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matlab/index.org
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@ -193,7 +193,7 @@ The true sensor dynamics $G_i(s)$ is then described by eqref:eq:sensor_dynamics_
The weights $W_i(s)$ representing the dynamical uncertainty are defined below and their magnitude is shown in Figure [[fig:sensors_uncertainty_weights]]. The weights $W_i(s)$ representing the dynamical uncertainty are defined below and their magnitude is shown in Figure [[fig:sensors_uncertainty_weights]].
#+begin_src matlab #+begin_src matlab
W1 = createWeight('n', 2, 'w0', 2*pi*3, 'G0', 2, 'G1', 0.1, 'Gc', 1) * ... W1 = createWeight('n', 2, 'w0', 2*pi*3, 'G0', 2, 'G1', 0.1, 'Gc', 1) * ...
createWeight('n', 2, 'w0', 2*pi*1e3, 'G0', 1, 'G1', 4/0.1, 'Gc', 1/0.1); createWeight('n', 2, 'w0', 2*pi*1e3, 'G0', 1, 'G1', 4/0.1, 'Gc', 1/0.1);
W2 = createWeight('n', 2, 'w0', 2*pi*1e2, 'G0', 0.05, 'G1', 4, 'Gc', 1); W2 = createWeight('n', 2, 'w0', 2*pi*1e2, 'G0', 0.05, 'G1', 4, 'Gc', 1);

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@ -14,6 +14,13 @@
#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@ #+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
#+AUTHOR: @@latex:thomas.dehaeze@esrf.fr@@ #+AUTHOR: @@latex:thomas.dehaeze@esrf.fr@@
#+AUTHOR: @@latex:}\and@@ #+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:\IEEEauthorblockN{Collette Christophe}@@
#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{BEAMS Department}\\@@ #+AUTHOR: @@latex:\IEEEauthorblockA{\textit{BEAMS Department}\\@@
#+AUTHOR: @@latex:\textit{Free University of Brussels}, Belgium\\@@ #+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 % Intended LaTeX compiler: pdflatex
\documentclass[conference]{IEEEtran} \documentclass[conference]{IEEEtran}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -34,8 +34,8 @@
\renewcommand{\citedash}{--} \renewcommand{\citedash}{--}
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
\usepackage{showframe} \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 }} \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{2020-10-25} \date{2020-10-26}
\title{Optimal and Robust Sensor Fusion} \title{Optimal and Robust Sensor Fusion}
\begin{document} \begin{document}
@ -50,7 +50,7 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{IEEEkeywords} \end{IEEEkeywords}
\section{Introduction} \section{Introduction}
\label{sec:org2a4e2c2} \label{sec:org2820158}
\label{sec:introduction} \label{sec:introduction}
\begin{itemize} \begin{itemize}
@ -61,11 +61,11 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{itemize} \end{itemize}
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis} \section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
\label{sec:orgb0fb3f0} \label{sec:org2513ad9}
\label{sec:optimal_fusion} \label{sec:optimal_fusion}
\subsection{Sensor Model} \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}). 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 \(G_i(s)\). 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} \end{figure}
\subsection{Sensor Fusion Architecture} \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}). 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. 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} \end{equation}
\subsection{Super Sensor Noise} \subsection{Super Sensor Noise}
\label{sec:orge42a7c0} \label{sec:org48c0d52}
Let's note \(n\) the super sensor noise. Let's note \(n\) the super sensor noise.
\begin{equation} \begin{equation}
n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2 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. 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} \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 \(\hat{x}\). 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 \(n\): 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} \end{figure}
\subsection{Example} \subsection{Example}
\label{sec:org4abe5c3} \label{sec:org99002de}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -232,7 +232,7 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
\end{figure} \end{figure}
\subsection{Robustness Problem} \subsection{Robustness Problem}
\label{sec:org1116fe0} \label{sec:org262893f}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -247,11 +247,11 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
\end{figure} \end{figure}
\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis} \section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
\label{sec:orgcf4e02a} \label{sec:org7c9047e}
\label{sec:robust_fusion} \label{sec:robust_fusion}
\subsection{Representation of Sensor Dynamical Uncertainty} \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 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. 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} \end{figure}
\subsection{Sensor Fusion Architecture} \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. 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: The super sensor estimate is then:
@ -296,7 +296,7 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
\end{figure} \end{figure}
\subsection{Super Sensor Dynamical Uncertainty} \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}. 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. 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} \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, 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. 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} \end{figure}
\subsection{Example} \subsection{Example}
\label{sec:orgfc0d330} \label{sec:org7df520f}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \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} \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} \label{sec:optimal_robust_fusion}
\subsection{Sensor with noise and model uncertainty} \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 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 \(N_i(s)\). 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} \end{figure}
\subsection{Sensor Fusion Architecture} \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. 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} \end{figure}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} \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. 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} \end{figure}
\subsection{Example} \subsection{Example}
\label{sec:orga68c808} \label{sec:org42ee165}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -506,27 +506,27 @@ The synthesis objective is to:
\end{figure} \end{figure}
\section{Experimental Validation} \section{Experimental Validation}
\label{sec:orga4af6ce} \label{sec:orge381a2a}
\label{sec:experimental_validation} \label{sec:experimental_validation}
\subsection{Experimental Setup} \subsection{Experimental Setup}
\label{sec:orgab10fd3} \label{sec:org473ab00}
\subsection{Sensor Noise and Dynamical Uncertainty} \subsection{Sensor Noise and Dynamical Uncertainty}
\label{sec:orgc6d5bae} \label{sec:orgebcb65d}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orga5c7815} \label{sec:org0259d19}
\subsection{Super Sensor Noise and Dynamical Uncertainty} \subsection{Super Sensor Noise and Dynamical Uncertainty}
\label{sec:orgd7da409} \label{sec:orgdb5d29f}
\section{Conclusion} \section{Conclusion}
\label{sec:org6eddbc8} \label{sec:org07df454}
\label{sec:conclusion} \label{sec:conclusion}
\section{Acknowledgment} \section{Acknowledgment}
\label{sec:org44ed488} \label{sec:org7b7e461}
\bibliography{ref} \bibliography{ref}
\end{document} \end{document}