencies as the other one goes to higher frequencies.
</p>
<p>
The poles of the coupling terms are the same as the poles of the diagonal terms. The magnitude of the coupling terms are increasing with the rotation speed.
</p>
<p>
As shown in the previous figures, the system with voice coil is much more sensitive to rotation speed.
</p>
</div>
</div>
<div id="outline-container-org2262aaa" class="outline-4">
<h4 id="org2262aaa"><span class="section-number-4">2.8.4</span> Campbell diagram</h4>
<div class="outline-text-4" id="text-2-8-4">
<p>
The poles of the system are computed for multiple values of the rotation frequency.
</p>
<div class="org-src-container">
<pre class="src src-matlab">m = mlight;
k = kvc;
c = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #6434A3;">*</span>sqrt<span style="color: #707183;">(</span>k<span style="color: #6434A3;">*</span>m<span style="color: #707183;">)</span>;
ws = linspace<span style="color: #707183;">(</span><span style="color: #D0372D;">0</span>, <span style="color: #D0372D;">10</span>, <span style="color: #D0372D;">100</span><span style="color: #707183;">)</span>; <span style="color: #8D8D84; font-style: italic;">% [rad/s]</span>
polesvc = zeros<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span>, length<span style="color: #7388D6;">(</span>ws<span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
<span style="color: #0000FF;">for</span> <span style="color: #BA36A5;">i</span> = <span style="color: #D0372D;">1</span><span style="color: #D0372D;">:length</span><span style="color: #707183;">(</span><span style="color: #D0372D;">ws</span><span style="color: #707183;">)</span>
polei = pole<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">/</span><span style="color: #7388D6;">(</span><span style="color: #909183;">(</span>m<span style="color: #6434A3;">*</span>s<span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span> <span style="color: #6434A3;">+</span> c<span style="color: #6434A3;">*</span>s <span style="color: #6434A3;">+</span> <span style="color: #709870;">(</span>k <span style="color: #6434A3;">-</span> m<span style="color: #6434A3;">*</span>ws<span style="color: #907373;">(</span><span style="color: #D0372D;">i</span><span style="color: #907373;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span><span style="color: #709870;">)</span><span style="color: #909183;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span> <span style="color: #6434A3;">+</span> <span style="color: #909183;">(</span><span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span>m<span style="color: #6434A3;">*</span>ws<span style="color: #709870;">(</span><span style="color: #D0372D;">i</span><span style="color: #709870;">)</span><span style="color: #6434A3;">*</span>s<span style="color: #909183;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
polesvc<span style="color: #707183;">(</span><span style="color: #6434A3;">:</span>, <span style="color: #D0372D;">i</span><span style="color: #707183;">)</span> = sort<span style="color: #707183;">(</span>polei<span style="color: #7388D6;">(</span>imag<span style="color: #909183;">(</span>polei<span style="color: #909183;">)</span> <span style="color: #6434A3;">&gt;</span> <span style="color: #D0372D;">0</span><span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
<span style="color: #0000FF;">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">m = mlight;
k = kpz;
c = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #6434A3;">*</span>sqrt<span style="color: #707183;">(</span>k<span style="color: #6434A3;">*</span>m<span style="color: #707183;">)</span>;
ws = linspace<span style="color: #707183;">(</span><span style="color: #D0372D;">0</span>, <span style="color: #D0372D;">1000</span>, <span style="color: #D0372D;">100</span><span style="color: #707183;">)</span>; <span style="color: #8D8D84; font-style: italic;">% [rad/s]</span>
polespz = zeros<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span>, length<span style="color: #7388D6;">(</span>ws<span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
<span style="color: #0000FF;">for</span> <span style="color: #BA36A5;">i</span> = <span style="color: #D0372D;">1</span><span style="color: #D0372D;">:length</span><span style="color: #707183;">(</span><span style="color: #D0372D;">ws</span><span style="color: #707183;">)</span>
polei = pole<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">/</span><span style="color: #7388D6;">(</span><span style="color: #909183;">(</span>m<span style="color: #6434A3;">*</span>s<span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span> <span style="color: #6434A3;">+</span> c<span style="color: #6434A3;">*</span>s <span style="color: #6434A3;">+</span> <span style="color: #709870;">(</span>k <span style="color: #6434A3;">-</span> m<span style="color: #6434A3;">*</span>ws<span style="color: #907373;">(</span><span style="color: #D0372D;">i</span><span style="color: #907373;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span><span style="color: #709870;">)</span><span style="color: #909183;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span> <span style="color: #6434A3;">+</span> <span style="color: #909183;">(</span><span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span>m<span style="color: #6434A3;">*</span>ws<span style="color: #709870;">(</span><span style="color: #D0372D;">i</span><span style="color: #709870;">)</span><span style="color: #6434A3;">*</span>s<span style="color: #909183;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
polespz<span style="color: #707183;">(</span><span style="color: #6434A3;">:</span>, <span style="color: #D0372D;">i</span><span style="color: #707183;">)</span> = sort<span style="color: #707183;">(</span>polei<span style="color: #7388D6;">(</span>imag<span style="color: #909183;">(</span>polei<span style="color: #909183;">)</span> <span style="color: #6434A3;">&gt;</span> <span style="color: #D0372D;">0</span><span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
<span style="color: #0000FF;">end</span>
</pre>
</div>
<p>
We then plot the real and imaginary part of the poles as a function of the rotation frequency (figures <a href="#org2e762b4">8</a> and <a href="#orgf969e06">9</a>).
</p>
<p>
When the real part of one pole becomes positive, the system goes unstable.
</p>
<p>
For the voice coil (figure <a href="#org2e762b4">8</a>), the system is unstable when the rotation speed is above 5 rad/s. The real and imaginary part of the poles of the system with piezoelectric actuators are changing much less (figure <a href="#orgf969e06">9</a>).
</p>
<div id="org2e762b4" class="figure">
<p><img src="Figures/poles_w_vc.png" alt="poles_w_vc.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Real and Imaginary part of the poles of the system as a function of the rotation speed - Voice Coil and light sample</p>
</div>
<div id="orgf969e06" class="figure">
<p><img src="Figures/poles_w_pz.png" alt="poles_w_pz.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Real and Imaginary part of the poles of the system as a function of the rotation speed - Voice Coil and light sample</p>
</div>
</div>
</div>
</div>
</div>
<div id="outline-container-orga07d0dd" class="outline-2">
<h2 id="orga07d0dd"><span class="section-number-2">3</span> Control Strategies</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org3747048"></a>
</p>
</div>
<div id="outline-container-orgeb25ab0" class="outline-3">
<h3 id="orgeb25ab0"><span class="section-number-3">3.1</span> Measurement in the fixed reference frame</h3>
<div class="outline-text-3" id="text-3-1">
<p>
First, let's consider a measurement in the fixed referenced frame.
@@ -923,14 +1084,14 @@ Finally, the control low \(K\) links the position errors \([\epsilon_u, \epsilon
</p>
<p>
The block diagram is shown on figure <a href="#org859df00">4</a>.
The block diagram is shown on figure <a href="#orgf3c5c1c">10</a>.
</p>
<div id="org859df00" class="figure">
<div id="orgf3c5c1c" class="figure">
<p><img src="./Figures/control_measure_fixed_2dof.png" alt="control_measure_fixed_2dof.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Control with a measure from fixed frame</p>
<p><span class="figure-number">Figure 10: </span>Control with a measure from fixed frame</p>
</div>
<p>
@@ -943,22 +1104,22 @@ One question we wish to answer is: is \(G(\theta) J(\theta) = G(\theta_0) J(\the
</div>
</div>
<div id="outline-container-org08a5499" class="outline-3">
<h3 id="org08a5499"><span class="section-number-3">3.2</span> Measurement in the rotating frame</h3>
<div id="outline-container-org9456905" class="outline-3">
<h3 id="org9456905"><span class="section-number-3">3.2</span> Measurement in the rotating frame</h3>
<div class="outline-text-3" id="text-3-2">
<p>
Let's consider that the measurement is made in the rotating reference frame.
</p>
<p>
The corresponding block diagram is shown figure <a href="#org36bbc4f">5</a>
The corresponding block diagram is shown figure <a href="#org6cffd33">11</a>
</p>
<div id="org36bbc4f" class="figure">
<div id="org6cffd33" class="figure">
<p><img src="./Figures/control_measure_rotating_2dof.png" alt="control_measure_rotating_2dof.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Control with a measure from rotating frame</p>
<p><span class="figure-number">Figure 11: </span>Control with a measure from rotating frame</p>
</div>
<p>
@@ -968,17 +1129,23 @@ The loop gain is \(L = G K\).
</div>
</div>
<div id="outline-container-org5b0bef3" class="outline-2">
<h2 id="org5b0bef3"><span class="section-number-2">4</span> Multi Body Model - Simscape</h2>
<div id="outline-container-orgab7ac9c" class="outline-2">
<h2 id="orgab7ac9c"><span class="section-number-2">4</span> Multi Body Model - Simscape</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org8ef210c"></a>
<a id="org149db50"></a>
</p>
</div>
<div id="outline-container-org13aaa95" class="outline-3">
<h3 id="org13aaa95"><span class="section-number-3">4.1</span> Parameter for the Simscape simulations</h3>
<div class="outline-text-3" id="text-4-1">
<div id="outline-container-org123b2ae" class="outline-3">
<h3 id="org123b2ae"><span class="section-number-3">4.1</span> Initialize</h3>
</div>
<div id="outline-container-orgc1906bb" class="outline-3">
<h3 id="orgc1906bb"><span class="section-number-3">4.2</span> Parameter for the Simscape simulations</h3>
<div class="outline-text-3" id="text-4-2">
<p>
First we define the parameters that must be defined in order to run the Simscape simulation.
</p>
<div class="org-src-container">
<pre class="src src-matlab">w = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>; <span style="color: #8D8D84; font-style: italic;">% Rotation speed [rad/s]</span>
@@ -992,6 +1159,9 @@ cTuv = <span style="color: #D0372D;">0</span>; <span style="color: #8D8D84; font
</pre>
</div>
<p>
Then, we defined parameters that will be used in the following analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">mlight = <span style="color: #D0372D;">5</span>; <span style="color: #8D8D84; font-style: italic;">% Mass for light sample [kg]</span>
mheavy = <span style="color: #D0372D;">55</span>; <span style="color: #8D8D84; font-style: italic;">% Mass for heavy sample [kg]</span>
@@ -1003,17 +1173,24 @@ kvc = <span style="color: #D0372D;">1e3</span>; <span style="color: #8D8D84; fon
kpz = <span style="color: #D0372D;">1e8</span>; <span style="color: #8D8D84; font-style: italic;">% Piezo Stiffness [N/m]</span>
d = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">01</span>; <span style="color: #8D8D84; font-style: italic;">% Maximum excentricity from rotational axis [m]</span>
freqs = logspace<span style="color: #707183;">(</span><span style="color: #6434A3;">-</span><span style="color: #D0372D;">2</span>, <span style="color: #D0372D;">3</span>, <span style="color: #D0372D;">1000</span><span style="color: #707183;">)</span>; <span style="color: #8D8D84; font-style: italic;">% Frequency vector for analysis [Hz]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgd334995" class="outline-3">
<h3 id="orgd334995"><span class="section-number-3">4.2</span> Identification in the rotating referenced frame</h3>
<div class="outline-text-3" id="text-4-2">
<div id="outline-container-org255159f" class="outline-3">
<h3 id="org255159f"><span class="section-number-3">4.3</span> Identification in the rotating referenced frame</h3>
<div class="outline-text-3" id="text-4-3">
<p>
We initialize the inputs and outputs of the system to identify.
We initialize the inputs and outputs of the system to identify:
</p>
<ul class="org-ul">
<li>Inputs: \(f_u\) and \(f_v\)</li>
<li>Outputs: \(d_u\) and \(d_v\)</li>
</ul>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Options for Linearized</span>
options = linearizeOptions;
@@ -1068,21 +1245,38 @@ Gvc_heavy.InputName = <span style="color: #707183;">{</span><span style="color:
Gvc_heavy.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-org5f1926d" class="outline-3">
<h3 id="org5f1926d"><span class="section-number-3">4.4</span> Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</h3>
<div class="outline-text-3" id="text-4-4">
<p>
Finally, we plot the coupling ratio in both case (figure <a href="#orgded0015">6</a>).
We obtain the same result than the analytical case (figures <a href="#org2b9a0e8">2</a> and <a href="#org24d5dc4">3</a>).
From the previous identification, we plot the coupling ratio in both case (figure <a href="#orgcd55860">12</a>).
We obtain the same result than the analytical case (figures <a href="#orgeb8c982">2</a> and <a href="#orga3125c6">3</a>).
</p>
<div id="orgded0015" class="figure">
<div id="orgcd55860" class="figure">
<p><img src="Figures/coupling_ration_light_heavy.png" alt="coupling_ration_light_heavy.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-orge5f2b9f" class="outline-3">
<h3 id="orge5f2b9f"><span class="section-number-3">4.5</span> Plant Control</h3>
<div class="outline-text-3" id="text-4-5">
<p>
The goal is the study control problems due to the coupling that appears because of the rotation.
</p>
<p>
First, we identify the system when the rotation speed is null and then when the rotation speed is equal to 60rpm.
</p>
<p>
The actuators are voice coil with some damping.
</p>
<div class="figure">
@@ -1135,9 +1329,9 @@ Plot the ratio between the main transfer function and the coupling term:
</div>
</div>
<div id="outline-container-org5cb3ac6" class="outline-4">
<h4 id="org5cb3ac6"><span class="section-number-4">4.2.1</span> Low rotation speed and High rotation speed</h4>
<div class="outline-text-4" id="text-4-2-1">
<div id="outline-container-orgb9cef97" class="outline-4">
<h4 id="orgb9cef97"><span class="section-number-4">4.5.1</span> Low rotation speed and High rotation speed</h4>
<div class="outline-text-4" id="text-4-5-1">
<div class="org-src-container">
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">60</span>; angle_e = <span style="color: #D0372D;">0</span>;
G_low = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
@@ -1161,9 +1355,9 @@ bode<span style="color: #707183;">(</span>G_low, G_high<span style="color: #7071
</div>
</div>
<div id="outline-container-orgb159f85" class="outline-3">
<h3 id="orgb159f85"><span class="section-number-3">4.3</span> Identification in the fixed frame</h3>
<div class="outline-text-3" id="text-4-3">
<div id="outline-container-org09ff6ca" class="outline-3">
<h3 id="org09ff6ca"><span class="section-number-3">4.6</span> Identification in the fixed frame</h3>
<div class="outline-text-3" id="text-4-6">
<p>
Let's define some options as well as the inputs and outputs for linearization.
</p>
@@ -1243,9 +1437,9 @@ bode<span style="color: #707183;">(</span>Ge<span style="color: #707183;">)</spa
</div>
</div>
<div id="outline-container-org6b50e4b" class="outline-3">
<h3 id="org6b50e4b"><span class="section-number-3">4.4</span> Identification from actuator forces to displacement in the fixed frame</h3>
<div class="outline-text-3" id="text-4-4">
<div id="outline-container-org588dae5" class="outline-3">
<h3 id="org588dae5"><span class="section-number-3">4.7</span> Identification from actuator forces to displacement in the fixed frame</h3>
<div class="outline-text-3" id="text-4-7">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Options for Linearized</span>
options = linearizeOptions;
@@ -1302,48 +1496,48 @@ exportFig<span style="color: #707183;">(</span><span style="color: #008000;">'G_
</div>
</div>
<div id="outline-container-org6a8d002" class="outline-3">
<h3 id="org6a8d002"><span class="section-number-3">4.5</span> Effect of the rotating Speed</h3>
<div class="outline-text-3" id="text-4-5">
<div id="outline-container-org53255e3" class="outline-3">
<h3 id="org53255e3"><span class="section-number-3">4.8</span> Effect of the rotating Speed</h3>
<div class="outline-text-3" id="text-4-8">
<p>
<a id="org5ada9df"></a>
<a id="org09b2961"></a>
</p>
</div>
<div id="outline-container-org4a07d2b" class="outline-4">
<h4 id="org4a07d2b"><span class="section-number-4">4.5.1</span> <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</h4>
<div id="outline-container-org14c5fe5" class="outline-4">
<h4 id="org14c5fe5"><span class="section-number-4">4.8.1</span> <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</h4>
</div>
<div id="outline-container-org01d22ae" class="outline-4">
<h4 id="org01d22ae"><span class="section-number-4">4.5.2</span> <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</h4>
<div id="outline-container-org5347efa" class="outline-4">
<h4 id="org5347efa"><span class="section-number-4">4.8.2</span> <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</h4>
</div>
</div>
<div id="outline-container-org6cdc442" class="outline-3">
<h3 id="org6cdc442"><span class="section-number-3">4.6</span> Effect of the X-Y stage stiffness</h3>
<div class="outline-text-3" id="text-4-6">
<div id="outline-container-orgd2cb6ed" class="outline-3">
<h3 id="orgd2cb6ed"><span class="section-number-3">4.9</span> Effect of the X-Y stage stiffness</h3>
<div class="outline-text-3" id="text-4-9">
<p>
<a id="org377008c"></a>
<a id="org2bcac98"></a>
</p>
</div>
<div id="outline-container-org74a0c06" class="outline-4">
<h4 id="org74a0c06"><span class="section-number-4">4.6.1</span> <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</h4>
<div id="outline-container-org177c370" class="outline-4">
<h4 id="org177c370"><span class="section-number-4">4.9.1</span> <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</h4>
</div>
</div>
</div>
<div id="outline-container-orge84791a" class="outline-2">
<h2 id="orge84791a"><span class="section-number-2">5</span> Control Implementation</h2>
<div id="outline-container-org4965ab2" class="outline-2">
<h2 id="org4965ab2"><span class="section-number-2">5</span> Control Implementation</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="orgd9942b8"></a>
<a id="org9e7daf4"></a>
</p>
</div>
<div id="outline-container-org86d67af" class="outline-3">
<h3 id="org86d67af"><span class="section-number-3">5.1</span> Measurement in the fixed reference frame</h3>
<div id="outline-container-org9f42bc5" class="outline-3">
<h3 id="org9f42bc5"><span class="section-number-3">5.1</span> Measurement in the fixed reference frame</h3>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-01-23 mer. 15:21</p>
<p class="date">Created: 2019-01-24 jeu. 14:05</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
rotating_frame.org
+453 -9
View File
@@ -33,7 +33,7 @@ In section [[sec:simscape]], the analytical study will be validated using a mult
Finally, in section [[sec:control]], the control strategies are implemented using Simulink and Simscape and compared.
* System
* System Description and Analysis
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle system_numerical_analysis.m
:END:
@@ -290,6 +290,7 @@ Similarly we can obtain $d_v$ function of $F_u$ and $F_v$:
The two previous equations can be written in a matrix form:
#+begin_important
#+NAME: eq:coupledplant
\begin{equation}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
\frac{1}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}
@@ -308,8 +309,7 @@ We plot on the same graph $\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \ome
- with the light sample (figure [[fig:coupling_light]]).
- with the heavy sample (figure [[fig:coupling_heavy]]).
#+HEADER: :exports none :results silent
#+begin_src matlab
#+begin_src matlab :exports none :results silent
f = logspace(-1, 3, 1000);
figure;
@@ -335,8 +335,7 @@ We plot on the same graph $\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \ome
#+RESULTS:
[[file:Figures/coupling_light.png]]
#+HEADER: :exports none :results silent
#+begin_src matlab
#+begin_src matlab :exports none :results silent
figure;
hold on;
plot(f, abs(-mheavy*(2*pi*f).^2 + kvc - mheavy * wheavy^2)./abs(2*mheavy*wheavy*2*pi*f), 'DisplayName', 'Voice Coil')
@@ -405,6 +404,342 @@ From this analysis, we can determine the lowest practical stiffness that is poss
|-------------+-------+-------|
| k min [N/m] | 2199 | 89 |
** Effect of rotation speed on the plant
As shown in equation [[eq:coupledplant]], the plant changes with the rotation speed $\omega_0$.
Then, we compute the bode plot of the direct term and coupling term for multiple rotating speed.
Then we compare the result between voice coil and piezoelectric actuators.
*** Voice coil actuator
#+begin_src matlab :exports none :results silent
m = mlight;
k = kvc;
#+end_src
#+begin_src matlab :exports none :results silent
ws = [0, 1, 10, 60]*2*pi/60; % [rmp]
Gs = {zeros(1, length(ws))};
Gcs = {zeros(1, length(ws))};
for i = 1:length(ws)
w = ws(i);
Gs(i) = {(m*s^2 + (k-m*w^2))/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2)};
Gcs(i) = {(2*m*w*s)/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2)};
end
#+end_src
#+begin_src matlab :exports none :results silent
freqs = logspace(-2, 1, 1000);
figure;
ax1 = subaxis(2,1,1);
hold on;
for i = 1:length(ws)
plot(freqs, abs(squeeze(freqresp(Gs{i}, freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi));
end
hold off;
xlim([freqs(1), freqs(end)]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude [m/N]');
legend('Location', 'southwest');
ax2 = subaxis(2,1,2);
hold on;
for i = 1:length(ws)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}, freqs, 'Hz'))));
end
hold off;
yticks(-180:90:180);
ylim([-180 180]);
xlim([freqs(1), freqs(end)]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x');
#+end_src
#+HEADER: :tangle no :exports results :results file :noweb yes
#+HEADER: :var filepath="Figures/G_ws_vc.png" :var figsize="wide-tall"
#+begin_src matlab
<<plt-matlab>>
#+end_src
#+NAME: fig:G_ws_vc
#+CAPTION: Bode plot of the direct transfer function term (from $F_u$ to $D_u$) for multiple rotation speed - Voice coil
#+RESULTS:
[[file:Figures/G_ws_vc.png]]
#+begin_src matlab :exports none :results silent
freqs = logspace(-2, 1, 1000);
figure;
ax1 = subaxis(2,1,1);
hold on;
for i = 1:length(ws)
plot(freqs, abs(squeeze(freqresp(Gcs{i}, freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi));
end
hold off;
xlim([freqs(1), freqs(end)]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude [m/N]');
legend('Location', 'southwest');
ax2 = subaxis(2,1,2);
hold on;
for i = 1:length(ws)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gcs{i}, freqs, 'Hz'))));
end
hold off;
yticks(-180:90:180);
ylim([-180 180]);
xlim([freqs(1), freqs(end)]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x');
#+end_src
#+HEADER: :tangle no :exports results :results file :noweb yes
#+HEADER: :var filepath="Figures/Gc_ws_vc.png" :var figsize="wide-tall"
#+begin_src matlab
<<plt-matlab>>
#+end_src
#+NAME: fig:Gc_ws_vc
#+CAPTION: Bode plot of the coupling transfer function term (from $F_u$ to $D_v$) for multiple rotation speed - Voice coil
#+RESULTS:
[[file:Figures/Gc_ws_vc.png]]
*** Piezoelectric actuator
#+begin_src matlab :exports none :results silent
m = mlight;
k = kpz;
#+end_src
#+begin_src matlab :exports none :results silent
ws = [0, 1, 10, 60]*2*pi/60; % [rmp]
Gs = {zeros(1, length(ws))};
Gcs = {zeros(1, length(ws))};
for i = 1:length(ws)
w = ws(i);
Gs(i) = {(m*s^2 + (k-m*w^2))/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2)};
Gcs(i) = {(2*m*w*s)/((m*s^2 + (k - m*w^2))^2 + (2*m*w*s)^2)};
end
#+end_src
#+begin_src matlab :exports none :results silent
freqs = logspace(2, 3, 1000);
figure;
ax1 = subaxis(2,1,1);
hold on;
for i = 1:length(ws)
plot(freqs, abs(squeeze(freqresp(Gs{i}, freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi));
end
hold off;
xlim([freqs(1), freqs(end)]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude [m/N]');
legend('Location', 'southwest');
ax2 = subaxis(2,1,2);
hold on;
for i = 1:length(ws)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}, freqs, 'Hz'))));
end
hold off;
yticks(-180:90:180);
ylim([-180 180]);
xlim([freqs(1), freqs(end)]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x');
#+end_src
#+HEADER: :tangle no :exports results :results file :noweb yes
#+HEADER: :var filepath="Figures/G_ws_pz.png" :var figsize="wide-tall"
#+begin_src matlab
<<plt-matlab>>
#+end_src
#+NAME: fig:G_ws_pz
#+CAPTION: Bode plot of the direct transfer function term (from $F_u$ to $D_u$) for multiple rotation speed - Piezoelectric actuator
#+RESULTS:
[[file:Figures/G_ws_pz.png]]
#+begin_src matlab :exports none :results silent
figure;
ax1 = subaxis(2,1,1);
hold on;
for i = 1:length(ws)
plot(freqs, abs(squeeze(freqresp(Gcs{i}, freqs, 'Hz'))), 'DisplayName', sprintf('w = %.0f [rpm]', ws(i)*60/2/pi));
end
hold off;
xlim([freqs(1), freqs(end)]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude [m/N]');
legend('Location', 'southwest');
ax2 = subaxis(2,1,2);
hold on;
for i = 1:length(ws)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gcs{i}, freqs, 'Hz'))));
end
hold off;
yticks(-180:90:180);
ylim([-180 180]);
xlim([freqs(1), freqs(end)]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x');
#+end_src
#+HEADER: :tangle no :exports results :results file :noweb yes
#+HEADER: :var filepath="Figures/Gc_ws_pz.png" :var figsize="wide-tall"
#+begin_src matlab
<<plt-matlab>>
#+end_src
#+NAME: fig:Gc_ws_pz
#+CAPTION: Bode plot of the coupling transfer function term (from $F_u$ to $D_v$) for multiple rotation speed - Piezoelectric actuator
#+RESULTS:
[[file:Figures/Gc_ws_pz.png]]
*** Analysis
When the rotation speed is null, the coupling terms are equal to zero and the diagonal terms corresponds to one degree of freedom mass spring system.
When the rotation speed in not null, the resonance frequency is duplicated into two pairs of complex conjugate poles.
As the rotation speed increases, one of the two resonant frequency goes to lower frequencies as the other one goes to higher frequencies.
The poles of the coupling terms are the same as the poles of the diagonal terms. The magnitude of the coupling terms are increasing with the rotation speed.
As shown in the previous figures, the system with voice coil is much more sensitive to rotation speed.
*** Campbell diagram
The poles of the system are computed for multiple values of the rotation frequency.
#+begin_src matlab :results silent :exports code
m = mlight;
k = kvc;
c = 0.1*sqrt(k*m);
ws = linspace(0, 10, 100); % [rad/s]
polesvc = zeros(2, length(ws));
for i = 1:length(ws)
polei = pole(1/((m*s^2 + c*s + (k - m*ws(i)^2))^2 + (2*m*ws(i)*s)^2));
polesvc(:, i) = sort(polei(imag(polei) > 0));
end
#+end_src
#+begin_src matlab :results silent :exports code
m = mlight;
k = kpz;
c = 0.1*sqrt(k*m);
ws = linspace(0, 1000, 100); % [rad/s]
polespz = zeros(2, length(ws));
for i = 1:length(ws)
polei = pole(1/((m*s^2 + c*s + (k - m*ws(i)^2))^2 + (2*m*ws(i)*s)^2));
polespz(:, i) = sort(polei(imag(polei) > 0));
end
#+end_src
We then plot the real and imaginary part of the poles as a function of the rotation frequency (figures [[fig:poles_w_vc]] and [[fig:poles_w_pz]]).
When the real part of one pole becomes positive, the system goes unstable.
For the voice coil (figure [[fig:poles_w_vc]]), the system is unstable when the rotation speed is above 5 rad/s. The real and imaginary part of the poles of the system with piezoelectric actuators are changing much less (figure [[fig:poles_w_pz]]).
#+begin_src matlab :results silent :exports none
figure;
% Amplitude
ax1 = subplot(1,2,1);
hold on;
plot(ws, real(polesvc(1, :)))
set(gca,'ColorOrderIndex',1)
plot(ws, real(polesvc(2, :)))
plot(ws, zeros(size(ws)), 'k--')
hold off;
xlabel('Rotation Frequency [rad/s]');
ylabel('Pole Real Part');
ax2 = subplot(1,2,2);
hold on;
plot(ws, imag(polesvc(1, :)))
set(gca,'ColorOrderIndex',1)
plot(ws, -imag(polesvc(1, :)))
set(gca,'ColorOrderIndex',1)
plot(ws, imag(polesvc(2, :)))
set(gca,'ColorOrderIndex',1)
plot(ws, -imag(polesvc(2, :)))
hold off;
xlabel('Rotation Frequency [rad/s]');
ylabel('Pole Imaginary Part');
#+end_src
#+HEADER: :tangle no :exports results :results file :noweb yes
#+HEADER: :var filepath="Figures/poles_w_vc.png" :var figsize="wide-tall"
#+begin_src matlab
<<plt-matlab>>
#+end_src
#+NAME: fig:poles_w_vc
#+CAPTION: Real and Imaginary part of the poles of the system as a function of the rotation speed - Voice Coil and light sample
#+RESULTS:
[[file:Figures/poles_w_vc.png]]
#+begin_src matlab :results silent :exports none
figure;
% Amplitude
ax1 = subplot(1,2,1);
hold on;
plot(ws, real(polespz(1, :)))
set(gca,'ColorOrderIndex',1)
plot(ws, real(polespz(2, :)))
plot(ws, zeros(size(ws)), 'k--')
hold off;
xlabel('Rotation Frequency [rad/s]');
ylabel('Pole Real Part');
ax2 = subplot(1,2,2);
hold on;
plot(ws, imag(polespz(1, :)))
set(gca,'ColorOrderIndex',1)
plot(ws, -imag(polespz(1, :)))
set(gca,'ColorOrderIndex',1)
plot(ws, imag(polespz(2, :)))
set(gca,'ColorOrderIndex',1)
plot(ws, -imag(polespz(2, :)))
hold off;
xlabel('Rotation Frequency [rad/s]');
ylabel('Pole Imaginary Part');
#+end_src
#+HEADER: :tangle no :exports results :results file :noweb yes
#+HEADER: :var filepath="Figures/poles_w_pz.png" :var figsize="wide-tall"
#+begin_src matlab
<<plt-matlab>>
#+end_src
#+NAME: fig:poles_w_pz
#+CAPTION: Real and Imaginary part of the poles of the system as a function of the rotation speed - Voice Coil and light sample
#+RESULTS:
[[file:Figures/poles_w_pz.png]]
* Control Strategies
<<sec:control_strategies>>
** Measurement in the fixed reference frame
@@ -445,6 +780,7 @@ The loop gain is $L = G K$.
:END:
<<sec:simscape>>
** Initialize
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
load('./mat/parameters.mat');
@@ -455,6 +791,7 @@ The loop gain is $L = G K$.
#+end_src
** Parameter for the Simscape simulations
First we define the parameters that must be defined in order to run the Simscape simulation.
#+begin_src matlab :exports code :results silent
w = 2*pi; % Rotation speed [rad/s]
@@ -467,6 +804,7 @@ The loop gain is $L = G K$.
cTuv = 0; % Damping of the Tuv stage [N/(m/s)]
#+end_src
Then, we defined parameters that will be used in the following analysis.
#+begin_src matlab :exports code :results silent
mlight = 5; % Mass for light sample [kg]
mheavy = 55; % Mass for heavy sample [kg]
@@ -478,11 +816,15 @@ The loop gain is $L = G K$.
kpz = 1e8; % Piezo Stiffness [N/m]
d = 0.01; % Maximum excentricity from rotational axis [m]
freqs = logspace(-2, 3, 1000); % Frequency vector for analysis [Hz]
#+end_src
** Identification in the rotating referenced frame
We initialize the inputs and outputs of the system to identify:
- Inputs: $f_u$ and $f_v$
- Outputs: $d_u$ and $d_v$
We initialize the inputs and outputs of the system to identify.
#+begin_src matlab :exports code :results silent
%% Options for Linearized
options = linearizeOptions;
@@ -531,11 +873,11 @@ Then we identify the system with an heavy mass and low speed.
Gvc_heavy.OutputName = {'Du', 'Dv'};
#+end_src
Finally, we plot the coupling ratio in both case (figure [[fig:coupling_ration_light_heavy]]).
** Coupling ratio between $f_{uv}$ and $d_{uv}$
From the previous identification, we plot the coupling ratio in both case (figure [[fig:coupling_ration_light_heavy]]).
We obtain the same result than the analytical case (figures [[fig:coupling_light]] and [[fig:coupling_heavy]]).
#+begin_src matlab :results silent :exports none
freqs = logspace(-2, 3, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gvc_light('Du', 'Fu'), freqs, 'Hz')))./abs(squeeze(freqresp(Gvc_light('Dv', 'Fu'), freqs, 'Hz'))));
@@ -560,6 +902,108 @@ We obtain the same result than the analytical case (figures [[fig:coupling_light
#+RESULTS:
[[file:Figures/coupling_ration_light_heavy.png]]
** Plant Control
The goal is the study control problems due to the coupling that appears because of the rotation.
#+begin_src matlab :exports none :results silent
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'rotating_frame';
%% Input/Output definition
io(1) = linio([mdl, '/fu'], 1, 'input');
io(2) = linio([mdl, '/fv'], 1, 'input');
io(3) = linio([mdl, '/du'], 1, 'output');
io(4) = linio([mdl, '/dv'], 1, 'output');
#+end_src
First, we identify the system when the rotation speed is null and then when the rotation speed is equal to 60rpm.
The actuators are voice coil with some damping.
#+begin_src matlab :exports none :results silent
w = 0; % Rotation speed [rad/s]
m = mlight; % mass of the sample [kg]
kTuv = kvc;
% cTuv = 0.1*sqrt(kTuv*m);
cTuv = 0;
G = linearize(mdl, io, 0.1);
G.InputName = {'Fu', 'Fv'};
G.OutputName = {'Du', 'Dv'};
#+end_src
#+begin_src matlab :exports none :results silent
w = 0.1; % Rotation speed [rad/s]
m = mlight; % mass of the sample [kg]
kTuv = kvc;
% cTuv = 0.1*sqrt(kTuv*m);
cTuv = 0;
Gt = linearize(mdl, io, 0.1);
Gt.InputName = {'Fu', 'Fv'};
Gt.OutputName = {'Du', 'Dv'};
#+end_src
#+begin_src matlab :exports none :results silent
figure;
bode(G, 'r-', Gt, 'b--')
legend({'G - w = 0', 'G - w = 60rpm'}, 'Location', 'southwest');
#+end_src
#+HEADER: :tangle no :exports results :results file :noweb yes
#+HEADER: :var filepath="Figures/coupling_ration_light_heavy.png" :var figsize="wide-tall"
#+begin_src matlab
<<plt-matlab>>
#+end_src
#+begin_src matlab :exports none :results silent
figure;
% Amplitude
ax1 = subaxis(2,1,1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gvc_light('du', 'fu'), freqs, 'Hz'))), '-');
set(gca,'xscale','log'); set(gca,'yscale','log');
ylabel('Amplitude [m/N]');
set(gca, 'XTickLabel',[]);
hold off;
% Phase
ax2 = subaxis(2,1,2);
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gvc_light('du', 'fu'), freqs, 'Hz')))), '-');
set(gca,'xscale','log');
yticks(-180:180:180);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :exports none :results silent
figure;
% Amplitude
ax1 = subaxis(2,1,1);
hold on;
plot(freqs, abs(squeeze(freqresp(-Gvc_light('dv', 'fv'), freqs, 'Hz'))), '-');
set(gca,'xscale','log'); set(gca,'yscale','log');
ylabel('Amplitude [m/N]');
set(gca, 'XTickLabel',[]);
hold off;
% Phase
ax2 = subaxis(2,1,2);
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(-Gvc_light('dv', 'fv'), freqs, 'Hz')))), '-');
set(gca,'xscale','log');
yticks(-180:180:180);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
#+end_src