svd-control/index.org

#+TITLE: SVD Control
:DRAWER:
#+STARTUP: overview

#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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#+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}")
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#+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;
  %P.PhaseVisible = 'off';
#+end_src

Frequency vector.
#+begin_src matlab
  w = 2*pi*logspace(-1,2,1000); % [rad/s]
#+end_src

*** generation of the state space model
#+begin_src matlab
  M = [m 0 0
       0 m 0
       0 0 I];

  %Jacobian of the bottom sensor
  Js1 = [1 0  h/2
         0 1 -l/2];
  %Jacobian of the top sensor
  Js2 = [1 0 -h/2
         0 1  0];

  %Jacobian of the actuators
  Ja = [1 0  ha   % Left horizontal actuator
        0 1 -la   % Left vertical actuator
        0 1  la]; % Right vertical actuator
  Jta = Ja';
  K = k*Jta*Ja;
  C = c*Jta*Ja;

  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)];

  % BB = [zeros(3,3)
  %     M\Jta ];
  %
  % CC = [Ja zeros(3)];
  % DD = zeros(3,3);

  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)];

  system_dec = ss(AA,BB,CC,DD);
#+end_src

- 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 :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]]