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<title>HAC-LAC applied on the Simscape Model</title>
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<h1 class="title">HAC-LAC applied on the Simscape Model</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org689cb19">1. Initialization</a></li>
<li><a href="#org24d1d7f">2. Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)</a>
<ul>
<li><a href="#org68cbbe7">2.1. Identification</a></li>
<li><a href="#org7c1c629">2.2. Plant</a></li>
<li><a href="#org719e252">2.3. Root Locus</a></li>
<li><a href="#org2c42b90">2.4. Controller and Loop Gain</a></li>
</ul>
</li>
<li><a href="#org195a29c">3. Uncertainty Improvements thanks to the LAC control</a></li>
<li><a href="#org81738ac">4. High Authority Control - \(\bm{K}_\mathcal{X}\)</a>
<ul>
<li><a href="#org94cb2c9">4.1. Identification of the damped plant</a></li>
<li><a href="#org2a269d5">4.2. Controller Design</a></li>
</ul>
</li>
<li><a href="#org1171960">5. Simulation</a></li>
<li><a href="#orgad3e1c7">6. Results</a></li>
</ul>
</div>
</div>
<p>
The position \(\bm{\mathcal{X}}\) of the Sample with respect to the granite is measured.
</p>
<p>
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.
</p>
<div id="org270f668" class="figure">
<p><img src="figs/hac_lac_control_schematic.png" alt="hac_lac_control_schematic.png" />
</p>
<p><span class="figure-number">Figure 1: </span>HAC-LAC Control Architecture used for the Control of the NASS</p>
</div>
<div id="outline-container-org689cb19" class="outline-2">
<h2 id="org689cb19"><span class="section-number-2">1</span> Initialization</h2>
<div class="outline-text-2" id="text-1">
<p>
We initialize all the stages with the default parameters.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
</pre>
</div>
<p>
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
initializeSample(<span class="org-string">'mass'</span>, 1);
</pre>
</div>
<p>
We set the references that corresponds to a tomography experiment.
</p>
<div class="org-src-container">
<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);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> initializeDisturbances();
</pre>
</div>
<p>
Open Loop.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
<p>
And we put some gravity.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
</pre>
</div>
<p>
We log the signals.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org24d1d7f" class="outline-2">
<h2 id="org24d1d7f"><span class="section-number-2">2</span> Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)</h2>
<div class="outline-text-2" id="text-2">
<p>
The first loop closed corresponds to a direct velocity feedback loop.
</p>
<p>
The design of the associated decentralized controller is explained in <a href="control_active_damping.html">this</a> file.
</p>
</div>
<div id="outline-container-org68cbbe7" class="outline-3">
<h3 id="org68cbbe7"><span class="section-number-3">2.1</span> Identification</h3>
<div class="outline-text-3" id="text-2-1">
<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">'nass_model'</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">'/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>
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>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_dvf = linearize(mdl, io, 0);
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>};
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>};
</pre>
</div>
</div>
</div>
<div id="outline-container-org7c1c629" class="outline-3">
<h3 id="org7c1c629"><span class="section-number-3">2.2</span> Plant</h3>
</div>
<div id="outline-container-org719e252" class="outline-3">
<h3 id="org719e252"><span class="section-number-3">2.3</span> Root Locus</h3>
</div>
<div id="outline-container-org2c42b90" class="outline-3">
<h3 id="org2c42b90"><span class="section-number-3">2.4</span> Controller and Loop Gain</h3>
<div class="outline-text-3" id="text-2-4">
<div class="org-src-container">
<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);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> K_dvf = <span class="org-type">-</span>K_dvf<span class="org-type">*</span>eye(6);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org195a29c" class="outline-2">
<h2 id="org195a29c"><span class="section-number-2">3</span> Uncertainty Improvements thanks to the LAC control</h2>
<div class="outline-text-2" id="text-3">
<div class="org-src-container">
<pre class="src src-matlab"> K_dvf_backup = K_dvf;
initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> masses = [1, 10, 50]; <span class="org-comment">% [kg]</span>
</pre>
</div>
<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">'nass_model'</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">'/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>
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>
</pre>
</div>
</div>
</div>
<div id="outline-container-org81738ac" class="outline-2">
<h2 id="org81738ac"><span class="section-number-2">4</span> High Authority Control - \(\bm{K}_\mathcal{X}\)</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org94cb2c9" class="outline-3">
<h3 id="org94cb2c9"><span class="section-number-3">4.1</span> Identification of the damped plant</h3>
<div class="outline-text-3" id="text-4-1">
<div class="org-src-container">
<pre class="src src-matlab"> Kx = tf(zeros(6));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
</pre>
</div>
<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">'nass_model'</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">'/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>
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>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, 0);
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>};
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>};
</pre>
</div>
<p>
The minus sine is put here because there is already a minus sign included due to the computation of the position error.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>);
Gx = <span class="org-type">-</span>G<span class="org-type">*</span>inv(nano_hexapod.kinematics.J<span class="org-type">'</span>);
Gx.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-org2a269d5" class="outline-3">
<h3 id="org2a269d5"><span class="section-number-3">4.2</span> Controller Design</h3>
<div class="outline-text-3" id="text-4-2">
<p>
The controller consists of:
</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>
<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>15; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
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.kinematics.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-org1171960" class="outline-2">
<h2 id="org1171960"><span class="section-number-2">5</span> Simulation</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/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">'2'</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"> hac_dvf = simout;
save(<span class="org-string">'./mat/tomo_exp_hac_lac.mat'</span>, <span class="org-string">'hac_dvf'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgad3e1c7" class="outline-2">
<h2 id="orgad3e1c7"><span class="section-number-2">6</span> Results</h2>
<div class="outline-text-2" id="text-6">
<p>
Let’s load the simulation when no control is applied.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> load(<span class="org-string">'./mat/experiment_tomography.mat'</span>, <span class="org-string">'tomo_align_dist'</span>);
load(<span class="org-string">'./mat/tomo_exp_hac_lac.mat'</span>, <span class="org-string">'hac_dvf'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-02-20 sam. 23:08</p>
</div>
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