From 5dfe5ee15a3e828542ba2dedd2637f25fa3d2d08 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 28 Apr 2021 16:00:12 +0200 Subject: [PATCH] Add journal files --- journal/elsarticle-num.bst | 1508 +++++++++++++++++ journal/elsarticle.cls | 1056 ++++++++++++ journal/elsarticle.pdf | Bin 0 -> 569010 bytes journal/figs/comp_filter_three_hinf.pdf | Bin 0 -> 54635 bytes journal/figs/comp_fir_ligo_hinf.pdf | Bin 0 -> 132665 bytes journal/figs/fusion_super_sensor.pdf | Bin 0 -> 69318 bytes journal/figs/h_infinity_robust_fusion.pdf | Bin 0 -> 51363 bytes journal/figs/hinf_synthesis_results.pdf | Bin 0 -> 103869 bytes journal/figs/hinf_three_synthesis_results.pdf | Bin 0 -> 123478 bytes journal/figs/ligo_weights.pdf | Bin 0 -> 76877 bytes .../sensor_fusion_dynamic_uncertainty.pdf | Bin 0 -> 68963 bytes journal/figs/uncertainty_set_super_sensor.pdf | Bin 0 -> 52505 bytes journal/figs/weight_formula.pdf | Bin 0 -> 64326 bytes journal/guidelines.pdf | Bin 0 -> 207953 bytes journal/journal.org | 516 ++++++ journal/journal.pdf | Bin 0 -> 695705 bytes journal/journal.tex | 448 +++++ journal/ref.bib | 556 ++++++ 18 files changed, 4084 insertions(+) create mode 100644 journal/elsarticle-num.bst create mode 100644 journal/elsarticle.cls create mode 100644 journal/elsarticle.pdf create mode 100644 journal/figs/comp_filter_three_hinf.pdf create mode 100644 journal/figs/comp_fir_ligo_hinf.pdf create mode 100644 journal/figs/fusion_super_sensor.pdf create mode 100644 journal/figs/h_infinity_robust_fusion.pdf create mode 100644 journal/figs/hinf_synthesis_results.pdf create mode 100644 journal/figs/hinf_three_synthesis_results.pdf create mode 100644 journal/figs/ligo_weights.pdf create mode 100644 journal/figs/sensor_fusion_dynamic_uncertainty.pdf create mode 100644 journal/figs/uncertainty_set_super_sensor.pdf create mode 100644 journal/figs/weight_formula.pdf create mode 100644 journal/guidelines.pdf create mode 100644 journal/journal.org create mode 100644 journal/journal.pdf create mode 100644 journal/journal.tex create mode 100644 journal/ref.bib diff --git a/journal/elsarticle-num.bst b/journal/elsarticle-num.bst new file mode 100644 index 0000000..7c26b98 --- /dev/null +++ b/journal/elsarticle-num.bst @@ -0,0 +1,1508 @@ +%% +%% This is file `elsarticle-num.bst' (Version 2.1), +%% +%% Copyright 2007-2019 Elsevier Ltd +%% +%% This file is part of the 'Elsarticle Bundle'. +%% --------------------------------------------- +%% +%% It may be distributed under the conditions of the LaTeX Project Public +%% License, either version 1.2 of this license or (at your option) any +%% later version. The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.2 or later is part of all distributions of LaTeX +%% version 1999/12/01 or later. +%% +%% +%% $Id: elsarticle-num.bst 174 2019-04-05 09:50:32Z rishi $ +%% +%% $URL: http://lenova.river-valley.com/svn/elsarticle/trunk/elsarticle-num.bst $ +%% +%% ---------------------------------------- + +ENTRY + { address + author + booktitle + chapter + edition + editor + howpublished + institution + journal + key + month + note + number + organization + pages + publisher + school + series + title + type + volume + year + eprint % urlbst + doi % urlbst + url % urlbst + lastchecked % urlbst + } + {} + { label } + +INTEGERS { output.state before.all mid.sentence after.sentence after.block } + +STRINGS { urlintro eprinturl eprintprefix doiprefix doiurl openinlinelink closeinlinelink } % urlbst... +INTEGERS { hrefform inlinelinks makeinlinelink addeprints adddoiresolver } +FUNCTION {init.urlbst.variables} +{ + "Available from: " 'urlintro := % prefix before URL + "http://arxiv.org/abs/" 'eprinturl := % prefix to make URL from eprint ref + "arXiv:" 'eprintprefix := % text prefix printed before eprint ref + "https://doi.org/" 'doiurl := % prefix to make URL from DOI + "doi:" 'doiprefix := % text prefix printed before DOI ref + #1 'addeprints := % 0=no eprints; 1=include eprints + #1 'adddoiresolver := % 0=no DOI resolver; 1=include it + #2 'hrefform := % 0=no crossrefs; 1=hypertex xrefs; 2=hyperref refs + #1 'inlinelinks := % 0=URLs explicit; 1=URLs attached to titles + % the following are internal state variables, not config constants + #0 'makeinlinelink := % state variable managed by setup.inlinelink + "" 'openinlinelink := % ditto + "" 'closeinlinelink := % ditto +} +INTEGERS { + bracket.state + outside.brackets + open.brackets + within.brackets + close.brackets +} +FUNCTION {init.state.consts} +{ #0 'outside.brackets := % urlbst + #1 'open.brackets := + #2 'within.brackets := + #3 'close.brackets := + + #0 'before.all := + #1 'mid.sentence := + #2 'after.sentence := + #3 'after.block := +} + +STRINGS { s t } + +FUNCTION {output.nonnull.original} +{ 's := + output.state mid.sentence = + { ", " * write$ } + { output.state after.block = + { add.period$ write$ + newline$ + "\newblock " write$ + } + { output.state before.all = + 'write$ + { add.period$ " " * write$ } + if$ + } + if$ + mid.sentence 'output.state := + } + if$ + s +} + +FUNCTION {setup.inlinelink} +{ makeinlinelink + { hrefform #1 = % hypertex + { "\special {html: }{" * 'openinlinelink := + "\special {html:}" 'closeinlinelink := + } + { hrefform #2 = % hyperref + { "\href{" url * "}{" * 'openinlinelink := + "}" 'closeinlinelink := + } + 'skip$ + if$ % hrefform #2 = + } + if$ % hrefform #1 = + #0 'makeinlinelink := + } + 'skip$ + if$ % makeinlinelink +} +FUNCTION {add.inlinelink} +{ openinlinelink empty$ + 'skip$ + { openinlinelink swap$ * closeinlinelink * + "" 'openinlinelink := + } + if$ +} +FUNCTION {output.nonnull} +{ % Save the thing we've been asked to output + 's := + % If the bracket-state is close.brackets, then add a close-bracket to + % what is currently at the top of the stack, and set bracket.state + % to outside.brackets + bracket.state close.brackets = + { "]" * + outside.brackets 'bracket.state := + } + 'skip$ + if$ + bracket.state outside.brackets = + { % We're outside all brackets -- this is the normal situation. + % Write out what's currently at the top of the stack, using the + % original output.nonnull function. + s + add.inlinelink + output.nonnull.original % invoke the original output.nonnull + } + { % Still in brackets. Add open-bracket or (continuation) comma, add the + % new text (in s) to the top of the stack, and move to the close-brackets + % state, ready for next time (unless inbrackets resets it). If we come + % into this branch, then output.state is carefully undisturbed. + bracket.state open.brackets = + { " [" * } + { ", " * } % bracket.state will be within.brackets + if$ + s * + close.brackets 'bracket.state := + } + if$ +} + +FUNCTION {inbrackets} +{ bracket.state close.brackets = + { within.brackets 'bracket.state := } % reset the state: not open nor closed + { open.brackets 'bracket.state := } + if$ +} + +FUNCTION {format.lastchecked} +{ lastchecked empty$ + { "" } + { inbrackets "cited " lastchecked * } + if$ +} + +FUNCTION {output} +{ duplicate$ empty$ + 'pop$ + 'output.nonnull + if$ +} + +FUNCTION {output.check} +{ 't := + duplicate$ empty$ + { pop$ "empty " t * " in " * cite$ * warning$ } + 'output.nonnull + if$ +} + +FUNCTION {fin.entry.original} +{ add.period$ + write$ + newline$ +} + +FUNCTION {new.block} +{ output.state before.all = + 'skip$ + { after.block 'output.state := } + if$ +} + +FUNCTION {new.sentence} +{ output.state after.block = + 'skip$ + { output.state before.all = + 'skip$ + { after.sentence 'output.state := } + if$ + } + if$ +} + +FUNCTION {add.blank} +{ " " * before.all 'output.state := +} + +FUNCTION {date.block} +{ + add.blank +} + +FUNCTION {not} +{ { #0 } + { #1 } + if$ +} + +FUNCTION {and} +{ 'skip$ + { pop$ #0 } + if$ +} + +FUNCTION {or} +{ { pop$ #1 } + 'skip$ + if$ +} + +FUNCTION {new.block.checka} +{ empty$ + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.block.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.sentence.checka} +{ empty$ + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {new.sentence.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {field.or.null} +{ duplicate$ empty$ + { pop$ "" } + 'skip$ + if$ +} + +FUNCTION {emphasize} +{ skip$ } + +FUNCTION {capitalize} +{ "u" change.case$ "t" change.case$ } + +FUNCTION {space.word} +{ " " swap$ * " " * } + + % Here are the language-specific definitions for explicit words. + % Each function has a name bbl.xxx where xxx is the English word. + % The language selected here is ENGLISH +FUNCTION {bbl.and} +{ "and"} + +FUNCTION {bbl.etal} +{ "et~al." } + +FUNCTION {bbl.editors} +{ "Eds." } + +FUNCTION {bbl.editor} +{ "Ed." } + +FUNCTION {bbl.edby} +{ "edited by" } + +FUNCTION {bbl.edition} +{ "Edition" } + +FUNCTION {bbl.volume} +{ "Vol." } + +FUNCTION {bbl.of} +{ "of" } + +FUNCTION {bbl.number} +{ "no." } + +FUNCTION {bbl.nr} +{ "no." } + +FUNCTION {bbl.in} +{ "in" } + +FUNCTION {bbl.pages} +{ "pp." } + +FUNCTION {bbl.page} +{ "p." } + +FUNCTION {bbl.chapter} +{ "Ch." } + +FUNCTION {bbl.techrep} +{ "Tech. Rep." } + +FUNCTION {bbl.mthesis} +{ "Master's thesis" } + +FUNCTION {bbl.phdthesis} +{ "Ph.D. thesis" } + +FUNCTION {bbl.first} +{ "1st" } + +FUNCTION {bbl.second} +{ "2nd" } + +FUNCTION {bbl.third} +{ "3rd" } + +FUNCTION {bbl.fourth} +{ "4th" } + +FUNCTION {bbl.fifth} +{ "5th" } + +FUNCTION {bbl.st} +{ "st" } + +FUNCTION {bbl.nd} +{ "nd" } + +FUNCTION {bbl.rd} +{ "rd" } + +FUNCTION {bbl.th} +{ "th" } + +MACRO {jan} {"Jan."} + +MACRO {feb} {"Feb."} + +MACRO {mar} {"Mar."} + +MACRO {apr} {"Apr."} + +MACRO {may} {"May"} + +MACRO {jun} {"Jun."} + +MACRO {jul} {"Jul."} + +MACRO {aug} {"Aug."} + +MACRO {sep} {"Sep."} + +MACRO {oct} {"Oct."} + +MACRO {nov} {"Nov."} + +MACRO {dec} {"Dec."} + +FUNCTION {eng.ord} +{ duplicate$ "1" swap$ * + #-2 #1 substring$ "1" = + { bbl.th * } + { duplicate$ #-1 #1 substring$ + duplicate$ "1" = + { pop$ bbl.st * } + { duplicate$ "2" = + { pop$ bbl.nd * } + { "3" = + { bbl.rd * } + { bbl.th * } + if$ + } + if$ + } + if$ + } + if$ +} + +MACRO {acmcs} {"ACM Comput. Surv."} + +MACRO {acta} {"Acta Inf."} + +MACRO {cacm} {"Commun. ACM"} + +MACRO {ibmjrd} {"IBM J. Res. Dev."} + +MACRO {ibmsj} {"IBM Syst.~J."} + +MACRO {ieeese} {"IEEE Trans. Softw. Eng."} + +MACRO {ieeetc} {"IEEE Trans. Comput."} + +MACRO {ieeetcad} + {"IEEE Trans. Comput.-Aided Design Integrated Circuits"} + +MACRO {ipl} {"Inf. Process. Lett."} + +MACRO {jacm} {"J.~ACM"} + +MACRO {jcss} {"J.~Comput. Syst. Sci."} + +MACRO {scp} {"Sci. Comput. Programming"} + +MACRO {sicomp} {"SIAM J. Comput."} + +MACRO {tocs} {"ACM Trans. Comput. Syst."} + +MACRO {tods} {"ACM Trans. Database Syst."} + +MACRO {tog} {"ACM Trans. Gr."} + +MACRO {toms} {"ACM Trans. Math. Softw."} + +MACRO {toois} {"ACM Trans. Office Inf. Syst."} + +MACRO {toplas} {"ACM Trans. Prog. Lang. Syst."} + +MACRO {tcs} {"Theoretical Comput. Sci."} + +FUNCTION {write.url} +{ url empty$ + { skip$ } + { "\newline\urlprefix\url{" url * "}" * write$ newline$ } + if$ +} + +INTEGERS { nameptr namesleft numnames } + +FUNCTION {format.names} +{ 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr + "{f.~}{vv~}{ll}{, jj}" format.name$ + 't := + nameptr #1 > + { + namesleft #1 > + { ", " * t * } + { + "," * + s nameptr "{ll}" format.name$ duplicate$ "others" = + { 't := } + { pop$ } + if$ + t "others" = + { + " " * bbl.etal * + } + { " " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} +FUNCTION {format.names.ed} +{ format.names } +FUNCTION {format.authors} +{ author empty$ + { "" } + { author format.names } + if$ +} + +FUNCTION {format.editors} +{ editor empty$ + { "" } + { editor format.names + editor num.names$ #1 > + { " (" * bbl.editors * ")" * } + { " (" * bbl.editor * ")" * } + if$ + } + if$ +} + +FUNCTION {format.in.editors} +{ editor empty$ + { "" } + { editor format.names.ed + editor num.names$ #1 > + { " (" * bbl.editors * ")" * } + { " (" * bbl.editor * ")" * } + if$ + } + if$ +} + +FUNCTION {format.note} +{ + note empty$ + { "" } + { note #1 #1 substring$ + duplicate$ "{" = + 'skip$ + { output.state mid.sentence = + { "l" } + { "u" } + if$ + change.case$ + } + if$ + note #2 global.max$ substring$ * + } + if$ +} + +FUNCTION {format.title} +{ title empty$ + { "" } + { title "t" change.case$ + } + if$ +} + +FUNCTION {output.bibitem.original} +{ newline$ + "\bibitem{" write$ + cite$ write$ + "}" write$ + newline$ + "" + before.all 'output.state := +} + +FUNCTION {n.dashify} +{ + 't := + "" + { t empty$ not } + { t #1 #1 substring$ "-" = + { t #1 #2 substring$ "--" = not + { "--" * + t #2 global.max$ substring$ 't := + } + { { t #1 #1 substring$ "-" = } + { "-" * + t #2 global.max$ substring$ 't := + } + while$ + } + if$ + } + { t #1 #1 substring$ * + t #2 global.max$ substring$ 't := + } + if$ + } + while$ +} + +FUNCTION {word.in} +{ bbl.in + ":" * + " " * } + +FUNCTION {format.date} +{ year empty$ + { month empty$ + { "" } + { "there's a month but no year in " cite$ * warning$ + month + } + if$ + } + { month empty$ + 'year + { month " " * year * } + if$ + } + if$ + duplicate$ empty$ + 'skip$ + { + before.all 'output.state := + " (" swap$ * ")" * + } + if$ +} + +FUNCTION{format.year} +{ year duplicate$ empty$ + { "empty year in " cite$ * warning$ pop$ "" } + { "(" swap$ * ")" * } + if$ +} + +FUNCTION {format.btitle} +{ title +} + +FUNCTION {tie.or.space.connect} +{ duplicate$ text.length$ #3 < + { "~" } + { " " } + if$ + swap$ * * +} + +FUNCTION {either.or.check} +{ empty$ + 'pop$ + { "can't use both " swap$ * " fields in " * cite$ * warning$ } + if$ +} + +FUNCTION {format.bvolume} +{ volume empty$ + { "" } + { bbl.volume volume tie.or.space.connect + series empty$ + 'skip$ + { bbl.of space.word * series emphasize * } + if$ + "volume and number" number either.or.check + } + if$ +} + +FUNCTION {format.number.series} +{ volume empty$ + { number empty$ + { series field.or.null } + { output.state mid.sentence = + { bbl.number } + { bbl.number capitalize } + if$ + number tie.or.space.connect + series empty$ + { "there's a number but no series in " cite$ * warning$ } + { bbl.in space.word * series * } + if$ + } + if$ + } + { "" } + if$ +} + +FUNCTION {is.num} +{ chr.to.int$ + duplicate$ "0" chr.to.int$ < not + swap$ "9" chr.to.int$ > not and +} + +FUNCTION {extract.num} +{ duplicate$ 't := + "" 's := + { t empty$ not } + { t #1 #1 substring$ + t #2 global.max$ substring$ 't := + duplicate$ is.num + { s swap$ * 's := } + { pop$ "" 't := } + if$ + } + while$ + s empty$ + 'skip$ + { pop$ s } + if$ +} + +FUNCTION {convert.edition} +{ edition extract.num "l" change.case$ 's := + s "first" = s "1" = or + { bbl.first 't := } + { s "second" = s "2" = or + { bbl.second 't := } + { s "third" = s "3" = or + { bbl.third 't := } + { s "fourth" = s "4" = or + { bbl.fourth 't := } + { s "fifth" = s "5" = or + { bbl.fifth 't := } + { s #1 #1 substring$ is.num + { s eng.ord 't := } + { edition 't := } + if$ + } + if$ + } + if$ + } + if$ + } + if$ + } + if$ + t +} + +FUNCTION {format.edition} +{ edition empty$ + { "" } + { output.state mid.sentence = + { convert.edition "l" change.case$ " " * bbl.edition * } + { convert.edition "t" change.case$ " " * bbl.edition * } + if$ + } + if$ +} + +INTEGERS { multiresult } + +FUNCTION {multi.page.check} +{ 't := + #0 'multiresult := + { multiresult not + t empty$ not + and + } + { t #1 #1 substring$ + duplicate$ "-" = + swap$ duplicate$ "," = + swap$ "+" = + or or + { #1 'multiresult := } + { t #2 global.max$ substring$ 't := } + if$ + } + while$ + multiresult +} + +FUNCTION {format.pages} +{ pages empty$ + { "" } + { pages multi.page.check + { bbl.pages pages n.dashify tie.or.space.connect } + { bbl.page pages tie.or.space.connect } + if$ + } + if$ +} + +FUNCTION {format.journal.pages} +{ pages empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.pages } + { + " " * + format.year * " " * + pages n.dashify * + } + if$ + } + if$ +} + +FUNCTION {format.vol.num.pages} +{ + % volume field.or.null + " " + volume empty$ + { pop$ "" } + { volume * } + if$ + number empty$ + 'skip$ + { + "~(" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ +} + +FUNCTION {format.chapter.pages} +{ chapter empty$ + { "" } + { type empty$ + { bbl.chapter } + { type "l" change.case$ } + if$ + chapter tie.or.space.connect + } + if$ +} + +FUNCTION {format.in.ed.booktitle} +{ booktitle empty$ + { "" } + { editor empty$ + { word.in booktitle * } + { word.in format.in.editors * ", " * + booktitle * } + if$ + } + if$ +} + +FUNCTION {empty.misc.check} +{ author empty$ title empty$ howpublished empty$ + month empty$ year empty$ note empty$ + and and and and and + { "all relevant fields are empty in " cite$ * warning$ } + 'skip$ + if$ +} + +FUNCTION {format.thesis.type} +{ type empty$ + 'skip$ + { pop$ + type "t" change.case$ + } + if$ +} + +FUNCTION {format.tr.number} +{ type empty$ + { bbl.techrep } + 'type + if$ + number empty$ + { "t" change.case$ } + { number tie.or.space.connect } + if$ +} + +FUNCTION {format.article.crossref} +{ + key empty$ + { journal empty$ + { "need key or journal for " cite$ * " to crossref " * crossref * + warning$ + "" + } + { word.in journal emphasize * } + if$ + } + { word.in key * " " *} + if$ + " \cite{" * crossref * "}" * +} + +FUNCTION {format.crossref.editor} +{ editor #1 "{vv~}{ll}" format.name$ + editor num.names$ duplicate$ + #2 > + { pop$ + " " * bbl.etal * + } + { #2 < + 'skip$ + { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = + { + " " * bbl.etal * + } + { bbl.and space.word * editor #2 "{vv~}{ll}" format.name$ + * } + if$ + } + if$ + } + if$ +} + +FUNCTION {format.book.crossref} +{ volume empty$ + { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ + word.in + } + { bbl.volume volume tie.or.space.connect + bbl.of space.word * + } + if$ + editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { series empty$ + { "need editor, key, or series for " cite$ * " to crossref " * + crossref * warning$ + "" * + } + { series emphasize * } + if$ + } + { key * } + if$ + } + { format.crossref.editor * } + if$ + " \cite{" * crossref * "}" * +} + +FUNCTION {format.incoll.inproc.crossref} +{ + editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { booktitle empty$ + { "need editor, key, or booktitle for " cite$ * " to crossref " * + crossref * warning$ + "" + } + { word.in booktitle * } + if$ + } + { word.in key * " " *} + if$ + } + { word.in format.crossref.editor * " " *} + if$ + " \cite{" * crossref * "}" * +} + +FUNCTION {format.org.or.pub} +{ 't := + "" + year empty$ + { "empty year in " cite$ * warning$ } + 'skip$ + if$ + address empty$ t empty$ and + year empty$ and + 'skip$ + { + t empty$ + { address empty$ + 'skip$ + { address * } + if$ + } + { t * + address empty$ + 'skip$ + { ", " * address * } + if$ + } + if$ + year empty$ + 'skip$ + { t empty$ address empty$ and + 'skip$ + { ", " * } + if$ + year * + } + if$ + } + if$ +} + +FUNCTION {format.publisher.address} +{ publisher empty$ + { "empty publisher in " cite$ * warning$ + "" + } + { publisher } + if$ + format.org.or.pub +} + +FUNCTION {format.organization.address} +{ organization empty$ + { "" } + { organization } + if$ + format.org.or.pub +} + +FUNCTION {make.href.null} +{ + pop$ +} +FUNCTION {make.href.hypertex} +{ + "\special {html: }" * swap$ * + "\special {html:}" * +} +FUNCTION {make.href.hyperref} +{ + "\href {" swap$ * "} {\path{" * swap$ * "}}" * +} +FUNCTION {make.href} +{ hrefform #2 = + 'make.href.hyperref % hrefform = 2 + { hrefform #1 = + 'make.href.hypertex % hrefform = 1 + 'make.href.null % hrefform = 0 (or anything else) + if$ + } + if$ +} + +FUNCTION {format.url} +{ inlinelinks #1 = url empty$ or + { "" } + { hrefform #1 = + { % special case -- add HyperTeX specials + urlintro "\url{" url * "}" * url make.href.hypertex * } + { urlintro "\url{" * url * "}" * } + if$ + } + if$ +} + +FUNCTION {format.eprint} +{ eprint empty$ + { "" } + { eprintprefix eprint * eprinturl eprint * make.href } + if$ +} + +FUNCTION {format.doi} +{ doi empty$ + { "" } + { doiprefix doi * doiurl doi * make.href } + if$ +} + +FUNCTION {output.url} +{ url empty$ + 'skip$ + { new.block + format.url output + format.lastchecked output + } + if$ +} + +FUNCTION {output.web.refs} +{ + new.block + output.url + addeprints eprint empty$ not and + { format.eprint output.nonnull } + 'skip$ + if$ + adddoiresolver doi empty$ not and + { format.doi output.nonnull } + 'skip$ + if$ +} + +FUNCTION {output.bibitem} +{ outside.brackets 'bracket.state := + output.bibitem.original + inlinelinks url empty$ not and + { #1 'makeinlinelink := } + { #0 'makeinlinelink := } + if$ +} + +FUNCTION {fin.entry} +{ output.web.refs % urlbst + makeinlinelink % ooops, it appears we didn't have a title for inlinelink + { setup.inlinelink % add some artificial link text here, as a fallback + "[link]" output.nonnull } + 'skip$ + if$ + bracket.state close.brackets = % urlbst + { "]" * } + 'skip$ + if$ + fin.entry.original +} + +FUNCTION {webpage} +{ output.bibitem + author empty$ + { editor empty$ + 'skip$ % author and editor both optional + { format.editors output.nonnull } + if$ + } + { editor empty$ + { format.authors output.nonnull } + { "can't use both author and editor fields in " cite$ * warning$ } + if$ + } + if$ + new.block + title empty$ 'skip$ 'setup.inlinelink if$ + format.title "title" output.check + inbrackets "online" output + new.block + year empty$ + 'skip$ + { format.date "year" output.check } + if$ + % We don't need to output the URL details ('lastchecked' and 'url'), + % because fin.entry does that for us, using output.web.refs. The only + % reason we would want to put them here is if we were to decide that + % they should go in front of the rather miscellaneous information in 'note'. + new.block + note output + fin.entry +} + +FUNCTION {article} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + crossref missing$ + { journal + "journal" output.check + % add.blank + before.all 'output.state := + format.vol.num.pages output + } + { format.article.crossref output.nonnull + format.pages output + } + if$ + format.journal.pages + format.note output + pages empty$ + { format.date "year" output.check } + 'skip$ + if$ + fin.entry + write.url +} + +FUNCTION {book} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.btitle "title" output.check + crossref missing$ + { format.edition output + format.bvolume output + format.number.series output + format.publisher.address output + } + { + format.book.crossref output.nonnull + } + if$ + format.note output + fin.entry + write.url +} + +FUNCTION {booklet} +{ output.bibitem + format.authors output + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + howpublished output + address output + format.note output + format.date "year" output.check + fin.entry + write.url +} + +FUNCTION {inbook} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.btitle "title" output.check + crossref missing$ + { + format.edition output + format.bvolume output + format.number.series output + format.publisher.address output + format.chapter.pages "chapter and pages" output.check + } + { + format.chapter.pages "chapter and pages" output.check + format.book.crossref output.nonnull + } + if$ + format.pages "pages" output.check + format.note output + fin.entry + write.url +} + +FUNCTION {incollection} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.edition output + format.bvolume output + format.number.series output + format.publisher.address output + format.chapter.pages output + } + { format.incoll.inproc.crossref output.nonnull + format.chapter.pages output + } + if$ + format.pages "pages" output.check + format.note output + fin.entry + write.url +} + +FUNCTION {inproceedings} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.edition output + format.bvolume output + format.number.series output + publisher empty$ + { format.organization.address output } + { organization output + format.publisher.address output + } + if$ + } + { format.incoll.inproc.crossref output.nonnull + } + if$ + format.pages "pages" output.check + format.note output + fin.entry + write.url +} + +FUNCTION {conference} { inproceedings } + +FUNCTION {manual} +{ output.bibitem + author empty$ + { organization empty$ + 'skip$ + { organization output.nonnull + address output + } + if$ + } + { format.authors output.nonnull } + if$ + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.btitle "title" output.check + author empty$ + { organization empty$ + { + address output + } + 'skip$ + if$ + } + { + organization output + address output + } + if$ + format.edition output + format.note output + format.date "year" output.check + fin.entry + write.url +} + +FUNCTION {mastersthesis} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + bbl.mthesis format.thesis.type output.nonnull + school "school" output.check + address output + format.note output + format.date "year" output.check + fin.entry + write.url +} + +FUNCTION {misc} +{ output.bibitem + format.authors output + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title output + howpublished output + format.note output + format.date "year" output.check + fin.entry + write.url + empty.misc.check +} + +FUNCTION {phdthesis} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + bbl.phdthesis format.thesis.type output.nonnull + school "school" output.check + address output + format.note output + format.date "year" output.check + fin.entry + write.url +} + +FUNCTION {proceedings} +{ output.bibitem + editor empty$ + { organization output } + { format.editors output.nonnull } + if$ + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.btitle "title" output.check + format.bvolume output + format.number.series output + editor empty$ + { publisher empty$ + 'skip$ + { + format.publisher.address output + } + if$ + } + { publisher empty$ + { + format.organization.address output } + { + organization output + format.publisher.address output + } + if$ + } + if$ + format.note output + fin.entry + write.url +} + +FUNCTION {techreport} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + format.tr.number output.nonnull + institution "institution" output.check + address output + format.note output + format.date "year" output.check + fin.entry + write.url +} + +FUNCTION {unpublished} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + format.note "note" output.check + format.date "year" output.check + fin.entry + write.url +} + +FUNCTION {default.type} { misc } + +READ + +STRINGS { longest.label } + +INTEGERS { number.label longest.label.width } + +FUNCTION {initialize.longest.label} +{ "" 'longest.label := + #1 'number.label := + #0 'longest.label.width := +} + +FUNCTION {longest.label.pass} +{ number.label int.to.str$ 'label := + number.label #1 + 'number.label := + label width$ longest.label.width > + { label 'longest.label := + label width$ 'longest.label.width := + } + 'skip$ + if$ +} + +EXECUTE {initialize.longest.label} + +ITERATE {longest.label.pass} + +FUNCTION {begin.bib} +{ preamble$ empty$ + 'skip$ + { preamble$ write$ newline$ } + if$ + "\begin{thebibliography}{" longest.label * "}" * + write$ newline$ + "\expandafter\ifx\csname url\endcsname\relax" + write$ newline$ + " \def\url#1{\texttt{#1}}\fi" + write$ newline$ + "\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi" + write$ newline$ + "\expandafter\ifx\csname href\endcsname\relax" + write$ newline$ + " \def\href#1#2{#2} \def\path#1{#1}\fi" + write$ newline$ +} + +EXECUTE {begin.bib} + +EXECUTE {init.urlbst.variables} +EXECUTE {init.state.consts} + +ITERATE {call.type$} + +FUNCTION {end.bib} +{ newline$ + "\end{thebibliography}" write$ newline$ +} + +EXECUTE {end.bib} +%% End of customized bst file +%% +%% End of file `elsarticle-num.bst'. + diff --git a/journal/elsarticle.cls b/journal/elsarticle.cls new file mode 100644 index 0000000..53a1072 --- /dev/null +++ b/journal/elsarticle.cls @@ -0,0 +1,1056 @@ +%% +%% This is file `elsarticle.cls', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% elsarticle.dtx (with options: `class') +%% +%% Copyright 2007-2019 Elsevier Ltd +%% +%% This file is part of the 'Elsarticle Bundle'. +%% ------------------------------------------- +%% +%% It may be distributed under the conditions of the LaTeX Project Public +%% License, either version 1.2 of this license or (at your option) any +%% later version. The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.2 or later is part of all distributions of LaTeX +%% version 1999/12/01 or later. +%% +%% The list of all files belonging to the 'Elsarticle Bundle' is +%% given in the file `manifest.txt'. +%% +%% +%% +%% + \def\RCSfile{elsarticle}% + \def\RCSversion{3.2}% + \def\RCSdate{2019/02/25}% + \def\@shortjnl{\relax} + \def\@journal{Elsevier Ltd} + \def\@company{Elsevier Ltd} + \def\@issn{000-0000} + \def\@shortjid{elsarticle} +\NeedsTeXFormat{LaTeX2e}[1995/12/01] +\ProvidesClass{\@shortjid}[\RCSdate, \RCSversion: \@journal] +\def\ABD{\AtBeginDocument} +\newif\ifpreprint \preprintfalse +\newif\ifnonatbib \nonatbibfalse +\newif\iflongmktitle \longmktitlefalse +\newif\ifnopreprintline \nopreprintlinefalse +\newif\ifdoubleblind \doubleblindfalse + +\def\@blstr{1} +\newdimen\@bls +\@bls=\baselineskip + +\def\@finalWarning{% + *****************************************************\MessageBreak + This document is typeset in the CRC style which\MessageBreak + is not suitable for submission.\MessageBreak + \MessageBreak + Please typeset again using 'preprint' option\MessageBreak + for creating PDF suitable for submission.\MessageBreak + ******************************************************\MessageBreak +} + +\DeclareOption{preprint}{\global\preprinttrue + \gdef\@blstr{1}\xdef\jtype{0}% + \AtBeginDocument{\@twosidefalse\@mparswitchfalse}} +\DeclareOption{nopreprintline}{\global\nopreprintlinetrue} +\DeclareOption{final}{\gdef\@blstr{1}\global\preprintfalse} +\DeclareOption{review}{\global\preprinttrue\gdef\@blstr{1.5}} +\DeclareOption{authoryear}{\xdef\@biboptions{round,authoryear}} +\DeclareOption{number}{\xdef\@biboptions{numbers}} +\DeclareOption{numbers}{\xdef\@biboptions{numbers}} +\DeclareOption{nonatbib}{\global\nonatbibtrue} +\DeclareOption{longtitle}{\global\longmktitletrue} +\DeclareOption{5p}{\xdef\jtype{5}\global\preprintfalse + \ExecuteOptions{twocolumn}} + \def\jtype{0} +\DeclareOption{3p}{\xdef\jtype{3}\global\preprintfalse} +\DeclareOption{1p}{\xdef\jtype{1}\global\preprintfalse + \AtBeginDocument{\@twocolumnfalse}} +\DeclareOption{times}{\IfFileExists{txfonts.sty}% + {\AtEndOfClass{\RequirePackage{txfonts}% + \gdef\ttdefault{cmtt}% + \let\iint\relax + \let\iiint\relax + \let\iiiint\relax + \let\idotsint\relax + \let\openbox\relax}}{\AtEndOfClass{\RequirePackage{times}}}} + +\DeclareOption{endfloat}{\IfFileExists{endfloat.sty} + {\AtEndOfClass{\RequirePackage[markers]{endfloat}}}{}} +\DeclareOption{endfloats}{\IfFileExists{endfloat.sty} + {\AtEndOfClass{\RequirePackage[markers]{endfloat}}}{}} +\DeclareOption{numafflabel} + {\AtBeginDocument{\def\theaffn{\arabic{affn}}}} %*% +\DeclareOption{lefttitle} + {\AtBeginDocument{\def\elsarticletitlealign{flushleft}}} %*% +\DeclareOption{centertitle} + {\AtBeginDocument{\def\elsarticletitlealign{center}}} %*% +\DeclareOption{reversenotenum} + {\AtBeginDocument{\def\theaffn{\arabic{affn}} + \def\thefnote{\alph{fnote}}}} +\DeclareOption{doubleblind}{\doubleblindtrue} + +\ExecuteOptions{a4paper,10pt,oneside,onecolumn,number,preprint,centertitle} +\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}} +\ProcessOptions +\LoadClass{article} +\RequirePackage{graphicx} +\let\comma\@empty +\let\tnotesep\@empty +\let\@title\@empty + +\def\useelstitle{} + +\def\title#1{\g@addto@macro\@title{#1% + \global\let\tnoteref\@gobble}% + \g@addto@macro\useelstitle{#1}} + +\def\elsLabel#1{\@bsphack\protected@write\@auxout{}% + {\string\Newlabel{#1}{\@currentlabel}}\@esphack} +\def\Newlabel#1#2{\expandafter\xdef\csname X@#1\endcsname{#2}} + +\def\elsRef#1{\@ifundefined{X@#1}{0}{\csname X@#1\endcsname}} + +\let\@tnotemark\@empty + +\ifdoubleblind + \def\tnotemark[#1]{} +\else + \def\tnotemark[#1]{\@for\mytmark:=#1\do{% + \expandafter\ifcase\elsRef{\mytmark}\or$^{\star}$\or + $^{,\star\star}$\fi + }% +} +\fi + +\def\tnoteref#1{\tnotemark[{#1}]} +\let\@tnotes\@empty +\newcounter{tnote} +\def\tnotetext[#1]#2{\g@addto@macro\@tnotes{% + \stepcounter{tnote}\elsLabel{#1}% + \def\thefootnote{\ifcase\c@tnote\or$\star$\or$\star\star$\fi}% + \footnotetext{#2}}} + +\let\@nonumnotes\@empty +\def\nonumnote#1{\g@addto@macro\@nonumnotes{% + \let\thefootnote\relax\footnotetext{#1}}} + +\newcounter{fnote} +\def\thefnote{\arabic{fnote}} +\def\fnmark[#1]{\let\comma\@empty + \def\@fnmark{\@for\@@fnmark:=#1\do{% + \edef\fnotenum{\@ifundefined{X@\@@fnmark}{1}{\elsRef{\@@fnmark}}}% + \unskip\comma\fnotenum\let\comma,}}% +} + +\def\fnref#1{\fnmark[#1]} + +\let\@fnotes\@empty\let\@fnmark\@empty +\def\fntext[#1]#2{\g@addto@macro\@fnotes{% + \refstepcounter{fnote}\elsLabel{#1}% + \def\thefootnote{\c@fnote}% + \global\setcounter{footnote}{\c@fnote}% + \footnotetext{#2}}} + +\def\cormark[#1]{\edef\cnotenum{\elsRef{#1}}% + \unskip\textsuperscript{\sep\ifcase\cnotenum\or + $\ast$\or$\ast\ast$\fi\hspace{-1pt}}\let\sep=,} + +\let\@cormark\@empty +\let\@cornotes\@empty +\newcounter{cnote} +\def\cortext[#1]#2{\g@addto@macro\@cornotes{% + \refstepcounter{cnote}\elsLabel{#1}% + \def\thefootnote{\ifcase\thecnote\or$\ast$\or + $\ast\ast$\fi}% + \footnotetext{#2}}} + +\let\@corref\@empty +\def\corref#1{\edef\cnotenum{\elsRef{#1}}% + \edef\@corref{\ifcase\cnotenum\or + $\ast$\or$\ast\ast$\fi\hskip-1pt}} + +\def\resetTitleCounters{\c@cnote=0 + \c@fnote=0 \c@tnote=0 \c@footnote=0} + +\let\eadsep\@empty +\def\@elseads{} +\let\@elsuads\@empty +\let\@cormark\@empty +\def\hashchar{\expandafter\@gobble\string\~} +\def\underscorechar{\expandafter\@gobble\string\_} +\def\lbracechar{\expandafter\@gobble\string\{} +\def\rbracechar{\expandafter\@gobble\string\}} + +\gdef\ead{\@ifnextchar[{\@uad}{\@ead}} +\gdef\@ead#1{\bgroup + \def\_{\underscorechar}% + \def\{{\lbracechar}% + \def~{\hashchar}% + \def\}{\rbracechar}% + \edef\tmp{\the\@eadauthor}% + \immediate\write\@auxout{\string\emailauthor + {#1}{\expandafter\strip@prefix\meaning\tmp}}% + \egroup +} +\newcounter{ead} +\gdef\emailauthor#1#2{\stepcounter{ead}% + \g@addto@macro\@elseads{\raggedright% + \let\corref\@gobble\def\@@tmp{#1}% + \eadsep{\ttfamily\expandafter\strip@prefix\meaning\@@tmp} + (#2)\def\eadsep{\unskip,\space}}% +} +\gdef\@uad[#1]#2{\bgroup + \def~{\hashchar}% + \def\_{\underscorechar}% + \def~{\hashchar}% + \def\}{\rbracechar}% + \edef\tmp{\the\@eadauthor} + \immediate\write\@auxout{\string\urlauthor + {#2}{\expandafter\strip@prefix\meaning\tmp}}% + \egroup +} +\gdef\urlauthor#1#2{\g@addto@macro\@elsuads{\let\corref\@gobble% + \def\@@tmp{#1}\raggedright\eadsep + {\ttfamily\expandafter\strip@prefix\meaning\@@tmp}\space(#2)% + \def\eadsep{\unskip,\space}}% +} + +\def\elsauthors{} +\def\useauthors{} +\def\elsprelimauthors{} + +\def\pprinttitle{} +\let\authorsep\@empty +\let\prelimauthorsep\@empty +\let\sep\@empty +\newcounter{author} +\def\author{\@ifnextchar[{\@@author}{\@author}} + +\newtoks\@eadauthor +\def\@@author[#1]#2{% + \g@addto@macro\elsprelimauthors{% + \prelimauthorsep#2% + \def\prelimauthorsep{\unskip,\space}}% + \g@addto@macro\elsauthors{% + \def\baselinestretch{1}% + \authorsep#2\unskip\textsuperscript{%#1% + \@for\@@affmark:=#1\do{% + \edef\affnum{\@ifundefined{X@\@@affmark}{1}{\elsRef{\@@affmark}}}% + \unskip\sep\affnum\let\sep=,}% + \ifx\@fnmark\@empty\else\unskip\sep\@fnmark\let\sep=,\fi + \ifx\@corref\@empty\else\unskip\sep\@corref\let\sep=,\fi + }% + \def\authorsep{\unskip,\space}% + \global\let\sep\@empty\global\let\@corref\@empty + \global\let\@fnmark\@empty}% + \@eadauthor={#2}% + \g@addto@macro\useauthors{#2; }% +} + +\def\@author#1{% + \g@addto@macro\elsprelimauthors{% + \prelimauthorsep#1% + \def\prelimauthorsep{\unskip,\space}}% + \g@addto@macro\elsauthors{\normalsize% + \def\baselinestretch{1}% + \upshape\authorsep#1\unskip\textsuperscript{% + \ifx\@fnmark\@empty\else\unskip\sep\@fnmark\let\sep=,\fi + \ifx\@corref\@empty\else\unskip\sep\@corref\let\sep=,\fi + }% + \def\authorsep{\unskip,\space}% + \global\let\@fnmark\@empty + \global\let\@corref\@empty \global\let\sep\@empty}% + \@eadauthor={#1}% + \g@addto@macro\useauthors{#1; }% +} + +\AtBeginDocument{% + \@ifpackageloaded{hyperref}{% + \expandafter\gdef\csname Hy@title\endcsname{\useelstitle}% + \expandafter\gdef\csname Hy@author\endcsname{\useauthors}% + }{} +} + +\def\elsaddress{} +\def\addsep{\par\vskip6pt} +\def\address{\@ifnextchar[{\@@address}{\@address}} + +\def\@alph#1{% + \ifcase#1\or a\or b\or c\or d\or e\or f\or g\or h\or i\or j\or k\or + l\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or x\or + y\or z% + \or aa\or ab\or ac\or ad\or ae\or af\or ag\or ah\or ai\or aj\or + ak\or al\or am\or an\or ao\or ap\or aq\or ar\or as\or at\or au\or + av\or aw\or ax\or ay\or az% + \or ba\or bb\or bc\or bd\or be\or bf\or bg\or bh\or bi\or bj\or + bk\or bl\or bm\or bn\or bo\or bp\or bq\or br\or bs\or bt\or bu\or + bv\or bw\or bx\or by\or bz% + \or ca\or cb\or cc\or cd\or ce\or cf\or cg\or ch\or ci\or cj\or + ck\or cl\or cm\or cn\or co\or cp\or cq\or cr\or cs\or ct\or cu\or + cv\or cw\or cx\or cy\or cz% + \or da\or db\or dc\or dd\or de\or df\or dg\or dh\or di\or dj\or + dk\or dl\or dm\or dn\or do\or dp\or dq\or dr\or ds\or dt\or du\or + dv\or dw\or dx\or dy\or dz% + \or ea\or eb\or ec\or ed\or ee\or ef\or eg\or eh\or ei\or ej\or + ek\or el\or em\or en\or eo\or ep\or eq\or er\or es\or et\or eu\or + ev\or ew\or ex\or ey\or ez% + \or fa\or fb\or fc\or fd\or fe\or ff\or fg\or fh\or fi\or fj\or + fk\or fl\or fm\or fn\or fo\or fp\or fq\or fr\or fs\or ft\or fu\or + fv\or fw\or fx\or fy\or fz% + \or ga\or gb\or gc\or gd\or ge\or gf\or gg\or gh\or gi\or gj\or + gk\or gl\or gm\or gn\or go\or gp\or gq\or gr\or gs\or gt\or gu\or + gv\or gw\or gx\or gy\or gz% + \else\@ctrerr\fi} + +\newcounter{affn} +\renewcommand\theaffn{\alph{affn}} + +\long\def\@@address[#1]#2{\g@addto@macro\elsaddress{% + \def\baselinestretch{1}% + \refstepcounter{affn} + \xdef\@currentlabel{\theaffn} + \elsLabel{#1}% + \textsuperscript{\theaffn}#2\par}} + +\long\def\@address#1{\g@addto@macro\elsauthors{% + \def\baselinestretch{1}% + \addsep\footnotesize\itshape#1\def\addsep{\par\vskip6pt}% + \def\authorsep{\par\vskip8pt}}} + +\newbox\absbox +\let\@elsarticleabstitle\@empty %*% +\def\abstracttitle#1{\gdef\@elsarticleabstitle{#1}} %*% +\abstracttitle{Abstract} %*% +\renewenvironment{abstract}{\global\setbox\absbox=\vbox\bgroup + \hsize=\textwidth\def\baselinestretch{1}% + \noindent\unskip\textbf{\@elsarticleabstitle} %*% + \par\medskip\noindent\unskip\ignorespaces} + {\egroup} + +\newbox\elsarticlehighlightsbox +\newbox\elsarticlegrabsbox +\def\@elsarticlehighlightstitle{Highlights} +\def\@elsarticlegrabstitle{Graphical Abstract} +\newif\ifelsprelimpagegrabs\global\elsprelimpagegrabsfalse +\newif\ifelsprelimpagehl\global\elsprelimpagehlfalse +\def\elsarticleprelims{% + \ifelsprelimpagegrabs\relax% + \thispagestyle{empty}% + \unvbox\elsarticlegrabsbox% + \pagebreak\clearpage% + \fi% + \ifelsprelimpagehl\relax% + \thispagestyle{empty} + \unvbox\elsarticlehighlightsbox% + \pagebreak\clearpage% + \setcounter{page}{1}% + \fi% +} +\newenvironment{highlights}{% + \global\elsprelimpagehltrue% + \global\setbox\elsarticlehighlightsbox=\vbox\bgroup + \hsize=\textwidth\def\baselinestretch{1}% + \noindent\unskip{\Large\@elsarticlehighlightstitle}%*% + \par\vskip12pt\noindent\unskip\ignorespaces\textbf{\@title}% + \ifx\elsprelimauthors\@empty\relax\else% + \par\vskip6pt\noindent\unskip\ignorespaces\elsprelimauthors% + \fi% + \par\medskip\noindent\unskip\ignorespaces + \begin{itemize} + } + {\end{itemize} + \egroup} +\newenvironment{graphicalabstract}{% + \global\elsprelimpagegrabstrue% + \global\setbox\elsarticlegrabsbox=\vbox\bgroup + \hsize=\textwidth\def\baselinestretch{1}% + \noindent\unskip{\Large\@elsarticlegrabstitle}%*% + \par\vskip12pt\noindent\unskip\ignorespaces\textbf{\@title}% + \ifx\elsprelimauthors\@empty\relax\else% + \par\vskip6pt\noindent\unskip\ignorespaces\elsprelimauthors% + \fi% + \par\medskip\noindent\unskip\ignorespaces} + {\egroup} + +\newbox\keybox +\let\@elsarticlekwdtitle\@empty %*% +\def\keywordtitle#1{\gdef\@elsarticlekwdtitle{#1}} %*% +\def\keywordtitlesep#1{\gdef\@elsarticlekeywordtitlesep{#1}} %*% +\keywordtitle{Keywords} %*% +\keywordtitlesep{:\ } +\def\keyword{% + \def\sep{\unskip, }% + \def\MSC{\@ifnextchar[{\@MSC}{\@MSC[2000]}} + \def\@MSC[##1]{\par\leavevmode\hbox {\it ##1~MSC:\space}}% + \def\PACS{\par\leavevmode\hbox {\it PACS:\space}}% + \def\JEL{\par\leavevmode\hbox {\it JEL:\space}}% + \global\setbox\keybox=\vbox\bgroup\hsize=\textwidth + \normalsize\normalfont\def\baselinestretch{1} + \parskip\z@ + \noindent\textit{\@elsarticlekwdtitle\@elsarticlekeywordtitlesep} + \raggedright % Keywords are not justified. + \ignorespaces} +\def\endkeyword{\par \egroup} + +\newdimen\Columnwidth +\Columnwidth=\columnwidth + +\def\printFirstPageNotes{% + \iflongmktitle + \let\columnwidth=\textwidth + \fi +\ifdoubleblind +\else + \ifx\@tnotes\@empty\else\@tnotes\fi + \ifx\@nonumnotes\@empty\else\@nonumnotes\fi + \ifx\@cornotes\@empty\else\@cornotes\fi + \ifx\@elseads\@empty\relax\else + \let\thefootnote\relax + \footnotetext{\ifnum\theead=1\relax + \textit{Email address:\space}\else + \textit{Email addresses:\space}\fi + \@elseads}\fi + \ifx\@elsuads\@empty\relax\else + \let\thefootnote\relax + \footnotetext{\textit{URL:\space}% + \@elsuads}\fi +\fi + \ifx\@fnotes\@empty\else\@fnotes\fi + \iflongmktitle\if@twocolumn + \let\columnwidth=\Columnwidth\fi\fi +} + +%% Pushing text to begin on newpage %*% +\def\newpage@after@title{title} +\def\newpage@after@author{author} +\def\newpage@after@abstract{abstract} +\def\newpageafter#1% + {\gdef\@elsarticlenewpageafter{#1}} + +\long\def\pprintMaketitle{\clearpage + \iflongmktitle\if@twocolumn\let\columnwidth=\textwidth\fi\fi + \resetTitleCounters + \def\baselinestretch{1}% + \printFirstPageNotes + \begin{\elsarticletitlealign}% + \thispagestyle{pprintTitle}% + \def\baselinestretch{1}% + \Large\@title\par\vskip18pt% + \ifx\@elsarticlenewpageafter\newpage@after@title% %*% + \newpage + \fi% + \ifdoubleblind + \vspace*{2pc} + \else + \normalsize\elsauthors\par\vskip10pt + \footnotesize\itshape\elsaddress\par\vskip36pt + \fi + \ifx\@elsarticlenewpageafter\newpage@after@author% %*% + \newpage + \fi% + \hrule\vskip12pt + \ifvoid\absbox\else\unvbox\absbox\par\vskip10pt\fi + \ifvoid\keybox\else\unvbox\keybox\par\vskip10pt\fi + \hrule\vskip12pt + \ifx\@elsarticlenewpageafter\newpage@after@abstract% %*% + \newpage + \fi% + \end{\elsarticletitlealign}% + \gdef\thefootnote{\arabic{footnote}}% + } + +\def\printWarning{% + \mbox{}\par\vfill\par\bgroup + \fboxsep12pt\fboxrule1pt + \hspace*{.18\textwidth} + \fcolorbox{gray50}{gray10}{\box\warnbox} + \egroup\par\vfill\thispagestyle{empty} + \setcounter{page}{0} + \clearpage} + +\long\def\finalMaketitle{% + \resetTitleCounters + \def\baselinestretch{1}% + \MaketitleBox + \thispagestyle{pprintTitle}% + \gdef\thefootnote{\arabic{footnote}}% + } + +\long\def\MaketitleBox{% + \resetTitleCounters + \def\baselinestretch{1}% + \begin{\elsarticletitlealign}% + \def\baselinestretch{1}% + \Large\@title\par\vskip18pt + \ifdoubleblind + \vspace*{2pc} + \else + \normalsize\elsauthors\par\vskip10pt + \footnotesize\itshape\elsaddress\par\vskip36pt + \fi + \hrule\vskip12pt + \ifvoid\absbox\else\unvbox\absbox\par\vskip10pt\fi + \ifvoid\keybox\else\unvbox\keybox\par\vskip10pt\fi + \hrule\vskip12pt + \end{\elsarticletitlealign}% +} + +\def\FNtext#1{\par\bgroup\footnotesize#1\egroup} +\newdimen\space@left +\def\alarm#1{\typeout{******************************}% + \typeout{#1}% + \typeout{******************************}% +} + +\def\titlespancalculator#1#2#3#4{% + % break count + \@tempcnta=#4\relax% + % pagebreakcount increment + \advance\@tempcnta by 1\relax% + % title page height + \@tempdima=#1\relax% + % Page height - title page notes height (only for first break) + % Page height - textheight (for remaining breaks) + % Page height - title page notes height + \@tempdimb=#2\relax% + % Remaining title page height + \advance\@tempdima -\the\@tempdimb% + % Checks if remaining title page + % height less than textheight + \ifdim\the\@tempdima>#3\relax% + \titlespancalculator% + {\the\@tempdima}{#3}{#3}{\the\@tempcnta}%Break again + \else% + % Save break count and exit. + \xdef\savetitlepagespan{\the\@tempcnta}% + \fi% +}% + +\long\def\myfor#1#2#3{% + \@tempcnta=#1\relax% + \ifnum#1<#2\relax% + \advance\@tempcnta by 1\relax% + #3% + \myfor{\the\@tempcnta}{#2}{#3}% + \fi} + +\long\def\getSpaceLeft{%\global\@twocolumnfalse% + \global\setbox0=\vbox{\hsize=\textwidth\MaketitleBox}% + \global\setbox1=\vbox{\hsize=\textwidth + \let\footnotetext\FNtext + \printFirstPageNotes}% + \xdef\noteheight{\the\ht1}% + \xdef\titleheight{\the\ht0}% + \@tempdima=\vsize + \advance\@tempdima-\noteheight + \advance\@tempdima-1\baselineskip + \xdef\savefpageheight{\the\@tempdima}% + \setbox2=\vbox{\titlespancalculator{\titleheight}% + {\savefpageheight}{\textheight}{0}}% +} + + \skip\footins=24pt + +\newbox\els@boxa +\newbox\els@boxb + +\ifpreprint + \def\maketitle{\elsarticleprelims\pprintMaketitle} + \else + \ifnum\jtype=1 + \def\maketitle{% + \elsarticleprelims% + \iflongmktitle\getSpaceLeft + \ifdim\noteheight>0pt% + \advance\@tempdima-1.35\baselineskip + \fi% + \global\setbox\els@boxa=\vsplit0 to \@tempdima + \box\els@boxa\par\resetTitleCounters + \thispagestyle{pprintTitle}% + \printFirstPageNotes + \ifnum\savetitlepagespan>1\relax% + \myfor{2}{\savetitlepagespan}{% + \global\setbox\els@boxb=\vsplit0 to \textheight%\@tempdima + \box\els@boxb} + \else% + \fi% + \box0% + \else + \finalMaketitle\printFirstPageNotes + \fi + \gdef\thefootnote{\arabic{footnote}}}% + \else + \ifnum\jtype=5 + \def\maketitle{% + \elsarticleprelims% + \iflongmktitle\getSpaceLeft + \ifdim\noteheight>0pt% + \advance\@tempdima-1.35\baselineskip + \fi% + \global\setbox\els@boxa=\vsplit0 to \@tempdima + \box\els@boxa\par\resetTitleCounters + \thispagestyle{pprintTitle}% + \printFirstPageNotes + \ifnum\savetitlepagespan>1\relax% + \myfor{2}{\savetitlepagespan}{% + \global\setbox\els@boxb=\vsplit0 to \textheight%\@tempdima + \twocolumn[\box\els@boxb]} + \else% + \fi% + \twocolumn[\box0]%\printFirstPageNotes + \else + \twocolumn[\finalMaketitle]\printFirstPageNotes + \fi + \gdef\thefootnote{\arabic{footnote}}} + \else + \if@twocolumn + \def\maketitle{% + \elsarticleprelims% + \iflongmktitle\getSpaceLeft + \ifdim\noteheight>0pt% + \advance\@tempdima-1.35\baselineskip + \fi% + \global\setbox\els@boxa=\vsplit0 to \@tempdima + \box\els@boxa\par\resetTitleCounters + \thispagestyle{pprintTitle}% + \printFirstPageNotes + \ifnum\savetitlepagespan>1\relax% + \myfor{2}{\savetitlepagespan}{% + \global\setbox\els@boxb=\vsplit0 to \textheight%\@tempdima + \twocolumn[\box\els@boxb]} + \else% + \fi% + \twocolumn[\box0]% + \else + \twocolumn[\finalMaketitle]\printFirstPageNotes + \fi + \gdef\thefootnote{\arabic{footnote}}}% + \else + \def\maketitle{% + \elsarticleprelims% + \iflongmktitle\getSpaceLeft + \ifdim\noteheight>0pt% + \advance\@tempdima-1.35\baselineskip + \fi% + \global\setbox\els@boxa=\vsplit0 to \@tempdima + \box\els@boxa\par\resetTitleCounters + \thispagestyle{pprintTitle}% + \printFirstPageNotes + \ifnum\savetitlepagespan>1\relax% + \myfor{2}{\savetitlepagespan}{% + \global\setbox\els@boxb=\vsplit0 to \textheight%\@tempdima + \box\els@boxb} + \else% + \fi% + \box0% + \else + \elsarticleprelims% + \finalMaketitle\printFirstPageNotes + \fi + \gdef\thefootnote{\arabic{footnote}}}% + \fi + \fi + \fi +\fi + +\let\@elsarticlemyfooter\@empty +\let\@elsarticlemyfooteralign\@empty +\def\@elsarticlemyfooteralignleft{L} +\def\@elsarticlemyfooteralignright{R} +\def\@elsarticlemyfooteraligncenter{C} + +\def\myfooter[#1]#2 %*% + {\gdef\@elsarticlemyfooteralign{#1} + \gdef\@elsarticlemyfooter{#2}} + +\def\myfooterfont#1{\gdef\@myfooterfont{#1}} +\myfooterfont{\footnotesize\itshape} +\def\ps@pprintTitle{% + \let\@oddhead\@empty + \let\@evenhead\@empty + \def\@oddfoot + {\hbox to \textwidth% + {\ifnopreprintline\relax\else + \@myfooterfont% + \ifx\@elsarticlemyfooteralign\@elsarticlemyfooteraligncenter% + \hfil\@elsarticlemyfooter\hfil% + \else% + \ifx\@elsarticlemyfooteralign\@elsarticlemyfooteralignleft% + \@elsarticlemyfooter\hfill{}% + \else% + \ifx\@elsarticlemyfooteralign\@elsarticlemyfooteralignright% + {}\hfill\@elsarticlemyfooter% + \else% + Preprint submitted to \ifx\@journal\@empty% + Elsevier% + \else\@journal\fi\hfill\@date\fi% + \fi% + \fi% + \fi% + } + }% + \let\@evenfoot\@oddfoot} +\def\@seccntDot{.} +\def\@seccntformat#1{\csname the#1\endcsname\@seccntDot\hskip 0.5em} + +\renewcommand\section{\@startsection {section}{1}{\z@}% + {18\p@ \@plus 6\p@ \@minus 3\p@}% + {9\p@ \@plus 6\p@ \@minus 3\p@}% + {\normalsize\bfseries\boldmath}} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {12\p@ \@plus 6\p@ \@minus 3\p@}% + {3\p@ \@plus 6\p@ \@minus 3\p@}% + {\normalfont\normalsize\itshape}} +\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {12\p@ \@plus 6\p@ \@minus 3\p@}% + {\p@}% + {\normalfont\normalsize\itshape}} + +\def\paragraph{\secdef{\els@aparagraph}{\els@bparagraph}} +\def\els@aparagraph[#1]#2{\elsparagraph[#1]{#2.}} +\def\els@bparagraph#1{\elsparagraph*{#1.}} + +\newcommand\elsparagraph{\@startsection{paragraph}{4}{0\z@}% + {10\p@ \@plus 6\p@ \@minus 3\p@}% + {-6\p@}% + {\normalfont\itshape}} +\newdimen\leftMargin +\leftMargin=2em +\newtoks\@enLab %\newtoks\@enfont +\def\@enQmark{?} +\def\@enLabel#1#2{% + \edef\@enThe{\noexpand#1{\@enumctr}}% + \@enLab\expandafter{\the\@enLab\csname the\@enumctr\endcsname}% + \@enloop} +\def\@enSpace{\afterassignment\@enSp@ce\let\@tempa= } +\def\@enSp@ce{\@enLab\expandafter{\the\@enLab\space}\@enloop} +\def\@enGroup#1{\@enLab\expandafter{\the\@enLab{#1}}\@enloop} +\def\@enOther#1{\@enLab\expandafter{\the\@enLab#1}\@enloop} +\def\@enloop{\futurelet\@entemp\@enloop@} +\def\@enloop@{% + \ifx A\@entemp \def\@tempa{\@enLabel\Alph }\else + \ifx a\@entemp \def\@tempa{\@enLabel\alph }\else + \ifx i\@entemp \def\@tempa{\@enLabel\roman }\else + \ifx I\@entemp \def\@tempa{\@enLabel\Roman }\else + \ifx 1\@entemp \def\@tempa{\@enLabel\arabic}\else + \ifx \@sptoken\@entemp \let\@tempa\@enSpace \else + \ifx \bgroup\@entemp \let\@tempa\@enGroup \else + \ifx \@enum@\@entemp \let\@tempa\@gobble \else + \let\@tempa\@enOther + \fi\fi\fi\fi\fi\fi\fi\fi + \@tempa} +\newlength{\@sep} \newlength{\@@sep} +\setlength{\@sep}{.5\baselineskip plus.2\baselineskip + minus.2\baselineskip} +\setlength{\@@sep}{.1\baselineskip plus.01\baselineskip + minus.05\baselineskip} +\providecommand{\sfbc}{\rmfamily\upshape} +\providecommand{\sfn}{\rmfamily\upshape} +\def\@enfont{\ifnum \@enumdepth >1\let\@nxt\sfn \else\let\@nxt\sfbc \fi\@nxt} +\def\enumerate{% + \ifnum \@enumdepth >3 \@toodeep\else + \advance\@enumdepth \@ne + \edef\@enumctr{enum\romannumeral\the\@enumdepth}\fi + \@ifnextchar[{\@@enum@}{\@enum@}} +\def\@@enum@[#1]{% + \@enLab{}\let\@enThe\@enQmark + \@enloop#1\@enum@ + \ifx\@enThe\@enQmark\@warning{The counter will not be printed.% + ^^J\space\@spaces\@spaces\@spaces The label is: \the\@enLab}\fi + \expandafter\edef\csname label\@enumctr\endcsname{\the\@enLab}% + \expandafter\let\csname the\@enumctr\endcsname\@enThe + \csname c@\@enumctr\endcsname7 + \expandafter\settowidth + \csname leftmargin\romannumeral\@enumdepth\endcsname + {\the\@enLab\hskip\labelsep}% + \@enum@} +\def\@enum@{\list{{\@enfont\csname label\@enumctr\endcsname}}% + {\usecounter{\@enumctr}\def\makelabel##1{\hss\llap{##1}}% + \ifnum \@enumdepth>1\setlength{\topsep}{\@@sep}\else + \setlength{\topsep}{\@sep}\fi + \ifnum \@enumdepth>1\setlength{\itemsep}{0pt plus1pt minus1pt}% + \else \setlength{\itemsep}{\@@sep}\fi + %\setlength\leftmargin{\leftMargin}%%%{1.8em} + \setlength{\parsep}{0pt plus1pt minus1pt}% + \setlength{\parskip}{0pt plus1pt minus1pt} + }} + +\def\endenumerate{\par\ifnum \@enumdepth >1\addvspace{\@@sep}\else + \addvspace{\@sep}\fi \endlist} + +\def\sitem{\@noitemargtrue\@item[\@itemlabel *]} + +\def\itemize{\@ifnextchar[{\@Itemize}{\@Itemize[]}} + +\def\@Itemize[#1]{\def\next{#1}% + \ifnum \@itemdepth >\thr@@\@toodeep\else + \advance\@itemdepth\@ne + \ifx\next\@empty\else\expandafter\def\csname + labelitem\romannumeral\the\@itemdepth\endcsname{#1}\fi% + \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}% + \expandafter\list\csname\@itemitem\endcsname + {\def\makelabel##1{\hss\llap{##1}}}% + \fi} +\def\newdefinition#1{% + \@ifnextchar[{\@odfn{#1}}{\@ndfn{#1}}}%] +\def\@ndfn#1#2{% + \@ifnextchar[{\@xndfn{#1}{#2}}{\@yndfn{#1}{#2}}} +\def\@xndfn#1#2[#3]{% + \expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}\@newctr{#1}[#3]% + \expandafter\xdef\csname the#1\endcsname{% + \expandafter\noexpand\csname the#3\endcsname \@dfncountersep + \@dfncounter{#1}}% + \global\@namedef{#1}{\@dfn{#1}{#2}}% + \global\@namedef{end#1}{\@enddefinition}}} +\def\@yndfn#1#2{% + \expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}% + \expandafter\xdef\csname the#1\endcsname{\@dfncounter{#1}}% + \global\@namedef{#1}{\@dfn{#1}{#2}}% + \global\@namedef{end#1}{\@enddefinition}}} +\def\@odfn#1[#2]#3{% + \@ifundefined{c@#2}{\@nocounterr{#2}}% + {\expandafter\@ifdefinable\csname #1\endcsname + {\global\@namedef{the#1}{\@nameuse{the#2}} + \global\@namedef{#1}{\@dfn{#2}{#3}}% + \global\@namedef{end#1}{\@enddefinition}}}} +\def\@dfn#1#2{% + \refstepcounter{#1}% + \@ifnextchar[{\@ydfn{#1}{#2}}{\@xdfn{#1}{#2}}} +\def\@xdfn#1#2{% + \@begindefinition{#2}{\csname the#1\endcsname}\ignorespaces} +\def\@ydfn#1#2[#3]{% + \@opargbegindefinition{#2}{\csname the#1\endcsname}{#3}\ignorespaces} +\def\@dfncounter#1{\noexpand\arabic{#1}} +\def\@dfncountersep{.} +\def\@begindefinition#1#2{\trivlist + \item[\hskip\labelsep{\bfseries #1\ #2.}]\upshape} +\def\@opargbegindefinition#1#2#3{\trivlist + \item[\hskip\labelsep{\bfseries #1\ #2\ (#3).}]\upshape} +\def\@enddefinition{\endtrivlist} + +\def\@begintheorem#1#2{\trivlist + \let\baselinestretch\@blstr + \item[\hskip \labelsep{\bfseries #1\ #2.}]\itshape} +\def\@opargbegintheorem#1#2#3{\trivlist + \let\baselinestretch\@blstr + \item[\hskip \labelsep{\bfseries #1\ #2\ (#3).}]\itshape} + +\def\newproof#1{% + \@ifnextchar[{\@oprf{#1}}{\@nprf{#1}}} +\def\@nprf#1#2{% + \@ifnextchar[{\@xnprf{#1}{#2}}{\@ynprf{#1}{#2}}} +\def\@xnprf#1#2[#3]{% + \expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}\@newctr{#1}[#3]% + \expandafter\xdef\csname the#1\endcsname{% + \expandafter\noexpand\csname the#3\endcsname \@prfcountersep + \@prfcounter{#1}}% + \global\@namedef{#1}{\@prf{#1}{#2}}% + \global\@namedef{end#1}{\@endproof}}} +\def\@ynprf#1#2{% + \expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}% + \expandafter\xdef\csname the#1\endcsname{\@prfcounter{#1}}% + \global\@namedef{#1}{\@prf{#1}{#2}}% + \global\@namedef{end#1}{\@endproof}}} +\def\@oprf#1[#2]#3{% + \@ifundefined{c@#2}{\@nocounterr{#2}}% + {\expandafter\@ifdefinable\csname #1\endcsname + {\global\@namedef{the#1}{\@nameuse{the#2}}% + \global\@namedef{#1}{\@prf{#2}{#3}}% + \global\@namedef{end#1}{\@endproof}}}} +\def\@prf#1#2{% + \refstepcounter{#1}% + \@ifnextchar[{\@yprf{#1}{#2}}{\@xprf{#1}{#2}}} +\def\@xprf#1#2{% + \@beginproof{#2}{\csname the#1\endcsname}\ignorespaces} +\def\@yprf#1#2[#3]{% + \@opargbeginproof{#2}{\csname the#1\endcsname}{#3}\ignorespaces} +\def\@prfcounter#1{\noexpand\arabic{#1}} +\def\@prfcountersep{.} +\def\@beginproof#1#2{\trivlist\let\baselinestretch\@blstr + \item[\hskip \labelsep{\scshape #1.}]\rmfamily} +\def\@opargbeginproof#1#2#3{\trivlist\let\baselinestretch\@blstr + \item[\hskip \labelsep{\scshape #1\ (#3).}]\rmfamily} +\def\@endproof{\endtrivlist} +\newcommand*{\qed}{\hbox{}\hfill$\Box$} + +\@ifundefined{@biboptions}{\xdef\@biboptions{numbers}}{} +\InputIfFileExists{\jobname.spl}{}{} +\ifnonatbib\relax\else + \RequirePackage[\@biboptions]{natbib} +\fi +\newwrite\splwrite +\immediate\openout\splwrite=\jobname.spl +\def\biboptions#1{\def\next{#1}\immediate\write\splwrite{% + \string\g@addto@macro\string\@biboptions{% + ,\expandafter\strip@prefix\meaning\next}}} + +\let\baselinestretch=\@blstr +\ifnum\jtype=1 + \RequirePackage{geometry} + \geometry{twoside, + paperwidth=210mm, + paperheight=297mm, + textheight=562pt, + textwidth=384pt, + centering, + headheight=50pt, + headsep=12pt, + footskip=12pt, + footnotesep=24pt plus 2pt minus 12pt, + } + \global\let\bibfont=\footnotesize + \global\bibsep=0pt + \if@twocolumn\global\@twocolumnfalse\fi +\else\ifnum\jtype=3 + \RequirePackage{geometry} + \geometry{twoside, + paperwidth=210mm, + paperheight=297mm, + textheight=622pt, + textwidth=468pt, + centering, + headheight=50pt, + headsep=12pt, + footskip=18pt, + footnotesep=24pt plus 2pt minus 12pt, + columnsep=2pc + } + \global\let\bibfont=\footnotesize + \global\bibsep=0pt + \if@twocolumn\input{fleqn.clo}\fi +\else\ifnum\jtype=5 + \RequirePackage{geometry} + \geometry{twoside, + paperwidth=210mm, + paperheight=297mm, + textheight=682pt, + textwidth=522pt, + centering, + headheight=50pt, + headsep=12pt, + footskip=18pt, + footnotesep=24pt plus 2pt minus 12pt, + columnsep=18pt + }% + \global\let\bibfont=\footnotesize + \global\bibsep=0pt + \input{fleqn.clo} + \global\@twocolumntrue +%% +%% End of option '5p' +%% +\fi\fi\fi +\def\journal#1{\gdef\@journal{#1}} + \let\@journal\@empty +\newenvironment{frontmatter}{}{\maketitle} + +\long\def\@makecaption#1#2{% + \vskip\abovecaptionskip\footnotesize + \sbox\@tempboxa{#1: #2}% + \ifdim \wd\@tempboxa >\hsize + #1: #2\par + \else + \global \@minipagefalse + \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}% + \fi + \vskip\belowcaptionskip} + +\AtBeginDocument{\@ifpackageloaded{hyperref} + {\def\@linkcolor{blue} + \def\@anchorcolor{blue} + \def\@citecolor{blue} + \def\@filecolor{blue} + \def\@urlcolor{blue} + \def\@menucolor{blue} + \def\@pagecolor{blue} +\begingroup + \@makeother\`% + \@makeother\=% + \edef\x{% + \edef\noexpand\x{% + \endgroup + \noexpand\toks@{% + \catcode 96=\noexpand\the\catcode`\noexpand\`\relax + \catcode 61=\noexpand\the\catcode`\noexpand\=\relax + }% + }% + \noexpand\x + }% +\x +\@makeother\` +\@makeother\= +}{}} +%% +\def\appendixname{Appendix } +\renewcommand\appendix{\par + \setcounter{section}{0}% + \setcounter{subsection}{0}% + \setcounter{equation}{0} + \gdef\thefigure{\@Alph\c@section.\arabic{figure}}% + \gdef\thetable{\@Alph\c@section.\arabic{table}}% + \gdef\thesection{\appendixname~\@Alph\c@section}% + \@addtoreset{equation}{section}% + \gdef\theequation{\@Alph\c@section.\arabic{equation}}% + \addtocontents{toc}{\string\let\string\numberline\string\tmptocnumberline}{}{} +} + +%%%% \numberline width calculation for appendix. +\newdimen\appnamewidth +\def\tmptocnumberline#1{% + \setbox0=\hbox{\appendixname} + \appnamewidth=\wd0 + \addtolength\appnamewidth{2.5pc} + \hb@xt@\appnamewidth{#1\hfill} +} + +%% Added for work with amsrefs.sty + +\@ifpackageloaded{amsrefs}% + {} + {%\let\bibsection\relax% + \AtBeginDocument{\def\cites@b#1#2,#3{% + \begingroup[% + \toks@{\InnerCite{#2}#1}% + \ifx\@empty#3\@xp\@gobble\fi + \cites@c#3% +}}} +%% +%% Added for avoiding clash with cleveref.sty +\@ifpackageloaded{cleveref}% + {} + {\def\tnotetext[#1]#2{\g@addto@macro\@tnotes{% + \refstepcounter{tnote}% + \immediate\write\@auxout{\string\Newlabel{#1}{\thetnote}} + \def\thefootnote{\ifcase\c@tnote\or$\star$\or$\star\star$\fi}% + \footnotetext{#2}}} +%%% + \def\fntext[#1]#2{\g@addto@macro\@fnotes{% + \refstepcounter{fnote}% + \immediate\write\@auxout{\string\Newlabel{#1}{\thefnote}} + \def\thefootnote{\thefnote}% + \global\setcounter{footnote}{\c@fnote}% + \footnotetext{#2}}} +%%% + \def\cortext[#1]#2{\g@addto@macro\@cornotes{% + \refstepcounter{cnote}% + 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Email Address: dehaeze.thomas@gmail.com} + +#+LATEX_HEADER_EXTRA: \address[a1]{European Synchrotron Radiation Facility, Grenoble, France} +#+LATEX_HEADER_EXTRA: \address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.} +#+LATEX_HEADER_EXTRA: \address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.} +#+LATEX_HEADER_EXTRA: \address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.} + +#+LATEX_HEADER: \usepackage[utf8]{inputenc} +#+LATEX_HEADER: \usepackage[T1]{fontenc} +#+LATEX_HEADER: \usepackage{graphicx} +#+LATEX_HEADER: \usepackage{grffile} +#+LATEX_HEADER: \usepackage{rotating} +#+LATEX_HEADER: \usepackage[normalem]{ulem} +#+LATEX_HEADER: \usepackage{capt-of} +#+LATEX_HEADER: \usepackage{hyperref} +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{array} +#+LATEX_HEADER: \usepackage{amsmath,amssymb,amsfonts} +#+LATEX_HEADER: \usepackage{algorithmic} +#+LATEX_HEADER: \usepackage{textcomp} +#+LATEX_HEADER: \usepackage{cases} +#+LATEX_HEADER: \usepackage{tabularx,siunitx,booktabs} +#+LATEX_HEADER: \usepackage{algorithmic} +#+LATEX_HEADER: \usepackage{import} +#+LATEX_HEADER: \usepackage{hyperref} +#+LATEX_HEADER: \usepackage[hyperref]{xcolor} +#+LATEX_HEADER: \hypersetup{colorlinks=true} +#+LATEX_HEADER: \usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry} +:END: + +* Build :noexport: +#+NAME: startblock +#+BEGIN_SRC emacs-lisp :results none +(add-to-list 'org-latex-classes + '("elsarticle" + "\\documentclass{elsarticle}" + ("\\section{%s}" . "\\section*{%s}") + ("\\subsection{%s}" . "\\subsection*{%s}") + ("\\subsubsection{%s}" . "\\subsubsection*{%s}") + ("\\paragraph{%s}" . "\\paragraph*{%s}") + ("\\subparagraph{%s}" . "\\subparagraph*{%s}")) + ) + +(defun delete-org-comments (backend) + (loop for comment in (reverse (org-element-map (org-element-parse-buffer) + 'comment 'identity)) + do + (setf (buffer-substring (org-element-property :begin comment) + (org-element-property :end comment)) + ""))) + +;; add to export hook +(add-hook 'org-export-before-processing-hook 'delete-org-comments) + +;; Remove hypersetup +(setq org-latex-with-hyperref nil) +#+END_SRC + +* =hypersetup= :ignore: + +\hypersetup{allcolors=teal} + +* Abstract and Keywords :ignore: +#+begin_frontmatter +#+begin_abstract +Sensor have limited bandwidth and are accurate only in a certain frequency band. +In many applications, the signals of different sensor are fused together in order to either enhance the stability or improve the operational bandwidth of the system. +The sensor signals can be fused using complementary filters. +The tuning of complementary filters is a complex task and is the subject of this paper. +The filters needs to meet design specifications while satisfying the complementary property. +This paper presents a framework to shape the norm of complementary filters using the $\mathcal{H}_\infty$ norm minimization. +The design specifications are imposed as constraints in the optimization problem by appropriate selection of weighting functions. +The proposed method is quite general and easily extendable to cases where more than two sensors are fused. +Finally, the proposed method is applied to the design of complementary filter design for active vibration isolation of the Laser Interferometer Gravitation-wave Observatory (LIGO). +#+end_abstract + +#+begin_keyword +Sensor fusion \sep{} Optimal filters \sep{} $\mathcal{H}_\infty$ synthesis \sep{} Vibration isolation \sep{} Precision +#+end_keyword +#+end_frontmatter + +* Introduction +<> +** New introduction :ignore: +*** Introduction to Sensor Fusion :ignore: + +- cite:bendat57_optim_filter_indep_measur_two roots of sensor fusion + +*** Advantages of Sensor Fusion :ignore: + +- Increase the bandwidth: cite:zimmermann92_high_bandw_orien_measur_contr +- Increased robustness: cite:collette15_sensor_fusion_method_high_perfor +- Decrease the noise: + +*** Applications :ignore: + +- UAV: cite:pascoal99_navig_system_desig_using_time, cite:jensen13_basic_uas +- Gravitational wave observer: cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system + +*** Kalman Filtering or Complementary filters :ignore: + +- cite:brown72_integ_navig_system_kalman_filter alternate form of complementary filters => Kalman filtering +- cite:higgins75_compar_compl_kalman_filter Compare Kalman Filtering with sensor fusion using complementary filters +- cite:robert12_introd_random_signal_applied_kalman advantage of complementary filters over Kalman filtering + +*** Design Methods of Complementary filters :ignore: + +- cite:pascoal99_navig_system_desig_using_time use LMI to generate complementary filters +- cite:plummer06_optim_compl_filter_their_applic_motion_measur use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density) +- cite:jensen13_basic_uas design of complementary filters with classical control theory +- cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system: FIR + convex optimization +- 3 complementary filters: cite:becker15_compl_filter_desig_three_frequen_bands +- Analytical methods: + - first order: cite:corke04_inert_visual_sensin_system_small_auton_helic + - second order: cite:baerveldt97_low_cost_low_weigh_attit, cite:stoten01_fusion_kinet_data_using_compos_filter, cite:jensen13_basic_uas + - higher order: cite:shaw90_bandw_enhan_posit_measur_using_measur_accel, cite:zimmermann92_high_bandw_orien_measur_contr, cite:collette15_sensor_fusion_method_high_perfor, cite:matichard15_seism_isolat_advan_ligo +- Analog complementary filters: cite:yong16_high_speed_vertic_posit_stage, cite:moore19_capac_instr_sensor_fusion_high_bandw_nanop + +*** Problematics / gap in the research :ignore: + +- Robustness problems: cite:zimmermann92_high_bandw_orien_measur_contr change of phase near the merging frequency +- Trial and error +- Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available. + +*** Describe the paper itself / the problem which is addressed :ignore: + +*** Introduce Each part of the paper :ignore: + +** Old Introduction :ignore:noexport: +*** Establish the importance of the research topic :ignore: +# What are Complementary Filters +A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies. +These filters are used when two or more sensors are measuring the same physical quantity with different noise characteristics. Unreliable frequencies of each sensor are filtered out by the complementary filters and then combined to form a super sensor giving a better estimate of the physical quantity over a wider bandwidth. +This technique is called sensor fusion and is used in many applications.\par + +*** Applications of complementary filtering :ignore: +# Improve bandwidth for UAV +In cite:zimmermann92_high_bandw_orien_measur_contr,corke04_inert_visual_sensin_system_small_auton_helic, various sensors (accelerometers, gyroscopes, vision sensors, etc.) are merged using complementary filters for the attitude estimation of Unmanned Aerial Vehicles (UAV). +# Improving the control robustness +In cite:collette15_sensor_fusion_method_high_perfor, several sensor fusion configurations using different types of sensors are discussed in order to increase the control bandwidth of active vibration isolation systems. +# Merging of different sensor types +Furthermore, sensor fusion is used in the isolation systems of the Laser Interferometer Gravitational-Wave Observator (LIGO) to merge inertial sensors with relative sensors +cite:matichard15_seism_isolat_advan_ligo,hua04_polyp_fir_compl_filter_contr_system. \par + +*** Current design methods for complementary filters :ignore: +# Why Design of Complementary Filter is important +As the super sensor noise characteristics largely depend on the complementary filter norms, their proper design is of primary importance for sensor fusion. +# Discuss the different approach to complementary filter design +In cite:corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas, first and second order analytical formulas of complementary filters have been presented. +# Third Order and Higher orders +Higher order complementary filters have been used in +cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,collette15_sensor_fusion_method_high_perfor. +# Alternate Formulation +In cite:jensen13_basic_uas, the sensitivity and complementary sensitivity transfer functions of a feedback architecture have been proposed to be used as complementary filters. The design of such filters can then benefit from the classical control theory developments. +# LMI / convex Optimization +Linear Matrix Inequalities (LMIs) are used in cite:pascoal99_navig_system_desig_using_time for the synthesis of complementary filters satisfying some frequency-like performance. +# FIR Filters +Finally, a synthesis method of high order Finite Impulse Response (FIR) complementary filters using convex optimization has been developed in cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system. \par + +*** Describe a gap in the research :ignore: +# There is a need for easy synthesis methods for complementary filters +Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available. + +*** Describe the paper itself / the problem which is addressed :ignore: +# In this paper, we propose a synthesis method for the shaping of complementary filters using the $\mathcal{H}_\infty$ norm.\par +This paper presents a new design method of complementary filters based on $\mathcal{H}_\infty$ synthesis. +This design method permits to easily shape the norms of the generated filters.\par + +*** Introduce Each part of the paper :ignore: +The section ref:sec:requirements gives a brief overview of sensor fusion using complementary filters and explains how the typical requirements for such fusion can be expressed as upper bounds on the filters norms. +In section ref:sec:hinf_method, a new design method for the shaping of complementary filters using $\mathcal{H}_\infty$ synthesis is proposed. +In section ref:sec:application_ligo, the method is used to design complex complementary filters that are used for sensor fusion at the LIGO. +Our conclusions are drawn in the final section. + +* Complementary Filters Requirements +<> +** Sensor Fusion Architecture +<> + +Let's consider two sensors measuring the same physical quantity $x$ with dynamics $G_1(s)$ and $G_2(s)$, and with uncorrelated noise characteristics $n_1$ and $n_2$. + +The signals from both sensors are fed into two complementary filters $H_1(s)$ and $H_2(s)$ and then combined to yield an estimate $\hat{x}$ of $x$ as shown in Fig. ref:fig:fusion_super_sensor. +#+name: eq:comp_filter_estimate +\begin{equation} + \hat{x} = \left(G_1 H_1 + G_2 H_2\right) x + H_1 n_1 + H_2 n_2 +\end{equation} + +#+name: fig:fusion_super_sensor +#+caption: Sensor fusion architecture +#+attr_latex: :scale 1 +[[file:figs/fusion_super_sensor.pdf]] + +The complementary property of $H_1(s)$ and $H_2(s)$ implies that their transfer function sum is equal to one at all frequencies eqref:eq:comp_filter. +#+name: eq:comp_filter +\begin{equation} + H_1(s) + H_2(s) = 1 +\end{equation} + +** Noise Sensor Filtering +<> + +Let's first consider sensors with perfect dynamics +#+name: eq:perfect_dynamics +\begin{equation} + G_1(s) = G_2(s) = 1 +\end{equation} + +The estimate $\hat{x}$ is then described by +#+name: eq:estimate_perfect_dyn +\begin{equation} + \hat{x} = x + H_1 n_1 + H_2 n_2 +\end{equation} + +From eqref:eq:estimate_perfect_dyn, the complementary filters $H_1(s)$ and $H_2(s)$ are shown to only operate on the sensor's noise. +Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured. + +Let's define the estimation error $\delta x$ by eqref:eq:estimate_error. +#+name: eq:estimate_error +\begin{equation} + \delta x \triangleq \hat{x} - x = H_1 n_1 + H_2 n_2 +\end{equation} + +As shown in eqref:eq:noise_filtering_psd, the Power Spectral Density (PSD) of the estimation error $\Phi_{\delta x}$ depends both on the norm of the two complementary filters and on the PSD of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$. +#+name: eq:noise_filtering_psd +\begin{equation} + \Phi_{\delta x} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2} +\end{equation} + +Usually, the two sensors have high noise levels over distinct frequency regions. +In order to lower the noise of the super sensor, the value of the norm $|H_1|$ has to be lowered when $\Phi_{n_1}$ is larger than $\Phi_{n_2}$ and that of $|H_2|$ lowered when $\Phi_{n_2}$ is larger than $\Phi_{n_1}$. + +** Robustness of the Fusion +<> + +In practical systems the sensor dynamics is not perfect and eqref:eq:perfect_dynamics is not verified. +In such case, one can use an inversion filter $\hat{G}_i^{-1}(s)$ to normalize the sensor dynamics, where $\hat{G}_i(s)$ is an estimate of the sensor dynamics $G_i(s)$. +However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and $\hat{G}_i^{-1}(s) G_i(s) \neq 1$. + +Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. ref:fig:sensor_fusion_dynamic_uncertainty where $\Delta_i$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $|w_i(s)|$ is a weight representing the magnitude of the uncertainty. + +#+name: fig:sensor_fusion_dynamic_uncertainty +#+caption: Sensor fusion architecture with sensor dynamics uncertainty +#+attr_latex: :scale 1 +[[file:figs/sensor_fusion_dynamic_uncertainty.pdf]] + +The super sensor dynamics eqref:eq:super_sensor_dyn_uncertainty is no longer equal to $1$ and now depends on the sensor dynamics uncertainty weights $w_i(s)$ as well as on the complementary filters $H_i(s)$. +#+name: eq:super_sensor_dyn_uncertainty +\begin{equation} + \frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s) +\end{equation} + +The uncertainty region of the super sensor can be represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ as shown in Fig. ref:fig:uncertainty_set_super_sensor. + +#+name: fig:uncertainty_set_super_sensor +#+caption: Uncertainty region of the super sensor dynamics in the complex plane (solid circle). The contribution of both sensors 1 and 2 to the uncertainty are represented respectively by a dotted and a dashed circle +#+attr_latex: :scale 1 +[[file:figs/uncertainty_set_super_sensor.pdf]] + +The maximum phase added $\Delta\phi(\omega)$ by the super sensor dynamics at frequency $\omega$ is then +#+name: eq:max_phase_uncertainty +\begin{equation} + \Delta\phi(\omega) = \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big) +\end{equation} + +As it is generally desired to limit the maximum phase added by the super sensor, $H_1(s)$ and $H_2(s)$ should be designed such that eqref:eq:max_uncertainty_super_sensor is satisfied. +#+name: eq:max_uncertainty_super_sensor +\begin{equation} + \max_\omega \big( \left|w_1 H_1\right| + \left|w_2 H_2\right|\big) < \sin\left( \Delta \phi_\text{max} \right) +\end{equation} +where $\Delta \phi_\text{max}$ is the maximum allowed added phase. + +Thus the norm of the complementary filter $|H_i|$ should be made small at frequencies where $|w_i|$ is large. + +* Complementary Filters Shaping using $\mathcal{H}_\infty$ Synthesis +<> +** Introduction :ignore: +As shown in Sec. ref:sec:requirements, the performance and robustness of the sensor fusion architecture depends on the complementary filters norms. +Therefore, the development of a synthesis method of complementary filters that allows the shaping of their norm is necessary. + +** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis +<> +The synthesis objective is to shape the norm of two filters $H_1(s)$ and $H_2(s)$ while ensuring their complementary property eqref:eq:comp_filter. +This is equivalent as to finding stable transfer functions $H_1(s)$ and $H_2(s)$ such that conditions eqref:eq:comp_filter_problem_form are satisfied. +#+name: eq:comp_filter_problem_form +\begin{subequations} + \begin{align} + & H_1(s) + H_2(s) = 1 \label{eq:hinf_cond_complementarity} \\ + & |H_1(j\omega)| \le \frac{1}{|W_1(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h1} \\ + & |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2} + \end{align} +\end{subequations} +where $W_1(s)$ and $W_2(s)$ are two weighting transfer functions that are chosen to shape the norms of the corresponding filters. + +In order to express this optimization problem as a standard $\mathcal{H}_\infty$ problem, the architecture shown in Fig. ref:fig:h_infinity_robust_fusion is used where the generalized plant $P$ is described by eqref:eq:generalized_plant. +#+name: eq:generalized_plant +\begin{equation} + \begin{bmatrix} z_1 \\ z_2 \\ v \end{bmatrix} = P(s) \begin{bmatrix} w\\u \end{bmatrix}; \quad P(s) = \begin{bmatrix}W_1(s) & -W_1(s) \\ 0 & W_2(s) \\ 1 & 0 \end{bmatrix} +\end{equation} + +#+name: fig:h_infinity_robust_fusion +#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters +#+attr_latex: :scale 1 +[[file:figs/h_infinity_robust_fusion.pdf]] + +The $\mathcal{H}_\infty$ filter design problem is then to find a stable filter $H_2(s)$ which based on $v$, generates a signal $u$ such that the $\mathcal{H}_\infty$ norm from $w$ to $[z_1, \ z_2]$ is less than one eqref:eq:hinf_syn_obj. +#+name: eq:hinf_syn_obj +\begin{equation} + \left\|\begin{matrix} \left[1 - H_2(s)\right] W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 +\end{equation} + +This is equivalent to having eqref:eq:hinf_problem by defining $H_1(s)$ as the complementary filter of $H_2(s)$ eqref:eq:definition_H1. +#+name: eq:hinf_problem +\begin{equation} + \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 +\end{equation} + +#+name: eq:definition_H1 +\begin{equation} + H_1(s) \triangleq 1 - H_2(s) +\end{equation} + +The complementary condition eqref:eq:hinf_cond_complementarity is ensured by eqref:eq:definition_H1. +The conditions eqref:eq:hinf_cond_h1 and eqref:eq:hinf_cond_h2 on the filters shapes are satisfied by eqref:eq:hinf_problem. +Therefore, all the conditions eqref:eq:comp_filter_problem_form are satisfied using this synthesis method based on $\mathcal{H}_\infty$ synthesis, and thus it permits to shape complementary filters as desired. + +** Weighting Functions Design +<> +The proper design of the weighting functions is of primary importance for the success of the presented complementary filters $\mathcal{H}_\infty$ synthesis. + +First, only proper, stable and minimum phase transfer functions should be used. +Second, the order of the weights should stay reasonably small in order to reduce the computational costs associated with the solving of the optimization problem and for the physical implementation of the filters (the order of the synthesized filters being equal to the sum of the weighting functions order). +Third, one should not forget the fundamental limitations imposed by the complementary property eqref:eq:comp_filter. +This implies for instance that $|H_1(j\omega)|$ and $|H_2(j\omega)|$ cannot be made small at the same time. + +# Explain why we propose such weighting function +When designing complementary filters, it is usually desired to specify the slope of the filter, its crossover frequency and its gain at low and high frequency. +To help with the design of the weighting functions such that the above specification can be easily expressed, the following formula is proposed. +#+name: eq:weight_formula +\begin{equation} + W(s) = \left( \frac{ + \hfill{} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}} + }{ + \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}} + }\right)^n +\end{equation} +The parameters permit to specify: +- the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$ +- the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$ +- the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$ +- the absolute slope between high and low frequency: $n$ + +The parameters $G_0$, $G_c$ and $G_\infty$ should either satisfy condition eqref:eq:cond_formula_1 or eqref:eq:cond_formula_2. +#+name: eq:condition_params_formula +\begin{subequations} + \begin{align} + G_0 < 1 < G_\infty \text{ and } G_0 < G_c < G_\infty \label{eq:cond_formula_1}\\ + G_\infty < 1 < G_0 \text{ and } G_\infty < G_c < G_0 \label{eq:cond_formula_2} + \end{align} +\end{subequations} + +The general shape of a weighting function generated using eqref:eq:weight_formula is shown in Fig. ref:fig:weight_formula. + +#+name: fig:weight_formula +#+caption: Magnitude of a weighting function generated using the proposed formula eqref:eq:weight_formula, $G_0 = 1e^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$ +#+attr_latex: :scale 1 +[[file:figs/weight_formula.pdf]] + +** Validation of the proposed synthesis method +<> +Let's validate the proposed design method of complementary filters with a simple example where two complementary filters $H_1(s)$ and $H_2(s)$ have to be designed such that: +- the merging frequency is around $\SI{10}{Hz}$ +- the slope of $|H_1(j\omega)|$ is $-2$ above $\SI{10}{Hz}$ +- the slope of $|H_2(j\omega)|$ is $+3$ below $\SI{10}{Hz}$ +- the gain of both filters is equal to $10^{-3}$ away from the merging frequency + +The weighting functions $W_1(s)$ and $W_2(s)$ are designed using eqref:eq:weight_formula. +The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_synthesis_results. + +#+name: tab:weights_params +#+caption: Parameters used for $W_1(s)$ and $W_2(s)$ +#+ATTR_LATEX: :environment tabularx :width 0.5\linewidth :align Xcc +#+ATTR_LATEX: :center t :booktabs t :float t +| Parameter | $W_1(s)$ | $W_2(s)$ | +|------------------------+----------+----------| +| $G_0$ | $0.1$ | $1000$ | +| $G_\infty$ | $1000$ | $0.1$ | +| $\omega_c$ [$\si{Hz}$] | $11$ | $10$ | +| $G_c$ | $0.5$ | $0.5$ | +| $n$ | $2$ | $3$ | + +The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_synthesis_results and their transfer functions in the Laplace domain are given below. +\begin{align*} + H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\ + H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)} +\end{align*} + +#+name: fig:hinf_synthesis_results +#+caption: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis +#+attr_latex: :scale 1 +[[file:figs/hinf_synthesis_results.pdf]] + +** Synthesis of Three Complementary Filters +<> + +*** Why it is used sometimes :ignore: +Some applications may require to merge more than two sensors. +In such a case, it is necessary to design as many complementary filters as the number of sensors used. + +*** Mathematical Problem :ignore: +The synthesis problem is then to compute $n$ stable transfer functions $H_i(s)$ such that eqref:eq:hinf_problem_gen is satisfied. +#+name: eq:hinf_problem_gen +\begin{subequations} + \begin{align} + & \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\ + & \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:hinf_cond_perf_gen} + \end{align} +\end{subequations} + +*** H-Infinity Architecture :ignore: +The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. ref:fig:comp_filter_three_hinf. + +The $\mathcal{H}_\infty$ synthesis objective applied on $P(s)$ is to design two stable filters $H_2(s)$ and $H_3(s)$ such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2, \ z_3]$ is less than one eqref:eq:hinf_syn_obj_three. +#+name: eq:hinf_syn_obj_three +\begin{equation} + \left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1 +\end{equation} + +#+name: fig:comp_filter_three_hinf +#+caption: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters +#+attr_latex: :scale 1 +[[file:figs/comp_filter_three_hinf.pdf]] + +By choosing $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\infty$ synthesis solves the design problem eqref:eq:hinf_problem_gen. \par + +*** Example of generated complementary filters :ignore: +An example is given to validate the method where three sensors are used in different frequency bands (up to $\SI{1}{Hz}$, from $1$ to $\SI{10}{Hz}$ and above $\SI{10}{Hz}$ respectively). +Three weighting functions are designed using eqref:eq:weight_formula and shown by dashed curves in Fig. ref:fig:hinf_three_synthesis_results. +The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_three_synthesis_results. + +#+name: fig:hinf_three_synthesis_results +#+caption: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis +#+attr_latex: :scale 1 +[[file:figs/hinf_three_synthesis_results.pdf]] + +* Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO +<> +** Introduction :ignore: +Several complementary filters are used in the active isolation system at the LIGO cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system. +The requirements on those filters are very tight and thus their design is complex. +The approach used in cite:hua05_low_ligo for their design is to write the synthesis of complementary FIR filters as a convex optimization problem. +The obtained FIR filters are compliant with the requirements. However they are of very high order so their implementation is quite complex. + +The effectiveness of the proposed method is demonstrated by designing complementary filters with the same requirements as the one described in cite:hua05_low_ligo. + +** Complementary Filters Specifications +<> +The specifications for one pair of complementary filters used at the LIGO are summarized below (for further details, refer to cite:hua04_polyp_fir_compl_filter_contr_system) and shown in Fig. ref:fig:ligo_weights: +- From $0$ to $\SI{0.008}{Hz}$, the magnitude of the filter's transfer function should be less or equal to $8 \times 10^{-4}$ +- Between $\SI{0.008}{Hz}$ to $\SI{0.04}{Hz}$, the filter should attenuate the input signal proportional to frequency cubed +- Between $\SI{0.04}{Hz}$ to $\SI{0.1}{Hz}$, the magnitude of the transfer function should be less than $3$ +- Above $\SI{0.1}{Hz}$, the magnitude of the complementary filter should be less than $0.045$ + +** Weighting Functions Design +<> +The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem. +However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters. + +A Type I Chebyshev filter of order $20$ is used as the weighting transfer function $w_L(s)$ corresponding to the low pass filter. +For the one corresponding to the high pass filter $w_H(s)$, a $7^{\text{th}}$ order transfer function is designed. +The magnitudes of the weighting functions are shown in Fig. ref:fig:ligo_weights. + +#+name: fig:ligo_weights +#+caption: Specifications and weighting functions magnitudes +#+attr_latex: :scale 1 +[[file:figs/ligo_weights.pdf]] + +** $\mathcal{H}_\infty$ Synthesis +<> +$\mathcal{H}_\infty$ synthesis is performed using the architecture shown in Fig. ref:eq:generalized_plant. +The complementary filters obtained are of order $27$. +In Fig. ref:fig:comp_fir_ligo_hinf, their bode plot is compared with the FIR filters of order 512 obtained in cite:hua05_low_ligo. +They are found to be very close to each other and this shows the effectiveness of the proposed synthesis method. + +#+name: fig:comp_fir_ligo_hinf +#+caption: Comparison of the FIR filters (solid) designed in cite:hua05_low_ligo with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed) +#+attr_latex: :scale 1 +[[file:figs/comp_fir_ligo_hinf.pdf]] + +* Conclusion +<> +This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor. +Typical specification on the super sensor noise and on the robustness of the sensor fusion has been shown to be linked to the norm of the complementary filters. +Therefore, a synthesis method that permits the shaping of the complementary filters norms has been proposed and has been successfully applied for the design of complex filters. +Future work will aim at further developing this synthesis method for the robust and optimal synthesis of complementary filters used in sensor fusion. + +* Acknowledgment +:PROPERTIES: +:UNNUMBERED: t +:END: +This research benefited from a FRIA grant from the French Community of Belgium. + +* Bibliography :ignore: +\bibliographystyle{elsarticle-num} +\bibliography{ref} diff --git a/journal/journal.pdf b/journal/journal.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0e5ec3d1f0fd0e213e63fceba5492098fac4e9a3 GIT binary patch literal 695705 zcmcecV~}jomaWUSZQHhO+qP}nwolnMPT96j**@i0_jccoe)0OpjsEd6V((mI=Zf5G zM`Yw2Ys`;SK}3v}k&YFLbZ%*64T_VIfPui?$O?*w2Z~j6Q7cjyE?jD}x z`E^X@6VwMD2!IqMY`)q@#)Ekxk%9+{0vd+12Z_x>enZ&mUs{qZ1l)~fH*!cYM_01< zd!4^PCrB?gQeZ3OH!{F_pbg#gj5Wrn>C$(a8ipKh`IDlDe&{Z2#6-;_)h@C;NT=8wwlGwQFw#Hh2F)liRNo6Zwbj}96!njpZvL8Ma za7dtr3XagG)f^s^24pljg$Nu|HvSVM8uH{-L*{ttPW?bXt#8;3iQ>>=ISDHJe2d^` 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0000000..b8f84a4 --- /dev/null +++ b/journal/journal.tex @@ -0,0 +1,448 @@ +% Created 2021-04-28 mer. 15:56 +% Intended LaTeX compiler: pdflatex +\documentclass[preprint, sort&compress]{elsarticle} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} +\usepackage{grffile} +\usepackage{rotating} +\usepackage[normalem]{ulem} +\usepackage{capt-of} +\usepackage{hyperref} +\usepackage{bm} +\usepackage{array} +\usepackage{amsmath,amssymb,amsfonts} +\usepackage{algorithmic} +\usepackage{textcomp} +\usepackage{cases} +\usepackage{tabularx,siunitx,booktabs} +\usepackage{algorithmic} +\usepackage{import} +\usepackage{hyperref} +\usepackage[hyperref]{xcolor} +\hypersetup{colorlinks=true} +\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry} +\journal{Mechanical Systems and Signal Processing} +\author[a1,a2]{Thomas Dehaeze\corref{cor1}} +\author[a3,a4]{Mohit Verma} +\author[a2,a4]{Christophe Collette} +\cortext[cor1]{Corresponding author. Email Address: dehaeze.thomas@gmail.com} +\address[a1]{European Synchrotron Radiation Facility, Grenoble, France} +\address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.} +\address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.} +\address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.} +\date{} +\title{A new method of designing complementary filters for sensor fusion using \(\mathcal{H}_\infty\) synthesis} +\begin{document} + + +\hypersetup{allcolors=teal} + +\begin{frontmatter} +\begin{abstract} +Sensor have limited bandwidth and are accurate only in a certain frequency band. +In many applications, the signals of different sensor are fused together in order to either enhance the stability or improve the operational bandwidth of the system. +The sensor signals can be fused using complementary filters. +The tuning of complementary filters is a complex task and is the subject of this paper. +The filters needs to meet design specifications while satisfying the complementary property. +This paper presents a framework to shape the norm of complementary filters using the \(\mathcal{H}_\infty\) norm minimization. +The design specifications are imposed as constraints in the optimization problem by appropriate selection of weighting functions. +The proposed method is quite general and easily extendable to cases where more than two sensors are fused. +Finally, the proposed method is applied to the design of complementary filter design for active vibration isolation of the Laser Interferometer Gravitation-wave Observatory (LIGO). +\end{abstract} + +\begin{keyword} +Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \sep{} Vibration isolation \sep{} Precision +\end{keyword} +\end{frontmatter} + +\section{Introduction} +\label{sec:orgf244196} +\label{sec:introduction} +\begin{itemize} +\item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion +\end{itemize} +\begin{itemize} +\item Increase the bandwidth: \cite{zimmermann92_high_bandw_orien_measur_contr} +\item Increased robustness: \cite{collette15_sensor_fusion_method_high_perfor} +\item Decrease the noise: +\end{itemize} +\begin{itemize} +\item UAV: \cite{pascoal99_navig_system_desig_using_time}, \cite{jensen13_basic_uas} +\item Gravitational wave observer: \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system} +\end{itemize} +\begin{itemize} +\item \cite{brown72_integ_navig_system_kalman_filter} alternate form of complementary filters => Kalman filtering +\item \cite{higgins75_compar_compl_kalman_filter} Compare Kalman Filtering with sensor fusion using complementary filters +\item \cite{robert12_introd_random_signal_applied_kalman} advantage of complementary filters over Kalman filtering +\end{itemize} +\begin{itemize} +\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters +\item \cite{plummer06_optim_compl_filter_their_applic_motion_measur} use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density) +\item \cite{jensen13_basic_uas} design of complementary filters with classical control theory +\item \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}: FIR + convex optimization +\item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands} +\item Analytical methods: +\begin{itemize} +\item first order: \cite{corke04_inert_visual_sensin_system_small_auton_helic} +\item second order: \cite{baerveldt97_low_cost_low_weigh_attit}, \cite{stoten01_fusion_kinet_data_using_compos_filter}, \cite{jensen13_basic_uas} +\item higher order: \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}, \cite{zimmermann92_high_bandw_orien_measur_contr}, \cite{collette15_sensor_fusion_method_high_perfor}, \cite{matichard15_seism_isolat_advan_ligo} +\end{itemize} +\item Analog complementary filters: \cite{yong16_high_speed_vertic_posit_stage}, \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop} +\end{itemize} +\begin{itemize} +\item Robustness problems: \cite{zimmermann92_high_bandw_orien_measur_contr} change of phase near the merging frequency +\item Trial and error +\item Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available. +\end{itemize} + +\section{Complementary Filters Requirements} +\label{sec:org2279a0f} +\label{sec:requirements} +\subsection{Sensor Fusion Architecture} +\label{sec:org60aa99d} +\label{sec:sensor_fusion} + +Let's consider two sensors measuring the same physical quantity \(x\) with dynamics \(G_1(s)\) and \(G_2(s)\), and with uncorrelated noise characteristics \(n_1\) and \(n_2\). + +The signals from both sensors are fed into two complementary filters \(H_1(s)\) and \(H_2(s)\) and then combined to yield an estimate \(\hat{x}\) of \(x\) as shown in Fig. \ref{fig:fusion_super_sensor}. +\begin{equation} +\label{eq:comp_filter_estimate} + \hat{x} = \left(G_1 H_1 + G_2 H_2\right) x + H_1 n_1 + H_2 n_2 +\end{equation} + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/fusion_super_sensor.pdf} +\caption{\label{fig:fusion_super_sensor}Sensor fusion architecture} +\end{figure} + +The complementary property of \(H_1(s)\) and \(H_2(s)\) implies that their transfer function sum is equal to one at all frequencies \eqref{eq:comp_filter}. +\begin{equation} +\label{eq:comp_filter} + H_1(s) + H_2(s) = 1 +\end{equation} + +\subsection{Noise Sensor Filtering} +\label{sec:orgf9a3723} +\label{sec:noise_filtering} + +Let's first consider sensors with perfect dynamics +\begin{equation} +\label{eq:perfect_dynamics} + G_1(s) = G_2(s) = 1 +\end{equation} + +The estimate \(\hat{x}\) is then described by +\begin{equation} +\label{eq:estimate_perfect_dyn} + \hat{x} = x + H_1 n_1 + H_2 n_2 +\end{equation} + +From \eqref{eq:estimate_perfect_dyn}, the complementary filters \(H_1(s)\) and \(H_2(s)\) are shown to only operate on the sensor's noise. +Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured. + +Let's define the estimation error \(\delta x\) by \eqref{eq:estimate_error}. +\begin{equation} +\label{eq:estimate_error} + \delta x \triangleq \hat{x} - x = H_1 n_1 + H_2 n_2 +\end{equation} + +As shown in \eqref{eq:noise_filtering_psd}, the Power Spectral Density (PSD) of the estimation error \(\Phi_{\delta x}\) depends both on the norm of the two complementary filters and on the PSD of the noise sources \(\Phi_{n_1}\) and \(\Phi_{n_2}\). +\begin{equation} +\label{eq:noise_filtering_psd} + \Phi_{\delta x} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2} +\end{equation} + +Usually, the two sensors have high noise levels over distinct frequency regions. +In order to lower the noise of the super sensor, the value of the norm \(|H_1|\) has to be lowered when \(\Phi_{n_1}\) is larger than \(\Phi_{n_2}\) and that of \(|H_2|\) lowered when \(\Phi_{n_2}\) is larger than \(\Phi_{n_1}\). + +\subsection{Robustness of the Fusion} +\label{sec:org227af67} +\label{sec:fusion_robustness} + +In practical systems the sensor dynamics is not perfect and \eqref{eq:perfect_dynamics} is not verified. +In such case, one can use an inversion filter \(\hat{G}_i^{-1}(s)\) to normalize the sensor dynamics, where \(\hat{G}_i(s)\) is an estimate of the sensor dynamics \(G_i(s)\). +However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and \(\hat{G}_i^{-1}(s) G_i(s) \neq 1\). + +Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. \ref{fig:sensor_fusion_dynamic_uncertainty} where \(\Delta_i\) is any stable transfer function satisfying \(|\Delta_i(j\omega)| \le 1,\ \forall\omega\), and \(|w_i(s)|\) is a weight representing the magnitude of the uncertainty. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/sensor_fusion_dynamic_uncertainty.pdf} +\caption{\label{fig:sensor_fusion_dynamic_uncertainty}Sensor fusion architecture with sensor dynamics uncertainty} +\end{figure} + +The super sensor dynamics \eqref{eq:super_sensor_dyn_uncertainty} is no longer equal to \(1\) and now depends on the sensor dynamics uncertainty weights \(w_i(s)\) as well as on the complementary filters \(H_i(s)\). +\begin{equation} +\label{eq:super_sensor_dyn_uncertainty} + \frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s) +\end{equation} + +The uncertainty region of the super sensor can be represented in the complex plane by a circle centered on \(1\) with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) as shown in Fig. \ref{fig:uncertainty_set_super_sensor}. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/uncertainty_set_super_sensor.pdf} +\caption{\label{fig:uncertainty_set_super_sensor}Uncertainty region of the super sensor dynamics in the complex plane (solid circle). The contribution of both sensors 1 and 2 to the uncertainty are represented respectively by a dotted and a dashed circle} +\end{figure} + +The maximum phase added \(\Delta\phi(\omega)\) by the super sensor dynamics at frequency \(\omega\) is then +\begin{equation} +\label{eq:max_phase_uncertainty} + \Delta\phi(\omega) = \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big) +\end{equation} + +As it is generally desired to limit the maximum phase added by the super sensor, \(H_1(s)\) and \(H_2(s)\) should be designed such that \eqref{eq:max_uncertainty_super_sensor} is satisfied. +\begin{equation} +\label{eq:max_uncertainty_super_sensor} + \max_\omega \big( \left|w_1 H_1\right| + \left|w_2 H_2\right|\big) < \sin\left( \Delta \phi_\text{max} \right) +\end{equation} +where \(\Delta \phi_\text{max}\) is the maximum allowed added phase. + +Thus the norm of the complementary filter \(|H_i|\) should be made small at frequencies where \(|w_i|\) is large. + +\section{Complementary Filters Shaping using \(\mathcal{H}_\infty\) Synthesis} +\label{sec:org10fbb17} +\label{sec:hinf_method} +As shown in Sec. \ref{sec:requirements}, the performance and robustness of the sensor fusion architecture depends on the complementary filters norms. +Therefore, the development of a synthesis method of complementary filters that allows the shaping of their norm is necessary. +\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis} +\label{sec:org933b14f} +\label{sec:hinf_synthesis} +The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:comp_filter}. +This is equivalent as to finding stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions \eqref{eq:comp_filter_problem_form} are satisfied. +\begin{subequations} +\label{eq:comp_filter_problem_form} + \begin{align} + & H_1(s) + H_2(s) = 1 \label{eq:hinf_cond_complementarity} \\ + & |H_1(j\omega)| \le \frac{1}{|W_1(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h1} \\ + & |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2} + \end{align} +\end{subequations} +where \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are chosen to shape the norms of the corresponding filters. + +In order to express this optimization problem as a standard \(\mathcal{H}_\infty\) problem, the architecture shown in Fig. \ref{fig:h_infinity_robust_fusion} is used where the generalized plant \(P\) is described by \eqref{eq:generalized_plant}. +\begin{equation} +\label{eq:generalized_plant} + \begin{bmatrix} z_1 \\ z_2 \\ v \end{bmatrix} = P(s) \begin{bmatrix} w\\u \end{bmatrix}; \quad P(s) = \begin{bmatrix}W_1(s) & -W_1(s) \\ 0 & W_2(s) \\ 1 & 0 \end{bmatrix} +\end{equation} + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/h_infinity_robust_fusion.pdf} +\caption{\label{fig:h_infinity_robust_fusion}Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters} +\end{figure} + +The \(\mathcal{H}_\infty\) filter design problem is then to find a stable filter \(H_2(s)\) which based on \(v\), generates a signal \(u\) such that the \(\mathcal{H}_\infty\) norm from \(w\) to \([z_1, \ z_2]\) is less than one \eqref{eq:hinf_syn_obj}. +\begin{equation} +\label{eq:hinf_syn_obj} + \left\|\begin{matrix} \left[1 - H_2(s)\right] W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 +\end{equation} + +This is equivalent to having \eqref{eq:hinf_problem} by defining \(H_1(s)\) as the complementary filter of \(H_2(s)\) \eqref{eq:definition_H1}. +\begin{equation} +\label{eq:hinf_problem} + \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 +\end{equation} + +\begin{equation} +\label{eq:definition_H1} + H_1(s) \triangleq 1 - H_2(s) +\end{equation} + +The complementary condition \eqref{eq:hinf_cond_complementarity} is ensured by \eqref{eq:definition_H1}. +The conditions \eqref{eq:hinf_cond_h1} and \eqref{eq:hinf_cond_h2} on the filters shapes are satisfied by \eqref{eq:hinf_problem}. +Therefore, all the conditions \eqref{eq:comp_filter_problem_form} are satisfied using this synthesis method based on \(\mathcal{H}_\infty\) synthesis, and thus it permits to shape complementary filters as desired. + +\subsection{Weighting Functions Design} +\label{sec:org3bb6eca} +\label{sec:hinf_weighting_func} +The proper design of the weighting functions is of primary importance for the success of the presented complementary filters \(\mathcal{H}_\infty\) synthesis. + +First, only proper, stable and minimum phase transfer functions should be used. +Second, the order of the weights should stay reasonably small in order to reduce the computational costs associated with the solving of the optimization problem and for the physical implementation of the filters (the order of the synthesized filters being equal to the sum of the weighting functions order). +Third, one should not forget the fundamental limitations imposed by the complementary property \eqref{eq:comp_filter}. +This implies for instance that \(|H_1(j\omega)|\) and \(|H_2(j\omega)|\) cannot be made small at the same time. + +When designing complementary filters, it is usually desired to specify the slope of the filter, its crossover frequency and its gain at low and high frequency. +To help with the design of the weighting functions such that the above specification can be easily expressed, the following formula is proposed. +\begin{equation} +\label{eq:weight_formula} + W(s) = \left( \frac{ + \hfill{} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}} + }{ + \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}} + }\right)^n +\end{equation} +The parameters permit to specify: +\begin{itemize} +\item the low frequency gain: \(G_0 = lim_{\omega \to 0} |W(j\omega)|\) +\item the high frequency gain: \(G_\infty = lim_{\omega \to \infty} |W(j\omega)|\) +\item the absolute gain at \(\omega_0\): \(G_c = |W(j\omega_0)|\) +\item the absolute slope between high and low frequency: \(n\) +\end{itemize} + +The parameters \(G_0\), \(G_c\) and \(G_\infty\) should either satisfy condition \eqref{eq:cond_formula_1} or \eqref{eq:cond_formula_2}. +\begin{subequations} +\label{eq:condition_params_formula} + \begin{align} + G_0 < 1 < G_\infty \text{ and } G_0 < G_c < G_\infty \label{eq:cond_formula_1}\\ + G_\infty < 1 < G_0 \text{ and } G_\infty < G_c < G_0 \label{eq:cond_formula_2} + \end{align} +\end{subequations} + +The general shape of a weighting function generated using \eqref{eq:weight_formula} is shown in Fig. \ref{fig:weight_formula}. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/weight_formula.pdf} +\caption{\label{fig:weight_formula}Magnitude of a weighting function generated using the proposed formula \eqref{eq:weight_formula}, \(G_0 = 1e^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\)} +\end{figure} + +\subsection{Validation of the proposed synthesis method} +\label{sec:org5517901} +\label{sec:hinf_example} +Let's validate the proposed design method of complementary filters with a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that: +\begin{itemize} +\item the merging frequency is around \(\SI{10}{Hz}\) +\item the slope of \(|H_1(j\omega)|\) is \(-2\) above \(\SI{10}{Hz}\) +\item the slope of \(|H_2(j\omega)|\) is \(+3\) below \(\SI{10}{Hz}\) +\item the gain of both filters is equal to \(10^{-3}\) away from the merging frequency +\end{itemize} + +The weighting functions \(W_1(s)\) and \(W_2(s)\) are designed using \eqref{eq:weight_formula}. +The parameters used are summarized in table \ref{tab:weights_params} and the magnitude of the weighting functions is shown in Fig. \ref{fig:hinf_synthesis_results}. + +\begin{table}[htbp] +\caption{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\)} +\centering +\begin{tabularx}{0.5\linewidth}{Xcc} +\toprule +Parameter & \(W_1(s)\) & \(W_2(s)\)\\ +\midrule +\(G_0\) & \(0.1\) & \(1000\)\\ +\(G_\infty\) & \(1000\) & \(0.1\)\\ +\(\omega_c\) [\(\si{Hz}\)] & \(11\) & \(10\)\\ +\(G_c\) & \(0.5\) & \(0.5\)\\ +\(n\) & \(2\) & \(3\)\\ +\bottomrule +\end{tabularx} +\end{table} + +The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_synthesis_results} and their transfer functions in the Laplace domain are given below. +\begin{align*} + H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\ + H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)} +\end{align*} + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/hinf_synthesis_results.pdf} +\caption{\label{fig:hinf_synthesis_results}Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis} +\end{figure} + +\subsection{Synthesis of Three Complementary Filters} +\label{sec:org1fd1484} +\label{sec:hinf_three_comp_filters} +Some applications may require to merge more than two sensors. +In such a case, it is necessary to design as many complementary filters as the number of sensors used. +The synthesis problem is then to compute \(n\) stable transfer functions \(H_i(s)\) such that \eqref{eq:hinf_problem_gen} is satisfied. +\begin{subequations} +\label{eq:hinf_problem_gen} + \begin{align} + & \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\ + & \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:hinf_cond_perf_gen} + \end{align} +\end{subequations} +The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. \ref{fig:comp_filter_three_hinf}. + +The \(\mathcal{H}_\infty\) synthesis objective applied on \(P(s)\) is to design two stable filters \(H_2(s)\) and \(H_3(s)\) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2, \ z_3]\) is less than one \eqref{eq:hinf_syn_obj_three}. +\begin{equation} +\label{eq:hinf_syn_obj_three} + \left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1 +\end{equation} + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/comp_filter_three_hinf.pdf} +\caption{\label{fig:comp_filter_three_hinf}Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters} +\end{figure} + +By choosing \(H_1(s) \triangleq 1 - H_2(s) - H_3(s)\), the proposed \(\mathcal{H}_\infty\) synthesis solves the design problem \eqref{eq:hinf_problem_gen}. \par +An example is given to validate the method where three sensors are used in different frequency bands (up to \(\SI{1}{Hz}\), from \(1\) to \(\SI{10}{Hz}\) and above \(\SI{10}{Hz}\) respectively). +Three weighting functions are designed using \eqref{eq:weight_formula} and shown by dashed curves in Fig. \ref{fig:hinf_three_synthesis_results}. +The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_three_synthesis_results}. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/hinf_three_synthesis_results.pdf} +\caption{\label{fig:hinf_three_synthesis_results}Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis} +\end{figure} + +\section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO} +\label{sec:org925754e} +\label{sec:application_ligo} +Several complementary filters are used in the active isolation system at the LIGO \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}. +The requirements on those filters are very tight and thus their design is complex. +The approach used in \cite{hua05_low_ligo} for their design is to write the synthesis of complementary FIR filters as a convex optimization problem. +The obtained FIR filters are compliant with the requirements. However they are of very high order so their implementation is quite complex. + +The effectiveness of the proposed method is demonstrated by designing complementary filters with the same requirements as the one described in \cite{hua05_low_ligo}. +\subsection{Complementary Filters Specifications} +\label{sec:org4ade5f6} +\label{sec:ligo_specifications} +The specifications for one pair of complementary filters used at the LIGO are summarized below (for further details, refer to \cite{hua04_polyp_fir_compl_filter_contr_system}) and shown in Fig. \ref{fig:ligo_weights}: +\begin{itemize} +\item From \(0\) to \(\SI{0.008}{Hz}\), the magnitude of the filter's transfer function should be less or equal to \(8 \times 10^{-4}\) +\item Between \(\SI{0.008}{Hz}\) to \(\SI{0.04}{Hz}\), the filter should attenuate the input signal proportional to frequency cubed +\item Between \(\SI{0.04}{Hz}\) to \(\SI{0.1}{Hz}\), the magnitude of the transfer function should be less than \(3\) +\item Above \(\SI{0.1}{Hz}\), the magnitude of the complementary filter should be less than \(0.045\) +\end{itemize} + +\subsection{Weighting Functions Design} +\label{sec:org8c85120} +\label{sec:ligo_weights} +The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem. +However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters. + +A Type I Chebyshev filter of order \(20\) is used as the weighting transfer function \(w_L(s)\) corresponding to the low pass filter. +For the one corresponding to the high pass filter \(w_H(s)\), a \(7^{\text{th}}\) order transfer function is designed. +The magnitudes of the weighting functions are shown in Fig. \ref{fig:ligo_weights}. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/ligo_weights.pdf} +\caption{\label{fig:ligo_weights}Specifications and weighting functions magnitudes} +\end{figure} + +\subsection{\(\mathcal{H}_\infty\) Synthesis} +\label{sec:org89d27e1} +\label{sec:ligo_results} +\(\mathcal{H}_\infty\) synthesis is performed using the architecture shown in Fig. \ref{eq:generalized_plant}. +The complementary filters obtained are of order \(27\). +In Fig. \ref{fig:comp_fir_ligo_hinf}, their bode plot is compared with the FIR filters of order 512 obtained in \cite{hua05_low_ligo}. +They are found to be very close to each other and this shows the effectiveness of the proposed synthesis method. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=1]{figs/comp_fir_ligo_hinf.pdf} +\caption{\label{fig:comp_fir_ligo_hinf}Comparison of the FIR filters (solid) designed in \cite{hua05_low_ligo} with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed)} +\end{figure} + +\section{Conclusion} +\label{sec:org153df0f} +\label{sec:conclusion} +This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor. +Typical specification on the super sensor noise and on the robustness of the sensor fusion has been shown to be linked to the norm of the complementary filters. +Therefore, a synthesis method that permits the shaping of the complementary filters norms has been proposed and has been successfully applied for the design of complex filters. +Future work will aim at further developing this synthesis method for the robust and optimal synthesis of complementary filters used in sensor fusion. + +\section*{Acknowledgment} +\label{sec:org6c8e8b0} +This research benefited from a FRIA grant from the French Community of Belgium. + +\bibliographystyle{elsarticle-num} +\bibliography{ref} +\end{document} diff --git a/journal/ref.bib b/journal/ref.bib new file mode 100644 index 0000000..ab551cf --- /dev/null +++ b/journal/ref.bib @@ 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