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SVD Control

Table of Contents

1 Gravimeter - Simscape Model

1.1 Introduction

gravimeter_model.png

Figure 1: Model of the gravimeter

1.2 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]

1.3 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_tf.png

Figure 2: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers

1.4 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_tf_g.png

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

1.5 Analytical Model

1.5.1 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]

1.5.2 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.

1.5.3 Comparison with the Simscape Model

gravimeter_analytical_system_open_loop_models.png

Figure 4: Comparison of the analytical and the Simscape models

1.5.4 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');

1.5.5 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.');

1.5.6 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')

1.5.7 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);

2 Gravimeter - Functions

2.1 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

2.2 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

3 Stewart Platform - Simscape Model

In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 5.

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

SP_assembly.png

Figure 5: Stewart Platform CAD View

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

  • Section 3.1: The parameters of the Simscape model of the Stewart platform are defined
  • Section 3.2: The plant is identified from the Simscape model and the system coupling is shown
  • Section 3.3: The plant is first decoupled using the Jacobian
  • Section 3.4: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)
  • Section 3.5: The decoupling is performed thanks to the SVD
  • Section 3.6: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
  • Section 3.7: The dynamics of the decoupled plants are shown
  • Section 3.8: A diagonal controller is defined to control the decoupled plant
  • Section 3.9: Finally, the closed loop system properties are studied

3.1 Simscape Model - Parameters

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));

3.2 Identification of the plant

The plant shown in Figure 6 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.

stewart_platform_plant.png

Figure 6: Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G\end{bmatrix}\). \(D_w\) 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'};

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 7.

One can easily see that the system is strongly coupled.

stewart_platform_coupled_plant.png

Figure 7: Magnitude of all 36 elements of the transfer function matrix \(\bm{G}\)

3.3 Physical Decoupling using the Jacobian

Consider the control architecture shown in Figure 8. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.

plant_decouple_jacobian.png

Figure 8: 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 = G*blkdiag(eye(6), inv(J'));
Gx.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
                 'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

3.4 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 \(G(j\omega_c)\) 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]

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);

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))));
Table 1: Real approximate of \(G\) at the decoupling frequency \(\omega_c\)
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

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

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

3.5 SVD Decoupling

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

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

The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 9.

plant_decouple_svd.png

Figure 9: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

The decoupled plant is then: \[ G_{SVD}(s) = U^T G(s) V \]

3.6 Verification of the decoupling using the “Gershgorin Radii”

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

This is computed over the following frequencies. 

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

simscape_model_gershgorin_radii.png

Figure 10: Gershgorin Radii of the Coupled and Decoupled plants

3.7 Obtained Decoupled Plants

The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 11.

simscape_model_decoupled_plant_svd.png

Figure 11: 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 12.

simscape_model_decoupled_plant_jacobian.png

Figure 12: 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)

3.8 Diagonal Controller

The controller \(K_c\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\). 

wc = 2*pi*0.1; % Crossover Frequency [rad/s]
C_g = 50; % DC Gain

Kc = eye(6)*C_g/(s+wc);

The control diagram for the centralized control is shown in Figure 13.

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.

centralized_control.png

Figure 13: Control Diagram for the Centralized control

The feedback system is computed as shown below. 

G_cen = feedback(G, inv(J')*Kc, [7:12], [1:6]);

The SVD control architecture is shown in Figure 14. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

svd_control.png

Figure 14: Control Diagram for the SVD control

The feedback system is computed as shown below. 

G_svd = feedback(G, pinv(V')*Kc*pinv(U), [7:12], [1:6]);

3.9 Closed-Loop system Performances

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 15.

stewart_platform_simscape_cl_transmissibility.png

Figure 15: Obtained Transmissibility

Author: Dehaeze Thomas

Created: 2020-11-06 ven. 16:58