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@@ -24,7 +24,6 @@
 
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 * Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
 <<sec:optimal_fusion>>
 
+** Sensor Model
+
 ** Sensor Fusion Architecture
 
 #+name: fig:sensor_fusion_noise_arch
@@ -96,10 +99,70 @@
 #+attr_latex: :scale 1
 [[file:figs/sensor_fusion_noise_arch.pdf]]
 
+Let note $\Phi$ the PSD.
+$\tilde{n}_1$ and $\tilde{n}_2$ are white noise with unitary power spectral density:
+\begin{equation}
+  \Phi_{\tilde{n}_i}(\omega) = 1
+\end{equation}
+
+\begin{equation}
+  \hat{x} = \left( H_1 \hat{G}_1^{-1} G_1 + H_2 \hat{G}_2^{-1} G_2 \right) x + \left( H_1 \hat{G}_1^{-1} N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} N_2 \right) \tilde{n}_2
+\end{equation}
+
+Suppose the sensor dynamical model $\hat{G}_i$ is perfect:
+\begin{equation}
+  \hat{G}_i = G_i
+\end{equation}
+
+Complementary Filters
+\begin{equation}
+  H_1(s) + H_2(s) = 1
+\end{equation}
+
+
+\begin{equation}
+  \hat{x} = x + \left( H_1 \hat{G}_1^{-1} N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} N_2 \right) \tilde{n}_2
+\end{equation}
+
+Perfect dynamics + filter noise
+
+
 ** Super Sensor Noise
 
+Let's note $n$ the super sensor noise.
+
+Its PSD is determined by:
+\begin{equation}
+  \Phi_n(\omega) = \left| H_1 \hat{G}_1^{-1} N_1 \right|^2 + \left| H_2 \hat{G}_2^{-1} N_2 \right|^2
+\end{equation}
+
 ** $\mathcal{H}_2$ Synthesis of Complementary Filters
 
+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 noise  $n$ of the estimation $\hat{x}$.
+
+And the goal is the minimize the Root Mean Square (RMS) value of $n$:
+#+name: eq:rms_value_estimation
+\begin{equation}
+  \sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega}
+\end{equation}
+
+
+The goal is to minimize the $\mathcal{H}_2$ norm between $w$ and $[z_1\ z_2]$:
+\begin{equation}
+  \left\| \begin{matrix} w/z_1 \\ w/z_2 \end{matrix} \right\|_2 = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 (1 - H_2) \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2
+\end{equation}
+
+By defining:
+\begin{equation}
+  H_1 = 1 - H_2
+\end{equation}
+
+\begin{equation}
+  \left\| \begin{matrix} w/z_1 \\ w/z_2 \end{matrix} \right\|_2 = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2 = \sqrt{\int_0^\infty \left| H_1 \hat{G}_1^{-1} N_1 \right|^2 + \left| H_2 \hat{G}_2^{-1} N_2 \right|^2 d\omega} = \sigma_n
+\end{equation}
+
+The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RMS value of the super sensor noise.
+
 #+name: fig:h_two_optimal_fusion
 #+caption: Figure caption
 #+attr_latex: :scale 1