Add equivalent super sensor analysis

matlab/index.org

@@ -819,7 +819,89 @@ We see that the blue complementary filters with a lower maximum norm permits to
 [[file:figs/tf_super_sensor_comp.png]]
 
 ** TODO More Complete example with model uncertainty
-* Complementary filters using analytical formula
+* Equivalent Super Sensor
+
+The goal here is to find the parameters of a single sensor that would best represent a super sensor.
+
+** Sensor Fusion Architecture
+Let consider figure [[fig:sensor_fusion_full]] where two sensors are merged.
+The dynamic uncertainty of each sensor is represented by a weight $w_i(s)$, the frequency characteristics each of the sensor noise is represented by the weights $N_i(s)$.
+The noise sources $\tilde{n}_i$ are considered to be white noise: $\Phi_{\tilde{n}_i}(\omega) = 1, \ \forall\omega$.
+
+#+name: fig:sensor_fusion_full
+#+caption: Sensor fusion architecture ([[./figs/sensor_fusion_full.png][png]], [[./figs/sensor_fusion_full.pdf][pdf]]).
+#+RESULTS:
+[[file:figs-tikz/sensor_fusion_full.png]]
+
+
+\begin{align*}
+  \hat{x} &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 + \Big(H_1(s) \big(1 + w_1(s) \Delta_1(s)\big) + H_2(s) \big(1 + w_2(s) \Delta_2(s)\big)\Big) x \\
+          &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 + \big(1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s)\big) x
+\end{align*}
+
+To the dynamics of the super sensor is:
+\begin{equation}
+  \frac{\hat{x}}{x} = 1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s)
+\end{equation}
+
+And the noise of the super sensor is:
+\begin{equation}
+  n_{ss} = H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2
+\end{equation}
+
+** Equivalent Configuration
+We try to determine $w_{ss}(s)$ and $N_{ss}(s)$ such that the sensor on figure [[fig:sensor_fusion_equivalent]] is equivalent to the super sensor of figure [[fig:sensor_fusion_full]].
+
+#+name: fig:sensor_fusion_equivalent
+#+caption: Equivalent Super Sensor ([[./figs/sensor_fusion_equivalent.png][png]], [[./figs/sensor_fusion_equivalent.pdf][pdf]]).
+#+RESULTS:
+[[file:figs-tikz/sensor_fusion_equivalent.png]]
+
+** Model the uncertainty of the super sensor
+At each frequency $\omega$, the uncertainty set of the super sensor shown on figure [[fig:sensor_fusion_full]] is a circle centered on $1$ with a radius equal to $|H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|$ on the complex plane.
+The uncertainty set of the sensor shown on figure [[fig:sensor_fusion_equivalent]] is a circle centered on $1$ with a radius equal to $|w_{ss}(j\omega)|$ on the complex plane.
+
+Ideally, we want to find a weight $w_{ss}(s)$ so that:
+#+begin_important
+  \[ |w_{ss}(j\omega)| = |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \]
+#+end_important
+
+** Model the noise of the super sensor
+The PSD of the estimation $\hat{x}$ when $x = 0$ of the configuration shown on figure [[fig:sensor_fusion_full]] is:
+\begin{align*}
+  \Phi_{\hat{x}}(\omega) &= | H_1(j\omega) N_1(j\omega) |^2 \Phi_{\tilde{n}_1} + | H_2(j\omega) N_2(j\omega) |^2 \Phi_{\tilde{n}_2} \\
+                         &= | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2
+\end{align*}
+
+The PSD of the estimation $\hat{x}$ when $x = 0$ of the configuration shown on figure [[fig:sensor_fusion_equivalent]] is:
+\begin{align*}
+  \Phi_{\hat{x}}(\omega) &= | N_{ss}(j\omega) |^2 \Phi_{\tilde{n}} \\
+                         &= | N_{ss}(j\omega) |^2
+\end{align*}
+
+Ideally, we want to find a weight $N_{ss}(s)$ such that:
+#+begin_important
+  \[ |N_{ss}(j\omega)|^2 = | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \]
+#+end_important
+
+** First guess
+We could choose
+\begin{align*}
+  w_{ss}(s) &= H_1(s) w_1(s) + H_2(s) w_2(s) \\
+  N_{ss}(s) &= H_1(s) N_1(s) + H_2(s) N_2(s)
+\end{align*}
+
+But we would have:
+\begin{align*}
+  |w_{ss}(j\omega)| &= |H_1(j\omega) w_1(j\omega) + H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \\
+                    &\neq |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega
+\end{align*}
+and
+\begin{align*}
+  |N_{ss}(j\omega)|^2 &= | H_1(j\omega) N_1(j\omega) + H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \\
+                      &\neq | H_1(j\omega) N_1(j\omega)|^2 + |H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \\
+\end{align*}
+
 * Methods of complementary filter synthesis
 ** Complementary filters using analytical formula
   :PROPERTIES: