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@@ -14,6 +14,13 @@
#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
#+AUTHOR: @@latex:thomas.dehaeze@esrf.fr@@
#+AUTHOR: @@latex:}\and@@
#+AUTHOR: @@latex:\IEEEauthorblockN{Verma Mohit}@@
#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{BEAMS Department}\\@@
#+AUTHOR: @@latex:\textit{Free University of Brussels}, Belgium\\@@
#+AUTHOR: @@latex:\textit{Precision Mechatronics Laboratory} \\@@
#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
#+AUTHOR: @@latex:mohitverma.serc@csir.res.in@@
#+AUTHOR: @@latex:}\and@@
#+AUTHOR: @@latex:\IEEEauthorblockN{Collette Christophe}@@
#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{BEAMS Department}\\@@
#+AUTHOR: @@latex:\textit{Free University of Brussels}, Belgium\\@@

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% Created 2020-10-25 dim. 10:05
% Created 2020-10-26 lun. 18:25
% Intended LaTeX compiler: pdflatex
\documentclass[conference]{IEEEtran}
\usepackage[utf8]{inputenc}
@@ -34,8 +34,8 @@
\renewcommand{\citedash}{--}
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
\usepackage{showframe}
\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 }}
\date{2020-10-25}
\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{Verma Mohit} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ mohitverma.serc@csir.res.in }\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 }}
\date{2020-10-26}
\title{Optimal and Robust Sensor Fusion}
\begin{document}
@@ -50,7 +50,7 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{IEEEkeywords}
\section{Introduction}
\label{sec:org2a4e2c2}
\label{sec:org2820158}
\label{sec:introduction}
\begin{itemize}
@@ -61,11 +61,11 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{itemize}
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
\label{sec:orgb0fb3f0}
\label{sec:org2513ad9}
\label{sec:optimal_fusion}
\subsection{Sensor Model}
\label{sec:org9e4a17b}
\label{sec:orgbcc6cb6}
Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model_noise}).
The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function \(G_i(s)\).
@@ -101,7 +101,7 @@ In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:orge7841b3}
\label{sec:orgdb526ec}
Let's now consider two sensors measuring the same physical quantity \(x\) but with different dynamics \((G_1, G_2)\) and noise characteristics \((N_1, N_2)\) (Figure \ref{fig:sensor_fusion_noise_arch}).
The noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) are considered to be uncorrelated.
@@ -138,7 +138,7 @@ In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the n
\end{equation}
\subsection{Super Sensor Noise}
\label{sec:orge42a7c0}
\label{sec:org48c0d52}
Let's note \(n\) the super sensor noise.
\begin{equation}
n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
@@ -152,7 +152,7 @@ As the noise of both sensors are considered to be uncorrelated, the PSD of the s
It is clear that the PSD of the super sensor depends on the norm of the complementary filters.
\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
\label{sec:org150fd28}
\label{sec:org0d9384e}
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 \(n\):
@@ -196,7 +196,7 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
\end{figure}
\subsection{Example}
\label{sec:org4abe5c3}
\label{sec:org99002de}
\begin{figure}[htbp]
\centering
@@ -232,7 +232,7 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
\end{figure}
\subsection{Robustness Problem}
\label{sec:org1116fe0}
\label{sec:org262893f}
\begin{figure}[htbp]
\centering
@@ -247,11 +247,11 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
\end{figure}
\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
\label{sec:orgcf4e02a}
\label{sec:org7c9047e}
\label{sec:robust_fusion}
\subsection{Representation of Sensor Dynamical Uncertainty}
\label{sec:org45ee620}
\label{sec:org7bd4379}
In Section \ref{sec:optimal_fusion}, the model \(\hat{G}_i(s)\) of the sensor was considered to be perfect.
In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
@@ -271,7 +271,7 @@ The sensor can then be represented as shown in Figure \ref{fig:sensor_model_unce
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:orgec549bc}
\label{sec:org0b8ce2b}
Let's consider the sensor fusion architecture shown in Figure \ref{fig:sensor_fusion_arch_uncertainty} where the dynamical uncertainties of both sensors are included.
The super sensor estimate is then:
@@ -296,7 +296,7 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
\end{figure}
\subsection{Super Sensor Dynamical Uncertainty}
\label{sec:org6867184}
\label{sec:org725af92}
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 \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure \ref{fig:uncertainty_set_super_sensor}.
@@ -311,7 +311,7 @@ And we can see that the dynamical uncertainty of the super sensor is equal to th
At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
\label{sec:org9cbbe5b}
\label{sec:org941ed72}
In order for the fusion to be ``robust'', meaning no phase drop will be induced in the super sensor dynamics,
The goal is to design two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the super sensor noise uncertainty is kept reasonably small.
@@ -357,7 +357,7 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\end{figure}
\subsection{Example}
\label{sec:orgfc0d330}
\label{sec:org7df520f}
\begin{figure}[htbp]
\centering
@@ -392,11 +392,11 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:org81d3977}
\label{sec:org75a038a}
\label{sec:optimal_robust_fusion}
\subsection{Sensor with noise and model uncertainty}
\label{sec:orgcd51fc4}
\label{sec:org3810d6b}
We wish now to combine the two previous synthesis, that is to say
The sensors are now modelled by a white noise with unitary PSD \(\tilde{n}_i\) shaped by a LTI transfer function \(N_i(s)\).
@@ -417,7 +417,7 @@ Multiplying by the inverse of the nominal model of the sensor dynamics gives an
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:org32c4c98}
\label{sec:org3758b1e}
For reason of space, the blocks \(\hat{G}_i\) and \(\hat{G}_i^{-1}\) are omitted.
@@ -444,7 +444,7 @@ The estimate \(\hat{x}\) of \(x\)
\end{figure}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:org73e0335}
\label{sec:org06317f4}
The synthesis objective is to generate two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the uncertainty associated with the super sensor is kept reasonably small and such that the RMS value of super sensors noise is minimized.
@@ -479,7 +479,7 @@ The synthesis objective is to:
\end{figure}
\subsection{Example}
\label{sec:orga68c808}
\label{sec:org42ee165}
\begin{figure}[htbp]
\centering
@@ -506,27 +506,27 @@ The synthesis objective is to:
\end{figure}
\section{Experimental Validation}
\label{sec:orga4af6ce}
\label{sec:orge381a2a}
\label{sec:experimental_validation}
\subsection{Experimental Setup}
\label{sec:orgab10fd3}
\label{sec:org473ab00}
\subsection{Sensor Noise and Dynamical Uncertainty}
\label{sec:orgc6d5bae}
\label{sec:orgebcb65d}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orga5c7815}
\label{sec:org0259d19}
\subsection{Super Sensor Noise and Dynamical Uncertainty}
\label{sec:orgd7da409}
\label{sec:orgdb5d29f}
\section{Conclusion}
\label{sec:org6eddbc8}
\label{sec:org07df454}
\label{sec:conclusion}
\section{Acknowledgment}
\label{sec:org44ed488}
\label{sec:org7b7e461}
\bibliography{ref}
\end{document}