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Analysis + Control of Simscape Model

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Thomas Dehaeze
2020-09-21 18:03:40 +02:00
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>SVD Control</title>
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@@ -15,6 +15,14 @@
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<div id="org-div-home-and-up">
@@ -27,27 +35,52 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org09b41c5">1. Simscape Model - Gravimeter</a>
<li><a href="#orge03ef95">1. Gravimeter - Simscape Model</a>
<ul>
<li><a href="#orgaf12c1d">1.1. Simulink</a></li>
<li><a href="#org94cda63">1.1. Simulink</a></li>
</ul>
</li>
<li><a href="#org84efeb7">2. Simscape Model - Stewart Platform</a>
<li><a href="#org01f2bcf">2. Stewart Platform - Simscape Model</a>
<ul>
<li><a href="#org157458d">2.1. Jacobian</a></li>
<li><a href="#org8947fec">2.2. Simulink</a></li>
<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 &ldquo;Gershgorin Radii&rdquo;</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>
</ul>
</li>
<li><a href="#orgdcb6e90">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 &ldquo;Gershgorin Radii&rdquo;</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>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org09b41c5" class="outline-2">
<h2 id="org09b41c5"><span class="section-number-2">1</span> Simscape Model - Gravimeter</h2>
<div id="outline-container-orge03ef95" class="outline-2">
<h2 id="orge03ef95"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-orgaf12c1d" class="outline-3">
<h3 id="orgaf12c1d"><span class="section-number-3">1.1</span> Simulink</h3>
<div id="outline-container-org94cda63" class="outline-3">
<h3 id="org94cda63"><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')
@@ -89,7 +122,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
<div id="org1c9b0ec" class="figure">
<div id="org57d8f45" 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>
@@ -98,17 +131,16 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
</div>
</div>
<div id="outline-container-org84efeb7" class="outline-2">
<h2 id="org84efeb7"><span class="section-number-2">2</span> Simscape Model - Stewart Platform</h2>
<div id="outline-container-org01f2bcf" class="outline-2">
<h2 id="org01f2bcf"><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-org157458d" class="outline-3">
<h3 id="org157458d"><span class="section-number-3">2.1</span> Jacobian</h3>
<div id="outline-container-org5d90a14" class="outline-3">
<h3 id="org5d90a14"><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 &ldquo;joints&rdquo; (points of force application) are estimated and the Jacobian computed.
</p>
<div class="org-src-container">
<pre class="src src-matlab">open('stewart_platform/drone_platform_jacobian.slx');
</pre>
@@ -146,8 +178,8 @@ save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
</div>
</div>
<div id="outline-container-org8947fec" class="outline-3">
<h3 id="org8947fec"><span class="section-number-3">2.2</span> Simulink</h3>
<div id="outline-container-org7bbb169" class="outline-3">
<h3 id="org7bbb169"><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');
@@ -175,8 +207,12 @@ We load the Jacobian.
<pre class="src src-matlab">load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
</pre>
</div>
</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 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.
</p>
@@ -186,64 +222,554 @@ mdl = 'drone_platform';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
</pre>
</div>
<p>
There are 24 states (6dof for the bottom platform + 6dof for the top platform).
</p>
<div class="org-src-container">
<pre class="src src-matlab">size(G)
</pre>
</div>
<pre class="example">
State-space model with 6 outputs, 6 inputs, and 12 states.
State-space model with 6 outputs, 12 inputs, and 24 states.
</pre>
<div class="org-src-container">
<pre class="src src-matlab">% G = G*blkdiag(inv(J), eye(6));
% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
</pre>
</div>
<p>
Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gx = -G*inv(J');
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
<pre class="src src-matlab">Gx = G*blkdiag(eye(6), inv(J'));
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
Gl = -J*G;
Gl = J*G;
Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
</pre>
</div>
</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 class="outline-text-3" id="text-2-4">
<div id="orgc94fa6a" class="figure">
<div id="orga7d2bfa" 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="org5e7bd8e" class="figure">
<div id="orge8ecc72" 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="orgce0e5a7" class="figure">
<div id="orga068faf" 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">
<p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Transmissibility</p>
</div>
</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 class="outline-text-3" id="text-2-5">
<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 \(\omega_c\).
</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);
</pre>
</div>
<p>
The real approximation is computed as follows:
</p>
<div class="org-src-container">
<pre class="src src-matlab">D = pinv(real(H1'*H1));
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
</pre>
</div>
</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 &ldquo;Gershgorin Radii&rdquo;</h3>
<div class="outline-text-3" id="text-2-6">
<p>
First, the Singular Value Decomposition of \(H_1\) is performed:
\[ H_1 = U \Sigma V^H \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">[U,S,V] = svd(H1);
</pre>
</div>
<p>
Then, the &ldquo;Gershgorin Radii&rdquo; is computed for the plant \(G_c(s)\) and the &ldquo;SVD Decoupled Plant&rdquo; \(G_d(s)\):
\[ G_d(s) = U^T G_c(s) V \]
</p>
<p>
It is done over the following frequencies.
</p>
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(-1,2,1000); % [Hz]
</pre>
</div>
<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
end
</pre>
</div>
<div class="org-src-container">
<pre class="src 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
end
</pre>
</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 class="outline-text-3" id="text-2-7">
<p>
Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
\[ G_d(s) = U^T G_c(s) V \]
</p>
</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 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 \(C_g\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">wc = 2*pi*0.1; % Crossover Frequency [rad/s]
C_g = 50; % DC Gain
K = eye(6)*C_g/(s+wc);
</pre>
</div>
</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 class="outline-text-3" id="text-2-9">
<p>
The control diagram for the centralized control is shown below.
</p>
<p>
The controller \(K_c\) is &ldquo;working&rdquo; in an cartesian frame.
The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.
</p>
<div class="figure">
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
</pre>
</div>
</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 class="outline-text-3" id="text-2-10">
<p>
The SVD control architecture is shown below.
The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
</p>
<div class="figure">
<p><img src="figs/svd_control.png" alt="svd_control.png" />
</p>
</div>
<p>
SVD Control
</p>
<div class="org-src-container">
<pre class="src src-matlab">G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
</pre>
</div>
</div>
</div>
<div id="outline-container-org89ccc9f" class="outline-3">
<h3 id="org89ccc9f"><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>.
</p>
<div id="orgfaedd1c" 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>
</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 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 class="outline-text-3" id="text-3-1">
<div class="org-src-container">
<pre class="src src-matlab">L = 0.055;
Zc = 0;
m = 0.2;
k = 1e3;
c = 2*0.1*sqrt(k*m);
Rx = 0.04;
Rz = 0.04;
Ix = m*Rx^2;
Iy = m*Rx^2;
Iz = m*Rz^2;
</pre>
</div>
</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 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;
0 1 0 -Zc 0 0;
0 0 1 0 0 0;
0 -Zc 0 Rx^2+Zc^2 0 0;
Zc 0 0 0 Rx^2+Zc^2 0;
0 0 0 0 0 Rz^2];
</pre>
</div>
</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 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;
sqrt(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;
sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);
0 0 L L -L -L;
-L*2/sqrt(3) -L*2/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3);
L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2)];
</pre>
</div>
</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 class="outline-text-3" id="text-3-4">
<div class="org-src-container">
<pre class="src src-matlab">kv = k/3; % [N/m]
kh = 0.5*k/3; % [N/m]
K = diag([3*kh,3*kh,3*kv,3*kv*Rx^2/2,3*kv*Rx^2/2,3*kh*Rx^2]); % Stiffness Matrix
C = c*K/100000; % Damping Matrix
</pre>
</div>
</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 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];
Bw = [zeros(6); -eye(6)];
Bu = [zeros(6); M\Bj];
Co = [-M\K -M\C];
D = [zeros(6) M\Bj];
ST = ss(A,[Bw Bu],Co,D);
</pre>
</div>
<ul class="org-ul">
<li>OUT 1-6: 6 dof</li>
<li>IN 1-6 : ground displacement in the directions of the legs</li>
<li>IN 7-12: forces in the actuators.</li>
</ul>
<div class="org-src-container">
<pre class="src src-matlab">ST.StateName = {'x';'y';'z';'theta_x';'theta_y';'theta_z';...
'dx';'dy';'dz';'dtheta_x';'dtheta_y';'dtheta_z'};
ST.InputName = {'w1';'w2';'w3';'w4';'w5';'w6';...
'u1';'u2';'u3';'u4';'u5';'u6'};
ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
</pre>
</div>
</div>
</div>
<div id="outline-container-org769c38a" class="outline-3">
<h3 id="org769c38a"><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)];
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">figure
subplot(231)
bodemag(TR(1,1),opts);
subplot(232)
bodemag(TR(2,2),opts);
subplot(233)
bodemag(TR(3,3),opts);
subplot(234)
bodemag(TR(4,4),opts);
subplot(235)
bodemag(TR(5,5),opts);
subplot(236)
bodemag(TR(6,6),opts);
</pre>
</div>
<div id="org55a5d25" 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>
</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 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
dec_fr = 20;
H1 = evalfr(sys1,j*2*pi*dec_fr);
H2 = H1;
D = pinv(real(H2'*H2));
H1 = inv(D*real(H2'*diag(exp(j*angle(diag(H2*D*H2.'))/2)))) ;
[U,S,V] = svd(H1);
wf = logspace(-1,2,1000);
for i = 1:length(wf)
H = abs(evalfr(sys1,j*2*pi*wf(i)));
H_dec = abs(evalfr(U'*sys1*V,j*2*pi*wf(i)));
for j = 1:size(H,2)
g_r1(i,j) = (sum(H(j,:))-H(j,j))/H(j,j);
g_r2(i,j) = (sum(H_dec(j,:))-H_dec(j,j))/H_dec(j,j);
% keyboard
end
g_lim(i) = 0.5;
end
</pre>
</div>
</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 &ldquo;Gershgorin Radii&rdquo;</h3>
<div class="outline-text-3" id="text-3-8">
<div class="org-src-container">
<pre class="src src-matlab">figure;
title('Coupled plant')
loglog(wf,g_r1(:,1),wf,g_r1(:,2),wf,g_r1(:,3),wf,g_r1(:,4),wf,g_r1(:,5),wf,g_r1(:,6),wf,g_lim,'--');
legend('$a_x$','$a_y$','$a_z$','$\theta_x$','$\theta_y$','$\theta_z$','Limit');
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
</pre>
</div>
<div id="org7d8bf66" 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>
</div>
<div class="org-src-container">
<pre class="src src-matlab">figure;
title('Decoupled plant (10 Hz)')
loglog(wf,g_r2(:,1),wf,g_r2(:,2),wf,g_r2(:,3),wf,g_r2(:,4),wf,g_r2(:,5),wf,g_r2(:,6),wf,g_lim,'--');
legend('$S_1$','$S_2$','$S_3$','$S_4$','$S_5$','$S_6$','Limit');
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
</pre>
</div>
<div id="org319f0f6" 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>
</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 class="outline-text-3" id="text-3-9">
<div class="org-src-container">
<pre class="src src-matlab">figure;
bodemag(U'*sys1*V,opts)
</pre>
</div>
<div id="org057e23e" 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>
</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 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]
c_gain = 50; %
cont = eye(6)*c_gain/(s+fc);
</pre>
</div>
</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 class="outline-text-3" id="text-3-11">
<div class="org-src-container">
<pre class="src src-matlab">FEEDIN = [7:12]; % Input of controller
FEEDOUT = [1:6]; % Output of controller
</pre>
</div>
<p>
Centralized Control
</p>
<div class="org-src-container">
<pre class="src src-matlab">STcen = feedback(ST, inv(Bj)*cont, FEEDIN, FEEDOUT);
TRcen = STcen*[eye(6); zeros(6)];
</pre>
</div>
<p>
SVD Control
</p>
<div class="org-src-container">
<pre class="src src-matlab">STsvd = feedback(ST, pinv(V')*cont*pinv(U), FEEDIN, FEEDOUT);
TRsvd = STsvd*[eye(6); zeros(6)];
</pre>
</div>
</div>
</div>
<div id="outline-container-org419f877" class="outline-3">
<h3 id="org419f877"><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
subplot(231)
bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
legend('OL','Centralized','SVD')
subplot(232)
bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
legend('OL','Centralized','SVD')
subplot(233)
bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
legend('OL','Centralized','SVD')
subplot(234)
bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
legend('OL','Centralized','SVD')
subplot(235)
bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
legend('OL','Centralized','SVD')
subplot(236)
bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
legend('OL','Centralized','SVD')
</pre>
</div>
<div id="orgbde8c92" 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>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-09-21 lun. 13:14</p>
<p class="date">Created: 2020-09-21 lun. 18:03</p>
</div>
</body>
</html>