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@@ -143,22 +143,21 @@ The goal is to design $H_1(s)$ and $H_2(s)$ such that the effect of the noise so
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And the goal is the minimize the Root Mean Square (RMS) value of $n$:
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#+name: eq:rms_value_estimation
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\begin{equation}
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\sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega}
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\sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2
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\end{equation}
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The goal is to minimize the $\mathcal{H}_2$ norm between $w$ and $[z_1\ z_2]$:
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\begin{equation}
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\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
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\end{equation}
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By defining:
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\begin{equation}
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H_1 = 1 - H_2
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\end{equation}
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Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ and such that $\left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2$ is minimized.
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\begin{equation}
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\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
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \begin{bmatrix}
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\hat{G}_1^{-1} N_1 & -\hat{G}_1^{-1} N_1 \\
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0 & \hat{G}_2^{-1} N_2 \\
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1 & 0
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\end{bmatrix} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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\end{equation}
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The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RMS value of the super sensor noise.
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@@ -177,6 +176,21 @@ The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RM
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** Representation of Sensor Dynamical Uncertainty
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** Sensor Fusion Architecture
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\begin{equation}
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\hat{x} = \left( H_1 \hat{G}_1^{-1} (1 + w_1 \Delta_1) G_1 + H_2 \hat{G}_2^{-1} (1 + w_2 \Delta_2) G_2 \right) x
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\end{equation}
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with $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$.
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Suppose the model inversion is equal to the nominal model:
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\begin{equation}
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\hat{G}_i = G_i
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\end{equation}
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\begin{equation}
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\hat{x} = \left( 1 + H_1 w_1 \Delta_1 + H_2 w_2 \Delta_2 \right) x
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\end{equation}
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#+name: fig:sensor_fusion_arch_uncertainty
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#+caption: Figure caption
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#+attr_latex: :scale 1
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@@ -184,6 +198,8 @@ The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RM
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** Super Sensor Dynamical Uncertainty
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The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency $\omega$ is bounded in the complex plane by a circle centered on 1 and with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$.
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#+name: fig:uncertainty_set_super_sensor
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#+caption: Figure caption
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#+attr_latex: :scale 1
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@@ -191,6 +207,8 @@ The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RM
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** $\mathcal{H_\infty}$ Synthesis of Complementary Filters
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In order to minimize the super sensor dynamical uncertainty
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#+name: fig:h_infinity_robust_fusion
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#+caption: Figure caption
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#+attr_latex: :scale 1
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paper/paper.tex
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paper/paper.tex
@@ -1,4 +1,4 @@
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% Created 2020-08-17 lun. 18:06
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% Created 2020-09-22 mar. 09:51
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% Intended LaTeX compiler: pdflatex
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\documentclass[conference]{IEEEtran}
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\usepackage[utf8]{inputenc}
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@@ -22,7 +22,6 @@
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\usepackage{siunitx}
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\IEEEoverridecommandlockouts
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\usepackage{cite}
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\usepackage{showframe}
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\usepackage{amsmath,amssymb,amsfonts}
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\usepackage{algorithmic}
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\usepackage{graphicx}
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@@ -34,8 +33,9 @@
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\usepackage{import, hyperref}
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\renewcommand{\citedash}{--}
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\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
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\usepackage{showframe}
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\author{\IEEEauthorblockN{Dehaeze Thomas} \IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\ Grenoble, France\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ thomas.dehaeze@esrf.fr }\and \IEEEauthorblockN{Collette Christophe} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ ccollett@ulb.ac.be }}
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\date{2020-08-17}
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\date{2020-09-22}
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\title{Robust and Optimal Sensor Fusion}
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\begin{document}
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@@ -50,15 +50,18 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{IEEEkeywords}
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\section{Introduction}
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\label{sec:orgc7e68f1}
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\label{sec:org2c6d9ef}
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\label{sec:introduction}
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\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
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\label{sec:org9dea05a}
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\label{sec:org5aa0717}
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\label{sec:optimal_fusion}
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\subsection{Sensor Model}
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\label{sec:org8f1053d}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org02189c8}
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\label{sec:orgc40deb4}
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\begin{figure}[htbp]
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\centering
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@@ -66,11 +69,70 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\caption{\label{fig:sensor_fusion_noise_arch}Figure caption}
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\end{figure}
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Let note \(\Phi\) the PSD.
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\(\tilde{n}_1\) and \(\tilde{n}_2\) are white noise with unitary power spectral density:
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\begin{equation}
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\Phi_{\tilde{n}_i}(\omega) = 1
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\end{equation}
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\begin{equation}
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\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
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\end{equation}
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Suppose the sensor dynamical model \(\hat{G}_i\) is perfect:
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\begin{equation}
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\hat{G}_i = G_i
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\end{equation}
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Complementary Filters
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\begin{equation}
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H_1(s) + H_2(s) = 1
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\end{equation}
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\begin{equation}
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\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
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\end{equation}
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Perfect dynamics + filter noise
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\subsection{Super Sensor Noise}
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\label{sec:org75bf123}
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\label{sec:orgf4b6ca9}
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Let's note \(n\) the super sensor noise.
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Its PSD is determined by:
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\begin{equation}
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\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
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\end{equation}
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\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
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\label{sec:org2372e4d}
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\label{sec:org5773772}
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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 \(\hat{x}\).
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And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
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\begin{equation}
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\label{eq:rms_value_estimation}
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\sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2
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\end{equation}
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Thus, the goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) and such that \(\left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2\) is minimized.
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\begin{equation}
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \begin{bmatrix}
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\hat{G}_1^{-1} N_1 & -\hat{G}_1^{-1} N_1 \\
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0 & \hat{G}_2^{-1} N_2 \\
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1 & 0
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\end{bmatrix} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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\end{equation}
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The \(\mathcal{H}_2\) synthesis of the complementary filters thus minimized the RMS value of the super sensor noise.
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\begin{figure}[htbp]
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\centering
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@@ -79,17 +141,33 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{figure}
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\subsection{Example}
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\label{sec:orgf58d3c3}
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\label{sec:orged06a27}
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\subsection{Robustness Problem}
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\label{sec:org25143ef}
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\label{sec:org62b375f}
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\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
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\label{sec:orge0ab922}
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\label{sec:orgef03e7c}
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\label{sec:robust_fusion}
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\subsection{Representation of Sensor Dynamical Uncertainty}
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\label{sec:orgde1ca0e}
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\label{sec:org9c9762b}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org9572e70}
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\begin{equation}
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\hat{x} = \left( H_1 \hat{G}_1^{-1} (1 + w_1 \Delta_1) G_1 + H_2 \hat{G}_2^{-1} (1 + w_2 \Delta_2) G_2 \right) x
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\end{equation}
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with \(\Delta_i\) is any transfer function satisfying \(\| \Delta_i \|_\infty < 1\).
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Suppose the model inversion is equal to the nominal model:
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\begin{equation}
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\hat{G}_i = G_i
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\end{equation}
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\begin{equation}
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\hat{x} = \left( 1 + H_1 w_1 \Delta_1 + H_2 w_2 \Delta_2 \right) x
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\end{equation}
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\begin{figure}[htbp]
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\centering
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@@ -98,7 +176,9 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{figure}
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\subsection{Super Sensor Dynamical Uncertainty}
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\label{sec:org7f51f8d}
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\label{sec:orgb9ee83e}
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The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\).
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\begin{figure}[htbp]
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\centering
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@@ -107,7 +187,9 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{figure}
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\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
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\label{sec:org04f04f3}
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\label{sec:orgf4e3c8e}
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In order to minimize the super sensor dynamical uncertainty
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\begin{figure}[htbp]
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\centering
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@@ -116,14 +198,14 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{figure}
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\subsection{Example}
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\label{sec:orgbeb8891}
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\label{sec:org4f663bc}
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\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:orgf739e93}
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\label{sec:org150b612}
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\label{sec:optimal_robust_fusion}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org8fde0b7}
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\label{sec:org9bc69b7}
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\begin{figure}[htbp]
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\centering
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@@ -132,10 +214,10 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{figure}
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\subsection{Synthesis Objective}
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\label{sec:org91a1cb3}
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\label{sec:orgbc5ac30}
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\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:orgcd9ab73}
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\label{sec:org541ef02}
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\begin{figure}[htbp]
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\centering
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@@ -144,30 +226,30 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{figure}
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\subsection{Example}
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\label{sec:orgc2f161d}
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\label{sec:org046c2e2}
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\section{Experimental Validation}
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\label{sec:org9c8a2db}
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\label{sec:org1bb9cff}
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\label{sec:experimental_validation}
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\subsection{Experimental Setup}
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\label{sec:orga1abc9b}
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\label{sec:org2c63393}
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\subsection{Sensor Noise and Dynamical Uncertainty}
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\label{sec:org9d92c2b}
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\label{sec:orgb0c6496}
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\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:orgbda5f50}
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\label{sec:orgfb3986f}
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\subsection{Super Sensor Noise and Dynamical Uncertainty}
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\label{sec:orgb1aa4c9}
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\label{sec:orgfd5c11e}
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\section{Conclusion}
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\label{sec:org468c862}
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\label{sec:orgda418fa}
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\label{sec:conclusion}
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\section{Acknowledgment}
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\label{sec:org07d8818}
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\label{sec:orgabdae67}
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\bibliography{ref}
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\end{document}
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