%% 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 = 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) % #+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) 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(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 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 % 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) % 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);