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Correct wrong computation of Gershgorin radii

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Thomas Dehaeze 2020-09-21 18:54:41 +02:00
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9 changed files with 299 additions and 161 deletions
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-09-21 lun. 18:03 --> <!-- 2020-09-21 lun. 18:53 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>SVD Control</title> <title>SVD Control</title>
<meta name="generator" content="Org mode" /> <meta name="generator" content="Org mode" />
@ -35,52 +35,52 @@
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#orge03ef95">1. Gravimeter - Simscape Model</a> <li><a href="#orgcdbafa9">1. Gravimeter - Simscape Model</a>
<ul> <ul>
<li><a href="#org94cda63">1.1. Simulink</a></li> <li><a href="#orge0e53b8">1.1. Simulink</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org01f2bcf">2. Stewart Platform - Simscape Model</a> <li><a href="#orgbbb84cc">2. Stewart Platform - Simscape Model</a>
<ul> <ul>
<li><a href="#org5d90a14">2.1. Jacobian</a></li> <li><a href="#org22e9f4a">2.1. Jacobian</a></li>
<li><a href="#org7bbb169">2.2. Simscape Model</a></li> <li><a href="#org692c2cc">2.2. Simscape Model</a></li>
<li><a href="#org2a265c4">2.3. Identification of the plant</a></li> <li><a href="#orga491806">2.3. Identification of the plant</a></li>
<li><a href="#orgfa83a84">2.4. Obtained Dynamics</a></li> <li><a href="#orgac4aba9">2.4. Obtained Dynamics</a></li>
<li><a href="#org92dd977">2.5. Real Approximation of \(G\) at the decoupling frequency</a></li> <li><a href="#org1e57236">2.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
<li><a href="#orgebf7751">2.6. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li> <li><a href="#org0172b58">2.6. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li>
<li><a href="#orge21a525">2.7. Decoupled Plant</a></li> <li><a href="#org37cffae">2.7. Decoupled Plant</a></li>
<li><a href="#org4c1f528">2.8. Diagonal Controller</a></li> <li><a href="#org868aae1">2.8. Diagonal Controller</a></li>
<li><a href="#org4f88748">2.9. Centralized Control</a></li> <li><a href="#org14f6c79">2.9. Centralized Control</a></li>
<li><a href="#org6eac181">2.10. SVD Control</a></li> <li><a href="#orgdfd243c">2.10. SVD Control</a></li>
<li><a href="#org89ccc9f">2.11. Results</a></li> <li><a href="#orge500424">2.11. Results</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgdcb6e90">3. Stewart Platform - Analytical Model</a> <li><a href="#org95fefe5">3. Stewart Platform - Analytical Model</a>
<ul> <ul>
<li><a href="#orgeb4b14b">3.1. Characteristics</a></li> <li><a href="#org79b474c">3.1. Characteristics</a></li>
<li><a href="#orgeff797b">3.2. Mass Matrix</a></li> <li><a href="#org8f55e38">3.2. Mass Matrix</a></li>
<li><a href="#org7027995">3.3. Jacobian Matrix</a></li> <li><a href="#orgecf190a">3.3. Jacobian Matrix</a></li>
<li><a href="#org51bab7b">3.4. Stifnness matrix and Damping matrix</a></li> <li><a href="#org7260a03">3.4. Stifnness matrix and Damping matrix</a></li>
<li><a href="#orga9e6cf5">3.5. State Space System</a></li> <li><a href="#orga9cf2e9">3.5. State Space System</a></li>
<li><a href="#org769c38a">3.6. Transmissibility</a></li> <li><a href="#org41defbd">3.6. Transmissibility</a></li>
<li><a href="#org24eb81f">3.7. Real approximation of \(G(j\omega)\) at decoupling frequency</a></li> <li><a href="#org5767eaf">3.7. Real approximation of \(G(j\omega)\) at decoupling frequency</a></li>
<li><a href="#org824e380">3.8. Coupled and Decoupled Plant &ldquo;Gershgorin Radii&rdquo;</a></li> <li><a href="#orgae9dbc4">3.8. Coupled and Decoupled Plant &ldquo;Gershgorin Radii&rdquo;</a></li>
<li><a href="#org8e5d2c7">3.9. Decoupled Plant</a></li> <li><a href="#orgc43f18b">3.9. Decoupled Plant</a></li>
<li><a href="#org102382b">3.10. Controller</a></li> <li><a href="#orgf75bbde">3.10. Controller</a></li>
<li><a href="#org27bf3be">3.11. Closed Loop System</a></li> <li><a href="#org275519b">3.11. Closed Loop System</a></li>
<li><a href="#org419f877">3.12. Results</a></li> <li><a href="#org16451fc">3.12. Results</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-orge03ef95" class="outline-2"> <div id="outline-container-orgcdbafa9" class="outline-2">
<h2 id="orge03ef95"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2> <h2 id="orgcdbafa9"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-org94cda63" class="outline-3"> <div id="outline-container-orge0e53b8" class="outline-3">
<h3 id="org94cda63"><span class="section-number-3">1.1</span> Simulink</h3> <h3 id="orge0e53b8"><span class="section-number-3">1.1</span> Simulink</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">open('gravimeter.slx') <pre class="src src-matlab">open('gravimeter.slx')
@ -122,7 +122,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
<div id="org57d8f45" class="figure"> <div id="org3d20c51" class="figure">
<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" /> <p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p> <p><span class="figure-number">Figure 1: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
@ -131,12 +131,12 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org01f2bcf" class="outline-2"> <div id="outline-container-orgbbb84cc" class="outline-2">
<h2 id="org01f2bcf"><span class="section-number-2">2</span> Stewart Platform - Simscape Model</h2> <h2 id="orgbbb84cc"><span class="section-number-2">2</span> Stewart Platform - Simscape Model</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
</div> </div>
<div id="outline-container-org5d90a14" class="outline-3"> <div id="outline-container-org22e9f4a" class="outline-3">
<h3 id="org5d90a14"><span class="section-number-3">2.1</span> Jacobian</h3> <h3 id="org22e9f4a"><span class="section-number-3">2.1</span> Jacobian</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
First, the position of the &ldquo;joints&rdquo; (points of force application) are estimated and the Jacobian computed. First, the position of the &ldquo;joints&rdquo; (points of force application) are estimated and the Jacobian computed.
@ -178,8 +178,8 @@ save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
</div> </div>
</div> </div>
<div id="outline-container-org7bbb169" class="outline-3"> <div id="outline-container-org692c2cc" class="outline-3">
<h3 id="org7bbb169"><span class="section-number-3">2.2</span> Simscape Model</h3> <h3 id="org692c2cc"><span class="section-number-3">2.2</span> Simscape Model</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">open('stewart_platform/drone_platform.slx'); <pre class="src src-matlab">open('stewart_platform/drone_platform.slx');
@ -210,8 +210,8 @@ We load the Jacobian.
</div> </div>
</div> </div>
<div id="outline-container-org2a265c4" class="outline-3"> <div id="outline-container-orga491806" class="outline-3">
<h3 id="org2a265c4"><span class="section-number-3">2.3</span> Identification of the plant</h3> <h3 id="orga491806"><span class="section-number-3">2.3</span> Identification of the plant</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform. The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
@ -268,32 +268,32 @@ Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
</div> </div>
</div> </div>
<div id="outline-container-orgfa83a84" class="outline-3"> <div id="outline-container-orgac4aba9" class="outline-3">
<h3 id="orgfa83a84"><span class="section-number-3">2.4</span> Obtained Dynamics</h3> <h3 id="orgac4aba9"><span class="section-number-3">2.4</span> Obtained Dynamics</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<div id="orga7d2bfa" class="figure"> <div id="org24b69fe" class="figure">
<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" /> <p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p> <p><span class="figure-number">Figure 2: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
</div> </div>
<div id="orge8ecc72" class="figure"> <div id="orgba8e960" class="figure">
<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" /> <p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p> <p><span class="figure-number">Figure 3: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
</div> </div>
<div id="orga068faf" class="figure"> <div id="orga66f388" class="figure">
<p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" /> <p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p> <p><span class="figure-number">Figure 4: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p>
</div> </div>
<div id="orgf48c4d4" class="figure"> <div id="orga7254f2" class="figure">
<p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" /> <p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Transmissibility</p> <p><span class="figure-number">Figure 5: </span>Transmissibility</p>
@ -301,20 +301,19 @@ Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
</div> </div>
</div> </div>
<div id="outline-container-org92dd977" class="outline-3"> <div id="outline-container-org1e57236" class="outline-3">
<h3 id="org92dd977"><span class="section-number-3">2.5</span> Real Approximation of \(G\) at the decoupling frequency</h3> <h3 id="org1e57236"><span class="section-number-3">2.5</span> Real Approximation of \(G\) at the decoupling frequency</h3>
<div class="outline-text-3" id="text-2-5"> <div class="outline-text-3" id="text-2-5">
<p> <p>
Let&rsquo;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 \(\omega_c\). Let&rsquo;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> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">wc = 2*pi*20; % Decoupling frequency [rad/s] <pre class="src src-matlab">wc = 2*pi*20; % Decoupling frequency [rad/s]
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
</pre>
</div>
<div class="org-src-container"> Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
<pre class="src src-matlab">H1 = evalfr(Gc, j*wc); {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
H1 = evalfr(Gc, j*wc);
</pre> </pre>
</div> </div>
@ -329,8 +328,8 @@ H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
</div> </div>
</div> </div>
<div id="outline-container-orgebf7751" class="outline-3"> <div id="outline-container-org0172b58" class="outline-3">
<h3 id="orgebf7751"><span class="section-number-3">2.6</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3> <h3 id="org0172b58"><span class="section-number-3">2.6</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3>
<div class="outline-text-3" id="text-2-6"> <div class="outline-text-3" id="text-2-6">
<p> <p>
First, the Singular Value Decomposition of \(H_1\) is performed: First, the Singular Value Decomposition of \(H_1\) is performed:
@ -348,49 +347,90 @@ Then, the &ldquo;Gershgorin Radii&rdquo; is computed for the plant \(G_c(s)\) an
</p> </p>
<p> <p>
It is done over the following frequencies. This is computed over the following frequencies.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(-1,2,1000); % [Hz] <pre class="src src-matlab">freqs = logspace(-2, 2, 1000); % [Hz]
</pre> </pre>
</div> </div>
<p>
Gershgorin Radii for the coupled plant:
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">for i = 1:length(freqs) <pre class="src src-matlab">Gr_coupled = zeros(length(freqs), size(Gc,2));
H = abs(evalfr(Gc, j*2*pi*freqs(i)));
for j = 1:size(H,2) H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
g_r1(i,j) = (sum(H(j,:)) - H(j,j))/H(j,j); for out_i = 1:size(Gc,2)
end Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end end
</pre> </pre>
</div> </div>
<p>
Gershgorin Radii for the decoupled plant using SVD:
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Gd = U'*Gc*V; <pre class="src src-matlab">Gd = U'*Gc*V;
Gr_decoupled = zeros(length(freqs), size(Gd,2));
for i = 1:length(freqs) H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
H_dec = abs(evalfr(Gd, j*2*pi*freqs(i))); for out_i = 1:size(Gd,2)
for j = 1:size(H,2) Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
g_r2(i,j) = (sum(H_dec(j,:)) - H_dec(j,j))/H_dec(j,j);
end
end end
</pre> </pre>
</div> </div>
<p>
Gershgorin Radii for the decoupled plant using the Jacobian:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gj = Gc*inv(J');
Gr_jacobian = zeros(length(freqs), size(Gj,2));
H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
for out_i = 1:size(Gj,2)
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end
</pre>
</div>
<div id="orgca6454b" class="figure">
<p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Gershgorin Radii of the Coupled and Decoupled plants</p>
</div>
</div> </div>
</div> </div>
<div id="outline-container-orge21a525" class="outline-3"> <div id="outline-container-org37cffae" class="outline-3">
<h3 id="orge21a525"><span class="section-number-3">2.7</span> Decoupled Plant</h3> <h3 id="org37cffae"><span class="section-number-3">2.7</span> Decoupled Plant</h3>
<div class="outline-text-3" id="text-2-7"> <div class="outline-text-3" id="text-2-7">
<p> <p>
Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\). Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
\[ G_d(s) = U^T G_c(s) V \] \[ G_d(s) = U^T G_c(s) V \]
</p> </p>
<div id="org5478477" 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 7: </span>Decoupled Plant using SVD</p>
</div>
<div id="org8fa8056" 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 8: </span>Decoupled Plant using the Jacobian</p>
</div>
</div> </div>
</div> </div>
<div id="outline-container-org4c1f528" class="outline-3"> <div id="outline-container-org868aae1" class="outline-3">
<h3 id="org4c1f528"><span class="section-number-3">2.8</span> Diagonal Controller</h3> <h3 id="org868aae1"><span class="section-number-3">2.8</span> Diagonal Controller</h3>
<div class="outline-text-3" id="text-2-8"> <div class="outline-text-3" id="text-2-8">
<p> <p>
The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\). The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\).
@ -406,8 +446,8 @@ K = eye(6)*C_g/(s+wc);
</div> </div>
</div> </div>
<div id="outline-container-org4f88748" class="outline-3"> <div id="outline-container-org14f6c79" class="outline-3">
<h3 id="org4f88748"><span class="section-number-3">2.9</span> Centralized Control</h3> <h3 id="org14f6c79"><span class="section-number-3">2.9</span> Centralized Control</h3>
<div class="outline-text-3" id="text-2-9"> <div class="outline-text-3" id="text-2-9">
<p> <p>
The control diagram for the centralized control is shown below. The control diagram for the centralized control is shown below.
@ -431,8 +471,8 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
</div> </div>
</div> </div>
<div id="outline-container-org6eac181" class="outline-3"> <div id="outline-container-orgdfd243c" class="outline-3">
<h3 id="org6eac181"><span class="section-number-3">2.10</span> SVD Control</h3> <h3 id="orgdfd243c"><span class="section-number-3">2.10</span> SVD Control</h3>
<div class="outline-text-3" id="text-2-10"> <div class="outline-text-3" id="text-2-10">
<p> <p>
The SVD control architecture is shown below. The SVD control architecture is shown below.
@ -455,29 +495,56 @@ SVD Control
</div> </div>
</div> </div>
<div id="outline-container-org89ccc9f" class="outline-3"> <div id="outline-container-orge500424" class="outline-3">
<h3 id="org89ccc9f"><span class="section-number-3">2.11</span> Results</h3> <h3 id="orge500424"><span class="section-number-3">2.11</span> Results</h3>
<div class="outline-text-3" id="text-2-11"> <div class="outline-text-3" id="text-2-11">
<p> <p>
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#orgfaedd1c">8</a>. Let&rsquo;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
1
</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="#orgcffbfc0">11</a>.
</p> </p>
<div id="orgfaedd1c" class="figure"> <div id="orgcffbfc0" class="figure">
<p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" /> <p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Obtained Transmissibility</p> <p><span class="figure-number">Figure 11: </span>Obtained Transmissibility</p>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgdcb6e90" class="outline-2"> <div id="outline-container-org95fefe5" class="outline-2">
<h2 id="orgdcb6e90"><span class="section-number-2">3</span> Stewart Platform - Analytical Model</h2> <h2 id="org95fefe5"><span class="section-number-2">3</span> Stewart Platform - Analytical Model</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
</div> </div>
<div id="outline-container-orgeb4b14b" class="outline-3"> <div id="outline-container-org79b474c" class="outline-3">
<h3 id="orgeb4b14b"><span class="section-number-3">3.1</span> Characteristics</h3> <h3 id="org79b474c"><span class="section-number-3">3.1</span> Characteristics</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">L = 0.055; <pre class="src src-matlab">L = 0.055;
@ -496,8 +563,8 @@ Iz = m*Rz^2;
</div> </div>
</div> </div>
<div id="outline-container-orgeff797b" class="outline-3"> <div id="outline-container-org8f55e38" class="outline-3">
<h3 id="orgeff797b"><span class="section-number-3">3.2</span> Mass Matrix</h3> <h3 id="org8f55e38"><span class="section-number-3">3.2</span> Mass Matrix</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">M = m*[1 0 0 0 Zc 0; <pre class="src src-matlab">M = m*[1 0 0 0 Zc 0;
@ -511,8 +578,8 @@ Iz = m*Rz^2;
</div> </div>
</div> </div>
<div id="outline-container-org7027995" class="outline-3"> <div id="outline-container-orgecf190a" class="outline-3">
<h3 id="org7027995"><span class="section-number-3">3.3</span> Jacobian Matrix</h3> <h3 id="orgecf190a"><span class="section-number-3">3.3</span> Jacobian Matrix</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Bj=1/sqrt(6)*[ 1 1 -2 1 1 -2; <pre class="src src-matlab">Bj=1/sqrt(6)*[ 1 1 -2 1 1 -2;
@ -526,8 +593,8 @@ Iz = m*Rz^2;
</div> </div>
</div> </div>
<div id="outline-container-org51bab7b" class="outline-3"> <div id="outline-container-org7260a03" class="outline-3">
<h3 id="org51bab7b"><span class="section-number-3">3.4</span> Stifnness matrix and Damping matrix</h3> <h3 id="org7260a03"><span class="section-number-3">3.4</span> Stifnness matrix and Damping matrix</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">kv = k/3; % [N/m] <pre class="src src-matlab">kv = k/3; % [N/m]
@ -541,8 +608,8 @@ C = c*K/100000; % Damping Matrix
</div> </div>
</div> </div>
<div id="outline-container-orga9e6cf5" class="outline-3"> <div id="outline-container-orga9cf2e9" class="outline-3">
<h3 id="orga9e6cf5"><span class="section-number-3">3.5</span> State Space System</h3> <h3 id="orga9cf2e9"><span class="section-number-3">3.5</span> State Space System</h3>
<div class="outline-text-3" id="text-3-5"> <div class="outline-text-3" id="text-3-5">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">A = [zeros(6) eye(6); -M\K -M\C]; <pre class="src src-matlab">A = [zeros(6) eye(6); -M\K -M\C];
@ -571,8 +638,8 @@ ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
</div> </div>
</div> </div>
<div id="outline-container-org769c38a" class="outline-3"> <div id="outline-container-org41defbd" class="outline-3">
<h3 id="org769c38a"><span class="section-number-3">3.6</span> Transmissibility</h3> <h3 id="org41defbd"><span class="section-number-3">3.6</span> Transmissibility</h3>
<div class="outline-text-3" id="text-3-6"> <div class="outline-text-3" id="text-3-6">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">TR=ST*[eye(6); zeros(6)]; <pre class="src src-matlab">TR=ST*[eye(6); zeros(6)];
@ -597,16 +664,16 @@ bodemag(TR(6,6),opts);
</div> </div>
<div id="org55a5d25" class="figure"> <div id="orgf939ecb" class="figure">
<p><img src="figs/stewart_platform_analytical_transmissibility.png" alt="stewart_platform_analytical_transmissibility.png" /> <p><img src="figs/stewart_platform_analytical_transmissibility.png" alt="stewart_platform_analytical_transmissibility.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Transmissibility</p> <p><span class="figure-number">Figure 12: </span>Transmissibility</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org24eb81f" class="outline-3"> <div id="outline-container-org5767eaf" class="outline-3">
<h3 id="org24eb81f"><span class="section-number-3">3.7</span> Real approximation of \(G(j\omega)\) at decoupling frequency</h3> <h3 id="org5767eaf"><span class="section-number-3">3.7</span> Real approximation of \(G(j\omega)\) at decoupling frequency</h3>
<div class="outline-text-3" id="text-3-7"> <div class="outline-text-3" id="text-3-7">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs <pre class="src src-matlab">sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs
@ -634,8 +701,8 @@ end
</div> </div>
</div> </div>
<div id="outline-container-org824e380" class="outline-3"> <div id="outline-container-orgae9dbc4" class="outline-3">
<h3 id="org824e380"><span class="section-number-3">3.8</span> Coupled and Decoupled Plant &ldquo;Gershgorin Radii&rdquo;</h3> <h3 id="orgae9dbc4"><span class="section-number-3">3.8</span> Coupled and Decoupled Plant &ldquo;Gershgorin Radii&rdquo;</h3>
<div class="outline-text-3" id="text-3-8"> <div class="outline-text-3" id="text-3-8">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">figure; <pre class="src src-matlab">figure;
@ -647,10 +714,10 @@ xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
</div> </div>
<div id="org7d8bf66" class="figure"> <div id="org52a89d1" class="figure">
<p><img src="figs/gershorin_raddii_coupled_analytical.png" alt="gershorin_raddii_coupled_analytical.png" /> <p><img src="figs/gershorin_raddii_coupled_analytical.png" alt="gershorin_raddii_coupled_analytical.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Gershorin Raddi for the coupled plant</p> <p><span class="figure-number">Figure 13: </span>Gershorin Raddi for the coupled plant</p>
</div> </div>
<div class="org-src-container"> <div class="org-src-container">
@ -663,16 +730,16 @@ xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
</div> </div>
<div id="org319f0f6" class="figure"> <div id="org31edc7e" class="figure">
<p><img src="figs/gershorin_raddii_decoupled_analytical.png" alt="gershorin_raddii_decoupled_analytical.png" /> <p><img src="figs/gershorin_raddii_decoupled_analytical.png" alt="gershorin_raddii_decoupled_analytical.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>Gershorin Raddi for the decoupled plant</p> <p><span class="figure-number">Figure 14: </span>Gershorin Raddi for the decoupled plant</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org8e5d2c7" class="outline-3"> <div id="outline-container-orgc43f18b" class="outline-3">
<h3 id="org8e5d2c7"><span class="section-number-3">3.9</span> Decoupled Plant</h3> <h3 id="orgc43f18b"><span class="section-number-3">3.9</span> Decoupled Plant</h3>
<div class="outline-text-3" id="text-3-9"> <div class="outline-text-3" id="text-3-9">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">figure; <pre class="src src-matlab">figure;
@ -681,16 +748,16 @@ bodemag(U'*sys1*V,opts)
</div> </div>
<div id="org057e23e" class="figure"> <div id="org160c886" class="figure">
<p><img src="figs/stewart_platform_analytical_decoupled_plant.png" alt="stewart_platform_analytical_decoupled_plant.png" /> <p><img src="figs/stewart_platform_analytical_decoupled_plant.png" alt="stewart_platform_analytical_decoupled_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Decoupled Plant</p> <p><span class="figure-number">Figure 15: </span>Decoupled Plant</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org102382b" class="outline-3"> <div id="outline-container-orgf75bbde" class="outline-3">
<h3 id="org102382b"><span class="section-number-3">3.10</span> Controller</h3> <h3 id="orgf75bbde"><span class="section-number-3">3.10</span> Controller</h3>
<div class="outline-text-3" id="text-3-10"> <div class="outline-text-3" id="text-3-10">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">fc = 2*pi*0.1; % Crossover Frequency [rad/s] <pre class="src src-matlab">fc = 2*pi*0.1; % Crossover Frequency [rad/s]
@ -702,8 +769,8 @@ cont = eye(6)*c_gain/(s+fc);
</div> </div>
</div> </div>
<div id="outline-container-org27bf3be" class="outline-3"> <div id="outline-container-org275519b" class="outline-3">
<h3 id="org27bf3be"><span class="section-number-3">3.11</span> Closed Loop System</h3> <h3 id="org275519b"><span class="section-number-3">3.11</span> Closed Loop System</h3>
<div class="outline-text-3" id="text-3-11"> <div class="outline-text-3" id="text-3-11">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">FEEDIN = [7:12]; % Input of controller <pre class="src src-matlab">FEEDIN = [7:12]; % Input of controller
@ -731,8 +798,8 @@ TRsvd = STsvd*[eye(6); zeros(6)];
</div> </div>
</div> </div>
<div id="outline-container-org419f877" class="outline-3"> <div id="outline-container-org16451fc" class="outline-3">
<h3 id="org419f877"><span class="section-number-3">3.12</span> Results</h3> <h3 id="org16451fc"><span class="section-number-3">3.12</span> Results</h3>
<div class="outline-text-3" id="text-3-12"> <div class="outline-text-3" id="text-3-12">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">figure <pre class="src src-matlab">figure
@ -758,10 +825,10 @@ legend('OL','Centralized','SVD')
</div> </div>
<div id="orgbde8c92" class="figure"> <div id="org0df88b9" class="figure">
<p><img src="figs/stewart_platform_analytical_svd_cen_comp.png" alt="stewart_platform_analytical_svd_cen_comp.png" /> <p><img src="figs/stewart_platform_analytical_svd_cen_comp.png" alt="stewart_platform_analytical_svd_cen_comp.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Comparison of the obtained transmissibility for the centralized control and the SVD control</p> <p><span class="figure-number">Figure 16: </span>Comparison of the obtained transmissibility for the centralized control and the SVD control</p>
</div> </div>
</div> </div>
</div> </div>
@ -769,7 +836,7 @@ legend('OL','Centralized','SVD')
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-09-21 lun. 18:03</p> <p class="date">Created: 2020-09-21 lun. 18:53</p>
</div> </div>
</body> </body>
</html> </html>

151
index.org
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@ -649,10 +649,10 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
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 $\omega_c$. 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 $\omega_c$.
#+begin_src matlab #+begin_src matlab
wc = 2*pi*20; % Decoupling frequency [rad/s] wc = 2*pi*20; % Decoupling frequency [rad/s]
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
#+end_src
#+begin_src matlab Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
H1 = evalfr(Gc, j*wc); H1 = evalfr(Gc, j*wc);
#+end_src #+end_src
@ -673,61 +673,74 @@ First, the Singular Value Decomposition of $H_1$ is performed:
Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$: Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
\[ G_d(s) = U^T G_c(s) V \] \[ G_d(s) = U^T G_c(s) V \]
It is done over the following frequencies. This is computed over the following frequencies.
#+begin_src matlab #+begin_src matlab
freqs = logspace(-1,2,1000); % [Hz] freqs = logspace(-2, 2, 1000); % [Hz]
#+end_src #+end_src
Gershgorin Radii for the coupled plant:
#+begin_src matlab #+begin_src matlab
for i = 1:length(freqs) Gr_coupled = zeros(length(freqs), size(Gc,2));
H = abs(evalfr(Gc, j*2*pi*freqs(i)));
for j = 1:size(H,2) H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
g_r1(i,j) = (sum(H(j,:)) - H(j,j))/H(j,j); for out_i = 1:size(Gc,2)
end Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end end
#+end_src #+end_src
Gershgorin Radii for the decoupled plant using SVD:
#+begin_src matlab #+begin_src matlab
Gd = U'*Gc*V; Gd = U'*Gc*V;
Gr_decoupled = zeros(length(freqs), size(Gd,2));
for i = 1:length(freqs) H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
H_dec = abs(evalfr(Gd, j*2*pi*freqs(i))); for out_i = 1:size(Gd,2)
for j = 1:size(H,2) Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
g_r2(i,j) = (sum(H_dec(j,:)) - H_dec(j,j))/H_dec(j,j); end
end #+end_src
Gershgorin Radii for the decoupled plant using the Jacobian:
#+begin_src matlab
Gj = Gc*inv(J');
Gr_jacobian = zeros(length(freqs), size(Gj,2));
H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
for out_i = 1:size(Gj,2)
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end end
#+end_src #+end_src
#+begin_src matlab :exports results #+begin_src matlab :exports results
figure; figure;
hold on; hold on;
plot(freqs, g_r1(:,1), 'DisplayName', '$a_x$') plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
plot(freqs, g_r1(:,2), 'DisplayName', '$a_y$') plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
plot(freqs, g_r1(:,3), 'DisplayName', '$a_z$') plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
plot(freqs, g_r1(:,4), 'DisplayName', '$a_{R_x}$') for i = 2:6
plot(freqs, g_r1(:,5), 'DisplayName', '$a_{R_y}$') set(gca,'ColorOrderIndex',1)
plot(freqs, g_r1(:,6), 'DisplayName', '$a_{R_z}$') plot(freqs, Gr_coupled(:,i), 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2)
plot(freqs, Gr_decoupled(:,i), 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',3)
plot(freqs, Gr_jacobian(:,i), 'HandleVisibility', 'off');
end
plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit') plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
hold off; hold off;
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii') xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
legend('location', 'northeast');
#+end_src #+end_src
#+begin_src matlab :exports results #+begin_src matlab :tangle no :exports results :results file replace
figure; exportFig('figs/simscape_model_gershgorin_radii.pdf', 'width', 'full', 'height', 'full');
hold on;
plot(freqs, g_r2(:,1), 'DisplayName', '$a_x$')
plot(freqs, g_r2(:,2), 'DisplayName', '$a_y$')
plot(freqs, g_r2(:,3), 'DisplayName', '$a_z$')
plot(freqs, g_r2(:,4), 'DisplayName', '$a_{R_x}$')
plot(freqs, g_r2(:,5), 'DisplayName', '$a_{R_y}$')
plot(freqs, g_r2(:,6), 'DisplayName', '$a_{R_z}$')
plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
#+end_src #+end_src
#+name: fig:simscape_model_gershgorin_radii
#+caption: Gershgorin Radii of the Coupled and Decoupled plants
#+RESULTS:
[[file:figs/simscape_model_gershgorin_radii.png]]
** Decoupled Plant ** Decoupled Plant
Let's see the bode plot of the decoupled plant $G_d(s)$. Let's see the bode plot of the decoupled plant $G_d(s)$.
\[ G_d(s) = U^T G_c(s) V \] \[ G_d(s) = U^T G_c(s) V \]
@ -737,13 +750,13 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
figure; figure;
hold on; hold on;
for i = 1:6 for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gd(i, i), freqs, 'Hz'))), ... plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G(%i, %i)$', i, i)); 'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
end end
for i = 1:5 for in_i = 1:5
for j = i+1:6 for out_i = in_i+1:6
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
@ -753,6 +766,45 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
legend('location', 'southeast'); legend('location', 'southeast');
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/simscape_model_decoupled_plant_svd.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:simscape_model_decoupled_plant_svd
#+caption: Decoupled Plant using SVD
#+RESULTS:
[[file:figs/simscape_model_decoupled_plant_svd.png]]
#+begin_src matlab :exports results
freqs = logspace(-1, 2, 1000);
figure;
hold on;
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gj(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
end
for in_i = 1:5
for out_i = in_i+1:6
plot(freqs, abs(squeeze(freqresp(Gj(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); xlabel('Frequency [Hz]');
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/simscape_model_decoupled_plant_jacobian.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:simscape_model_decoupled_plant_jacobian
#+caption: Decoupled Plant using the Jacobian
#+RESULTS:
[[file:figs/simscape_model_decoupled_plant_jacobian.png]]
** Diagonal Controller ** Diagonal Controller
The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$. The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
@ -829,6 +881,25 @@ SVD Control
#+end_src #+end_src
** Results ** Results
Let's first verify the stability of the closed-loop systems:
#+begin_src matlab :results output replace text
isstable(G_cen)
#+end_src
#+RESULTS:
: ans =
: logical
: 1
#+begin_src matlab :results output replace text
isstable(G_svd)
#+end_src
#+RESULTS:
: ans =
: logical
: 1
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]]. The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
#+begin_src matlab :exports results #+begin_src matlab :exports results