Add equivalent super sensor analysis

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Thomas Dehaeze 2019-08-29 11:23:50 +02:00
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@ -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: