Delete unused sections (analytical models)
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@ -818,533 +818,6 @@ The obtained transmissibility in Open-loop, for the centralized control as well
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#+RESULTS:
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[[file:figs/gravimeter_platform_simscape_cl_transmissibility.png]]
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* Gravimeter - Analytical Model :noexport:
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** System Identification - With Gravity :noexport:
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#+begin_src matlab
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g = 9.80665; % Gravity [m/s2]
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#+end_src
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#+begin_src matlab
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Gg = linearize(mdl, io);
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Gg.InputName = {'F1', 'F2', 'F3'};
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Gg.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
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#+end_src
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We can now see that the system is unstable due to gravity.
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#+begin_src matlab :results output replace :exports results
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pole(Gg)
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#+end_src
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#+RESULTS:
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#+begin_example
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-7.49865861504606e-05 + 8.65948534948982i
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-7.49865861504606e-05 - 8.65948534948982i
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-4.76450798645977 + 0i
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4.7642612321107 + 0i
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-7.34348883628024e-05 + 4.29133825321225i
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-7.34348883628024e-05 - 4.29133825321225i
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#+end_example
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#+begin_src matlab :exports none
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freqs = logspace(-2, 2, 1000);
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figure;
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for in_i = 1:3
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for out_i = 1:4
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subplot(4, 3, 3*(out_i-1)+in_i);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
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plot(freqs, abs(squeeze(freqresp(Gg(out_i,in_i), freqs, 'Hz'))), '-');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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end
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end
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/open_loop_tf_g.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:open_loop_tf_g
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#+caption: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity
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#+RESULTS:
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[[file:figs/open_loop_tf_g.png]]
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** Parameters
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Bode options.
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#+begin_src matlab
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P = bodeoptions;
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P.FreqUnits = 'Hz';
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P.MagUnits = 'abs';
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P.MagScale = 'log';
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P.Grid = 'on';
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P.PhaseWrapping = 'on';
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P.Title.FontSize = 14;
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P.XLabel.FontSize = 14;
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P.YLabel.FontSize = 14;
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P.TickLabel.FontSize = 12;
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P.Xlim = [1e-1,1e2];
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P.MagLowerLimMode = 'manual';
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P.MagLowerLim= 1e-3;
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#+end_src
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Frequency vector.
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#+begin_src matlab
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w = 2*pi*logspace(-1,2,1000); % [rad/s]
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#+end_src
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** Generation of the State Space Model
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Mass matrix
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#+begin_src matlab
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M = [m 0 0
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0 m 0
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0 0 I];
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#+end_src
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Jacobian of the bottom sensor
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#+begin_src matlab
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Js1 = [1 0 h/2
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0 1 -l/2];
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#+end_src
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Jacobian of the top sensor
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#+begin_src matlab
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Js2 = [1 0 -h/2
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0 1 0];
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#+end_src
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Jacobian of the actuators
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#+begin_src matlab
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Ja = [1 0 ha % Left horizontal actuator
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0 1 -la % Left vertical actuator
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0 1 la]; % Right vertical actuator
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Jta = Ja';
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#+end_src
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Stiffness and Damping matrices
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#+begin_src matlab
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K = k*Jta*Ja;
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C = c*Jta*Ja;
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#+end_src
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State Space Matrices
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#+begin_src matlab
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E = [1 0 0
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0 1 0
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0 0 1]; %projecting ground motion in the directions of the legs
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AA = [zeros(3) eye(3)
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-M\K -M\C];
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BB = [zeros(3,6)
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M\Jta M\(k*Jta*E)];
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CC = [[Js1;Js2] zeros(4,3);
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zeros(2,6)
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(Js1+Js2)./2 zeros(2,3)
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(Js1-Js2)./2 zeros(2,3)
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(Js1-Js2)./(2*h) zeros(2,3)];
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DD = [zeros(4,6)
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zeros(2,3) eye(2,3)
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zeros(6,6)];
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#+end_src
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State Space model:
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- Input = three actuators and three ground motions
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- Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation
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#+begin_src matlab
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system_dec = ss(AA,BB,CC,DD);
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#+end_src
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#+begin_src matlab :results output replace
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size(system_dec)
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#+end_src
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#+RESULTS:
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: State-space model with 12 outputs, 6 inputs, and 6 states.
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** Comparison with the Simscape Model
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#+begin_src matlab :exports none
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freqs = logspace(-2, 2, 1000);
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figure;
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for in_i = 1:3
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for out_i = 1:4
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subplot(4, 3, 3*(out_i-1)+in_i);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
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plot(freqs, abs(squeeze(freqresp(system_dec(out_i,in_i)*s^2, freqs, 'Hz'))), '-');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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end
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end
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/gravimeter_analytical_system_open_loop_models.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:gravimeter_analytical_system_open_loop_models
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#+caption: Comparison of the analytical and the Simscape models
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#+RESULTS:
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[[file:figs/gravimeter_analytical_system_open_loop_models.png]]
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** Analysis
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#+begin_src matlab
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% figure
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% bode(system_dec,P);
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% return
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#+end_src
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#+begin_src matlab
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%% svd decomposition
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% system_dec_freq = freqresp(system_dec,w);
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% S = zeros(3,length(w));
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% for m = 1:length(w)
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% S(:,m) = svd(system_dec_freq(1:4,1:3,m));
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% end
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% figure
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% loglog(w./(2*pi), S);hold on;
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% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));
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% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
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% legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');
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% ylim([1e-8 1e-2]);
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%
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% %condition number
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% figure
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% loglog(w./(2*pi), S(1,:)./S(3,:));hold on;
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% % loglog(w./(2*pi), abs(Val(1,:)),w./(2*pi), abs(Val(2,:)),w./(2*pi), abs(Val(3,:)));
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% xlabel('Frequency [Hz]');ylabel('Condition number [-]');
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% % legend('\sigma_1','\sigma_2','\sigma_3');%,'\sigma_4','\sigma_5','\sigma_6');
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%
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% %performance indicator
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% system_dec_svd = freqresp(system_dec(1:4,1:3),2*pi*10);
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% [U,S,V] = svd(system_dec_svd);
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% 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);%
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% H_svd = pinv(V')*H_svd_OL*pinv(U);
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% % system_dec_control_svd_ = feedback(system_dec,g*pinv(V')*H*pinv(U));
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%
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% OL_dec = g_svd*H_svd*system_dec(1:4,1:3);
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% OL_freq = freqresp(OL_dec,w); % OL = G*H
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% CL_system = feedback(eye(3),-g_svd*H_svd*system_dec(1:4,1:3));
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% CL_freq = freqresp(CL_system,w); % CL = (1+G*H)^-1
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% % CL_system_2 = feedback(system_dec,H);
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% % CL_freq_2 = freqresp(CL_system_2,w); % CL = G/(1+G*H)
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% for i = 1:size(w,2)
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% OL(:,i) = svd(OL_freq(:,:,i));
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% CL (:,i) = svd(CL_freq(:,:,i));
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% %CL2 (:,i) = svd(CL_freq_2(:,:,i));
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% end
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%
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% un = ones(1,length(w));
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% figure
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% 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;%
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% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));
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% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));
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% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
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% legend('GH \sigma_{inf} +1 ','GH \sigma_{inf} -1','S 1/\sigma_{sup}');%,'\lambda_1','\lambda_2','\lambda_3');
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%
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% figure
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% 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;%
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% % loglog(w./(2*pi), 1./(CL(2,:)),w./(2*pi), 1./(CL(3,:)));
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% % semilogx(w./(2*pi), 1./(CL2(1,:)),w./(2*pi), 1./(CL2(2,:)),w./(2*pi), 1./(CL2(3,:)));
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% xlabel('Frequency [Hz]');ylabel('Singular Value [-]');
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% legend('GH \sigma_{sup} +1 ','GH \sigma_{sup} -1','S 1/\sigma_{inf}');%,'\lambda_1','\lambda_2','\lambda_3');
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#+end_src
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** Control Section
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#+begin_src matlab
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system_dec_10Hz = freqresp(system_dec,2*pi*10);
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system_dec_0Hz = freqresp(system_dec,0);
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system_decReal_10Hz = pinv(align(system_dec_10Hz));
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[Ureal,Sreal,Vreal] = svd(system_decReal_10Hz(1:4,1:3));
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normalizationMatrixReal = abs(pinv(Ureal)*system_dec_0Hz(1:4,1:3)*pinv(Vreal'));
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[U,S,V] = svd(system_dec_10Hz(1:4,1:3));
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normalizationMatrix = abs(pinv(U)*system_dec_0Hz(1:4,1:3)*pinv(V'));
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H_dec = ([zpk(-2*pi*5,-2*pi*30,30/5) 0 0 0
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0 zpk(-2*pi*4,-2*pi*20,20/4) 0 0
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0 0 0 zpk(-2*pi,-2*pi*10,10)]);
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H_cen_OL = [zpk(-2*pi,-2*pi*10,10) 0 0; 0 zpk(-2*pi,-2*pi*10,10) 0;
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0 0 zpk(-2*pi*5,-2*pi*30,30/5)];
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H_cen = pinv(Jta)*H_cen_OL*pinv([Js1; Js2]);
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% H_svd_OL = -[1/normalizationMatrix(1,1) 0 0 0
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% 0 1/normalizationMatrix(2,2) 0 0
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% 0 0 1/normalizationMatrix(3,3) 0];
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% H_svd_OL_real = -[1/normalizationMatrixReal(1,1) 0 0 0
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% 0 1/normalizationMatrixReal(2,2) 0 0
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% 0 0 1/normalizationMatrixReal(3,3) 0];
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H_svd_OL = -[1/normalizationMatrix(1,1)*zpk(-2*pi*10,-2*pi*60,60/10) 0 0 0
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0 1/normalizationMatrix(2,2)*zpk(-2*pi*5,-2*pi*30,30/5) 0 0
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0 0 1/normalizationMatrix(3,3)*zpk(-2*pi*2,-2*pi*10,10/2) 0];
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H_svd_OL_real = -[1/normalizationMatrixReal(1,1)*zpk(-2*pi*10,-2*pi*60,60/10) 0 0 0
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0 1/normalizationMatrixReal(2,2)*zpk(-2*pi*5,-2*pi*30,30/5) 0 0
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0 0 1/normalizationMatrixReal(3,3)*zpk(-2*pi*2,-2*pi*10,10/2) 0];
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% 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);
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% 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);%
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H_svd = pinv(V')*H_svd_OL*pinv(U);
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H_svd_real = pinv(Vreal')*H_svd_OL_real*pinv(Ureal);
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OL_dec = g*H_dec*system_dec(1:4,1:3);
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OL_cen = g*H_cen_OL*pinv([Js1; Js2])*system_dec(1:4,1:3)*pinv(Jta);
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OL_svd = 100*H_svd_OL*pinv(U)*system_dec(1:4,1:3)*pinv(V');
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OL_svd_real = 100*H_svd_OL_real*pinv(Ureal)*system_dec(1:4,1:3)*pinv(Vreal');
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#+end_src
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#+begin_src matlab
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% figure
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% bode(OL_dec,w,P);title('OL Decentralized');
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% figure
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% bode(OL_cen,w,P);title('OL Centralized');
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#+end_src
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#+begin_src matlab
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figure
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bode(g*system_dec(1:4,1:3),w,P);
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title('gain * Plant');
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#+end_src
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#+begin_src matlab
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figure
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bode(OL_svd,OL_svd_real,w,P);
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title('OL SVD');
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legend('SVD of Complex plant','SVD of real approximation of the complex plant')
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#+end_src
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#+begin_src matlab
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figure
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bode(system_dec(1:4,1:3),pinv(U)*system_dec(1:4,1:3)*pinv(V'),P);
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#+end_src
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#+begin_src matlab
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CL_dec = feedback(system_dec,g*H_dec,[1 2 3],[1 2 3 4]);
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CL_cen = feedback(system_dec,g*H_cen,[1 2 3],[1 2 3 4]);
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CL_svd = feedback(system_dec,100*H_svd,[1 2 3],[1 2 3 4]);
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CL_svd_real = feedback(system_dec,100*H_svd_real,[1 2 3],[1 2 3 4]);
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#+end_src
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#+begin_src matlab
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pzmap_testCL(system_dec,H_dec,g,[1 2 3],[1 2 3 4])
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title('Decentralized control');
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#+end_src
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#+begin_src matlab
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pzmap_testCL(system_dec,H_cen,g,[1 2 3],[1 2 3 4])
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title('Centralized control');
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#+end_src
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#+begin_src matlab
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pzmap_testCL(system_dec,H_svd,100,[1 2 3],[1 2 3 4])
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title('SVD control');
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#+end_src
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#+begin_src matlab
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pzmap_testCL(system_dec,H_svd_real,100,[1 2 3],[1 2 3 4])
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title('Real approximation SVD control');
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#+end_src
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#+begin_src matlab
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P.Ylim = [1e-8 1e-3];
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figure
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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);
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title('Motion/actuator')
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legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
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#+end_src
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#+begin_src matlab
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P.Ylim = [1e-5 1e1];
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figure
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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);
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title('Transmissibility');
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legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
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#+end_src
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#+begin_src matlab
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figure
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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);
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title('Transmissibility from half sum and half difference in the X direction');
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legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
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#+end_src
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#+begin_src matlab
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figure
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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);
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title('Transmissibility from half sum and half difference in the Z direction');
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legend('Control OFF','Decentralized control','Centralized control','SVD control','SVD control real appr.');
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#+end_src
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** Greshgorin radius
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#+begin_src matlab
|
||||
system_dec_freq = freqresp(system_dec,w);
|
||||
x1 = zeros(1,length(w));
|
||||
z1 = zeros(1,length(w));
|
||||
x2 = zeros(1,length(w));
|
||||
S1 = zeros(1,length(w));
|
||||
S2 = zeros(1,length(w));
|
||||
S3 = zeros(1,length(w));
|
||||
|
||||
for t = 1:length(w)
|
||||
x1(t) = (abs(system_dec_freq(1,2,t))+abs(system_dec_freq(1,3,t)))/abs(system_dec_freq(1,1,t));
|
||||
z1(t) = (abs(system_dec_freq(2,1,t))+abs(system_dec_freq(2,3,t)))/abs(system_dec_freq(2,2,t));
|
||||
x2(t) = (abs(system_dec_freq(3,1,t))+abs(system_dec_freq(3,2,t)))/abs(system_dec_freq(3,3,t));
|
||||
system_svd = pinv(Ureal)*system_dec_freq(1:4,1:3,t)*pinv(Vreal');
|
||||
S1(t) = (abs(system_svd(1,2))+abs(system_svd(1,3)))/abs(system_svd(1,1));
|
||||
S2(t) = (abs(system_svd(2,1))+abs(system_svd(2,3)))/abs(system_svd(2,2));
|
||||
S2(t) = (abs(system_svd(3,1))+abs(system_svd(3,2)))/abs(system_svd(3,3));
|
||||
end
|
||||
|
||||
limit = 0.5*ones(1,length(w));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
loglog(w./(2*pi),x1,w./(2*pi),z1,w./(2*pi),x2,w./(2*pi),limit,'--');
|
||||
legend('x_1','z_1','x_2','Limit');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Greshgorin radius [-]');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
loglog(w./(2*pi),S1,w./(2*pi),S2,w./(2*pi),S3,w./(2*pi),limit,'--');
|
||||
legend('S1','S2','S3','Limit');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Greshgorin radius [-]');
|
||||
% set(gcf,'color','w')
|
||||
#+end_src
|
||||
|
||||
** Injecting ground motion in the system to have the output
|
||||
#+begin_src matlab
|
||||
Fr = logspace(-2,3,1e3);
|
||||
w=2*pi*Fr*1i;
|
||||
%fit of the ground motion data in m/s^2/rtHz
|
||||
Fr_ground_x = [0.07 0.1 0.15 0.3 0.7 0.8 0.9 1.2 5 10];
|
||||
n_ground_x1 = [4e-7 4e-7 2e-6 1e-6 5e-7 5e-7 5e-7 1e-6 1e-5 3.5e-5];
|
||||
Fr_ground_v = [0.07 0.08 0.1 0.11 0.12 0.15 0.25 0.6 0.8 1 1.2 1.6 2 6 10];
|
||||
n_ground_v1 = [7e-7 7e-7 7e-7 1e-6 1.2e-6 1.5e-6 1e-6 9e-7 7e-7 7e-7 7e-7 1e-6 2e-6 1e-5 3e-5];
|
||||
|
||||
n_ground_x = interp1(Fr_ground_x,n_ground_x1,Fr,'linear');
|
||||
n_ground_v = interp1(Fr_ground_v,n_ground_v1,Fr,'linear');
|
||||
% figure
|
||||
% loglog(Fr,abs(n_ground_v),Fr_ground_v,n_ground_v1,'*');
|
||||
% xlabel('Frequency [Hz]');ylabel('ASD [m/s^2 /rtHz]');
|
||||
% return
|
||||
|
||||
%converting into PSD
|
||||
n_ground_x = (n_ground_x).^2;
|
||||
n_ground_v = (n_ground_v).^2;
|
||||
|
||||
%Injecting ground motion in the system and getting the outputs
|
||||
system_dec_f = (freqresp(system_dec,abs(w)));
|
||||
PHI = zeros(size(Fr,2),12,12);
|
||||
for p = 1:size(Fr,2)
|
||||
Sw=zeros(6,6);
|
||||
Iact = zeros(3,3);
|
||||
Sw(4,4) = n_ground_x(p);
|
||||
Sw(5,5) = n_ground_v(p);
|
||||
Sw(6,6) = n_ground_v(p);
|
||||
Sw(1:3,1:3) = Iact;
|
||||
PHI(p,:,:) = (system_dec_f(:,:,p))*Sw(:,:)*(system_dec_f(:,:,p))';
|
||||
end
|
||||
x1 = PHI(:,1,1);
|
||||
z1 = PHI(:,2,2);
|
||||
x2 = PHI(:,3,3);
|
||||
z2 = PHI(:,4,4);
|
||||
wx = PHI(:,5,5);
|
||||
wz = PHI(:,6,6);
|
||||
x12 = PHI(:,1,3);
|
||||
z12 = PHI(:,2,4);
|
||||
PHIwx = PHI(:,1,5);
|
||||
PHIwz = PHI(:,2,6);
|
||||
xsum = PHI(:,7,7);
|
||||
zsum = PHI(:,8,8);
|
||||
xdelta = PHI(:,9,9);
|
||||
zdelta = PHI(:,10,10);
|
||||
rot = PHI(:,11,11);
|
||||
#+end_src
|
||||
|
||||
* Gravimeter - Functions :noexport:
|
||||
: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
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle stewart_platform/simscape_model.m
|
||||
@ -2393,258 +1866,3 @@ The system is identified again:
|
||||
ylim([1e-3, 1e2]);
|
||||
#+end_src
|
||||
|
||||
* Stewart Platform - Analytical Model :noexport:
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle stewart_platform/analytical_model.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>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% Bode plot options
|
||||
opts = bodeoptions('cstprefs');
|
||||
opts.FreqUnits = 'Hz';
|
||||
opts.MagUnits = 'abs';
|
||||
opts.MagScale = 'log';
|
||||
opts.PhaseWrapping = 'on';
|
||||
opts.xlim = [1 1000];
|
||||
#+end_src
|
||||
|
||||
** Characteristics
|
||||
#+begin_src matlab
|
||||
L = 0.055; % Leg length [m]
|
||||
Zc = 0; % ?
|
||||
m = 0.2; % Top platform mass [m]
|
||||
k = 1e3; % Total vertical stiffness [N/m]
|
||||
c = 2*0.1*sqrt(k*m); % Damping ? [N/(m/s)]
|
||||
|
||||
Rx = 0.04; % ?
|
||||
Rz = 0.04; % ?
|
||||
Ix = m*Rx^2; % ?
|
||||
Iy = m*Rx^2; % ?
|
||||
Iz = m*Rz^2; % ?
|
||||
#+end_src
|
||||
|
||||
** Mass Matrix
|
||||
#+begin_src matlab
|
||||
M = m*[1 0 0 0 Zc 0;
|
||||
0 1 0 -Zc 0 0;
|
||||
0 0 1 0 0 0;
|
||||
0 -Zc 0 Rx^2+Zc^2 0 0;
|
||||
Zc 0 0 0 Rx^2+Zc^2 0;
|
||||
0 0 0 0 0 Rz^2];
|
||||
#+end_src
|
||||
|
||||
** Jacobian Matrix
|
||||
#+begin_src matlab
|
||||
Bj=1/sqrt(6)*[ 1 1 -2 1 1 -2;
|
||||
sqrt(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;
|
||||
sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);
|
||||
0 0 L L -L -L;
|
||||
-L*2/sqrt(3) -L*2/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3);
|
||||
L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2)];
|
||||
#+end_src
|
||||
|
||||
** Stifnness and Damping matrices
|
||||
#+begin_src matlab
|
||||
kv = k/3; % Vertical Stiffness of the springs [N/m]
|
||||
kh = 0.5*k/3; % Horizontal Stiffness of the springs [N/m]
|
||||
|
||||
K = diag([3*kh, 3*kh, 3*kv, 3*kv*Rx^2/2, 3*kv*Rx^2/2, 3*kh*Rx^2]); % Stiffness Matrix
|
||||
C = c*K/100000; % Damping Matrix
|
||||
#+end_src
|
||||
|
||||
** State Space System
|
||||
#+begin_src matlab
|
||||
A = [ zeros(6) eye(6); ...
|
||||
-M\K -M\C];
|
||||
Bw = [zeros(6); -eye(6)];
|
||||
Bu = [zeros(6); M\Bj];
|
||||
|
||||
Co = [-M\K -M\C];
|
||||
|
||||
D = [zeros(6) M\Bj];
|
||||
|
||||
ST = ss(A,[Bw Bu],Co,D);
|
||||
#+end_src
|
||||
|
||||
- OUT 1-6: 6 dof
|
||||
- IN 1-6 : ground displacement in the directions of the legs
|
||||
- IN 7-12: forces in the actuators.
|
||||
#+begin_src matlab
|
||||
ST.StateName = {'x';'y';'z';'theta_x';'theta_y';'theta_z';...
|
||||
'dx';'dy';'dz';'dtheta_x';'dtheta_y';'dtheta_z'};
|
||||
|
||||
ST.InputName = {'w1';'w2';'w3';'w4';'w5';'w6';...
|
||||
'u1';'u2';'u3';'u4';'u5';'u6'};
|
||||
|
||||
ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
|
||||
#+end_src
|
||||
|
||||
** Transmissibility
|
||||
#+begin_src matlab
|
||||
TR=ST*[eye(6); zeros(6)];
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
subplot(231)
|
||||
bodemag(TR(1,1));
|
||||
subplot(232)
|
||||
bodemag(TR(2,2));
|
||||
subplot(233)
|
||||
bodemag(TR(3,3));
|
||||
subplot(234)
|
||||
bodemag(TR(4,4));
|
||||
subplot(235)
|
||||
bodemag(TR(5,5));
|
||||
subplot(236)
|
||||
bodemag(TR(6,6));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/stewart_platform_analytical_transmissibility.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_platform_analytical_transmissibility
|
||||
#+caption: Transmissibility
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_analytical_transmissibility.png]]
|
||||
|
||||
** Real approximation of $G(j\omega)$ at decoupling frequency
|
||||
#+begin_src matlab
|
||||
sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs
|
||||
|
||||
dec_fr = 20;
|
||||
H1 = evalfr(sys1,j*2*pi*dec_fr);
|
||||
H2 = H1;
|
||||
D = pinv(real(H2'*H2));
|
||||
H1 = inv(D*real(H2'*diag(exp(j*angle(diag(H2*D*H2.'))/2)))) ;
|
||||
[U,S,V] = svd(H1);
|
||||
|
||||
wf = logspace(-1,2,1000);
|
||||
for i = 1:length(wf)
|
||||
H = abs(evalfr(sys1,j*2*pi*wf(i)));
|
||||
H_dec = abs(evalfr(U'*sys1*V,j*2*pi*wf(i)));
|
||||
for j = 1:size(H,2)
|
||||
g_r1(i,j) = (sum(H(j,:))-H(j,j))/H(j,j);
|
||||
g_r2(i,j) = (sum(H_dec(j,:))-H_dec(j,j))/H_dec(j,j);
|
||||
% keyboard
|
||||
end
|
||||
g_lim(i) = 0.5;
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Coupled and Decoupled Plant "Gershgorin Radii"
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
title('Coupled plant')
|
||||
loglog(wf,g_r1(:,1),wf,g_r1(:,2),wf,g_r1(:,3),wf,g_r1(:,4),wf,g_r1(:,5),wf,g_r1(:,6),wf,g_lim,'--');
|
||||
legend('$a_x$','$a_y$','$a_z$','$\theta_x$','$\theta_y$','$\theta_z$','Limit');
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/gershorin_raddii_coupled_analytical.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:gershorin_raddii_coupled_analytical
|
||||
#+caption: Gershorin Raddi for the coupled plant
|
||||
#+RESULTS:
|
||||
[[file:figs/gershorin_raddii_coupled_analytical.png]]
|
||||
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
title('Decoupled plant (10 Hz)')
|
||||
loglog(wf,g_r2(:,1),wf,g_r2(:,2),wf,g_r2(:,3),wf,g_r2(:,4),wf,g_r2(:,5),wf,g_r2(:,6),wf,g_lim,'--');
|
||||
legend('$S_1$','$S_2$','$S_3$','$S_4$','$S_5$','$S_6$','Limit');
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/gershorin_raddii_decoupled_analytical.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:gershorin_raddii_decoupled_analytical
|
||||
#+caption: Gershorin Raddi for the decoupled plant
|
||||
#+RESULTS:
|
||||
[[file:figs/gershorin_raddii_decoupled_analytical.png]]
|
||||
|
||||
** Decoupled Plant
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
bodemag(U'*sys1*V,opts)
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/stewart_platform_analytical_decoupled_plant.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_platform_analytical_decoupled_plant
|
||||
#+caption: Decoupled Plant
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_analytical_decoupled_plant.png]]
|
||||
|
||||
** Controller
|
||||
#+begin_src matlab
|
||||
fc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||
c_gain = 50; %
|
||||
|
||||
cont = eye(6)*c_gain/(s+fc);
|
||||
#+end_src
|
||||
|
||||
** Closed Loop System
|
||||
#+begin_src matlab
|
||||
FEEDIN = [7:12]; % Input of controller
|
||||
FEEDOUT = [1:6]; % Output of controller
|
||||
#+end_src
|
||||
|
||||
Centralized Control
|
||||
#+begin_src matlab
|
||||
STcen = feedback(ST, inv(Bj)*cont, FEEDIN, FEEDOUT);
|
||||
TRcen = STcen*[eye(6); zeros(6)];
|
||||
#+end_src
|
||||
|
||||
SVD Control
|
||||
#+begin_src matlab
|
||||
STsvd = feedback(ST, pinv(V')*cont*pinv(U), FEEDIN, FEEDOUT);
|
||||
TRsvd = STsvd*[eye(6); zeros(6)];
|
||||
#+end_src
|
||||
|
||||
** Results
|
||||
#+begin_src matlab
|
||||
figure
|
||||
subplot(231)
|
||||
bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(232)
|
||||
bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(233)
|
||||
bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(234)
|
||||
bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(235)
|
||||
bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(236)
|
||||
bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/stewart_platform_analytical_svd_cen_comp.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_platform_analytical_svd_cen_comp
|
||||
#+caption: Comparison of the obtained transmissibility for the centralized control and the SVD control
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_analytical_svd_cen_comp.png]]
|
||||
|
||||
Loading…
Reference in New Issue
Block a user