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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="#org40c86ca">1. Gravimeter - Simscape Model</a>
<ul>
<li><a href="#orgac27a65">1.1. Introduction</a></li>
<li><a href="#org991b9ad">1.2. Simscape Model - Parameters</a></li>
<li><a href="#org7417c14">1.3. System Identification - Without Gravity</a></li>
<li><a href="#org3ac74c3">1.4. System Identification - With Gravity</a></li>
<li><a href="#org13de6f7">1.5. Analytical Model</a>
<ul>
<li><a href="#orgef157da">1.5.1. Parameters</a></li>
<li><a href="#orgb72d17d">1.5.2. Generation of the State Space Model</a></li>
<li><a href="#org3b77585">1.5.3. Comparison with the Simscape Model</a></li>
<li><a href="#org2f7cb8f">1.5.4. Analysis</a></li>
<li><a href="#org218243e">1.5.5. Control Section</a></li>
<li><a href="#orgad11a63">1.5.6. Greshgorin radius</a></li>
<li><a href="#orga23d907">1.5.7. Injecting ground motion in the system to have the output</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org23fa18d">2. Gravimeter - Functions</a>
<ul>
<li><a href="#org81c3333">2.1. <code>align</code></a></li>
<li><a href="#org8b6878d">2.2. <code>pzmap_testCL</code></a></li>
</ul>
</li>
<li><a href="#org50746f8">3. Stewart Platform - Simscape Model</a>
<ul>
<li><a href="#orga12724f">3.1. Simscape Model - Parameters</a></li>
<li><a href="#org820527f">3.2. Identification of the plant</a></li>
<li><a href="#orga58761b">3.3. Obtained Dynamics</a></li>
<li><a href="#orgb3d55c6">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li>
<li><a href="#org2f2890a">3.5. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li>
<li><a href="#org70b5fa2">3.6. Decoupled Plant</a></li>
<li><a href="#orgc23974f">3.7. Diagonal Controller</a></li>
<li><a href="#org6e4ced6">3.8. Closed-Loop system Performances</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org40c86ca" class="outline-2">
<h2 id="org40c86ca"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-orgac27a65" class="outline-3">
<h3 id="orgac27a65"><span class="section-number-3">1.1</span> Introduction</h3>
<div class="outline-text-3" id="text-1-1">
<div id="orgfaa8196" class="figure">
<p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p>
</div>
</div>
</div>
<div id="outline-container-org991b9ad" class="outline-3">
<h3 id="org991b9ad"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
<div class="outline-text-3" id="text-1-2">
<div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'gravimeter.slx'</span>)
</pre>
</div>
<p>
Parameters
</p>
<div class="org-src-container">
<pre class="src src-matlab">l = 1.0; <span class="org-comment">% Length of the mass [m]</span>
la = 0.5; <span class="org-comment">% Position of Act. [m]</span>
h = 3.4; <span class="org-comment">% Height of the mass [m]</span>
ha = 1.7; <span class="org-comment">% Position of Act. [m]</span>
m = 400; <span class="org-comment">% Mass [kg]</span>
I = 115; <span class="org-comment">% Inertia [kg m^2]</span>
k = 15e3; <span class="org-comment">% Actuator Stiffness [N/m]</span>
c = 0.03; <span class="org-comment">% Actuator Damping [N/(m/s)]</span>
deq = 0.2; <span class="org-comment">% Length of the actuators [m]</span>
g = 0; <span class="org-comment">% Gravity [m/s2]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org7417c14" class="outline-3">
<h3 id="org7417c14"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
<div class="outline-text-3" id="text-1-3">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'gravimeter'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/F1'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/F2'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/F3'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Acc_side'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Acc_side'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Acc_top'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Acc_top'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
G = linearize(mdl, io);
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>};
G.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string">'Az1'</span>, <span class="org-string">'Ax2'</span>, <span class="org-string">'Az2'</span>};
</pre>
</div>
<pre class="example" id="orgefbf7cd">
pole(G)
ans =
-0.000473481142385795 + 21.7596190728632i
-0.000473481142385795 - 21.7596190728632i
-7.49842879459172e-05 + 8.6593576906982i
-7.49842879459172e-05 - 8.6593576906982i
-5.1538686792578e-06 + 2.27025295182756i
-5.1538686792578e-06 - 2.27025295182756i
</pre>
<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="orgfe2be7d" class="figure">
<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
</div>
</div>
</div>
<div id="outline-container-org3ac74c3" class="outline-3">
<h3 id="org3ac74c3"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
<div class="outline-text-3" id="text-1-4">
<div class="org-src-container">
<pre class="src src-matlab">g = 9.80665; <span class="org-comment">% Gravity [m/s2]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Gg = linearize(mdl, io);
Gg.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>};
Gg.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string">'Az1'</span>, <span class="org-string">'Ax2'</span>, <span class="org-string">'Az2'</span>};
</pre>
</div>
<p>
We can now see that the system is unstable due to gravity.
</p>
<pre class="example" id="org9de3a30">
pole(Gg)
ans =
-10.9848275341252 + 0i
10.9838836405201 + 0i
-7.49855379478109e-05 + 8.65962885770051i
-7.49855379478109e-05 - 8.65962885770051i
-6.68819548733559e-06 + 0.832960422243848i
-6.68819548733559e-06 - 0.832960422243848i
</pre>
<div id="org295f713" class="figure">
<p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p>
</div>
</div>
</div>
<div id="outline-container-org13de6f7" class="outline-3">
<h3 id="org13de6f7"><span class="section-number-3">1.5</span> Analytical Model</h3>
<div class="outline-text-3" id="text-1-5">
</div>
<div id="outline-container-orgef157da" class="outline-4">
<h4 id="orgef157da"><span class="section-number-4">1.5.1</span> Parameters</h4>
<div class="outline-text-4" id="text-1-5-1">
<p>
Bode options.
</p>
<div class="org-src-container">
<pre class="src src-matlab">P = bodeoptions;
P.FreqUnits = <span class="org-string">'Hz'</span>;
P.MagUnits = <span class="org-string">'abs'</span>;
P.MagScale = <span class="org-string">'log'</span>;
P.Grid = <span class="org-string">'on'</span>;
P.PhaseWrapping = <span class="org-string">'on'</span>;
P.Title.FontSize = 14;
P.XLabel.FontSize = 14;
P.YLabel.FontSize = 14;
P.TickLabel.FontSize = 12;
P.Xlim = [1e<span class="org-type">-</span>1,1e2];
P.MagLowerLimMode = <span class="org-string">'manual'</span>;
P.MagLowerLim= 1e<span class="org-type">-</span>3;
</pre>
</div>
<p>
Frequency vector.
</p>
<div class="org-src-container">
<pre class="src src-matlab">w = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>logspace(<span class="org-type">-</span>1,2,1000); <span class="org-comment">% [rad/s]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb72d17d" class="outline-4">
<h4 id="orgb72d17d"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
<div class="outline-text-4" id="text-1-5-2">
<p>
Mass matrix
</p>
<div class="org-src-container">
<pre class="src src-matlab">M = [m 0 0
0 m 0
0 0 I];
</pre>
</div>
<p>
Jacobian of the bottom sensor
</p>
<div class="org-src-container">
<pre class="src src-matlab">Js1 = [1 0 h<span class="org-type">/</span>2
0 1 <span class="org-type">-</span>l<span class="org-type">/</span>2];
</pre>
</div>
<p>
Jacobian of the top sensor
</p>
<div class="org-src-container">
<pre class="src src-matlab">Js2 = [1 0 <span class="org-type">-</span>h<span class="org-type">/</span>2
0 1 0];
</pre>
</div>
<p>
Jacobian of the actuators
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ja = [1 0 ha <span class="org-comment">% Left horizontal actuator</span>
0 1 <span class="org-type">-</span>la <span class="org-comment">% Left vertical actuator</span>
0 1 la]; <span class="org-comment">% Right vertical actuator</span>
Jta = Ja<span class="org-type">'</span>;
</pre>
</div>
<p>
Stiffness and Damping matrices
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = k<span class="org-type">*</span>Jta<span class="org-type">*</span>Ja;
C = c<span class="org-type">*</span>Jta<span class="org-type">*</span>Ja;
</pre>
</div>
<p>
State Space Matrices
</p>
<div class="org-src-container">
<pre class="src src-matlab">E = [1 0 0
0 1 0
0 0 1]; <span class="org-comment">%projecting ground motion in the directions of the legs</span>
AA = [zeros(3) eye(3)
<span class="org-type">-</span>M<span class="org-type">\</span>K <span class="org-type">-</span>M<span class="org-type">\</span>C];
BB = [zeros(3,6)
M<span class="org-type">\</span>Jta M<span class="org-type">\</span>(k<span class="org-type">*</span>Jta<span class="org-type">*</span>E)];
CC = [[Js1;Js2] zeros(4,3);
zeros(2,6)
(Js1<span class="org-type">+</span>Js2)<span class="org-type">./</span>2 zeros(2,3)
(Js1<span class="org-type">-</span>Js2)<span class="org-type">./</span>2 zeros(2,3)
(Js1<span class="org-type">-</span>Js2)<span class="org-type">./</span>(2<span class="org-type">*</span>h) zeros(2,3)];
DD = [zeros(4,6)
zeros<span class="org-type">(2,3) eye(2,3)</span>
zeros<span class="org-type">(6,6)];</span>
</pre>
</div>
<p>
State Space model:
</p>
<ul class="org-ul">
<li>Input = three actuators and three ground motions</li>
<li>Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation</li>
</ul>
<div class="org-src-container">
<pre class="src src-matlab">system_dec = ss(AA,BB,CC,DD);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">size(system_dec)
</pre>
</div>
<pre class="example">
State-space model with 12 outputs, 6 inputs, and 6 states.
</pre>
</div>
</div>
<div id="outline-container-org3b77585" class="outline-4">
<h4 id="org3b77585"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
<div class="outline-text-4" id="text-1-5-3">
<div id="org8f52253" class="figure">
<p><img src="figs/gravimeter_analytical_system_open_loop_models.png" alt="gravimeter_analytical_system_open_loop_models.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Comparison of the analytical and the Simscape models</p>
</div>
</div>
</div>
<div id="outline-container-org2f7cb8f" class="outline-4">
<h4 id="org2f7cb8f"><span class="section-number-4">1.5.4</span> Analysis</h4>
<div class="outline-text-4" id="text-1-5-4">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% figure</span>
<span class="org-comment">% bode(system_dec,P);</span>
<span class="org-comment">% return</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% svd decomposition</span></span>
<span class="org-comment">% system_dec_freq = freqresp(system_dec,w);</span>
<span class="org-comment">% S = zeros(3,length(w));</span>
<span class="org-comment">% for </span><span class="org-variable-name"><span class="org-comment">m</span></span><span class="org-comment"> = </span><span class="org-constant"><span class="org-comment">1:length(w)</span></span>
<span class="org-comment">% S(:,m) = svd(system_dec_freq(1:4,1:3,m));</span>
<span class="org-comment">% end</span>
<span class="org-comment">% figure</span>
<span class="org-comment">% loglog(w./(2*pi), S);hold on;</span>
<span class="org-comment">% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));</span>
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');</span>
<span class="org-comment">% legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');</span>
<span class="org-comment">% ylim([1e-8 1e-2]);</span>
<span class="org-comment">%</span>
<span class="org-comment">% %condition number</span>
<span class="org-comment">% figure</span>
<span class="org-comment">% loglog(w./(2*pi), S(1,:)./S(3,:));hold on;</span>
<span class="org-comment">% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));</span>
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('Condition number [-]');</span>
<span class="org-comment">% % legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');</span>
<span class="org-comment">%</span>
<span class="org-comment">% %performance indicator</span>
<span class="org-comment">% system_dec_svd = freqresp(system_dec(1:4,1:3),2*pi*10);</span>
<span class="org-comment">% [U,S,V] = svd(system_dec_svd);</span>
<span class="org-comment">% H_svd_OL = -eye(3,4);%-[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 10*zpk(-2*pi*40,-2*pi*200,40/200) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];% - eye(3,4);%</span>
<span class="org-comment">% H_svd = pinv(V')*H_svd_OL*pinv(U);</span>
<span class="org-comment">% % system_dec_control_svd_ = feedback(system_dec,g*pinv(V')*H*pinv(U));</span>
<span class="org-comment">%</span>
<span class="org-comment">% OL_dec = g_svd*H_svd*system_dec(1:4,1:3);</span>
<span class="org-comment">% OL_freq = freqresp(OL_dec,w); % OL = G*H</span>
<span class="org-comment">% CL_system = feedback(eye(3),-g_svd*H_svd*system_dec(1:4,1:3));</span>
<span class="org-comment">% CL_freq = freqresp(CL_system,w); % CL = (1+G*H)^-1</span>
<span class="org-comment">% % CL_system_2 = feedback(system_dec,H);</span>
<span class="org-comment">% % CL_freq_2 = freqresp(CL_system_2,w); % CL = G/(1+G*H)</span>
<span class="org-comment">% for </span><span class="org-variable-name"><span class="org-comment">i</span></span><span class="org-comment"> = </span><span class="org-constant"><span class="org-comment">1:size(w,2)</span></span>
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">OL(:,i)</span></span><span class="org-comment"> = svd(OL_freq(:,:,i));</span>
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">CL </span></span><span class="org-comment">(:,i) = svd(CL_freq(:,:,i));</span>
<span class="org-comment">% %CL2 (:,i) = svd(CL_freq_2(:,:,i));</span>
<span class="org-comment">% end</span>
<span class="org-comment">%</span>
<span class="org-comment">% un = ones(1,length(w));</span>
<span class="org-comment">% figure</span>
<span class="org-comment">% loglog(w./(2*pi),OL(3,:)+1,'k',w./(2*pi),OL(3,:)-1,'b',w./(2*pi),1./CL(1,:),'r--',w./(2*pi),un,'k:');hold on;%</span>
<span class="org-comment">% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));</span>
<span class="org-comment">% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));</span>
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');</span>
<span class="org-comment">% legend('GH \sigma_{inf} +1 ','GH \sigma_{inf} -1','S 1/\sigma_{sup}');%,'\lambda_1','\lambda_2','\lambda_3');</span>
<span class="org-comment">%</span>
<span class="org-comment">% figure</span>
<span class="org-comment">% loglog(w./(2*pi),OL(1,:)+1,'k',w./(2*pi),OL(1,:)-1,'b',w./(2*pi),1./CL(3,:),'r--',w./(2*pi),un,'k:');hold on;%</span>
<span class="org-comment">% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));</span>
<span class="org-comment">% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));</span>
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');</span>
<span class="org-comment">% legend('GH \sigma_{sup} +1 ','GH \sigma_{sup} -1','S 1/\sigma_{inf}');%,'\lambda_1','\lambda_2','\lambda_3');</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org218243e" class="outline-4">
<h4 id="org218243e"><span class="section-number-4">1.5.5</span> Control Section</h4>
<div class="outline-text-4" id="text-1-5-5">
<div class="org-src-container">
<pre class="src src-matlab">system_dec_10Hz = freqresp(system_dec,2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10);
system_dec_0Hz = freqresp(system_dec,0);
system_decReal_10Hz = pinv(align(system_dec_10Hz));
[Ureal,Sreal,Vreal] = svd(system_decReal_10Hz(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3));
normalizationMatrixReal = abs(pinv(Ureal)<span class="org-type">*</span>system_dec_0Hz(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(Vreal<span class="org-type">'</span>));
[U,S,V] = svd(system_dec_10Hz(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3));
normalizationMatrix = abs(pinv(U)<span class="org-type">*</span>system_dec_0Hz(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(V<span class="org-type">'</span>));
H_dec = ([zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30,30<span class="org-type">/</span>5) 0 0 0
0 zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>4,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>20,20<span class="org-type">/</span>4) 0 0
0 0 0 zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span>,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10)]);
H_cen_OL = [zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span>,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10) 0 0; 0 zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span>,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10) 0;
0 0 zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30,30<span class="org-type">/</span>5)];
H_cen = pinv(Jta)<span class="org-type">*</span>H_cen_OL<span class="org-type">*</span>pinv([Js1; Js2]);
<span class="org-comment">% H_svd_OL = -[1/normalizationMatrix(1,1) 0 0 0</span>
<span class="org-comment">% 0 1/normalizationMatrix(2,2) 0 0</span>
<span class="org-comment">% 0 0 1/normalizationMatrix(3,3) 0];</span>
<span class="org-comment">% H_svd_OL_real = -[1/normalizationMatrixReal(1,1) 0 0 0</span>
<span class="org-comment">% 0 1/normalizationMatrixReal(2,2) 0 0</span>
<span class="org-comment">% 0 0 1/normalizationMatrixReal(3,3) 0];</span>
H_svd_OL = <span class="org-type">-</span>[1<span class="org-type">/</span>normalizationMatrix(1,1)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>60,60<span class="org-type">/</span>10) 0 0 0
0 1<span class="org-type">/</span>normalizationMatrix(2,2)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30,30<span class="org-type">/</span>5) 0 0
0 0 1<span class="org-type">/</span>normalizationMatrix(3,3)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>2,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10<span class="org-type">/</span>2) 0];
H_svd_OL_real = <span class="org-type">-</span>[1<span class="org-type">/</span>normalizationMatrixReal(1,1)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>60,60<span class="org-type">/</span>10) 0 0 0
0 1<span class="org-type">/</span>normalizationMatrixReal(2,2)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30,30<span class="org-type">/</span>5) 0 0
0 0 1<span class="org-type">/</span>normalizationMatrixReal(3,3)<span class="org-type">*</span>zpk(<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>2,<span class="org-type">-</span>2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10,10<span class="org-type">/</span>2) 0];
<span class="org-comment">% H_svd_OL_real = -[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 10*zpk(-2*pi*10,-2*pi*100,100/10) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];%-eye(3,4);</span>
<span class="org-comment">% H_svd_OL = -[zpk(-2*pi*10,-2*pi*40,40/10) 0 0 0; 0 zpk(-2*pi*4,-2*pi*20,4/20) 0 0; 0 0 zpk(-2*pi*2,-2*pi*10,10/2) 0];% - eye(3,4);%</span>
H_svd = pinv(V<span class="org-type">'</span>)<span class="org-type">*</span>H_svd_OL<span class="org-type">*</span>pinv(U);
H_svd_real = pinv(Vreal<span class="org-type">'</span>)<span class="org-type">*</span>H_svd_OL_real<span class="org-type">*</span>pinv(Ureal);
OL_dec = g<span class="org-type">*</span>H_dec<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3);
OL_cen = g<span class="org-type">*</span>H_cen_OL<span class="org-type">*</span>pinv([Js1; Js2])<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(Jta);
OL_svd = 100<span class="org-type">*</span>H_svd_OL<span class="org-type">*</span>pinv(U)<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(V<span class="org-type">'</span>);
OL_svd_real = 100<span class="org-type">*</span>H_svd_OL_real<span class="org-type">*</span>pinv(Ureal)<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(Vreal<span class="org-type">'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% figure</span>
<span class="org-comment">% bode(OL_dec,w,P);title('OL Decentralized');</span>
<span class="org-comment">% figure</span>
<span class="org-comment">% bode(OL_cen,w,P);title('OL Centralized');</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>
bode(g<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),w,P);
title(<span class="org-string">'gain * Plant'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>
bode(OL_svd,OL_svd_real,w,P);
title(<span class="org-string">'OL SVD'</span>);
legend(<span class="org-string">'SVD of Complex plant'</span>,<span class="org-string">'SVD of real approximation of the complex plant'</span>)
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>
bode(system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),pinv(U)<span class="org-type">*</span>system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3)<span class="org-type">*</span>pinv(V<span class="org-type">'</span>),P);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">CL_dec = feedback(system_dec,g<span class="org-type">*</span>H_dec,[1 2 3],[1 2 3 4]);
CL_cen = feedback(system_dec,g<span class="org-type">*</span>H_cen,[1 2 3],[1 2 3 4]);
CL_svd = feedback(system_dec,100<span class="org-type">*</span>H_svd,[1 2 3],[1 2 3 4]);
CL_svd_real = feedback(system_dec,100<span class="org-type">*</span>H_svd_real,[1 2 3],[1 2 3 4]);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">pzmap_testCL(system_dec,H_dec,g,[1 2 3],[1 2 3 4])
title(<span class="org-string">'Decentralized control'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">pzmap_testCL(system_dec,H_cen,g,[1 2 3],[1 2 3 4])
title(<span class="org-string">'Centralized control'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">pzmap_testCL(system_dec,H_svd,100,[1 2 3],[1 2 3 4])
title(<span class="org-string">'SVD control'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">pzmap_testCL(system_dec,H_svd_real,100,[1 2 3],[1 2 3 4])
title(<span class="org-string">'Real approximation SVD control'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">P.Ylim = [1e<span class="org-type">-</span>8 1e<span class="org-type">-</span>3];
<span class="org-type">figure</span>
bodemag(system_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),CL_dec(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),CL_cen(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),CL_svd(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),CL_svd_real(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3),P);
title(<span class="org-string">'Motion/actuator'</span>)
legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'Decentralized control'</span>,<span class="org-string">'Centralized control'</span>,<span class="org-string">'SVD control'</span>,<span class="org-string">'SVD control real appr.'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">P.Ylim = [1e<span class="org-type">-</span>5 1e1];
<span class="org-type">figure</span>
bodemag(system_dec(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),CL_dec(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),CL_cen(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),CL_svd(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),CL_svd_real(1<span class="org-type">:</span>4,4<span class="org-type">:</span>6),P);
title(<span class="org-string">'Transmissibility'</span>);
legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'Decentralized control'</span>,<span class="org-string">'Centralized control'</span>,<span class="org-string">'SVD control'</span>,<span class="org-string">'SVD control real appr.'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>
bodemag(system_dec([7 9],4<span class="org-type">:</span>6),CL_dec([7 9],4<span class="org-type">:</span>6),CL_cen([7 9],4<span class="org-type">:</span>6),CL_svd([7 9],4<span class="org-type">:</span>6),CL_svd_real([7 9],4<span class="org-type">:</span>6),P);
title(<span class="org-string">'Transmissibility from half sum and half difference in the X direction'</span>);
legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'Decentralized control'</span>,<span class="org-string">'Centralized control'</span>,<span class="org-string">'SVD control'</span>,<span class="org-string">'SVD control real appr.'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>
bodemag(system_dec([8 10],4<span class="org-type">:</span>6),CL_dec([8 10],4<span class="org-type">:</span>6),CL_cen([8 10],4<span class="org-type">:</span>6),CL_svd([8 10],4<span class="org-type">:</span>6),CL_svd_real([8 10],4<span class="org-type">:</span>6),P);
title(<span class="org-string">'Transmissibility from half sum and half difference in the Z direction'</span>);
legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'Decentralized control'</span>,<span class="org-string">'Centralized control'</span>,<span class="org-string">'SVD control'</span>,<span class="org-string">'SVD control real appr.'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgad11a63" class="outline-4">
<h4 id="orgad11a63"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
<div class="outline-text-4" id="text-1-5-6">
<div class="org-src-container">
<pre class="src src-matlab">system_dec_freq = freqresp(system_dec,w);
x1 = zeros(1,length(w));
z1 = zeros(1,length(w));
x2 = zeros(1,length(w));
S1 = zeros(1,length(w));
S2 = zeros(1,length(w));
S3 = zeros(1,length(w));
<span class="org-keyword">for</span> <span class="org-variable-name">t</span> = <span class="org-constant">1:length(w)</span>
x1(t) = (abs(system_dec_freq(1,2,t))<span class="org-type">+</span>abs(system_dec_freq(1,3,t)))<span class="org-type">/</span>abs(system_dec_freq(1,1,t));
z1(t) = (abs(system_dec_freq(2,1,t))<span class="org-type">+</span>abs(system_dec_freq(2,3,t)))<span class="org-type">/</span>abs(system_dec_freq(2,2,t));
x2(t) = (abs(system_dec_freq(3,1,t))<span class="org-type">+</span>abs(system_dec_freq(3,2,t)))<span class="org-type">/</span>abs(system_dec_freq(3,3,t));
system_svd = pinv(Ureal)<span class="org-type">*</span>system_dec_freq(1<span class="org-type">:</span>4,1<span class="org-type">:</span>3,t)<span class="org-type">*</span>pinv(Vreal<span class="org-type">'</span>);
S1(t) = (abs(system_svd(1,2))<span class="org-type">+</span>abs(system_svd(1,3)))<span class="org-type">/</span>abs(system_svd(1,1));
S2(t) = (abs(system_svd(2,1))<span class="org-type">+</span>abs(system_svd(2,3)))<span class="org-type">/</span>abs(system_svd(2,2));
S2(t) = (abs(system_svd(3,1))<span class="org-type">+</span>abs(system_svd(3,2)))<span class="org-type">/</span>abs(system_svd(3,3));
<span class="org-keyword">end</span>
limit = 0.5<span class="org-type">*</span>ones(1,length(w));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>
loglog(w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),x1,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),z1,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),x2,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),limit,<span class="org-string">'--'</span>);
legend(<span class="org-string">'x_1'</span>,<span class="org-string">'z_1'</span>,<span class="org-string">'x_2'</span>,<span class="org-string">'Limit'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>
loglog(w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),S1,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),S2,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),S3,w<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),limit,<span class="org-string">'--'</span>);
legend(<span class="org-string">'S1'</span>,<span class="org-string">'S2'</span>,<span class="org-string">'S3'</span>,<span class="org-string">'Limit'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
<span class="org-comment">% set(gcf,'color','w')</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orga23d907" class="outline-4">
<h4 id="orga23d907"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
<div class="outline-text-4" id="text-1-5-7">
<div class="org-src-container">
<pre class="src src-matlab">Fr = logspace(<span class="org-type">-</span>2,3,1e3);
w=2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>Fr<span class="org-type">*</span>1<span class="org-constant">i</span>;
<span class="org-comment">%fit of the ground motion data in m/s^2/rtHz</span>
Fr_ground_x = [0.07 0.1 0.15 0.3 0.7 0.8 0.9 1.2 5 10];
n_ground_x1 = [4e<span class="org-type">-</span>7 4e<span class="org-type">-</span>7 2e<span class="org-type">-</span>6 1e<span class="org-type">-</span>6 5e<span class="org-type">-</span>7 5e<span class="org-type">-</span>7 5e<span class="org-type">-</span>7 1e<span class="org-type">-</span>6 1e<span class="org-type">-</span>5 3.5e<span class="org-type">-</span>5];
Fr_ground_v = [0.07 0.08 0.1 0.11 0.12 0.15 0.25 0.6 0.8 1 1.2 1.6 2 6 10];
n_ground_v1 = [7e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 1e<span class="org-type">-</span>6 1.2e<span class="org-type">-</span>6 1.5e<span class="org-type">-</span>6 1e<span class="org-type">-</span>6 9e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 7e<span class="org-type">-</span>7 1e<span class="org-type">-</span>6 2e<span class="org-type">-</span>6 1e<span class="org-type">-</span>5 3e<span class="org-type">-</span>5];
n_ground_x = interp1(Fr_ground_x,n_ground_x1,Fr,<span class="org-string">'linear'</span>);
n_ground_v = interp1(Fr_ground_v,n_ground_v1,Fr,<span class="org-string">'linear'</span>);
<span class="org-comment">% figure</span>
<span class="org-comment">% loglog(Fr,abs(n_ground_v),Fr_ground_v,n_ground_v1,'*');</span>
<span class="org-comment">% xlabel('Frequency [Hz]');ylabel('ASD [m/s^2 /rtHz]');</span>
<span class="org-comment">% return</span>
<span class="org-comment">%converting into PSD</span>
n_ground_x = (n_ground_x)<span class="org-type">.^</span>2;
n_ground_v = (n_ground_v)<span class="org-type">.^</span>2;
<span class="org-comment">%Injecting ground motion in the system and getting the outputs</span>
system_dec_f = (freqresp(system_dec,abs(w)));
PHI = zeros(size(Fr,2),12,12);
<span class="org-keyword">for</span> <span class="org-variable-name">p</span> = <span class="org-constant">1:size(Fr,2)</span>
Sw=zeros(6,6);
Iact = zeros(3,3);
Sw<span class="org-type">(4,4) </span>= n_ground_x(p);
Sw<span class="org-type">(5,5) </span>= n_ground_v(p);
Sw<span class="org-type">(6,6) </span>= n_ground_v(p);
Sw<span class="org-type">(1:3,1:3) </span>= Iact;
PHI(p,<span class="org-type">:</span>,<span class="org-type">:</span>) = (system_dec_f(<span class="org-type">:</span>,<span class="org-type">:</span>,p))<span class="org-type">*</span>Sw(<span class="org-type">:</span>,<span class="org-type">:</span>)<span class="org-type">*</span>(system_dec_f(<span class="org-type">:</span>,<span class="org-type">:</span>,p))<span class="org-type">'</span>;
<span class="org-keyword">end</span>
x1 = PHI(<span class="org-type">:</span>,1,1);
z1 = PHI(<span class="org-type">:</span>,2,2);
x2 = PHI(<span class="org-type">:</span>,3,3);
z2 = PHI(<span class="org-type">:</span>,4,4);
wx = PHI(<span class="org-type">:</span>,5,5);
wz = PHI(<span class="org-type">:</span>,6,6);
x12 = PHI(<span class="org-type">:</span>,1,3);
z12 = PHI(<span class="org-type">:</span>,2,4);
PHIwx = PHI(<span class="org-type">:</span>,1,5);
PHIwz = PHI(<span class="org-type">:</span>,2,6);
xsum = PHI(<span class="org-type">:</span>,7,7);
zsum = PHI(<span class="org-type">:</span>,8,8);
xdelta = PHI(<span class="org-type">:</span>,9,9);
zdelta = PHI(<span class="org-type">:</span>,10,10);
rot = PHI(<span class="org-type">:</span>,11,11);
</pre>
</div>
</div>
</div>
</div>
</div>
<div id="outline-container-org23fa18d" class="outline-2">
<h2 id="org23fa18d"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org81c3333" class="outline-3">
<h3 id="org81c3333"><span class="section-number-3">2.1</span> <code>align</code></h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="org303d818"></a>
</p>
<p>
This Matlab function is accessible <a href="gravimeter/align.m">here</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[A]</span> = <span class="org-function-name">align</span>(<span class="org-variable-name">V</span>)
<span class="org-comment">%A!ALIGN(V) returns a constat matrix A which is the real alignment of the</span>
<span class="org-comment">%INVERSE of the complex input matrix V</span>
<span class="org-comment">%from Mohit slides</span>
<span class="org-keyword">if</span> (nargin <span class="org-type">==</span>0) <span class="org-type">||</span> (nargin <span class="org-type">&gt;</span> 1)
disp(<span class="org-string">'usage: mat_inv_real = align(mat)'</span>)
<span class="org-keyword">return</span>
<span class="org-keyword">end</span>
D = pinv(real(V<span class="org-type">'*</span>V));
A = D<span class="org-type">*</span>real(V<span class="org-type">'*</span>diag(exp(1<span class="org-constant">i</span> <span class="org-type">*</span> angle(diag(V<span class="org-type">*</span>D<span class="org-type">*</span>V<span class="org-type">.'</span>))<span class="org-type">/</span>2)));
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org8b6878d" class="outline-3">
<h3 id="org8b6878d"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
<div class="outline-text-3" id="text-2-2">
<p>
<a id="org7c6ecb8"></a>
</p>
<p>
This Matlab function is accessible <a href="gravimeter/pzmap_testCL.m">here</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[]</span> = <span class="org-function-name">pzmap_testCL</span>(<span class="org-variable-name">system</span>,<span class="org-variable-name">H</span>,<span class="org-variable-name">gain</span>,<span class="org-variable-name">feedin</span>,<span class="org-variable-name">feedout</span>)
<span class="org-comment">% evaluate and plot the pole-zero map for the closed loop system for</span>
<span class="org-comment">% different values of the gain</span>
[<span class="org-type">~</span>, n] = size(gain);
[m1, n1, <span class="org-type">~</span>] = size(H);
[<span class="org-type">~</span>,n2] = size(feedin);
<span class="org-type">figure</span>
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
<span class="org-comment">% if n1 == n2</span>
system_CL = feedback(system,gain(<span class="org-constant">i</span>)<span class="org-type">*</span>H,feedin,feedout);
[P,Z] = pzmap(system_CL);
plot(real(P(<span class="org-type">:</span>)),imag(P(<span class="org-type">:</span>)),<span class="org-string">'x'</span>,real(Z(<span class="org-type">:</span>)),imag(Z(<span class="org-type">:</span>)),<span class="org-string">'o'</span>);hold on
xlabel(<span class="org-string">'Real axis (s^{-1})'</span>);ylabel(<span class="org-string">'Imaginary Axis (s^{-1})'</span>);
<span class="org-comment">% clear P Z</span>
<span class="org-comment">% else</span>
<span class="org-comment">% system_CL = feedback(system,gain(i)*H(:,1+(i-1)*m1:m1+(i-1)*m1),feedin,feedout);</span>
<span class="org-comment">%</span>
<span class="org-comment">% [P,Z] = pzmap(system_CL);</span>
<span class="org-comment">% plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on</span>
<span class="org-comment">% xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');</span>
<span class="org-comment">% clear P Z</span>
<span class="org-comment">% end</span>
<span class="org-keyword">end</span>
str = {strcat(<span class="org-string">'gain = '</span> , num2str(gain(1)))}; <span class="org-comment">% at the end of first loop, z being loop output</span>
str = [str , strcat(<span class="org-string">'gain = '</span> , num2str(gain(1)))]; <span class="org-comment">% after 2nd loop</span>
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">2:n</span>
str = [str , strcat(<span class="org-string">'gain = '</span> , num2str(gain(<span class="org-constant">i</span>)))]; <span class="org-comment">% after 2nd loop</span>
str = [str , strcat(<span class="org-string">'gain = '</span> , num2str(gain(<span class="org-constant">i</span>)))]; <span class="org-comment">% after 2nd loop</span>
<span class="org-keyword">end</span>
legend(str{<span class="org-type">:</span>})
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org50746f8" class="outline-2">
<h2 id="org50746f8"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
<div class="outline-text-2" id="text-3">
<p>
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org9c6bf2d">5</a>.
</p>
<div id="org9c6bf2d" class="figure">
<p><img src="figs/SP_assembly.png" alt="SP_assembly.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Stewart Platform CAD View</p>
</div>
<p>
The analysis of the SVD control applied to the Stewart platform is performed in the following sections:
</p>
<ul class="org-ul">
<li>Section <a href="#org5932d29">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li>
<li>Section <a href="#org7980ba7">3.2</a>: The plant is identified from the Simscape model and the centralized plant is computed thanks to the Jacobian</li>
<li>Section <a href="#orgb9c44bf">3.3</a>: The identified Dynamics is shown</li>
<li>Section <a href="#orgea1a70b">3.4</a>: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)</li>
<li>Section <a href="#org0cd9585">3.5</a>: The decoupling is performed thanks to the SVD. The effectiveness of the decoupling is verified using the Gershorin Radii</li>
<li>Section <a href="#org6e20bec">3.6</a>: The dynamics of the decoupled plant is shown</li>
<li>Section <a href="#org7c9ebe2">3.7</a>: A diagonal controller is defined to control the decoupled plant</li>
<li>Section <a href="#orgfaeace7">3.8</a>: Finally, the closed loop system properties are studied</li>
</ul>
</div>
<div id="outline-container-orga12724f" class="outline-3">
<h3 id="orga12724f"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3>
<div class="outline-text-3" id="text-3-1">
<p>
<a id="org5932d29"></a>
</p>
<div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'drone_platform.slx'</span>);
</pre>
</div>
<p>
Definition of spring parameters
</p>
<div class="org-src-container">
<pre class="src src-matlab">kx = 0.5<span class="org-type">*</span>1e3<span class="org-type">/</span>3; <span class="org-comment">% [N/m]</span>
ky = 0.5<span class="org-type">*</span>1e3<span class="org-type">/</span>3;
kz = 1e3<span class="org-type">/</span>3;
cx = 0.025; <span class="org-comment">% [Nm/rad]</span>
cy = 0.025;
cz = 0.025;
</pre>
</div>
<p>
Gravity:
</p>
<div class="org-src-container">
<pre class="src src-matlab">g = 0;
</pre>
</div>
<p>
We load the Jacobian (previously computed from the geometry).
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./jacobian.mat'</span>, <span class="org-string">'Aa'</span>, <span class="org-string">'Ab'</span>, <span class="org-string">'As'</span>, <span class="org-string">'l'</span>, <span class="org-string">'J'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org820527f" class="outline-3">
<h3 id="org820527f"><span class="section-number-3">3.2</span> Identification of the plant</h3>
<div class="outline-text-3" id="text-3-2">
<p>
<a id="org7980ba7"></a>
</p>
<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"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'drone_platform'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/u'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Inertial Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
G = linearize(mdl, io);
G.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Rwx'</span>, <span class="org-string">'Rwy'</span>, <span class="org-string">'Rwz'</span>, ...
<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
G.OutputName = {<span class="org-string">'Ax'</span>, <span class="org-string">'Ay'</span>, <span class="org-string">'Az'</span>, <span class="org-string">'Arx'</span>, <span class="org-string">'Ary'</span>, <span class="org-string">'Arz'</span>};
</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>
<p>
The &ldquo;centralized&rdquo; plant \(\bm{G}_x\) is now computed (Figure <a href="#org249f9cd">6</a>).
</p>
<div id="org249f9cd" class="figure">
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Centralized control architecture</p>
</div>
<p>
Thanks to the Jacobian, we compute the transfer functions in the inertial frame (transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gx = G<span class="org-type">*</span>blkdiag(eye(6), inv(J<span class="org-type">'</span>));
Gx.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Rwx'</span>, <span class="org-string">'Rwy'</span>, <span class="org-string">'Rwz'</span>, ...
<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-orga58761b" class="outline-3">
<h3 id="orga58761b"><span class="section-number-3">3.3</span> Obtained Dynamics</h3>
<div class="outline-text-3" id="text-3-3">
<p>
<a id="orgb9c44bf"></a>
</p>
<div id="org6d21a96" class="figure">
<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
</div>
<div id="orge724936" class="figure">
<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
</div>
</div>
</div>
<div id="outline-container-orgb3d55c6" class="outline-3">
<h3 id="orgb3d55c6"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3>
<div class="outline-text-3" id="text-3-4">
<p>
<a id="orgea1a70b"></a>
</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 \(\omega_c\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30; <span class="org-comment">% Decoupling frequency [rad/s]</span>
Gc = G({<span class="org-string">'Ax'</span>, <span class="org-string">'Ay'</span>, <span class="org-string">'Az'</span>, <span class="org-string">'Arx'</span>, <span class="org-string">'Ary'</span>, <span class="org-string">'Arz'</span>}, ...
{<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}); <span class="org-comment">% Transfer function to find a real approximation</span>
H1 = evalfr(Gc, <span class="org-constant">j</span><span class="org-type">*</span>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<span class="org-type">'*</span>H1));
H1 = inv(D<span class="org-type">*</span>real(H1<span class="org-type">'*</span>diag(exp(<span class="org-constant">j</span><span class="org-type">*</span>angle(diag(H1<span class="org-type">*</span>D<span class="org-type">*</span>H1<span class="org-type">.'</span>))<span class="org-type">/</span>2))));
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Real approximate of \(G\) at the decoupling frequency \(\omega_c\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">4.4</td>
<td class="org-right">-2.1</td>
<td class="org-right">-2.1</td>
<td class="org-right">4.4</td>
<td class="org-right">-2.4</td>
<td class="org-right">-2.4</td>
</tr>
<tr>
<td class="org-right">-0.2</td>
<td class="org-right">-3.9</td>
<td class="org-right">3.9</td>
<td class="org-right">0.2</td>
<td class="org-right">-3.8</td>
<td class="org-right">3.8</td>
</tr>
<tr>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
</tr>
<tr>
<td class="org-right">-367.1</td>
<td class="org-right">-323.8</td>
<td class="org-right">323.8</td>
<td class="org-right">367.1</td>
<td class="org-right">43.3</td>
<td class="org-right">-43.3</td>
</tr>
<tr>
<td class="org-right">-162.0</td>
<td class="org-right">-237.0</td>
<td class="org-right">-237.0</td>
<td class="org-right">-162.0</td>
<td class="org-right">398.9</td>
<td class="org-right">398.9</td>
</tr>
<tr>
<td class="org-right">220.6</td>
<td class="org-right">-220.6</td>
<td class="org-right">220.6</td>
<td class="org-right">-220.6</td>
<td class="org-right">220.6</td>
<td class="org-right">-220.6</td>
</tr>
</tbody>
</table>
<p>
Note that the plant \(G\) at \(\omega_c\) is already an almost real matrix.
This can be seen on the Bode plots where the phase is close to 1.
This can be verified below where only the real value of \(G(\omega_c)\) is shown
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">4.4</td>
<td class="org-right">-2.1</td>
<td class="org-right">-2.1</td>
<td class="org-right">4.4</td>
<td class="org-right">-2.4</td>
<td class="org-right">-2.4</td>
</tr>
<tr>
<td class="org-right">-0.2</td>
<td class="org-right">-3.9</td>
<td class="org-right">3.9</td>
<td class="org-right">0.2</td>
<td class="org-right">-3.8</td>
<td class="org-right">3.8</td>
</tr>
<tr>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
<td class="org-right">3.4</td>
</tr>
<tr>
<td class="org-right">-367.1</td>
<td class="org-right">-323.8</td>
<td class="org-right">323.8</td>
<td class="org-right">367.1</td>
<td class="org-right">43.3</td>
<td class="org-right">-43.3</td>
</tr>
<tr>
<td class="org-right">-162.0</td>
<td class="org-right">-237.0</td>
<td class="org-right">-237.0</td>
<td class="org-right">-162.0</td>
<td class="org-right">398.9</td>
<td class="org-right">398.9</td>
</tr>
<tr>
<td class="org-right">220.6</td>
<td class="org-right">-220.6</td>
<td class="org-right">220.6</td>
<td class="org-right">-220.6</td>
<td class="org-right">220.6</td>
<td class="org-right">-220.6</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org2f2890a" class="outline-3">
<h3 id="org2f2890a"><span class="section-number-3">3.5</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3>
<div class="outline-text-3" id="text-3-5">
<p>
<a id="org0cd9585"></a>
</p>
<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>
This is computed over the following frequencies.
</p>
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(<span class="org-type">-</span>2, 2, 1000); <span class="org-comment">% [Hz]</span>
</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, <span class="org-string">'Hz'</span>)));
<span class="org-keyword">for</span> <span class="org-variable-name">out_i</span> = <span class="org-constant">1:size(Gc,2)</span>
Gr_coupled(<span class="org-type">:</span>, out_i) = squeeze((sum(H(out_i,<span class="org-type">:</span>,<span class="org-type">:</span>)) <span class="org-type">-</span> H(out_i,out_i,<span class="org-type">:</span>))<span class="org-type">./</span>H(out_i, out_i, <span class="org-type">:</span>));
<span class="org-keyword">end</span>
</pre>
</div>
<p>
Gershgorin Radii for the decoupled plant using SVD:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gd = U<span class="org-type">'*</span>Gc<span class="org-type">*</span>V;
Gr_decoupled = zeros(length(freqs), size(Gd,2));
H = abs(squeeze(freqresp(Gd, freqs, <span class="org-string">'Hz'</span>)));
<span class="org-keyword">for</span> <span class="org-variable-name">out_i</span> = <span class="org-constant">1:size(Gd,2)</span>
Gr_decoupled(<span class="org-type">:</span>, out_i) = squeeze((sum(H(out_i,<span class="org-type">:</span>,<span class="org-type">:</span>)) <span class="org-type">-</span> H(out_i,out_i,<span class="org-type">:</span>))<span class="org-type">./</span>H(out_i, out_i, <span class="org-type">:</span>));
<span class="org-keyword">end</span>
</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<span class="org-type">*</span>inv(J<span class="org-type">'</span>);
Gr_jacobian = zeros(length(freqs), size(Gj,2));
H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
<span class="org-keyword">for</span> <span class="org-variable-name">out_i</span> = <span class="org-constant">1:size(Gj,2)</span>
Gr_jacobian(<span class="org-type">:</span>, out_i) = squeeze((sum(H(out_i,<span class="org-type">:</span>,<span class="org-type">:</span>)) <span class="org-type">-</span> H(out_i,out_i,<span class="org-type">:</span>))<span class="org-type">./</span>H(out_i, out_i, <span class="org-type">:</span>));
<span class="org-keyword">end</span>
</pre>
</div>
<div id="org4e85b3b" class="figure">
<p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Gershgorin Radii of the Coupled and Decoupled plants</p>
</div>
</div>
</div>
<div id="outline-container-org70b5fa2" class="outline-3">
<h3 id="org70b5fa2"><span class="section-number-3">3.6</span> Decoupled Plant</h3>
<div class="outline-text-3" id="text-3-6">
<p>
<a id="org6e20bec"></a>
</p>
<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 id="org82227b9" 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 10: </span>Decoupled Plant using SVD</p>
</div>
<div id="org2cf3f8e" 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 11: </span>Decoupled Plant using the Jacobian</p>
</div>
</div>
</div>
<div id="outline-container-orgc23974f" class="outline-3">
<h3 id="orgc23974f"><span class="section-number-3">3.7</span> Diagonal Controller</h3>
<div class="outline-text-3" id="text-3-7">
<p>
<a id="org7c9ebe2"></a>
</p>
<p>
The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>0.1; <span class="org-comment">% Crossover Frequency [rad/s]</span>
C_g = 50; <span class="org-comment">% DC Gain</span>
K = eye(6)<span class="org-type">*</span>C_g<span class="org-type">/</span>(s<span class="org-type">+</span>wc);
</pre>
</div>
<p>
The control diagram for the centralized control is shown in Figure <a href="#org249f9cd">6</a>.
</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 id="org6e49f6b" class="figure">
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Control Diagram for the Centralized control</p>
</div>
<p>
The feedback system is computed as shown below.
</p>
<div class="org-src-container">
<pre class="src src-matlab">G_cen = feedback(G, inv(J<span class="org-type">'</span>)<span class="org-type">*</span>K, [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
</pre>
</div>
<p>
The SVD control architecture is shown in Figure <a href="#org98507fe">13</a>.
The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
</p>
<div id="org98507fe" class="figure">
<p><img src="figs/svd_control.png" alt="svd_control.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Control Diagram for the SVD control</p>
</div>
<p>
The feedback system is computed as shown below.
</p>
<div class="org-src-container">
<pre class="src src-matlab">G_svd = feedback(G, pinv(V<span class="org-type">'</span>)<span class="org-type">*</span>K<span class="org-type">*</span>pinv(U), [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
</pre>
</div>
</div>
</div>
<div id="outline-container-org6e4ced6" class="outline-3">
<h3 id="org6e4ced6"><span class="section-number-3">3.8</span> Closed-Loop system Performances</h3>
<div class="outline-text-3" id="text-3-8">
<p>
<a id="orgfaeace7"></a>
</p>
<p>
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
0
</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="#org500fc7e">14</a>.
</p>
<div id="org500fc7e" 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 14: </span>Obtained Transmissibility</p>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-06 ven. 12:22</p>
</div>
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