Add test on H2 synthesis
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#+TITLE: Robust and Optimal Sensor Fusion
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#+TITLE: Optimal and Robust Sensor Fusion
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:DRAWER:
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#+LATEX_CLASS: IEEEtran
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#+LATEX_CLASS_OPTIONS: [conference]
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@ -87,30 +87,68 @@
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* Introduction
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<<sec:introduction>>
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- Section ref:sec:optimal_fusion
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- Section ref:sec:robust_fusion
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- Section ref:sec:optimal_robust_fusion
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- Section ref:sec:experimental_validation
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* Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
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<<sec:optimal_fusion>>
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** Sensor Model
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Let's consider a sensor measuring a physical quantity $x$ (Figure [[fig:sensor_model]]).
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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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The noise of sensor can be described by the Power Spectral Density (PSD) $\Phi_{n_i}(\omega)$.
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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 $N_i(s)$:
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\begin{equation}
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\begin{aligned}
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\Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
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&= \left| N_i(j\omega) \right|^2
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\end{aligned}
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\end{equation}
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#+name: eq:unitary_sensor_noise_psd
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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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The output of the sensor $v_i$:
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\begin{equation}
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v_i = \left( G_i \right) x + \left( G_i N_i \right) \tilde{n}_i
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\end{equation}
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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]]):
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\begin{equation}
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\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
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\end{equation}
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#+name: fig:sensor_model
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#+caption: Sensor Model
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#+attr_latex: :scale 1
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[[file:figs/sensor_model.pdf]]
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** Sensor Fusion Architecture
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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 [[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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#+name: fig:sensor_fusion_noise_arch
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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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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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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 $x$.
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These two estimates are then filtered out by two filters $H_1$ and $H_2$ and summed to gives the super sensor estimate $\hat{x}$.
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\begin{equation}
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\begin{split}
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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} G_1 N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} G_2 N_2 \right) \tilde{n}_2
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\end{split}
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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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@ -118,50 +156,65 @@ Suppose the sensor dynamical model $\hat{G}_i$ is perfect:
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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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We considered here 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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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:
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\begin{equation}
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\hat{x} = x + \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
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\end{equation}
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Perfect dynamics + filter noise
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** Super Sensor Noise
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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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\end{equation}
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Its PSD is determined by:
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As the noise of both sensors are considered to be uncorrelated, the PSD of the super sensor noise is computed as follow:
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\begin{equation}
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\Phi_n(\omega) = \left| H_1 N_1 \right|^2 + \left| H_2 N_2 \right|^2
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\end{equation}
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** $\mathcal{H}_2$ Synthesis of Complementary Filters
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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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** $\mathcal{H}_2$ Synthesis of Complementary Filters
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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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#+name: eq:rms_value_estimation
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\begin{equation}
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\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
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\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
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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 $\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2$ is minimized.
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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 $\sigma_n$ is minimized.
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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]]):
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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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\end{pmatrix} = \underbrace{\begin{bmatrix}
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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} \begin{pmatrix}
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\end{bmatrix}}_{P_{\mathcal{H}_2}} \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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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:
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#+NAME: eq:H2_norm
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\begin{equation}
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\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
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\end{equation}
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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 $H_2(s)$:
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\begin{equation}
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H_1(s) = 1 - H_2(s)
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\end{equation}
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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.
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#+name: fig:h_two_optimal_fusion
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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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% Created 2020-09-22 mar. 10:15
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% Created 2020-09-22 mar. 11:10
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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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@ -36,7 +36,7 @@
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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{2020-09-22}
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\title{Robust and Optimal Sensor Fusion}
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\title{Optimal and Robust Sensor Fusion}
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\begin{document}
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\maketitle
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@ -50,18 +50,61 @@ 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:orgfaa194e}
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\label{sec:introduction}
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Section \ref{sec:optimal_fusion}
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Section \ref{sec:robust_fusion}
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Section \ref{sec:optimal_robust_fusion}
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Section \ref{sec:experimental_validation}
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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:org08f9f0e}
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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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\label{sec:orgaa5ec56}
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Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model}).
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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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The noise of sensor can be described by the Power Spectral Density (PSD) \(\Phi_{n_i}(\omega)\).
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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 \(N_i(s)\):
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\begin{equation}
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\begin{aligned}
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\Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
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&= \left| N_i(j\omega) \right|^2
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\end{aligned}
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\end{equation}
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\begin{equation}
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\label{eq:unitary_sensor_noise_psd}
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\Phi_{\tilde{n}_i}(\omega) = 1
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\end{equation}
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The output of the sensor \(v_i\):
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\begin{equation}
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v_i = \left( G_i \right) x + \left( G_i N_i \right) \tilde{n}_i
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\end{equation}
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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}):
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\begin{equation}
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\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
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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_model.pdf}
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\caption{\label{fig:sensor_model}Sensor Model}
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\end{figure}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org49f3948}
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\label{sec:org17e7387}
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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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\begin{figure}[htbp]
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\centering
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@ -69,18 +112,14 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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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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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 \(x\).
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These two estimates are then filtered out by two filters \(H_1\) and \(H_2\) and summed to gives the super sensor estimate \(\hat{x}\).
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\begin{equation}
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\begin{split}
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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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&+ \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
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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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@ -88,48 +127,49 @@ Suppose the sensor dynamical model \(\hat{G}_i\) is perfect:
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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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We considered here 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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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:
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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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\hat{x} = x + \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 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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\label{sec:orgb010f68}
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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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n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
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\end{equation}
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As the noise of both sensors are considered to be uncorrelated, the PSD of the super sensor noise is computed as follow:
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\begin{equation}
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\Phi_n(\omega) = \left| H_1 N_1 \right|^2 + \left| H_2 N_2 \right|^2
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\end{equation}
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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:orgeaad969}
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\label{sec:orgf1d735c}
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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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\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
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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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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.
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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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N_1 & N_1 \\
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0 & 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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@ -145,17 +185,17 @@ The \(\mathcal{H}_2\) synthesis of the complementary filters thus minimized the
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\end{figure}
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\subsection{Example}
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\label{sec:org50664f6}
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\label{sec:org074433c}
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\subsection{Robustness Problem}
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\label{sec:orgaa5f7af}
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\label{sec:org21dc09f}
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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:org2041184}
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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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\label{sec:orgfd12a50}
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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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@ -164,7 +204,7 @@ Suppose that the sensor dynamics \(G_i(s)\) can be modelled by a nominal d
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\subsection{Sensor Fusion Architecture}
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\label{sec:orgda3fb09}
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\label{sec:org11c9d00}
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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) \\
|
||||
@ -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}
|
||||
|
Loading…
Reference in New Issue
Block a user