Correct wrong computation of Gershgorin radii
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153
index.org
153
index.org
@@ -649,10 +649,10 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
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Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
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#+begin_src matlab
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wc = 2*pi*20; % Decoupling frequency [rad/s]
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Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
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#+end_src
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#+begin_src matlab
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Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
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{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
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H1 = evalfr(Gc, j*wc);
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#+end_src
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@@ -673,61 +673,74 @@ First, the Singular Value Decomposition of $H_1$ is performed:
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Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
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\[ G_d(s) = U^T G_c(s) V \]
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It is done over the following frequencies.
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This is computed over the following frequencies.
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#+begin_src matlab
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freqs = logspace(-1,2,1000); % [Hz]
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freqs = logspace(-2, 2, 1000); % [Hz]
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#+end_src
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Gershgorin Radii for the coupled plant:
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#+begin_src matlab
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for i = 1:length(freqs)
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H = abs(evalfr(Gc, j*2*pi*freqs(i)));
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for j = 1:size(H,2)
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g_r1(i,j) = (sum(H(j,:)) - H(j,j))/H(j,j);
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end
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Gr_coupled = zeros(length(freqs), size(Gc,2));
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H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
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for out_i = 1:size(Gc,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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#+end_src
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Gershgorin Radii for the decoupled plant using SVD:
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#+begin_src matlab
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Gd = U'*Gc*V;
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for i = 1:length(freqs)
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H_dec = abs(evalfr(Gd, j*2*pi*freqs(i)));
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for j = 1:size(H,2)
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g_r2(i,j) = (sum(H_dec(j,:)) - H_dec(j,j))/H_dec(j,j);
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end
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Gr_decoupled = zeros(length(freqs), size(Gd,2));
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H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
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for out_i = 1:size(Gd,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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#+end_src
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Gershgorin Radii for the decoupled plant using the Jacobian:
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#+begin_src matlab
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Gj = Gc*inv(J');
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Gr_jacobian = zeros(length(freqs), size(Gj,2));
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H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
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for out_i = 1:size(Gj,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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#+end_src
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#+begin_src matlab :exports results
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figure;
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hold on;
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plot(freqs, g_r1(:,1), 'DisplayName', '$a_x$')
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plot(freqs, g_r1(:,2), 'DisplayName', '$a_y$')
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plot(freqs, g_r1(:,3), 'DisplayName', '$a_z$')
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plot(freqs, g_r1(:,4), 'DisplayName', '$a_{R_x}$')
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plot(freqs, g_r1(:,5), 'DisplayName', '$a_{R_y}$')
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plot(freqs, g_r1(:,6), 'DisplayName', '$a_{R_z}$')
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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 i = 2:6
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set(gca,'ColorOrderIndex',1)
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plot(freqs, Gr_coupled(:,i), 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, Gr_decoupled(:,i), 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',3)
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plot(freqs, Gr_jacobian(:,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', 'northeast');
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#+end_src
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#+begin_src matlab :exports results
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figure;
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hold on;
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plot(freqs, g_r2(:,1), 'DisplayName', '$a_x$')
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plot(freqs, g_r2(:,2), 'DisplayName', '$a_y$')
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plot(freqs, g_r2(:,3), 'DisplayName', '$a_z$')
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plot(freqs, g_r2(:,4), 'DisplayName', '$a_{R_x}$')
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plot(freqs, g_r2(:,5), 'DisplayName', '$a_{R_y}$')
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plot(freqs, g_r2(:,6), 'DisplayName', '$a_{R_z}$')
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plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
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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)'); ylabel('Gershgorin Radii')
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/simscape_model_gershgorin_radii.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:simscape_model_gershgorin_radii
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#+caption: Gershgorin Radii of the Coupled and Decoupled plants
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#+RESULTS:
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[[file:figs/simscape_model_gershgorin_radii.png]]
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** Decoupled Plant
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Let's see the bode plot of the decoupled plant $G_d(s)$.
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\[ G_d(s) = U^T G_c(s) V \]
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@@ -737,13 +750,13 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
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figure;
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hold on;
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for i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gd(i, i), freqs, 'Hz'))), ...
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'DisplayName', sprintf('$G(%i, %i)$', i, i));
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for ch_i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
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'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
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end
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for i = 1:5
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
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for in_i = 1:5
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for out_i = in_i+1:6
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plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), '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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@@ -753,6 +766,45 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
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legend('location', 'southeast');
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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/simscape_model_decoupled_plant_svd.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:simscape_model_decoupled_plant_svd
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#+caption: Decoupled Plant using SVD
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#+RESULTS:
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[[file:figs/simscape_model_decoupled_plant_svd.png]]
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#+begin_src matlab :exports results
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freqs = logspace(-1, 2, 1000);
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figure;
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hold on;
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for ch_i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gj(ch_i, ch_i), freqs, 'Hz'))), ...
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'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
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end
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for in_i = 1:5
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for out_i = in_i+1:6
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plot(freqs, abs(squeeze(freqresp(Gj(out_i, in_i), freqs, 'Hz'))), '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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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude'); xlabel('Frequency [Hz]');
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legend('location', 'southeast');
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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/simscape_model_decoupled_plant_jacobian.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:simscape_model_decoupled_plant_jacobian
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#+caption: Decoupled Plant using the Jacobian
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#+RESULTS:
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[[file:figs/simscape_model_decoupled_plant_jacobian.png]]
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** Diagonal Controller
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The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
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@@ -829,6 +881,25 @@ SVD Control
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#+end_src
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** Results
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Let's first verify the stability of the closed-loop systems:
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#+begin_src matlab :results output replace text
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isstable(G_cen)
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#+end_src
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#+RESULTS:
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: ans =
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: logical
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: 1
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#+begin_src matlab :results output replace text
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isstable(G_svd)
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#+end_src
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#+RESULTS:
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: ans =
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: logical
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: 1
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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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#+begin_src matlab :exports results
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