Add equivalent super sensor analysis
This commit is contained in:
parent
e04f9df47a
commit
fc2acd4c64
@ -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:
|
||||
|
Loading…
Reference in New Issue
Block a user