269 lines
8.4 KiB
TeX
269 lines
8.4 KiB
TeX
% Created 2020-09-22 mar. 10:15
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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{2020-09-22}
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\title{Robust and Optimal Sensor Fusion}
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\begin{document}
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\maketitle
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\begin{abstract}
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Abstract text to be done
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\end{abstract}
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\begin{IEEEkeywords}
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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: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:org86da8fa}
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\label{sec:optimal_fusion}
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\subsection{Sensor Model}
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\label{sec:org60743ab}
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\subsection{Sensor Fusion Architecture}
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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}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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\(\tilde{n}_1\) and \(\tilde{n}_2\) are white noise with unitary power spectral density:
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\begin{equation}
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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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\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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\begin{equation}
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\hat{G}_i = G_i
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\end{equation}
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Complementary Filters
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\begin{equation}
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H_1(s) + H_2(s) = 1
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\end{equation}
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\begin{equation}
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\hat{x} = 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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\end{equation}
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Perfect dynamics + filter noise
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\subsection{Super Sensor Noise}
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\label{sec:org06ff958}
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Let's note \(n\) the super sensor noise.
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Its PSD is determined by:
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\begin{equation}
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\Phi_n(\omega) = \left| H_1 \hat{G}_1^{-1} N_1 \right|^2 + \left| H_2 \hat{G}_2^{-1} N_2 \right|^2
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\end{equation}
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\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
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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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And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
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\begin{equation}
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\label{eq:rms_value_estimation}
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\sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2
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\end{equation}
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Thus, the goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) and such that \(\left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2\) is minimized.
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\begin{equation}
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \begin{bmatrix}
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\hat{G}_1^{-1} N_1 & -\hat{G}_1^{-1} N_1 \\
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0 & \hat{G}_2^{-1} N_2 \\
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1 & 0
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\end{bmatrix} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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\end{equation}
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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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\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}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:org50664f6}
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\subsection{Robustness Problem}
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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: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: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:orgda3fb09}
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\begin{equation}
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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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Suppose the model inversion is equal to the nominal model:
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\begin{equation}
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\hat{G}_i = G_i
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\end{equation}
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\begin{equation}
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\hat{x} = \left( 1 + H_1 w_1 \Delta_1 + H_2 w_2 \Delta_2 \right) x
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\end{equation}
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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}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: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}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: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}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: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:orgfb16ef1}
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\label{sec:optimal_robust_fusion}
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\subsection{Sensor Fusion Architecture}
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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}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:org567ad90}
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\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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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}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:org3967eb3}
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\section{Experimental Validation}
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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:orgeaa87ec}
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\subsection{Sensor Noise and Dynamical Uncertainty}
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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:org1c2c752}
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\subsection{Super Sensor Noise and Dynamical Uncertainty}
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\label{sec:org06f5947}
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\section{Conclusion}
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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:org267a8aa}
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\bibliography{ref}
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\end{document}
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