1508 lines
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1508 lines
80 KiB
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<!-- 2020-10-13 mar. 14:53 -->
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<title>SVD Control</title>
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<div id="org-div-home-and-up">
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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">SVD Control</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org6dd65c1">1. Gravimeter - Simscape Model</a>
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<ul>
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<li><a href="#org85dbe5c">1.1. Introduction</a></li>
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<li><a href="#org0b31481">1.2. Simscape Model - Parameters</a></li>
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<li><a href="#org949338c">1.3. System Identification - Without Gravity</a></li>
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<li><a href="#org3e8d708">1.4. System Identification - With Gravity</a></li>
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<li><a href="#org8263a33">1.5. Analytical Model</a>
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<ul>
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<li><a href="#org5ce809b">1.5.1. Parameters</a></li>
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<li><a href="#org485b7e0">1.5.2. Generation of the State Space Model</a></li>
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<li><a href="#orgb77d12b">1.5.3. Comparison with the Simscape Model</a></li>
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<li><a href="#orgbede3a4">1.5.4. Analysis</a></li>
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<li><a href="#org00d06a7">1.5.5. Control Section</a></li>
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<li><a href="#org8d48657">1.5.6. Greshgorin radius</a></li>
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<li><a href="#org7348f99">1.5.7. Injecting ground motion in the system to have the output</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org534f1d2">2. Gravimeter - Functions</a>
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<ul>
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<li><a href="#org8fd3468">2.1. <code>align</code></a></li>
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<li><a href="#org7fc9d1b">2.2. <code>pzmap_testCL</code></a></li>
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</ul>
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</li>
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<li><a href="#orga726921">3. Stewart Platform - Simscape Model</a>
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<ul>
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<li><a href="#org0f4c378">3.1. Jacobian</a></li>
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<li><a href="#org8e93915">3.2. Simscape Model</a></li>
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<li><a href="#orga80ad9d">3.3. Identification of the plant</a></li>
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<li><a href="#org820395d">3.4. Obtained Dynamics</a></li>
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<li><a href="#org531c180">3.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
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<li><a href="#org04886ad">3.6. Verification of the decoupling using the “Gershgorin Radii”</a></li>
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<li><a href="#org96683a8">3.7. Decoupled Plant</a></li>
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<li><a href="#orgde9fab8">3.8. Diagonal Controller</a></li>
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<li><a href="#org47bbca8">3.9. Centralized Control</a></li>
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<li><a href="#org2c1e3f7">3.10. SVD Control</a></li>
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<li><a href="#orgd6985da">3.11. Results</a></li>
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</ul>
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</li>
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<li><a href="#org99c6262">4. Stewart Platform - Analytical Model</a>
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<ul>
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<li><a href="#org6e044dd">4.1. Characteristics</a></li>
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<li><a href="#org20b7c2e">4.2. Mass Matrix</a></li>
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<li><a href="#org2f016df">4.3. Jacobian Matrix</a></li>
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<li><a href="#org2c9ff6d">4.4. Stifnness and Damping matrices</a></li>
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<li><a href="#orgffba0a8">4.5. State Space System</a></li>
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<li><a href="#org42b1b07">4.6. Transmissibility</a></li>
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<li><a href="#org38c8159">4.7. Real approximation of \(G(j\omega)\) at decoupling frequency</a></li>
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<li><a href="#org477b3ce">4.8. Coupled and Decoupled Plant “Gershgorin Radii”</a></li>
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<li><a href="#orgde4eec1">4.9. Decoupled Plant</a></li>
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<li><a href="#org11b0182">4.10. Controller</a></li>
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<li><a href="#org5c893a8">4.11. Closed Loop System</a></li>
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<li><a href="#orgb1c0711">4.12. Results</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org6dd65c1" class="outline-2">
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<h2 id="org6dd65c1"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org85dbe5c" class="outline-3">
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<h3 id="org85dbe5c"><span class="section-number-3">1.1</span> Introduction</h3>
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<div class="outline-text-3" id="text-1-1">
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<div id="org02345c4" class="figure">
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<p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org0b31481" class="outline-3">
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<h3 id="org0b31481"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
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<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<pre class="src src-matlab">open(<span class="org-string">'gravimeter.slx'</span>)
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</pre>
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</div>
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<p>
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Parameters
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">l = 1.0; <span class="org-comment">% Length of the mass [m]</span>
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la = 0.5; <span class="org-comment">% Position of Act. [m]</span>
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h = 3.4; <span class="org-comment">% Height of the mass [m]</span>
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ha = 1.7; <span class="org-comment">% Position of Act. [m]</span>
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m = 400; <span class="org-comment">% Mass [kg]</span>
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I = 115; <span class="org-comment">% Inertia [kg m^2]</span>
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k = 15e3; <span class="org-comment">% Actuator Stiffness [N/m]</span>
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c = 0.03; <span class="org-comment">% Actuator Damping [N/(m/s)]</span>
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deq = 0.2; <span class="org-comment">% Length of the actuators [m]</span>
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g = 0; <span class="org-comment">% Gravity [m/s2]</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org949338c" class="outline-3">
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<h3 id="org949338c"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
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<div class="outline-text-3" id="text-1-3">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
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mdl = <span class="org-string">'gravimeter'</span>;
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
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clear io; io_i = 1;
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io(io_i) = linio([mdl, <span class="org-string">'/F1'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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io(io_i) = linio([mdl, <span class="org-string">'/F2'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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io(io_i) = linio([mdl, <span class="org-string">'/F3'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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io(io_i) = linio([mdl, <span class="org-string">'/Acc_side'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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io(io_i) = linio([mdl, <span class="org-string">'/Acc_side'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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io(io_i) = linio([mdl, <span class="org-string">'/Acc_top'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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io(io_i) = linio([mdl, <span class="org-string">'/Acc_top'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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G = linearize(mdl, io);
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G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>};
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G.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string">'Az1'</span>, <span class="org-string">'Ax2'</span>, <span class="org-string">'Az2'</span>};
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</pre>
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</div>
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<pre class="example">
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pole(G)
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ans =
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-0.000473481142385795 + 21.7596190728632i
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-0.000473481142385795 - 21.7596190728632i
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-7.49842879459172e-05 + 8.6593576906982i
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-7.49842879459172e-05 - 8.6593576906982i
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-5.1538686792578e-06 + 2.27025295182756i
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-5.1538686792578e-06 - 2.27025295182756i
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</pre>
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<p>
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The plant as 6 states as expected (2 translations + 1 rotation)
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">size(G)
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</pre>
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</div>
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<pre class="example">
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State-space model with 4 outputs, 3 inputs, and 6 states.
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</pre>
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<div id="orga082635" class="figure">
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<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org3e8d708" class="outline-3">
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<h3 id="org3e8d708"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
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<div class="outline-text-3" id="text-1-4">
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<div class="org-src-container">
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<pre class="src src-matlab">g = 9.80665; <span class="org-comment">% Gravity [m/s2]</span>
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">Gg = linearize(mdl, io);
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Gg.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>};
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Gg.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string">'Az1'</span>, <span class="org-string">'Ax2'</span>, <span class="org-string">'Az2'</span>};
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</pre>
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</div>
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<p>
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We can now see that the system is unstable due to gravity.
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</p>
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<pre class="example">
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pole(Gg)
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ans =
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-10.9848275341252 + 0i
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10.9838836405201 + 0i
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-7.49855379478109e-05 + 8.65962885770051i
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-7.49855379478109e-05 - 8.65962885770051i
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-6.68819548733559e-06 + 0.832960422243848i
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-6.68819548733559e-06 - 0.832960422243848i
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</pre>
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<div id="org1a94741" class="figure">
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<p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org8263a33" class="outline-3">
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<h3 id="org8263a33"><span class="section-number-3">1.5</span> Analytical Model</h3>
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<div class="outline-text-3" id="text-1-5">
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</div>
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<div id="outline-container-org5ce809b" class="outline-4">
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<h4 id="org5ce809b"><span class="section-number-4">1.5.1</span> Parameters</h4>
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<div class="outline-text-4" id="text-1-5-1">
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<p>
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Bode options.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">P = bodeoptions;
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P.FreqUnits = <span class="org-string">'Hz'</span>;
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P.MagUnits = <span class="org-string">'abs'</span>;
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P.MagScale = <span class="org-string">'log'</span>;
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P.Grid = <span class="org-string">'on'</span>;
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P.PhaseWrapping = <span class="org-string">'on'</span>;
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P.Title.FontSize = 14;
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P.XLabel.FontSize = 14;
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P.YLabel.FontSize = 14;
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P.TickLabel.FontSize = 12;
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P.Xlim = [1e<span class="org-type">-</span>1,1e2];
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P.MagLowerLimMode = <span class="org-string">'manual'</span>;
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P.MagLowerLim= 1e<span class="org-type">-</span>3;
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</pre>
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</div>
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<p>
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Frequency vector.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">w = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>logspace(<span class="org-type">-</span>1,2,1000); <span class="org-comment">% [rad/s]</span>
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</pre>
|
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</div>
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</div>
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</div>
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<div id="outline-container-org485b7e0" class="outline-4">
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<h4 id="org485b7e0"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
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<div class="outline-text-4" id="text-1-5-2">
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<p>
|
|
Mass matrix
|
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</p>
|
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<div class="org-src-container">
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<pre class="src src-matlab">M = [m 0 0
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|
0 m 0
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|
0 0 I];
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</pre>
|
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</div>
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<p>
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|
Jacobian of the bottom sensor
|
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</p>
|
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<div class="org-src-container">
|
|
<pre class="src src-matlab">Js1 = [1 0 h<span class="org-type">/</span>2
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0 1 <span class="org-type">-</span>l<span class="org-type">/</span>2];
|
|
</pre>
|
|
</div>
|
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<p>
|
|
Jacobian of the top sensor
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</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Js2 = [1 0 <span class="org-type">-</span>h<span class="org-type">/</span>2
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0 1 0];
|
|
</pre>
|
|
</div>
|
|
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<p>
|
|
Jacobian of the actuators
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</p>
|
|
<div class="org-src-container">
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|
<pre class="src src-matlab">Ja = [1 0 ha <span class="org-comment">% Left horizontal actuator</span>
|
|
0 1 <span class="org-type">-</span>la <span class="org-comment">% Left vertical actuator</span>
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0 1 la]; <span class="org-comment">% Right vertical actuator</span>
|
|
Jta = Ja<span class="org-type">'</span>;
|
|
</pre>
|
|
</div>
|
|
|
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<p>
|
|
Stiffness and Damping matrices
|
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</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = k<span class="org-type">*</span>Jta<span class="org-type">*</span>Ja;
|
|
C = c<span class="org-type">*</span>Jta<span class="org-type">*</span>Ja;
|
|
</pre>
|
|
</div>
|
|
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<p>
|
|
State Space Matrices
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">E = [1 0 0
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|
0 1 0
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0 0 1]; <span class="org-comment">%projecting ground motion in the directions of the legs</span>
|
|
|
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AA = [zeros(3) eye(3)
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<span class="org-type">-</span>M<span class="org-type">\</span>K <span class="org-type">-</span>M<span class="org-type">\</span>C];
|
|
|
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BB = [zeros(3,6)
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M<span class="org-type">\</span>Jta M<span class="org-type">\</span>(k<span class="org-type">*</span>Jta<span class="org-type">*</span>E)];
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CC = [[Js1;Js2] zeros(4,3);
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zeros(2,6)
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(Js1<span class="org-type">+</span>Js2)<span class="org-type">./</span>2 zeros(2,3)
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(Js1<span class="org-type">-</span>Js2)<span class="org-type">./</span>2 zeros(2,3)
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(Js1<span class="org-type">-</span>Js2)<span class="org-type">./</span>(2<span class="org-type">*</span>h) zeros(2,3)];
|
|
|
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DD = [zeros(4,6)
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zeros<span class="org-type">(2,3) eye(2,3)</span>
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zeros<span class="org-type">(6,6)];</span>
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</pre>
|
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</div>
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<p>
|
|
State Space model:
|
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</p>
|
|
<ul class="org-ul">
|
|
<li>Input = three actuators and three ground motions</li>
|
|
<li>Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation</li>
|
|
</ul>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">system_dec = ss(AA,BB,CC,DD);
|
|
</pre>
|
|
</div>
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|
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">size(system_dec)
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
State-space model with 12 outputs, 6 inputs, and 6 states.
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb77d12b" class="outline-4">
|
|
<h4 id="orgb77d12b"><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="org7bbc6ef" 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>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbede3a4" class="outline-4">
|
|
<h4 id="orgbede3a4"><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>
|
|
<span class="org-comment">% bode(system_dec,P);</span>
|
|
<span class="org-comment">% return</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% svd decomposition</span></span>
|
|
<span class="org-comment">% system_dec_freq = freqresp(system_dec,w);</span>
|
|
<span class="org-comment">% S = zeros(3,length(w));</span>
|
|
<span class="org-comment">% for </span><span class="org-variable-name"><span class="org-comment">m</span></span><span class="org-comment"> = </span><span class="org-constant"><span class="org-comment">1:length(w)</span></span>
|
|
<span class="org-comment">% S(:,m) = svd(system_dec_freq(1:4,1:3,m));</span>
|
|
<span class="org-comment">% end</span>
|
|
<span class="org-comment">% figure</span>
|
|
<span class="org-comment">% loglog(w./(2*pi), S);hold on;</span>
|
|
<span class="org-comment">% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));</span>
|
|
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');</span>
|
|
<span class="org-comment">% legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');</span>
|
|
<span class="org-comment">% ylim([1e-8 1e-2]);</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% %condition number</span>
|
|
<span class="org-comment">% figure</span>
|
|
<span class="org-comment">% loglog(w./(2*pi), S(1,:)./S(3,:));hold on;</span>
|
|
<span class="org-comment">% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));</span>
|
|
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('Condition number [-]');</span>
|
|
<span class="org-comment">% % legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% %performance indicator</span>
|
|
<span class="org-comment">% system_dec_svd = freqresp(system_dec(1:4,1:3),2*pi*10);</span>
|
|
<span class="org-comment">% [U,S,V] = svd(system_dec_svd);</span>
|
|
<span class="org-comment">% H_svd_OL = -eye(3,4);%-[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 10*zpk(-2*pi*40,-2*pi*200,40/200) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];% - eye(3,4);%</span>
|
|
<span class="org-comment">% H_svd = pinv(V')*H_svd_OL*pinv(U);</span>
|
|
<span class="org-comment">% % system_dec_control_svd_ = feedback(system_dec,g*pinv(V')*H*pinv(U));</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% OL_dec = g_svd*H_svd*system_dec(1:4,1:3);</span>
|
|
<span class="org-comment">% OL_freq = freqresp(OL_dec,w); % OL = G*H</span>
|
|
<span class="org-comment">% CL_system = feedback(eye(3),-g_svd*H_svd*system_dec(1:4,1:3));</span>
|
|
<span class="org-comment">% CL_freq = freqresp(CL_system,w); % CL = (1+G*H)^-1</span>
|
|
<span class="org-comment">% % CL_system_2 = feedback(system_dec,H);</span>
|
|
<span class="org-comment">% % CL_freq_2 = freqresp(CL_system_2,w); % CL = G/(1+G*H)</span>
|
|
<span class="org-comment">% for </span><span class="org-variable-name"><span class="org-comment">i</span></span><span class="org-comment"> = </span><span class="org-constant"><span class="org-comment">1:size(w,2)</span></span>
|
|
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">OL(:,i)</span></span><span class="org-comment"> = svd(OL_freq(:,:,i));</span>
|
|
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">CL </span></span><span class="org-comment">(:,i) = svd(CL_freq(:,:,i));</span>
|
|
<span class="org-comment">% %CL2 (:,i) = svd(CL_freq_2(:,:,i));</span>
|
|
<span class="org-comment">% end</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% un = ones(1,length(w));</span>
|
|
<span class="org-comment">% figure</span>
|
|
<span class="org-comment">% loglog(w./(2*pi),OL(3,:)+1,'k',w./(2*pi),OL(3,:)-1,'b',w./(2*pi),1./CL(1,:),'r--',w./(2*pi),un,'k:');hold on;%</span>
|
|
<span class="org-comment">% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));</span>
|
|
<span class="org-comment">% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));</span>
|
|
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');</span>
|
|
<span class="org-comment">% legend('GH \sigma_{inf} +1 ','GH \sigma_{inf} -1','S 1/\sigma_{sup}');%,'\lambda_1','\lambda_2','\lambda_3');</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% figure</span>
|
|
<span class="org-comment">% loglog(w./(2*pi),OL(1,:)+1,'k',w./(2*pi),OL(1,:)-1,'b',w./(2*pi),1./CL(3,:),'r--',w./(2*pi),un,'k:');hold on;%</span>
|
|
<span class="org-comment">% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));</span>
|
|
<span class="org-comment">% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));</span>
|
|
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');</span>
|
|
<span class="org-comment">% legend('GH \sigma_{sup} +1 ','GH \sigma_{sup} -1','S 1/\sigma_{inf}');%,'\lambda_1','\lambda_2','\lambda_3');</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org00d06a7" class="outline-4">
|
|
<h4 id="org00d06a7"><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);
|
|
system_dec_0Hz = freqresp(system_dec,0);
|
|
|
|
system_decReal_10Hz = pinv(align(system_dec_10Hz));
|
|
[Ureal,Sreal,Vreal] = svd(system_decReal_10Hz(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3));
|
|
normalizationMatrixReal = abs(pinv(Ureal)<span class="org-type">*</span>system_dec_0Hz(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(Vreal<span class="org-type">'</span>));
|
|
|
|
[U,S,V] = svd(system_dec_10Hz(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3));
|
|
normalizationMatrix = abs(pinv(U)<span class="org-type">*</span>system_dec_0Hz(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(V<span class="org-type">'</span>));
|
|
|
|
H_dec = ([zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30,30<span class="org-type">/</span>5) 0 0 0
|
|
0 zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>4,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>20,20<span class="org-type">/</span>4) 0 0
|
|
0 0 0 zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span>,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10)]);
|
|
H_cen_OL = [zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span>,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10) 0 0; 0 zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span>,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10) 0;
|
|
0 0 zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30,30<span class="org-type">/</span>5)];
|
|
H_cen = pinv(Jta)<span class="org-type">*</span>H_cen_OL<span class="org-type">*</span>pinv([Js1; Js2]);
|
|
<span class="org-comment">% H_svd_OL = -[1/normalizationMatrix(1,1) 0 0 0</span>
|
|
<span class="org-comment">% 0 1/normalizationMatrix(2,2) 0 0</span>
|
|
<span class="org-comment">% 0 0 1/normalizationMatrix(3,3) 0];</span>
|
|
<span class="org-comment">% H_svd_OL_real = -[1/normalizationMatrixReal(1,1) 0 0 0</span>
|
|
<span class="org-comment">% 0 1/normalizationMatrixReal(2,2) 0 0</span>
|
|
<span class="org-comment">% 0 0 1/normalizationMatrixReal(3,3) 0];</span>
|
|
H_svd_OL = <span class="org-type">-</span>[1<span class="org-type">/</span>normalizationMatrix(1,1)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>60,60<span class="org-type">/</span>10) 0 0 0
|
|
0 1<span class="org-type">/</span>normalizationMatrix(2,2)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30,30<span class="org-type">/</span>5) 0 0
|
|
0 0 1<span class="org-type">/</span>normalizationMatrix(3,3)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>2,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10<span class="org-type">/</span>2) 0];
|
|
H_svd_OL_real = <span class="org-type">-</span>[1<span class="org-type">/</span>normalizationMatrixReal(1,1)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>60,60<span class="org-type">/</span>10) 0 0 0
|
|
0 1<span class="org-type">/</span>normalizationMatrixReal(2,2)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30,30<span class="org-type">/</span>5) 0 0
|
|
0 0 1<span class="org-type">/</span>normalizationMatrixReal(3,3)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>2,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10<span class="org-type">/</span>2) 0];
|
|
<span class="org-comment">% H_svd_OL_real = -[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 10*zpk(-2*pi*10,-2*pi*100,100/10) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];%-eye(3,4);</span>
|
|
<span class="org-comment">% H_svd_OL = -[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 zpk(-2*pi*4,-2*pi*20,4/20) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];% - eye(3,4);%</span>
|
|
H_svd = pinv(V<span class="org-type">'</span>)<span class="org-type">*</span>H_svd_OL<span class="org-type">*</span>pinv(U);
|
|
H_svd_real = pinv(Vreal<span class="org-type">'</span>)<span class="org-type">*</span>H_svd_OL_real<span class="org-type">*</span>pinv(Ureal);
|
|
|
|
OL_dec = g<span class="org-type">*</span>H_dec<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3);
|
|
OL_cen = g<span class="org-type">*</span>H_cen_OL<span class="org-type">*</span>pinv([Js1; Js2])<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(Jta);
|
|
OL_svd = 100<span class="org-type">*</span>H_svd_OL<span class="org-type">*</span>pinv(U)<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(V<span class="org-type">'</span>);
|
|
OL_svd_real = 100<span class="org-type">*</span>H_svd_OL_real<span class="org-type">*</span>pinv(Ureal)<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(Vreal<span class="org-type">'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-comment">% figure</span>
|
|
<span class="org-comment">% bode(OL_dec,w,P);title('OL Decentralized');</span>
|
|
<span class="org-comment">% figure</span>
|
|
<span class="org-comment">% bode(OL_cen,w,P);title('OL Centralized');</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>
|
|
bode(g<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),w,P);
|
|
title(<span class="org-string">'gain * Plant'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>
|
|
bode(OL_svd,OL_svd_real,w,P);
|
|
title(<span class="org-string">'OL SVD'</span>);
|
|
legend(<span class="org-string">'SVD of Complex plant'</span>,<span class="org-string">'SVD of real approximation of the complex plant'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>
|
|
bode(system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),pinv(U)<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(V<span class="org-type">'</span>),P);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">CL_dec = feedback(system_dec,g<span class="org-type">*</span>H_dec,[1 2 3],[1 2 3 4]);
|
|
CL_cen = feedback(system_dec,g<span class="org-type">*</span>H_cen,[1 2 3],[1 2 3 4]);
|
|
CL_svd = feedback(system_dec,100<span class="org-type">*</span>H_svd,[1 2 3],[1 2 3 4]);
|
|
CL_svd_real = feedback(system_dec,100<span class="org-type">*</span>H_svd_real,[1 2 3],[1 2 3 4]);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">pzmap_testCL(system_dec,H_dec,g,[1 2 3],[1 2 3 4])
|
|
title(<span class="org-string">'Decentralized control'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">pzmap_testCL(system_dec,H_cen,g,[1 2 3],[1 2 3 4])
|
|
title(<span class="org-string">'Centralized control'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">pzmap_testCL(system_dec,H_svd,100,[1 2 3],[1 2 3 4])
|
|
title(<span class="org-string">'SVD control'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">pzmap_testCL(system_dec,H_svd_real,100,[1 2 3],[1 2 3 4])
|
|
title(<span class="org-string">'Real approximation SVD control'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">P.Ylim = [1e<span class="org-type">-</span>8 1e<span class="org-type">-</span>3];
|
|
<span class="org-type">figure</span>
|
|
bodemag(system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),CL_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),CL_cen(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),CL_svd(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),CL_svd_real(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),P);
|
|
title(<span class="org-string">'Motion/actuator'</span>)
|
|
legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'Decentralized control'</span>,<span class="org-string">'Centralized control'</span>,<span class="org-string">'SVD control'</span>,<span class="org-string">'SVD control real appr.'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">P.Ylim = [1e<span class="org-type">-</span>5 1e1];
|
|
<span class="org-type">figure</span>
|
|
bodemag(system_dec(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),CL_dec(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),CL_cen(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),CL_svd(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),CL_svd_real(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),P);
|
|
title(<span class="org-string">'Transmissibility'</span>);
|
|
legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'Decentralized control'</span>,<span class="org-string">'Centralized control'</span>,<span class="org-string">'SVD control'</span>,<span class="org-string">'SVD control real appr.'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>
|
|
bodemag(system_dec([7 9],4<span class="org-type">:</span>6),CL_dec([7 9],4<span class="org-type">:</span>6),CL_cen([7 9],4<span class="org-type">:</span>6),CL_svd([7 9],4<span class="org-type">:</span>6),CL_svd_real([7 9],4<span class="org-type">:</span>6),P);
|
|
title(<span class="org-string">'Transmissibility from half sum and half difference in the X direction'</span>);
|
|
legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'Decentralized control'</span>,<span class="org-string">'Centralized control'</span>,<span class="org-string">'SVD control'</span>,<span class="org-string">'SVD control real appr.'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>
|
|
bodemag(system_dec([8 10],4<span class="org-type">:</span>6),CL_dec([8 10],4<span class="org-type">:</span>6),CL_cen([8 10],4<span class="org-type">:</span>6),CL_svd([8 10],4<span class="org-type">:</span>6),CL_svd_real([8 10],4<span class="org-type">:</span>6),P);
|
|
title(<span class="org-string">'Transmissibility from half sum and half difference in the Z direction'</span>);
|
|
legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'Decentralized control'</span>,<span class="org-string">'Centralized control'</span>,<span class="org-string">'SVD control'</span>,<span class="org-string">'SVD control real appr.'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8d48657" class="outline-4">
|
|
<h4 id="org8d48657"><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);
|
|
x1 = zeros(1,length(w));
|
|
z1 = zeros(1,length(w));
|
|
x2 = zeros(1,length(w));
|
|
S1 = zeros(1,length(w));
|
|
S2 = zeros(1,length(w));
|
|
S3 = zeros(1,length(w));
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">t</span> = <span class="org-constant">1:length(w)</span>
|
|
x1(t) = (abs(system_dec_freq(1,2,t))<span class="org-type">+</span>abs(system_dec_freq(1,3,t)))<span class="org-type">/</span>abs(system_dec_freq(1,1,t));
|
|
z1(t) = (abs(system_dec_freq(2,1,t))<span class="org-type">+</span>abs(system_dec_freq(2,3,t)))<span class="org-type">/</span>abs(system_dec_freq(2,2,t));
|
|
x2(t) = (abs(system_dec_freq(3,1,t))<span class="org-type">+</span>abs(system_dec_freq(3,2,t)))<span class="org-type">/</span>abs(system_dec_freq(3,3,t));
|
|
system_svd = pinv(Ureal)<span class="org-type">*</span>system_dec_freq(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3,t)<span class="org-type">*</span>pinv(Vreal<span class="org-type">'</span>);
|
|
S1(t) = (abs(system_svd(1,2))<span class="org-type">+</span>abs(system_svd(1,3)))<span class="org-type">/</span>abs(system_svd(1,1));
|
|
S2(t) = (abs(system_svd(2,1))<span class="org-type">+</span>abs(system_svd(2,3)))<span class="org-type">/</span>abs(system_svd(2,2));
|
|
S2(t) = (abs(system_svd(3,1))<span class="org-type">+</span>abs(system_svd(3,2)))<span class="org-type">/</span>abs(system_svd(3,3));
|
|
<span class="org-keyword">end</span>
|
|
|
|
limit = 0.5<span class="org-type">*</span>ones(1,length(w));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>
|
|
loglog(w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),x1,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),z1,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),x2,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),limit,<span class="org-string">'--'</span>);
|
|
legend(<span class="org-string">'x_1'</span>,<span class="org-string">'z_1'</span>,<span class="org-string">'x_2'</span>,<span class="org-string">'Limit'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
|
|
ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>
|
|
loglog(w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),S1,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),S2,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),S3,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),limit,<span class="org-string">'--'</span>);
|
|
legend(<span class="org-string">'S1'</span>,<span class="org-string">'S2'</span>,<span class="org-string">'S3'</span>,<span class="org-string">'Limit'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
|
|
ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
|
|
<span class="org-comment">% set(gcf,'color','w')</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7348f99" class="outline-4">
|
|
<h4 id="org7348f99"><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);
|
|
w=2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>Fr<span class="org-type">*</span>1<span class="org-constant">i</span>;
|
|
<span class="org-comment">%fit of the ground motion data in m/s^2/rtHz</span>
|
|
Fr_ground_x = [0.07 0.1 0.15 0.3 0.7 0.8 0.9 1.2 5 10];
|
|
n_ground_x1 = [4e<span class="org-type">-</span>7 4e<span class="org-type">-</span>7 2e<span class="org-type">-</span>6 1e<span class="org-type">-</span>6 5e<span class="org-type">-</span>7 5e<span class="org-type">-</span>7 5e<span class="org-type">-</span>7 1e<span class="org-type">-</span>6 1e<span class="org-type">-</span>5 3.5e<span class="org-type">-</span>5];
|
|
Fr_ground_v = [0.07 0.08 0.1 0.11 0.12 0.15 0.25 0.6 0.8 1 1.2 1.6 2 6 10];
|
|
n_ground_v1 = [7e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 1e<span class="org-type">-</span>6 1.2e<span class="org-type">-</span>6 1.5e<span class="org-type">-</span>6 1e<span class="org-type">-</span>6 9e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 1e<span class="org-type">-</span>6 2e<span class="org-type">-</span>6 1e<span class="org-type">-</span>5 3e<span class="org-type">-</span>5];
|
|
|
|
n_ground_x = interp1(Fr_ground_x,n_ground_x1,Fr,<span class="org-string">'linear'</span>);
|
|
n_ground_v = interp1(Fr_ground_v,n_ground_v1,Fr,<span class="org-string">'linear'</span>);
|
|
<span class="org-comment">% figure</span>
|
|
<span class="org-comment">% loglog(Fr,abs(n_ground_v),Fr_ground_v,n_ground_v1,'*');</span>
|
|
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('ASD [m/s^2 /rtHz]');</span>
|
|
<span class="org-comment">% return</span>
|
|
|
|
<span class="org-comment">%converting into PSD</span>
|
|
n_ground_x = (n_ground_x)<span class="org-type">.^</span>2;
|
|
n_ground_v = (n_ground_v)<span class="org-type">.^</span>2;
|
|
|
|
<span class="org-comment">%Injecting ground motion in the system and getting the outputs</span>
|
|
system_dec_f = (freqresp(system_dec,abs(w)));
|
|
PHI = zeros(size(Fr,2),12,12);
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">p</span> = <span class="org-constant">1:size(Fr,2)</span>
|
|
Sw=zeros(6,6);
|
|
Iact = zeros(3,3);
|
|
Sw<span class="org-type">(4,4) </span>= n_ground_x(p);
|
|
Sw<span class="org-type">(5,5) </span>= n_ground_v(p);
|
|
Sw<span class="org-type">(6,6) </span>= n_ground_v(p);
|
|
Sw<span class="org-type">(1:3,1:3) </span>= Iact;
|
|
PHI(p,<span class="org-type">:</span>,<span class="org-type">:</span>) = (system_dec_f(<span class="org-type">:</span>,<span class="org-type">:</span>,p))<span class="org-type">*</span>Sw(<span class="org-type">:</span>,<span class="org-type">:</span>)<span class="org-type">*</span>(system_dec_f(<span class="org-type">:</span>,<span class="org-type">:</span>,p))<span class="org-type">'</span>;
|
|
<span class="org-keyword">end</span>
|
|
x1 = PHI(<span class="org-type">:</span>,1,1);
|
|
z1 = PHI(<span class="org-type">:</span>,2,2);
|
|
x2 = PHI(<span class="org-type">:</span>,3,3);
|
|
z2 = PHI(<span class="org-type">:</span>,4,4);
|
|
wx = PHI(<span class="org-type">:</span>,5,5);
|
|
wz = PHI(<span class="org-type">:</span>,6,6);
|
|
x12 = PHI(<span class="org-type">:</span>,1,3);
|
|
z12 = PHI(<span class="org-type">:</span>,2,4);
|
|
PHIwx = PHI(<span class="org-type">:</span>,1,5);
|
|
PHIwz = PHI(<span class="org-type">:</span>,2,6);
|
|
xsum = PHI(<span class="org-type">:</span>,7,7);
|
|
zsum = PHI(<span class="org-type">:</span>,8,8);
|
|
xdelta = PHI(<span class="org-type">:</span>,9,9);
|
|
zdelta = PHI(<span class="org-type">:</span>,10,10);
|
|
rot = PHI(<span class="org-type">:</span>,11,11);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org534f1d2" class="outline-2">
|
|
<h2 id="org534f1d2"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
</div>
|
|
<div id="outline-container-org8fd3468" class="outline-3">
|
|
<h3 id="org8fd3468"><span class="section-number-3">2.1</span> <code>align</code></h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
<a id="org95a25f3"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="gravimeter/align.m">here</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[A]</span> = <span class="org-function-name">align</span>(<span class="org-variable-name">V</span>)
|
|
<span class="org-comment">%A!ALIGN(V) returns a constat matrix A which is the real alignment of the</span>
|
|
<span class="org-comment">%INVERSE of the complex input matrix V</span>
|
|
<span class="org-comment">%from Mohit slides</span>
|
|
|
|
<span class="org-keyword">if</span> (nargin <span class="org-type">==</span>0) <span class="org-type">||</span> (nargin <span class="org-type">></span> 1)
|
|
disp(<span class="org-string">'usage: mat_inv_real = align(mat)'</span>)
|
|
<span class="org-keyword">return</span>
|
|
<span class="org-keyword">end</span>
|
|
|
|
D = pinv(real(V<span class="org-type">'*</span>V));
|
|
A = D<span class="org-type">*</span>real(V<span class="org-type">'*</span>diag(exp(1<span class="org-constant">i</span> <span class="org-type">*</span> angle(diag(V<span class="org-type">*</span>D<span class="org-type">*</span>V<span class="org-type">.'</span>))<span class="org-type">/</span>2)));
|
|
|
|
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org7fc9d1b" class="outline-3">
|
|
<h3 id="org7fc9d1b"><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="orge776d7f"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="gravimeter/pzmap_testCL.m">here</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[]</span> = <span class="org-function-name">pzmap_testCL</span>(<span class="org-variable-name">system</span>,<span class="org-variable-name">H</span>,<span class="org-variable-name">gain</span>,<span class="org-variable-name">feedin</span>,<span class="org-variable-name">feedout</span>)
|
|
<span class="org-comment">% evaluate and plot the pole-zero map for the closed loop system for</span>
|
|
<span class="org-comment">% different values of the gain</span>
|
|
|
|
[<span class="org-type">~</span>, n] = size(gain);
|
|
[m1, n1, <span class="org-type">~</span>] = size(H);
|
|
[<span class="org-type">~</span>,n2] = size(feedin);
|
|
|
|
<span class="org-type">figure</span>
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
|
|
<span class="org-comment">% if n1 == n2</span>
|
|
system_CL = feedback(system,gain(<span class="org-constant">i</span>)<span class="org-type">*</span>H,feedin,feedout);
|
|
|
|
[P,Z] = pzmap(system_CL);
|
|
plot(real(P(<span class="org-type">:</span>)),imag(P(<span class="org-type">:</span>)),<span class="org-string">'x'</span>,real(Z(<span class="org-type">:</span>)),imag(Z(<span class="org-type">:</span>)),<span class="org-string">'o'</span>);hold on
|
|
xlabel(<span class="org-string">'Real axis (s^{-1})'</span>);ylabel(<span class="org-string">'Imaginary Axis (s^{-1})'</span>);
|
|
<span class="org-comment">% clear P Z</span>
|
|
<span class="org-comment">% else</span>
|
|
<span class="org-comment">% system_CL = feedback(system,gain(i)*H(:,1+(i-1)*m1:m1+(i-1)*m1),feedin,feedout);</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% [P,Z] = pzmap(system_CL);</span>
|
|
<span class="org-comment">% plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on</span>
|
|
<span class="org-comment">% xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');</span>
|
|
<span class="org-comment">% clear P Z</span>
|
|
<span class="org-comment">% end</span>
|
|
<span class="org-keyword">end</span>
|
|
str = {strcat(<span class="org-string">'gain = '</span> , num2str(gain(1)))}; <span class="org-comment">% at the end of first loop, z being loop output</span>
|
|
str = [str , strcat(<span class="org-string">'gain = '</span> , num2str(gain(1)))]; <span class="org-comment">% after 2nd loop</span>
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">2:n</span>
|
|
str = [str , strcat(<span class="org-string">'gain = '</span> , num2str(gain(<span class="org-constant">i</span>)))]; <span class="org-comment">% after 2nd loop</span>
|
|
str = [str , strcat(<span class="org-string">'gain = '</span> , num2str(gain(<span class="org-constant">i</span>)))]; <span class="org-comment">% after 2nd loop</span>
|
|
<span class="org-keyword">end</span>
|
|
legend(str{<span class="org-type">:</span>})
|
|
<span class="org-keyword">end</span>
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga726921" class="outline-2">
|
|
<h2 id="orga726921"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
</div>
|
|
<div id="outline-container-org0f4c378" class="outline-3">
|
|
<h3 id="org0f4c378"><span class="section-number-3">3.1</span> Jacobian</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
First, the position of the “joints” (points of force application) are estimated and the Jacobian computed.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">open(<span class="org-string">'drone_platform_jacobian.slx'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'drone_platform_jacobian'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Aa = [a1.Data(1,<span class="org-type">:</span>);
|
|
a2.Data(1,<span class="org-type">:</span>);
|
|
a3.Data(1,<span class="org-type">:</span>);
|
|
a4.Data(1,<span class="org-type">:</span>);
|
|
a5.Data(1,<span class="org-type">:</span>);
|
|
a6.Data(1,<span class="org-type">:</span>)]<span class="org-type">'</span>;
|
|
|
|
Ab = [b1.Data(1,<span class="org-type">:</span>);
|
|
b2.Data(1,<span class="org-type">:</span>);
|
|
b3.Data(1,<span class="org-type">:</span>);
|
|
b4.Data(1,<span class="org-type">:</span>);
|
|
b5.Data(1,<span class="org-type">:</span>);
|
|
b6.Data(1,<span class="org-type">:</span>)]<span class="org-type">'</span>;
|
|
|
|
As = (Ab <span class="org-type">-</span> Aa)<span class="org-type">./</span>vecnorm(Ab <span class="org-type">-</span> Aa);
|
|
|
|
l = vecnorm(Ab <span class="org-type">-</span> Aa)<span class="org-type">'</span>;
|
|
|
|
J = [As<span class="org-type">'</span> , cross(Ab, As)<span class="org-type">'</span>];
|
|
|
|
save(<span class="org-string">'./jacobian.mat'</span>, <span class="org-string">'Aa'</span>, <span class="org-string">'Ab'</span>, <span class="org-string">'As'</span>, <span class="org-string">'l'</span>, <span class="org-string">'J'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8e93915" class="outline-3">
|
|
<h3 id="org8e93915"><span class="section-number-3">3.2</span> Simscape Model</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">open(<span class="org-string">'drone_platform.slx'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Definition of spring parameters
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">kx = 0.5<span class="org-type">*</span>1e3<span class="org-type">/</span>3; <span class="org-comment">% [N/m]</span>
|
|
ky = 0.5<span class="org-type">*</span>1e3<span class="org-type">/</span>3;
|
|
kz = 1e3<span class="org-type">/</span>3;
|
|
|
|
cx = 0.025; <span class="org-comment">% [Nm/rad]</span>
|
|
cy = 0.025;
|
|
cz = 0.025;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We load the Jacobian.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./jacobian.mat'</span>, <span class="org-string">'Aa'</span>, <span class="org-string">'Ab'</span>, <span class="org-string">'As'</span>, <span class="org-string">'l'</span>, <span class="org-string">'J'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga80ad9d" class="outline-3">
|
|
<h3 id="orga80ad9d"><span class="section-number-3">3.3</span> Identification of the plant</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
|
|
</p>
|
|
<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>
|
|
mdl = <span class="org-string">'drone_platform'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/u'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Inertial Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
|
|
|
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>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
There are 24 states (6dof for the bottom platform + 6dof for the top platform).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">size(G)
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
State-space model with 6 outputs, 12 inputs, and 24 states.
|
|
</pre>
|
|
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-comment">% G = G*blkdiag(inv(J), eye(6));</span>
|
|
<span class="org-comment">% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...</span>
|
|
<span class="org-comment">% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
|
|
</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>};
|
|
|
|
Gl = J<span class="org-type">*</span>G;
|
|
Gl.OutputName = {<span class="org-string">'A1'</span>, <span class="org-string">'A2'</span>, <span class="org-string">'A3'</span>, <span class="org-string">'A4'</span>, <span class="org-string">'A5'</span>, <span class="org-string">'A6'</span>};
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org820395d" class="outline-3">
|
|
<h3 id="org820395d"><span class="section-number-3">3.4</span> Obtained Dynamics</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
|
|
<div id="orgf45efb1" class="figure">
|
|
<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
|
|
</div>
|
|
|
|
|
|
<div id="org7a9f376" class="figure">
|
|
<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
|
|
</div>
|
|
|
|
|
|
<div id="org01ccd4c" class="figure">
|
|
<p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p>
|
|
</div>
|
|
|
|
|
|
<div id="org18ecae5" class="figure">
|
|
<p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Transmissibility</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org531c180" class="outline-3">
|
|
<h3 id="org531c180"><span class="section-number-3">3.5</span> Real Approximation of \(G\) at the decoupling frequency</h3>
|
|
<div class="outline-text-3" id="text-3-5">
|
|
<p>
|
|
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 \(\omega_c\).
|
|
</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>20; <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);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The real approximation is computed as follows:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">D = pinv(real(H1<span class="org-type">'*</span>H1));
|
|
H1 = inv(D<span class="org-type">*</span>real(H1<span class="org-type">'*</span>diag(exp(<span class="org-constant">j</span><span class="org-type">*</span>angle(diag(H1<span class="org-type">*</span>D<span class="org-type">*</span>H1<span class="org-type">.'</span>))<span class="org-type">/</span>2))));
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org04886ad" class="outline-3">
|
|
<h3 id="org04886ad"><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>
|
|
First, the Singular Value Decomposition of \(H_1\) is performed:
|
|
\[ H_1 = U \Sigma V^H \]
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">[U,S,V] = svd(H1);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then, the “Gershgorin Radii” is computed for the plant \(G_c(s)\) and the “SVD Decoupled Plant” \(G_d(s)\):
|
|
\[ G_d(s) = U^T G_c(s) V \]
|
|
</p>
|
|
|
|
<p>
|
|
This is computed over the following frequencies.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = logspace(<span class="org-type">-</span>2, 2, 1000); <span class="org-comment">% [Hz]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Gershgorin Radii for the coupled plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gr_coupled = zeros(length(freqs), size(Gc,2));
|
|
|
|
H = abs(squeeze(freqresp(Gc, freqs, <span class="org-string">'Hz'</span>)));
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">out_i</span> = <span class="org-constant">1:size(Gc,2)</span>
|
|
Gr_coupled(<span class="org-type">:</span>, out_i) = squeeze((sum(H(out_i,<span class="org-type">:</span>,<span class="org-type">:</span>)) <span class="org-type">-</span> H(out_i,out_i,<span class="org-type">:</span>))<span class="org-type">./</span>H(out_i, out_i, <span class="org-type">:</span>));
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Gershgorin Radii for the decoupled plant using SVD:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gd = U<span class="org-type">'*</span>Gc<span class="org-type">*</span>V;
|
|
Gr_decoupled = zeros(length(freqs), size(Gd,2));
|
|
|
|
H = abs(squeeze(freqresp(Gd, freqs, <span class="org-string">'Hz'</span>)));
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">out_i</span> = <span class="org-constant">1:size(Gd,2)</span>
|
|
Gr_decoupled(<span class="org-type">:</span>, out_i) = squeeze((sum(H(out_i,<span class="org-type">:</span>,<span class="org-type">:</span>)) <span class="org-type">-</span> H(out_i,out_i,<span class="org-type">:</span>))<span class="org-type">./</span>H(out_i, out_i, <span class="org-type">:</span>));
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Gershgorin Radii for the decoupled plant using the Jacobian:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gj = Gc<span class="org-type">*</span>inv(J<span class="org-type">'</span>);
|
|
Gr_jacobian = zeros(length(freqs), size(Gj,2));
|
|
|
|
H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">out_i</span> = <span class="org-constant">1:size(Gj,2)</span>
|
|
Gr_jacobian(<span class="org-type">:</span>, out_i) = squeeze((sum(H(out_i,<span class="org-type">:</span>,<span class="org-type">:</span>)) <span class="org-type">-</span> H(out_i,out_i,<span class="org-type">:</span>))<span class="org-type">./</span>H(out_i, out_i, <span class="org-type">:</span>));
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org98d0c86" class="figure">
|
|
<p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Gershgorin Radii of the Coupled and Decoupled plants</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org96683a8" class="outline-3">
|
|
<h3 id="org96683a8"><span class="section-number-3">3.7</span> Decoupled Plant</h3>
|
|
<div class="outline-text-3" id="text-3-7">
|
|
<p>
|
|
Let’s see the bode plot of the decoupled plant \(G_d(s)\).
|
|
\[ G_d(s) = U^T G_c(s) V \]
|
|
</p>
|
|
|
|
|
|
<div id="org2351e85" 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 10: </span>Decoupled Plant using SVD</p>
|
|
</div>
|
|
|
|
|
|
<div id="org6699d5a" 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 11: </span>Decoupled Plant using the Jacobian</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgde9fab8" class="outline-3">
|
|
<h3 id="orgde9fab8"><span class="section-number-3">3.8</span> Diagonal Controller</h3>
|
|
<div class="outline-text-3" id="text-3-8">
|
|
<p>
|
|
The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\).
|
|
</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>0.1; <span class="org-comment">% Crossover Frequency [rad/s]</span>
|
|
C_g = 50; <span class="org-comment">% DC Gain</span>
|
|
|
|
K = eye(6)<span class="org-type">*</span>C_g<span class="org-type">/</span>(s<span class="org-type">+</span>wc);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org47bbca8" class="outline-3">
|
|
<h3 id="org47bbca8"><span class="section-number-3">3.9</span> Centralized Control</h3>
|
|
<div class="outline-text-3" id="text-3-9">
|
|
<p>
|
|
The control diagram for the centralized control is shown below.
|
|
</p>
|
|
|
|
<p>
|
|
The controller \(K_c\) is “working” in an cartesian frame.
|
|
The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.
|
|
</p>
|
|
|
|
|
|
<div class="figure">
|
|
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
|
|
</p>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_cen = feedback(G, inv(J<span class="org-type">'</span>)<span class="org-type">*</span>K, [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2c1e3f7" class="outline-3">
|
|
<h3 id="org2c1e3f7"><span class="section-number-3">3.10</span> SVD Control</h3>
|
|
<div class="outline-text-3" id="text-3-10">
|
|
<p>
|
|
The SVD control architecture is shown below.
|
|
The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
|
|
</p>
|
|
|
|
|
|
<div class="figure">
|
|
<p><img src="figs/svd_control.png" alt="svd_control.png" />
|
|
</p>
|
|
</div>
|
|
|
|
<p>
|
|
SVD Control
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_svd = feedback(G, pinv(V<span class="org-type">'</span>)<span class="org-type">*</span>K<span class="org-type">*</span>pinv(U), [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd6985da" class="outline-3">
|
|
<h3 id="orgd6985da"><span class="section-number-3">3.11</span> Results</h3>
|
|
<div class="outline-text-3" id="text-3-11">
|
|
<p>
|
|
Let’s first verify the stability of the closed-loop systems:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable(G_cen)
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
ans =
|
|
logical
|
|
1
|
|
</pre>
|
|
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable(G_svd)
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
ans =
|
|
logical
|
|
0
|
|
</pre>
|
|
|
|
|
|
<p>
|
|
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org0856618">14</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org0856618" 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 14: </span>Obtained Transmissibility</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org99c6262" class="outline-2">
|
|
<h2 id="org99c6262"><span class="section-number-2">4</span> Stewart Platform - Analytical Model</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
</div>
|
|
<div id="outline-container-org6e044dd" class="outline-3">
|
|
<h3 id="org6e044dd"><span class="section-number-3">4.1</span> Characteristics</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">L = 0.055; <span class="org-comment">% Leg length [m]</span>
|
|
Zc = 0; <span class="org-comment">% ?</span>
|
|
m = 0.2; <span class="org-comment">% Top platform mass [m]</span>
|
|
k = 1e3; <span class="org-comment">% Total vertical stiffness [N/m]</span>
|
|
c = 2<span class="org-type">*</span>0.1<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m); <span class="org-comment">% Damping ? [N/(m/s)]</span>
|
|
|
|
Rx = 0.04; <span class="org-comment">% ?</span>
|
|
Rz = 0.04; <span class="org-comment">% ?</span>
|
|
Ix = m<span class="org-type">*</span>Rx<span class="org-type">^</span>2; <span class="org-comment">% ?</span>
|
|
Iy = m<span class="org-type">*</span>Rx<span class="org-type">^</span>2; <span class="org-comment">% ?</span>
|
|
Iz = m<span class="org-type">*</span>Rz<span class="org-type">^</span>2; <span class="org-comment">% ?</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org20b7c2e" class="outline-3">
|
|
<h3 id="org20b7c2e"><span class="section-number-3">4.2</span> Mass Matrix</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">M = m<span class="org-type">*</span>[1 0 0 0 Zc 0;
|
|
0 1 0 <span class="org-type">-</span>Zc 0 0;
|
|
0 0 1 0 0 0;
|
|
0 <span class="org-type">-</span>Zc 0 Rx<span class="org-type">^</span>2<span class="org-type">+</span>Zc<span class="org-type">^</span>2 0 0;
|
|
Zc 0 0 0 Rx<span class="org-type">^</span>2<span class="org-type">+</span>Zc<span class="org-type">^</span>2 0;
|
|
0 0 0 0 0 Rz<span class="org-type">^</span>2];
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2f016df" class="outline-3">
|
|
<h3 id="org2f016df"><span class="section-number-3">4.3</span> Jacobian Matrix</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Bj=1<span class="org-type">/</span>sqrt(6)<span class="org-type">*</span>[ 1 1 <span class="org-type">-</span>2 1 1 <span class="org-type">-</span>2;
|
|
sqrt<span class="org-type">(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;</span>
|
|
sqrt<span class="org-type">(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);</span>
|
|
0 0 L L <span class="org-type">-</span>L <span class="org-type">-</span>L;
|
|
<span class="org-type">-</span>L<span class="org-type">*</span>2<span class="org-type">/</span>sqrt(3) <span class="org-type">-</span>L<span class="org-type">*</span>2<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3);
|
|
L<span class="org-type">*</span>sqrt(2) <span class="org-type">-</span>L<span class="org-type">*</span>sqrt(2) L<span class="org-type">*</span>sqrt(2) <span class="org-type">-</span>L<span class="org-type">*</span>sqrt(2) L<span class="org-type">*</span>sqrt(2) <span class="org-type">-</span>L<span class="org-type">*</span>sqrt(2)];
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2c9ff6d" class="outline-3">
|
|
<h3 id="org2c9ff6d"><span class="section-number-3">4.4</span> Stifnness and Damping matrices</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">kv = k<span class="org-type">/</span>3; <span class="org-comment">% Vertical Stiffness of the springs [N/m]</span>
|
|
kh = 0.5<span class="org-type">*</span>k<span class="org-type">/</span>3; <span class="org-comment">% Horizontal Stiffness of the springs [N/m]</span>
|
|
|
|
K = diag([3<span class="org-type">*</span>kh, 3<span class="org-type">*</span>kh, 3<span class="org-type">*</span>kv, 3<span class="org-type">*</span>kv<span class="org-type">*</span>Rx<span class="org-type">^</span>2<span class="org-type">/</span>2, 3<span class="org-type">*</span>kv<span class="org-type">*</span>Rx<span class="org-type">^</span>2<span class="org-type">/</span>2, 3<span class="org-type">*</span>kh<span class="org-type">*</span>Rx<span class="org-type">^</span>2]); <span class="org-comment">% Stiffness Matrix</span>
|
|
C = c<span class="org-type">*</span>K<span class="org-type">/</span>100000; <span class="org-comment">% Damping Matrix</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgffba0a8" class="outline-3">
|
|
<h3 id="orgffba0a8"><span class="section-number-3">4.5</span> State Space System</h3>
|
|
<div class="outline-text-3" id="text-4-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">A = [ zeros(6) eye(6); ...
|
|
<span class="org-type">-</span>M<span class="org-type">\</span>K <span class="org-type">-</span>M<span class="org-type">\</span>C];
|
|
Bw = [zeros(6); <span class="org-type">-</span>eye(6)];
|
|
Bu = [zeros(6); M<span class="org-type">\</span>Bj];
|
|
|
|
Co = [<span class="org-type">-</span>M<span class="org-type">\</span>K <span class="org-type">-</span>M<span class="org-type">\</span>C];
|
|
|
|
D = [zeros(6) M<span class="org-type">\</span>Bj];
|
|
|
|
ST = ss(A,[Bw Bu],Co,D);
|
|
</pre>
|
|
</div>
|
|
|
|
<ul class="org-ul">
|
|
<li>OUT 1-6: 6 dof</li>
|
|
<li>IN 1-6 : ground displacement in the directions of the legs</li>
|
|
<li>IN 7-12: forces in the actuators.</li>
|
|
</ul>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ST.StateName = {<span class="org-string">'x'</span>;<span class="org-string">'y'</span>;<span class="org-string">'z'</span>;<span class="org-string">'theta_x'</span>;<span class="org-string">'theta_y'</span>;<span class="org-string">'theta_z'</span>;...
|
|
<span class="org-string">'dx'</span>;<span class="org-string">'dy'</span>;<span class="org-string">'dz'</span>;<span class="org-string">'dtheta_x'</span>;<span class="org-string">'dtheta_y'</span>;<span class="org-string">'dtheta_z'</span>};
|
|
|
|
ST.InputName = {<span class="org-string">'w1'</span>;<span class="org-string">'w2'</span>;<span class="org-string">'w3'</span>;<span class="org-string">'w4'</span>;<span class="org-string">'w5'</span>;<span class="org-string">'w6'</span>;...
|
|
<span class="org-string">'u1'</span>;<span class="org-string">'u2'</span>;<span class="org-string">'u3'</span>;<span class="org-string">'u4'</span>;<span class="org-string">'u5'</span>;<span class="org-string">'u6'</span>};
|
|
|
|
ST.OutputName = {<span class="org-string">'ax'</span>;<span class="org-string">'ay'</span>;<span class="org-string">'az'</span>;<span class="org-string">'atheta_x'</span>;<span class="org-string">'atheta_y'</span>;<span class="org-string">'atheta_z'</span>};
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org42b1b07" class="outline-3">
|
|
<h3 id="org42b1b07"><span class="section-number-3">4.6</span> Transmissibility</h3>
|
|
<div class="outline-text-3" id="text-4-6">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">TR=ST<span class="org-type">*</span>[eye(6); zeros(6)];
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>
|
|
subplot(231)
|
|
bodemag(TR(1,1));
|
|
subplot(232)
|
|
bodemag(TR(2,2));
|
|
subplot(233)
|
|
bodemag(TR(3,3));
|
|
subplot(234)
|
|
bodemag(TR(4,4));
|
|
subplot(235)
|
|
bodemag(TR(5,5));
|
|
subplot(236)
|
|
bodemag(TR(6,6));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgd9b6731" class="figure">
|
|
<p><img src="figs/stewart_platform_analytical_transmissibility.png" alt="stewart_platform_analytical_transmissibility.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Transmissibility</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org38c8159" class="outline-3">
|
|
<h3 id="org38c8159"><span class="section-number-3">4.7</span> Real approximation of \(G(j\omega)\) at decoupling frequency</h3>
|
|
<div class="outline-text-3" id="text-4-7">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sys1 = ST<span class="org-type">*</span>[zeros(6); eye(6)]; <span class="org-comment">% take only the forces inputs</span>
|
|
|
|
dec_fr = 20;
|
|
H1 = evalfr(sys1,<span class="org-constant">j</span><span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>dec_fr);
|
|
H2 = H1;
|
|
D = pinv(real(H2<span class="org-type">'*</span>H2));
|
|
H1 = inv(D<span class="org-type">*</span>real(H2<span class="org-type">'*</span>diag(exp(<span class="org-constant">j</span><span class="org-type">*</span>angle(diag(H2<span class="org-type">*</span>D<span class="org-type">*</span>H2<span class="org-type">.'</span>))<span class="org-type">/</span>2)))) ;
|
|
[U,S,V] = svd(H1);
|
|
|
|
wf = logspace(<span class="org-type">-</span>1,2,1000);
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(wf)</span>
|
|
H = abs(evalfr(sys1,<span class="org-constant">j</span><span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wf(<span class="org-constant">i</span>)));
|
|
H_dec = abs(evalfr(U<span class="org-type">'*</span>sys1<span class="org-type">*</span>V,<span class="org-constant">j</span><span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wf(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:size(H,2)</span>
|
|
g_r1(<span class="org-constant">i</span>,<span class="org-constant">j</span>) = (sum(H(<span class="org-constant">j</span>,<span class="org-type">:</span>))<span class="org-type">-</span>H(<span class="org-constant">j</span>,<span class="org-constant">j</span>))<span class="org-type">/</span>H(<span class="org-constant">j</span>,<span class="org-constant">j</span>);
|
|
g_r2(<span class="org-constant">i</span>,<span class="org-constant">j</span>) = (sum(H_dec(<span class="org-constant">j</span>,<span class="org-type">:</span>))<span class="org-type">-</span>H_dec(<span class="org-constant">j</span>,<span class="org-constant">j</span>))<span class="org-type">/</span>H_dec(<span class="org-constant">j</span>,<span class="org-constant">j</span>);
|
|
<span class="org-comment">% keyboard</span>
|
|
<span class="org-keyword">end</span>
|
|
g_lim(<span class="org-constant">i</span>) = 0.5;
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org477b3ce" class="outline-3">
|
|
<h3 id="org477b3ce"><span class="section-number-3">4.8</span> Coupled and Decoupled Plant “Gershgorin Radii”</h3>
|
|
<div class="outline-text-3" id="text-4-8">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
title(<span class="org-string">'Coupled plant'</span>)
|
|
loglog(wf,g_r1(<span class="org-type">:</span>,1),wf,g_r1(<span class="org-type">:</span>,2),wf,g_r1(<span class="org-type">:</span>,3),wf,g_r1(<span class="org-type">:</span>,4),wf,g_r1(<span class="org-type">:</span>,5),wf,g_r1(<span class="org-type">:</span>,6),wf,g_lim,<span class="org-string">'--'</span>);
|
|
legend(<span class="org-string">'$a_x$'</span>,<span class="org-string">'$a_y$'</span>,<span class="org-string">'$a_z$'</span>,<span class="org-string">'$\theta_x$'</span>,<span class="org-string">'$\theta_y$'</span>,<span class="org-string">'$\theta_z$'</span>,<span class="org-string">'Limit'</span>);
|
|
xlabel(<span class="org-string">'Frequency (Hz)'</span>); ylabel(<span class="org-string">'Gershgorin Radii'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org1416731" class="figure">
|
|
<p><img src="figs/gershorin_raddii_coupled_analytical.png" alt="gershorin_raddii_coupled_analytical.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Gershorin Raddi for the coupled plant</p>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
title(<span class="org-string">'Decoupled plant (10 Hz)'</span>)
|
|
loglog(wf,g_r2(<span class="org-type">:</span>,1),wf,g_r2(<span class="org-type">:</span>,2),wf,g_r2(<span class="org-type">:</span>,3),wf,g_r2(<span class="org-type">:</span>,4),wf,g_r2(<span class="org-type">:</span>,5),wf,g_r2(<span class="org-type">:</span>,6),wf,g_lim,<span class="org-string">'--'</span>);
|
|
legend(<span class="org-string">'$S_1$'</span>,<span class="org-string">'$S_2$'</span>,<span class="org-string">'$S_3$'</span>,<span class="org-string">'$S_4$'</span>,<span class="org-string">'$S_5$'</span>,<span class="org-string">'$S_6$'</span>,<span class="org-string">'Limit'</span>);
|
|
xlabel(<span class="org-string">'Frequency (Hz)'</span>); ylabel(<span class="org-string">'Gershgorin Radii'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgdfe4880" class="figure">
|
|
<p><img src="figs/gershorin_raddii_decoupled_analytical.png" alt="gershorin_raddii_decoupled_analytical.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Gershorin Raddi for the decoupled plant</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgde4eec1" class="outline-3">
|
|
<h3 id="orgde4eec1"><span class="section-number-3">4.9</span> Decoupled Plant</h3>
|
|
<div class="outline-text-3" id="text-4-9">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
bodemag(U<span class="org-type">'*</span>sys1<span class="org-type">*</span>V,opts)
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orge835f28" class="figure">
|
|
<p><img src="figs/stewart_platform_analytical_decoupled_plant.png" alt="stewart_platform_analytical_decoupled_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Decoupled Plant</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org11b0182" class="outline-3">
|
|
<h3 id="org11b0182"><span class="section-number-3">4.10</span> Controller</h3>
|
|
<div class="outline-text-3" id="text-4-10">
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<div class="org-src-container">
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<pre class="src src-matlab">fc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>0.1; <span class="org-comment">% Crossover Frequency [rad/s]</span>
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c_gain = 50; <span class="org-comment">%</span>
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cont = eye(6)<span class="org-type">*</span>c_gain<span class="org-type">/</span>(s<span class="org-type">+</span>fc);
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org5c893a8" class="outline-3">
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<h3 id="org5c893a8"><span class="section-number-3">4.11</span> Closed Loop System</h3>
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<div class="outline-text-3" id="text-4-11">
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<div class="org-src-container">
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<pre class="src src-matlab">FEEDIN = [7<span class="org-type">:</span>12]; <span class="org-comment">% Input of controller</span>
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FEEDOUT = [1<span class="org-type">:</span>6]; <span class="org-comment">% Output of controller</span>
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</pre>
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</div>
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<p>
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Centralized Control
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">STcen = feedback(ST, inv(Bj)<span class="org-type">*</span>cont, FEEDIN, FEEDOUT);
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TRcen = STcen<span class="org-type">*</span>[eye(6); zeros(6)];
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</pre>
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</div>
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<p>
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SVD Control
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">STsvd = feedback(ST, pinv(V<span class="org-type">'</span>)<span class="org-type">*</span>cont<span class="org-type">*</span>pinv(U), FEEDIN, FEEDOUT);
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TRsvd = STsvd<span class="org-type">*</span>[eye(6); zeros(6)];
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</pre>
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</div>
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|
</div>
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|
</div>
|
|
|
|
<div id="outline-container-orgb1c0711" class="outline-3">
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<h3 id="orgb1c0711"><span class="section-number-3">4.12</span> Results</h3>
|
|
<div class="outline-text-3" id="text-4-12">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>
|
|
subplot(231)
|
|
bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
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legend(<span class="org-string">'OL'</span>,<span class="org-string">'Centralized'</span>,<span class="org-string">'SVD'</span>)
|
|
subplot(232)
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bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
|
|
legend(<span class="org-string">'OL'</span>,<span class="org-string">'Centralized'</span>,<span class="org-string">'SVD'</span>)
|
|
subplot(233)
|
|
bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
|
|
legend(<span class="org-string">'OL'</span>,<span class="org-string">'Centralized'</span>,<span class="org-string">'SVD'</span>)
|
|
subplot(234)
|
|
bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
|
|
legend(<span class="org-string">'OL'</span>,<span class="org-string">'Centralized'</span>,<span class="org-string">'SVD'</span>)
|
|
subplot(235)
|
|
bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
|
|
legend(<span class="org-string">'OL'</span>,<span class="org-string">'Centralized'</span>,<span class="org-string">'SVD'</span>)
|
|
subplot(236)
|
|
bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
|
|
legend(<span class="org-string">'OL'</span>,<span class="org-string">'Centralized'</span>,<span class="org-string">'SVD'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgb680082" class="figure">
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|
<p><img src="figs/stewart_platform_analytical_svd_cen_comp.png" alt="stewart_platform_analytical_svd_cen_comp.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Comparison of the obtained transmissibility for the centralized control and the SVD control</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
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</div>
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|
</div>
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|
<div id="postamble" class="status">
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|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-10-13 mar. 14:53</p>
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</div>
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</body>
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</html>
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