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Change the controller tuning for Stewart platform

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Thomas Dehaeze
2020-11-06 18:00:30 +01:00
parent 028102a43a
commit c0ea426854
7 changed files with 20515 additions and 23455 deletions
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@@ -1229,9 +1229,7 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
** Diagonal Controller
<<sec:stewart_diagonal_control>>
The controller $K_c$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
#+begin_src matlab
#+begin_src matlab :exports none :tangle no
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
C_g = 50; % DC Gain
@@ -1268,11 +1266,6 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
#+RESULTS:
[[file:figs/centralized_control.png]]
The feedback system is computed as shown below.
#+begin_src matlab
G_cen = feedback(G, inv(J')*Kc, [7:12], [1:6]);
#+end_src
The SVD control architecture is shown in Figure [[fig:svd_control]].
The matrices $U$ and $V$ are used to decoupled the plant $G$.
#+begin_src latex :file svd_control.pdf :tangle no :exports results
@@ -1301,17 +1294,89 @@ The matrices $U$ and $V$ are used to decoupled the plant $G$.
#+RESULTS:
[[file:figs/svd_control.png]]
The feedback system is computed as shown below.
#+begin_src matlab
G_svd = feedback(G, pinv(V')*Kc*pinv(U), [7:12], [1:6]);
#+end_src
Let's look at the Loop Gains.
We choose the controller to be a low pass filter:
\[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]
$G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
#+begin_src matlab
L_svd =
wc = 2*pi*80;
w0 = 2*pi*0.1;
#+end_src
#+begin_src matlab
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_cen = K_cen*Gx;
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
#+end_src
#+begin_src matlab
K_svd = diag(1./diag(abs(evalfr(Gd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_svd = K_svd*Gd;
G_svd = feedback(G, pinv(V')*K_svd*pinv(U), [7:12], [1:6]);
#+end_src
The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]].
#+begin_src matlab :exports none
freqs = logspace(-1, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
for i_in_out = 2:6
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
'DisplayName', '$L_{J}(i,i)$');
for i_in_out = 2:6
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
legend('location', 'northwest');
ylim([5e-2, 2e3])
% Phase
ax2 = nexttile;
hold on;
for i_in_out = 1:6
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
end
set(gca,'ColorOrderIndex',2)
for i_in_out = 1:6
set(gca,'ColorOrderIndex',2)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180:90:360]);
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_comp_loop_gain_diagonal.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:stewart_comp_loop_gain_diagonal
#+caption: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one
#+RESULTS:
[[file:figs/stewart_comp_loop_gain_diagonal.png]]
** Closed-Loop system Performances
<<sec:stewart_closed_loop_results>>
@@ -1389,7 +1454,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
linkaxes([ax1,ax2,ax3,ax4],'xy');
xlim([freqs(1), freqs(end)]);
ylim([1e-5, 1e2]);
ylim([1e-3, 1e2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace