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<h1 class="title">SVD Control</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org1e5473a">1. Gravimeter - Simscape Model</a>
<ul>
<li><a href="#orgaaa0a01">1.1. Simulink</a></li>
</ul>
</li>
<li><a href="#orgc639438">2. Stewart Platform - Simscape Model</a>
<ul>
<li><a href="#orgd3669fa">2.1. Jacobian</a></li>
<li><a href="#org132134e">2.2. Simscape Model</a></li>
<li><a href="#org93250af">2.3. Identification of the plant</a></li>
<li><a href="#org7c5a6ec">2.4. Obtained Dynamics</a></li>
<li><a href="#orgcb31a63">2.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
<li><a href="#orge3bd56a">2.6. Verification of the decoupling using the “Gershgorin Radii”</a></li>
<li><a href="#org6ec08f8">2.7. Decoupled Plant</a></li>
<li><a href="#orgbbbbe29">2.8. Diagonal Controller</a></li>
<li><a href="#org1826150">2.9. Centralized Control</a></li>
<li><a href="#org0a68dcb">2.10. SVD Control</a></li>
<li><a href="#org5019def">2.11. Results</a></li>
</ul>
</li>
<li><a href="#orgb4abc3a">3. Stewart Platform - Analytical Model</a>
<ul>
<li><a href="#org2e00dda">3.1. Characteristics</a></li>
<li><a href="#org0d2bbce">3.2. Mass Matrix</a></li>
<li><a href="#orgb89db41">3.3. Jacobian Matrix</a></li>
<li><a href="#org6f5c365">3.4. Stifnness matrix and Damping matrix</a></li>
<li><a href="#org27a9531">3.5. State Space System</a></li>
<li><a href="#org3ec37d3">3.6. Transmissibility</a></li>
<li><a href="#org7a0bae6">3.7. Real approximation of \(G(j\omega)\) at decoupling frequency</a></li>
<li><a href="#org024989e">3.8. Coupled and Decoupled Plant “Gershgorin Radii”</a></li>
<li><a href="#org47772f8">3.9. Decoupled Plant</a></li>
<li><a href="#org3fd828d">3.10. Controller</a></li>
<li><a href="#org9bccdc5">3.11. Closed Loop System</a></li>
<li><a href="#org31917c6">3.12. Results</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org1e5473a" class="outline-2">
<h2 id="org1e5473a"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-orgaaa0a01" class="outline-3">
<h3 id="orgaaa0a01"><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')
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">%% Name of the Simulink File
mdl = 'gravimeter';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
</pre>
</div>
<p>
The plant as 6 states as expected (2 translations + 1 rotation)
</p>
<div class="org-src-container">
<pre class="src src-matlab">size(G)
</pre>
</div>
<pre class="example">
State-space model with 4 outputs, 3 inputs, and 6 states.
</pre>
<div id="org2d37a75" 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>
</div>
</div>
</div>
</div>
<div id="outline-container-orgc639438" class="outline-2">
<h2 id="orgc639438"><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-orgd3669fa" class="outline-3">
<h3 id="orgd3669fa"><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.
</p>
<div class="org-src-container">
<pre class="src src-matlab">open('stewart_platform/drone_platform_jacobian.slx');
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">sim('drone_platform_jacobian');
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Aa = [a1.Data(1,:);
a2.Data(1,:);
a3.Data(1,:);
a4.Data(1,:);
a5.Data(1,:);
a6.Data(1,:)]';
Ab = [b1.Data(1,:);
b2.Data(1,:);
b3.Data(1,:);
b4.Data(1,:);
b5.Data(1,:);
b6.Data(1,:)]';
As = (Ab - Aa)./vecnorm(Ab - Aa);
l = vecnorm(Ab - Aa)';
J = [As' , cross(Ab, As)'];
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
</pre>
</div>
</div>
</div>
<div id="outline-container-org132134e" class="outline-3">
<h3 id="org132134e"><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');
</pre>
</div>
<p>
Definition of spring parameters
</p>
<div class="org-src-container">
<pre class="src src-matlab">kx = 50; % [N/m]
ky = 50;
kz = 50;
cx = 0.025; % [Nm/rad]
cy = 0.025;
cz = 0.025;
</pre>
</div>
<p>
We load the Jacobian.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
</pre>
</div>
</div>
</div>
<div id="outline-container-org93250af" class="outline-3">
<h3 id="org93250af"><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>
<div class="org-src-container">
<pre class="src src-matlab">%% Name of the Simulink File
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 = {'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, 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*blkdiag(eye(6), inv(J'));
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
Gl = J*G;
Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
</pre>
</div>
</div>
</div>
<div id="outline-container-org7c5a6ec" class="outline-3">
<h3 id="org7c5a6ec"><span class="section-number-3">2.4</span> Obtained Dynamics</h3>
<div class="outline-text-3" id="text-2-4">
<div id="org15e7322" 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="org46f6f81" 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="org873da45" 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="org8bfc6d1" 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-orgcb31a63" class="outline-3">
<h3 id="orgcb31a63"><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
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-orge3bd56a" class="outline-3">
<h3 id="orge3bd56a"><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:
\[ H_1 = U \Sigma V^H \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">[U,S,V] = svd(H1);
</pre>
</div>
<p>
Then, the “Gershgorin Radii” is computed for the plant \(G_c(s)\) and the “SVD Decoupled Plant” \(G_d(s)\):
\[ G_d(s) = U^T G_c(s) V \]
</p>
<p>
This is computed over the following frequencies.
</p>
<div class="org-src-container">
<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">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));
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="orgb8cf952" 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-org6ec08f8" class="outline-3">
<h3 id="org6ec08f8"><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="orge07e98d" 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="org2f051b6" 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-orgbbbbe29" class="outline-3">
<h3 id="orgbbbbe29"><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-org1826150" class="outline-3">
<h3 id="org1826150"><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 “working” 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-org0a68dcb" class="outline-3">
<h3 id="org0a68dcb"><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-org5019def" class="outline-3">
<h3 id="org5019def"><span class="section-number-3">2.11</span> Results</h3>
<div class="outline-text-3" id="text-2-11">
<p>
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="#org3004af5">11</a>.
</p>
<div id="org3004af5" 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 11: </span>Obtained Transmissibility</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgb4abc3a" class="outline-2">
<h2 id="orgb4abc3a"><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-org2e00dda" class="outline-3">
<h3 id="org2e00dda"><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-org0d2bbce" class="outline-3">
<h3 id="org0d2bbce"><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-orgb89db41" class="outline-3">
<h3 id="orgb89db41"><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-org6f5c365" class="outline-3">
<h3 id="org6f5c365"><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-org27a9531" class="outline-3">
<h3 id="org27a9531"><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-org3ec37d3" class="outline-3">
<h3 id="org3ec37d3"><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="orgff4f271" class="figure">
<p><img src="figs/stewart_platform_analytical_transmissibility.png" alt="stewart_platform_analytical_transmissibility.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Transmissibility</p>
</div>
</div>
</div>
<div id="outline-container-org7a0bae6" class="outline-3">
<h3 id="org7a0bae6"><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-org024989e" class="outline-3">
<h3 id="org024989e"><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;
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="org15871e4" class="figure">
<p><img src="figs/gershorin_raddii_coupled_analytical.png" alt="gershorin_raddii_coupled_analytical.png" />
</p>
<p><span class="figure-number">Figure 13: </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="org951fc6d" class="figure">
<p><img src="figs/gershorin_raddii_decoupled_analytical.png" alt="gershorin_raddii_decoupled_analytical.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Gershorin Raddi for the decoupled plant</p>
</div>
</div>
</div>
<div id="outline-container-org47772f8" class="outline-3">
<h3 id="org47772f8"><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="org961889e" 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 15: </span>Decoupled Plant</p>
</div>
</div>
</div>
<div id="outline-container-org3fd828d" class="outline-3">
<h3 id="org3fd828d"><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-org9bccdc5" class="outline-3">
<h3 id="org9bccdc5"><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-org31917c6" class="outline-3">
<h3 id="org31917c6"><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="org636fa6f" 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 16: </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. 18:57</p>
</div>
</body>
</html>