gravimeter/align.m

@@ -1,15 +0,0 @@
-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

gravimeter/pzmap_testCL.m

@@ -1,34 +0,0 @@
-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

gravimeter/script.m

@@ -1,492 +0,0 @@
-%% Clear Workspace and Close figures
-clear; close all; clc;
-
-%% Intialize Laplace variable
-s = zpk('s');
-
-addpath('gravimeter');
-
-% Simscape Model - Parameters
-
-open('gravimeter.slx')
-
-
-
-% Parameters
-
-l  = 0.5; % Length of the mass [m]
-la = 0.5; % Position of Act. [m]
-
-h  = 1.7; % 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)
-
-
-
-% #+RESULTS:
-% #+begin_example
-% pole(G)
-% ans =
-%       -0.000473481142385801 +      21.7596190728632i
-%       -0.000473481142385801 -      21.7596190728632i
-%       -7.49842879459177e-05 +       8.6593576906982i
-%       -7.49842879459177e-05 -       8.6593576906982i
-%       -5.15386867925747e-06 +      2.27025295182755i
-%       -5.15386867925747e-06 -      2.27025295182755i
-% #+end_example
-
-% The plant as 6 states as expected (2 translations + 1 rotation)
-
-size(G)
-
-
-
-% #+RESULTS:
-% : State-space model with 4 outputs, 3 inputs, and 6 states.
-
-
-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
-
-% 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)
-
-
-
-% #+RESULTS:
-% #+begin_example
-% pole(G)
-% ans =
-%           -10.9848275341276 +                     0i
-%            10.9838836405193 +                     0i
-%       -7.49855396089326e-05 +      8.65962885769976i
-%       -7.49855396089326e-05 -      8.65962885769976i
-%       -6.68819341967921e-06 +      0.83296042226902i
-%       -6.68819341967921e-06 -      0.83296042226902i
-% #+end_example
-
-
-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
-
-% Parameters
-% Control parameters
-
-g = 1e5;
-g_svd = 1e5;
-
-
-
-% System parameters
-
-w0 = 2*pi*.5; % MinusK BM1 tablle
-
-l  = 0.8; % [m]
-la = l;   % [m]
-
-h  = 1.7; % [m]
-ha = h;   % [m]
-
-m = 70; % [kg]
-
-k = 3e3; % [N/m]
-I = 10;  % [kg m^2]
-
-
-
-% 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;
-%P.PhaseVisible = 'off';
-
-
-
-% Frequency vector.
-
-w = 2*pi*logspace(-1,2,1000); % [rad/s]
-
-% generation of the state space model
-
-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/2 %Left horizontal actuator
-               %1 0 h/2 %Right horizontal actuator
-      0 1 -la/2 %Left vertical actuator
-      0 1 la/2]; %Right vertical actuator
-Jta = Ja';
-K = k*Jta*Ja;
-C = 0.06*k*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);
-
-
-
-% - 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
-
-
-size(system_dec)
-
-% Comparison with the Simscape Model
-
-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
-
-% 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);

index.org

@@ -44,9 +44,6 @@
 :END:
 
 * Gravimeter - Simscape Model
-:PROPERTIES:
-:header-args:matlab+: :tangle gravimeter/script.m
-:END:
 ** 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>>
@@ -56,35 +53,11 @@
   <<matlab-init>>
 #+end_src
 
-#+begin_src matlab
+** Simulink
-  addpath('gravimeter');
-#+end_src
-
-** Simscape Model - Parameters
 #+begin_src matlab
   open('gravimeter.slx')
 #+end_src
 
-Parameters
-#+begin_src matlab
-  l  = 0.5; % Length of the mass [m]
-  la = 0.5; % Position of Act. [m]
-
-  h  = 1.7; % 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';
@@ -104,23 +77,8 @@ Parameters
   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 =
-      -0.000473481142385801 +      21.7596190728632i
-      -0.000473481142385801 -      21.7596190728632i
-      -7.49842879459177e-05 +       8.6593576906982i
-      -7.49842879459177e-05 -       8.6593576906982i
-      -5.15386867925747e-06 +      2.27025295182755i
-      -5.15386867925747e-06 -      2.27025295182755i
-#+end_example
-
 The plant as 6 states as expected (2 translations + 1 rotation)
+
 #+begin_src matlab :results output replace
   size(G)
 #+end_src
@@ -150,108 +108,181 @@ The plant as 6 states as expected (2 translations + 1 rotation)
 #+RESULTS:
 [[file:figs/open_loop_tf.png]]
 
-** System Identification - With Gravity
+** Matlab Code                                                     :noexport:
 #+begin_src matlab
-  g = 9.80665; % Gravity [m/s2]
+  clc;
-#+end_src
+  % close all
 
-#+begin_src matlab
+  g = 100000;
-  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(G)
-ans =
-          -10.9848275341276 +                     0i
-           10.9838836405193 +                     0i
-      -7.49855396089326e-05 +      8.65962885769976i
-      -7.49855396089326e-05 -      8.65962885769976i
-      -6.68819341967921e-06 +      0.83296042226902i
-      -6.68819341967921e-06 -      0.83296042226902i
-#+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
-Control parameters
-#+begin_src matlab
-  g = 1e5;
-  g_svd = 1e5;
-#+end_src
-
-System parameters
-#+begin_src matlab
   w0 = 2*pi*.5; % MinusK BM1 tablle
+  l = 0.5; %[m]
+  la = 1; %[m]
+  h = 1.7; %[m]
+  ha = 1.7;% %[m]
+  m = 400; %[kg]
+  k = 15e3;%[N/m]
+  kv = k;
+  kh = 15e3;
+  I = 115;%[kg m^2]
+          % c = 0.06;
+          % l = 0.4719; %[m]
+          % la = .477; %[m]
+          % h = 1.8973; %[m]
+          % ha = 1.9060;% %[m]
+          % m = 98.1421; %[kg]
+          % k = 14512;%[N/m]
+          % I = 28.5372;%[kg m^2]
+  cv = 0.03;
+  ch = 0.03;
 
-  l  = 0.8; % [m]
+  %% System definition
-  la = l;   % [m]
+  [Fr, x1, z1, x2, z2, wx, wz, x12, z12, PHIwx, PHIwz,xsum,zsum,xdelta,zdelta,rot]...
+      = modelGeneration(m,I,k,h,ha,l,la,cv,ch,kv,kh);
 
-  h  = 1.7; % [m]
+  %% Bode options
-  ha = h;   % [m]
-
-  m = 70; % [kg]
- 
-  k = 3e3; % [N/m]
-  I = 10;  % [kg m^2]
-#+end_src
-
-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';
+  w = 2*pi*logspace(-1,2,1000);
+
+  %% curves points
+  % slide 4
+  F_sl4 = [2e-1 4e-1 7e-1 1 2 3 5];
+  Amp_sl4 = [ 1 2 4 2.5 1 7e-1 7e-1];
+  F_sl4_phase = [2e-1 4e-1 7e-1 1 ];
+  Phase_sl4 = (180/pi).*[0 0 -0.5 -1.7];
+
+  %slide 6
+  F_sl6 = [2e-1 4e-1 1 2 3 5];
+  Amp_sl6 = [1 1 6e-1 2e-1 3e-1 3e-1];
+  F_sl6_phase = [2e-1 4e-1 1 ];
+  Phase_sl6 = (180/pi).*[0 0 0 ];
+
+  %slide 9
+  F_sl9 = [2.5e-1 4e-1 6e-1 1 1.7 2.2 3 5 10];
+  Amp_sl9 = [3 6 1 5e-1 1 2 7e-1 2.5e-1 7e-2];
+  Phase_sl9 = (180/pi)*[0 -1 -pi 0 -1 -1.5 -pi -pi -pi];
+
+  % slide 14
+  F_sl14 = [ 2e-1 4e-1 6e-1 8e-1 1 2 3 5 10];
+  Amp_sl14 = [9e-1 1.5 1.2 0.35 .3 1.2 .3 .1 5e-2];
+  F_sl14_phase = [ 2e-1 4e-1 6e-1 8e-1 ];
+  Phase_sl14 = (180/pi).*[0 0 -1.7 -2];
+
+  %rotation
+  F_rot = [1e-1 2e-1 4e-1 5e-1 7e-1 1 2 3 6.5 10 20];
+  Amp_rot = [7e-8 2.2e-7 3e-7 1e-7 2e-8 9e-9 3e-8 9e-9 1e-9 4e-10 8e-11];
+
+  %% Plots
+  % %slide 3
+  % figure
+  % loglog(Fr,abs(x2).^.5,Fr,abs(x1).^.5,Fr,abs(xsum).^.5,Fr,abs(xdelta).^.5)
+  % xlabel('Frequency [Hz]');ylabel('Acceleration [m/s^2/rtHz]')
+  % legend('Top sensor','Bottom sensor','Half sum','Half difference');
+  % title('Horizontal')
+  % xlim([7e-2 1e1]);
+
+  %slide 4
+  figure
+  subplot 211
+  loglog(Fr, abs(x12)./abs(x1));hold on;
+  loglog(F_sl4,Amp_sl4,'*');
+  xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
+  title('X direction Top/bottom sensor');
+  xlim([7e-2 1e1]);
+  subplot 212
+  semilogx(Fr, (180/pi).*angle(x12./abs(x1)));hold on;
+  loglog(F_sl4_phase,Phase_sl4,'*');
+  xlabel('Frequency [Hz]');ylabel('Phase [deg]');
+  xlim([7e-2 1e1]);
+
+  %slide 6
+  figure
+  subplot 211
+  loglog(Fr, abs(z12)./abs(z1));hold on;
+  loglog(F_sl6,Amp_sl6,'*');
+  xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
+  title('Z direction Top/bottom sensor');
+  xlim([7e-2 1e1]);
+  subplot 212
+  semilogx(Fr, (180/pi).*angle(z12./abs(z1)));hold on;
+  loglog(F_sl6_phase,Phase_sl6,'*');
+  xlabel('Frequency [Hz]');ylabel('Phase [deg]');
+  xlim([7e-2 1e1]);ylim([-180 180]);
+
+  % %slide 6
+  % figure
+  % loglog(Fr,abs(z2).^.5,Fr,abs(z1).^.5,Fr,abs(zsum).^.5,Fr,abs(zdelta).^.5)
+  % xlabel('Frequency [Hz]');ylabel('Acceleration [m/s^2/rtHz]')
+  % legend('Top sensor','Bottom sensor','Half sum','Half difference');
+  % title('Vertical')
+  % xlim([7e-2 1e1]);
+
+  %slide 9
+  figure
+  subplot 211
+  loglog(Fr, abs(PHIwx)./abs(wx));hold on;
+  loglog(F_sl9,Amp_sl9,'*');
+  xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
+  title('X direction bottom/ground sensor');
+  xlim([7e-2 1e1]);
+  ylim([0.01 10]);
+  subplot 212
+  semilogx(Fr, (180/pi).*angle(PHIwx./abs(wx)));hold on;
+  loglog(F_sl9,Phase_sl9,'*');
+  xlabel('Frequency [Hz]');ylabel('Phase [deg]');
+  xlim([7e-2 1e1]);
+
+  % %slide 8
+  % figure
+  % loglog(Fr,abs(wx).^.5,Fr,abs(x1).^.5,'-.',Fr,abs(x2).^.5,'.');
+  % grid on;xlabel('Frequency [Hz]');
+  % ylabel('ASD [m/s^2/rtHz]');
+  % xlim([7e-2 1e1]);
+  % legend('Ground','Bottom sensor','Top sensor');
+  % title('Horizontal');
+  %
+  % %slide 13
+  % figure
+  % loglog(Fr,abs(wz).^.5,Fr,abs(z1).^.5,'-.',Fr,abs(z2).^.5,'.');
+  % grid on;xlabel('Frequency [Hz]');
+  % ylabel('ASD [m/s^2/rtHz]');
+  % xlim([7e-2 1e1]);
+  % legend('Ground','Bottom sensor','Top sensor');
+  % title('Vertical');
+
+  %slide 14
+  figure
+  subplot 211
+  loglog(Fr, abs(PHIwz)./abs(wz));hold on;
+  loglog(F_sl14,Amp_sl14,'*');
+  xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
+  title('Z direction bottom/ground sensor');
+  xlim([7e-2 1e1]);
+  ylim([0.01 10]);
+  subplot 212
+  semilogx(Fr, (180/pi).*angle(PHIwz./abs(wz)));hold on;
+  loglog(F_sl14_phase,Phase_sl14,'*');
+  xlabel('Frequency [Hz]');ylabel('Phase [deg]');
+  xlim([7e-2 1e1]);
+
+  %rotation
+  figure
+  loglog(Fr,abs(rot).^.5./((2*pi*Fr').^2),F_rot,Amp_rot,'*');
+  xlabel('Frequency [Hz]');ylabel('Rotation [rad/rtHz]')
+  xlim([7e-2 1e1]);
 #+end_src
 
-Frequency vector.
+** Model Generation                                                :noexport:
-#+begin_src matlab
-  w = 2*pi*logspace(-1,2,1000); % [rad/s]
-#+end_src
-
-*** generation of the state space model
 #+begin_src matlab
+  function [Fr, x1, z1, x2, z2, wx, wz, x12, z12, PHIwx, PHIwz,xsum,zsum,xdelta,zdelta,rot] = modelGeneration(m,I,k,h,ha,l,la,dampv,damph,kv,kh)
+      %% generation of the state space model
       M = [m 0 0
            0 m 0
            0 0 I];
@@ -268,9 +299,14 @@ Frequency vector.
                      %1 0 h/2 %Right horizontal actuator
             0 1 -la/2 %Left vertical actuator
             0 1 la/2]; %Right vertical actuator
+      Jah = [1 0 ha/2];
+      Jav = [0 1 -la/2 %Left vertical actuator
+             0 1 la/2]; %Right vertical actuator
       Jta = Ja';
-  K = k*Jta*Ja;
+      Jtah = Jah';
-  C = 0.06*k*Jta*Ja;
+      Jtav = Jav';
+      K = kv*Jtav*Jav + kh*Jtah*Jah;
+      C = dampv*kv*Jtav*Jav+damph*kh*Jtah*Jah;
 
       E = [1 0 0
            0 1 0
@@ -282,12 +318,6 @@ Frequency vector.
       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)
@@ -299,272 +329,11 @@ Frequency vector.
             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
 
-- Input = three actuators and three ground motions
+      %% Injecting ground motion in the system to have the output
-- 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
@@ -613,83 +382,6 @@ Frequency vector.
       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)
@@ -1283,7 +975,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
 #+RESULTS:
 [[file:figs/stewart_platform_simscape_cl_transmissibility.png]]
 
-* Stewart Platform - Analytical Model                               :noexport:
+* Stewart Platform - Analytical 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>>