Close loop performance - SISO control

rotating_frame.org

@@ -1092,19 +1092,64 @@ However, when we look at the poles of the closed loop with a diagonal controller
 | -1.1837+0.0041777i |
 | -1.1837-0.0041777i |
 
-*** TODO Close loop performance
-First we compute the close loop transfer functions.
+*** Close loop performance
+First, we create the closed loop systems. Then, we plot the transfer function from the reference signals $[\epsilon_u, \epsilon_v]$ to the output $[d_u, d_v]$ (figure [[fig:perfcomp]]).
 #+begin_src matlab :results none :exports code
-  Gcl = feedback(Gvc, K);
-  Gtcl = feedback(Gtvc, K);
+  K.InputName = 'e';
+  K.OutputName = 'u';
+
+  Gtvc.InputName = 'u';
+  Gtvc.OutputName = 'y';
+
+  Gvc.InputName = 'u';
+  Gvc.OutputName = 'y';
+
+  Sum = sumblk('e = r-y', 2);
+
+  Tvc = connect(Gvc, K, Sum, 'r', 'y');
+  Ttvc = connect(Gtvc, K, Sum, 'r', 'y');
 #+end_src
 
-
 #+begin_src matlab :results none :exports code
+  freqs = logspace(-2, 2, 1000);
+
   figure;
-  bode(Gcl, Gtcl)
+  ax1 = subplot(1,2,1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Tvc(1, 1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Ttvc(1, 1), freqs, 'Hz'))));
+  hold off;
+  xlim([freqs(1), freqs(end)]);
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude [m/N]');
+  legend({'w = 0 [rpm]', 'w = 60 [rpm]'}, 'Location', 'southwest')
+  title('$G_{r_u \to d_u}$')
+
+  ax2 = subplot(1,2,2);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Tvc(1, 2), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Ttvc(1, 2), freqs, 'Hz'))));
+  hold off;
+  xlim([freqs(1), freqs(end)]);
+  ylim([1e-5, 1]);
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]');
+  title('$G_{r_u \to d_v}$')
+
+  linkaxes([ax1,ax2],'x');
 #+end_src
 
+#+HEADER: :tangle no :exports results :results file :noweb yes
+#+HEADER: :var filepath="Figures/perfconp.png" :var figsize="full-tall"
+#+begin_src matlab
+  <<plt-matlab>>
+#+end_src
+
+#+NAME: fig:perfcomp
+#+CAPTION: Close loop performance for $\omega = 0$ and $\omega = 60 rpm$
+#+RESULTS:
+[[file:Figures/perfconp.png]]
+
 *** Effect of rotation speed
 We first identify the system (voice coil and light mass) for multiple rotation speed.
 Then we compute the bode plot of the diagonal element (figure [[fig:Guu_ws]]) and of the coupling element (figure [[fig:Guv_ws]]).
@@ -1112,7 +1157,7 @@ Then we compute the bode plot of the diagonal element (figure [[fig:Guu_ws]]) an
 As the rotation frequency increases:
 - one pole goes to lower frequencies while the other goes to higher frequencies
 - one zero appears between the two poles
-- the zero disappears when $\omega > \sqrt{\frac{k}{m}}$ and the low frequency pole becomes unstable (positive real part)
+- [ ] the zero disappears when $\omega > \sqrt{\frac{k}{m}}$ and the low frequency pole becomes unstable (positive real part)
 
 To stabilize the unstable pole, we need a control bandwidth of at least twice of frequency of the unstable pole.
 
@@ -1214,7 +1259,6 @@ To stabilize the unstable pole, we need a control bandwidth of at least twice of
 [[file:Figures/Guv_ws.png]]
 
 ** TODO Plant Control - MIMO approach
-
 ** test
 
 #+begin_src matlab :exports none :results silent