Add figure caption

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Thomas Dehaeze 2020-09-22 10:15:26 +02:00
parent f252a26a4c
commit 86ad85f17c
3 changed files with 71 additions and 45 deletions

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@ -95,7 +95,7 @@
** Sensor Fusion Architecture ** Sensor Fusion Architecture
#+name: fig:sensor_fusion_noise_arch #+name: fig:sensor_fusion_noise_arch
#+caption: Figure caption #+caption: Sensor Fusion Architecture with sensor noise
#+attr_latex: :scale 1 #+attr_latex: :scale 1
[[file:figs/sensor_fusion_noise_arch.pdf]] [[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 \Phi_{\tilde{n}_i}(\omega) = 1
\end{equation} \end{equation}
\begin{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} \end{equation}
Suppose the sensor dynamical model $\hat{G}_i$ is perfect: 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. 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 #+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 #+attr_latex: :scale 1
[[file:figs/h_two_optimal_fusion.pdf]] [[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 ** 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 ** Sensor Fusion Architecture
\begin{equation} \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} \end{equation}
with $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$. 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} \end{equation}
#+name: fig:sensor_fusion_arch_uncertainty #+name: fig:sensor_fusion_arch_uncertainty
#+caption: Figure caption #+caption: Sensor Fusion Architecture with sensor model uncertainty
#+attr_latex: :scale 1 #+attr_latex: :scale 1
[[file:figs/sensor_fusion_arch_uncertainty.pdf]] [[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 $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$. 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 #+name: fig:uncertainty_set_super_sensor
#+caption: Figure caption #+caption: Super Sensor model uncertainty displayed in the complex plane
#+attr_latex: :scale 1 #+attr_latex: :scale 1
[[file:figs/uncertainty_set_super_sensor.pdf]] [[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 In order to minimize the super sensor dynamical uncertainty
#+name: fig:h_infinity_robust_fusion #+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 #+attr_latex: :scale 1
[[file:figs/h_infinity_robust_fusion.pdf]] [[file:figs/h_infinity_robust_fusion.pdf]]
@ -222,7 +235,7 @@ In order to minimize the super sensor dynamical uncertainty
** Sensor Fusion Architecture ** Sensor Fusion Architecture
#+name: fig:sensor_fusion_arch_full #+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 #+attr_latex: :scale 1
[[file:figs/sensor_fusion_arch_full.pdf]] [[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 ** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
#+name: fig:mixed_h2_hinf_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 #+attr_latex: :scale 1
[[file:figs/mixed_h2_hinf_synthesis.pdf]] [[file:figs/mixed_h2_hinf_synthesis.pdf]]

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@ -1,4 +1,4 @@
% Created 2020-09-22 mar. 09:51 % Created 2020-09-22 mar. 10:15
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[conference]{IEEEtran} \documentclass[conference]{IEEEtran}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -50,23 +50,23 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{IEEEkeywords} \end{IEEEkeywords}
\section{Introduction} \section{Introduction}
\label{sec:org2c6d9ef} \label{sec:org4ebc807}
\label{sec:introduction} \label{sec:introduction}
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis} \section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
\label{sec:org5aa0717} \label{sec:org86da8fa}
\label{sec:optimal_fusion} \label{sec:optimal_fusion}
\subsection{Sensor Model} \subsection{Sensor Model}
\label{sec:org8f1053d} \label{sec:org60743ab}
\subsection{Sensor Fusion Architecture} \subsection{Sensor Fusion Architecture}
\label{sec:orgc40deb4} \label{sec:org49f3948}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/sensor_fusion_noise_arch.pdf} \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} \end{figure}
Let note \(\Phi\) the PSD. Let note \(\Phi\) the PSD.
@ -75,8 +75,12 @@ Let note \(\Phi\) the PSD.
\Phi_{\tilde{n}_i}(\omega) = 1 \Phi_{\tilde{n}_i}(\omega) = 1
\end{equation} \end{equation}
\begin{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} \end{equation}
Suppose the sensor dynamical model \(\hat{G}_i\) is perfect: Suppose the sensor dynamical model \(\hat{G}_i\) is perfect:
@ -98,7 +102,7 @@ Perfect dynamics + filter noise
\subsection{Super Sensor Noise} \subsection{Super Sensor Noise}
\label{sec:orgf4b6ca9} \label{sec:org06ff958}
Let's note \(n\) the super sensor noise. Let's note \(n\) the super sensor noise.
@ -108,7 +112,7 @@ Its PSD is determined by:
\end{equation} \end{equation}
\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters} \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 \(\hat{x}\). 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] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/h_two_optimal_fusion.pdf} \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} \end{figure}
\subsection{Example} \subsection{Example}
\label{sec:orged06a27} \label{sec:org50664f6}
\subsection{Robustness Problem} \subsection{Robustness Problem}
\label{sec:org62b375f} \label{sec:orgaa5f7af}
\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis} \section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
\label{sec:orgef03e7c} \label{sec:org88ac630}
\label{sec:robust_fusion} \label{sec:robust_fusion}
\subsection{Representation of Sensor Dynamical Uncertainty} \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} \subsection{Sensor Fusion Architecture}
\label{sec:org9572e70} \label{sec:orgda3fb09}
\begin{equation} \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} \end{equation}
with \(\Delta_i\) is any transfer function satisfying \(\| \Delta_i \|_\infty < 1\). 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] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/sensor_fusion_arch_uncertainty.pdf} \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} \end{figure}
\subsection{Super Sensor Dynamical Uncertainty} \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 \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\). 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] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/uncertainty_set_super_sensor.pdf} \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} \end{figure}
\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters} \subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
\label{sec:orgf4e3c8e} \label{sec:orgbb494ca}
In order to minimize the super sensor dynamical uncertainty In order to minimize the super sensor dynamical uncertainty
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/h_infinity_robust_fusion.pdf} \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} \end{figure}
\subsection{Example} \subsection{Example}
\label{sec:org4f663bc} \label{sec:orgad1fefd}
\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:org150b612} \label{sec:orgfb16ef1}
\label{sec:optimal_robust_fusion} \label{sec:optimal_robust_fusion}
\subsection{Sensor Fusion Architecture} \subsection{Sensor Fusion Architecture}
\label{sec:org9bc69b7} \label{sec:orgd611f0b}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/sensor_fusion_arch_full.pdf} \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} \end{figure}
\subsection{Synthesis Objective} \subsection{Synthesis Objective}
\label{sec:orgbc5ac30} \label{sec:org567ad90}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:org541ef02} \label{sec:org42ee907}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/mixed_h2_hinf_synthesis.pdf} \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} \end{figure}
\subsection{Example} \subsection{Example}
\label{sec:org046c2e2} \label{sec:org3967eb3}
\section{Experimental Validation} \section{Experimental Validation}
\label{sec:org1bb9cff} \label{sec:org06c0515}
\label{sec:experimental_validation} \label{sec:experimental_validation}
\subsection{Experimental Setup} \subsection{Experimental Setup}
\label{sec:org2c63393} \label{sec:orgeaa87ec}
\subsection{Sensor Noise and Dynamical Uncertainty} \subsection{Sensor Noise and Dynamical Uncertainty}
\label{sec:orgb0c6496} \label{sec:orgad4e45c}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orgfb3986f} \label{sec:org1c2c752}
\subsection{Super Sensor Noise and Dynamical Uncertainty} \subsection{Super Sensor Noise and Dynamical Uncertainty}
\label{sec:orgfd5c11e} \label{sec:org06f5947}
\section{Conclusion} \section{Conclusion}
\label{sec:orgda418fa} \label{sec:orgfb9928f}
\label{sec:conclusion} \label{sec:conclusion}
\section{Acknowledgment} \section{Acknowledgment}
\label{sec:orgabdae67} \label{sec:org267a8aa}
\bibliography{ref} \bibliography{ref}
\end{document} \end{document}