Better Matlab notations
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<head>
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<!-- 2020-11-09 lun. 14:36 -->
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<title>SVD Control</title>
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|
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<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org8480cb8">1. Gravimeter - Simscape Model</a>
|
||||
<li><a href="#org4262bdd">1. Gravimeter - Simscape Model</a>
|
||||
<ul>
|
||||
<li><a href="#org669566a">1.1. Introduction</a></li>
|
||||
<li><a href="#org513c970">1.2. Simscape Model - Parameters</a></li>
|
||||
<li><a href="#org7e0371b">1.3. System Identification - Without Gravity</a></li>
|
||||
<li><a href="#org9dfc541">1.4. System Identification - With Gravity</a></li>
|
||||
<li><a href="#org06067ff">1.5. Analytical Model</a>
|
||||
<li><a href="#org46fc636">1.1. Introduction</a></li>
|
||||
<li><a href="#org137a1ed">1.2. Simscape Model - Parameters</a></li>
|
||||
<li><a href="#org08acfbd">1.3. System Identification - Without Gravity</a></li>
|
||||
<li><a href="#orge4c219d">1.4. System Identification - With Gravity</a></li>
|
||||
<li><a href="#org744c6c9">1.5. Analytical Model</a>
|
||||
<ul>
|
||||
<li><a href="#org063c200">1.5.1. Parameters</a></li>
|
||||
<li><a href="#orgec24c80">1.5.2. Generation of the State Space Model</a></li>
|
||||
<li><a href="#org1590891">1.5.3. Comparison with the Simscape Model</a></li>
|
||||
<li><a href="#orgb615e54">1.5.4. Analysis</a></li>
|
||||
<li><a href="#org2243155">1.5.5. Control Section</a></li>
|
||||
<li><a href="#orgd28ecdb">1.5.6. Greshgorin radius</a></li>
|
||||
<li><a href="#org24f83eb">1.5.7. Injecting ground motion in the system to have the output</a></li>
|
||||
<li><a href="#orga42f590">1.5.1. Parameters</a></li>
|
||||
<li><a href="#org288ddf0">1.5.2. Generation of the State Space Model</a></li>
|
||||
<li><a href="#orgcd68a21">1.5.3. Comparison with the Simscape Model</a></li>
|
||||
<li><a href="#orga3239b9">1.5.4. Analysis</a></li>
|
||||
<li><a href="#orgda0f1ad">1.5.5. Control Section</a></li>
|
||||
<li><a href="#org7ffae54">1.5.6. Greshgorin radius</a></li>
|
||||
<li><a href="#org72dd1a0">1.5.7. Injecting ground motion in the system to have the output</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgc1560cf">2. Gravimeter - Functions</a>
|
||||
<li><a href="#org01bdedf">2. Gravimeter - Functions</a>
|
||||
<ul>
|
||||
<li><a href="#org814c22c">2.1. <code>align</code></a></li>
|
||||
<li><a href="#orga936ee3">2.2. <code>pzmap_testCL</code></a></li>
|
||||
<li><a href="#org4647e37">2.1. <code>align</code></a></li>
|
||||
<li><a href="#orga0981c0">2.2. <code>pzmap_testCL</code></a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgf783e5e">3. Stewart Platform - Simscape Model</a>
|
||||
<li><a href="#orgd6f892a">3. Stewart Platform - Simscape Model</a>
|
||||
<ul>
|
||||
<li><a href="#org698a574">3.1. Simscape Model - Parameters</a></li>
|
||||
<li><a href="#orgdfc6136">3.2. Identification of the plant</a></li>
|
||||
<li><a href="#orgadaff5c">3.3. Physical Decoupling using the Jacobian</a></li>
|
||||
<li><a href="#org6ba1c1a">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li>
|
||||
<li><a href="#org7e2a42e">3.5. SVD Decoupling</a></li>
|
||||
<li><a href="#orgc6f3016">3.6. Verification of the decoupling using the “Gershgorin Radii”</a></li>
|
||||
<li><a href="#orgedf5c94">3.7. Obtained Decoupled Plants</a></li>
|
||||
<li><a href="#orgff44b51">3.8. Diagonal Controller</a></li>
|
||||
<li><a href="#org949d9ca">3.9. Closed-Loop system Performances</a></li>
|
||||
<li><a href="#org98f27a1">3.1. Simscape Model - Parameters</a></li>
|
||||
<li><a href="#orgfc4057f">3.2. Identification of the plant</a></li>
|
||||
<li><a href="#org06bff3b">3.3. Physical Decoupling using the Jacobian</a></li>
|
||||
<li><a href="#org7208fcb">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li>
|
||||
<li><a href="#orgdcfefc4">3.5. SVD Decoupling</a></li>
|
||||
<li><a href="#orgeedb4ac">3.6. Verification of the decoupling using the “Gershgorin Radii”</a></li>
|
||||
<li><a href="#orga3edea8">3.7. Obtained Decoupled Plants</a></li>
|
||||
<li><a href="#orgb371cb1">3.8. Diagonal Controller</a></li>
|
||||
<li><a href="#orgb6d90eb">3.9. Closed-Loop system Performances</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8480cb8" class="outline-2">
|
||||
<h2 id="org8480cb8"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
|
||||
<div id="outline-container-org4262bdd" class="outline-2">
|
||||
<h2 id="org4262bdd"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
</div>
|
||||
<div id="outline-container-org669566a" class="outline-3">
|
||||
<h3 id="org669566a"><span class="section-number-3">1.1</span> Introduction</h3>
|
||||
<div id="outline-container-org46fc636" class="outline-3">
|
||||
<h3 id="org46fc636"><span class="section-number-3">1.1</span> Introduction</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
|
||||
<div id="org1f9eedf" class="figure">
|
||||
<div id="orgca5b956" class="figure">
|
||||
<p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p>
|
||||
@@ -93,8 +93,8 @@
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org513c970" class="outline-3">
|
||||
<h3 id="org513c970"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
|
||||
<div id="outline-container-org137a1ed" class="outline-3">
|
||||
<h3 id="org137a1ed"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
|
||||
<div class="outline-text-3" id="text-1-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">open(<span class="org-string">'gravimeter.slx'</span>)
|
||||
@@ -125,8 +125,8 @@ g = 0; <span class="org-comment">% Gravity [m/s2]</span>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7e0371b" class="outline-3">
|
||||
<h3 id="org7e0371b"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
|
||||
<div id="outline-container-org08acfbd" class="outline-3">
|
||||
<h3 id="org08acfbd"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
@@ -148,7 +148,7 @@ G.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string">
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example" id="org6eb6401">
|
||||
<pre class="example" id="org2c5c71d">
|
||||
pole(G)
|
||||
ans =
|
||||
-0.000473481142385795 + 21.7596190728632i
|
||||
@@ -173,7 +173,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
|
||||
|
||||
|
||||
|
||||
<div id="org3874001" class="figure">
|
||||
<div id="orgddb1793" class="figure">
|
||||
<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
|
||||
@@ -181,8 +181,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9dfc541" class="outline-3">
|
||||
<h3 id="org9dfc541"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
|
||||
<div id="outline-container-orge4c219d" class="outline-3">
|
||||
<h3 id="orge4c219d"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
|
||||
<div class="outline-text-3" id="text-1-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">g = 9.80665; <span class="org-comment">% Gravity [m/s2]</span>
|
||||
@@ -199,7 +199,7 @@ Gg.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string"
|
||||
<p>
|
||||
We can now see that the system is unstable due to gravity.
|
||||
</p>
|
||||
<pre class="example" id="orga5f4271">
|
||||
<pre class="example" id="org78beae2">
|
||||
pole(Gg)
|
||||
ans =
|
||||
-10.9848275341252 + 0i
|
||||
@@ -211,7 +211,7 @@ ans =
|
||||
</pre>
|
||||
|
||||
|
||||
<div id="org5e8aee0" class="figure">
|
||||
<div id="org70961c1" class="figure">
|
||||
<p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p>
|
||||
@@ -219,12 +219,12 @@ ans =
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org06067ff" class="outline-3">
|
||||
<h3 id="org06067ff"><span class="section-number-3">1.5</span> Analytical Model</h3>
|
||||
<div id="outline-container-org744c6c9" class="outline-3">
|
||||
<h3 id="org744c6c9"><span class="section-number-3">1.5</span> Analytical Model</h3>
|
||||
<div class="outline-text-3" id="text-1-5">
|
||||
</div>
|
||||
<div id="outline-container-org063c200" class="outline-4">
|
||||
<h4 id="org063c200"><span class="section-number-4">1.5.1</span> Parameters</h4>
|
||||
<div id="outline-container-orga42f590" class="outline-4">
|
||||
<h4 id="orga42f590"><span class="section-number-4">1.5.1</span> Parameters</h4>
|
||||
<div class="outline-text-4" id="text-1-5-1">
|
||||
<p>
|
||||
Bode options.
|
||||
@@ -256,8 +256,8 @@ Frequency vector.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgec24c80" class="outline-4">
|
||||
<h4 id="orgec24c80"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
|
||||
<div id="outline-container-org288ddf0" class="outline-4">
|
||||
<h4 id="org288ddf0"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
|
||||
<div class="outline-text-4" id="text-1-5-2">
|
||||
<p>
|
||||
Mass matrix
|
||||
@@ -358,11 +358,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1590891" class="outline-4">
|
||||
<h4 id="org1590891"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
|
||||
<div id="outline-container-orgcd68a21" class="outline-4">
|
||||
<h4 id="orgcd68a21"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
|
||||
<div class="outline-text-4" id="text-1-5-3">
|
||||
|
||||
<div id="orgfa66619" class="figure">
|
||||
<div id="orgacf77cc" class="figure">
|
||||
<p><img src="figs/gravimeter_analytical_system_open_loop_models.png" alt="gravimeter_analytical_system_open_loop_models.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Comparison of the analytical and the Simscape models</p>
|
||||
@@ -370,8 +370,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb615e54" class="outline-4">
|
||||
<h4 id="orgb615e54"><span class="section-number-4">1.5.4</span> Analysis</h4>
|
||||
<div id="outline-container-orga3239b9" class="outline-4">
|
||||
<h4 id="orga3239b9"><span class="section-number-4">1.5.4</span> Analysis</h4>
|
||||
<div class="outline-text-4" id="text-1-5-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-comment">% figure</span>
|
||||
@@ -439,8 +439,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2243155" class="outline-4">
|
||||
<h4 id="org2243155"><span class="section-number-4">1.5.5</span> Control Section</h4>
|
||||
<div id="outline-container-orgda0f1ad" class="outline-4">
|
||||
<h4 id="orgda0f1ad"><span class="section-number-4">1.5.5</span> Control Section</h4>
|
||||
<div class="outline-text-4" id="text-1-5-5">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">system_dec_10Hz = freqresp(system_dec,2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10);
|
||||
@@ -580,8 +580,8 @@ legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'D
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd28ecdb" class="outline-4">
|
||||
<h4 id="orgd28ecdb"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
|
||||
<div id="outline-container-org7ffae54" class="outline-4">
|
||||
<h4 id="org7ffae54"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
|
||||
<div class="outline-text-4" id="text-1-5-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">system_dec_freq = freqresp(system_dec,w);
|
||||
@@ -627,8 +627,8 @@ ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org24f83eb" class="outline-4">
|
||||
<h4 id="org24f83eb"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
|
||||
<div id="outline-container-org72dd1a0" class="outline-4">
|
||||
<h4 id="org72dd1a0"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
|
||||
<div class="outline-text-4" id="text-1-5-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Fr = logspace(<span class="org-type">-</span>2,3,1e3);
|
||||
@@ -684,15 +684,15 @@ rot = PHI(<span class="org-type">:</span>,11,11);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc1560cf" class="outline-2">
|
||||
<h2 id="orgc1560cf"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
|
||||
<div id="outline-container-org01bdedf" class="outline-2">
|
||||
<h2 id="org01bdedf"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
</div>
|
||||
<div id="outline-container-org814c22c" class="outline-3">
|
||||
<h3 id="org814c22c"><span class="section-number-3">2.1</span> <code>align</code></h3>
|
||||
<div id="outline-container-org4647e37" class="outline-3">
|
||||
<h3 id="org4647e37"><span class="section-number-3">2.1</span> <code>align</code></h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
<a id="org3643797"></a>
|
||||
<a id="org787b0b4"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -721,11 +721,11 @@ This Matlab function is accessible <a href="gravimeter/align.m">here</a>.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orga936ee3" class="outline-3">
|
||||
<h3 id="orga936ee3"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
|
||||
<div id="outline-container-orga0981c0" class="outline-3">
|
||||
<h3 id="orga0981c0"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
<a id="org7c6bace"></a>
|
||||
<a id="org6adb39c"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -774,11 +774,11 @@ This Matlab function is accessible <a href="gravimeter/pzmap_testCL.m">here</a>.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf783e5e" class="outline-2">
|
||||
<h2 id="orgf783e5e"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
|
||||
<div id="outline-container-orgd6f892a" class="outline-2">
|
||||
<h2 id="orgd6f892a"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org325e3af">5</a>.
|
||||
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org2113119">5</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -791,7 +791,7 @@ Some notes about the system:
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org325e3af" class="figure">
|
||||
<div id="org2113119" class="figure">
|
||||
<p><img src="figs/SP_assembly.png" alt="SP_assembly.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Stewart Platform CAD View</p>
|
||||
@@ -801,22 +801,22 @@ Some notes about the system:
|
||||
The analysis of the SVD control applied to the Stewart platform is performed in the following sections:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Section <a href="#org46c7682">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li>
|
||||
<li>Section <a href="#orgacf8e97">3.2</a>: The plant is identified from the Simscape model and the system coupling is shown</li>
|
||||
<li>Section <a href="#orgf5489c1">3.3</a>: The plant is first decoupled using the Jacobian</li>
|
||||
<li>Section <a href="#orgccd7599">3.4</a>: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)</li>
|
||||
<li>Section <a href="#org5abe937">3.5</a>: The decoupling is performed thanks to the SVD</li>
|
||||
<li>Section <a href="#org7691f01">3.6</a>: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii</li>
|
||||
<li>Section <a href="#orgabc21ab">3.7</a>: The dynamics of the decoupled plants are shown</li>
|
||||
<li>Section <a href="#org9620c1c">3.8</a>: A diagonal controller is defined to control the decoupled plant</li>
|
||||
<li>Section <a href="#org823e1cb">3.9</a>: Finally, the closed loop system properties are studied</li>
|
||||
<li>Section <a href="#org9eff470">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li>
|
||||
<li>Section <a href="#orgb8efc36">3.2</a>: The plant is identified from the Simscape model and the system coupling is shown</li>
|
||||
<li>Section <a href="#org9d45510">3.3</a>: The plant is first decoupled using the Jacobian</li>
|
||||
<li>Section <a href="#orgbe757a9">3.4</a>: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)</li>
|
||||
<li>Section <a href="#orgb593bce">3.5</a>: The decoupling is performed thanks to the SVD</li>
|
||||
<li>Section <a href="#org9c68bed">3.6</a>: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii</li>
|
||||
<li>Section <a href="#orgc065295">3.7</a>: The dynamics of the decoupled plants are shown</li>
|
||||
<li>Section <a href="#orgaf53d60">3.8</a>: A diagonal controller is defined to control the decoupled plant</li>
|
||||
<li>Section <a href="#org60a86ad">3.9</a>: Finally, the closed loop system properties are studied</li>
|
||||
</ul>
|
||||
</div>
|
||||
<div id="outline-container-org698a574" class="outline-3">
|
||||
<h3 id="org698a574"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3>
|
||||
<div id="outline-container-org98f27a1" class="outline-3">
|
||||
<h3 id="org98f27a1"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
<a id="org46c7682"></a>
|
||||
<a id="org9eff470"></a>
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">open(<span class="org-string">'drone_platform.slx'</span>);
|
||||
@@ -864,14 +864,14 @@ Kc = tf(zeros(6));
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org48cc1aa" class="figure">
|
||||
<div id="orgf541900" class="figure">
|
||||
<p><img src="figs/stewart_simscape.png" alt="stewart_simscape.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>General view of the Simscape Model</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd93f514" class="figure">
|
||||
<div id="orge4629ec" class="figure">
|
||||
<p><img src="figs/stewart_platform_details.png" alt="stewart_platform_details.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Simscape model of the Stewart platform</p>
|
||||
@@ -879,15 +879,15 @@ Kc = tf(zeros(6));
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdfc6136" class="outline-3">
|
||||
<h3 id="orgdfc6136"><span class="section-number-3">3.2</span> Identification of the plant</h3>
|
||||
<div id="outline-container-orgfc4057f" class="outline-3">
|
||||
<h3 id="orgfc4057f"><span class="section-number-3">3.2</span> Identification of the plant</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
<a id="orgacf8e97"></a>
|
||||
<a id="orgb8efc36"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The plant shown in Figure <a href="#org6611cbe">8</a> is identified from the Simscape model.
|
||||
The plant shown in Figure <a href="#orge3a32c6">8</a> is identified from the Simscape model.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -903,10 +903,10 @@ The outputs are the 6 accelerations measured by the inertial unit.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org6611cbe" class="figure">
|
||||
<div id="orge3a32c6" class="figure">
|
||||
<p><img src="figs/stewart_platform_plant.png" alt="stewart_platform_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform</p>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
@@ -923,6 +923,11 @@ G = linearize(mdl, io);
|
||||
G.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Rwx'</span>, <span class="org-string">'Rwy'</span>, <span class="org-string">'Rwz'</span>, ...
|
||||
<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Ax'</span>, <span class="org-string">'Ay'</span>, <span class="org-string">'Az'</span>, <span class="org-string">'Arx'</span>, <span class="org-string">'Ary'</span>, <span class="org-string">'Arz'</span>};
|
||||
|
||||
<span class="org-comment">% Plant</span>
|
||||
Gu = G(<span class="org-type">:</span>, {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>});
|
||||
<span class="org-comment">% Disturbance dynamics</span>
|
||||
Gd = G(<span class="org-type">:</span>, {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Rwx'</span>, <span class="org-string">'Rwy'</span>, <span class="org-string">'Rwz'</span>});
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -940,7 +945,7 @@ State-space model with 6 outputs, 12 inputs, and 24 states.
|
||||
|
||||
|
||||
<p>
|
||||
The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure <a href="#org3e2e269">9</a>.
|
||||
The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure <a href="#org602aa13">9</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -948,25 +953,25 @@ One can easily see that the system is strongly coupled.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org3e2e269" class="figure">
|
||||
<div id="org602aa13" class="figure">
|
||||
<p><img src="figs/stewart_platform_coupled_plant.png" alt="stewart_platform_coupled_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Magnitude of all 36 elements of the transfer function matrix \(\bm{G}\)</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Magnitude of all 36 elements of the transfer function matrix \(G_u\)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgadaff5c" class="outline-3">
|
||||
<h3 id="orgadaff5c"><span class="section-number-3">3.3</span> Physical Decoupling using the Jacobian</h3>
|
||||
<div id="outline-container-org06bff3b" class="outline-3">
|
||||
<h3 id="org06bff3b"><span class="section-number-3">3.3</span> Physical Decoupling using the Jacobian</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
<a id="orgf5489c1"></a>
|
||||
Consider the control architecture shown in Figure <a href="#orgeef0f77">10</a>.
|
||||
<a id="org9d45510"></a>
|
||||
Consider the control architecture shown in Figure <a href="#org1c673db">10</a>.
|
||||
The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgeef0f77" class="figure">
|
||||
<div id="org1c673db" class="figure">
|
||||
<p><img src="figs/plant_decouple_jacobian.png" alt="plant_decouple_jacobian.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)</p>
|
||||
@@ -982,31 +987,27 @@ We define a new plant:
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Gx = G<span class="org-type">*</span>blkdiag(eye(6), inv(J<span class="org-type">'</span>));
|
||||
Gx.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Rwx'</span>, <span class="org-string">'Rwy'</span>, <span class="org-string">'Rwz'</span>, ...
|
||||
<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
<pre class="src src-matlab">Gx = Gu<span class="org-type">*</span>inv(J<span class="org-type">'</span>);
|
||||
Gx.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6ba1c1a" class="outline-3">
|
||||
<h3 id="org6ba1c1a"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3>
|
||||
<div id="outline-container-org7208fcb" class="outline-3">
|
||||
<h3 id="org7208fcb"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<p>
|
||||
<a id="orgccd7599"></a>
|
||||
<a id="orgbe757a9"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).
|
||||
Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_u(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30; <span class="org-comment">% Decoupling frequency [rad/s]</span>
|
||||
|
||||
Gc = G({<span class="org-string">'Ax'</span>, <span class="org-string">'Ay'</span>, <span class="org-string">'Az'</span>, <span class="org-string">'Arx'</span>, <span class="org-string">'Ary'</span>, <span class="org-string">'Arz'</span>}, ...
|
||||
{<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}); <span class="org-comment">% Transfer function to find a real approximation</span>
|
||||
|
||||
H1 = evalfr(Gc, <span class="org-constant">j</span><span class="org-type">*</span>wc);
|
||||
H1 = evalfr(Gu, <span class="org-constant">j</span><span class="org-type">*</span>wc);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1094,9 +1095,9 @@ H1 = inv(D<span class="org-type">*</span>real(H1<span class="org-type">'*</span>
|
||||
|
||||
|
||||
<p>
|
||||
Note that the plant \(G\) at \(\omega_c\) is already an almost real matrix.
|
||||
Note that the plant \(G_u\) at \(\omega_c\) is already an almost real matrix.
|
||||
This can be seen on the Bode plots where the phase is close to 1.
|
||||
This can be verified below where only the real value of \(G(\omega_c)\) is shown
|
||||
This can be verified below where only the real value of \(G_u(\omega_c)\) is shown
|
||||
</p>
|
||||
|
||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
@@ -1174,11 +1175,11 @@ This can be verified below where only the real value of \(G(\omega_c)\) is shown
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7e2a42e" class="outline-3">
|
||||
<h3 id="org7e2a42e"><span class="section-number-3">3.5</span> SVD Decoupling</h3>
|
||||
<div id="outline-container-orgdcfefc4" class="outline-3">
|
||||
<h3 id="orgdcfefc4"><span class="section-number-3">3.5</span> SVD Decoupling</h3>
|
||||
<div class="outline-text-3" id="text-3-5">
|
||||
<p>
|
||||
<a id="org5abe937"></a>
|
||||
<a id="orgb593bce"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -1187,16 +1188,16 @@ First, the Singular Value Decomposition of \(H_1\) is performed:
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">[U,S,V] = svd(H1);
|
||||
<pre class="src src-matlab">[U,<span class="org-type">~</span>,V] = svd(H1);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure <a href="#orga151aa3">11</a>.
|
||||
The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure <a href="#orgfbe015c">11</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orga151aa3" class="figure">
|
||||
<div id="orgfbe015c" class="figure">
|
||||
<p><img src="figs/plant_decouple_svd.png" alt="plant_decouple_svd.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition</p>
|
||||
@@ -1204,22 +1205,32 @@ The obtained matrices \(U\) and \(V\) are used to decouple the system as shown i
|
||||
|
||||
<p>
|
||||
The decoupled plant is then:
|
||||
\[ G_{SVD}(s) = U^{-1} G(s) V^{-H} \]
|
||||
\[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Gsvd = inv(U)<span class="org-type">*</span>Gu<span class="org-type">*</span>inv(V<span class="org-type">'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc6f3016" class="outline-3">
|
||||
<h3 id="orgc6f3016"><span class="section-number-3">3.6</span> Verification of the decoupling using the “Gershgorin Radii”</h3>
|
||||
<div id="outline-container-orgeedb4ac" class="outline-3">
|
||||
<h3 id="orgeedb4ac"><span class="section-number-3">3.6</span> Verification of the decoupling using the “Gershgorin Radii”</h3>
|
||||
<div class="outline-text-3" id="text-3-6">
|
||||
<p>
|
||||
<a id="org7691f01"></a>
|
||||
<a id="org9c68bed"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The “Gershgorin Radii” is computed for the coupled plant \(G(s)\), for the “Jacobian plant” \(G_x(s)\) and the “SVD Decoupled Plant” \(G_{SVD}(s)\):
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The “Gershgorin Radii” of a matrix \(S\) is defined by:
|
||||
\[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This is computed over the following frequencies.
|
||||
</p>
|
||||
@@ -1229,7 +1240,7 @@ This is computed over the following frequencies.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgea46431" class="figure">
|
||||
<div id="org0864583" class="figure">
|
||||
<p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Gershgorin Radii of the Coupled and Decoupled plants</p>
|
||||
@@ -1237,30 +1248,30 @@ This is computed over the following frequencies.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgedf5c94" class="outline-3">
|
||||
<h3 id="orgedf5c94"><span class="section-number-3">3.7</span> Obtained Decoupled Plants</h3>
|
||||
<div id="outline-container-orga3edea8" class="outline-3">
|
||||
<h3 id="orga3edea8"><span class="section-number-3">3.7</span> Obtained Decoupled Plants</h3>
|
||||
<div class="outline-text-3" id="text-3-7">
|
||||
<p>
|
||||
<a id="orgabc21ab"></a>
|
||||
<a id="orgc065295"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure <a href="#org0947ccc">13</a>.
|
||||
The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure <a href="#org57489d0">13</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org0947ccc" class="figure">
|
||||
<div id="org57489d0" class="figure">
|
||||
<p><img src="figs/simscape_model_decoupled_plant_svd.png" alt="simscape_model_decoupled_plant_svd.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Decoupled Plant using SVD</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure <a href="#org14484c8">14</a>.
|
||||
Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure <a href="#orga4b3cd1">14</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org14484c8" class="figure">
|
||||
<div id="orga4b3cd1" class="figure">
|
||||
<p><img src="figs/simscape_model_decoupled_plant_jacobian.png" alt="simscape_model_decoupled_plant_jacobian.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)</p>
|
||||
@@ -1268,15 +1279,12 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgff44b51" class="outline-3">
|
||||
<h3 id="orgff44b51"><span class="section-number-3">3.8</span> Diagonal Controller</h3>
|
||||
<div id="outline-container-orgb371cb1" class="outline-3">
|
||||
<h3 id="orgb371cb1"><span class="section-number-3">3.8</span> Diagonal Controller</h3>
|
||||
<div class="outline-text-3" id="text-3-8">
|
||||
<p>
|
||||
<a id="org9620c1c"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The control diagram for the centralized control is shown in Figure <a href="#org656626f">15</a>.
|
||||
<a id="orgaf53d60"></a>
|
||||
The control diagram for the centralized control is shown in Figure <a href="#org457c7b6">15</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -1285,19 +1293,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org656626f" class="figure">
|
||||
<div id="org457c7b6" class="figure">
|
||||
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Control Diagram for the Centralized control</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The SVD control architecture is shown in Figure <a href="#org0f3cfd0">16</a>.
|
||||
The SVD control architecture is shown in Figure <a href="#org84af546">16</a>.
|
||||
The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org0f3cfd0" class="figure">
|
||||
<div id="org84af546" class="figure">
|
||||
<p><img src="figs/svd_control.png" alt="svd_control.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Control Diagram for the SVD control</p>
|
||||
@@ -1314,31 +1322,31 @@ We choose the controller to be a low pass filter:
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>80;
|
||||
w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>0.1;
|
||||
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>80; <span class="org-comment">% Crossover Frequency [rad/s]</span>
|
||||
w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>0.1; <span class="org-comment">% Controller Pole [rad/s]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">K_cen = diag(1<span class="org-type">./</span>diag(abs(evalfr(Gx(1<span class="org-type">:</span>6, 7<span class="org-type">:</span>12), <span class="org-constant">j</span><span class="org-type">*</span>wc))))<span class="org-type">*</span>(1<span class="org-type">/</span>abs(evalfr(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0), <span class="org-constant">j</span><span class="org-type">*</span>wc)))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0);
|
||||
L_cen = K_cen<span class="org-type">*</span>Gx(1<span class="org-type">:</span>6, 7<span class="org-type">:</span>12);
|
||||
<pre class="src src-matlab">K_cen = diag(1<span class="org-type">./</span>diag(abs(evalfr(Gx, <span class="org-constant">j</span><span class="org-type">*</span>wc))))<span class="org-type">*</span>(1<span class="org-type">/</span>abs(evalfr(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0), <span class="org-constant">j</span><span class="org-type">*</span>wc)))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0);
|
||||
L_cen = K_cen<span class="org-type">*</span>Gx;
|
||||
G_cen = feedback(G, pinv(J<span class="org-type">'</span>)<span class="org-type">*</span>K_cen, [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">K_svd = diag(1<span class="org-type">./</span>diag(abs(evalfr(Gd, <span class="org-constant">j</span><span class="org-type">*</span>wc))))<span class="org-type">*</span>(1<span class="org-type">/</span>abs(evalfr(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0), <span class="org-constant">j</span><span class="org-type">*</span>wc)))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0);
|
||||
L_svd = K_svd<span class="org-type">*</span>Gd;
|
||||
<pre class="src src-matlab">K_svd = diag(1<span class="org-type">./</span>diag(abs(evalfr(Gsvd, <span class="org-constant">j</span><span class="org-type">*</span>wc))))<span class="org-type">*</span>(1<span class="org-type">/</span>abs(evalfr(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0), <span class="org-constant">j</span><span class="org-type">*</span>wc)))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0);
|
||||
L_svd = K_svd<span class="org-type">*</span>Gsvd;
|
||||
G_svd = feedback(G, inv(V<span class="org-type">'</span>)<span class="org-type">*</span>K_svd<span class="org-type">*</span>inv(U), [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The obtained diagonal elements of the loop gains are shown in Figure <a href="#orgc313067">17</a>.
|
||||
The obtained diagonal elements of the loop gains are shown in Figure <a href="#org51e7e94">17</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgc313067" class="figure">
|
||||
<div id="org51e7e94" class="figure">
|
||||
<p><img src="figs/stewart_comp_loop_gain_diagonal.png" alt="stewart_comp_loop_gain_diagonal.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one</p>
|
||||
@@ -1346,11 +1354,11 @@ The obtained diagonal elements of the loop gains are shown in Figure <a href="#o
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org949d9ca" class="outline-3">
|
||||
<h3 id="org949d9ca"><span class="section-number-3">3.9</span> Closed-Loop system Performances</h3>
|
||||
<div id="outline-container-orgb6d90eb" class="outline-3">
|
||||
<h3 id="orgb6d90eb"><span class="section-number-3">3.9</span> Closed-Loop system Performances</h3>
|
||||
<div class="outline-text-3" id="text-3-9">
|
||||
<p>
|
||||
<a id="org823e1cb"></a>
|
||||
<a id="org60a86ad"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -1381,11 +1389,11 @@ ans =
|
||||
|
||||
|
||||
<p>
|
||||
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org32234b0">18</a>.
|
||||
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org5c79b0b">18</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org32234b0" class="figure">
|
||||
<div id="org5c79b0b" class="figure">
|
||||
<p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Obtained Transmissibility</p>
|
||||
@@ -1396,7 +1404,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-11-09 lun. 10:54</p>
|
||||
<p class="date">Created: 2020-11-09 lun. 14:36</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
144
index.org
144
index.org
@@ -713,7 +713,7 @@ The analysis of the SVD control applied to the Stewart platform is performed in
|
||||
- Section [[sec:stewart_diagonal_control]]: A diagonal controller is defined to control the decoupled plant
|
||||
- Section [[sec:stewart_closed_loop_results]]: Finally, the closed loop system properties are studied
|
||||
|
||||
** Matlab Init :noexport:ignore:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
@@ -820,7 +820,7 @@ The outputs are the 6 accelerations measured by the inertial unit.
|
||||
|
||||
#+begin_src latex :file stewart_platform_plant.pdf :tangle no :exports results
|
||||
\begin{tikzpicture}
|
||||
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$};
|
||||
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
|
||||
\node[above] at (G.north) {$\bm{G}$};
|
||||
|
||||
% Inputs of the controllers
|
||||
@@ -835,7 +835,7 @@ The outputs are the 6 accelerations measured by the inertial unit.
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_platform_plant
|
||||
#+caption: Considered plant $\bm{G} = \begin{bmatrix}G_d\\G\end{bmatrix}$. $D_w$ is the translation/rotation of the support, $\tau$ the actuator forces, $a$ the acceleration/angular acceleration of the top platform
|
||||
#+caption: Considered plant $\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}$. $D_w$ is the translation/rotation of the support, $\tau$ the actuator forces, $a$ the acceleration/angular acceleration of the top platform
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_plant.png]]
|
||||
|
||||
@@ -853,6 +853,11 @@ The outputs are the 6 accelerations measured by the inertial unit.
|
||||
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
|
||||
|
||||
% Plant
|
||||
Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
|
||||
% Disturbance dynamics
|
||||
Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
|
||||
#+end_src
|
||||
|
||||
There are 24 states (6dof for the bottom platform + 6dof for the top platform).
|
||||
@@ -876,15 +881,15 @@ One can easily see that the system is strongly coupled.
|
||||
hold on;
|
||||
for i_in = 1:6
|
||||
for i_out = [1:i_in-1, i_in+1:6]
|
||||
plot(freqs, abs(squeeze(freqresp(G(i_out, 6+i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
plot(freqs, abs(squeeze(freqresp(G(i_out, 6+i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
'DisplayName', '$G(i,j)\ i \neq j$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
'DisplayName', '$G_u(i,j)\ i \neq j$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
for i_in_out = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G(i_in_out, 6+i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G(%d,%d)$', i_in_out, i_in_out));
|
||||
plot(freqs, abs(squeeze(freqresp(Gu(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_u(%d,%d)$', i_in_out, i_in_out));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
@@ -898,7 +903,7 @@ One can easily see that the system is strongly coupled.
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_platform_coupled_plant
|
||||
#+caption: Magnitude of all 36 elements of the transfer function matrix $\bm{G}$
|
||||
#+caption: Magnitude of all 36 elements of the transfer function matrix $G_u$
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_coupled_plant.png]]
|
||||
|
||||
@@ -909,22 +914,16 @@ The Jacobian matrix is used to transform forces/torques applied on the top platf
|
||||
|
||||
#+begin_src latex :file plant_decouple_jacobian.pdf :tangle no :exports results
|
||||
\begin{tikzpicture}
|
||||
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$};
|
||||
|
||||
% Inputs of the controllers
|
||||
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
|
||||
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
|
||||
|
||||
\node[block, left=0.6 of inputu] (J) {$J^{-T}$};
|
||||
\node[block] (G) {$G_u$};
|
||||
\node[block, left=0.6 of G] (J) {$J^{-T}$};
|
||||
|
||||
% Connections and labels
|
||||
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
|
||||
\draw[->] (G.east) -- ++( 0.8, 0) node[above left]{$a$};
|
||||
\draw[->] (J.east) -- (inputu) node[above left]{$\tau$};
|
||||
\draw[<-] (J.west) -- ++(-0.8, 0) node[above right]{$\mathcal{F}$};
|
||||
\draw[<-] (J.west) -- ++(-1.0, 0) node[above right]{$\mathcal{F}$};
|
||||
\draw[->] (J.east) -- (G.west) node[above left]{$\tau$};
|
||||
\draw[->] (G.east) -- ++( 1.0, 0) node[above left]{$a$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(J.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=8pt] (Gx) {};
|
||||
\node[fit={(J.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
|
||||
\node[below right] at (Gx.north west) {$\bm{G}_x$};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
@@ -941,22 +940,18 @@ We define a new plant:
|
||||
$G_x(s)$ correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.
|
||||
|
||||
#+begin_src matlab
|
||||
Gx = G*blkdiag(eye(6), inv(J'));
|
||||
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
Gx = Gu*inv(J');
|
||||
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
#+end_src
|
||||
|
||||
** Real Approximation of $G$ at the decoupling frequency
|
||||
<<sec:stewart_real_approx>>
|
||||
|
||||
Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
|
||||
Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_u(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
|
||||
#+begin_src matlab
|
||||
wc = 2*pi*30; % Decoupling frequency [rad/s]
|
||||
|
||||
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
|
||||
{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
|
||||
|
||||
H1 = evalfr(Gc, j*wc);
|
||||
H1 = evalfr(Gu, j*wc);
|
||||
#+end_src
|
||||
|
||||
The real approximation is computed as follows:
|
||||
@@ -979,12 +974,12 @@ The real approximation is computed as follows:
|
||||
| 220.6 | -220.6 | 220.6 | -220.6 | 220.6 | -220.6 |
|
||||
|
||||
|
||||
Note that the plant $G$ at $\omega_c$ is already an almost real matrix.
|
||||
Note that the plant $G_u$ at $\omega_c$ is already an almost real matrix.
|
||||
This can be seen on the Bode plots where the phase is close to 1.
|
||||
This can be verified below where only the real value of $G(\omega_c)$ is shown
|
||||
This can be verified below where only the real value of $G_u(\omega_c)$ is shown
|
||||
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no
|
||||
data2orgtable(real(evalfr(Gc, j*wc)), {}, {}, ' %.1f ');
|
||||
data2orgtable(real(evalfr(Gu, j*wc)), {}, {}, ' %.1f ');
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
@@ -1002,31 +997,26 @@ First, the Singular Value Decomposition of $H_1$ is performed:
|
||||
\[ H_1 = U \Sigma V^H \]
|
||||
|
||||
#+begin_src matlab
|
||||
[U,S,V] = svd(H1);
|
||||
[U,~,V] = svd(H1);
|
||||
#+end_src
|
||||
|
||||
The obtained matrices $U$ and $V$ are used to decouple the system as shown in Figure [[fig:plant_decouple_svd]].
|
||||
|
||||
#+begin_src latex :file plant_decouple_svd.pdf :tangle no :exports results
|
||||
\begin{tikzpicture}
|
||||
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$};
|
||||
\node[block] (G) {$G_u$};
|
||||
|
||||
% Inputs of the controllers
|
||||
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
|
||||
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
|
||||
|
||||
\node[block, left=0.6 of inputu] (V) {$V^{-T}$};
|
||||
\node[block, left=0.6 of G.west] (V) {$V^{-T}$};
|
||||
\node[block, right=0.6 of G.east] (U) {$U^{-1}$};
|
||||
|
||||
% Connections and labels
|
||||
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
|
||||
\draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$};
|
||||
\draw[->] (V.east) -- (G.west) node[above left]{$\tau$};
|
||||
\draw[->] (G.east) -- (U.west) node[above left]{$a$};
|
||||
\draw[->] (U.east) -- ++( 0.8, 0) node[above left]{$y$};
|
||||
\draw[->] (V.east) -- (inputu) node[above left]{$\tau$};
|
||||
\draw[<-] (V.west) -- ++(-0.8, 0) node[above right]{$u$};
|
||||
\draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=8pt] (Gsvd) {};
|
||||
\node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {};
|
||||
\node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
@@ -1038,13 +1028,20 @@ The obtained matrices $U$ and $V$ are used to decouple the system as shown in Fi
|
||||
[[file:figs/plant_decouple_svd.png]]
|
||||
|
||||
The decoupled plant is then:
|
||||
\[ G_{SVD}(s) = U^{-1} G(s) V^{-H} \]
|
||||
\[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
|
||||
|
||||
#+begin_src matlab
|
||||
Gsvd = inv(U)*Gu*inv(V');
|
||||
#+end_src
|
||||
|
||||
** Verification of the decoupling using the "Gershgorin Radii"
|
||||
<<sec:comp_decoupling>>
|
||||
|
||||
The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$:
|
||||
|
||||
The "Gershgorin Radii" of a matrix $S$ is defined by:
|
||||
\[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
|
||||
|
||||
This is computed over the following frequencies.
|
||||
#+begin_src matlab
|
||||
freqs = logspace(-2, 2, 1000); % [Hz]
|
||||
@@ -1052,29 +1049,23 @@ This is computed over the following frequencies.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
% Gershgorin Radii for the coupled plant:
|
||||
Gr_coupled = zeros(length(freqs), size(Gc,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gc,2)
|
||||
Gr_coupled = zeros(length(freqs), size(Gu,2));
|
||||
H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gu,2)
|
||||
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
% Gershgorin Radii for the decoupled plant using SVD:
|
||||
Gd = inv(U)*Gc*inv(V');
|
||||
Gr_decoupled = zeros(length(freqs), size(Gd,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gd,2)
|
||||
Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
|
||||
H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gsvd,2)
|
||||
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
% Gershgorin Radii for the decoupled plant using the Jacobian:
|
||||
Gj = Gc*inv(J');
|
||||
Gr_jacobian = zeros(length(freqs), size(Gj,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
|
||||
|
||||
for out_i = 1:size(Gj,2)
|
||||
Gr_jacobian = zeros(length(freqs), size(Gx,2));
|
||||
H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gx,2)
|
||||
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
#+end_src
|
||||
@@ -1126,15 +1117,15 @@ The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown i
|
||||
hold on;
|
||||
for i_in = 1:6
|
||||
for i_out = [1:i_in-1, i_in+1:6]
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
|
||||
plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
|
||||
'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
for ch_i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
|
||||
plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
|
||||
end
|
||||
hold off;
|
||||
@@ -1147,7 +1138,7 @@ The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown i
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for ch_i = 1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
@@ -1180,7 +1171,7 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
|
||||
hold on;
|
||||
for i_in = 1:6
|
||||
for i_out = [1:i_in-1, i_in+1:6]
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(i_out, 6+i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
@@ -1228,14 +1219,6 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
|
||||
|
||||
** Diagonal Controller
|
||||
<<sec:stewart_diagonal_control>>
|
||||
|
||||
#+begin_src matlab :exports none :tangle no
|
||||
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||
C_g = 50; % DC Gain
|
||||
|
||||
Kc = eye(6)*C_g/(s+wc);
|
||||
#+end_src
|
||||
|
||||
The control diagram for the centralized control is shown in Figure [[fig:centralized_control]].
|
||||
|
||||
The controller $K_c$ is "working" in an cartesian frame.
|
||||
@@ -1243,7 +1226,7 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
|
||||
|
||||
#+begin_src latex :file centralized_control.pdf :tangle no :exports results
|
||||
\begin{tikzpicture}
|
||||
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$};
|
||||
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
|
||||
\node[above] at (G.north) {$\bm{G}$};
|
||||
\node[block, below right=0.6 and -0.5 of G] (K) {$K_c$};
|
||||
\node[block, below left= 0.6 and -0.5 of G] (J) {$J^{-T}$};
|
||||
@@ -1271,7 +1254,8 @@ The matrices $U$ and $V$ are used to decoupled the plant $G$.
|
||||
|
||||
#+begin_src latex :file svd_control.pdf :tangle no :exports results
|
||||
\begin{tikzpicture}
|
||||
\node[block={2cm}{1.5cm}] (G) {$G$};
|
||||
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
|
||||
\node[above] at (G.north) {$\bm{G}$};
|
||||
\node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$};
|
||||
\node[block, below=0.6 of G] (K) {$K_{\text{SVD}}$};
|
||||
\node[block, below left= 0.6 and 0 of G] (V) {$V^{-T}$};
|
||||
@@ -1302,19 +1286,19 @@ We choose the controller to be a low pass filter:
|
||||
$G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
|
||||
|
||||
#+begin_src matlab
|
||||
wc = 2*pi*80;
|
||||
w0 = 2*pi*0.1;
|
||||
wc = 2*pi*80; % Crossover Frequency [rad/s]
|
||||
w0 = 2*pi*0.1; % Controller Pole [rad/s]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
K_cen = diag(1./diag(abs(evalfr(Gx(1:6, 7:12), j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
|
||||
L_cen = K_cen*Gx(1:6, 7:12);
|
||||
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
|
||||
L_cen = K_cen*Gx;
|
||||
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
K_svd = diag(1./diag(abs(evalfr(Gd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
|
||||
L_svd = K_svd*Gd;
|
||||
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
|
||||
L_svd = K_svd*Gsvd;
|
||||
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
|
||||
#+end_src
|
||||
|
||||
|
||||
@@ -6,42 +6,14 @@ s = zpk('s');
|
||||
|
||||
addpath('STEP');
|
||||
|
||||
% Jacobian
|
||||
% First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
|
||||
|
||||
open('drone_platform_jacobian.slx');
|
||||
|
||||
sim('drone_platform_jacobian');
|
||||
|
||||
Aa = [a1.Data(1,:);
|
||||
a2.Data(1,:);
|
||||
a3.Data(1,:);
|
||||
a4.Data(1,:);
|
||||
a5.Data(1,:);
|
||||
a6.Data(1,:)]';
|
||||
|
||||
Ab = [b1.Data(1,:);
|
||||
b2.Data(1,:);
|
||||
b3.Data(1,:);
|
||||
b4.Data(1,:);
|
||||
b5.Data(1,:);
|
||||
b6.Data(1,:)]';
|
||||
|
||||
As = (Ab - Aa)./vecnorm(Ab - Aa);
|
||||
|
||||
l = vecnorm(Ab - Aa)';
|
||||
|
||||
J = [As' , cross(Ab, As)'];
|
||||
|
||||
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||
|
||||
% Simscape Model
|
||||
% Simscape Model - Parameters
|
||||
% <<sec:stewart_simscape>>
|
||||
|
||||
open('drone_platform.slx');
|
||||
|
||||
|
||||
|
||||
% Definition of spring parameters
|
||||
% Definition of spring parameters:
|
||||
|
||||
kx = 0.5*1e3/3; % [N/m]
|
||||
ky = 0.5*1e3/3;
|
||||
@@ -51,31 +23,53 @@ cx = 0.025; % [Nm/rad]
|
||||
cy = 0.025;
|
||||
cz = 0.025;
|
||||
|
||||
|
||||
|
||||
% Gravity:
|
||||
|
||||
g = 0;
|
||||
|
||||
|
||||
|
||||
% We load the Jacobian.
|
||||
% We load the Jacobian (previously computed from the geometry):
|
||||
|
||||
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||
|
||||
% Identification of the plant
|
||||
% The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
|
||||
|
||||
|
||||
% We initialize other parameters:
|
||||
|
||||
U = eye(6);
|
||||
V = eye(6);
|
||||
Kc = tf(zeros(6));
|
||||
|
||||
|
||||
|
||||
% #+name: fig:stewart_platform_plant
|
||||
% #+caption: Considered plant $\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}$. $D_w$ is the translation/rotation of the support, $\tau$ the actuator forces, $a$ the acceleration/angular acceleration of the top platform
|
||||
% #+RESULTS:
|
||||
% [[file:figs/stewart_platform_plant.png]]
|
||||
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'drone_platform';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion
|
||||
io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces
|
||||
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
|
||||
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
|
||||
|
||||
% Plant
|
||||
Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
|
||||
% Disturbance dynamics
|
||||
Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
|
||||
|
||||
|
||||
|
||||
% There are 24 states (6dof for the bottom platform + 6dof for the top platform).
|
||||
@@ -87,176 +81,59 @@ size(G)
|
||||
% #+RESULTS:
|
||||
% : State-space model with 6 outputs, 12 inputs, and 24 states.
|
||||
|
||||
% The elements of the transfer matrix $\bm{G}$ corresponding to the transfer function from actuator forces $\tau$ to the measured acceleration $a$ are shown in Figure [[fig:stewart_platform_coupled_plant]].
|
||||
|
||||
% G = G*blkdiag(inv(J), eye(6));
|
||||
% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
|
||||
% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
|
||||
|
||||
|
||||
% Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
|
||||
|
||||
Gx = G*blkdiag(eye(6), inv(J'));
|
||||
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
|
||||
% Gl = J*G;
|
||||
% Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
|
||||
|
||||
% Obtained Dynamics
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:stewart_platform_translations
|
||||
% #+caption: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform
|
||||
% #+RESULTS:
|
||||
% [[file:figs/stewart_platform_translations.png]]
|
||||
% One can easily see that the system is strongly coupled.
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
% Magnitude
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:stewart_platform_rotations
|
||||
% #+caption: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform
|
||||
% #+RESULTS:
|
||||
% [[file:figs/stewart_platform_rotations.png]]
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for out_i = 1:5
|
||||
for in_i = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', out_i), sprintf('F%i', in_i)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||
end
|
||||
for i_in = 1:6
|
||||
for i_out = [1:i_in-1, i_in+1:6]
|
||||
plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
for ch_i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
'DisplayName', '$G_u(i,j)\ i \neq j$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
for i_in_out = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gu(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_u(%d,%d)$', i_in_out, i_in_out));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for ch_i = 1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
ylim([1e-2, 1e5]);
|
||||
legend('location', 'northwest');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:stewart_platform_legs
|
||||
% #+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
|
||||
% #+name: fig:plant_decouple_jacobian
|
||||
% #+caption: Decoupled plant $\bm{G}_x$ using the Jacobian matrix $J$
|
||||
% #+RESULTS:
|
||||
% [[file:figs/stewart_platform_legs.png]]
|
||||
% [[file:figs/plant_decouple_jacobian.png]]
|
||||
|
||||
% We define a new plant:
|
||||
% \[ G_x(s) = G(s) J^{-T} \]
|
||||
|
||||
% $G_x(s)$ correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$');
|
||||
% set(gca,'ColorOrderIndex',1)
|
||||
% plot(freqs, abs(squeeze(freqresp(TR(1,1), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
% plot(freqs, abs(squeeze(freqresp(TR(2,2), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
% plot(freqs, abs(squeeze(freqresp(TR(3,3), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - Translations'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$');
|
||||
% set(gca,'ColorOrderIndex',1)
|
||||
% plot(freqs, abs(squeeze(freqresp(TR(4,4), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
% plot(freqs, abs(squeeze(freqresp(TR(5,5), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
% plot(freqs, abs(squeeze(freqresp(TR(6,6), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - Rotations'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
Gx = Gu*inv(J');
|
||||
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
|
||||
% Real Approximation of $G$ at the decoupling frequency
|
||||
% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
|
||||
% <<sec:stewart_real_approx>>
|
||||
|
||||
% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_u(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
|
||||
|
||||
wc = 2*pi*30; % Decoupling frequency [rad/s]
|
||||
|
||||
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
|
||||
{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
|
||||
|
||||
H1 = evalfr(Gc, j*wc);
|
||||
H1 = evalfr(Gu, j*wc);
|
||||
|
||||
|
||||
|
||||
@@ -265,55 +142,58 @@ H1 = evalfr(Gc, j*wc);
|
||||
D = pinv(real(H1'*H1));
|
||||
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
|
||||
|
||||
% Verification of the decoupling using the "Gershgorin Radii"
|
||||
% SVD Decoupling
|
||||
% <<sec:stewart_svd_decoupling>>
|
||||
|
||||
% First, the Singular Value Decomposition of $H_1$ is performed:
|
||||
% \[ H_1 = U \Sigma V^H \]
|
||||
|
||||
|
||||
[U,S,V] = svd(H1);
|
||||
[U,~,V] = svd(H1);
|
||||
|
||||
|
||||
|
||||
% Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
|
||||
% \[ G_d(s) = U^T G_c(s) V \]
|
||||
% #+name: fig:plant_decouple_svd
|
||||
% #+caption: Decoupled plant $\bm{G}_{SVD}$ using the Singular Value Decomposition
|
||||
% #+RESULTS:
|
||||
% [[file:figs/plant_decouple_svd.png]]
|
||||
|
||||
% The decoupled plant is then:
|
||||
% \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
|
||||
|
||||
|
||||
Gsvd = inv(U)*Gu*inv(V');
|
||||
|
||||
% Verification of the decoupling using the "Gershgorin Radii"
|
||||
% <<sec:comp_decoupling>>
|
||||
|
||||
% The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$:
|
||||
|
||||
% The "Gershgorin Radii" of a matrix $S$ is defined by:
|
||||
% \[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
|
||||
|
||||
% This is computed over the following frequencies.
|
||||
|
||||
freqs = logspace(-2, 2, 1000); % [Hz]
|
||||
|
||||
|
||||
|
||||
% Gershgorin Radii for the coupled plant:
|
||||
|
||||
Gr_coupled = zeros(length(freqs), size(Gc,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gc,2)
|
||||
Gr_coupled = zeros(length(freqs), size(Gu,2));
|
||||
H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gu,2)
|
||||
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
|
||||
|
||||
% Gershgorin Radii for the decoupled plant using SVD:
|
||||
|
||||
Gd = U'*Gc*V;
|
||||
Gr_decoupled = zeros(length(freqs), size(Gd,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gd,2)
|
||||
Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
|
||||
H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gsvd,2)
|
||||
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
|
||||
|
||||
% Gershgorin Radii for the decoupled plant using the Jacobian:
|
||||
|
||||
Gj = Gc*inv(J');
|
||||
Gr_jacobian = zeros(length(freqs), size(Gj,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
|
||||
|
||||
for out_i = 1:size(Gj,2)
|
||||
Gr_jacobian = zeros(length(freqs), size(Gx,2));
|
||||
H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gx,2)
|
||||
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
@@ -334,31 +214,55 @@ plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
legend('location', 'northeast');
|
||||
legend('location', 'northwest');
|
||||
ylim([1e-3, 1e3]);
|
||||
|
||||
% Decoupled Plant
|
||||
% Let's see the bode plot of the decoupled plant $G_d(s)$.
|
||||
% \[ G_d(s) = U^T G_c(s) V \]
|
||||
% Obtained Decoupled Plants
|
||||
% <<sec:stewart_decoupled_plant>>
|
||||
|
||||
% The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:simscape_model_decoupled_plant_svd]].
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
for ch_i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
|
||||
for i_in = 1:6
|
||||
for i_out = [1:i_in-1, i_in+1:6]
|
||||
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
for in_i = 1:5
|
||||
for out_i = in_i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
|
||||
'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
for ch_i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast');
|
||||
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'northwest');
|
||||
ylim([1e-1, 1e5])
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for ch_i = 1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
|
||||
|
||||
@@ -367,53 +271,135 @@ legend('location', 'southeast');
|
||||
% #+RESULTS:
|
||||
% [[file:figs/simscape_model_decoupled_plant_svd.png]]
|
||||
|
||||
% Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure [[fig:simscape_model_decoupled_plant_jacobian]].
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
for ch_i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gj(ch_i, ch_i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
|
||||
end
|
||||
for in_i = 1:5
|
||||
for out_i = in_i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gj(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
for i_in = 1:6
|
||||
for i_out = [1:i_in-1, i_in+1:6]
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
|
||||
'DisplayName', '$G_x(i,j),\ i \neq j$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = A_z/F_z$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$G_x(4,4) = A_{R_x}/M_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$G_x(5,5) = A_{R_y}/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$G_x(6,6) = A_{R_z}/M_z$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast');
|
||||
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'northwest');
|
||||
ylim([1e-2, 2e6])
|
||||
|
||||
% Diagonal Controller
|
||||
% The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([0, 180]);
|
||||
yticks([0:45:360]);
|
||||
|
||||
|
||||
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||
C_g = 50; % DC Gain
|
||||
|
||||
K = eye(6)*C_g/(s+wc);
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% [[file:figs/centralized_control.png]]
|
||||
|
||||
|
||||
G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:svd_control
|
||||
% #+caption: Control Diagram for the SVD control
|
||||
% #+RESULTS:
|
||||
% [[file:figs/svd_control.png]]
|
||||
|
||||
% SVD Control
|
||||
|
||||
G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
|
||||
% We choose the controller to be a low pass filter:
|
||||
% \[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]
|
||||
|
||||
% $G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
|
||||
|
||||
|
||||
wc = 2*pi*80; % Crossover Frequency [rad/s]
|
||||
w0 = 2*pi*0.1; % Controller Pole [rad/s]
|
||||
|
||||
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
|
||||
L_cen = K_cen*Gx;
|
||||
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
|
||||
|
||||
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
|
||||
L_svd = K_svd*Gsvd;
|
||||
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
|
||||
|
||||
|
||||
|
||||
% The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]].
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
|
||||
for i_in_out = 2:6
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
|
||||
'DisplayName', '$L_{J}(i,i)$');
|
||||
for i_in_out = 2:6
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'northwest');
|
||||
ylim([5e-2, 2e3])
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for i_in_out = 1:6
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
for i_in_out = 1:6
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Closed-Loop system Performances
|
||||
% <<sec:stewart_closed_loop_results>>
|
||||
|
||||
% Results
|
||||
% Let's first verify the stability of the closed-loop systems:
|
||||
|
||||
isstable(G_cen)
|
||||
@@ -433,69 +419,61 @@ isstable(G_svd)
|
||||
% #+RESULTS:
|
||||
% : ans =
|
||||
% : logical
|
||||
% : 0
|
||||
% : 1
|
||||
|
||||
% The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
|
||||
|
||||
|
||||
freqs = logspace(-3, 3, 1000);
|
||||
freqs = logspace(-2, 2, 1000);
|
||||
|
||||
figure
|
||||
figure;
|
||||
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
ax1 = subplot(2, 3, 1);
|
||||
ax1 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD');
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), '--', 'DisplayName', 'SVD');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $D_x/D_{w,x}$'); xlabel('Frequency [Hz]');
|
||||
ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southwest');
|
||||
|
||||
ax2 = subplot(2, 3, 2);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $D_y/D_{w,y}$'); xlabel('Frequency [Hz]');
|
||||
|
||||
ax3 = subplot(2, 3, 3);
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))), '--');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $D_z/D_{w,z}$'); xlabel('Frequency [Hz]');
|
||||
ylabel('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax4 = subplot(2, 3, 4);
|
||||
ax3 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $R_x/R_{w,x}$'); xlabel('Frequency [Hz]');
|
||||
|
||||
ax5 = subplot(2, 3, 5);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))), '--');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))), '--');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
|
||||
ylabel('$R_x/R_{w,x}$, $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
|
||||
|
||||
ax6 = subplot(2, 3, 6);
|
||||
ax4 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))), '--');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
|
||||
ylabel('$R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
||||
linkaxes([ax1,ax2,ax3,ax4],'xy');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-3, 1e2]);
|
||||
|
||||
Reference in New Issue
Block a user