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Thomas Dehaeze
2021-05-20 11:14:26 +02:00
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commit c114cdbc58
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@@ -196,9 +196,11 @@ is given by $$\label{eq:noise_filtering_psd}
\Phi_{\delta x} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2}$$
where, $\Phi_{\delta x}$ is the PSD of estimation error, $\Phi_{n_1}$
and $\Phi_{n_2}$ are the PSDs of the noise associated with the
individual sensor. It can be seen that the estimation error's PSD
individual sensor.
It can be seen that the estimation error's PSD
depends on the PSD of the noise in individual sensor as well as the norm
of the complementary filters. Therefore, by properly shaping the norm of
of the complementary filters.
Therefore, by properly shaping the norm of
the complementary filters, it is possible to minimize the noise of the
super sensor noise.
@@ -236,19 +238,8 @@ The dynamics of the super sensor now depends on the weighting functions
($w_1(s),w_2(s)$) and the complementary filters ($H_1(s),H_2(s)$).
The robust stability of the fusion can be studied graphically (refer
Figure [[#fig:uncertainty_set_super_sensor][3]]). The frequency response
of the fusion output is plotted in a complex plane. The unity transfer
function leads to a point $(1,0)$ located on the real axis. The
uncertainty associated with first sensor at a particular frequency is
represented by a circle with the center at (1,0) and radius $|w_1H_1|$.
The uncertainty associated with the second is also represented using a
circle centered at any point on the circle representing uncertainty
associated with the first sensor and radius equal to $|w_2H_2|$.
Therefore, the overall uncertainty of the fusion is represented with a
circle centered at (1,0) and radius equal to $|w_1H_1|+|w_2H_2|$. The
maximum phase difference that can result from the fusion is found by
drawing a tangent from the origin to the uncertainty circle of super
sensor. Mathematically, the maximum phase difference at frequency
Figure [[#fig:uncertainty_set_super_sensor][3]]). The frequency response of the fusion output is plotted in a complex plane. The unity transfer function leads to a point $(1,0)$ located on the real axis. The uncertainty associated with first sensor at a particular frequency is represented by a circle with the center at (1,0) and radius $|w_1H_1|$. The uncertainty associated with the second is also represented using a circle centered at any point on the circle representing uncertainty associated with the first sensor and radius equal to $|w_2H_2|$. Therefore, the overall uncertainty of the fusion is represented with a circle centered at (1,0) and radius equal to $|w_1H_1|+|w_2H_2|$.
Mathematically, the maximum phase difference at frequency
$\omega$ that can result from fusion is given by
$$\label{eq:max_phase_uncertainty}
\Delta\phi(\omega) = \arcsin\left( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \right)$$
@@ -271,10 +262,8 @@ transfer functions ($|w_i|$) representing sensor uncertainty is large.
* Design formulation using $\mathcal{H}_\infty$ synthesis
** Introduction :ignore:
In this section, the shaping of complementary filters is expressed as an
optimal $\mathcal{H}_{\infty}$ synthesis problem. The synthesis goal is
to shape the frequency response of the filters such that they satisfy
the design requirements presented in Section [[*Complementary filters
In this section, the shaping of complementary filters is expressed as an optimal $\mathcal{H}_{\infty}$ synthesis problem.
The synthesis goal is to shape the frequency response of the filters such that they satisfy the design requirements presented in Section [[*Complementary filters
requirements][2]].
** Synthesis problem formulation