Add figure caption
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@ -95,7 +95,7 @@
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** Sensor Fusion Architecture
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#+name: fig:sensor_fusion_noise_arch
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#+caption: Figure caption
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#+caption: Sensor Fusion Architecture with sensor noise
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#+attr_latex: :scale 1
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[[file:figs/sensor_fusion_noise_arch.pdf]]
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@ -105,8 +105,12 @@ $\tilde{n}_1$ and $\tilde{n}_2$ are white noise with unitary power spectral dens
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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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\begin{split}
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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 \\
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&+ \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{split}
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\end{equation}
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Suppose the sensor dynamical model $\hat{G}_i$ is perfect:
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@ -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
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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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#+name: fig:h_two_optimal_fusion
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#+caption: Figure caption
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#+caption: Generalized plant $P_{\mathcal{H}_2}$ used for the $\mathcal{H}_2$ synthesis of complementary filters
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#+attr_latex: :scale 1
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[[file:figs/h_two_optimal_fusion.pdf]]
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@ -176,9 +180,18 @@ 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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Suppose that the sensor dynamics $G_i(s)$ can be modelled by a nominal d
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\begin{equation}
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G_i(s) = \hat{G}_i(s) \left( 1 + w_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
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\end{equation}
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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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\begin{split}
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\hat{x} = \Big( {} & H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + w_1 \Delta_1) \\
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+ & H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + w_2 \Delta_2) \Big) x
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\end{split}
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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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@ -192,7 +205,7 @@ Suppose the model inversion is equal to the nominal model:
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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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#+caption: Sensor Fusion Architecture with sensor model uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_fusion_arch_uncertainty.pdf]]
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@ -201,7 +214,7 @@ Suppose the model inversion is equal to the nominal model:
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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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#+caption: Super Sensor model uncertainty displayed in the complex plane
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#+attr_latex: :scale 1
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[[file:figs/uncertainty_set_super_sensor.pdf]]
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@ -210,7 +223,7 @@ The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency
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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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#+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters
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#+attr_latex: :scale 1
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[[file:figs/h_infinity_robust_fusion.pdf]]
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@ -222,7 +235,7 @@ In order to minimize the super sensor dynamical uncertainty
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** Sensor Fusion Architecture
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#+name: fig:sensor_fusion_arch_full
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#+caption: Figure caption
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#+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_fusion_arch_full.pdf]]
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@ -231,7 +244,7 @@ In order to minimize the super sensor dynamical uncertainty
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** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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#+name: fig:mixed_h2_hinf_synthesis
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#+caption: Figure caption
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#+caption: Generalized plant $P_{\mathcal{H}_2/\matlcal{H}_\infty}$ used for the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of complementary filters
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#+attr_latex: :scale 1
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[[file:figs/mixed_h2_hinf_synthesis.pdf]]
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BIN
paper/paper.pdf
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paper/paper.pdf
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@ -1,4 +1,4 @@
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% Created 2020-09-22 mar. 09:51
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% Created 2020-09-22 mar. 10:15
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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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@ -50,23 +50,23 @@ 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:org2c6d9ef}
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\label{sec:org4ebc807}
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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:org5aa0717}
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\label{sec:org86da8fa}
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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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\label{sec:org60743ab}
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\subsection{Sensor Fusion Architecture}
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\label{sec:orgc40deb4}
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\label{sec:org49f3948}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensor_fusion_noise_arch.pdf}
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\caption{\label{fig:sensor_fusion_noise_arch}Figure caption}
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\caption{\label{fig:sensor_fusion_noise_arch}Sensor Fusion Architecture with sensor noise}
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\end{figure}
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Let note \(\Phi\) the PSD.
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@ -75,8 +75,12 @@ Let note \(\Phi\) the PSD.
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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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\begin{split}
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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 \\
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&+ \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{split}
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\end{equation}
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Suppose the sensor dynamical model \(\hat{G}_i\) is perfect:
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@ -98,7 +102,7 @@ Perfect dynamics + filter noise
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\subsection{Super Sensor Noise}
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\label{sec:orgf4b6ca9}
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\label{sec:org06ff958}
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Let's note \(n\) the super sensor noise.
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@ -108,7 +112,7 @@ Its PSD is determined by:
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\end{equation}
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\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
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\label{sec:org5773772}
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\label{sec:orgeaad969}
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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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@ -137,26 +141,35 @@ The \(\mathcal{H}_2\) synthesis of the complementary filters thus minimized the
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/h_two_optimal_fusion.pdf}
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\caption{\label{fig:h_two_optimal_fusion}Figure caption}
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\caption{\label{fig:h_two_optimal_fusion}Generalized plant \(P_{\mathcal{H}_2}\) used for the \(\mathcal{H}_2\) synthesis of complementary filters}
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\end{figure}
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\subsection{Example}
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\label{sec:orged06a27}
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\label{sec:org50664f6}
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\subsection{Robustness Problem}
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\label{sec:org62b375f}
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\label{sec:orgaa5f7af}
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\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
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\label{sec:orgef03e7c}
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\label{sec:org88ac630}
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\label{sec:robust_fusion}
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\subsection{Representation of Sensor Dynamical Uncertainty}
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\label{sec:org9c9762b}
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\label{sec:orgde90433}
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Suppose that the sensor dynamics \(G_i(s)\) can be modelled by a nominal d
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\begin{equation}
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G_i(s) = \hat{G}_i(s) \left( 1 + w_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
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\end{equation}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org9572e70}
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\label{sec:orgda3fb09}
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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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\begin{split}
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\hat{x} = \Big( {} & H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + w_1 \Delta_1) \\
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+ & H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + w_2 \Delta_2) \Big) x
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\end{split}
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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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@ -172,84 +185,84 @@ Suppose the model inversion is equal to the nominal model:
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensor_fusion_arch_uncertainty.pdf}
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\caption{\label{fig:sensor_fusion_arch_uncertainty}Figure caption}
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\caption{\label{fig:sensor_fusion_arch_uncertainty}Sensor Fusion Architecture with sensor model uncertainty}
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\end{figure}
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\subsection{Super Sensor Dynamical Uncertainty}
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\label{sec:orgb9ee83e}
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\label{sec:orgc9ca84c}
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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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\includegraphics[scale=1]{figs/uncertainty_set_super_sensor.pdf}
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\caption{\label{fig:uncertainty_set_super_sensor}Figure caption}
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\caption{\label{fig:uncertainty_set_super_sensor}Super Sensor model uncertainty displayed in the complex plane}
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\end{figure}
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\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
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\label{sec:orgf4e3c8e}
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\label{sec:orgbb494ca}
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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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\includegraphics[scale=1]{figs/h_infinity_robust_fusion.pdf}
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\caption{\label{fig:h_infinity_robust_fusion}Figure caption}
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\caption{\label{fig:h_infinity_robust_fusion}Generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) synthesis of complementary filters}
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\end{figure}
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\subsection{Example}
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\label{sec:org4f663bc}
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\label{sec:orgad1fefd}
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\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:org150b612}
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\label{sec:orgfb16ef1}
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\label{sec:optimal_robust_fusion}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org9bc69b7}
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\label{sec:orgd611f0b}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensor_fusion_arch_full.pdf}
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\caption{\label{fig:sensor_fusion_arch_full}Figure caption}
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\caption{\label{fig:sensor_fusion_arch_full}Super Sensor Fusion with both sensor noise and sensor model uncertainty}
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\end{figure}
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\subsection{Synthesis Objective}
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\label{sec:orgbc5ac30}
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\label{sec:org567ad90}
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\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:org541ef02}
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\label{sec:org42ee907}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/mixed_h2_hinf_synthesis.pdf}
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\caption{\label{fig:mixed_h2_hinf_synthesis}Figure caption}
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\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}
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\end{figure}
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\subsection{Example}
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\label{sec:org046c2e2}
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\label{sec:org3967eb3}
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\section{Experimental Validation}
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\label{sec:org1bb9cff}
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\label{sec:org06c0515}
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\label{sec:experimental_validation}
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\subsection{Experimental Setup}
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\label{sec:org2c63393}
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\label{sec:orgeaa87ec}
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\subsection{Sensor Noise and Dynamical Uncertainty}
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\label{sec:orgb0c6496}
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\label{sec:orgad4e45c}
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\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:orgfb3986f}
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\label{sec:org1c2c752}
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\subsection{Super Sensor Noise and Dynamical Uncertainty}
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\label{sec:orgfd5c11e}
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\label{sec:org06f5947}
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\section{Conclusion}
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\label{sec:orgda418fa}
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\label{sec:orgfb9928f}
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\label{sec:conclusion}
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\section{Acknowledgment}
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\label{sec:orgabdae67}
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\label{sec:org267a8aa}
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\bibliography{ref}
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\end{document}
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