Add figure about RGA
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stewart_platform/jacobian.mat
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stewart_platform/jacobian.mat
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@@ -6,6 +6,8 @@ s = zpk('s');
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addpath('STEP');
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freqs = logspace(-1, 2, 1000);
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% Simscape Model - Parameters
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% <<sec:stewart_simscape>>
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@@ -25,6 +27,12 @@ cz = 0.025;
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% We suppose the sensor is perfectly positioned.
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sens_pos_error = zeros(3,1);
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% Gravity:
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g = 0;
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@@ -33,7 +41,7 @@ g = 0;
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% We load the Jacobian (previously computed from the geometry):
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load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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@@ -86,8 +94,6 @@ size(G)
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% One can easily see that the system is strongly coupled.
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freqs = logspace(-1, 2, 1000);
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figure;
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% Magnitude
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@@ -174,8 +180,6 @@ Gsvd = inv(U)*Gu*inv(V');
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% This is computed over the following frequencies.
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freqs = logspace(-2, 2, 1000); % [Hz]
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% Gershgorin Radii for the coupled plant:
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Gr_coupled = zeros(length(freqs), size(Gu,2));
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H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
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@@ -210,21 +214,99 @@ for in_i = 2:6
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set(gca,'ColorOrderIndex',3)
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plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
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end
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plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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hold off;
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xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
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legend('location', 'northwest');
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ylim([1e-3, 1e3]);
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% Verification of the decoupling using the "Relative Gain Array"
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% The relative gain array (RGA) is defined as:
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% \begin{equation}
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% \Lambda\big(G(s)\big) = G(s) \times \big( G(s)^{-1} \big)^T
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% \end{equation}
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% where $\times$ denotes an element by element multiplication and $G(s)$ is an $n \times n$ square transfer matrix.
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% The obtained RGA elements are shown in Figure [[fig:simscape_model_rga]].
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% Relative Gain Array for the coupled plant:
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RGA_coupled = zeros(length(freqs), size(Gu,1), size(Gu,2));
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Gu_inv = inv(Gu);
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for f_i = 1:length(freqs)
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RGA_coupled(f_i, :, :) = abs(evalfr(Gu, j*2*pi*freqs(f_i)).*evalfr(Gu_inv, j*2*pi*freqs(f_i))');
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end
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% Relative Gain Array for the decoupled plant using SVD:
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RGA_svd = zeros(length(freqs), size(Gsvd,1), size(Gsvd,2));
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Gsvd_inv = inv(Gsvd);
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for f_i = 1:length(freqs)
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RGA_svd(f_i, :, :) = abs(evalfr(Gsvd, j*2*pi*freqs(f_i)).*evalfr(Gsvd_inv, j*2*pi*freqs(f_i))');
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end
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% Relative Gain Array for the decoupled plant using the Jacobian:
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RGA_x = zeros(length(freqs), size(Gx,1), size(Gx,2));
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Gx_inv = inv(Gx);
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for f_i = 1:length(freqs)
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RGA_x(f_i, :, :) = abs(evalfr(Gx, j*2*pi*freqs(f_i)).*evalfr(Gx_inv, j*2*pi*freqs(f_i))');
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end
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figure;
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tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile;
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hold on;
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for i_in = 1:6
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for i_out = [1:i_in-1, i_in+1:6]
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plot(freqs, RGA_svd(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
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'HandleVisibility', 'off');
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end
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end
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plot(freqs, RGA_svd(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
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'DisplayName', '$RGA_{SVD}(i,j),\ i \neq j$');
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plot(freqs, RGA_svd(:, 1, 1), 'k-', ...
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'DisplayName', '$RGA_{SVD}(i,i)$');
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for ch_i = 1:6
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plot(freqs, RGA_svd(:, ch_i, ch_i), 'k-', ...
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'HandleVisibility', 'off');
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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('Magnitude'); xlabel('Frequency [Hz]');
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legend('location', 'southwest');
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ax2 = nexttile;
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hold on;
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for i_in = 1:6
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for i_out = [1:i_in-1, i_in+1:6]
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plot(freqs, RGA_x(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
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'HandleVisibility', 'off');
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end
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end
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plot(freqs, RGA_x(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
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'DisplayName', '$RGA_{X}(i,j),\ i \neq j$');
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plot(freqs, RGA_x(:, 1, 1), 'k-', ...
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'DisplayName', '$RGA_{X}(i,i)$');
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for ch_i = 1:6
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plot(freqs, RGA_x(:, ch_i, ch_i), 'k-', ...
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'HandleVisibility', 'off');
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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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xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
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legend('location', 'southwest');
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linkaxes([ax1,ax2],'y');
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ylim([1e-5, 1e1]);
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% Obtained Decoupled Plants
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% <<sec:stewart_decoupled_plant>>
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% The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:simscape_model_decoupled_plant_svd]].
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freqs = logspace(-1, 2, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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@@ -274,8 +356,6 @@ linkaxes([ax1,ax2],'x');
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% Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure [[fig:simscape_model_decoupled_plant_jacobian]].
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freqs = logspace(-1, 2, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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@@ -350,8 +430,6 @@ G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
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% The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]].
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freqs = logspace(-1, 2, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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@@ -424,7 +502,140 @@ isstable(G_svd)
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% The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
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freqs = logspace(-2, 2, 1000);
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figure;
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tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
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plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
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plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), '--', 'DisplayName', 'SVD');
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
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legend('location', 'southwest');
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ax2 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/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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ylabel('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
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ax3 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))), '--');
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/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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ylabel('$R_x/R_{w,x}$, $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
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ax4 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/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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ylabel('$R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
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linkaxes([ax1,ax2,ax3,ax4],'xy');
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xlim([freqs(1), freqs(end)]);
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ylim([1e-3, 1e2]);
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% Small error on the sensor location :no_export:
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% Let's now consider a small position error of the sensor:
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sens_pos_error = [105 5 -1]*1e-3; % [m]
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% The system is identified again:
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%% Name of the Simulink File
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mdl = 'drone_platform';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion
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io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces
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io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
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G = linearize(mdl, io);
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G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
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'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
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% Plant
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Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
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% Disturbance dynamics
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Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
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Gx = Gu*inv(J');
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Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gsvd = inv(U)*Gu*inv(V');
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% Gershgorin Radii for the coupled plant:
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Gr_coupled = zeros(length(freqs), size(Gu,2));
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H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
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for out_i = 1:size(Gu,2)
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Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
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end
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% Gershgorin Radii for the decoupled plant using SVD:
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Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
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H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
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for out_i = 1:size(Gsvd,2)
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Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
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end
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% Gershgorin Radii for the decoupled plant using the Jacobian:
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Gr_jacobian = zeros(length(freqs), size(Gx,2));
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H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
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for out_i = 1:size(Gx,2)
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Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
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end
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figure;
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hold on;
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plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
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plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
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plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
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for in_i = 2:6
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set(gca,'ColorOrderIndex',1)
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plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',3)
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plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
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end
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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hold off;
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xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
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legend('location', 'northwest');
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ylim([1e-3, 1e3]);
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L_cen = K_cen*Gx;
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G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
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L_svd = K_svd*Gsvd;
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G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
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isstable(G_cen)
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isstable(G_svd)
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figure;
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tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
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