Rename paper files
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journal/.latexmkrc
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journal/.latexmkrc
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#!/bin/env perl
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# Shebang is only to get syntax highlighting right across GitLab, GitHub and IDEs.
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# This file is not meant to be run, but read by `latexmk`.
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# ======================================================================================
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# Perl `latexmk` configuration file
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# ======================================================================================
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# ======================================================================================
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# PDF Generation/Building/Compilation
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# ======================================================================================
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@default_files=('dehaeze22_optim_robus_compl_filte.tex');
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# PDF-generating modes are:
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# 1: pdflatex, as specified by $pdflatex variable (still largely in use)
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# 2: postscript conversion, as specified by the $ps2pdf variable (useless)
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# 3: dvi conversion, as specified by the $dvipdf variable (useless)
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# 4: lualatex, as specified by the $lualatex variable (best)
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# 5: xelatex, as specified by the $xelatex variable (second best)
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$pdf_mode = 1;
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# Treat undefined references and citations as well as multiply defined references as
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# ERRORS instead of WARNINGS.
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# This is only checked in the *last* run, since naturally, there are undefined references
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# in initial runs.
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# This setting is potentially annoying when debugging/editing, but highly desirable
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# in the CI pipeline, where such a warning should result in a failed pipeline, since the
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# final document is incomplete/corrupted.
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#
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# However, I could not eradicate all warnings, so that `latexmk` currently fails with
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# this option enabled.
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# Specifically, `microtype` fails together with `fontawesome`/`fontawesome5`, see:
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# https://tex.stackexchange.com/a/547514/120853
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# The fix in that answer did not help.
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# Setting `verbose=silent` to mute `microtype` warnings did not work.
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# Switching between `fontawesome` and `fontawesome5` did not help.
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$warnings_as_errors = 0;
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# Show used CPU time. Looks like: https://tex.stackexchange.com/a/312224/120853
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$show_time = 1;
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# Default is 5; we seem to need more owed to the complexity of the document.
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# Actual documents probably don't need this many since they won't use all features,
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# plus won't be compiling from cold each time.
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$max_repeat=7;
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# --shell-escape option (execution of code outside of latex) is required for the
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#'svg' package.
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# It converts raw SVG files to the PDF+PDF_TEX combo using InkScape.
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#
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# SyncTeX allows to jump between source (code) and output (PDF) in IDEs with support
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# (many have it). A value of `1` is enabled (gzipped), `-1` is enabled but uncompressed,
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# `0` is off.
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# Testing in VSCode w/ LaTeX Workshop only worked for the compressed version.
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# Adjust this as needed. Of course, only relevant for local use, no effect on a remote
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# CI pipeline (except for slower compilation, probably).
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#
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# %O and %S will forward Options and the Source file, respectively, given to latexmk.
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#
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# `set_tex_cmds` applies to all *latex commands (latex, xelatex, lualatex, ...), so
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# no need to specify these each. This allows to simply change `$pdf_mode` to get a
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# different engine. Check if this works with `latexmk --commands`.
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set_tex_cmds("--shell-escape -interaction=nonstopmode --synctex=1 %O %S");
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# Use default pdf viewer
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$pdf_previewer = 'zathura';
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# option 2 is same as 1 (run biber when necessary), but also deletes the
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# regeneratable bbl-file in a clenaup (`latexmk -c`). Do not use if original
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# bib file is not available!
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$bibtex_use = 1; # default: 1
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# Change default `biber` call, help catch errors faster/clearer. See
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# https://web.archive.org/web/20200526101657/https://www.semipol.de/2018/06/12/latex-best-practices.html#database-entries
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$bibtex = "bibtex %S";
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# ======================================================================================
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# Auxiliary Files
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# ======================================================================================
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# Let latexmk know about generated files, so they can be used to detect if a
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# rerun is required, or be deleted in a cleanup.
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# loe: List of Examples (KOMAScript)
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# lol: List of Listings (`listings` and `minted` packages)
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# run.xml: biber runs
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# glg: glossaries log
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# glstex: generated from glossaries-extra
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push @generated_exts, 'loe', 'lol', 'run.xml', 'glg', 'glstex';
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# Also delete the *.glstex files from package glossaries-extra. Problem is,
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# that that package generates files of the form "basename-digit.glstex" if
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# multiple glossaries are present. Latexmk looks for "basename.glstex" and so
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# does not find those. For that purpose, use wildcard.
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# Also delete files generated by gnuplot/pgfplots contour plots
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# (.dat, .script, .table).
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$clean_ext = "%R-*.glstex %R_contourtmp*.*";
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6347
journal/IEEEtran.cls
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journal/IEEEtran.cls
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journal/dehaeze22_optim_robus_compl_filte.org
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journal/dehaeze22_optim_robus_compl_filte.org
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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: [10pt,final,journal,a4paper]
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#+OPTIONS: toc:nil todo:nil
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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{Verma Mohit}@@
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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:mohitverma.serc@csir.res.in@@
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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: \usepackage{cite}
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#+LATEX_HEADER: \usepackage{amsmath,amssymb,amsfonts}
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#+LATEX_HEADER: \usepackage{algorithmic}
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#+LATEX_HEADER: \usepackage{graphicx}
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#+LATEX_HEADER: \usepackage{textcomp}
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#+LATEX_HEADER: \usepackage{xcolor}
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#+LATEX_HEADER: \usepackage{cases}
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#+LATEX_HEADER: \usepackage{tabularx,siunitx,booktabs}
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#+LATEX_HEADER: \usepackage{algorithmic}
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#+LATEX_HEADER: \usepackage{import, hyperref}
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#+LATEX_HEADER: \renewcommand{\citedash}{--}
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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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:END:
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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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;; Remove automatic org headings
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(defun my-latex-filter-removeOrgAutoLabels (text backend info)
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"Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings."
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(when (org-export-derived-backend-p backend 'latex)
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(replace-regexp-in-string "\\\\label{sec:org[a-f0-9]+}\n" "" text)))
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(add-to-list 'org-export-filter-headline-functions
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'my-latex-filter-removeOrgAutoLabels)
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;; Automatic delete org org-comments
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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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cite:mahony08_nonlin_compl_filter_special_orthog_group
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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 ref:fig:sensor_model_noise).
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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 ref:fig:sensor_model_noise):
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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_noise
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#+caption: Sensor Model
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#+attr_latex: :scale 1
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[[file:figs/sensor_model_noise.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 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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#+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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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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\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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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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** 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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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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** $\mathcal{H}_2$ 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_{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 $\sigma_n$ 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 ref: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} = \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}}_{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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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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||||
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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 $H_2(s)$:
|
||||
\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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#+attr_latex: :scale 1
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[[file:figs/h_two_optimal_fusion.pdf]]
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** Example
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||||
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#+name: fig:sensors_nominal_dynamics
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#+caption: Sensor nominal dynamics from the velocity of the object to the output voltage
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#+attr_latex: :scale 1
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[[file:figs/sensors_nominal_dynamics.pdf]]
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#+name: fig:sensors_noise
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#+caption: Amplitude spectral density of the sensors $\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|$
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#+attr_latex: :scale 1
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[[file:figs/sensors_noise.pdf]]
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#+name: fig:htwo_comp_filters
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#+caption: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis
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#+attr_latex: :scale 1
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[[file:figs/htwo_comp_filters.pdf]]
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||||
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#+name: fig:psd_sensors_htwo_synthesis
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#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal
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#+attr_latex: :scale 1
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[[file:figs/psd_sensors_htwo_synthesis.pdf]]
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#+name: fig:super_sensor_time_domain_h2
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#+caption: Noise of individual sensors and noise of the super sensor
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||||
#+attr_latex: :scale 1
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[[file:figs/super_sensor_time_domain_h2.pdf]]
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** Robustness Problem
|
||||
|
||||
#+name: fig:sensors_nominal_dynamics_and_uncertainty
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#+caption: Nominal Sensor Dynamics $\hat{G}_i$ (solid lines) as well as the spread of the dynamical uncertainty (background color)
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||||
#+attr_latex: :scale 1
|
||||
[[file:figs/sensors_nominal_dynamics_and_uncertainty.pdf]]
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#+name: fig:super_sensor_dynamical_uncertainty_H2
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||||
#+caption: Super sensor dynamical uncertainty when using the $\mathcal{H}_2$ Synthesis
|
||||
#+attr_latex: :scale 1
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||||
[[file:figs/super_sensor_dynamical_uncertainty_H2.pdf]]
|
||||
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||||
* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
|
||||
<<sec:robust_fusion>>
|
||||
|
||||
** Representation of Sensor Dynamical Uncertainty
|
||||
|
||||
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.
|
||||
|
||||
The Uncertainty on the sensor dynamics $G_i(s)$ is here modelled by (input) multiplicative uncertainty:
|
||||
\begin{equation}
|
||||
G_i(s) = \hat{G}_i(s) \left( 1 + W_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
|
||||
\end{equation}
|
||||
where $\hat{G}_i(s)$ is the nominal model, $W_i$ a weight representing the size of the uncertainty at each frequency, and $\Delta_i$ is any complex perturbation such that $\left\| \Delta_i \right\|_\infty < 1$.
|
||||
|
||||
The sensor can then be represented as shown in Figure ref:fig:sensor_model_uncertainty.
|
||||
|
||||
#+name: fig:sensor_model_uncertainty
|
||||
#+caption: Sensor Model including Dynamical Uncertainty
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/sensor_model_uncertainty.pdf]]
|
||||
|
||||
** Sensor Fusion Architecture
|
||||
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:
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\hat{x} &= \Big( H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + W_1 \Delta_1) \\
|
||||
& \quad + H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + W_2 \Delta_2) \Big) x \\
|
||||
&= \Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
with $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$.
|
||||
|
||||
As $H_1$ and $H_2$ are complementary filters, we finally have:
|
||||
\begin{equation}
|
||||
\hat{x} = \left( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \right) x, \quad \|\Delta_i\|_\infty<1
|
||||
\end{equation}
|
||||
|
||||
#+name: fig:sensor_fusion_arch_uncertainty
|
||||
#+caption: Sensor Fusion Architecture with sensor model uncertainty
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/sensor_fusion_arch_uncertainty.pdf]]
|
||||
|
||||
** Super Sensor Dynamical Uncertainty
|
||||
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.
|
||||
|
||||
|
||||
And we can see that the dynamical uncertainty of the super sensor is equal to the sum of the individual sensor uncertainties filtered out by the complementary filters.
|
||||
|
||||
#+name: fig:uncertainty_set_super_sensor
|
||||
#+caption: Super Sensor model uncertainty displayed in the complex plane
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/uncertainty_set_super_sensor.pdf]]
|
||||
|
||||
# Some comments on the weights
|
||||
At frequencies where $\left|W_i(j\omega)\right| > 1$ the uncertainty exceeds $100\%$ and sensor fusion is impossible.
|
||||
|
||||
** $\mathcal{H_\infty}$ Synthesis of Complementary Filters
|
||||
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.
|
||||
|
||||
To define what by "small" we mean, we use a weighting filter $W_u(s)$ such that the synthesis objective is:
|
||||
\begin{equation}
|
||||
\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}
|
||||
|
||||
# Comments on the choice of the weights => we cannot ask for less uncertainty than both sensors
|
||||
|
||||
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}
|
||||
|
||||
This problem can thus be dealt with an $\mathcal{H}_\infty$ synthesis problem by considering the following generalized plant (Figure ref:fig:h_infinity_robust_fusion):
|
||||
\begin{equation}
|
||||
\begin{pmatrix}
|
||||
z_1 \\ z_2 \\ v
|
||||
\end{pmatrix} = \underbrace{\begin{bmatrix}
|
||||
W_u W_1 & -W_u W_1 \\
|
||||
0 & W_u W_2 \\
|
||||
1 & 0
|
||||
\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
|
||||
w \\ u
|
||||
\end{pmatrix}
|
||||
\end{equation}
|
||||
|
||||
Applying the $\mathcal{H}_\infty$ synthesis on $P_{\mathcal{H}_\infty}$ will generate a filter $H_2(s)$ such that the $\mathcal{H}_\infty$ norm from $w$ to $(z_1,z_2)$ is minimized:
|
||||
#+NAME: eq:Hinf_norm
|
||||
\begin{equation}
|
||||
\left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_\infty = \left\| \begin{matrix} W_u W_1 (1 - H_2) \\ W_u W_2 H_2 \end{matrix} \right\|_\infty
|
||||
\end{equation}
|
||||
|
||||
The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_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}
|
||||
|
||||
#+name: fig:h_infinity_robust_fusion
|
||||
#+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/h_infinity_robust_fusion.pdf]]
|
||||
|
||||
** 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>>
|
||||
|
||||
** Sensor with noise and model uncertainty
|
||||
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}
|
||||
|
||||
#+name: fig:sensor_model_noise_uncertainty
|
||||
#+caption: Sensor Model including Noise and Dynamical Uncertainty
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/sensor_model_noise_uncertainty.pdf]]
|
||||
|
||||
** Sensor Fusion Architecture
|
||||
|
||||
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 $x$
|
||||
|
||||
#+name: fig:sensor_fusion_arch_full
|
||||
#+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/sensor_fusion_arch_full.pdf]]
|
||||
|
||||
** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
|
||||
|
||||
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:
|
||||
- Keep the $\mathcal{H}_\infty$ norm from $w$ to $(z_{\infty,1}, z_{\infty,2})$ below $1$
|
||||
- Minimize the $\mathcal{H}_2$ norm from $w$ to $(z_{2,1}, z_{2,2})$
|
||||
|
||||
#+name: fig:mixed_h2_hinf_synthesis
|
||||
#+caption: Generalized plant $P_{\mathcal{H}_2/\matlcal{H}_\infty}$ used for the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/mixed_h2_hinf_synthesis.pdf]]
|
||||
|
||||
** 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>>
|
||||
|
||||
** Experimental Setup
|
||||
|
||||
** Sensor Noise and Dynamical Uncertainty
|
||||
|
||||
** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
|
||||
|
||||
** Super Sensor Noise and Dynamical Uncertainty
|
||||
|
||||
* Conclusion
|
||||
<<sec:conclusion>>
|
||||
|
||||
* Acknowledgment
|
||||
|
||||
* Bibliography :ignore:
|
||||
\bibliographystyle{IEEEtran}
|
||||
\bibliography{ref}
|
||||
|
||||
* Local Variables :noexport:
|
||||
# Local Variables:
|
||||
# org-latex-packages-alist: nil
|
||||
# End:
|
||||
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|
||||
% Created 2021-09-10 ven. 10:25
|
||||
% Intended LaTeX compiler: pdflatex
|
||||
\documentclass[final,journal,a4paper]{IEEEtran}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage{graphicx}
|
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\usepackage{grffile}
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\usepackage{amsmath}
|
||||
\usepackage{textcomp}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{capt-of}
|
||||
\usepackage{hyperref}
|
||||
\IEEEoverridecommandlockouts
|
||||
\usepackage{cite}
|
||||
\usepackage{amsmath,amssymb,amsfonts}
|
||||
\usepackage{algorithmic}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{textcomp}
|
||||
\usepackage{xcolor}
|
||||
\usepackage{cases}
|
||||
\usepackage{tabularx,siunitx,booktabs}
|
||||
\usepackage{algorithmic}
|
||||
\usepackage{import, hyperref}
|
||||
\renewcommand{\citedash}{--}
|
||||
\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{Verma Mohit} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ mohitverma.serc@csir.res.in }\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{2021-09-10}
|
||||
\title{Optimal and Robust Sensor Fusion}
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
\begin{abstract}
|
||||
Abstract text to be done
|
||||
\end{abstract}
|
||||
|
||||
\begin{IEEEkeywords}
|
||||
Complementary Filters, Sensor Fusion, H-Infinity Synthesis
|
||||
\end{IEEEkeywords}
|
||||
|
||||
\section{Introduction}
|
||||
\label{sec:introduction}
|
||||
|
||||
\cite{mahony08_nonlin_compl_filter_special_orthog_group}
|
||||
|
||||
\begin{itemize}
|
||||
\item Section \ref{sec:optimal_fusion}
|
||||
\item Section \ref{sec:robust_fusion}
|
||||
\item Section \ref{sec:optimal_robust_fusion}
|
||||
\item Section \ref{sec:experimental_validation}
|
||||
\end{itemize}
|
||||
|
||||
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
|
||||
\label{sec:optimal_fusion}
|
||||
|
||||
\subsection{Sensor Model}
|
||||
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 \(G_i(s)\).
|
||||
|
||||
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_noise}):
|
||||
\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,scale=1]{figs/sensor_model_noise.pdf}
|
||||
\caption{\label{fig:sensor_model_noise}Sensor Model}
|
||||
\end{figure}
|
||||
|
||||
\subsection{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 \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
|
||||
\includegraphics[scale=1,scale=1]{figs/sensor_fusion_noise_arch.pdf}
|
||||
\caption{\label{fig:sensor_fusion_noise_arch}Sensor Fusion Architecture with sensor noise}
|
||||
\end{figure}
|
||||
|
||||
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}
|
||||
\end{equation}
|
||||
|
||||
Suppose the sensor dynamical model \(\hat{G}_i\) is perfect:
|
||||
\begin{equation}
|
||||
\hat{G}_i = G_i
|
||||
\end{equation}
|
||||
|
||||
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}
|
||||
|
||||
\subsection{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}
|
||||
|
||||
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}
|
||||
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_{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 \(\sigma_n\) is minimized.
|
||||
|
||||
This can be cast into an \(\mathcal{H}_2\) synthesis problem by considering the following generalized plant (also represented in Figure \ref{fig:h_two_optimal_fusion}):
|
||||
\begin{equation}
|
||||
\begin{pmatrix}
|
||||
z_1 \\ z_2 \\ v
|
||||
\end{pmatrix} = \underbrace{\begin{bmatrix}
|
||||
N_1 & -N_1 \\
|
||||
0 & N_2 \\
|
||||
1 & 0
|
||||
\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
|
||||
w \\ u
|
||||
\end{pmatrix}
|
||||
\end{equation}
|
||||
|
||||
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:
|
||||
\begin{equation}
|
||||
\label{eq:H2_norm}
|
||||
\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.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,scale=1]{figs/h_two_optimal_fusion.pdf}
|
||||
\caption{\label{fig:h_two_optimal_fusion}Generalized plant \(P_{\mathcal{H}_2}\) used for the \(\mathcal{H}_2\) synthesis of complementary filters}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Example}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,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,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,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,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,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}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,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,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:robust_fusion}
|
||||
|
||||
\subsection{Representation of Sensor Dynamical Uncertainty}
|
||||
|
||||
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.
|
||||
|
||||
The Uncertainty on the sensor dynamics \(G_i(s)\) is here modelled by (input) multiplicative uncertainty:
|
||||
\begin{equation}
|
||||
G_i(s) = \hat{G}_i(s) \left( 1 + W_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
|
||||
\end{equation}
|
||||
where \(\hat{G}_i(s)\) is the nominal model, \(W_i\) a weight representing the size of the uncertainty at each frequency, and \(\Delta_i\) is any complex perturbation such that \(\left\| \Delta_i \right\|_\infty < 1\).
|
||||
|
||||
The sensor can then be represented as shown in Figure \ref{fig:sensor_model_uncertainty}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,scale=1]{figs/sensor_model_uncertainty.pdf}
|
||||
\caption{\label{fig:sensor_model_uncertainty}Sensor Model including Dynamical Uncertainty}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Sensor Fusion Architecture}
|
||||
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:
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\hat{x} &= \Big( H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + W_1 \Delta_1) \\
|
||||
& \quad + H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + W_2 \Delta_2) \Big) x \\
|
||||
&= \Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
with \(\Delta_i\) is any transfer function satisfying \(\| \Delta_i \|_\infty < 1\).
|
||||
|
||||
As \(H_1\) and \(H_2\) are complementary filters, we finally have:
|
||||
\begin{equation}
|
||||
\hat{x} = \left( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \right) x, \quad \|\Delta_i\|_\infty<1
|
||||
\end{equation}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,scale=1]{figs/sensor_fusion_arch_uncertainty.pdf}
|
||||
\caption{\label{fig:sensor_fusion_arch_uncertainty}Sensor Fusion Architecture with sensor model uncertainty}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Super Sensor Dynamical Uncertainty}
|
||||
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}.
|
||||
|
||||
|
||||
And we can see that the dynamical uncertainty of the super sensor is equal to the sum of the individual sensor uncertainties filtered out by the complementary filters.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,scale=1]{figs/uncertainty_set_super_sensor.pdf}
|
||||
\caption{\label{fig:uncertainty_set_super_sensor}Super Sensor model uncertainty displayed in the complex plane}
|
||||
\end{figure}
|
||||
|
||||
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}
|
||||
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.
|
||||
|
||||
To define what by ``small'' we mean, we use a weighting filter \(W_u(s)\) such that the synthesis objective is:
|
||||
\begin{equation}
|
||||
\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 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}
|
||||
|
||||
This problem can thus be dealt with an \(\mathcal{H}_\infty\) synthesis problem by considering the following generalized plant (Figure \ref{fig:h_infinity_robust_fusion}):
|
||||
\begin{equation}
|
||||
\begin{pmatrix}
|
||||
z_1 \\ z_2 \\ v
|
||||
\end{pmatrix} = \underbrace{\begin{bmatrix}
|
||||
W_u W_1 & -W_u W_1 \\
|
||||
0 & W_u W_2 \\
|
||||
1 & 0
|
||||
\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
|
||||
w \\ u
|
||||
\end{pmatrix}
|
||||
\end{equation}
|
||||
|
||||
Applying the \(\mathcal{H}_\infty\) synthesis on \(P_{\mathcal{H}_\infty}\) will generate a filter \(H_2(s)\) such that the \(\mathcal{H}_\infty\) norm from \(w\) to \((z_1,z_2)\) is minimized:
|
||||
\begin{equation}
|
||||
\label{eq:Hinf_norm}
|
||||
\left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_\infty = \left\| \begin{matrix} W_u W_1 (1 - H_2) \\ W_u W_2 H_2 \end{matrix} \right\|_\infty
|
||||
\end{equation}
|
||||
|
||||
The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_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}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,scale=1]{figs/h_infinity_robust_fusion.pdf}
|
||||
\caption{\label{fig:h_infinity_robust_fusion}Generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) synthesis of complementary filters}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Example}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,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,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,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,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,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:optimal_robust_fusion}
|
||||
|
||||
\subsection{Sensor with noise and model uncertainty}
|
||||
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
|
||||
\includegraphics[scale=1,scale=1]{figs/sensor_model_noise_uncertainty.pdf}
|
||||
\caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Sensor Fusion Architecture}
|
||||
|
||||
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
|
||||
\includegraphics[scale=1,scale=1]{figs/sensor_fusion_arch_full.pdf}
|
||||
\caption{\label{fig:sensor_fusion_arch_full}Super Sensor Fusion with both sensor noise and sensor model uncertainty}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
||||
|
||||
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
|
||||
\includegraphics[scale=1,scale=1]{figs/mixed_h2_hinf_synthesis.pdf}
|
||||
\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}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Example}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,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,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,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}
|
||||
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||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,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:experimental_validation}
|
||||
|
||||
\subsection{Experimental Setup}
|
||||
|
||||
\subsection{Sensor Noise and Dynamical Uncertainty}
|
||||
|
||||
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
||||
|
||||
\subsection{Super Sensor Noise and Dynamical Uncertainty}
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:conclusion}
|
||||
|
||||
\section{Acknowledgment}
|
||||
|
||||
\bibliographystyle{IEEEtran}
|
||||
\bibliography{ref}
|
||||
\end{document}
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journal/ref.bib
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@@ -0,0 +1,171 @@
|
||||
@article{collette15_sensor_fusion_method_high_perfor,
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keywords = {},
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|
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|
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@phdthesis{hua05_low_ligo,
|
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|
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}
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@inproceedings{pascoal99_navig_system_desig_using_time,
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Reference in New Issue
Block a user