Correct wrong computation of Gershgorin radii
This commit is contained in:
parent
a5ac2205ed
commit
ade5587ee7
BIN
figs/simscape_model_decoupled_plant_jacobian.pdf
Normal file
BIN
figs/simscape_model_decoupled_plant_jacobian.pdf
Normal file
Binary file not shown.
BIN
figs/simscape_model_decoupled_plant_jacobian.png
Normal file
BIN
figs/simscape_model_decoupled_plant_jacobian.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 174 KiB |
BIN
figs/simscape_model_decoupled_plant_svd.pdf
Normal file
BIN
figs/simscape_model_decoupled_plant_svd.pdf
Normal file
Binary file not shown.
BIN
figs/simscape_model_decoupled_plant_svd.png
Normal file
BIN
figs/simscape_model_decoupled_plant_svd.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 173 KiB |
BIN
figs/simscape_model_gershgorin_radii.pdf
Normal file
BIN
figs/simscape_model_gershgorin_radii.pdf
Normal file
Binary file not shown.
BIN
figs/simscape_model_gershgorin_radii.png
Normal file
BIN
figs/simscape_model_gershgorin_radii.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 256 KiB |
@ -3,7 +3,7 @@
|
||||
1 0 obj
|
||||
<<
|
||||
/Producer (Apache FOP Version 2.4.0-SNAPSHOT: PDFDocumentGraphics2D)
|
||||
/CreationDate (D:20200921180142+02'00')
|
||||
/CreationDate (D:20200921185333+02'00')
|
||||
>>
|
||||
endobj
|
||||
2 0 obj
|
||||
@ -3274,7 +3274,7 @@ trailer
|
||||
<<
|
||||
/Root 330 0 R
|
||||
/Info 1 0 R
|
||||
/ID [<71179BCA24FE8C45DDCC5D8276904472> <71179BCA24FE8C45DDCC5D8276904472>]
|
||||
/ID [<9EA9F41BDF4EDA41D5FDA346759D12CF> <9EA9F41BDF4EDA41D5FDA346759D12CF>]
|
||||
/Size 331
|
||||
>>
|
||||
startxref
|
||||
|
||||
303
index.html
303
index.html
@ -3,7 +3,7 @@
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2020-09-21 lun. 18:03 -->
|
||||
<!-- 2020-09-21 lun. 18:53 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<title>SVD Control</title>
|
||||
<meta name="generator" content="Org mode" />
|
||||
@ -35,52 +35,52 @@
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#orge03ef95">1. Gravimeter - Simscape Model</a>
|
||||
<li><a href="#orgcdbafa9">1. Gravimeter - Simscape Model</a>
|
||||
<ul>
|
||||
<li><a href="#org94cda63">1.1. Simulink</a></li>
|
||||
<li><a href="#orge0e53b8">1.1. Simulink</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org01f2bcf">2. Stewart Platform - Simscape Model</a>
|
||||
<li><a href="#orgbbb84cc">2. Stewart Platform - Simscape Model</a>
|
||||
<ul>
|
||||
<li><a href="#org5d90a14">2.1. Jacobian</a></li>
|
||||
<li><a href="#org7bbb169">2.2. Simscape Model</a></li>
|
||||
<li><a href="#org2a265c4">2.3. Identification of the plant</a></li>
|
||||
<li><a href="#orgfa83a84">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="#orgebf7751">2.6. Verification of the decoupling using the “Gershgorin Radii”</a></li>
|
||||
<li><a href="#orge21a525">2.7. Decoupled Plant</a></li>
|
||||
<li><a href="#org4c1f528">2.8. Diagonal Controller</a></li>
|
||||
<li><a href="#org4f88748">2.9. Centralized Control</a></li>
|
||||
<li><a href="#org6eac181">2.10. SVD Control</a></li>
|
||||
<li><a href="#org89ccc9f">2.11. Results</a></li>
|
||||
<li><a href="#org22e9f4a">2.1. Jacobian</a></li>
|
||||
<li><a href="#org692c2cc">2.2. Simscape Model</a></li>
|
||||
<li><a href="#orga491806">2.3. Identification of the plant</a></li>
|
||||
<li><a href="#orgac4aba9">2.4. Obtained Dynamics</a></li>
|
||||
<li><a href="#org1e57236">2.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
|
||||
<li><a href="#org0172b58">2.6. Verification of the decoupling using the “Gershgorin Radii”</a></li>
|
||||
<li><a href="#org37cffae">2.7. Decoupled Plant</a></li>
|
||||
<li><a href="#org868aae1">2.8. Diagonal Controller</a></li>
|
||||
<li><a href="#org14f6c79">2.9. Centralized Control</a></li>
|
||||
<li><a href="#orgdfd243c">2.10. SVD Control</a></li>
|
||||
<li><a href="#orge500424">2.11. Results</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgdcb6e90">3. Stewart Platform - Analytical Model</a>
|
||||
<li><a href="#org95fefe5">3. Stewart Platform - Analytical Model</a>
|
||||
<ul>
|
||||
<li><a href="#orgeb4b14b">3.1. Characteristics</a></li>
|
||||
<li><a href="#orgeff797b">3.2. Mass Matrix</a></li>
|
||||
<li><a href="#org7027995">3.3. Jacobian Matrix</a></li>
|
||||
<li><a href="#org51bab7b">3.4. Stifnness matrix and Damping matrix</a></li>
|
||||
<li><a href="#orga9e6cf5">3.5. State Space System</a></li>
|
||||
<li><a href="#org769c38a">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="#org824e380">3.8. Coupled and Decoupled Plant “Gershgorin Radii”</a></li>
|
||||
<li><a href="#org8e5d2c7">3.9. Decoupled Plant</a></li>
|
||||
<li><a href="#org102382b">3.10. Controller</a></li>
|
||||
<li><a href="#org27bf3be">3.11. Closed Loop System</a></li>
|
||||
<li><a href="#org419f877">3.12. Results</a></li>
|
||||
<li><a href="#org79b474c">3.1. Characteristics</a></li>
|
||||
<li><a href="#org8f55e38">3.2. Mass Matrix</a></li>
|
||||
<li><a href="#orgecf190a">3.3. Jacobian Matrix</a></li>
|
||||
<li><a href="#org7260a03">3.4. Stifnness matrix and Damping matrix</a></li>
|
||||
<li><a href="#orga9cf2e9">3.5. State Space System</a></li>
|
||||
<li><a href="#org41defbd">3.6. Transmissibility</a></li>
|
||||
<li><a href="#org5767eaf">3.7. Real approximation of \(G(j\omega)\) at decoupling frequency</a></li>
|
||||
<li><a href="#orgae9dbc4">3.8. Coupled and Decoupled Plant “Gershgorin Radii”</a></li>
|
||||
<li><a href="#orgc43f18b">3.9. Decoupled Plant</a></li>
|
||||
<li><a href="#orgf75bbde">3.10. Controller</a></li>
|
||||
<li><a href="#org275519b">3.11. Closed Loop System</a></li>
|
||||
<li><a href="#org16451fc">3.12. Results</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge03ef95" class="outline-2">
|
||||
<h2 id="orge03ef95"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
|
||||
<div id="outline-container-orgcdbafa9" class="outline-2">
|
||||
<h2 id="orgcdbafa9"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
</div>
|
||||
<div id="outline-container-org94cda63" class="outline-3">
|
||||
<h3 id="org94cda63"><span class="section-number-3">1.1</span> Simulink</h3>
|
||||
<div id="outline-container-orge0e53b8" class="outline-3">
|
||||
<h3 id="orge0e53b8"><span class="section-number-3">1.1</span> Simulink</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<div class="org-src-container">
|
||||
<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>
|
||||
<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 id="outline-container-org01f2bcf" class="outline-2">
|
||||
<h2 id="org01f2bcf"><span class="section-number-2">2</span> Stewart Platform - Simscape Model</h2>
|
||||
<div id="outline-container-orgbbb84cc" class="outline-2">
|
||||
<h2 id="orgbbb84cc"><span class="section-number-2">2</span> Stewart Platform - Simscape Model</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
</div>
|
||||
<div id="outline-container-org5d90a14" class="outline-3">
|
||||
<h3 id="org5d90a14"><span class="section-number-3">2.1</span> Jacobian</h3>
|
||||
<div id="outline-container-org22e9f4a" class="outline-3">
|
||||
<h3 id="org22e9f4a"><span class="section-number-3">2.1</span> Jacobian</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
First, the position of the “joints” (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 id="outline-container-org7bbb169" class="outline-3">
|
||||
<h3 id="org7bbb169"><span class="section-number-3">2.2</span> Simscape Model</h3>
|
||||
<div id="outline-container-org692c2cc" class="outline-3">
|
||||
<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="org-src-container">
|
||||
<pre class="src src-matlab">open('stewart_platform/drone_platform.slx');
|
||||
@ -210,8 +210,8 @@ We load the Jacobian.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2a265c4" class="outline-3">
|
||||
<h3 id="org2a265c4"><span class="section-number-3">2.3</span> Identification of the plant</h3>
|
||||
<div id="outline-container-orga491806" class="outline-3">
|
||||
<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">
|
||||
<p>
|
||||
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 id="outline-container-orgfa83a84" class="outline-3">
|
||||
<h3 id="orgfa83a84"><span class="section-number-3">2.4</span> Obtained Dynamics</h3>
|
||||
<div id="outline-container-orgac4aba9" class="outline-3">
|
||||
<h3 id="orgac4aba9"><span class="section-number-3">2.4</span> Obtained Dynamics</h3>
|
||||
<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>
|
||||
<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 id="orge8ecc72" class="figure">
|
||||
<div id="orgba8e960" class="figure">
|
||||
<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
|
||||
</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 id="orga068faf" class="figure">
|
||||
<div id="orga66f388" class="figure">
|
||||
<p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" />
|
||||
</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 id="orgf48c4d4" class="figure">
|
||||
<div id="orga7254f2" class="figure">
|
||||
<p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" />
|
||||
</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 id="outline-container-org92dd977" class="outline-3">
|
||||
<h3 id="org92dd977"><span class="section-number-3">2.5</span> Real Approximation of \(G\) at the decoupling frequency</h3>
|
||||
<div id="outline-container-org1e57236" class="outline-3">
|
||||
<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">
|
||||
<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*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">
|
||||
<pre class="src src-matlab">H1 = evalfr(Gc, j*wc);
|
||||
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);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -329,8 +328,8 @@ H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgebf7751" class="outline-3">
|
||||
<h3 id="orgebf7751"><span class="section-number-3">2.6</span> Verification of the decoupling using the “Gershgorin Radii”</h3>
|
||||
<div id="outline-container-org0172b58" class="outline-3">
|
||||
<h3 id="org0172b58"><span class="section-number-3">2.6</span> Verification of the decoupling using the “Gershgorin Radii”</h3>
|
||||
<div class="outline-text-3" id="text-2-6">
|
||||
<p>
|
||||
First, the Singular Value Decomposition of \(H_1\) is performed:
|
||||
@ -348,49 +347,90 @@ Then, the “Gershgorin Radii” is computed for the plant \(G_c(s)\) an
|
||||
</p>
|
||||
|
||||
<p>
|
||||
It is done over the following frequencies.
|
||||
This is computed over the following frequencies.
|
||||
</p>
|
||||
<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>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Gershgorin Radii for the coupled plant:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">for i = 1:length(freqs)
|
||||
H = abs(evalfr(Gc, j*2*pi*freqs(i)));
|
||||
for j = 1:size(H,2)
|
||||
g_r1(i,j) = (sum(H(j,:)) - H(j,j))/H(j,j);
|
||||
end
|
||||
<pre class="src src-matlab">Gr_coupled = zeros(length(freqs), size(Gc,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gc,2)
|
||||
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Gershgorin Radii for the decoupled plant using SVD:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Gd = U'*Gc*V;
|
||||
Gr_decoupled = zeros(length(freqs), size(Gd,2));
|
||||
|
||||
for i = 1:length(freqs)
|
||||
H_dec = abs(evalfr(Gd, j*2*pi*freqs(i)));
|
||||
for j = 1:size(H,2)
|
||||
g_r2(i,j) = (sum(H_dec(j,:)) - H_dec(j,j))/H_dec(j,j);
|
||||
end
|
||||
H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gd,2)
|
||||
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
</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*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 id="outline-container-orge21a525" class="outline-3">
|
||||
<h3 id="orge21a525"><span class="section-number-3">2.7</span> Decoupled Plant</h3>
|
||||
<div id="outline-container-org37cffae" class="outline-3">
|
||||
<h3 id="org37cffae"><span class="section-number-3">2.7</span> Decoupled Plant</h3>
|
||||
<div class="outline-text-3" id="text-2-7">
|
||||
<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="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 id="outline-container-org4c1f528" class="outline-3">
|
||||
<h3 id="org4c1f528"><span class="section-number-3">2.8</span> Diagonal Controller</h3>
|
||||
<div id="outline-container-org868aae1" class="outline-3">
|
||||
<h3 id="org868aae1"><span class="section-number-3">2.8</span> Diagonal Controller</h3>
|
||||
<div class="outline-text-3" id="text-2-8">
|
||||
<p>
|
||||
The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\).
|
||||
@ -406,8 +446,8 @@ K = eye(6)*C_g/(s+wc);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4f88748" class="outline-3">
|
||||
<h3 id="org4f88748"><span class="section-number-3">2.9</span> Centralized Control</h3>
|
||||
<div id="outline-container-org14f6c79" class="outline-3">
|
||||
<h3 id="org14f6c79"><span class="section-number-3">2.9</span> Centralized Control</h3>
|
||||
<div class="outline-text-3" id="text-2-9">
|
||||
<p>
|
||||
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 id="outline-container-org6eac181" class="outline-3">
|
||||
<h3 id="org6eac181"><span class="section-number-3">2.10</span> SVD Control</h3>
|
||||
<div id="outline-container-orgdfd243c" class="outline-3">
|
||||
<h3 id="orgdfd243c"><span class="section-number-3">2.10</span> SVD Control</h3>
|
||||
<div class="outline-text-3" id="text-2-10">
|
||||
<p>
|
||||
The SVD control architecture is shown below.
|
||||
@ -455,29 +495,56 @@ SVD Control
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org89ccc9f" class="outline-3">
|
||||
<h3 id="org89ccc9f"><span class="section-number-3">2.11</span> Results</h3>
|
||||
<div id="outline-container-orge500424" class="outline-3">
|
||||
<h3 id="orge500424"><span class="section-number-3">2.11</span> Results</h3>
|
||||
<div class="outline-text-3" id="text-2-11">
|
||||
<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’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>
|
||||
|
||||
|
||||
<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>
|
||||
<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 id="outline-container-orgdcb6e90" class="outline-2">
|
||||
<h2 id="orgdcb6e90"><span class="section-number-2">3</span> Stewart Platform - Analytical Model</h2>
|
||||
<div id="outline-container-org95fefe5" class="outline-2">
|
||||
<h2 id="org95fefe5"><span class="section-number-2">3</span> Stewart Platform - Analytical Model</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
</div>
|
||||
<div id="outline-container-orgeb4b14b" class="outline-3">
|
||||
<h3 id="orgeb4b14b"><span class="section-number-3">3.1</span> Characteristics</h3>
|
||||
<div id="outline-container-org79b474c" class="outline-3">
|
||||
<h3 id="org79b474c"><span class="section-number-3">3.1</span> Characteristics</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">L = 0.055;
|
||||
@ -496,8 +563,8 @@ Iz = m*Rz^2;
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgeff797b" class="outline-3">
|
||||
<h3 id="orgeff797b"><span class="section-number-3">3.2</span> Mass Matrix</h3>
|
||||
<div id="outline-container-org8f55e38" class="outline-3">
|
||||
<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="org-src-container">
|
||||
<pre class="src src-matlab">M = m*[1 0 0 0 Zc 0;
|
||||
@ -511,8 +578,8 @@ Iz = m*Rz^2;
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7027995" class="outline-3">
|
||||
<h3 id="org7027995"><span class="section-number-3">3.3</span> Jacobian Matrix</h3>
|
||||
<div id="outline-container-orgecf190a" class="outline-3">
|
||||
<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="org-src-container">
|
||||
<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 id="outline-container-org51bab7b" class="outline-3">
|
||||
<h3 id="org51bab7b"><span class="section-number-3">3.4</span> Stifnness matrix and Damping matrix</h3>
|
||||
<div id="outline-container-org7260a03" class="outline-3">
|
||||
<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="org-src-container">
|
||||
<pre class="src src-matlab">kv = k/3; % [N/m]
|
||||
@ -541,8 +608,8 @@ C = c*K/100000; % Damping Matrix
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga9e6cf5" class="outline-3">
|
||||
<h3 id="orga9e6cf5"><span class="section-number-3">3.5</span> State Space System</h3>
|
||||
<div id="outline-container-orga9cf2e9" class="outline-3">
|
||||
<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="org-src-container">
|
||||
<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 id="outline-container-org769c38a" class="outline-3">
|
||||
<h3 id="org769c38a"><span class="section-number-3">3.6</span> Transmissibility</h3>
|
||||
<div id="outline-container-org41defbd" class="outline-3">
|
||||
<h3 id="org41defbd"><span class="section-number-3">3.6</span> Transmissibility</h3>
|
||||
<div class="outline-text-3" id="text-3-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">TR=ST*[eye(6); zeros(6)];
|
||||
@ -597,16 +664,16 @@ bodemag(TR(6,6),opts);
|
||||
</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>
|
||||
<p><span class="figure-number">Figure 9: </span>Transmissibility</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Transmissibility</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org24eb81f" class="outline-3">
|
||||
<h3 id="org24eb81f"><span class="section-number-3">3.7</span> Real approximation of \(G(j\omega)\) at decoupling frequency</h3>
|
||||
<div id="outline-container-org5767eaf" class="outline-3">
|
||||
<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="org-src-container">
|
||||
<pre class="src src-matlab">sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs
|
||||
@ -634,8 +701,8 @@ end
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org824e380" class="outline-3">
|
||||
<h3 id="org824e380"><span class="section-number-3">3.8</span> Coupled and Decoupled Plant “Gershgorin Radii”</h3>
|
||||
<div id="outline-container-orgae9dbc4" class="outline-3">
|
||||
<h3 id="orgae9dbc4"><span class="section-number-3">3.8</span> Coupled and Decoupled Plant “Gershgorin Radii”</h3>
|
||||
<div class="outline-text-3" id="text-3-8">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">figure;
|
||||
@ -647,10 +714,10 @@ xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
</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>
|
||||
<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 class="org-src-container">
|
||||
@ -663,16 +730,16 @@ xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
</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>
|
||||
<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 id="outline-container-org8e5d2c7" class="outline-3">
|
||||
<h3 id="org8e5d2c7"><span class="section-number-3">3.9</span> Decoupled Plant</h3>
|
||||
<div id="outline-container-orgc43f18b" class="outline-3">
|
||||
<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="org-src-container">
|
||||
<pre class="src src-matlab">figure;
|
||||
@ -681,16 +748,16 @@ bodemag(U'*sys1*V,opts)
|
||||
</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>
|
||||
<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 id="outline-container-org102382b" class="outline-3">
|
||||
<h3 id="org102382b"><span class="section-number-3">3.10</span> Controller</h3>
|
||||
<div id="outline-container-orgf75bbde" class="outline-3">
|
||||
<h3 id="orgf75bbde"><span class="section-number-3">3.10</span> Controller</h3>
|
||||
<div class="outline-text-3" id="text-3-10">
|
||||
<div class="org-src-container">
|
||||
<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 id="outline-container-org27bf3be" class="outline-3">
|
||||
<h3 id="org27bf3be"><span class="section-number-3">3.11</span> Closed Loop System</h3>
|
||||
<div id="outline-container-org275519b" class="outline-3">
|
||||
<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="org-src-container">
|
||||
<pre class="src src-matlab">FEEDIN = [7:12]; % Input of controller
|
||||
@ -731,8 +798,8 @@ TRsvd = STsvd*[eye(6); zeros(6)];
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org419f877" class="outline-3">
|
||||
<h3 id="org419f877"><span class="section-number-3">3.12</span> Results</h3>
|
||||
<div id="outline-container-org16451fc" class="outline-3">
|
||||
<h3 id="org16451fc"><span class="section-number-3">3.12</span> Results</h3>
|
||||
<div class="outline-text-3" id="text-3-12">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">figure
|
||||
@ -758,10 +825,10 @@ legend('OL','Centralized','SVD')
|
||||
</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>
|
||||
<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>
|
||||
@ -769,7 +836,7 @@ legend('OL','Centralized','SVD')
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<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>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
153
index.org
153
index.org
@ -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 $\omega_c$.
|
||||
#+begin_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
|
||||
#+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);
|
||||
#+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" $G_d(s)$:
|
||||
\[ G_d(s) = U^T G_c(s) V \]
|
||||
|
||||
It is done over the following frequencies.
|
||||
This is computed over the following frequencies.
|
||||
#+begin_src matlab
|
||||
freqs = logspace(-1,2,1000); % [Hz]
|
||||
freqs = logspace(-2, 2, 1000); % [Hz]
|
||||
#+end_src
|
||||
|
||||
Gershgorin Radii for the coupled plant:
|
||||
#+begin_src matlab
|
||||
for i = 1:length(freqs)
|
||||
H = abs(evalfr(Gc, j*2*pi*freqs(i)));
|
||||
for j = 1:size(H,2)
|
||||
g_r1(i,j) = (sum(H(j,:)) - H(j,j))/H(j,j);
|
||||
end
|
||||
Gr_coupled = zeros(length(freqs), size(Gc,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gc,2)
|
||||
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
#+end_src
|
||||
|
||||
Gershgorin Radii for the decoupled plant using SVD:
|
||||
#+begin_src matlab
|
||||
Gd = U'*Gc*V;
|
||||
|
||||
for i = 1:length(freqs)
|
||||
H_dec = abs(evalfr(Gd, j*2*pi*freqs(i)));
|
||||
for j = 1:size(H,2)
|
||||
g_r2(i,j) = (sum(H_dec(j,:)) - H_dec(j,j))/H_dec(j,j);
|
||||
end
|
||||
Gr_decoupled = zeros(length(freqs), size(Gd,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gd,2)
|
||||
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
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_src
|
||||
|
||||
#+begin_src matlab :exports results
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, g_r1(:,1), 'DisplayName', '$a_x$')
|
||||
plot(freqs, g_r1(:,2), 'DisplayName', '$a_y$')
|
||||
plot(freqs, g_r1(:,3), 'DisplayName', '$a_z$')
|
||||
plot(freqs, g_r1(:,4), 'DisplayName', '$a_{R_x}$')
|
||||
plot(freqs, g_r1(:,5), 'DisplayName', '$a_{R_y}$')
|
||||
plot(freqs, g_r1(:,6), 'DisplayName', '$a_{R_z}$')
|
||||
plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
|
||||
plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
|
||||
plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
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')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
legend('location', 'northeast');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports results
|
||||
figure;
|
||||
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')
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/simscape_model_gershgorin_radii.pdf', 'width', 'full', 'height', 'full');
|
||||
#+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
|
||||
Let's see the bode plot of the decoupled plant $G_d(s)$.
|
||||
\[ G_d(s) = U^T G_c(s) V \]
|
||||
@ -737,13 +750,13 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(i, i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$G(%i, %i)$', i, i));
|
||||
for ch_i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
|
||||
end
|
||||
for i = 1:5
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
for in_i = 1:5
|
||||
for out_i = in_i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
@ -753,6 +766,45 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
|
||||
legend('location', 'southeast');
|
||||
#+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
|
||||
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
|
||||
|
||||
** 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]].
|
||||
|
||||
#+begin_src matlab :exports results
|
||||
|
||||
Loading…
Reference in New Issue
Block a user