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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="#org4f444af">1. Gravimeter - Simscape Model</a>
<ul>
<li><a href="#org9c89f1b">1.1. Introduction</a></li>
<li><a href="#org5c18570">1.2. Simscape Model - Parameters</a></li>
<li><a href="#org890db47">1.3. System Identification - Without Gravity</a></li>
<li><a href="#org8166361">1.4. System Identification - With Gravity</a></li>
<li><a href="#org1769947">1.5. Analytical Model</a>
<ul>
<li><a href="#org6553335">1.5.1. Parameters</a></li>
<li><a href="#orgb9d8709">1.5.2. generation of the state space model</a></li>
<li><a href="#org756ddd7">1.5.3. Comparison with the Simscape Model</a></li>
<li><a href="#orgbf8fd91">1.5.4. Analysis</a></li>
<li><a href="#org8a51806">1.5.5. Control Section</a></li>
<li><a href="#org28606d1">1.5.6. Greshgorin radius</a></li>
<li><a href="#org6605d22">1.5.7. Injecting ground motion in the system to have the output</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org569be4b">2. Gravimeter - Functions</a>
<ul>
<li><a href="#orgb5adbf9">2.1. <code>align</code></a></li>
<li><a href="#orgdc0b26f">2.2. <code>pzmap_testCL</code></a></li>
</ul>
</li>
<li><a href="#org6b34180">3. Stewart Platform - Simscape Model</a>
<ul>
<li><a href="#org3ce25f0">3.1. Jacobian</a></li>
<li><a href="#org7544d0c">3.2. Simscape Model</a></li>
<li><a href="#orgf131c5c">3.3. Identification of the plant</a></li>
<li><a href="#org9c8d3f6">3.4. Obtained Dynamics</a></li>
<li><a href="#org3cbc2c6">3.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
<li><a href="#org2ed1102">3.6. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li>
<li><a href="#org89bbbb9">3.7. Decoupled Plant</a></li>
<li><a href="#org16afb20">3.8. Diagonal Controller</a></li>
<li><a href="#orgae885a2">3.9. Centralized Control</a></li>
<li><a href="#org31ce44b">3.10. SVD Control</a></li>
<li><a href="#org6d772a3">3.11. Results</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org4f444af" class="outline-2">
<h2 id="org4f444af"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org9c89f1b" class="outline-3">
<h3 id="org9c89f1b"><span class="section-number-3">1.1</span> Introduction</h3>
<div class="outline-text-3" id="text-1-1">
<div id="orgfe166e2" 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-org5c18570" class="outline-3">
<h3 id="org5c18570"><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-org890db47" class="outline-3">
<h3 id="org890db47"><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">
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="org87b0848" 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-org8166361" class="outline-3">
<h3 id="org8166361"><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">
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="org69c29e0" 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-org1769947" class="outline-3">
<h3 id="org1769947"><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-org6553335" class="outline-4">
<h4 id="org6553335"><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;
<span class="org-comment">%P.PhaseVisible = 'off';</span>
</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-orgb9d8709" class="outline-4">
<h4 id="orgb9d8709"><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">
<div class="org-src-container">
<pre class="src src-matlab">M = [m 0 0
0 m 0
0 0 I];
<span class="org-comment">%Jacobian of the bottom sensor</span>
Js1 = [1 0 h<span class="org-type">/</span>2
0 1 <span class="org-type">-</span>l<span class="org-type">/</span>2];
<span class="org-comment">%Jacobian of the top sensor</span>
Js2 = [1 0 <span class="org-type">-</span>h<span class="org-type">/</span>2
0 1 0];
<span class="org-comment">%Jacobian of the actuators</span>
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>;
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;
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)];
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">BB </span></span><span class="org-comment">= [zeros(3,3)</span>
<span class="org-comment">% M\Jta ];</span>
<span class="org-comment">%</span>
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">CC </span></span><span class="org-comment">= [Ja zeros(3)];</span>
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">DD </span></span><span class="org-comment">= zeros(3,3);</span>
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>
system_dec = ss(AA,BB,CC,DD);
</pre>
</div>
<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">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-org756ddd7" class="outline-4">
<h4 id="org756ddd7"><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="orgcea07c6" 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-orgbf8fd91" class="outline-4">
<h4 id="orgbf8fd91"><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-org8a51806" class="outline-4">
<h4 id="org8a51806"><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-org28606d1" class="outline-4">
<h4 id="org28606d1"><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-org6605d22" class="outline-4">
<h4 id="org6605d22"><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-org569be4b" class="outline-2">
<h2 id="org569be4b"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-orgb5adbf9" class="outline-3">
<h3 id="orgb5adbf9"><span class="section-number-3">2.1</span> <code>align</code></h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="org3b1a721"></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-orgdc0b26f" class="outline-3">
<h3 id="orgdc0b26f"><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="org18df03d"></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-org6b34180" class="outline-2">
<h2 id="org6b34180"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-org3ce25f0" class="outline-3">
<h3 id="org3ce25f0"><span class="section-number-3">3.1</span> Jacobian</h3>
<div class="outline-text-3" id="text-3-1">
<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(<span class="org-string">'stewart_platform/drone_platform_jacobian.slx'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'drone_platform_jacobian'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Aa = [a1.Data(1,<span class="org-type">:</span>);
a2.Data(1,<span class="org-type">:</span>);
a3.Data(1,<span class="org-type">:</span>);
a4.Data(1,<span class="org-type">:</span>);
a5.Data(1,<span class="org-type">:</span>);
a6.Data(1,<span class="org-type">:</span>)]<span class="org-type">'</span>;
Ab = [b1.Data(1,<span class="org-type">:</span>);
b2.Data(1,<span class="org-type">:</span>);
b3.Data(1,<span class="org-type">:</span>);
b4.Data(1,<span class="org-type">:</span>);
b5.Data(1,<span class="org-type">:</span>);
b6.Data(1,<span class="org-type">:</span>)]<span class="org-type">'</span>;
As = (Ab <span class="org-type">-</span> Aa)<span class="org-type">./</span>vecnorm(Ab <span class="org-type">-</span> Aa);
l = vecnorm(Ab <span class="org-type">-</span> Aa)<span class="org-type">'</span>;
J = [As<span class="org-type">'</span> , cross(Ab, As)<span class="org-type">'</span>];
save(<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-org7544d0c" class="outline-3">
<h3 id="org7544d0c"><span class="section-number-3">3.2</span> Simscape Model</h3>
<div class="outline-text-3" id="text-3-2">
<div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'stewart_platform/drone_platform.slx'</span>);
</pre>
</div>
<p>
Definition of spring parameters
</p>
<div class="org-src-container">
<pre class="src src-matlab">kx = 50; <span class="org-comment">% [N/m]</span>
ky = 50;
kz = 50;
cx = 0.025; <span class="org-comment">% [Nm/rad]</span>
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(<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-orgf131c5c" class="outline-3">
<h3 id="orgf131c5c"><span class="section-number-3">3.3</span> Identification of the plant</h3>
<div class="outline-text-3" id="text-3-3">
<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"><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>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% G = G*blkdiag(inv(J), eye(6));</span>
<span class="org-comment">% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...</span>
<span class="org-comment">% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};</span>
</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<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>};
Gl = J<span class="org-type">*</span>G;
Gl.OutputName = {<span class="org-string">'A1'</span>, <span class="org-string">'A2'</span>, <span class="org-string">'A3'</span>, <span class="org-string">'A4'</span>, <span class="org-string">'A5'</span>, <span class="org-string">'A6'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-org9c8d3f6" class="outline-3">
<h3 id="org9c8d3f6"><span class="section-number-3">3.4</span> Obtained Dynamics</h3>
<div class="outline-text-3" id="text-3-4">
<div id="orgbe1afe0" class="figure">
<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
</div>
<div id="orgb7a99f5" class="figure">
<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
</div>
<div id="orgf03b0e7" class="figure">
<p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p>
</div>
<div id="orga22b4e1" class="figure">
<p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Transmissibility</p>
</div>
</div>
</div>
<div id="outline-container-org3cbc2c6" class="outline-3">
<h3 id="org3cbc2c6"><span class="section-number-3">3.5</span> Real Approximation of \(G\) at the decoupling frequency</h3>
<div class="outline-text-3" id="text-3-5">
<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<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>20; <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>
</div>
</div>
<div id="outline-container-org2ed1102" class="outline-3">
<h3 id="org2ed1102"><span class="section-number-3">3.6</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3>
<div class="outline-text-3" id="text-3-6">
<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>
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="org040d1c3" 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-org89bbbb9" class="outline-3">
<h3 id="org89bbbb9"><span class="section-number-3">3.7</span> Decoupled Plant</h3>
<div class="outline-text-3" id="text-3-7">
<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 id="orge51c3a9" 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="orgf570a47" 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-org16afb20" class="outline-3">
<h3 id="org16afb20"><span class="section-number-3">3.8</span> Diagonal Controller</h3>
<div class="outline-text-3" id="text-3-8">
<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<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>
</div>
</div>
<div id="outline-container-orgae885a2" class="outline-3">
<h3 id="orgae885a2"><span class="section-number-3">3.9</span> Centralized Control</h3>
<div class="outline-text-3" id="text-3-9">
<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<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>
</div>
</div>
<div id="outline-container-org31ce44b" class="outline-3">
<h3 id="org31ce44b"><span class="section-number-3">3.10</span> SVD Control</h3>
<div class="outline-text-3" id="text-3-10">
<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<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-org6d772a3" class="outline-3">
<h3 id="org6d772a3"><span class="section-number-3">3.11</span> Results</h3>
<div class="outline-text-3" id="text-3-11">
<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
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="#orgd6d0049">14</a>.
</p>
<div id="orgd6d0049" 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: 2020-10-05 lun. 18:06</p>
</div>
</body>
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