535 lines
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535 lines
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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">HAC-LAC applied on the Simscape Model</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="#org9a782be">1. Initialization</a></li>
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<li><a href="#org5f71356">2. Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)</a>
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<ul>
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<li><a href="#orgd91db57">2.1. Identification</a></li>
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<li><a href="#org09c8990">2.2. Plant</a></li>
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<li><a href="#orgc21ad83">2.3. Root Locus</a></li>
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<li><a href="#orgd18d476">2.4. Controller and Loop Gain</a></li>
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</ul>
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</li>
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<li><a href="#orgbba86f0">3. High Authority Control - \(\bm{K}_\mathcal{X}\)</a>
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<ul>
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<li><a href="#orgdd75cdd">3.1. Identification of the damped plant</a></li>
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<li><a href="#orgf70f020">3.2. Controller Design</a></li>
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</ul>
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</li>
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<li><a href="#org5a1507e">4. Simulation</a></li>
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<li><a href="#org9498b7b">5. Results</a></li>
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</ul>
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</div>
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</div>
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<p>
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The position \(\bm{\mathcal{X}}\) of the Sample with respect to the granite is measured.
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</p>
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<p>
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It is then compare to the wanted position of the Sample \(\bm{r}_\mathcal{X}\) in order to obtain the position error \(\bm{\epsilon}_\mathcal{X}\) of the Sample with respect to a frame attached to the Stewart top platform.
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</p>
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<div class="figure">
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<p><img src="figs/hac_lac_control_schematic.png" alt="hac_lac_control_schematic.png" />
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</p>
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</div>
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<div id="outline-container-org9a782be" class="outline-2">
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<h2 id="org9a782be"><span class="section-number-2">1</span> Initialization</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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We initialize all the stages with the default parameters.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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</pre>
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</div>
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<p>
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The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
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initializeSample(<span class="org-string">'mass'</span>, 1);
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</pre>
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</div>
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<p>
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We set the references that corresponds to a tomography experiment.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, 1);
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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">initializeDisturbances();
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</pre>
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</div>
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<p>
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Open Loop.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
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</pre>
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</div>
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<p>
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And we put some gravity.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
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</pre>
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</div>
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<p>
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We log the signals.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</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-org5f71356" class="outline-2">
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<h2 id="org5f71356"><span class="section-number-2">2</span> Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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The first loop closed corresponds to a direct velocity feedback loop.
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</p>
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<p>
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The design of the associated decentralized controller is explained in <a href="active_damping.html">this</a> file.
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</p>
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</div>
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<div id="outline-container-orgd91db57" class="outline-3">
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<h3 id="orgd91db57"><span class="section-number-3">2.1</span> Identification</h3>
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<div class="outline-text-3" id="text-2-1">
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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">'nass_model'</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">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Inputs</span>
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io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station'</span>], 3, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Dnlm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Motion Outputs</span>
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
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G_dvf = linearize(mdl, io, 0);
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G_dvf.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>};
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G_dvf.OutputName = {<span class="org-string">'Dnlm1'</span>, <span class="org-string">'Dnlm2'</span>, <span class="org-string">'Dnlm3'</span>, <span class="org-string">'Dnlm4'</span>, <span class="org-string">'Dnlm5'</span>, <span class="org-string">'Dnlm6'</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-org09c8990" class="outline-3">
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<h3 id="org09c8990"><span class="section-number-3">2.2</span> Plant</h3>
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</div>
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<div id="outline-container-orgc21ad83" class="outline-3">
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<h3 id="orgc21ad83"><span class="section-number-3">2.3</span> Root Locus</h3>
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</div>
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<div id="outline-container-orgd18d476" class="outline-3">
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<h3 id="orgd18d476"><span class="section-number-3">2.4</span> Controller and Loop Gain</h3>
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<div class="outline-text-3" id="text-2-4">
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<div class="org-src-container">
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<pre class="src src-matlab">K_dvf = s<span class="org-type">*</span>15000<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10000);
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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">K_dvf = <span class="org-type">-</span>K_dvf<span class="org-type">*</span>eye(6);
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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>
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<div id="outline-container-orgbba86f0" class="outline-2">
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<h2 id="orgbba86f0"><span class="section-number-2">3</span> High Authority Control - \(\bm{K}_\mathcal{X}\)</h2>
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<div class="outline-text-2" id="text-3">
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</div>
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<div id="outline-container-orgdd75cdd" class="outline-3">
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<h3 id="orgdd75cdd"><span class="section-number-3">3.1</span> Identification of the damped plant</h3>
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<div class="outline-text-3" id="text-3-1">
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<div class="org-src-container">
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<pre class="src src-matlab">Kx = tf(zeros(6));
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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">initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</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"><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">'nass_model'</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">'/Controller'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Inputs</span>
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io(io_i) = linio([mdl, <span class="org-string">'/Tracking Error'</span>], 1, <span class="org-string">'output'</span>, [], <span class="org-string">'En'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position Errror</span>
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
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G = linearize(mdl, io, 0);
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G.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>};
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G.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">'Ey'</span>, <span class="org-string">'Ez'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
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</pre>
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</div>
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<p>
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The minus sine is put here because there is already a minus sign included due to the computation of the position error.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>);
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|
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Gx = <span class="org-type">-</span>G<span class="org-type">*</span>inv(nano_hexapod.J<span class="org-type">'</span>);
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Gx.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
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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-orgf70f020" class="outline-3">
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<h3 id="orgf70f020"><span class="section-number-3">3.2</span> Controller Design</h3>
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|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
The controller consists of:
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|
</p>
|
|
<ul class="org-ul">
|
|
<li>A pure integrator</li>
|
|
<li>A Second integrator up to half the wanted bandwidth</li>
|
|
<li>A Lead around the cross-over frequency</li>
|
|
<li>A low pass filter with a cut-off equal to two times the wanted bandwidth</li>
|
|
</ul>
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|
|
|
<div class="org-src-container">
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|
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>15; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
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|
|
h = 1.5; <span class="org-comment">% Lead parameter</span>
|
|
|
|
Kx = (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> wc<span class="org-type">/</span>s <span class="org-type">*</span> ((s<span class="org-type">/</span>wc<span class="org-type">*</span>2 <span class="org-type">+</span> 1)<span class="org-type">/</span>(s<span class="org-type">/</span>wc<span class="org-type">*</span>2)) <span class="org-type">*</span> (1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>2));
|
|
|
|
<span class="org-comment">% Normalization of the gain of have a loop gain of 1 at frequency wc</span>
|
|
Kx = Kx<span class="org-type">.*</span>diag(1<span class="org-type">./</span>diag(abs(freqresp(Gx<span class="org-type">*</span>Kx, wc))));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable(feedback(Gx<span class="org-type">*</span>Kx, eye(6), <span class="org-type">-</span>1))
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kx = inv(nano_hexapod.J<span class="org-type">'</span>)<span class="org-type">*</span>Kx;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable(feedback(G<span class="org-type">*</span>Kx, eye(6), 1))
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5a1507e" class="outline-2">
|
|
<h2 id="org5a1507e"><span class="section-number-2">4</span> Simulation</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
|
|
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'1.5'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we simulate the system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'./mat/tomo_exp_hac_lac.mat'</span>, <span class="org-string">'simout'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9498b7b" class="outline-2">
|
|
<h2 id="org9498b7b"><span class="section-number-2">5</span> Results</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./mat/tomo_exp_hac_lac.mat'</span>, <span class="org-string">'simout'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-03-13 ven. 17:39</p>
|
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</div>
|
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</body>
|
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</html>
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