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Add equations - Section 1

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Thomas Dehaeze 2020-08-19 10:43:56 +02:00
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#+LATEX_HEADER: \IEEEoverridecommandlockouts
#+LATEX_HEADER: \usepackage{cite}
#+LATEX_HEADER: \usepackage{showframe}
#+LATEX_HEADER: \usepackage{amsmath,amssymb,amsfonts}
#+LATEX_HEADER: \usepackage{algorithmic}
#+LATEX_HEADER: \usepackage{graphicx}
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#+LATEX_HEADER: \usepackage{import, hyperref}
#+LATEX_HEADER: \renewcommand{\citedash}{--}
#+LATEX_HEADER_EXTRA: \usepackage{showframe}
#+LATEX_HEADER: \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
\bibliographystyle{IEEEtran}
@ -89,6 +90,8 @@
* 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