svd-control/index.org

SVD Control

• Gravimeter - Simscape Model
  • Introduction
  • Simscape Model - Parameters
  • System Identification - Without Gravity
  • System Identification - With Gravity
  • Analytical Model
    • Parameters
    • Generation of the State Space Model
    • Comparison with the Simscape Model
    • Analysis
    • Control Section
    • Greshgorin radius
    • Injecting ground motion in the system to have the output
• Gravimeter - Functions
  • align
  • pzmap_testCL
• Stewart Platform - Simscape Model
  • Introduction
  • Simscape Model - Parameters
  • Identification of the plant
  • Physical Decoupling using the Jacobian
  • Real Approximation of $G$ at the decoupling frequency
  • SVD Decoupling
  • Verification of the decoupling using the "Gershgorin Radii"
  • Obtained Decoupled Plants
  • Diagonal Controller
  • Closed-Loop system Performances

Gravimeter - Simscape Model

Introduction

Model of the gravimeter

Simscape Model - Parameters

  open('gravimeter.slx')

Parameters

  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]

System Identification - Without Gravity

  %% 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'};

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

The plant as 6 states as expected (2 translations + 1 rotation)

  size(G)

State-space model with 4 outputs, 3 inputs, and 6 states.

Open Loop Transfer Function from 3 Actuators to 4 Accelerometers

System Identification - With Gravity

  g = 9.80665; % Gravity [m/s2]

  Gg = linearize(mdl, io);
  Gg.InputName  = {'F1', 'F2', 'F3'};
  Gg.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};

We can now see that the system is unstable due to gravity.

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

Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity

Analytical Model

Parameters

Bode options.

  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;

Frequency vector.

  w = 2*pi*logspace(-1,2,1000); % [rad/s]

Generation of the State Space Model

Mass matrix

  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';

Stiffness and Damping matrices

  K = k*Jta*Ja;
  C = c*Jta*Ja;

State Space Matrices

  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)];

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

  system_dec = ss(AA,BB,CC,DD);

  size(system_dec)

State-space model with 12 outputs, 6 inputs, and 6 states.

Comparison with the Simscape Model

Comparison of the analytical and the Simscape models

Analysis

  % figure
  % bode(system_dec,P);
  % return

  %% 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');

Control Section

  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');

  % figure
  % bode(OL_dec,w,P);title('OL Decentralized');
  % figure
  % bode(OL_cen,w,P);title('OL Centralized');

  figure
  bode(g*system_dec(1:4,1:3),w,P);
  title('gain * Plant');

  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')

  figure
  bode(system_dec(1:4,1:3),pinv(U)*system_dec(1:4,1:3)*pinv(V'),P);

  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]);

  pzmap_testCL(system_dec,H_dec,g,[1 2 3],[1 2 3 4])
  title('Decentralized control');

  pzmap_testCL(system_dec,H_cen,g,[1 2 3],[1 2 3 4])
  title('Centralized control');

  pzmap_testCL(system_dec,H_svd,100,[1 2 3],[1 2 3 4])
  title('SVD control');

  pzmap_testCL(system_dec,H_svd_real,100,[1 2 3],[1 2 3 4])
  title('Real approximation SVD control');

  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.');

  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.');

  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.');

  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.');

Greshgorin radius

  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));

  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 [-]');

  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')

Injecting ground motion in the system to have the output

  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);

Gravimeter - Functions

align

<<sec:align>>

This Matlab function is accessible here.

  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

pzmap_testCL

<<sec:pzmap_testCL>>

This Matlab function is accessible here.

  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

Stewart Platform - Simscape Model

Introduction   ignore

In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure fig:SP_assembly.

Some notes about the system:

• 6 voice coils actuators are used to control the motion of the top platform.
• the motion of the top platform is measured with a 6-axis inertial unit (3 acceleration + 3 angular accelerations)
• the control objective is to isolate the top platform from vibrations coming from the bottom platform

Stewart Platform CAD View

The analysis of the SVD control applied to the Stewart platform is performed in the following sections:

• Section sec:stewart_simscape: The parameters of the Simscape model of the Stewart platform are defined
• Section sec:stewart_identification: The plant is identified from the Simscape model and the system coupling is shown
• Section sec:stewart_jacobian_decoupling: The plant is first decoupled using the Jacobian
• Section sec:stewart_real_approx: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)
• Section sec:stewart_svd_decoupling: The decoupling is performed thanks to the SVD
• Section sec:comp_decoupling: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
• Section sec:stewart_decoupled_plant: The dynamics of the decoupled plants are shown
• Section sec:stewart_diagonal_control: A diagonal controller is defined to control the decoupled plant
• Section sec:stewart_closed_loop_results: Finally, the closed loop system properties are studied

Simscape Model - Parameters

<<sec:stewart_simscape>>

  open('drone_platform.slx');

Definition of spring parameters:

  kx = 0.5*1e3/3; % [N/m]
  ky = 0.5*1e3/3;
  kz = 1e3/3;

  cx = 0.025; % [Nm/rad]
  cy = 0.025;
  cz = 0.025;

Gravity:

  g = 0;

We load the Jacobian (previously computed from the geometry):

  load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');

We initialize other parameters:

  U = eye(6);
  V = eye(6);
  Kc = tf(zeros(6));

General view of the Simscape Model

Simscape model of the Stewart platform

Identification of the plant

<<sec:stewart_identification>>

The plant shown in Figure fig:stewart_platform_plant is identified from the Simscape model.

The inputs are:

• $D_w$ translation and rotation of the bottom platform (with respect to the center of mass of the top platform)
• $\tau$ the 6 forces applied by the voice coils

The outputs are the 6 accelerations measured by the inertial unit.

Considered plant $\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}$. $D_w$ is the translation/rotation of the support, $\tau$ the actuator forces, $a$ the acceleration/angular acceleration of the top platform

  %% 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; % Ground Motion
  io(io_i) = linio([mdl, '/V-T'],             1, 'openinput');  io_i = io_i + 1; % Actuator Forces
  io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration

  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'};

  % Plant
  Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
  % Disturbance dynamics
  Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});

There are 24 states (6dof for the bottom platform + 6dof for the top platform).

  size(G)

State-space model with 6 outputs, 12 inputs, and 24 states.

The elements of the transfer matrix $\bm{G}$ corresponding to the transfer function from actuator forces $\tau$ to the measured acceleration $a$ are shown in Figure fig:stewart_platform_coupled_plant.

One can easily see that the system is strongly coupled.

Magnitude of all 36 elements of the transfer function matrix $G_u$

Physical Decoupling using the Jacobian

<<sec:stewart_jacobian_decoupling>> Consider the control architecture shown in Figure fig:plant_decouple_jacobian. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.

Decoupled plant $\bm{G}_x$ using the Jacobian matrix $J$

We define a new plant: \[ G_x(s) = G(s) J^{-T} \]

$G_x(s)$ correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.

  Gx = Gu*inv(J');
  Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

Real Approximation of $G$ at the decoupling frequency

<<sec:stewart_real_approx>>

Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_u(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.

  wc = 2*pi*30; % Decoupling frequency [rad/s]

  H1 = evalfr(Gu, j*wc);

The real approximation is computed as follows:

  D = pinv(real(H1'*H1));
  H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));

4.4	-2.1	-2.1	4.4	-2.4	-2.4
-0.2	-3.9	3.9	0.2	-3.8	3.8
3.4	3.4	3.4	3.4	3.4	3.4
-367.1	-323.8	323.8	367.1	43.3	-43.3
-162.0	-237.0	-237.0	-162.0	398.9	398.9
220.6	-220.6	220.6	-220.6	220.6	-220.6

Real approximate of $G$ at the decoupling frequency $\omega_c$

Note that the plant $G_u$ at $\omega_c$ is already an almost real matrix. This can be seen on the Bode plots where the phase is close to 1. This can be verified below where only the real value of $G_u(\omega_c)$ is shown

4.4	-2.1	-2.1	4.4	-2.4	-2.4
-0.2	-3.9	3.9	0.2	-3.8	3.8
3.4	3.4	3.4	3.4	3.4	3.4
-367.1	-323.8	323.8	367.1	43.3	-43.3
-162.0	-237.0	-237.0	-162.0	398.9	398.9
220.6	-220.6	220.6	-220.6	220.6	-220.6

SVD Decoupling

<<sec:stewart_svd_decoupling>>

First, the Singular Value Decomposition of $H_1$ is performed: \[ H_1 = U \Sigma V^H \]

  [U,~,V] = svd(H1);

The obtained matrices $U$ and $V$ are used to decouple the system as shown in Figure fig:plant_decouple_svd.

Decoupled plant $\bm{G}_{SVD}$ using the Singular Value Decomposition

The decoupled plant is then: \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]

  Gsvd = inv(U)*Gu*inv(V');

Verification of the decoupling using the "Gershgorin Radii"

<<sec:comp_decoupling>>

The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$:

The "Gershgorin Radii" of a matrix $S$ is defined by: \[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]

This is computed over the following frequencies.

  freqs = logspace(-2, 2, 1000); % [Hz]

Gershgorin Radii of the Coupled and Decoupled plants

Obtained Decoupled Plants

<<sec:stewart_decoupled_plant>>

The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure fig:simscape_model_decoupled_plant_svd.

Decoupled Plant using SVD

Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure fig:simscape_model_decoupled_plant_jacobian.

Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

Diagonal Controller

<<sec:stewart_diagonal_control>> The control diagram for the centralized control is shown in Figure fig:centralized_control.

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.

Control Diagram for the Centralized control

The SVD control architecture is shown in Figure fig:svd_control. The matrices $U$ and $V$ are used to decoupled the plant $G$.

Control Diagram for the SVD control

We choose the controller to be a low pass filter: \[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]

$G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$

  wc = 2*pi*80;  % Crossover Frequency [rad/s]
  w0 = 2*pi*0.1; % Controller Pole [rad/s]

  K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
  L_cen = K_cen*Gx;
  G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);

  K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
  L_svd = K_svd*Gsvd;
  G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);

The obtained diagonal elements of the loop gains are shown in Figure fig:stewart_comp_loop_gain_diagonal.

Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

Closed-Loop system Performances

<<sec:stewart_closed_loop_results>>

Let's first verify the stability of the closed-loop systems:

  isstable(G_cen)

ans =
  logical
   1

  isstable(G_svd)

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.

Obtained Transmissibility