From 86ad85f17ced6129ed65ed42cb520fbde73d946c Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 22 Sep 2020 10:15:26 +0200 Subject: [PATCH] Add figure caption --- paper/paper.org | 31 +++++++++++++----- paper/paper.pdf | Bin 250212 -> 305579 bytes paper/paper.tex | 85 ++++++++++++++++++++++++++++-------------------- 3 files changed, 71 insertions(+), 45 deletions(-) diff --git a/paper/paper.org b/paper/paper.org index f237e61..42886de 100644 --- a/paper/paper.org +++ b/paper/paper.org @@ -95,7 +95,7 @@ ** Sensor Fusion Architecture #+name: fig:sensor_fusion_noise_arch -#+caption: Figure caption +#+caption: Sensor Fusion Architecture with sensor noise #+attr_latex: :scale 1 [[file:figs/sensor_fusion_noise_arch.pdf]] @@ -105,8 +105,12 @@ $\tilde{n}_1$ and $\tilde{n}_2$ are white noise with unitary power spectral dens \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 +\begin{split} + \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{split} \end{equation} Suppose the sensor dynamical model $\hat{G}_i$ is perfect: @@ -163,7 +167,7 @@ Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1 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 +#+caption: Generalized plant $P_{\mathcal{H}_2}$ used for the $\mathcal{H}_2$ synthesis of complementary filters #+attr_latex: :scale 1 [[file:figs/h_two_optimal_fusion.pdf]] @@ -176,9 +180,18 @@ The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RM ** Representation of Sensor Dynamical Uncertainty +Suppose that the sensor dynamics $G_i(s)$ can be modelled by a nominal d +\begin{equation} + G_i(s) = \hat{G}_i(s) \left( 1 + w_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega +\end{equation} + + ** Sensor Fusion Architecture \begin{equation} - \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 +\begin{split} + \hat{x} = \Big( {} & H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + w_1 \Delta_1) \\ + + & H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + w_2 \Delta_2) \Big) x +\end{split} \end{equation} with $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$. @@ -192,7 +205,7 @@ Suppose the model inversion is equal to the nominal model: \end{equation} #+name: fig:sensor_fusion_arch_uncertainty -#+caption: Figure caption +#+caption: Sensor Fusion Architecture with sensor model uncertainty #+attr_latex: :scale 1 [[file:figs/sensor_fusion_arch_uncertainty.pdf]] @@ -201,7 +214,7 @@ Suppose the model inversion is equal to the nominal model: 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 $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$. #+name: fig:uncertainty_set_super_sensor -#+caption: Figure caption +#+caption: Super Sensor model uncertainty displayed in the complex plane #+attr_latex: :scale 1 [[file:figs/uncertainty_set_super_sensor.pdf]] @@ -210,7 +223,7 @@ The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency In order to minimize the super sensor dynamical uncertainty #+name: fig:h_infinity_robust_fusion -#+caption: Figure caption +#+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters #+attr_latex: :scale 1 [[file:figs/h_infinity_robust_fusion.pdf]] @@ -222,7 +235,7 @@ In order to minimize the super sensor dynamical uncertainty ** Sensor Fusion Architecture #+name: fig:sensor_fusion_arch_full -#+caption: Figure caption +#+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty #+attr_latex: :scale 1 [[file:figs/sensor_fusion_arch_full.pdf]] @@ -231,7 +244,7 @@ In order to minimize the super sensor dynamical uncertainty ** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis #+name: fig:mixed_h2_hinf_synthesis -#+caption: Figure caption +#+caption: Generalized plant $P_{\mathcal{H}_2/\matlcal{H}_\infty}$ used for the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of complementary filters #+attr_latex: :scale 1 [[file:figs/mixed_h2_hinf_synthesis.pdf]] diff --git a/paper/paper.pdf b/paper/paper.pdf index 15d7ccbda511db47a22a9f90ce5a2ae215931c0a..c10dc2978038ec2b6d49c789195b2d45b94605e0 100644 GIT binary patch delta 105639 zcmZ^qb8seI*Y0E6<~`BGwr$(CGZRhj*uG=iHYT<;u`}Vsc5>$VzVp2G)~P!G^;NyQ zch|06Yp=DgU+)*7RxV)F>BB>EurPy6plE=b`Z{ieTo``KwQ2keI`HTlQWdJ$+uz2Q zWyc2`W9cG^bgK69okLyr!rsr<+YX^n)0-%dhS>JNVkEy8zMjGuXLU_DKb#?b&p~CH 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\section{Introduction} -\label{sec:org2c6d9ef} +\label{sec:org4ebc807} \label{sec:introduction} \section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis} -\label{sec:org5aa0717} +\label{sec:org86da8fa} \label{sec:optimal_fusion} \subsection{Sensor Model} -\label{sec:org8f1053d} +\label{sec:org60743ab} \subsection{Sensor Fusion Architecture} -\label{sec:orgc40deb4} +\label{sec:org49f3948} \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/sensor_fusion_noise_arch.pdf} -\caption{\label{fig:sensor_fusion_noise_arch}Figure caption} +\caption{\label{fig:sensor_fusion_noise_arch}Sensor Fusion Architecture with sensor noise} \end{figure} Let note \(\Phi\) the PSD. @@ -75,8 +75,12 @@ Let note \(\Phi\) the PSD. \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 +\begin{split} + \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{split} \end{equation} Suppose the sensor dynamical model \(\hat{G}_i\) is perfect: @@ -98,7 +102,7 @@ Perfect dynamics + filter noise \subsection{Super Sensor Noise} -\label{sec:orgf4b6ca9} +\label{sec:org06ff958} Let's note \(n\) the super sensor noise. @@ -108,7 +112,7 @@ Its PSD is determined by: \end{equation} \subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters} -\label{sec:org5773772} +\label{sec:orgeaad969} 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 \(\hat{x}\). @@ -137,26 +141,35 @@ The \(\mathcal{H}_2\) synthesis of the complementary filters thus minimized the \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/h_two_optimal_fusion.pdf} -\caption{\label{fig:h_two_optimal_fusion}Figure caption} +\caption{\label{fig:h_two_optimal_fusion}Generalized plant \(P_{\mathcal{H}_2}\) used for the \(\mathcal{H}_2\) synthesis of complementary filters} \end{figure} \subsection{Example} -\label{sec:orged06a27} +\label{sec:org50664f6} \subsection{Robustness Problem} -\label{sec:org62b375f} +\label{sec:orgaa5f7af} \section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis} -\label{sec:orgef03e7c} +\label{sec:org88ac630} \label{sec:robust_fusion} \subsection{Representation of Sensor Dynamical Uncertainty} -\label{sec:org9c9762b} +\label{sec:orgde90433} + +Suppose that the sensor dynamics \(G_i(s)\) can be modelled by a nominal d +\begin{equation} + G_i(s) = \hat{G}_i(s) \left( 1 + w_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega +\end{equation} + \subsection{Sensor Fusion Architecture} -\label{sec:org9572e70} +\label{sec:orgda3fb09} \begin{equation} - \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 +\begin{split} + \hat{x} = \Big( {} & H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + w_1 \Delta_1) \\ + + & H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + w_2 \Delta_2) \Big) x +\end{split} \end{equation} with \(\Delta_i\) is any transfer function satisfying \(\| \Delta_i \|_\infty < 1\). @@ -172,84 +185,84 @@ Suppose the model inversion is equal to the nominal model: \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/sensor_fusion_arch_uncertainty.pdf} -\caption{\label{fig:sensor_fusion_arch_uncertainty}Figure caption} +\caption{\label{fig:sensor_fusion_arch_uncertainty}Sensor Fusion Architecture with sensor model uncertainty} \end{figure} \subsection{Super Sensor Dynamical Uncertainty} -\label{sec:orgb9ee83e} +\label{sec:orgc9ca84c} 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 \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\). \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/uncertainty_set_super_sensor.pdf} -\caption{\label{fig:uncertainty_set_super_sensor}Figure caption} +\caption{\label{fig:uncertainty_set_super_sensor}Super Sensor model uncertainty displayed in the complex plane} \end{figure} \subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters} -\label{sec:orgf4e3c8e} +\label{sec:orgbb494ca} In order to minimize the super sensor dynamical uncertainty \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/h_infinity_robust_fusion.pdf} -\caption{\label{fig:h_infinity_robust_fusion}Figure caption} +\caption{\label{fig:h_infinity_robust_fusion}Generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) synthesis of complementary filters} \end{figure} \subsection{Example} -\label{sec:org4f663bc} +\label{sec:orgad1fefd} \section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} -\label{sec:org150b612} +\label{sec:orgfb16ef1} \label{sec:optimal_robust_fusion} \subsection{Sensor Fusion Architecture} -\label{sec:org9bc69b7} +\label{sec:orgd611f0b} \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/sensor_fusion_arch_full.pdf} -\caption{\label{fig:sensor_fusion_arch_full}Figure caption} +\caption{\label{fig:sensor_fusion_arch_full}Super Sensor Fusion with both sensor noise and sensor model uncertainty} \end{figure} \subsection{Synthesis Objective} -\label{sec:orgbc5ac30} +\label{sec:org567ad90} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} -\label{sec:org541ef02} +\label{sec:org42ee907} \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/mixed_h2_hinf_synthesis.pdf} -\caption{\label{fig:mixed_h2_hinf_synthesis}Figure caption} +\caption{\label{fig:mixed_h2_hinf_synthesis}Generalized plant \(P_{\mathcal{H}_2/\matlcal{H}_\infty}\) used for the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of complementary filters} \end{figure} \subsection{Example} -\label{sec:org046c2e2} +\label{sec:org3967eb3} \section{Experimental Validation} -\label{sec:org1bb9cff} +\label{sec:org06c0515} \label{sec:experimental_validation} \subsection{Experimental Setup} -\label{sec:org2c63393} +\label{sec:orgeaa87ec} \subsection{Sensor Noise and Dynamical Uncertainty} -\label{sec:orgb0c6496} +\label{sec:orgad4e45c} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} -\label{sec:orgfb3986f} +\label{sec:org1c2c752} \subsection{Super Sensor Noise and Dynamical Uncertainty} -\label{sec:orgfd5c11e} +\label{sec:org06f5947} \section{Conclusion} -\label{sec:orgda418fa} +\label{sec:orgfb9928f} \label{sec:conclusion} \section{Acknowledgment} -\label{sec:orgabdae67} +\label{sec:org267a8aa} \bibliography{ref} \end{document}