268 lines
8.5 KiB
Org Mode
268 lines
8.5 KiB
Org Mode
#+TITLE: Robust and Optimal 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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#+STARTUP: overview
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#+DATE: {{{time(%Y-%m-%d)}}}
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#+AUTHOR: @@latex:\IEEEauthorblockN{Dehaeze Thomas}@@
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#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\@@
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#+AUTHOR: @@latex:Grenoble, France\\@@
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#+AUTHOR: @@latex:\textit{Precision Mechatronics Laboratory} \\@@
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#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
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#+AUTHOR: @@latex:thomas.dehaeze@esrf.fr@@
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#+AUTHOR: @@latex:}\and@@
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#+AUTHOR: @@latex:\IEEEauthorblockN{Collette Christophe}@@
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#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{BEAMS Department}\\@@
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#+AUTHOR: @@latex:\textit{Free University of Brussels}, Belgium\\@@
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#+AUTHOR: @@latex:\textit{Precision Mechatronics Laboratory} \\@@
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#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
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#+AUTHOR: @@latex:ccollett@ulb.ac.be@@
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#+AUTHOR: @@latex:}@@
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#+LATEX_HEADER: \IEEEoverridecommandlockouts
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#+LATEX_HEADER_EXTRA: \usepackage{showframe}
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#+LATEX_HEADER: \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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\bibliographystyle{IEEEtran}
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:END:
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* LaTeX Config :noexport:
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#+begin_src latex :tangle config.tex
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#+end_src
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* Build :noexport:
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#+NAME: startblock
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#+BEGIN_SRC emacs-lisp :results none
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(add-to-list 'org-latex-classes
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'("IEEEtran"
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"\\documentclass{IEEEtran}"
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("\\section{%s}" . "\\section*{%s}")
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("\\subsection{%s}" . "\\subsection*{%s}")
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("\\subsubsection{%s}" . "\\subsubsection*{%s}")
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("\\paragraph{%s}" . "\\paragraph*{%s}")
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("\\subparagraph{%s}" . "\\subparagraph*{%s}"))
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)
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(defun delete-org-comments (backend)
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(loop for comment in (reverse (org-element-map (org-element-parse-buffer)
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'comment 'identity))
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do
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(setf (buffer-substring (org-element-property :begin comment)
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(org-element-property :end comment))
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"")))
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;; add to export hook
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(add-hook 'org-export-before-processing-hook 'delete-org-comments)
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;; Remove hypersetup
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(setq org-latex-with-hyperref nil)
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#+END_SRC
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* Abstract :ignore:
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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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* Keywords :ignore:
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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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* Introduction
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<<sec:introduction>>
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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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** Sensor Fusion Architecture
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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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\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} 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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\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 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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Its PSD is determined by:
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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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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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\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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\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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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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\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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#+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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#+attr_latex: :scale 1
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[[file:figs/h_two_optimal_fusion.pdf]]
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** Example
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** Robustness Problem
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* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
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<<sec:robust_fusion>>
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** Representation of Sensor Dynamical Uncertainty
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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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** Sensor Fusion Architecture
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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) \\ + & 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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#+name: fig:sensor_fusion_arch_uncertainty
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#+caption: Sensor Fusion Architecture with sensor model uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_fusion_arch_uncertainty.pdf]]
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** Super Sensor Dynamical Uncertainty
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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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#+name: fig:uncertainty_set_super_sensor
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#+caption: Super Sensor model uncertainty displayed in the complex plane
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#+attr_latex: :scale 1
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[[file:figs/uncertainty_set_super_sensor.pdf]]
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** $\mathcal{H_\infty}$ Synthesis of Complementary Filters
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In order to minimize the super sensor dynamical uncertainty
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#+name: fig:h_infinity_robust_fusion
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#+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters
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#+attr_latex: :scale 1
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[[file:figs/h_infinity_robust_fusion.pdf]]
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** Example
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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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** Sensor Fusion Architecture
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#+name: fig:sensor_fusion_arch_full
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#+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_fusion_arch_full.pdf]]
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** Synthesis Objective
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** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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#+name: fig:mixed_h2_hinf_synthesis
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#+caption: 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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#+attr_latex: :scale 1
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[[file:figs/mixed_h2_hinf_synthesis.pdf]]
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** Example
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* Experimental Validation
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<<sec:experimental_validation>>
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** Experimental Setup
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** Sensor Noise and Dynamical Uncertainty
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** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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** Super Sensor Noise and Dynamical Uncertainty
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* Conclusion
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<<sec:conclusion>>
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* Acknowledgment
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* Bibliography :ignore:
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\bibliography{ref}
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