Work on mixed synthesis

matlab/index.org

@@ -332,10 +332,6 @@ All the dynamical systems representing the sensors are saved for further use.
   PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
   PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
   PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
-
-  CPS_S1 = cumtrapz(freqs, PSD_S1);
-  CPS_S2 = cumtrapz(freqs, PSD_S2);
-  CPS_H2 = cumtrapz(freqs, PSD_H2);
 #+end_src
 
 #+begin_src matlab
@@ -603,20 +599,9 @@ The Power Spectral Density of the individual sensors' noise $\Phi_{n_1}, \Phi_{n
            abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
 #+end_src
 
-The corresponding Cumulative Power Spectrum $\Gamma_{n_1}$, $\Gamma_{n_2}$ and $\Gamma_{n_{\mathcal{H}_2}}$ (cumulative integration of the PSD eqref:eq:CPS_definition) are computed below and shown in Figure [[fig:cps_h2_synthesis]].
-#+begin_src matlab
-  CPS_S1 = cumtrapz(freqs, PSD_S1);
-  CPS_S2 = cumtrapz(freqs, PSD_S2);
-  CPS_H2 = cumtrapz(freqs, PSD_H2);
-#+end_src
-
-\begin{equation}
-  \Gamma_n (\omega) = \int_0^\omega \Phi_n(\nu) d\nu \label{eq:CPS_definition}
-\end{equation}
-
 The RMS value of the individual sensors and of the super sensor are listed in Table [[tab:rms_noise_H2]].
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-  data2orgtable([sqrt(CPS_S1(end)); sqrt(CPS_S2(end)); sqrt(CPS_H2(end))], {'$\sigma_{n_1}$', '$\sigma_{n_2}$', '$\sigma_{n_{\mathcal{H}_2}}$'}, {'RMS value $[m/s]$'}, ' %.3f ');
+  data2orgtable([sqrt(trapz(freqs, PSD_S1)); sqrt(trapz(freqs, PSD_S2)); sqrt(trapz(freqs, PSD_H2))], {'$\sigma_{n_1}$', '$\sigma_{n_2}$', '$\sigma_{n_{\mathcal{H}_2}}$'}, {'RMS value $[m/s]$'}, ' %.3f ');
 #+end_src
 
 #+name: tab:rms_noise_H2
@@ -637,7 +622,7 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
   plot(freqs, PSD_S2, '-',  'DisplayName', '$\Phi_{n_2}$');
   plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{n_{\mathcal{H}_2}}$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$(m/s)^2/Hz$]');
+  xlabel('Frequency [Hz]'); ylabel('Power Spectral Density $\left[ \frac{(m/s)^2}{Hz} \right]$');
   hold off;
   xlim([freqs(1), freqs(end)]);
   legend('location', 'northeast');
@@ -652,28 +637,6 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
 #+RESULTS:
 [[file:figs/psd_sensors_htwo_synthesis.png]]
 
-#+begin_src matlab :exports none
-  figure;
-  hold on;
-  plot(freqs, CPS_S1, '-',  'DisplayName', '$\Gamma_{n_1}$');
-  plot(freqs, CPS_S2, '-',  'DisplayName', '$\Gamma_{n_2}$');
-  plot(freqs, CPS_H2, 'k-', 'DisplayName', '$\Gamma_{n_{\mathcal{H}_2}}$');
-  set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum $[(m/s)^2]$');
-  hold off;
-  xlim([2*freqs(1), freqs(end)]);
-  legend('location', 'southeast');
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/cps_h2_synthesis.pdf', 'width', 'wide', 'height', 'normal');
-#+end_src
-
-#+name: fig:cps_h2_synthesis
-#+caption: Cumulative Power Spectrum of individual sensors and super sensor using the $\mathcal{H}_2$ synthesis
-#+RESULTS:
-[[file:figs/cps_h2_synthesis.png]]
-
 A time domain simulation is now performed.
 The measured velocity $x$ is set to be a sweep sine with an amplitude of $0.1\ [m/s]$.
 The velocity estimates from the two sensors and from the super sensors are shown in Figure [[fig:super_sensor_time_domain_h2]].
@@ -1109,13 +1072,6 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i
 
   PSD_H2   = abs(squeeze(freqresp(N1*H2_filters.H1, freqs, 'Hz'))).^2 + ...
              abs(squeeze(freqresp(N2*H2_filters.H2, freqs, 'Hz'))).^2;
-  CPS_H2   = cumtrapz(freqs, PSD_H2);
-#+end_src
-
-#+begin_src matlab :exports none
-  CPS_S2   = cumtrapz(freqs, PSD_S2);
-  CPS_S1   = cumtrapz(freqs, PSD_S1);
-  CPS_Hinf = cumtrapz(freqs, PSD_Hinf);
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1126,7 +1082,7 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i
   plot(freqs, PSD_H2,   'k-',  'DisplayName', '$\Phi_{n_{\mathcal{H}_2}}$');
   plot(freqs, PSD_Hinf, 'k--', 'DisplayName', '$\Phi_{n_{\mathcal{H}_\infty}}$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$(m/s)^2/Hz$]');
+  xlabel('Frequency [Hz]'); ylabel('Power Spectral Density $\left[ \frac{(m/s)^2}{Hz} \right]$');
   hold off;
   xlim([freqs(1), freqs(end)]);
   legend('location', 'northeast');
@@ -1142,7 +1098,7 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i
 [[file:figs/psd_sensors_hinf_synthesis.png]]
 
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-  data2orgtable([sqrt(CPS_H2(end)), sqrt(CPS_Hinf(end))]', {'Optimal: $\mathcal{H}_2$', 'Robust: $\mathcal{H}_\infty$'}, {'RMS [m/s]'}, ' %.1e ');
+  data2orgtable([sqrt(trapz(freqs, PSD_H2)), sqrt(trapz(freqs, PSD_Hinf))]', {'Optimal: $\mathcal{H}_2$', 'Robust: $\mathcal{H}_\infty$'}, {'RMS [m/s]'}, ' %.1e ');
 #+end_src
 
 #+name: tab:rms_noise_comp_H2_Hinf
@@ -1168,17 +1124,19 @@ However, the RMS of the super sensor noise is not optimized as it was the case w
 <<sec:mixed_synthesis_sensor_fusion>>
 ** Introduction                                                      :ignore:
 
+The (optima) $\mathcal{H}_2$ synthesis and the (robust) $\mathcal{H}_\infty$ synthesis are now combined to form an Optimal and Robust synthesis of complementary filters for sensor fusion.
+
+The sensor fusion architecture is shown in Figure [[fig:sensor_fusion_arch_full]] ($\hat{G}_i$ are omitted for space reasons).
+
 #+name: fig:sensor_fusion_arch_full
 #+caption: Sensor fusion architecture with sensor dynamics uncertainty
 [[file:figs-tikz/sensor_fusion_arch_full.png]]
 
 The goal is to design complementary filters such that:
-- the maximum uncertainty of the super sensor is bounded
+- the maximum uncertainty of the super sensor is bounded to acceptable values (defined by $W_u(s)$)
 - the RMS value of the super sensor noise is minimized
 
-To do so, we can use the Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis.
+To do so, we can use the Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis presented in Section [[sec:H2_Hinf_synthesis]].
-
-The Matlab function for that is =h2hinfsyn= ([[https://fr.mathworks.com/help/robust/ref/h2hinfsyn.html][doc]]).
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -1195,53 +1153,44 @@ The Matlab function for that is =h2hinfsyn= ([[https://fr.mathworks.com/help/rob
 #+end_src
 
 ** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis
+<<sec:H2_Hinf_synthesis>>
+
 The synthesis architecture that is used here is shown in Figure [[fig:mixed_h2_hinf_synthesis]].
 
-The controller $K$ is synthesized such that it:
+The filter $H_2(s)$ is synthesized such that it:
-- Keeps the $\mathcal{H}_\infty$ norm $G$ of the transfer function from $w$ to $z_\infty$ bellow some specified value
+- keeps the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $z_{\mathcal{H}_\infty}$ bellow some specified value
-- Keeps the $\mathcal{H}_2$ norm $H$ of the transfer function from $w$ to $z_2$ bellow some specified value
+- minimizes the $\mathcal{H}_2$ norm of the transfer function from $w$ to $z_{\mathcal{H}_2}$
-- Minimizes a trade-off criterion of the form $W_1 G^2 + W_2 H^2$ where $W_1$ and $W_2$ are specified values
 
 #+name: fig:mixed_h2_hinf_synthesis
 #+caption: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
 [[file:figs-tikz/mixed_h2_hinf_synthesis.png]]
 
-Here, we define $P$ such that:
+Let's see that
-\begin{align*}
+with $H_1(s)= 1 - H_2(s)$
-  \left\| \frac{z_\infty}{w} \right\|_\infty &= \left\| \begin{matrix}W_1(s) H_1(s) \\ W_2(s) H_2(s)\end{matrix} \right\|_\infty \\
+\begin{align}
-  \left\| \frac{z_2}{w} \right\|_2           &= \left\| \begin{matrix}N_1(s) H_1(s) \\ N_2(s) H_2(s)\end{matrix} \right\|_2
+  \left\| \frac{z_\infty}{w} \right\|_\infty &= \left\| \begin{matrix}H_1(s) W_1(s) W_u(s)\\ H_2(s) W_2(s) W_u(s)\end{matrix} \right\|_\infty \\
-\end{align*}
+  \left\| \frac{z_2}{w} \right\|_2           &= \left\| \begin{matrix}H_1(s) N_1(s)       \\ H_2(s) N_2(s)\end{matrix} \right\|_2 = \sigma_n
+\end{align}
 
-Then:
+The generalized plant $P_{\mathcal{H}_2/\mathcal{H}_\infty}$ is defined below
-- we specify the maximum value for the $\mathcal{H}_\infty$ norm between $w$ and $z_\infty$ to be $1$
-- we don't specify any maximum value for the $\mathcal{H}_2$ norm between $w$ and $z_2$
-- we choose $W_1 = 0$ and $W_2 = 1$ such that the objective is to minimize the $\mathcal{H}_2$ norm between $w$ and $z_2$
-
-The synthesis objective is to have:
-\[ \left\| \frac{z_\infty}{w} \right\|_\infty = \left\| \begin{matrix}W_1(s) H_1(s) \\ W_2(s) H_2(s)\end{matrix} \right\|_\infty < 1 \]
-and to minimize:
-\[ \left\| \frac{z_2}{w} \right\|_2 = \left\| \begin{matrix}N_1(s) H_1(s) \\ N_2(s) H_2(s)\end{matrix} \right\|_2 \]
-which is what we wanted.
-
-We define the generalized plant that will be used for the mixed synthesis.
 #+begin_src matlab
   W1u = ss(W2*Wu); W2u = ss(W1*Wu); % Weight on the uncertainty
   W1n = ss(N2); W2n = ss(N1); % Weight on the noise
 
-  P = [W1u -W1u;
+  P = [Wu*W1 -Wu*W1;
-       0    W2u;
+       0      Wu*W2;
-       W1n -W1n;
+       N1    -N1;
-       0    W2n;
+       0      N2;
-       1    0];
+       1      0];
 #+end_src
 
-The mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis is performed below.
+And the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis is performed.
 #+begin_src matlab
-  Nmeas = 1; Ncon = 1; Nz2 = 2;
+  [H2, ~] = h2hinfsyn(ss(P), 1, 1, 2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on');
+#+end_src
 
-  [H1, ~, normz, ~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on');
+#+begin_src matlab
-
+  H1 = 1 - H2;
-  H2 = 1 - H1;
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1256,15 +1205,12 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt
 
   ax1 = subplot(2,1,1);
   hold on;
-  set(gca,'ColorOrderIndex',1)
+  plot(freqs, 1./abs(squeeze(freqresp(Wu*W1, freqs, 'Hz'))), '--', 'DisplayName', '$1/|W_uW_1|$');
-  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_1$');
+  plot(freqs, 1./abs(squeeze(freqresp(Wu*W2, freqs, 'Hz'))), '--', 'DisplayName', '$1/|W_uW_2|$');
-  set(gca,'ColorOrderIndex',2)
-  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_2$');
 
   set(gca,'ColorOrderIndex',1)
-  plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
+  plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
-  set(gca,'ColorOrderIndex',2)
+  plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
-  plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
 
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -1275,10 +1221,8 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt
 
   ax2 = subplot(2,1,2);
   hold on;
-  set(gca,'ColorOrderIndex',1)
-  plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
-  set(gca,'ColorOrderIndex',2)
   plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
   hold off;
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   set(gca, 'XScale', 'log');
@@ -1286,7 +1230,6 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt
 
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
-  xticks([0.1, 1, 10, 100, 1000]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
@@ -1299,12 +1242,17 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt
 [[file:figs/htwo_hinf_comp_filters.png]]
 
 ** Obtained Super Sensor's noise
-The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_sensors_htwo_hinf_synthesis]] and Figure [[fig:cps_h2_hinf_synthesis]] respectively.
+The Power Spectral Density of the super sensor's noise is shown in Figure [[fig:psd_sensors_htwo_hinf_synthesis]].
 
-#+begin_src matlab :exports none
+A time domain simulation is shown in Figure [[fig:super_sensor_time_domain_h2_hinf]].
-  % The filters are loaded
+
-  H2_filters      = load('./mat/H2_filters.mat',      'H2', 'H1');
+The RMS values of the super sensor noise for the presented three synthesis are listed in Table [[tab:rms_noise_comp]].
-  Hinf_filters    = load('./mat/Hinf_filters.mat',    'H2', 'H1');
+
+#+begin_src matlab
+  PSD_S2     = abs(squeeze(freqresp(N2,    freqs, 'Hz'))).^2;
+  PSD_S1     = abs(squeeze(freqresp(N1,    freqs, 'Hz'))).^2;
+  PSD_H2Hinf = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2 + ...
+               abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1312,7 +1260,6 @@ The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_sensor
 
   PSD_H2   = abs(squeeze(freqresp(N1*H2_filters.H1, freqs, 'Hz'))).^2 + ...
              abs(squeeze(freqresp(N2*H2_filters.H2, freqs, 'Hz'))).^2;
-  CPS_H2   = cumtrapz(freqs, PSD_H2);
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1320,17 +1267,6 @@ The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_sensor
 
   PSD_Hinf   = abs(squeeze(freqresp(N1*Hinf_filters.H1, freqs, 'Hz'))).^2 + ...
                abs(squeeze(freqresp(N2*Hinf_filters.H2, freqs, 'Hz'))).^2;
-  CPS_Hinf   = cumtrapz(freqs, PSD_Hinf);
-#+end_src
-
-#+begin_src matlab
-  PSD_S2     = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
-  PSD_S1     = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
-  PSD_H2Hinf = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
-
-  CPS_S2     = cumtrapz(freqs, PSD_S2);
-  CPS_S1     = cumtrapz(freqs, PSD_S1);
-  CPS_H2Hinf = cumtrapz(freqs, PSD_H2Hinf);
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1357,32 +1293,63 @@ The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_sensor
 #+RESULTS:
 [[file:figs/psd_sensors_htwo_hinf_synthesis.png]]
 
+#+begin_src matlab :exports none
+  Fs = 1e4;  % Sampling Frequency [Hz]
+  Ts = 1/Fs; % Sampling Time [s]
+
+  t = 0:Ts:2; % Time Vector [s]
+
+  v = 0.1*sin((10*t).*t)'; % Velocity measured [m/s]
+
+  % Generate noises in velocity corresponding to sensor 1 and 2:
+  n1 = lsim(N1, sqrt(Fs/2)*randn(length(t), 1), t);
+  n2 = lsim(N2, sqrt(Fs/2)*randn(length(t), 1), t);
+#+end_src
+
 #+begin_src matlab :exports none
   figure;
   hold on;
-  plot(freqs, CPS_S1,     '-',   'DisplayName', '$\Gamma_{n_1}$');
+  set(gca,'ColorOrderIndex',2)
-  plot(freqs, CPS_S2,     '-',   'DisplayName', '$\Gamma_{n_2}$');
+  plot(t, v + n2, 'DisplayName', '$\hat{x}_2$');
-  plot(freqs, CPS_H2,     'k-',  'DisplayName', '$\Gamma_{n_{\mathcal{H}_2}}$');
+  set(gca,'ColorOrderIndex',1)
-  plot(freqs, CPS_Hinf,   'k--', 'DisplayName', '$\Gamma_{n_{\mathcal{H}_\infty}}$');
+  plot(t, v + n1, 'DisplayName', '$\hat{x}_1$');
-  plot(freqs, CPS_H2Hinf, 'k-.', 'DisplayName', '$\Gamma_{n_{\mathcal{H}_2/\mathcal{H}_\infty}}$');
+  set(gca,'ColorOrderIndex',3)
-  set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
+  plot(t, v + (lsim(H1, n1, t) + lsim(H2, n2, t)), 'DisplayName', '$\hat{x}$');
-  xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
+  plot(t, v, 'k--', 'DisplayName', '$x$');
   hold off;
-  xlim([2*freqs(1), freqs(end)]);
+  xlabel('Time [s]'); ylabel('Velocity [m/s]');
-  legend('location', 'southeast');
+  legend('location', 'southwest');
+  ylim([-0.3, 0.3]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/cps_h2_hinf_synthesis.pdf', 'width', 'wide', 'height', 'normal');
+  exportFig('figs/super_sensor_time_domain_h2_hinf.pdf', 'width', 'wide', 'height', 'normal');
 #+end_src
 
-#+name: fig:cps_h2_hinf_synthesis
+#+name: fig:super_sensor_time_domain_h2_hinf
-#+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
+#+caption: Noise of individual sensors and noise of the super sensor
 #+RESULTS:
-[[file:figs/cps_h2_hinf_synthesis.png]]
+[[file:figs/super_sensor_time_domain_h2_hinf.png]]
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([sqrt(trapz(freqs, PSD_H2)), sqrt(trapz(freqs, PSD_Hinf)), sqrt(trapz(freqs, PSD_H2Hinf))]', ...
+                {'Optimal: $\mathcal{H}_2$', 'Robust: $\mathcal{H}_\infty$', 'Mixed: $\mathcal{H}_2/\mathcal{H}_\infty$'}, ...
+                {'RMS [m/s]'}, ' %.1e ');
+#+end_src
+
+#+name: tab:rms_noise_comp
+#+caption: Comparison of the obtained RMS noise of the super sensor
+#+attr_latex: :environment tabular :align cc
+#+attr_latex: :center t :booktabs t :float t
+#+RESULTS:
+|                                           | RMS [m/s] |
+|-------------------------------------------+-----------|
+| Optimal: $\mathcal{H}_2$                  |    0.0027 |
+| Robust: $\mathcal{H}_\infty$              |     0.041 |
+| Mixed: $\mathcal{H}_2/\mathcal{H}_\infty$ |      0.01 |
 
 ** Obtained Super Sensor's Uncertainty
-The uncertainty on the super sensor's dynamics is shown in Figure
+The uncertainty on the super sensor's dynamics is shown in Figure [[fig:super_sensor_dynamical_uncertainty_Htwo_Hinf]].
 
 #+begin_src matlab :exports none
   Dphi_Wu = 180/pi*asin(abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))));
@@ -1395,18 +1362,21 @@ The uncertainty on the super sensor's dynamics is shown in Figure
   % Magnitude
   ax1 = subplot(2,1,1);
   hold on;
-  plotMagUncertainty(W1, freqs, 'color_i', 1);
+  plotMagUncertainty(W1, freqs, 'color_i', 1, 'DisplayName', '$1 + W_1 \Delta_1$');
-  plotMagUncertainty(W2, freqs, 'color_i', 2);
+  plotMagUncertainty(W2, freqs, 'color_i', 2, 'DisplayName', '$1 + W_2 \Delta_2$');
-  p = patch([freqs flip(freqs)], [1 + abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))+abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))); flip(max(1 - abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))-abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))), 0.001))], 'w');
+  plot(freqs, 1 + abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))+abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))), 'k-', ...
-  p.EdgeColor = 'black'; p.FaceAlpha = 0;
+       'DisplayName', '$1 + W_1 \Delta_1 + W_2 \Delta_2$')
-  plot(freqs, 1 + abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))), 'r--', ...
+  plot(freqs, max(1 - abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))-abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))), 0.001), 'k-', ...
-       'DisplayName', '$W_u$')
+       'HandleVisibility', 'off');
-  plot(freqs, 1 - abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))), 'r--', ...
+  plot(freqs, 1 + abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))), 'k--', ...
+       'DisplayName', '$1 + W_u^{-1}\Delta$')
+  plot(freqs, 1 - abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))), 'k--', ...
        'HandleVisibility', 'off')
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   set(gca, 'XTickLabel',[]);
   ylabel('Magnitude');
   ylim([1e-2, 1e1]);
+  legend('location', 'southeast');
   hold off;
 
   % Phase
@@ -1414,10 +1384,10 @@ The uncertainty on the super sensor's dynamics is shown in Figure
   hold on;
   plotPhaseUncertainty(W1, freqs, 'color_i', 1);
   plotPhaseUncertainty(W2, freqs, 'color_i', 2);
-  p = patch([freqs flip(freqs)], [Dphi_ss; flip(-Dphi_ss)], 'w');
+  plot(freqs,  Dphi_ss, 'k-');
-  p.EdgeColor = 'black'; p.FaceAlpha = 0;
+  plot(freqs, -Dphi_ss, 'k-');
-  plot(freqs,  Dphi_Wu, 'r--');
+  plot(freqs,  Dphi_Wu, 'k--');
-  plot(freqs, -Dphi_Wu, 'r--');
+  plot(freqs, -Dphi_Wu, 'k--');
   set(gca,'xscale','log');
   yticks(-180:90:180);
   ylim([-180 180]);
@@ -1427,41 +1397,17 @@ The uncertainty on the super sensor's dynamics is shown in Figure
   xlim([freqs(1), freqs(end)]);
 #+end_src
 
-** Comparison Hinf H2 H2/Hinf
+#+begin_src matlab :tangle no :exports results :results file replace
-#+begin_src matlab
+  exportFig('figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf', 'width', 'full', 'height', 'full');
-  H2_filters      = load('./mat/H2_filters.mat',      'H2', 'H1');
-  Hinf_filters    = load('./mat/Hinf_filters.mat',    'H2', 'H1');
-  H2_Hinf_filters = load('./mat/H2_Hinf_filters.mat', 'H2', 'H1');
 #+end_src
 
-#+begin_src matlab
+#+name: fig:super_sensor_dynamical_uncertainty_Htwo_Hinf
-  PSD_H2 = abs(squeeze(freqresp(N2*H2_filters.H2, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N1*H2_filters.H1, freqs, 'Hz'))).^2;
+#+caption: Super sensor dynamical uncertainty (solid curve) when using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
-  CPS_H2 = cumtrapz(freqs, PSD_H2);
-
-  PSD_Hinf = abs(squeeze(freqresp(N2*Hinf_filters.H2, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N1*Hinf_filters.H1, freqs, 'Hz'))).^2;
-  CPS_Hinf = cumtrapz(freqs, PSD_Hinf);
-
-  PSD_H2Hinf = abs(squeeze(freqresp(N2*H2_Hinf_filters.H2, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N1*H2_Hinf_filters.H1, freqs, 'Hz'))).^2;
-  CPS_H2Hinf = cumtrapz(freqs, PSD_H2Hinf);
-#+end_src
-
-#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-  data2orgtable([sqrt(CPS_H2(end)), sqrt(CPS_Hinf(end)), sqrt(CPS_H2Hinf(end))]', {'Optimal: $\mathcal{H}_2$', 'Robust: $\mathcal{H}_\infty$', 'Mixed: $\mathcal{H}_2/\mathcal{H}_\infty$'}, {'RMS [m/s]'}, ' %.1e ');
-#+end_src
-
-#+name: tab:rms_noise_comp
-#+caption: Comparison of the obtained RMS noise of the super sensor
-#+attr_latex: :environment tabular :align cc
-#+attr_latex: :center t :booktabs t :float t
 #+RESULTS:
-|                                           | RMS [m/s] |
+[[file:figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.png]]
-|-------------------------------------------+-----------|
-| Optimal: $\mathcal{H}_2$                  |    0.0012 |
-| Robust: $\mathcal{H}_\infty$              |     0.041 |
-| Mixed: $\mathcal{H}_2/\mathcal{H}_\infty$ |     0.011 |
 
 ** Conclusion
-This synthesis methods allows both to:
+The mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of the complementary filters allows to:
 - limit the dynamical uncertainty of the super sensor
 - minimize the RMS value of the estimation