Reworked all the analysis about optimal sensor fusion

matlab/index.org

@@ -86,61 +86,65 @@ The complementary filters have to be designed in order to minimize the effect no
 #+end_src
 
 ** Architecture
-Let's consider the sensor fusion architecture shown on figure [[fig:fusion_two_noisy_sensors_with_dyn]] where two sensors 1 and 2 are measuring the same quantity $x$ with different noise characteristics determined by $W_1$ and $W_2$.
+Let's consider the sensor fusion architecture shown on figure [[fig:fusion_two_noisy_sensors_weights]] where two sensors (sensor 1 and sensor 2) are measuring the same quantity $x$ with different noise characteristics determined by $N_1(s)$ and $N_2(s)$.
 
-$n_1$ and $n_2$ are white noise (constant power spectral density over all frequencies).
+$\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise:
+#+name: eq:normalized_noise
+\begin{equation}
+  \Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_1}(\omega) = 1
+\end{equation}
 
-#+name: fig:fusion_two_noisy_sensors_with_dyn
+#+name: fig:fusion_two_noisy_sensors_weights
 #+caption: Fusion of two sensors
-[[file:figs-tikz/fusion_two_noisy_sensors_with_dyn.png]]
+[[file:figs-tikz/fusion_two_noisy_sensors_weights.png]]
 
-We consider that the two sensor dynamics $G_1$ and $G_2$ are ideal ($G_1 = G_2 = 1$). We obtain the architecture of figure [[fig:fusion_two_noisy_sensors]].
+We consider that the two sensor dynamics $G_1(s)$ and $G_2(s)$ are ideal:
+#+name: eq:idea_dynamics
+\begin{equation}
+  G_1(s) = G_2(s) = 1
+\end{equation}
 
-#+name: fig:fusion_two_noisy_sensors
+We obtain the architecture of figure [[fig:sensor_fusion_noisy_perfect_dyn]].
+
+#+name: fig:sensor_fusion_noisy_perfect_dyn
 #+caption: Fusion of two sensors with ideal dynamics
-[[file:figs-tikz/fusion_two_noisy_sensors.png]]
+[[file:figs-tikz/sensor_fusion_noisy_perfect_dyn.png]]
 
-$H_1$ and $H_2$ are complementary filters ($H_1 + H_2 = 1$). The goal is to design $H_1$ and $H_2$ such that the effect of the noise sources $n_1$ and $n_2$ has the smallest possible effect on the estimation $\hat{x}$.
+$H_1(s)$ and $H_2(s)$ are complementary filters:
+#+name: eq:comp_filters_property
+\begin{equation}
+  H_1(s) + H_2(s) = 1
+\end{equation}
+
+The goal is to design $H_1(s)$ and $H_2(s)$ such that the effect of the noise sources $\tilde{n}_1$ and $\tilde{n}_2$ has the smallest possible effect on the estimation $\hat{x}$.
 
 We have that the Power Spectral Density (PSD) of $\hat{x}$ is:
-\[ \Gamma_{\hat{x}} = |H_1 W_1|^2 \Gamma_{n_1} + |H_2 W_2|^2 \Gamma_{n_2} \]
+\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \]
 
 And the goal is the minimize the Root Mean Square (RMS) value of $\hat{x}$:
-\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Gamma_{\hat{x}}(\omega) d\omega} \]
+#+name: eq:rms_value_estimation
-
+\begin{equation}
-As $n_1$ and $n_2$ are white noise: $\Gamma_{n_1} = \Gamma_{n_2} = 1$ and we have:
+  \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega}
-\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 W_1|^2(\omega) + |H_2 W_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2 \]
+\end{equation}
-
-Thus, the goal is to design $H_1$ and $H_2$ such that $H_1 + H_2 = 1$ and such that $\left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2$ is minimized.
-
-For that, we will use the $\mathcal{H}_2$ Synthesis.
 
 ** Noise of the sensors
-Let's define the noise characteristics of the two sensors by choosing $W_1$ and $W_2$:
+Let's define the noise characteristics of the two sensors by choosing $N_1$ and $N_2$:
-- Sensor 1 characterized by $W_1$ has low noise at low frequency (for instance a geophone)
+- Sensor 1 characterized by $N_1(s)$ has low noise at low frequency (for instance a geophone)
-- Sensor 2 characterized by $W_2$ has low noise at high frequency (for instance an accelerometer)
+- Sensor 2 characterized by $N_2(s)$ has low noise at high frequency (for instance an accelerometer)
-
-#+begin_src matlab :exports none
-  omegac = 2*pi; G0 = 1e-2; Ginf = 1e-6;
-  W1 = ((sqrt(G0))/(s/omegac + 1))^2;
-
-  omegac = 100*2*pi; G0 = 1e-6; Ginf = 1e-2;
-  W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
-#+end_src
 
 #+begin_src matlab
   omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
-  W1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
+  N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
 
   omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
-  W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
+  N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
 #+end_src
 
 #+begin_src matlab :exports none
   figure;
   hold on;
-  plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), '-', 'DisplayName', '$W_1$');
+  plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$N_1$');
-  plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), '-', 'DisplayName', '$W_2$');
+  plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$N_2$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   xlabel('Frequency [Hz]'); ylabel('Magnitude');
   hold off;
@@ -149,15 +153,21 @@ Let's define the noise characteristics of the two sensors by choosing $W_1$ and
 #+end_src
 
 #+HEADER: :tangle no :exports results :results none :noweb yes
-#+begin_src matlab :var filepath="figs/nosie_characteristics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+#+begin_src matlab :var filepath="figs/noise_characteristics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
   <<plt-matlab>>
 #+end_src
 
-#+NAME: fig:nosie_characteristics_sensors
+#+NAME: fig:noise_characteristics_sensors
-#+CAPTION: Noise Characteristics of the two sensors ([[./figs/nosie_characteristics_sensors.png][png]], [[./figs/nosie_characteristics_sensors.pdf][pdf]])
+#+CAPTION: Noise Characteristics of the two sensors ([[./figs/noise_characteristics_sensors.png][png]], [[./figs/noise_characteristics_sensors.pdf][pdf]])
-[[file:figs/nosie_characteristics_sensors.png]]
+[[file:figs/noise_characteristics_sensors.png]]
 
 ** H-Two Synthesis
+As $\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise: $\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1$ and we have:
+\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 N_1|^2(\omega) + |H_2 N_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \]
+Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ and such that $\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2$ is minimized.
+
+For that, we use the $\mathcal{H}_2$ Synthesis.
+
 We use the generalized plant architecture shown on figure [[fig:h_infinity_optimal_comp_filters]].
 
 #+name: fig:h_infinity_optimal_comp_filters
@@ -165,16 +175,16 @@ We use the generalized plant architecture shown on figure [[fig:h_infinity_optim
 [[file:figs-tikz/h_infinity_optimal_comp_filters.png]]
 
 The transfer function from $[n_1, n_2]$ to $\hat{x}$ is:
-\[ \begin{bmatrix} W_1 H_1 \\ W_2 (1 - H_1) \end{bmatrix} \]
+\[ \begin{bmatrix} N_1 H_1 \\ N_2 (1 - H_1) \end{bmatrix} \]
 If we define $H_2 = 1 - H_1$, we obtain:
-\[ \begin{bmatrix} W_1 H_1 \\ W_2 H_2 \end{bmatrix} \]
+\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \]
 
 Thus, if we minimize the $\mathcal{H}_2$ norm of this transfer function, we minimize the RMS value of $\hat{x}$.
 
 We define the generalized plant $P$ on matlab as shown on figure [[fig:h_infinity_optimal_comp_filters]].
 #+begin_src matlab
-  P = [0   W2  1;
+  P = [0   N2  1;
-       W1 -W2  0];
+       N1 -N2  0];
 #+end_src
 
 And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command.
@@ -182,17 +192,18 @@ And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command.
   [H1, ~, gamma] = h2syn(P, 1, 1);
 #+end_src
 
-What is minimized is =norm([W1*H1,W2*H2], 2)=.
+Finally, we define $H_2(s) = 1 - H_1(s)$.
-
-Finally, we define $H_2 = 1 - H_1$.
 #+begin_src matlab
   H2 = 1 - H1;
 #+end_src
 
-** Analysis
+The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]].
-The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]]. The PSD of the [[fig:psd_sensors_htwo_synthesis]].
+
-Finally, the RMS value of $\hat{x}$ is shown on table [[tab:rms_results]].
+The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. [[fig:psd_sensors_htwo_synthesis]].
-The optimal sensor fusion has permitted to reduced the RMS value of the estimation error by a factor 8 compare to when using only one sensor.
+
+The Cumulative Power Spectrum (CPS) is shown on Fig. [[fig:cps_h2_synthesis]].
+
+The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors.
 
 #+begin_src matlab :exports none
   figure;
@@ -215,14 +226,20 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati
 #+CAPTION: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis ([[./figs/htwo_comp_filters.png][png]], [[./figs/htwo_comp_filters.pdf][pdf]])
 [[file:figs/htwo_comp_filters.png]]
 
+#+begin_src matlab
+  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;
+#+end_src
+
 #+begin_src matlab :exports none
   figure;
   hold on;
-  plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))).^2, '-',  'DisplayName', '$|W_1|^2$');
+  plot(freqs, PSD_S1, '-',  'DisplayName', '$\Phi_{\hat{x}_1}$');
-  plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))).^2, '-',  'DisplayName', '$|W_2|^2$');
+  plot(freqs, PSD_S2, '-',  'DisplayName', '$\Phi_{\hat{x}_2}$');
-  plot(freqs, abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|W_1H_1|^2+|W_2H_2|^2$');
+  plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
   hold off;
   xlim([freqs(1), freqs(end)]);
   legend('location', 'northeast');
@@ -237,77 +254,76 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati
 #+CAPTION: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal ([[./figs/psd_sensors_htwo_synthesis.png][png]], [[./figs/psd_sensors_htwo_synthesis.pdf][pdf]])
 [[file:figs/psd_sensors_htwo_synthesis.png]]
 
-#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-  data2orgtable([norm([W1], 2);norm([W2], 2);norm([W1*H1 + W2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e');
-#+end_src
-
-#+name: tab:rms_results
-#+caption: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters
-#+RESULTS:
-|                       | rms value |
-|-----------------------+-----------|
-| Sensor 1              |   1.1e-02 |
-| Sensor 2              |   1.3e-03 |
-| Optimal Sensor Fusion |   1.5e-04 |
-
-** H-Infinity Synthesis
-*** First idea
-Another objective that we may have is that the noise of the super sensor $n_{SS}$ is following the minimum of the noise of the two sensors $n_1$ and $n_2$:
-\[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \]
-
-In order to obtain that ideal case, we need that the complementary filters be designed such that:
-\begin{align*}
-  & H_1(j\omega) = 1 \text{ and } H_2(j\omega) = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\
-  & H_1(j\omega) = 0 \text{ and } H_2(j\omega) = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega)
-\end{align*}
-
-Which is indeed *impossible in practice*.
-
-We could try to use filters with the highest possible order to approximate that.
-However, we may have robustness and implementation problems.
-
 #+begin_src matlab
-  omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
+  CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1);
-  N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
+  CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2);
-
+  CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2);
-  omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
-  N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
-#+end_src
-
-#+begin_src matlab
-  n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2;
-  W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
-
-  n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2;
-  W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
 #+end_src
 
 #+begin_src matlab :exports none
   figure;
   hold on;
-  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1 = \frac{N_1}{N_2}$');
+  plot(freqs, CPS_S1, '-',  'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end))));
-  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2 = \frac{N_2}{N_1}$');
+  plot(freqs, CPS_S2, '-',  'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end))));
-  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end))));
-  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
   hold off;
-  xlim([freqs(1), freqs(end)]);
+  xlim([2e-1, freqs(end)]);
-  legend('location', 'northeast');
+  ylim([1e-10 1e-5]);
+  legend('location', 'southeast');
+#+end_src
+
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/cps_h2_synthesis.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+NAME: fig:cps_h2_synthesis
+#+CAPTION: Cumulative Power Spectrum of individual sensors and super sensor using the $\mathcal{H}_2$ synthesis ([[./figs/cps_h2_synthesis.png][png]], [[./figs/cps_h2_synthesis.pdf][pdf]])
+[[file:figs/cps_h2_synthesis.png]]
+
+** H-Infinity Synthesis - method A
+Another objective that we may have is that the noise of the super sensor $n_{SS}$ is following the minimum of the noise of the two sensors $n_1$ and $n_2$:
+\[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \]
+
+In order to obtain that ideal case, we need that the complementary filters be designed such that:
+\begin{align*}
+  & |H_1(j\omega)| = 1 \text{ and } |H_2(j\omega)| = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\
+  & |H_1(j\omega)| = 0 \text{ and } |H_2(j\omega)| = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega)
+\end{align*}
+
+Which is indeed impossible in practice.
+
+We could try to approach that with the $\mathcal{H}_\infty$ synthesis by using high order filters.
+
+As shown on Fig. [[fig:noise_characteristics_sensors]], the frequency where the two sensors have the same noise level is around 9Hz.
+We will thus choose weighting functions such that the merging frequency is around 9Hz.
+
+The weighting functions used as well as the obtained complementary filters are shown in Fig. [[fig:weights_comp_filters_Hinfa]].
+
+#+begin_src matlab
+  n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2;
+  W1a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
+
+  n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2;
+  W2a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
 #+end_src
 
 #+begin_src matlab
-  P = [W1 -W1;
+  P = [W1a -W1a;
-       0   W2;
+       0    W2a;
-       1   0];
+       1    0];
 #+end_src
 
 And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
 #+begin_src matlab :results output replace :exports both
-  [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+  [H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
 #+end_src
 
 #+RESULTS:
 #+begin_example
-[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+[H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
 Resetting value of Gamma min based on D_11, D_12, D_21 terms
 
 Test bounds:      0.1000 <  gamma  <=  10500.0000
@@ -343,7 +359,7 @@ Test bounds:      0.1000 <  gamma  <=  10500.0000
 #+end_example
 
 #+begin_src matlab
-  H1 = 1 - H2;
+  H1a = 1 - H2a;
 #+end_src
 
 #+begin_src matlab :exports none
@@ -352,14 +368,14 @@ Test bounds:      0.1000 <  gamma  <=  10500.0000
   ax1 = subplot(2,1,1);
   hold on;
   set(gca,'ColorOrderIndex',1)
-  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
+  plot(freqs, 1./abs(squeeze(freqresp(W1a, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
   set(gca,'ColorOrderIndex',2)
-  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
+  plot(freqs, 1./abs(squeeze(freqresp(W2a, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
 
   set(gca,'ColorOrderIndex',1)
-  plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
+  plot(freqs, abs(squeeze(freqresp(H1a, 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(H2a, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
 
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -371,9 +387,9 @@ Test bounds:      0.1000 <  gamma  <=  10500.0000
   ax2 = subplot(2,1,2);
   hold on;
   set(gca,'ColorOrderIndex',1)
-  plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H1a, freqs, 'Hz'))), '-');
   set(gca,'ColorOrderIndex',2)
-  plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H2a, freqs, 'Hz'))), '-');
   hold off;
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   set(gca, 'XScale', 'log');
@@ -384,135 +400,100 @@ Test bounds:      0.1000 <  gamma  <=  10500.0000
   xticks([0.1, 1, 10, 100, 1000]);
 #+end_src
 
-#+begin_src matlab :exports none
+#+HEADER: :tangle no :exports results :results none :noweb yes
-  figure;
+#+begin_src matlab :var filepath="figs/weights_comp_filters_Hinfa.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
-  hold on;
+  <<plt-matlab>>
-  plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2, '-',  'DisplayName', '$|N_1|^2$');
-  plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2, '-',  'DisplayName', '$|N_2|^2$');
-  plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|N_1H_1|^2+|N_2H_2|^2$');
-  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Magnitude');
-  hold off;
-  xlim([freqs(1), freqs(end)]);
-  legend('location', 'northeast');
 #+end_src
 
-#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+#+NAME: fig:weights_comp_filters_Hinfa
-  data2orgtable([norm([N1], 2);norm([N2], 2);norm([N1*H1 + N2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e');
+#+CAPTION: Weights and Complementary Fitlers obtained ([[./figs/weights_comp_filters_Hinfa.png][png]], [[./figs/weights_comp_filters_Hinfa.pdf][pdf]])
+[[file:figs/weights_comp_filters_Hinfa.png]]
+
+We then compute the Power Spectral Density as well as the Cumulative Power Spectrum.
+
+#+begin_src matlab
+  PSD_Ha = abs(squeeze(freqresp(N1*H1a, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2a, freqs, 'Hz'))).^2;
+  CPS_Ha = 1/pi*cumtrapz(2*pi*freqs, PSD_Ha);
 #+end_src
 
-#+RESULTS:
+** H-Infinity Synthesis - method B
-|                       | rms value |
-|-----------------------+-----------|
-| Sensor 1              |   1.1e-02 |
-| Sensor 2              |   1.3e-03 |
-| Optimal Sensor Fusion |   3.2e-04 |
-
-*** Second idea
 We have that:
-\[ \Phi_{n_{ss}} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2} \]
+\[ \Phi_{\hat{x}}(\omega) = \left|H_1(j\omega) N_1(j\omega)\right|^2 + \left|H_2(j\omega) N_2(j\omega)\right|^2 \]
 
-We model the Power Spectral Densities of the two sensors by weighting functions $N_1(s)$ and $N_2(s)$.
+Then, at frequencies where $|H_1(j\omega)| < |H_2(j\omega)|$ we would like that $|N_1(j\omega)| = 1$ and $|N_2(j\omega)| = 0$ as we discussed before.
-\[ \Phi_{n_{ss}} = \left|H_1 N_1\right|^2 + \left|H_2\right N_2|^2 \]
+Then $|H_1 N_1|^2 + |H_2 N_2|^2 = |N_1|^2$.
 
-Then, at frequencies where $|H_1| < |H_2|$ we would like that $|N_1| = 1$ and $|N_2| = 0$ as we discussed before.
+We know that this is impossible in practice. A more realistic choice is to design $H_2(s)$ such that when $|N_2(j\omega)| > |N_1(j\omega)|$, we have that:
-Then $|H_1 N_1|^2 + |H_2 N_2|^2 = |H_1 N_1|^2$.
-
-However, if we design $H_2$ such that when $|N_1| < |N_2|$, we have that:
 \[ |H_2 N_2|^2 = \epsilon |H_1 N_1|^2 \]
+
+Which is equivalent to have (by supposing $|H_1| \approx 1$):
+\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \]
+
 And we have:
 \begin{align*}
-  \Phi_{n_{ss}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\
+  \Phi_{\hat{x}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\
-                &= (1 + \epsilon)\left|H_1 N_1\right|^2 \\
+                &= (1 + \epsilon) \left| H_1 N_1 \right|^2 \\
-                &\approx \left|H_1 N_1\right|^2
+                &\approx \left|N_1\right|^2
 \end{align*}
 
-We have then that:
+Similarly, we design $H_1(s)$ such that at frequencies where $|N_1| > |N_2|$:
-\[ |H_2 N_2| = \sqrt{\epsilon} |H_1 N_1| \]
-We suppose that $|H_1| = 1$:
-\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \]
-We can use that for the maximum magnitude of $|H_2|$ at frequencies where $|N_1| < |N_2|$
-
-Similarly, we have that the maximum magnitude of $|H_1|$ at frequencies where $|N_1| > |N_2|$ is:
 \[ |H_1| = \sqrt{\epsilon} \frac{|N_2|}{|N_1|} \]
 
-We could take $\epsilon = 1$, then we increase the noise of the super sensor just by a factor $\sqrt{2}$ over the all bandwidth from the idea case.
+For instance, is we take $\epsilon = 1$, then the PSD of $\hat{x}$ is increased by just by a factor $\sqrt{2}$ over the all frequencies from the idea case.
 
-This could be used as weighting functions for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
+We use this as the weighting functions for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
+
+The weighting function and the obtained complementary filters are shown in Fig. [[fig:weights_comp_filters_Hinfb]].
 
 #+begin_src matlab
-  omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
+  epsilon = 2;
-  N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
 
-  omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
+  W1b = 1/epsilon*N1/N2;
-  N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
+  W2b = 1/epsilon*N2/N1;
+
+  W1b = W1b/(1 + s/2/pi/1000); % this is added so that it is proper
 #+end_src
 
 #+begin_src matlab
-  W1 = N1/N2;
+  P = [W1b -W1b;
-  W2 = N2/N1;
+       0    W2b;
-
+       1    0];
-  W1 = W1/(1 + s/2/pi/1000); % this is added so that it is proper
-#+end_src
-
-#+begin_src matlab :exports none
-  figure;
-  hold on;
-  plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1 = \frac{N_1}{N_2}$');
-  plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2 = \frac{N_2}{N_1}$');
-  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Magnitude');
-  hold off;
-  xlim([freqs(1), freqs(end)]);
-  legend('location', 'northeast');
-#+end_src
-
-#+begin_src matlab
-  P = [W1 -W1;
-       0   W2;
-       1   0];
 #+end_src
 
 And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
 #+begin_src matlab :results output replace :exports both
-  [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+  [H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
 #+end_src
 
 #+RESULTS:
 #+begin_example
-[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-Test bounds:      0.0000 <  gamma  <=   2625.0000
+Test bounds:      0.0000 <  gamma  <=     32.8125
 
   gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
-2.625e+03   1.8e+01 -2.3e-12   6.3e+00   -1.7e-14    0.0000    p
+   32.812   1.8e+01   3.4e-10   6.3e+00   -2.9e-13    0.0000    p
-1.313e+03   1.8e+01 -3.3e-12   6.3e+00   -5.9e-13    0.0000    p
+   16.406   1.8e+01   3.4e-10   6.3e+00   -1.2e-15    0.0000    p
-  656.250   1.8e+01 -1.8e-12   6.3e+00   -1.2e-13    0.0000    p
+    8.203   1.8e+01   3.3e-10   6.3e+00   -2.6e-13    0.0000    p
-  328.125   1.8e+01 -8.3e-13   6.3e+00   -7.5e-14    0.0000    p
+    4.102   1.8e+01   3.3e-10   6.3e+00   -2.1e-13    0.0000    p
-  164.063   1.8e+01 -3.8e-13   6.3e+00   -6.1e-14    0.0000    p
+    2.051   1.7e+01   3.4e-10   6.3e+00   -7.2e-16    0.0000    p
-   82.031   1.8e+01 -3.3e-22   6.3e+00   -5.4e-14    0.0000    p
+    1.025   1.6e+01 -1.3e+06#  6.3e+00   -8.3e-14    0.0000    f
-   41.016   1.8e+01 -3.4e-12   6.3e+00   -1.4e-16    0.0000    p
+    1.538   1.7e+01   3.4e-10   6.3e+00   -2.0e-13    0.0000    p
-   20.508   1.8e+01 -6.4e-14   6.3e+00   -4.4e-13    0.0000    p
+    1.282   1.7e+01   3.4e-10   6.3e+00   -7.9e-17    0.0000    p
-   10.254   1.8e+01 -1.4e-13   6.3e+00   -4.9e-14    0.0000    p
+    1.154   1.7e+01   3.6e-10   6.3e+00   -1.8e-13    0.0000    p
-    5.127   1.7e+01 -1.1e-12   6.3e+00   -1.6e-13    0.0000    p
+    1.089   1.7e+01 -3.4e+06#  6.3e+00   -1.7e-13    0.0000    f
-    2.563   1.7e+01 -6.5e-13   6.3e+00   -7.3e-14    0.0000    p
+    1.122   1.7e+01 -1.0e+07#  6.3e+00   -3.2e-13    0.0000    f
-    1.282   1.4e+01 -8.9e+04#  6.3e+00   -1.6e-14    0.0000    f
+    1.138   1.7e+01 -1.3e+08#  6.3e+00   -1.8e-13    0.0000    f
-    1.923   1.6e+01 -4.6e+05#  6.3e+00   -2.1e-13    0.0000    f
+    1.146   1.7e+01   3.2e-10   6.3e+00   -3.0e-13    0.0000    p
-    2.243   1.7e+01 -1.9e+06#  6.3e+00   -9.9e-14    0.0000    f
+    1.142   1.7e+01   5.5e-10   6.3e+00   -2.8e-13    0.0000    p
-    2.403   1.7e+01 -3.7e-11   6.3e+00   -2.1e-12    0.0000    p
+    1.140   1.7e+01 -1.5e-10   6.3e+00   -2.3e-13    0.0000    p
-    2.323   1.7e+01 -4.6e+06#  6.3e+00   -5.8e-14    0.0000    f
+    1.139   1.7e+01 -4.8e+08#  6.3e+00   -6.2e-14    0.0000    f
-    2.363   1.7e+01 -1.2e+07#  6.3e+00   -3.3e-14    0.0000    f
+    1.139   1.7e+01   1.3e-09   6.3e+00   -8.9e-17    0.0000    p
-    2.383   1.7e+01 -6.8e+07#  6.3e+00   -1.2e-13    0.0000    f
-    2.393   1.7e+01 -5.5e-11   6.3e+00   -1.8e-12    0.0000    p
-    2.388   1.7e+01   1.3e-22   6.3e+00   -2.5e-15    0.0000    p
-    2.386   1.7e+01 -1.5e+08#  6.3e+00   -1.4e-13    0.0000    f
-    2.387   1.7e+01 -4.0e+08#  6.3e+00   -1.0e-13    0.0000    f
-    2.388   1.7e+01 -2.1e+09#  6.3e+00   -4.2e-13    0.0000    f
 
- Gamma value achieved:     2.3882
+ Gamma value achieved:     1.1390
 #+end_example
 
 #+begin_src matlab
-  H1 = 1 - H2;
+  H1b = 1 - H2b;
 #+end_src
 
 #+begin_src matlab :exports none
@@ -521,14 +502,14 @@ Test bounds:      0.0000 <  gamma  <=   2625.0000
   ax1 = subplot(2,1,1);
   hold on;
   set(gca,'ColorOrderIndex',1)
-  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
+  plot(freqs, 1./abs(squeeze(freqresp(W1b, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
   set(gca,'ColorOrderIndex',2)
-  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
+  plot(freqs, 1./abs(squeeze(freqresp(W2b, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
 
   set(gca,'ColorOrderIndex',1)
-  plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
+  plot(freqs, abs(squeeze(freqresp(H1b, 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(H2b, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
 
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -540,9 +521,9 @@ Test bounds:      0.0000 <  gamma  <=   2625.0000
   ax2 = subplot(2,1,2);
   hold on;
   set(gca,'ColorOrderIndex',1)
-  plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H1b, freqs, 'Hz'))), '-');
   set(gca,'ColorOrderIndex',2)
-  plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H2b, freqs, 'Hz'))), '-');
   hold off;
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   set(gca, 'XScale', 'log');
@@ -553,32 +534,96 @@ Test bounds:      0.0000 <  gamma  <=   2625.0000
   xticks([0.1, 1, 10, 100, 1000]);
 #+end_src
 
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/weights_comp_filters_Hinfb.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+NAME: fig:weights_comp_filters_Hinfb
+#+CAPTION: Weights and Complementary Fitlers obtained ([[./figs/weights_comp_filters_Hinfb.png][png]], [[./figs/weights_comp_filters_Hinfb.pdf][pdf]])
+[[file:figs/weights_comp_filters_Hinfb.png]]
+
+#+begin_src matlab
+  PSD_Hb = abs(squeeze(freqresp(N1*H1b, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2b, freqs, 'Hz'))).^2;
+  CPS_Hb = 1/pi*cumtrapz(2*pi*freqs, PSD_Hb);
+#+end_src
+
+** Comparison of the methods
+The three methods are now compared.
+
+The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. [[fig:comparison_psd_noise]].
+
+The Cumulative Power Spectrum for the same sensors are shown on Fig. [[fig:comparison_cps_noise]].
+
+The RMS value of the obtained super sensors are shown on table [[tab:rms_results]].
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([norm([N1], 2) ; norm([N2], 2) ; norm([N1*H1, N2*H2], 2) ; norm([N1*H1a, N2*H2a], 2) ; norm([N1*H1b, N2*H2b], 2)], {'Sensor 1', 'Sensor 2', 'H2 Fusion', 'H-Infinity a', 'H-Infinity b'}, {'rms value'}, ' %.1e');
+#+end_src
+
+#+name: tab:rms_results
+#+caption: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters
+#+RESULTS:
+|              | rms value |
+|--------------+-----------|
+| Sensor 1     |   1.3e-03 |
+| Sensor 2     |   1.3e-03 |
+| H2 Fusion    |   1.2e-04 |
+| H-Infinity a |   2.4e-04 |
+| H-Infinity b |   1.4e-04 |
+
+
 #+begin_src matlab :exports none
   figure;
   hold on;
-  plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2, '-',  'DisplayName', '$|N_1|^2$');
+  plot(freqs, PSD_S1, '-',  'DisplayName', '$\Phi_{\hat{x}_1}$');
-  plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2, '-',  'DisplayName', '$|N_2|^2$');
+  plot(freqs, PSD_S2, '-',  'DisplayName', '$\Phi_{\hat{x}_2}$');
-  plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|N_1H_1|^2+|N_2H_2|^2$');
+  plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$');
+  plot(freqs, PSD_Ha, 'k--', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},a}$');
+  plot(freqs, PSD_Hb, 'k-.', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},b}$');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
   hold off;
   xlim([freqs(1), freqs(end)]);
   legend('location', 'northeast');
 #+end_src
 
-#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+#+HEADER: :tangle no :exports results :results none :noweb yes
-  data2orgtable([norm([N1], 2);norm([N2], 2);norm([N1*H1 + N2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e');
+#+begin_src matlab :var filepath="figs/comparison_psd_noise.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
 #+end_src
 
-#+RESULTS:
+#+NAME: fig:comparison_psd_noise
-|                       | rms value |
+#+CAPTION: Comparison of the obtained Power Spectral Density using the three methods ([[./figs/comparison_psd_noise.png][png]], [[./figs/comparison_psd_noise.pdf][pdf]])
-|-----------------------+-----------|
+[[file:figs/comparison_psd_noise.png]]
-| Sensor 1              |   1.1e-02 |
-| Sensor 2              |   1.3e-03 |
-| Optimal Sensor Fusion |   1.9e-04 |
 
-*** Conclusion
+#+begin_src matlab :exports none
-From the above two experiments with $\mathcal{H}_\infty$ it still seems that the $\mathcal{H}_2$ synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the $\mathcal{H}_2$ norm)
+  figure;
+  hold on;
+  plot(freqs, CPS_S1, '-',  'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end))));
+  plot(freqs, CPS_S2, '-',  'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end))));
+  plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end))));
+  plot(freqs, CPS_Ha, 'k--', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, a}} = %.1e$', sqrt(CPS_Ha(end))));
+  plot(freqs, CPS_Hb, 'k-.', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, b}} = %.1e$', sqrt(CPS_Hb(end))));
+  set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
+  hold off;
+  xlim([2e-1, freqs(end)]);
+  ylim([1e-10 1e-5]);
+  legend('location', 'southeast');
+#+end_src
+
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/comparison_cps_noise.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+NAME: fig:comparison_cps_noise
+#+CAPTION: Comparison of the obtained Cumulative Power Spectrum using the three methods ([[./figs/comparison_cps_noise.png][png]], [[./figs/comparison_cps_noise.pdf][pdf]])
+[[file:figs/comparison_cps_noise.png]]
+
+** Conclusion
+From the above complementary filter design with the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ synthesis, it still seems that the $\mathcal{H}_2$ synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the $\mathcal{H}_2$ norm).
 
 * Robustness to sensor dynamics uncertainty
   :PROPERTIES:

tikz/figs/fusion_two_noisy_sensors.svg

@@ -1,165 +1,264 @@
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+</svg>

tikz/index.org

@@ -108,7 +108,7 @@ Configuration file is accessible [[file:config.org][here]].
 #+begin_src latex :file h_infinity_optimal_comp_filters.pdf :tangle figs/h_infinity_optimal_comp_filters.tex
   <<tikz_default>>
   \begin{tikzpicture}
-     \node[block={5.0cm}{3.0cm}, dashed] (P) {};
+     \node[block={5.0cm}{3.0cm}, fill=black!20!white, dashed] (P) {};
      \node[above] at (P.north) {$P$};
 
      \coordinate[] (inputn1)  at ($(P.south west)!0.8!(P.north west) + (-\cdist, 0)$);
@@ -118,18 +118,18 @@ Configuration file is accessible [[file:config.org][here]].
      \coordinate[] (outputx) at ($(P.south east)!0.5!(P.north east) + ( \cdist, 0)$);
      \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( \cdist, 0)$);
 
-     \node[block, right=1.5 of inputn1] (W1){$W_1$};
+     \node[block, right=1.5 of inputn1] (N1){$N_1$};
-     \node[block, right=1.5 of inputn2] (W2){$W_2$};
+     \node[block, right=1.5 of inputn2] (N2){$N_2$};
-     \node[addb={+}{}{}{}{-}, right=of W1] (sub) {};
+     \node[addb={+}{}{}{}{-}, right=of N1] (sub) {};
-     \node[addb, right=2 of W2] (add) {};
+     \node[addb, right=2 of N2] (add) {};
 
      \node[block, below=of P] (H1) {$H_1$};
 
-     \draw[->] (inputn1)node[above right]{$n_1$} -- (W1.west);
+     \draw[->] (inputn1)node[above right]{$\tilde{n}_1$} -- (N1.west);
-     \draw[->] (inputn2)node[above right]{$n_2$} -- (W2.west);
+     \draw[->] (inputn2)node[above right]{$\tilde{n}_2$} -- (N2.west);
-     \draw[->] (W1) -- (sub.west);
+     \draw[->] (N1) -- (sub.west);
-     \draw[->] (W2) -| (sub.south);
+     \draw[->] (N2) -| (sub.south);
-     \draw[->] (W2-|sub.south) node[branch]{} -- (add.west);
+     \draw[->] (N2-|sub.south) node[branch]{} -- (add.west);
      \draw[->] (sub.east) -- ++(0.5, 0) |- ($(outputv) + (-0.3, 0)$) |- (H1.east);
      \draw[->] (H1.west) -| ($(inputu) + (0.3, 0)$) -| (add.south);
      \draw[->] (add.east) -- (outputx) node[above left]{$\hat{x}$};
@@ -142,28 +142,40 @@ Configuration file is accessible [[file:config.org][here]].
 [[file:figs/h_infinity_optimal_comp_filters.png]]
 * Fusion of two noisy sensors
 #+begin_src latex :file fusion_two_noisy_sensors.pdf :tangle figs/fusion_two_noisy_sensors.tex
+  <<tikz_default>>
+
   \begin{tikzpicture}
     \node[branch] (x) at (0, 0);
-    \node[addb, above right=1.5 and 1 of x](add1){};
+    \node[block, above right=0.5 and 0.5 of x](G1){$G_1(s)$};
-    \node[addb, below right=1.5 and 1 of x](add2){};
+    \node[block, below right=0.5 and 0.5 of x](G2){$G_2(s)$};
-    \node[block, above=0.5 of add1](W1){$W_1$};
+    \node[addb, right=0.8 of G1](add1){};
-    \node[block, above=0.5 of add2](W2){$W_2$};
+    \node[addb, right=0.8 of G2](add2){};
-    \node[block, right=1 of add1](H1){$H_1$};
+    \node[block, right=0.8 of add1](H1){$H_1(s)$};
-    \node[block, right=1 of add2](H2){$H_2$};
+    \node[block, right=0.8 of add2](H2){$H_2(s)$};
-    \node[addb, right=4 of x](add){};
+    \node[addb, right=5 of x](add){};
 
-    \draw[] ($(x)+(-1, 0)$) node[above right]{$x$} -- (x);
+    \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
-    \draw[->] (x) |- (add1.west);
+    \draw[->] (x.center) |- (G1.west);
-    \draw[->] (x) |- (add2.west);
+    \draw[->] (x.center) |- (G2.west);
-    \draw[->] (W1.south) -- (add1.north);
+    \draw[->] (G1.east) -- (add1.west);
-    \draw[->] (W2.south) -- (add2.north);
+    \draw[->] (G2.east) -- (add2.west);
-    \draw[<-] (W1.north) -- ++(0, 0.8)node[below right]{$n_1$};
+    \draw[<-] (add1.north) -- ++(0, 0.8)node[below right](n1){$n_1$};
-    \draw[<-] (W2.north) -- ++(0, 0.8)node[below right]{$n_2$};
+    \draw[<-] (add2.north) -- ++(0, 0.8)node[below right](n2){$n_2$};
     \draw[->] (add1.east) -- (H1.west);
     \draw[->] (add2.east) -- (H2.west);
     \draw[->] (H1) -| (add.north);
     \draw[->] (H2) -| (add.south);
-    \draw[->] (add.east) -- ++(1, 0) node[above left]{$\hat{x}$};
+    \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
+
+    \begin{scope}[on background layer]
+      \node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
+      \node[below left] at (supersensor.north east) {Super Sensor};
+
+      \node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
+      \node[below right] at (sensor1.north west) {Sensor 1};
+      \node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
+      \node[below right] at (sensor2.north west) {Sensor 2};
+    \end{scope}
   \end{tikzpicture}
 #+end_src
 
@@ -171,6 +183,97 @@ Configuration file is accessible [[file:config.org][here]].
 #+caption: Fusion of two noisy sensors ([[./figs/fusion_two_noisy_sensors.png][png]], [[./figs/fusion_two_noisy_sensors.pdf][pdf]], [[./figs/fusion_two_noisy_sensors.tex][tex]]).
 #+RESULTS:
 [[file:figs/fusion_two_noisy_sensors.png]]
+
+* Fusion of two noisy sensors with perfect dynamics
+#+begin_src latex :file sensor_fusion_noisy_perfect_dyn.pdf :tangle figs/sensor_fusion_noisy_perfect_dyn.tex
+  <<tikz_default>>
+  \begin{tikzpicture}
+    \node[branch] (x) at (0, 0);
+    \node[addb, above right=1.2 and 1.8 of x](addn1){};
+    \node[addb, below right=1.2 and 1.8 of x](addn2){};
+    \node[block, above=0.5 of addn1](N1) {$N_1(s)$};
+    \node[block, above=0.5 of addn2](N2) {$N_2(s)$};
+    \node[block, right=1.2 of addn1](H1){$H_1(s)$};
+    \node[block, right=1.2 of addn2](H2){$H_2(s)$};
+    \node[addb, right=4.5 of x](add){};
+
+    \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
+    \draw[->] (x.center) |- (addn1.west);
+    \draw[->] (x.center) |- (addn2.west);
+    \draw[->] (addn1.east) -- (H1.west)node[above left]{$\hat{x}_1$};
+    \draw[->] (addn2.east) -- (H2.west)node[above left]{$\hat{x}_2$};
+    \draw[->] ($(N1.north)+(0,0.7)$) node[below right](n1){$\tilde{n}_1$} -- (N1.north);
+    \draw[->] ($(N2.north)+(0,0.7)$) node[below right](n2){$\tilde{n}_2$} -- (N2.north);
+    \draw[->] (N1.south) -- (addn1.north)node[above right]{$n_1$};
+    \draw[->] (N2.south) -- (addn2.north)node[above right]{$n_2$};
+    \draw[->] (H1.east) -| (add.north);
+    \draw[->] (H2.east) -| (add.south);
+    \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
+
+    \begin{scope}[on background layer]
+      \node[fit={($(addn2.south-|x)+(-0.2, -0.4)$) ($(n1.north-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
+      \node[below left] at (supersensor.north east) {Super Sensor};
+      \node[block, fit={($(x|-addn1.south)+(0.2, -0.2)$) ($(n1.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
+      \node[below right] at (sensor1.north west) {Sensor 1};
+      \node[block, fit={($(x|-addn2.south)+(0.2, -0.2)$) ($(n2.north-|N2.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
+      \node[below right] at (sensor2.north west) {Sensor 2};
+    \end{scope}
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:sensor_fusion_noisy_perfect_dyn
+#+caption: Sensor fusion architecture with sensor dynamics uncertainty ([[./figs/sensor_fusion_noisy_perfect_dyn.png][png]], [[./figs/sensor_fusion_noisy_perfect_dyn.pdf][pdf]], [[./figs/sensor_fusion_noisy_perfect_dyn.tex][tex]]).
+#+RESULTS:
+[[file:figs/sensor_fusion_noisy_perfect_dyn.png]]
+
+* Fusion of two noisy sensors with sensors weights
+#+begin_src latex :file fusion_two_noisy_sensors_weights.pdf :tangle figs/fusion_two_noisy_sensors_weights.tex
+  <<tikz_default>>
+
+  \begin{tikzpicture}
+    \node[branch] (x) at (0, 0);
+    \node[block, above right=1 and 0.5 of x](G1){$G_1(s)$};
+    \node[block, below right=1 and 0.5 of x](G2){$G_2(s)$};
+    \node[addb, right=0.8 of G1](add1){};
+    \node[addb, right=0.8 of G2](add2){};
+    \node[block, above=0.5 of add1](N1){$N_1(s)$};
+    \node[block, above=0.5 of add2](N2){$N_2(s)$};
+    \node[block, right=0.9 of add1](H1){$H_1(s)$};
+    \node[block, right=0.9 of add2](H2){$H_2(s)$};
+    \node[addb, right=5 of x](add){};
+
+    \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
+    \draw[->] (x.center) |- (G1.west);
+    \draw[->] (x.center) |- (G2.west);
+    \draw[->] (G1.east) -- (add1.west);
+    \draw[->] (G2.east) -- (add2.west);
+    \draw[<-] (N1.north) -- ++(0, 0.6)node[below right](n1){$\tilde{n}_1$};
+    \draw[<-] (N2.north) -- ++(0, 0.6)node[below right](n2){$\tilde{n}_2$};
+    \draw[->] (N1.south) -- (add1.north)node[above right]{$n_1$};
+    \draw[->] (N2.south) -- (add2.north)node[above right]{$n_2$};
+    \draw[->] (add1.east) -- (H1.west);
+    \draw[->] (add2.east) -- (H2.west);
+    \draw[->] (H1) -| (add.north);
+    \draw[->] (H2) -| (add.south);
+    \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
+
+    \begin{scope}[on background layer]
+      \node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
+      \node[below left] at (supersensor.north east) {Super Sensor};
+
+      \node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
+      \node[below right] at (sensor1.north west) {Sensor 1};
+      \node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
+      \node[below right] at (sensor2.north west) {Sensor 2};
+    \end{scope}
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:fusion_two_noisy_sensors_weights
+#+caption: Fusion of two noisy sensors ([[./figs/fusion_two_noisy_sensors_weights.png][png]], [[./figs/fusion_two_noisy_sensors_weights.pdf][pdf]], [[./figs/fusion_two_noisy_sensors_weights.tex][tex]]).
+#+RESULTS:
+[[file:figs/fusion_two_noisy_sensors_weights.png]]
+
 * Fusion of two noisy signals
 #+begin_src latex :file fusion_two_signals.pdf :tangle figs/fusion_two_signals.tex
   \begin{tikzpicture}