282 lines
11 KiB
Mathematica
282 lines
11 KiB
Mathematica
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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open 'simscape/sim_nano_station_id.slx'
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% Control Design
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% Let's load the undamped plant:
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load('./active_damping/mat/plants.mat', 'G');
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% Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure [[fig:iff_plant]]).
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freqs = logspace(0, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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for i=1:6
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plot(freqs, abs(squeeze(freqresp(G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i=1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2],'x');
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% #+NAME: fig:iff_plant
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% #+CAPTION: Transfer function from forces applied in the legs to force sensor ([[./figs/iff_plant.png][png]], [[./figs/iff_plant.pdf][pdf]])
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% [[file:figs/iff_plant.png]]
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% The controller for each pair of actuator/sensor is:
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K_iff = -1000/s;
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% The corresponding loop gains are shown in figure [[fig:iff_open_loop]].
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freqs = logspace(0, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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for i=1:6
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plot(freqs, abs(squeeze(freqresp(K_iff*G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i=1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2],'x');
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% Identification of the damped plant
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% Let's initialize the system prior to identification.
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initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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initializeNanoHexapod(struct('actuator', 'piezo'));
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initializeSample(struct('mass', 50));
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% All the controllers are set to 0.
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K = tf(zeros(6));
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save('./mat/controllers.mat', 'K', '-append');
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K_iff = -K_iff*eye(6);
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save('./mat/controllers.mat', 'K_iff', '-append');
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K_rmc = tf(zeros(6));
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save('./mat/controllers.mat', 'K_rmc', '-append');
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K_dvf = tf(zeros(6));
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save('./mat/controllers.mat', 'K_dvf', '-append');
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% We identify the system dynamics now that the IFF controller is ON.
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G_iff = identifyPlant();
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% And we save the damped plant for further analysis
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save('./active_damping/mat/plants.mat', 'G_iff', '-append');
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% Sensitivity to disturbances
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% As shown on figure [[fig:sensitivity_dist_iff]]:
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% - The top platform of the nano-hexapod how behaves as a "free-mass".
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% - The transfer function from direct forces $F_s$ to the relative displacement $D$ is equivalent to the one of an isolated mass.
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% - The transfer function from ground motion $D_g$ to the relative displacement $D$ tends to the transfer function from $D_g$ to the displacement of the granite (the sample is being isolated thanks to IFF).
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% However, as the goal is to make the relative displacement $D$ as small as possible (e.g. to make the sample motion follows the granite motion), this is not a good thing.
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freqs = logspace(0, 3, 1000);
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figure;
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subplot(2, 1, 1);
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title('$D_g$ to $D$');
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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subplot(2, 1, 2);
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title('$F_s$ to $D$');
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{s,x}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{s,y}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{s,z}\right|$');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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% #+NAME: fig:sensitivity_dist_iff
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% #+CAPTION: Sensitivity to disturbance once the IFF controller is applied to the system ([[./figs/sensitivity_dist_iff.png][png]], [[./figs/sensitivity_dist_iff.pdf][pdf]])
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% [[file:figs/sensitivity_dist_iff.png]]
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% #+begin_warning
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% The order of the models are very high and thus the plots may be wrong.
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% For instance, the plots are not the same when using =minreal=.
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% #+end_warning
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freqs = logspace(0, 3, 1000);
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(minreal(prescale(G_iff.G_dist('Dz', 'Frzz'), {2*pi, 2*pi*1e3})), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(minreal(G_iff.G_dist('Dz', 'Ftyz')), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(minreal(G_iff.G_dist('Dx', 'Ftyx')), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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% Damped Plant
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% Now, look at the new damped plant to control.
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% It damps the plant (resonance of the nano hexapod as well as other resonances) as shown in figure [[fig:plant_iff_damped]].
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freqs = logspace(0, 3, 1000);
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figure;
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ax1 = subplot(2, 2, 1);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--');
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plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--');
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plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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ax2 = subplot(2, 2, 2);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--');
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plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--');
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plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]');
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ax3 = subplot(2, 2, 3);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend('location', 'northwest');
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ax4 = subplot(2, 2, 4);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend('location', 'northwest');
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linkaxes([ax1,ax2,ax3,ax4],'x');
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% #+NAME: fig:plant_iff_damped
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% #+CAPTION: Damped Plant after IFF is applied ([[./figs/plant_iff_damped.png][png]], [[./figs/plant_iff_damped.pdf][pdf]])
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% [[file:figs/plant_iff_damped.png]]
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% However, it increases coupling at low frequency (figure [[fig:plant_iff_coupling]]).
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freqs = logspace(0, 3, 1000);
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figure;
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for ix = 1:6
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for iy = 1:6
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subplot(6, 6, (ix-1)*6 + iy);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_cart(ix, iy), freqs, 'Hz'))), 'k-');
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plot(freqs, abs(squeeze(freqresp(G_iff.G_cart(ix, iy), freqs, 'Hz'))), 'k--');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylim([1e-12, 1e-5]);
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end
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end
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