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@ -221,13 +221,46 @@ We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ g
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** Example
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#+name: fig:figure_name
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#+caption: Figure caption
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#+name: fig:sensors_nominal_dynamics
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#+caption: Sensor nominal dynamics from the velocity of the object to the output voltage
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#+attr_latex: :scale 1
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[[file:figs/sensors_nominal_dynamics.pdf]]
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#+name: fig:sensors_noise
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#+caption: Amplitude spectral density of the sensors $\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|$
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#+attr_latex: :scale 1
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[[file:figs/sensors_noise.pdf]]
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#+name: fig:htwo_comp_filters
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#+caption: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis
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#+attr_latex: :scale 1
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[[file:figs/htwo_comp_filters.pdf]]
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#+name: fig:psd_sensors_htwo_synthesis
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#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal
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#+attr_latex: :scale 1
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[[file:figs/psd_sensors_htwo_synthesis.pdf]]
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#+name: fig:super_sensor_time_domain_h2
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#+caption: Noise of individual sensors and noise of the super sensor
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#+attr_latex: :scale 1
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[[file:figs/super_sensor_time_domain_h2.pdf]]
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** Robustness Problem
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#+name: fig:sensors_nominal_dynamics_and_uncertainty
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#+caption: Nominal Sensor Dynamics $\hat{G}_i$ (solid lines) as well as the spread of the dynamical uncertainty (background color)
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#+attr_latex: :scale 1
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[[file:figs/sensors_nominal_dynamics_and_uncertainty.pdf]]
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#+name: fig:super_sensor_dynamical_uncertainty_H2
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#+caption: Super sensor dynamical uncertainty when using the $\mathcal{H}_2$ Synthesis
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#+attr_latex: :scale 1
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[[file:figs/super_sensor_dynamical_uncertainty_H2.pdf]]
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* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
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<<sec:robust_fusion>>
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@ -334,6 +367,33 @@ The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$
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** Example
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#+name: fig:sensors_uncertainty_weights
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#+caption: Magnitude of the multiplicative uncertainty weights $|W_i(j\omega)|$
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#+attr_latex: :scale 1
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[[file:figs/sensors_uncertainty_weights.pdf]]
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#+name: fig:weight_uncertainty_bounds_Wu
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#+caption: Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)
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#+attr_latex: :scale 1
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[[file:figs/weight_uncertainty_bounds_Wu.pdf]]
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#+name: fig:hinf_comp_filters
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#+caption: Obtained complementary filters using the $\mathcal{H}_\infty$ Synthesis
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#+attr_latex: :scale 1
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[[file:figs/hinf_comp_filters.pdf]]
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#+name: fig:super_sensor_dynamical_uncertainty_Hinf
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#+caption: Super sensor dynamical uncertainty (solid curve) when using the $\mathcal{H}_\infty$ Synthesis
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#+attr_latex: :scale 1
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[[file:figs/super_sensor_dynamical_uncertainty_Hinf.pdf]]
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#+name: fig:psd_sensors_hinf_synthesis
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#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
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#+attr_latex: :scale 1
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[[file:figs/psd_sensors_hinf_synthesis.pdf]]
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* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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<<sec:optimal_robust_fusion>>
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@ -414,6 +474,26 @@ The synthesis objective is to:
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** Example
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#+name: fig:htwo_hinf_comp_filters
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#+caption: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
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#+attr_latex: :scale 1
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[[file:figs/htwo_hinf_comp_filters.pdf]]
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#+name: fig:psd_sensors_htwo_hinf_synthesis
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#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
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#+attr_latex: :scale 1
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[[file:figs/psd_sensors_htwo_hinf_synthesis.pdf]]
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#+name: fig:super_sensor_time_domain_h2_hinf
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#+caption: Noise of individual sensors and noise of the super sensor
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#+attr_latex: :scale 1
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[[file:figs/super_sensor_time_domain_h2_hinf.pdf]]
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#+name: fig:super_sensor_dynamical_uncertainty_Htwo_Hinf
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#+caption: Super sensor dynamical uncertainty (solid curve) when using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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#+attr_latex: :scale 1
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[[file:figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf]]
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* Experimental Validation
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<<sec:experimental_validation>>
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@ -1,4 +1,4 @@
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% Created 2020-09-23 mer. 14:15
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% Created 2020-10-05 lun. 15:33
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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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@ -35,7 +35,7 @@
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\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
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\usepackage{showframe}
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\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 }}
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\date{2020-09-23}
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\date{2020-10-05}
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\title{Optimal and Robust Sensor Fusion}
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\begin{document}
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@ -50,7 +50,7 @@ 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:org88afd51}
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\label{sec:org26a7400}
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\label{sec:introduction}
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\begin{itemize}
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@ -61,12 +61,11 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{itemize}
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\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
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\label{sec:org5853545}
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\label{sec:org49e80fd}
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\label{sec:optimal_fusion}
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\subsection{Sensor Model}
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\label{sec:org565ea86}
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\label{sec:org9555932}
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Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model_noise}).
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The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function \(G_i(s)\).
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@ -102,8 +101,7 @@ In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of
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\end{figure}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org1ae73e8}
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\label{sec:orga12ae12}
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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}).
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The noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) are considered to be uncorrelated.
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@ -140,7 +138,7 @@ In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the n
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\end{equation}
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\subsection{Super Sensor Noise}
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\label{sec:orgb2e8dd6}
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\label{sec:org924b750}
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Let's note \(n\) the super sensor noise.
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\begin{equation}
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n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
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@ -154,7 +152,7 @@ As the noise of both sensors are considered to be uncorrelated, the PSD of the s
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It is clear that the PSD of the super sensor depends on the norm of the complementary filters.
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\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
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\label{sec:orga4cf5f1}
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\label{sec:org042a601}
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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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And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
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@ -170,8 +168,8 @@ This can be cast into an \(\mathcal{H}_2\) synthesis problem by considering the
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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N_1 & N_1 \\
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0 & N_2 \\
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N_1 & -N_1 \\
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0 & N_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
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w \\ u
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@ -198,17 +196,62 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
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\end{figure}
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\subsection{Example}
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\label{sec:org74634c9}
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\label{sec:org98c54c2}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensors_nominal_dynamics.pdf}
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\caption{\label{fig:sensors_nominal_dynamics}Sensor nominal dynamics from the velocity of the object to the output voltage}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensors_noise.pdf}
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\caption{\label{fig:sensors_noise}Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/htwo_comp_filters.pdf}
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\caption{\label{fig:htwo_comp_filters}Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/psd_sensors_htwo_synthesis.pdf}
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\caption{\label{fig:psd_sensors_htwo_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/super_sensor_time_domain_h2.pdf}
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\caption{\label{fig:super_sensor_time_domain_h2}Noise of individual sensors and noise of the super sensor}
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\end{figure}
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\subsection{Robustness Problem}
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\label{sec:org5fda5c1}
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\label{sec:org81a0772}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensors_nominal_dynamics_and_uncertainty.pdf}
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\caption{\label{fig:sensors_nominal_dynamics_and_uncertainty}Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_H2.pdf}
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\caption{\label{fig:super_sensor_dynamical_uncertainty_H2}Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis}
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\end{figure}
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\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
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\label{sec:orgc88050f}
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\label{sec:org78ced60}
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\label{sec:robust_fusion}
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\subsection{Representation of Sensor Dynamical Uncertainty}
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\label{sec:orgb09aa5a}
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\label{sec:org9df3b01}
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In Section \ref{sec:optimal_fusion}, the model \(\hat{G}_i(s)\) of the sensor was considered to be perfect.
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In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
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@ -228,7 +271,7 @@ The sensor can then be represented as shown in Figure \ref{fig:sensor_model_unce
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\end{figure}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org1d92a74}
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\label{sec:orgf4531ff}
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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.
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The super sensor estimate is then:
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@ -253,8 +296,7 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
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\end{figure}
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\subsection{Super Sensor Dynamical Uncertainty}
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\label{sec:org81db1d8}
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\label{sec:orgf5bb33e}
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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 \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure \ref{fig:uncertainty_set_super_sensor}.
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@ -269,7 +311,7 @@ And we can see that the dynamical uncertainty of the super sensor is equal to th
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At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
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\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
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\label{sec:org0e2a7a8}
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\label{sec:orgf07efa7}
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In order for the fusion to be ``robust'', meaning no phase drop will be induced in the super sensor dynamics,
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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.
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@ -279,7 +321,7 @@ To define what by ``small'' we mean, we use a weighting filter \(W_u(s)\) such t
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\left| W_1(j\omega)H_1(j\omega) \right| + \left| W_2(j\omega)H_2(j\omega) \right| < \frac{1}{\left| W_u(j\omega) \right|}, \quad \forall \omega
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\end{equation}
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This is actually almost equivalent (to within a factor \(\sqrt{2}\)) equivalent as to have:
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This is actually almost equivalent as to have (within a factor \(\sqrt{2}\)):
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\begin{equation}
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\left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
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\end{equation}
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@ -289,9 +331,9 @@ This problem can thus be dealt with an \(\mathcal{H}_\infty\) synthesis problem
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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W_u W_1 & W_u W_1 \\
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0 & W_u W_2 \\
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1 & 0
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W_u W_1 & -W_u W_1 \\
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0 & W_u W_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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@ -315,15 +357,58 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
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\end{figure}
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\subsection{Example}
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\label{sec:org0122000}
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\label{sec:org0ca6ef9}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensors_uncertainty_weights.pdf}
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\caption{\label{fig:sensors_uncertainty_weights}Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/weight_uncertainty_bounds_Wu.pdf}
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\caption{\label{fig:weight_uncertainty_bounds_Wu}Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/hinf_comp_filters.pdf}
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\caption{\label{fig:hinf_comp_filters}Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Hinf.pdf}
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\caption{\label{fig:super_sensor_dynamical_uncertainty_Hinf}Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/psd_sensors_hinf_synthesis.pdf}
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\caption{\label{fig:psd_sensors_hinf_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the \(\mathcal{H}_\infty\) synthesis}
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\end{figure}
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\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:orgdf5a196}
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\label{sec:orgf642e73}
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\label{sec:optimal_robust_fusion}
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\subsection{Sensor Fusion Architecture}
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\label{sec:orge16b510}
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\subsection{Sensor with noise and model uncertainty}
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\label{sec:org8949812}
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We wish now to combine the two previous synthesis, that is to say
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The sensors are now modelled by a white noise with unitary PSD \(\tilde{n}_i\) shaped by a LTI transfer function \(N_i(s)\).
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The dynamical uncertainty of the sensor is modelled using multiplicative uncertainty
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\begin{equation}
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v_i = \hat{G}_i (1 + W_i \Delta_i) x + \hat{G_i} (1 + W_i \Delta_i) N_i \tilde{n}_i
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\end{equation}
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Multiplying by the inverse of the nominal model of the sensor dynamics gives an estimate \(\hat{x}_i\) of \(x\):
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\begin{equation}
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\hat{x} = (1 + W_i \Delta_i) x + (1 + W_i \Delta_i) N_i \tilde{n}_i
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\end{equation}
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\begin{figure}[htbp]
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\centering
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@ -331,6 +416,26 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
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\caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty}
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\end{figure}
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\subsection{Sensor Fusion Architecture}
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\label{sec:orgcbc3d54}
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For reason of space, the blocks \(\hat{G}_i\) and \(\hat{G}_i^{-1}\) are omitted.
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\hat{x} = &\Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x \\
|
||||
&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\hat{x} = &\Big( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \Big) x \\
|
||||
&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
The estimate \(\hat{x}\) of \(x\)
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -338,11 +443,34 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
|
||||
\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:orgb4b43b3}
|
||||
|
||||
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
||||
\label{sec:orgb9b52ad}
|
||||
\label{sec:org9d3f160}
|
||||
|
||||
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.
|
||||
|
||||
To specify how small we want the super sensor dynamic spread, we use a weighting filter \(W_u(s)\) as was done in Section \ref{sec:robust_fusion}.
|
||||
|
||||
|
||||
This synthesis problem can be solved using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis on the following generalized plant:
|
||||
\begin{equation}
|
||||
\begin{pmatrix}
|
||||
z_{\infty, 1} \\ z_{\infty, 2} \\ z_{2, 1} \\ z_{2, 2} \\ v
|
||||
\end{pmatrix} = \underbrace{\begin{bmatrix}
|
||||
W_u W_1 & W_u W_1 \\
|
||||
0 & W_u W_2 \\
|
||||
N_1 & N_1 \\
|
||||
0 & N_2 \\
|
||||
1 & 0
|
||||
\end{bmatrix}}_{P_{\mathcal{H}_2/\mathcal{H}_\infty}} \begin{pmatrix}
|
||||
w \\ u
|
||||
\end{pmatrix}
|
||||
\end{equation}
|
||||
|
||||
The synthesis objective is to:
|
||||
\begin{itemize}
|
||||
\item Keep the \(\mathcal{H}_\infty\) norm from \(w\) to \((z_{\infty,1}, z_{\infty,2})\) below \(1\)
|
||||
\item Minimize the \(\mathcal{H}_2\) norm from \(w\) to \((z_{2,1}, z_{2,2})\)
|
||||
\end{itemize}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -351,30 +479,54 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
|
||||
\end{figure}
|
||||
|
||||
\subsection{Example}
|
||||
\label{sec:orgc881f20}
|
||||
\label{sec:org85f304b}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/htwo_hinf_comp_filters.pdf}
|
||||
\caption{\label{fig:htwo_hinf_comp_filters}Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/psd_sensors_htwo_hinf_synthesis.pdf}
|
||||
\caption{\label{fig:psd_sensors_htwo_hinf_synthesis}Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/super_sensor_time_domain_h2_hinf.pdf}
|
||||
\caption{\label{fig:super_sensor_time_domain_h2_hinf}Noise of individual sensors and noise of the super sensor}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf}
|
||||
\caption{\label{fig:super_sensor_dynamical_uncertainty_Htwo_Hinf}Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
||||
\end{figure}
|
||||
|
||||
\section{Experimental Validation}
|
||||
\label{sec:org05b79a0}
|
||||
\label{sec:org49bf34a}
|
||||
\label{sec:experimental_validation}
|
||||
|
||||
\subsection{Experimental Setup}
|
||||
\label{sec:orgc3daf35}
|
||||
\label{sec:orgdd8fce6}
|
||||
|
||||
\subsection{Sensor Noise and Dynamical Uncertainty}
|
||||
\label{sec:org26fedf6}
|
||||
\label{sec:org21add72}
|
||||
|
||||
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
||||
\label{sec:org72f2969}
|
||||
\label{sec:org30521a3}
|
||||
|
||||
\subsection{Super Sensor Noise and Dynamical Uncertainty}
|
||||
\label{sec:orgf66f78b}
|
||||
\label{sec:org86cde79}
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:orge0f0a43}
|
||||
\label{sec:org16245b7}
|
||||
\label{sec:conclusion}
|
||||
|
||||
\section{Acknowledgment}
|
||||
\label{sec:orgb16559e}
|
||||
\label{sec:orgd992049}
|
||||
|
||||
\bibliography{ref}
|
||||
\end{document}
|
||||
|
||||
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Reference in New Issue
Block a user