Add figures to paper

paper/paper.org

@@ -221,13 +221,46 @@ We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ g
 
 ** Example
 
-#+name: fig:figure_name
+#+name: fig:sensors_nominal_dynamics
-#+caption: Figure caption
+#+caption: Sensor nominal dynamics from the velocity of the object to the output voltage
 #+attr_latex: :scale 1
 [[file:figs/sensors_nominal_dynamics.pdf]]
 
+#+name: fig:sensors_noise
+#+caption: Amplitude spectral density of the sensors $\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|$
+#+attr_latex: :scale 1
+[[file:figs/sensors_noise.pdf]]
+
+
+#+name: fig:htwo_comp_filters
+#+caption:  Obtained complementary filters using the $\mathcal{H}_2$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/htwo_comp_filters.pdf]]
+
+
+#+name: fig:psd_sensors_htwo_synthesis
+#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal
+#+attr_latex: :scale 1
+[[file:figs/psd_sensors_htwo_synthesis.pdf]]
+
+
+#+name: fig:super_sensor_time_domain_h2
+#+caption: Noise of individual sensors and noise of the super sensor
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_time_domain_h2.pdf]]
+
 ** Robustness Problem
 
+#+name: fig:sensors_nominal_dynamics_and_uncertainty
+#+caption: Nominal Sensor Dynamics $\hat{G}_i$ (solid lines) as well as the spread of the dynamical uncertainty (background color)
+#+attr_latex: :scale 1
+[[file:figs/sensors_nominal_dynamics_and_uncertainty.pdf]]
+
+#+name: fig:super_sensor_dynamical_uncertainty_H2
+#+caption: Super sensor dynamical uncertainty when using the $\mathcal{H}_2$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_dynamical_uncertainty_H2.pdf]]
+
 * Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
 <<sec:robust_fusion>>
 
@@ -334,6 +367,33 @@ The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$
 
 ** Example
 
+#+name: fig:sensors_uncertainty_weights
+#+caption: Magnitude of the multiplicative uncertainty weights $|W_i(j\omega)|$
+#+attr_latex: :scale 1
+[[file:figs/sensors_uncertainty_weights.pdf]]
+
+
+#+name: fig:weight_uncertainty_bounds_Wu
+#+caption: Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)
+#+attr_latex: :scale 1
+[[file:figs/weight_uncertainty_bounds_Wu.pdf]]
+
+#+name: fig:hinf_comp_filters
+#+caption: Obtained complementary filters using the $\mathcal{H}_\infty$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/hinf_comp_filters.pdf]]
+
+#+name: fig:super_sensor_dynamical_uncertainty_Hinf
+#+caption: Super sensor dynamical uncertainty (solid curve) when using the $\mathcal{H}_\infty$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_dynamical_uncertainty_Hinf.pdf]]
+
+#+name: fig:psd_sensors_hinf_synthesis
+#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
+#+attr_latex: :scale 1
+[[file:figs/psd_sensors_hinf_synthesis.pdf]]
+
+
 * Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
 <<sec:optimal_robust_fusion>>
 
@@ -414,6 +474,26 @@ The synthesis objective is to:
 
 ** Example
 
+#+name: fig:htwo_hinf_comp_filters
+#+caption: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
+#+attr_latex: :scale 1
+[[file:figs/htwo_hinf_comp_filters.pdf]]
+
+#+name: fig:psd_sensors_htwo_hinf_synthesis
+#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
+#+attr_latex: :scale 1
+[[file:figs/psd_sensors_htwo_hinf_synthesis.pdf]]
+
+#+name: fig:super_sensor_time_domain_h2_hinf
+#+caption: Noise of individual sensors and noise of the super sensor
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_time_domain_h2_hinf.pdf]]
+
+#+name: fig:super_sensor_dynamical_uncertainty_Htwo_Hinf
+#+caption: Super sensor dynamical uncertainty (solid curve) when using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
+#+attr_latex: :scale 1
+[[file:figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf]]
+
 * Experimental Validation
 <<sec:experimental_validation>>
 

paper/paper.tex

@@ -1,4 +1,4 @@
-% Created 2020-09-23 mer. 14:15
+% Created 2020-10-05 lun. 15:33
 % Intended LaTeX compiler: pdflatex
 \documentclass[conference]{IEEEtran}
 \usepackage[utf8]{inputenc}
@@ -35,7 +35,7 @@
 \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
 \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{2020-09-23}
+\date{2020-10-05}
 \title{Optimal and Robust Sensor Fusion}
 \begin{document}
 
@@ -50,7 +50,7 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
 \end{IEEEkeywords}
 
 \section{Introduction}
-\label{sec:org88afd51}
+\label{sec:org26a7400}
 \label{sec:introduction}
 
 \begin{itemize}
@@ -61,12 +61,11 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
 \end{itemize}
 
 \section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
-\label{sec:org5853545}
+\label{sec:org49e80fd}
 \label{sec:optimal_fusion}
 
 \subsection{Sensor Model}
-\label{sec:org565ea86}
+\label{sec:org9555932}
-
 Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model_noise}).
 The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function \(G_i(s)\).
 
@@ -102,8 +101,7 @@ In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of
 \end{figure}
 
 \subsection{Sensor Fusion Architecture}
-\label{sec:org1ae73e8}
+\label{sec:orga12ae12}
-
 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.
@@ -140,7 +138,7 @@ In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the n
 \end{equation}
 
 \subsection{Super Sensor Noise}
-\label{sec:orgb2e8dd6}
+\label{sec:org924b750}
 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
@@ -154,7 +152,7 @@ As the noise of both sensors are considered to be uncorrelated, the PSD of the s
 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:orga4cf5f1}
+\label{sec:org042a601}
 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 \(n\):
@@ -170,8 +168,8 @@ This can be cast into an \(\mathcal{H}_2\) synthesis problem by considering the
 \begin{pmatrix}
   z_1 \\ z_2 \\ v
 \end{pmatrix} = \underbrace{\begin{bmatrix}
-  N_1 & N_1 \\
+  N_1 & -N_1 \\
-  0   & N_2 \\
+  0   &  N_2 \\
   1   &  0
 \end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
   w \\ u
@@ -198,17 +196,62 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
 \end{figure}
 
 \subsection{Example}
-\label{sec:org74634c9}
+\label{sec:org98c54c2}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensors_nominal_dynamics.pdf}
+\caption{\label{fig:sensors_nominal_dynamics}Sensor nominal dynamics from the velocity of the object to the output voltage}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensors_noise.pdf}
+\caption{\label{fig:sensors_noise}Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)}
+\end{figure}
+
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/htwo_comp_filters.pdf}
+\caption{\label{fig:htwo_comp_filters}Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis}
+\end{figure}
+
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/psd_sensors_htwo_synthesis.pdf}
+\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}
+\end{figure}
+
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/super_sensor_time_domain_h2.pdf}
+\caption{\label{fig:super_sensor_time_domain_h2}Noise of individual sensors and noise of the super sensor}
+\end{figure}
 
 \subsection{Robustness Problem}
-\label{sec:org5fda5c1}
+\label{sec:org81a0772}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensors_nominal_dynamics_and_uncertainty.pdf}
+\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)}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_H2.pdf}
+\caption{\label{fig:super_sensor_dynamical_uncertainty_H2}Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis}
+\end{figure}
 
 \section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
-\label{sec:orgc88050f}
+\label{sec:org78ced60}
 \label{sec:robust_fusion}
 
 \subsection{Representation of Sensor Dynamical Uncertainty}
-\label{sec:orgb09aa5a}
+\label{sec:org9df3b01}
 
 In Section \ref{sec:optimal_fusion}, the model \(\hat{G}_i(s)\) of the sensor was considered to be perfect.
 In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
@@ -228,7 +271,7 @@ The sensor can then be represented as shown in Figure \ref{fig:sensor_model_unce
 \end{figure}
 
 \subsection{Sensor Fusion Architecture}
-\label{sec:org1d92a74}
+\label{sec:orgf4531ff}
 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.
 
 The super sensor estimate is then:
@@ -253,8 +296,7 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
 \end{figure}
 
 \subsection{Super Sensor Dynamical Uncertainty}
-\label{sec:org81db1d8}
+\label{sec:orgf5bb33e}
-
 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}.
 
 
@@ -269,7 +311,7 @@ And we can see that the dynamical uncertainty of the super sensor is equal to th
 At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
 
 \subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
-\label{sec:org0e2a7a8}
+\label{sec:orgf07efa7}
 In order for the fusion to be ``robust'', meaning no phase drop will be induced in the super sensor dynamics,
 
 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.
@@ -279,7 +321,7 @@ To define what by ``small'' we mean, we use a weighting filter \(W_u(s)\) such t
   \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
 \end{equation}
 
-This is actually almost equivalent (to within a factor \(\sqrt{2}\)) equivalent as to have:
+This is actually almost equivalent as to have (within a factor \(\sqrt{2}\)):
 \begin{equation}
   \left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
 \end{equation}
@@ -289,9 +331,9 @@ This problem can thus be dealt with an \(\mathcal{H}_\infty\) synthesis problem
 \begin{pmatrix}
   z_1 \\ z_2 \\ v
 \end{pmatrix} = \underbrace{\begin{bmatrix}
-  W_u W_1 & W_u W_1 \\
+  W_u W_1 & -W_u W_1 \\
-  0       & W_u W_2 \\
+  0       &  W_u W_2 \\
-  1       & 0
+  1       &  0
 \end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
   w \\ u
 \end{pmatrix}
@@ -315,15 +357,58 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
 \end{figure}
 
 \subsection{Example}
-\label{sec:org0122000}
+\label{sec:org0ca6ef9}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensors_uncertainty_weights.pdf}
+\caption{\label{fig:sensors_uncertainty_weights}Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)}
+\end{figure}
+
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/weight_uncertainty_bounds_Wu.pdf}
+\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)}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/hinf_comp_filters.pdf}
+\caption{\label{fig:hinf_comp_filters}Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Hinf.pdf}
+\caption{\label{fig:super_sensor_dynamical_uncertainty_Hinf}Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/psd_sensors_hinf_synthesis.pdf}
+\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}
+\end{figure}
 
 
 \section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
-\label{sec:orgdf5a196}
+\label{sec:orgf642e73}
 \label{sec:optimal_robust_fusion}
 
-\subsection{Sensor Fusion Architecture}
+\subsection{Sensor with noise and model uncertainty}
-\label{sec:orge16b510}
+\label{sec:org8949812}
+We wish now to combine the two previous synthesis, that is to say
+
+The sensors are now modelled by a white noise with unitary PSD \(\tilde{n}_i\) shaped by a LTI transfer function \(N_i(s)\).
+The dynamical uncertainty of the sensor is modelled using multiplicative uncertainty
+\begin{equation}
+  v_i = \hat{G}_i (1 + W_i \Delta_i) x + \hat{G_i} (1 + W_i \Delta_i) N_i \tilde{n}_i
+\end{equation}
+
+Multiplying by the inverse of the nominal model of the sensor dynamics gives an estimate \(\hat{x}_i\) of \(x\):
+\begin{equation}
+  \hat{x} = (1 + W_i \Delta_i) x + (1 + W_i \Delta_i) N_i \tilde{n}_i
+\end{equation}
 
 \begin{figure}[htbp]
 \centering
@@ -331,6 +416,26 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
 \caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty}
 \end{figure}
 
+\subsection{Sensor Fusion Architecture}
+\label{sec:orgcbc3d54}
+
+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}