From 333f814d47dfbf1e6a6e718b8a21fb4dd80ec84f Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 22 Sep 2020 11:29:13 +0200 Subject: [PATCH] Add test on H2 synthesis --- paper/paper.org | 97 ++++++++++++++++++++++++-------- paper/paper.pdf | Bin 305579 -> 310641 bytes paper/paper.tex | 146 ++++++++++++++++++++++++++++++------------------ 3 files changed, 168 insertions(+), 75 deletions(-) diff --git a/paper/paper.org b/paper/paper.org index 612f65b..0f49397 100644 --- a/paper/paper.org +++ b/paper/paper.org @@ -1,4 +1,4 @@ -#+TITLE: Robust and Optimal Sensor Fusion +#+TITLE: Optimal and Robust Sensor Fusion :DRAWER: #+LATEX_CLASS: IEEEtran #+LATEX_CLASS_OPTIONS: [conference] @@ -87,30 +87,68 @@ * Introduction <> +- Section ref:sec:optimal_fusion +- Section ref:sec:robust_fusion +- Section ref:sec:optimal_robust_fusion +- Section ref:sec:experimental_validation + * Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis <> ** Sensor Model +Let's consider a sensor measuring a physical quantity $x$ (Figure [[fig:sensor_model]]). +The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function $G_i(s)$. + +The noise of sensor can be described by the Power Spectral Density (PSD) $\Phi_{n_i}(\omega)$. + +This is approximated by shaping a white noise with unitary PSD $\tilde{n}_i$ eqref:eq:unitary_sensor_noise_psd with a LTI transfer function $N_i(s)$: +\begin{equation} +\begin{aligned} + \Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\ + &= \left| N_i(j\omega) \right|^2 +\end{aligned} +\end{equation} + +#+name: eq:unitary_sensor_noise_psd +\begin{equation} + \Phi_{\tilde{n}_i}(\omega) = 1 +\end{equation} + +The output of the sensor $v_i$: +\begin{equation} + v_i = \left( G_i \right) x + \left( G_i N_i \right) \tilde{n}_i +\end{equation} + +In order to obtain an estimate $\hat{x}_i$ of $x$, a model $\hat{G}_i$ of the (true) sensor dynamics $G_i$ is inverted and applied at the output (Figure [[fig:sensor_model]]): +\begin{equation} + \hat{x}_i = \left( \hat{G}_i^{-1} G_i \right) x + \left( \hat{G}_i^{-1} G_i N_i \right) \tilde{n}_i +\end{equation} + +#+name: fig:sensor_model +#+caption: Sensor Model +#+attr_latex: :scale 1 +[[file:figs/sensor_model.pdf]] + ** Sensor Fusion Architecture +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 [[fig:sensor_fusion_noise_arch]]). + +The noise sources $\tilde{n}_1$ and $\tilde{n}_2$ are considered to be uncorrelated. + #+name: fig:sensor_fusion_noise_arch #+caption: Sensor Fusion Architecture with sensor noise #+attr_latex: :scale 1 [[file:figs/sensor_fusion_noise_arch.pdf]] -Let note $\Phi$ the PSD. -$\tilde{n}_1$ and $\tilde{n}_2$ are white noise with unitary power spectral density: -\begin{equation} - \Phi_{\tilde{n}_i}(\omega) = 1 -\end{equation} - +The output of both sensors $(v1,v2)$ are then passed through the inverse of the sensor model to obtained two estimates $(\hat{x}_1, \hat{x}_2)$ of $x$. +These two estimates are then filtered out by two filters $H_1$ and $H_2$ and summed to gives the super sensor estimate $\hat{x}$. \begin{equation} -\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} G_1 N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} G_2 N_2 \right) \tilde{n}_2 -\end{split} + \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} G_1 N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} G_2 N_2 \right) \tilde{n}_2 + \end{split} \end{equation} Suppose the sensor dynamical model $\hat{G}_i$ is perfect: @@ -118,50 +156,65 @@ Suppose the sensor dynamical model $\hat{G}_i$ is perfect: \hat{G}_i = G_i \end{equation} -Complementary Filters +We considered here complementary filters: \begin{equation} H_1(s) + H_2(s) = 1 \end{equation} +In such case, the super sensor estimate $\hat{x}$ is equal to $x$ plus the noise of the individual sensors filtered out by the complementary filters: \begin{equation} \hat{x} = x + \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2 \end{equation} -Perfect dynamics + filter noise - ** Super Sensor Noise Let's note $n$ the super sensor noise. +\begin{equation} + n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2 +\end{equation} -Its PSD is determined by: +As the noise of both sensors are considered to be uncorrelated, the PSD of the super sensor noise is computed as follow: \begin{equation} \Phi_n(\omega) = \left| H_1 N_1 \right|^2 + \left| H_2 N_2 \right|^2 \end{equation} -** $\mathcal{H}_2$ Synthesis of Complementary Filters +It is clear that the PSD of the super sensor depends on the norm of the complementary filters. -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 $\hat{x}$. +** $\mathcal{H}_2$ Synthesis of Complementary Filters +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 $\hat{x}$. And the goal is the minimize the Root Mean Square (RMS) value of $n$: #+name: eq:rms_value_estimation \begin{equation} - \sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 + \sigma_{n} = \sqrt{\int_0^\infty \Phi_{n}(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \end{equation} -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 $\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2$ is minimized. +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 $\sigma_n$ is minimized. +This can be cast into an $\mathcal{H}_2$ synthesis problem by considering the following generalized plant (also represented in Figure [[fig:h_two_optimal_fusion]]): \begin{equation} \begin{pmatrix} z_1 \\ z_2 \\ v -\end{pmatrix} = \begin{bmatrix} +\end{pmatrix} = \underbrace{\begin{bmatrix} N_1 & N_1 \\ 0 & N_2 \\ 1 & 0 -\end{bmatrix} \begin{pmatrix} +\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix} w \\ u \end{pmatrix} \end{equation} -The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RMS value of the super sensor noise. +Applying the $\mathcal{H}_2$ synthesis on $P_{\mathcal{H}_2}$ will generate a filter $H_2(s)$ such that the $\mathcal{H}_2$ norm from $w$ to $(z_1,z_2)$ is minimized: +#+NAME: eq:H2_norm +\begin{equation} + \left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_2 = \left\| \begin{matrix} N_1 (1 - H_2) \\ N_2 H_2 \end{matrix} \right\|_2 +\end{equation} + +The $\mathcal{H}_2$ norm of Eq. eqref:eq:H2_norm is equals to $\sigma_n$ by defining $H_1(s)$ to be the complementary filter of $H_2(s)$: +\begin{equation} + H_1(s) = 1 - H_2(s) +\end{equation} + +We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ generates two complementary filters $H_1(s)$ and $H_2(s)$ such that the RMS value of super sensor noise is minimized. #+name: fig:h_two_optimal_fusion #+caption: Generalized plant $P_{\mathcal{H}_2}$ used for the $\mathcal{H}_2$ synthesis of complementary filters diff --git a/paper/paper.pdf b/paper/paper.pdf index c10dc2978038ec2b6d49c789195b2d45b94605e0..1ef6ea3b86060db78da6adaff4a73d2c1ef6fc65 100644 GIT binary patch delta 91999 zcmZsiV{|6}_T^*Swr$(CZQFjLj&0kvZQD-APIsJilIh=_|Gjh9%)F}FtMFZ=UFXzi zSM@T=uXmINNoXt%79u7hM-v+uK0X*mIdca~H!C7mPG+`bV=yYf4W72+&e)rNzsbjn zJ{fQ3hmNDOU)HyOyKt-^6+8D)9(7h z^W&RPg}nTQ!Mme}=@8; zDXylSep$FOAGvh_Xs4V2X?4-^1aKd3Bta(~r|IVtF-~ z)^7bnD5h%PH|uY58P;7BQ5L9-%MJS_dp2!UL&=zv;0b7XYm6QB^QL@u4X{IYK}0!BMILWflN*<|L) zU9h=cEf( zoAL>-dI$sZ25DBlxyPK7XZ2+e3ZH%y^>2ml$FzR!-KoLN`S2A=prS$JDgilIelmYI z=HQf2n752mT=r8RX#?H%Sn_CpxUp!}FQbZpcA6CRN~512_+;4u zaMrT{;M+$iT#sWaJRf08{qVhDR0q3K*R2!9PnK*Io+=>A*c+x07Md*v06n_6W-^#&KWpCiQ4Fcy z^-E3n7lqRx#bhvWT#->dT(29bg+7tN2}Oh1-48Vy}8N%Ae~16UY4CYwQ|iKx%P@Mgng&xD(BWPxLJ39Ul;H*9D+bewAqgeei-)406V216XLaQW+kvCu>KNAw_ 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zlv5_aIU{FqobBVvz220?<4m5jZk#c2mW(qCP7a)zc81;A0EH0>YU`t=C6X8jUR~tzn)14y2${Z2<?VzSON@UEQ2b*ILB%WFBbG!{UqoC{18Moe`M81#yK+GQDI;ZqZS#17`49G0+At! z;}+h1OV85|(~{A4JPXc&1;ErsW;N1(&d6NG3xGk4mjL4z8LPMoZh$p#6Wju~!5y#; zn6h{m+yhKhd;m82Zu%j32-H~hbj!liQN8YC3txRpm4cN&d6TE-cYTkPf7Ji|UR7b% zI;g5-Ho2^-Y__tcs>PqhWxLezG-IkBhkO4{?WRmJw)pgNlb+B*W9odPcohl nWqdXp{sUHJOsooJZe(+Ga%EwcQN08e3p6n}Hwq;sMNdWwwLVY8 diff --git a/paper/paper.tex b/paper/paper.tex index 1f1f8d8..ac080e5 100644 --- a/paper/paper.tex +++ b/paper/paper.tex @@ -1,4 +1,4 @@ -% Created 2020-09-22 mar. 10:15 +% Created 2020-09-22 mar. 11:10 % Intended LaTeX compiler: pdflatex \documentclass[conference]{IEEEtran} \usepackage[utf8]{inputenc} @@ -36,7 +36,7 @@ \usepackage{showframe} \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 }} \date{2020-09-22} -\title{Robust and Optimal Sensor Fusion} +\title{Optimal and Robust Sensor Fusion} \begin{document} \maketitle @@ -50,18 +50,61 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis \end{IEEEkeywords} \section{Introduction} -\label{sec:org4ebc807} +\label{sec:orgfaa194e} \label{sec:introduction} +Section \ref{sec:optimal_fusion} +Section \ref{sec:robust_fusion} +Section \ref{sec:optimal_robust_fusion} +Section \ref{sec:experimental_validation} + \section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis} -\label{sec:org86da8fa} +\label{sec:org08f9f0e} \label{sec:optimal_fusion} \subsection{Sensor Model} -\label{sec:org60743ab} +\label{sec:orgaa5ec56} + +Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model}). +The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function \(G_i(s)\). + +The noise of sensor can be described by the Power Spectral Density (PSD) \(\Phi_{n_i}(\omega)\). + +This is approximated by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_sensor_noise_psd} with a LTI transfer function \(N_i(s)\): +\begin{equation} +\begin{aligned} + \Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\ + &= \left| N_i(j\omega) \right|^2 +\end{aligned} +\end{equation} + +\begin{equation} +\label{eq:unitary_sensor_noise_psd} + \Phi_{\tilde{n}_i}(\omega) = 1 +\end{equation} + +The output of the sensor \(v_i\): +\begin{equation} + v_i = \left( G_i \right) x + \left( G_i N_i \right) \tilde{n}_i +\end{equation} + +In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of the (true) sensor dynamics \(G_i\) is inverted and applied at the output (Figure \ref{fig:sensor_model}): +\begin{equation} + \hat{x}_i = \left( \hat{G}_i^{-1} G_i \right) x + \left( \hat{G}_i^{-1} G_i N_i \right) \tilde{n}_i +\end{equation} + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1]{figs/sensor_model.pdf} +\caption{\label{fig:sensor_model}Sensor Model} +\end{figure} \subsection{Sensor Fusion Architecture} -\label{sec:org49f3948} +\label{sec:org17e7387} + +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}). + +The noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) are considered to be uncorrelated. \begin{figure}[htbp] \centering @@ -69,18 +112,14 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis \caption{\label{fig:sensor_fusion_noise_arch}Sensor Fusion Architecture with sensor noise} \end{figure} -Let note \(\Phi\) the PSD. -\(\tilde{n}_1\) and \(\tilde{n}_2\) are white noise with unitary power spectral density: -\begin{equation} - \Phi_{\tilde{n}_i}(\omega) = 1 -\end{equation} - +The output of both sensors \((v1,v2)\) are then passed through the inverse of the sensor model to obtained two estimates \((\hat{x}_1, \hat{x}_2)\) of \(x\). +These two estimates are then filtered out by two filters \(H_1\) and \(H_2\) and summed to gives the super sensor estimate \(\hat{x}\). \begin{equation} -\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} + \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} G_1 N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} G_2 N_2 \right) \tilde{n}_2 + \end{split} \end{equation} Suppose the sensor dynamical model \(\hat{G}_i\) is perfect: @@ -88,49 +127,50 @@ Suppose the sensor dynamical model \(\hat{G}_i\) is perfect: \hat{G}_i = G_i \end{equation} -Complementary Filters +We considered here complementary filters: \begin{equation} H_1(s) + H_2(s) = 1 \end{equation} - +In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the noise of the individual sensors filtered out by the complementary filters: \begin{equation} - \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 + \hat{x} = x + \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2 \end{equation} -Perfect dynamics + filter noise - - \subsection{Super Sensor Noise} -\label{sec:org06ff958} - +\label{sec:orgb010f68} Let's note \(n\) the super sensor noise. - -Its PSD is determined by: \begin{equation} - \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 + n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2 \end{equation} +As the noise of both sensors are considered to be uncorrelated, the PSD of the super sensor noise is computed as follow: +\begin{equation} + \Phi_n(\omega) = \left| H_1 N_1 \right|^2 + \left| H_2 N_2 \right|^2 +\end{equation} + +It is clear that the PSD of the super sensor depends on the norm of the complementary filters. + \subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters} -\label{sec:orgeaad969} +\label{sec:orgf1d735c} 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 \(\hat{x}\). And the goal is the minimize the Root Mean Square (RMS) value of \(n\): \begin{equation} \label{eq:rms_value_estimation} - \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 + \sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \end{equation} -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. +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 \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized. \begin{equation} \begin{pmatrix} z_1 \\ z_2 \\ v \end{pmatrix} = \begin{bmatrix} - \hat{G}_1^{-1} N_1 & -\hat{G}_1^{-1} N_1 \\ - 0 & \hat{G}_2^{-1} N_2 \\ - 1 & 0 + N_1 & N_1 \\ + 0 & N_2 \\ + 1 & 0 \end{bmatrix} \begin{pmatrix} w \\ u \end{pmatrix} @@ -145,17 +185,17 @@ The \(\mathcal{H}_2\) synthesis of the complementary filters thus minimized the \end{figure} \subsection{Example} -\label{sec:org50664f6} +\label{sec:org074433c} \subsection{Robustness Problem} -\label{sec:orgaa5f7af} +\label{sec:org21dc09f} \section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis} -\label{sec:org88ac630} +\label{sec:org2041184} \label{sec:robust_fusion} \subsection{Representation of Sensor Dynamical Uncertainty} -\label{sec:orgde90433} +\label{sec:orgfd12a50} Suppose that the sensor dynamics \(G_i(s)\) can be modelled by a nominal d \begin{equation} @@ -164,7 +204,7 @@ Suppose that the sensor dynamics \(G_i(s)\) can be modelled by a nominal d \subsection{Sensor Fusion Architecture} -\label{sec:orgda3fb09} +\label{sec:org11c9d00} \begin{equation} \begin{split} \hat{x} = \Big( {} & H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + w_1 \Delta_1) \\ @@ -189,7 +229,7 @@ Suppose the model inversion is equal to the nominal model: \end{figure} \subsection{Super Sensor Dynamical Uncertainty} -\label{sec:orgc9ca84c} +\label{sec:org6673a25} 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 \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\). @@ -200,7 +240,7 @@ The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at freque \end{figure} \subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters} -\label{sec:orgbb494ca} +\label{sec:org41ccb1e} In order to minimize the super sensor dynamical uncertainty @@ -211,14 +251,14 @@ In order to minimize the super sensor dynamical uncertainty \end{figure} \subsection{Example} -\label{sec:orgad1fefd} +\label{sec:orgba594da} \section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} -\label{sec:orgfb16ef1} +\label{sec:orgc07eeab} \label{sec:optimal_robust_fusion} \subsection{Sensor Fusion Architecture} -\label{sec:orgd611f0b} +\label{sec:orgddd6d33} \begin{figure}[htbp] \centering @@ -227,10 +267,10 @@ In order to minimize the super sensor dynamical uncertainty \end{figure} \subsection{Synthesis Objective} -\label{sec:org567ad90} +\label{sec:org79824da} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} -\label{sec:org42ee907} +\label{sec:org247ac1c} \begin{figure}[htbp] \centering @@ -239,30 +279,30 @@ In order to minimize the super sensor dynamical uncertainty \end{figure} \subsection{Example} -\label{sec:org3967eb3} +\label{sec:org7af2158} \section{Experimental Validation} -\label{sec:org06c0515} +\label{sec:orgb54c59b} \label{sec:experimental_validation} \subsection{Experimental Setup} -\label{sec:orgeaa87ec} +\label{sec:org40eadad} \subsection{Sensor Noise and Dynamical Uncertainty} -\label{sec:orgad4e45c} +\label{sec:orgb81743f} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} -\label{sec:org1c2c752} +\label{sec:orgb2232ac} \subsection{Super Sensor Noise and Dynamical Uncertainty} -\label{sec:org06f5947} +\label{sec:orgd80a558} \section{Conclusion} -\label{sec:orgfb9928f} +\label{sec:org0da5eb6} \label{sec:conclusion} \section{Acknowledgment} -\label{sec:org267a8aa} +\label{sec:orge5b9b80} \bibliography{ref} \end{document}