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#+TITLE: SVD Control
:DRAWER:
#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP: ../index.html
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
#+HTML_MATHJAX: align: center tagside: right font: TeX
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :tangle no
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:shell :eval no-export
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
* Gravimeter - Simscape Model
:PROPERTIES:
:header-args:matlab+: :tangle gravimeter/script.m
:END:
** Introduction
#+name: fig:gravimeter_model
#+caption: Model of the gravimeter
[[file:figs/gravimeter_model.png]]
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
addpath('gravimeter');
#+end_src
** Simscape Model - Parameters
#+begin_src matlab
open('gravimeter.slx')
#+end_src
Parameters
#+begin_src matlab
l = 1.0; % Length of the mass [m]
la = 0.5; % Position of Act. [m]
h = 3.4; % Height of the mass [m]
ha = 1.7; % Position of Act. [m]
m = 400; % Mass [kg]
I = 115; % Inertia [kg m^2]
k = 15e3; % Actuator Stiffness [N/m]
c = 0.03; % Actuator Damping [N/(m/s)]
deq = 0.2; % Length of the actuators [m]
g = 0; % Gravity [m/s2]
#+end_src
** System Identification - Without Gravity
#+begin_src matlab
%% Name of the Simulink File
mdl = 'gravimeter';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
#+end_src
#+begin_src matlab :results output replace :exports results
pole(G)
#+end_src
#+RESULTS:
#+begin_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
#+end_example
The plant as 6 states as expected (2 translations + 1 rotation)
#+begin_src matlab :results output replace
size(G)
#+end_src
#+RESULTS:
: State-space model with 4 outputs, 3 inputs, and 6 states.
#+begin_src matlab :exports none
freqs = logspace(-2, 2, 1000);
figure;
for in_i = 1:3
for out_i = 1:4
subplot(4, 3, 3*(out_i-1)+in_i);
plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
end
end
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/open_loop_tf.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:open_loop_tf
#+caption: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers
#+RESULTS:
[[file:figs/open_loop_tf.png]]
** System Identification - With Gravity
#+begin_src matlab
g = 9.80665; % Gravity [m/s2]
#+end_src
#+begin_src matlab
Gg = linearize(mdl, io);
Gg.InputName = {'F1', 'F2', 'F3'};
Gg.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
#+end_src
We can now see that the system is unstable due to gravity.
#+begin_src matlab :results output replace :exports results
pole(Gg)
#+end_src
#+RESULTS:
#+begin_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
#+end_example
#+begin_src matlab :exports none
freqs = logspace(-2, 2, 1000);
figure;
for in_i = 1:3
for out_i = 1:4
subplot(4, 3, 3*(out_i-1)+in_i);
hold on;
plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
plot(freqs, abs(squeeze(freqresp(Gg(out_i,in_i), freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
end
end
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/open_loop_tf_g.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:open_loop_tf_g
#+caption: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity
#+RESULTS:
[[file:figs/open_loop_tf_g.png]]
** Analytical Model
*** Parameters
Bode options.
#+begin_src matlab
P = bodeoptions;
P.FreqUnits = 'Hz';
P.MagUnits = 'abs';
P.MagScale = 'log';
P.Grid = 'on';
P.PhaseWrapping = 'on';
P.Title.FontSize = 14;
P.XLabel.FontSize = 14;
P.YLabel.FontSize = 14;
P.TickLabel.FontSize = 12;
P.Xlim = [1e-1,1e2];
P.MagLowerLimMode = 'manual';
P.MagLowerLim= 1e-3;
#+end_src
Frequency vector.
#+begin_src matlab
w = 2*pi*logspace(-1,2,1000); % [rad/s]
#+end_src
*** Generation of the State Space Model
Mass matrix
#+begin_src matlab
M = [m 0 0
0 m 0
0 0 I];
#+end_src
Jacobian of the bottom sensor
#+begin_src matlab
Js1 = [1 0 h/2
0 1 -l/2];
#+end_src
Jacobian of the top sensor
#+begin_src matlab
Js2 = [1 0 -h/2
0 1 0];
#+end_src
Jacobian of the actuators
#+begin_src matlab
Ja = [1 0 ha % Left horizontal actuator
0 1 -la % Left vertical actuator
0 1 la]; % Right vertical actuator
Jta = Ja';
#+end_src
Stiffness and Damping matrices
#+begin_src matlab
K = k*Jta*Ja;
C = c*Jta*Ja;
#+end_src
State Space Matrices
#+begin_src matlab
E = [1 0 0
0 1 0
0 0 1]; %projecting ground motion in the directions of the legs
AA = [zeros(3) eye(3)
-M\K -M\C];
BB = [zeros(3,6)
M\Jta M\(k*Jta*E)];
CC = [[Js1;Js2] zeros(4,3);
zeros(2,6)
(Js1+Js2)./2 zeros(2,3)
(Js1-Js2)./2 zeros(2,3)
(Js1-Js2)./(2*h) zeros(2,3)];
DD = [zeros(4,6)
zeros(2,3) eye(2,3)
zeros(6,6)];
#+end_src
State Space model:
- Input = three actuators and three ground motions
- Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation
#+begin_src matlab
system_dec = ss(AA,BB,CC,DD);
#+end_src
#+begin_src matlab :results output replace
size(system_dec)
#+end_src
#+RESULTS:
: State-space model with 12 outputs, 6 inputs, and 6 states.
*** Comparison with the Simscape Model
#+begin_src matlab :exports none
freqs = logspace(-2, 2, 1000);
figure;
for in_i = 1:3
for out_i = 1:4
subplot(4, 3, 3*(out_i-1)+in_i);
hold on;
plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
plot(freqs, abs(squeeze(freqresp(system_dec(out_i,in_i)*s^2, freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
end
end
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/gravimeter_analytical_system_open_loop_models.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:gravimeter_analytical_system_open_loop_models
#+caption: Comparison of the analytical and the Simscape models
#+RESULTS:
[[file:figs/gravimeter_analytical_system_open_loop_models.png]]
*** Analysis
#+begin_src matlab
% figure
% bode(system_dec,P);
% return
#+end_src
#+begin_src matlab
%% svd decomposition
% system_dec_freq = freqresp(system_dec,w);
% S = zeros(3,length(w));
% for m = 1:length(w)
% S(:,m) = svd(system_dec_freq(1:4,1:3,m));
% end
% figure
% loglog(w./(2*pi), S);hold on;
% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));
% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
% legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');
% ylim([1e-8 1e-2]);
%
% %condition number
% figure
% loglog(w./(2*pi), S(1,:)./S(3,:));hold on;
% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));
% xlabel('Frequency [Hz]');ylabel('Condition number [-]');
% % legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');
%
% %performance indicator
% system_dec_svd = freqresp(system_dec(1:4,1:3),2*pi*10);
% [U,S,V] = svd(system_dec_svd);
% 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);%
% H_svd = pinv(V')*H_svd_OL*pinv(U);
% % system_dec_control_svd_ = feedback(system_dec,g*pinv(V')*H*pinv(U));
%
% OL_dec = g_svd*H_svd*system_dec(1:4,1:3);
% OL_freq = freqresp(OL_dec,w); % OL = G*H
% CL_system = feedback(eye(3),-g_svd*H_svd*system_dec(1:4,1:3));
% CL_freq = freqresp(CL_system,w); % CL = (1+G*H)^-1
% % CL_system_2 = feedback(system_dec,H);
% % CL_freq_2 = freqresp(CL_system_2,w); % CL = G/(1+G*H)
% for i = 1:size(w,2)
% OL(:,i) = svd(OL_freq(:,:,i));
% CL (:,i) = svd(CL_freq(:,:,i));
% %CL2 (:,i) = svd(CL_freq_2(:,:,i));
% end
%
% un = ones(1,length(w));
% figure
% 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;%
% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));
% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));
% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
% legend('GH \sigma_{inf} +1 ','GH \sigma_{inf} -1','S 1/\sigma_{sup}');%,'\lambda_1','\lambda_2','\lambda_3');
%
% figure
% 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;%
% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));
% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));
% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
% legend('GH \sigma_{sup} +1 ','GH \sigma_{sup} -1','S 1/\sigma_{inf}');%,'\lambda_1','\lambda_2','\lambda_3');
#+end_src
*** Control Section
#+begin_src matlab
system_dec_10Hz = freqresp(system_dec,2*pi*10);
system_dec_0Hz = freqresp(system_dec,0);
system_decReal_10Hz = pinv(align(system_dec_10Hz));
[Ureal,Sreal,Vreal] = svd(system_decReal_10Hz(1:4,1:3));
normalizationMatrixReal = abs(pinv(Ureal)*system_dec_0Hz(1:4,1:3)*pinv(Vreal'));
[U,S,V] = svd(system_dec_10Hz(1:4,1:3));
normalizationMatrix = abs(pinv(U)*system_dec_0Hz(1:4,1:3)*pinv(V'));
H_dec = ([zpk(-2*pi*5,-2*pi*30,30/5) 0 0 0
0 zpk(-2*pi*4,-2*pi*20,20/4) 0 0
0 0 0 zpk(-2*pi,-2*pi*10,10)]);
H_cen_OL = [zpk(-2*pi,-2*pi*10,10) 0 0; 0 zpk(-2*pi,-2*pi*10,10) 0;
0 0 zpk(-2*pi*5,-2*pi*30,30/5)];
H_cen = pinv(Jta)*H_cen_OL*pinv([Js1; Js2]);
% H_svd_OL = -[1/normalizationMatrix(1,1) 0 0 0
% 0 1/normalizationMatrix(2,2) 0 0
% 0 0 1/normalizationMatrix(3,3) 0];
% H_svd_OL_real = -[1/normalizationMatrixReal(1,1) 0 0 0
% 0 1/normalizationMatrixReal(2,2) 0 0
% 0 0 1/normalizationMatrixReal(3,3) 0];
H_svd_OL = -[1/normalizationMatrix(1,1)*zpk(-2*pi*10,-2*pi*60,60/10) 0 0 0
0 1/normalizationMatrix(2,2)*zpk(-2*pi*5,-2*pi*30,30/5) 0 0
0 0 1/normalizationMatrix(3,3)*zpk(-2*pi*2,-2*pi*10,10/2) 0];
H_svd_OL_real = -[1/normalizationMatrixReal(1,1)*zpk(-2*pi*10,-2*pi*60,60/10) 0 0 0
0 1/normalizationMatrixReal(2,2)*zpk(-2*pi*5,-2*pi*30,30/5) 0 0
0 0 1/normalizationMatrixReal(3,3)*zpk(-2*pi*2,-2*pi*10,10/2) 0];
% 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);
% 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);%
H_svd = pinv(V')*H_svd_OL*pinv(U);
H_svd_real = pinv(Vreal')*H_svd_OL_real*pinv(Ureal);
OL_dec = g*H_dec*system_dec(1:4,1:3);
OL_cen = g*H_cen_OL*pinv([Js1; Js2])*system_dec(1:4,1:3)*pinv(Jta);
OL_svd = 100*H_svd_OL*pinv(U)*system_dec(1:4,1:3)*pinv(V');
OL_svd_real = 100*H_svd_OL_real*pinv(Ureal)*system_dec(1:4,1:3)*pinv(Vreal');
#+end_src
#+begin_src matlab
% figure
% bode(OL_dec,w,P);title('OL Decentralized');
% figure
% bode(OL_cen,w,P);title('OL Centralized');
#+end_src
#+begin_src matlab
figure
bode(g*system_dec(1:4,1:3),w,P);
title('gain * Plant');
#+end_src
#+begin_src matlab
figure
bode(OL_svd,OL_svd_real,w,P);
title('OL SVD');
legend('SVD of Complex plant','SVD of real approximation of the complex plant')
#+end_src
#+begin_src matlab
figure
bode(system_dec(1:4,1:3),pinv(U)*system_dec(1:4,1:3)*pinv(V'),P);
#+end_src
#+begin_src matlab
CL_dec = feedback(system_dec,g*H_dec,[1 2 3],[1 2 3 4]);
CL_cen = feedback(system_dec,g*H_cen,[1 2 3],[1 2 3 4]);
CL_svd = feedback(system_dec,100*H_svd,[1 2 3],[1 2 3 4]);
CL_svd_real = feedback(system_dec,100*H_svd_real,[1 2 3],[1 2 3 4]);
#+end_src
#+begin_src matlab
pzmap_testCL(system_dec,H_dec,g,[1 2 3],[1 2 3 4])
title('Decentralized control');
#+end_src
#+begin_src matlab
pzmap_testCL(system_dec,H_cen,g,[1 2 3],[1 2 3 4])
title('Centralized control');
#+end_src
#+begin_src matlab
pzmap_testCL(system_dec,H_svd,100,[1 2 3],[1 2 3 4])
title('SVD control');
#+end_src
#+begin_src matlab
pzmap_testCL(system_dec,H_svd_real,100,[1 2 3],[1 2 3 4])
title('Real approximation SVD control');
#+end_src
#+begin_src matlab
P.Ylim = [1e-8 1e-3];
figure
bodemag(system_dec(1:4,1:3),CL_dec(1:4,1:3),CL_cen(1:4,1:3),CL_svd(1:4,1:3),CL_svd_real(1:4,1:3),P);
title('Motion/actuator')
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
#+end_src
#+begin_src matlab
P.Ylim = [1e-5 1e1];
figure
bodemag(system_dec(1:4,4:6),CL_dec(1:4,4:6),CL_cen(1:4,4:6),CL_svd(1:4,4:6),CL_svd_real(1:4,4:6),P);
title('Transmissibility');
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
#+end_src
#+begin_src matlab
figure
bodemag(system_dec([7 9],4:6),CL_dec([7 9],4:6),CL_cen([7 9],4:6),CL_svd([7 9],4:6),CL_svd_real([7 9],4:6),P);
title('Transmissibility from half sum and half difference in the X direction');
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
#+end_src
#+begin_src matlab
figure
bodemag(system_dec([8 10],4:6),CL_dec([8 10],4:6),CL_cen([8 10],4:6),CL_svd([8 10],4:6),CL_svd_real([8 10],4:6),P);
title('Transmissibility from half sum and half difference in the Z direction');
legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
#+end_src
*** Greshgorin radius
#+begin_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));
for t = 1:length(w)
x1(t) = (abs(system_dec_freq(1,2,t))+abs(system_dec_freq(1,3,t)))/abs(system_dec_freq(1,1,t));
z1(t) = (abs(system_dec_freq(2,1,t))+abs(system_dec_freq(2,3,t)))/abs(system_dec_freq(2,2,t));
x2(t) = (abs(system_dec_freq(3,1,t))+abs(system_dec_freq(3,2,t)))/abs(system_dec_freq(3,3,t));
system_svd = pinv(Ureal)*system_dec_freq(1:4,1:3,t)*pinv(Vreal');
S1(t) = (abs(system_svd(1,2))+abs(system_svd(1,3)))/abs(system_svd(1,1));
S2(t) = (abs(system_svd(2,1))+abs(system_svd(2,3)))/abs(system_svd(2,2));
S2(t) = (abs(system_svd(3,1))+abs(system_svd(3,2)))/abs(system_svd(3,3));
end
limit = 0.5*ones(1,length(w));
#+end_src
#+begin_src matlab
figure
loglog(w./(2*pi),x1,w./(2*pi),z1,w./(2*pi),x2,w./(2*pi),limit,'--');
legend('x_1','z_1','x_2','Limit');
xlabel('Frequency [Hz]');
ylabel('Greshgorin radius [-]');
#+end_src
#+begin_src matlab
figure
loglog(w./(2*pi),S1,w./(2*pi),S2,w./(2*pi),S3,w./(2*pi),limit,'--');
legend('S1','S2','S3','Limit');
xlabel('Frequency [Hz]');
ylabel('Greshgorin radius [-]');
% set(gcf,'color','w')
#+end_src
*** Injecting ground motion in the system to have the output
#+begin_src matlab
Fr = logspace(-2,3,1e3);
w=2*pi*Fr*1i;
%fit of the ground motion data in m/s^2/rtHz
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-7 4e-7 2e-6 1e-6 5e-7 5e-7 5e-7 1e-6 1e-5 3.5e-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-7 7e-7 7e-7 1e-6 1.2e-6 1.5e-6 1e-6 9e-7 7e-7 7e-7 7e-7 1e-6 2e-6 1e-5 3e-5];
n_ground_x = interp1(Fr_ground_x,n_ground_x1,Fr,'linear');
n_ground_v = interp1(Fr_ground_v,n_ground_v1,Fr,'linear');
% figure
% loglog(Fr,abs(n_ground_v),Fr_ground_v,n_ground_v1,'*');
% xlabel('Frequency [Hz]');ylabel('ASD [m/s^2 /rtHz]');
% return
%converting into PSD
n_ground_x = (n_ground_x).^2;
n_ground_v = (n_ground_v).^2;
%Injecting ground motion in the system and getting the outputs
system_dec_f = (freqresp(system_dec,abs(w)));
PHI = zeros(size(Fr,2),12,12);
for p = 1:size(Fr,2)
Sw=zeros(6,6);
Iact = zeros(3,3);
Sw(4,4) = n_ground_x(p);
Sw(5,5) = n_ground_v(p);
Sw(6,6) = n_ground_v(p);
Sw(1:3,1:3) = Iact;
PHI(p,:,:) = (system_dec_f(:,:,p))*Sw(:,:)*(system_dec_f(:,:,p))';
end
x1 = PHI(:,1,1);
z1 = PHI(:,2,2);
x2 = PHI(:,3,3);
z2 = PHI(:,4,4);
wx = PHI(:,5,5);
wz = PHI(:,6,6);
x12 = PHI(:,1,3);
z12 = PHI(:,2,4);
PHIwx = PHI(:,1,5);
PHIwz = PHI(:,2,6);
xsum = PHI(:,7,7);
zsum = PHI(:,8,8);
xdelta = PHI(:,9,9);
zdelta = PHI(:,10,10);
rot = PHI(:,11,11);
#+end_src
* Gravimeter - Functions
:PROPERTIES:
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
** =align=
:PROPERTIES:
:header-args:matlab+: :tangle gravimeter/align.m
:END:
<<sec:align>>
This Matlab function is accessible [[file:gravimeter/align.m][here]].
#+begin_src matlab
function [A] = align(V)
%A!ALIGN(V) returns a constat matrix A which is the real alignment of the
%INVERSE of the complex input matrix V
%from Mohit slides
if (nargin ==0) || (nargin > 1)
disp('usage: mat_inv_real = align(mat)')
return
end
D = pinv(real(V'*V));
A = D*real(V'*diag(exp(1i * angle(diag(V*D*V.'))/2)));
end
#+end_src
** =pzmap_testCL=
:PROPERTIES:
:header-args:matlab+: :tangle gravimeter/pzmap_testCL.m
:END:
<<sec:pzmap_testCL>>
This Matlab function is accessible [[file:gravimeter/pzmap_testCL.m][here]].
#+begin_src matlab
function [] = pzmap_testCL(system,H,gain,feedin,feedout)
% evaluate and plot the pole-zero map for the closed loop system for
% different values of the gain
[~, n] = size(gain);
[m1, n1, ~] = size(H);
[~,n2] = size(feedin);
figure
for i = 1:n
% if n1 == n2
system_CL = feedback(system,gain(i)*H,feedin,feedout);
[P,Z] = pzmap(system_CL);
plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on
xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');
% clear P Z
% else
% system_CL = feedback(system,gain(i)*H(:,1+(i-1)*m1:m1+(i-1)*m1),feedin,feedout);
%
% [P,Z] = pzmap(system_CL);
% plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on
% xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');
% clear P Z
% end
end
str = {strcat('gain = ' , num2str(gain(1)))}; % at the end of first loop, z being loop output
str = [str , strcat('gain = ' , num2str(gain(1)))]; % after 2nd loop
for i = 2:n
str = [str , strcat('gain = ' , num2str(gain(i)))]; % after 2nd loop
str = [str , strcat('gain = ' , num2str(gain(i)))]; % after 2nd loop
end
legend(str{:})
end
#+end_src
* Stewart Platform - Simscape Model
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
** Jacobian
First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
#+begin_src matlab
open('stewart_platform/drone_platform_jacobian.slx');
#+end_src
#+begin_src matlab
sim('drone_platform_jacobian');
#+end_src
#+begin_src matlab
Aa = [a1.Data(1,:);
a2.Data(1,:);
a3.Data(1,:);
a4.Data(1,:);
a5.Data(1,:);
a6.Data(1,:)]';
Ab = [b1.Data(1,:);
b2.Data(1,:);
b3.Data(1,:);
b4.Data(1,:);
b5.Data(1,:);
b6.Data(1,:)]';
As = (Ab - Aa)./vecnorm(Ab - Aa);
l = vecnorm(Ab - Aa)';
J = [As' , cross(Ab, As)'];
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
#+end_src
** Simscape Model
#+begin_src matlab
open('stewart_platform/drone_platform.slx');
#+end_src
Definition of spring parameters
#+begin_src matlab
kx = 50; % [N/m]
ky = 50;
kz = 50;
cx = 0.025; % [Nm/rad]
cy = 0.025;
cz = 0.025;
#+end_src
We load the Jacobian.
#+begin_src matlab
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
#+end_src
** Identification of the plant
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
#+begin_src matlab
%% Name of the Simulink File
mdl = 'drone_platform';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
#+end_src
There are 24 states (6dof for the bottom platform + 6dof for the top platform).
#+begin_src matlab :results output replace
size(G)
#+end_src
#+RESULTS:
: State-space model with 6 outputs, 12 inputs, and 24 states.
#+begin_src matlab
% G = G*blkdiag(inv(J), eye(6));
% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
#+end_src
Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
#+begin_src matlab
Gx = G*blkdiag(eye(6), inv(J'));
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
Gl = J*G;
Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
#+end_src
** Obtained Dynamics
#+begin_src matlab :exports none
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_translations.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:stewart_platform_translations
#+caption: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform
#+RESULTS:
[[file:figs/stewart_platform_translations.png]]
#+begin_src matlab :exports none
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_rotations.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:stewart_platform_rotations
#+caption: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform
#+RESULTS:
[[file:figs/stewart_platform_rotations.png]]
#+begin_src matlab :exports none
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
end
for out_i = 1:5
for in_i = i+1:6
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', out_i), sprintf('F%i', in_i)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for ch_i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_legs.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:stewart_platform_legs
#+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
#+RESULTS:
[[file:figs/stewart_platform_legs.png]]
#+begin_src matlab :exports none
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - Translations'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - Rotations'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_transmissibility.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:stewart_platform_transmissibility
#+caption: Transmissibility
#+RESULTS:
[[file:figs/stewart_platform_transmissibility.png]]
** Real Approximation of $G$ at the decoupling frequency
Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
#+begin_src matlab
wc = 2*pi*20; % Decoupling frequency [rad/s]
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
H1 = evalfr(Gc, j*wc);
#+end_src
The real approximation is computed as follows:
#+begin_src matlab
D = pinv(real(H1'*H1));
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
#+end_src
** Verification of the decoupling using the "Gershgorin Radii"
First, the Singular Value Decomposition of $H_1$ is performed:
\[ H_1 = U \Sigma V^H \]
#+begin_src matlab
[U,S,V] = svd(H1);
#+end_src
Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
\[ G_d(s) = U^T G_c(s) V \]
This is computed over the following frequencies.
#+begin_src matlab
freqs = logspace(-2, 2, 1000); % [Hz]
#+end_src
Gershgorin Radii for the coupled plant:
#+begin_src matlab
Gr_coupled = zeros(length(freqs), size(Gc,2));
H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
for out_i = 1:size(Gc,2)
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end
#+end_src
Gershgorin Radii for the decoupled plant using SVD:
#+begin_src matlab
Gd = U'*Gc*V;
Gr_decoupled = zeros(length(freqs), size(Gd,2));
H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
for out_i = 1:size(Gd,2)
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end
#+end_src
Gershgorin Radii for the decoupled plant using the Jacobian:
#+begin_src matlab
Gj = Gc*inv(J');
Gr_jacobian = zeros(length(freqs), size(Gj,2));
H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
for out_i = 1:size(Gj,2)
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end
#+end_src
#+begin_src matlab :exports results
figure;
hold on;
plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
for in_i = 2:6
set(gca,'ColorOrderIndex',1)
plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2)
plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',3)
plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
end
plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
hold off;
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
legend('location', 'northeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/simscape_model_gershgorin_radii.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:simscape_model_gershgorin_radii
#+caption: Gershgorin Radii of the Coupled and Decoupled plants
#+RESULTS:
[[file:figs/simscape_model_gershgorin_radii.png]]
** Decoupled Plant
Let's see the bode plot of the decoupled plant $G_d(s)$.
\[ G_d(s) = U^T G_c(s) V \]
#+begin_src matlab :exports results
freqs = logspace(-1, 2, 1000);
figure;
hold on;
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
end
for in_i = 1:5
for out_i = in_i+1:6
plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); xlabel('Frequency [Hz]');
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/simscape_model_decoupled_plant_svd.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:simscape_model_decoupled_plant_svd
#+caption: Decoupled Plant using SVD
#+RESULTS:
[[file:figs/simscape_model_decoupled_plant_svd.png]]
#+begin_src matlab :exports results
freqs = logspace(-1, 2, 1000);
figure;
hold on;
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gj(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
end
for in_i = 1:5
for out_i = in_i+1:6
plot(freqs, abs(squeeze(freqresp(Gj(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); xlabel('Frequency [Hz]');
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/simscape_model_decoupled_plant_jacobian.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:simscape_model_decoupled_plant_jacobian
#+caption: Decoupled Plant using the Jacobian
#+RESULTS:
[[file:figs/simscape_model_decoupled_plant_jacobian.png]]
** Diagonal Controller
The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
#+begin_src matlab
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
C_g = 50; % DC Gain
K = eye(6)*C_g/(s+wc);
#+end_src
** Centralized Control
The control diagram for the centralized control is shown below.
The controller $K_c$ is "working" in an cartesian frame.
The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.
#+begin_src latex :file centralized_control.pdf
\begin{tikzpicture}
\node[block={2cm}{1.5cm}] (G) {$G$};
\node[block, below right=0.6 and -0.5 of G] (K) {$K_c$};
\node[block, below left= 0.6 and -0.5 of G] (J) {$J^{-T}$};
% Inputs of the controllers
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
% Connections and labels
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
\draw[->] (G.east) -- ++(2.0, 0) node[above left]{$a$};
\draw[->] ($(G.east)+(1.4, 0)$)node[branch]{} |- (K.east);
\draw[->] (K.west) -- (J.east) node[above right]{$\mathcal{F}$};
\draw[->] (J.west) -- ++(-0.6, 0) |- (inputu) node[above left]{$\tau$};
\end{tikzpicture}
#+end_src
#+RESULTS:
[[file:figs/centralized_control.png]]
#+begin_src matlab
G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
#+end_src
** SVD Control
The SVD control architecture is shown below.
The matrices $U$ and $V$ are used to decoupled the plant $G$.
#+begin_src latex :file svd_control.pdf
\begin{tikzpicture}
\node[block={2cm}{1.5cm}] (G) {$G$};
\node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$};
\node[block, below=0.6 of G] (K) {$K_{\text{SVD}}$};
\node[block, below left= 0.6 and 0 of G] (V) {$V^{-T}$};
% Inputs of the controllers
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
% Connections and labels
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
\draw[->] (G.east) -- ++(2.5, 0) node[above left]{$a$};
\draw[->] ($(G.east)+(2.0, 0)$) node[branch]{} |- (U.east);
\draw[->] (U.west) -- (K.east);
\draw[->] (K.west) -- (V.east);
\draw[->] (V.west) -- ++(-0.6, 0) |- (inputu) node[above left]{$\tau$};
\end{tikzpicture}
#+end_src
#+RESULTS:
[[file:figs/svd_control.png]]
SVD Control
#+begin_src matlab
G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
#+end_src
** Results
Let's first verify the stability of the closed-loop systems:
#+begin_src matlab :results output replace text
isstable(G_cen)
#+end_src
#+RESULTS:
: ans =
: logical
: 1
#+begin_src matlab :results output replace text
isstable(G_svd)
#+end_src
#+RESULTS:
: ans =
: logical
: 1
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
#+begin_src matlab :exports results
freqs = logspace(-3, 3, 1000);
figure
ax1 = subplot(2, 3, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_x/D_{w,x}$'); xlabel('Frequency [Hz]');
legend('location', 'southwest');
ax2 = subplot(2, 3, 2);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_y/D_{w,y}$'); xlabel('Frequency [Hz]');
ax3 = subplot(2, 3, 3);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_z/D_{w,z}$'); xlabel('Frequency [Hz]');
ax4 = subplot(2, 3, 4);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_x/R_{w,x}$'); xlabel('Frequency [Hz]');
ax5 = subplot(2, 3, 5);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
ax6 = subplot(2, 3, 6);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_simscape_cl_transmissibility.pdf', 'width', 1600, 'height', 'full');
#+end_src
#+name: fig:stewart_platform_simscape_cl_transmissibility
#+caption: Obtained Transmissibility
#+RESULTS:
[[file:figs/stewart_platform_simscape_cl_transmissibility.png]]
* Stewart Platform - Analytical Model :noexport:
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
%% Bode plot options
opts = bodeoptions('cstprefs');
opts.FreqUnits = 'Hz';
opts.MagUnits = 'abs';
opts.MagScale = 'log';
opts.PhaseWrapping = 'on';
opts.xlim = [1 1000];
#+end_src
** Characteristics
#+begin_src matlab
L = 0.055;
Zc = 0;
m = 0.2;
k = 1e3;
c = 2*0.1*sqrt(k*m);
Rx = 0.04;
Rz = 0.04;
Ix = m*Rx^2;
Iy = m*Rx^2;
Iz = m*Rz^2;
#+end_src
** Mass Matrix
#+begin_src matlab
M = m*[1 0 0 0 Zc 0;
0 1 0 -Zc 0 0;
0 0 1 0 0 0;
0 -Zc 0 Rx^2+Zc^2 0 0;
Zc 0 0 0 Rx^2+Zc^2 0;
0 0 0 0 0 Rz^2];
#+end_src
** Jacobian Matrix
#+begin_src matlab
Bj=1/sqrt(6)*[ 1 1 -2 1 1 -2;
sqrt(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;
sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);
0 0 L L -L -L;
-L*2/sqrt(3) -L*2/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3);
L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2)];
#+end_src
** Stifnness matrix and Damping matrix
#+begin_src matlab
kv = k/3; % [N/m]
kh = 0.5*k/3; % [N/m]
K = diag([3*kh,3*kh,3*kv,3*kv*Rx^2/2,3*kv*Rx^2/2,3*kh*Rx^2]); % Stiffness Matrix
C = c*K/100000; % Damping Matrix
#+end_src
** State Space System
#+begin_src matlab
A = [zeros(6) eye(6); -M\K -M\C];
Bw = [zeros(6); -eye(6)];
Bu = [zeros(6); M\Bj];
Co = [-M\K -M\C];
D = [zeros(6) M\Bj];
ST = ss(A,[Bw Bu],Co,D);
#+end_src
- OUT 1-6: 6 dof
- IN 1-6 : ground displacement in the directions of the legs
- IN 7-12: forces in the actuators.
#+begin_src matlab
ST.StateName = {'x';'y';'z';'theta_x';'theta_y';'theta_z';...
'dx';'dy';'dz';'dtheta_x';'dtheta_y';'dtheta_z'};
ST.InputName = {'w1';'w2';'w3';'w4';'w5';'w6';...
'u1';'u2';'u3';'u4';'u5';'u6'};
ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
#+end_src
** Transmissibility
#+begin_src matlab
TR=ST*[eye(6); zeros(6)];
#+end_src
#+begin_src matlab
figure
subplot(231)
bodemag(TR(1,1),opts);
subplot(232)
bodemag(TR(2,2),opts);
subplot(233)
bodemag(TR(3,3),opts);
subplot(234)
bodemag(TR(4,4),opts);
subplot(235)
bodemag(TR(5,5),opts);
subplot(236)
bodemag(TR(6,6),opts);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_analytical_transmissibility.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:stewart_platform_analytical_transmissibility
#+caption: Transmissibility
#+RESULTS:
[[file:figs/stewart_platform_analytical_transmissibility.png]]
** Real approximation of $G(j\omega)$ at decoupling frequency
#+begin_src matlab
sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs
dec_fr = 20;
H1 = evalfr(sys1,j*2*pi*dec_fr);
H2 = H1;
D = pinv(real(H2'*H2));
H1 = inv(D*real(H2'*diag(exp(j*angle(diag(H2*D*H2.'))/2)))) ;
[U,S,V] = svd(H1);
wf = logspace(-1,2,1000);
for i = 1:length(wf)
H = abs(evalfr(sys1,j*2*pi*wf(i)));
H_dec = abs(evalfr(U'*sys1*V,j*2*pi*wf(i)));
for j = 1:size(H,2)
g_r1(i,j) = (sum(H(j,:))-H(j,j))/H(j,j);
g_r2(i,j) = (sum(H_dec(j,:))-H_dec(j,j))/H_dec(j,j);
% keyboard
end
g_lim(i) = 0.5;
end
#+end_src
** Coupled and Decoupled Plant "Gershgorin Radii"
#+begin_src matlab
figure;
title('Coupled plant')
loglog(wf,g_r1(:,1),wf,g_r1(:,2),wf,g_r1(:,3),wf,g_r1(:,4),wf,g_r1(:,5),wf,g_r1(:,6),wf,g_lim,'--');
legend('$a_x$','$a_y$','$a_z$','$\theta_x$','$\theta_y$','$\theta_z$','Limit');
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/gershorin_raddii_coupled_analytical.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:gershorin_raddii_coupled_analytical
#+caption: Gershorin Raddi for the coupled plant
#+RESULTS:
[[file:figs/gershorin_raddii_coupled_analytical.png]]
#+begin_src matlab
figure;
title('Decoupled plant (10 Hz)')
loglog(wf,g_r2(:,1),wf,g_r2(:,2),wf,g_r2(:,3),wf,g_r2(:,4),wf,g_r2(:,5),wf,g_r2(:,6),wf,g_lim,'--');
legend('$S_1$','$S_2$','$S_3$','$S_4$','$S_5$','$S_6$','Limit');
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/gershorin_raddii_decoupled_analytical.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:gershorin_raddii_decoupled_analytical
#+caption: Gershorin Raddi for the decoupled plant
#+RESULTS:
[[file:figs/gershorin_raddii_decoupled_analytical.png]]
** Decoupled Plant
#+begin_src matlab
figure;
bodemag(U'*sys1*V,opts)
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_analytical_decoupled_plant.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:stewart_platform_analytical_decoupled_plant
#+caption: Decoupled Plant
#+RESULTS:
[[file:figs/stewart_platform_analytical_decoupled_plant.png]]
** Controller
#+begin_src matlab
fc = 2*pi*0.1; % Crossover Frequency [rad/s]
c_gain = 50; %
cont = eye(6)*c_gain/(s+fc);
#+end_src
** Closed Loop System
#+begin_src matlab
FEEDIN = [7:12]; % Input of controller
FEEDOUT = [1:6]; % Output of controller
#+end_src
Centralized Control
#+begin_src matlab
STcen = feedback(ST, inv(Bj)*cont, FEEDIN, FEEDOUT);
TRcen = STcen*[eye(6); zeros(6)];
#+end_src
SVD Control
#+begin_src matlab
STsvd = feedback(ST, pinv(V')*cont*pinv(U), FEEDIN, FEEDOUT);
TRsvd = STsvd*[eye(6); zeros(6)];
#+end_src
** Results
#+begin_src matlab
figure
subplot(231)
bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
legend('OL','Centralized','SVD')
subplot(232)
bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
legend('OL','Centralized','SVD')
subplot(233)
bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
legend('OL','Centralized','SVD')
subplot(234)
bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
legend('OL','Centralized','SVD')
subplot(235)
bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
legend('OL','Centralized','SVD')
subplot(236)
bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
legend('OL','Centralized','SVD')
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_analytical_svd_cen_comp.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:stewart_platform_analytical_svd_cen_comp
#+caption: Comparison of the obtained transmissibility for the centralized control and the SVD control
#+RESULTS:
[[file:figs/stewart_platform_analytical_svd_cen_comp.png]]
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