HAL Id: inria-00070310
https://hal.inria.fr/inria-00070310
Submitted on 19 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Time Supervision of Concurrent Systems using Symbolic Unfoldings of Time Petri Nets
Thomas Chatain, Claude Jard
To cite this version:
Thomas Chatain, Claude Jard. Time Supervision of Concurrent Systems using Symbolic Unfoldings
of Time Petri Nets. [Research Report] RR-5706, INRIA. 2005, pp.19. �inria-00070310�
ISRN INRIA/RR--5706--FR+ENG
a p p o r t
d e r e c h e r c h e
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Time Supervision of Concurrent Systems using Symbolic Unfoldings of Time Petri Nets
Thomas Chatain , Claude Jard
N˚5706
Septembre 2005
Systèmes communicants
+,'.-/1023/4 +56879!$:<;=!$
>?:@BADCBE"FG?:CIH3CBJLKMONPFGQRCBST:UWVXCBYZT[M\M
]_^a`cbedfgh`#ijkfflnmpoqihrsmtbe`
uwv jsxcgb#yzo{`cb v oq|Bijkf
} rk~n~Ij v bng v ghipg v ingmt 3%]_gh~abghf| v gskth%~prsgh`
"h3nkk jmnobej v omp v gZrsLbeofg ijkmIil vev ghmtb%` ^a` bghf`%o{`Griprsqqgmnomn¡berk`¢X£)¤¥m¦bno{`%~prs~Bg v
§
g¨©j vf%lnqrsbgOª©fjaag«|Irk`ghp¬`ln~Bg ve® oq`ojm<|_^fgZrsmp`%js¨*no{naghm¦` ber3beg"no{` bj v ^ v gZijkmI`cb v lpiboqjkm[¯
¨v jkf°g
®
gmtb±ª²gk£p£trs{r
v
f¬³jk|p`g
ve®
r3boqjkmI`£s´Ogz¨©jkqj
§
r`js¥irkqghGb
v
lngzijkmIil
vev
ghmpi^Grs~n~
v
jtrkilp`omn
beofg
u
gb
v
opmngb`µkbng#fjaagIag¶pmpgh`³g·_~po{iobq^beng*irklp`ersqobc^rkmpijmpil
vev gmpi^
v
ghqrsboqjkmp`:|Bgb
§
gghm
bengOjk|p`g
ve®
rs|nqgWg
®
gmtb`¯!~
v
j_nlpigZ¸|_^6bng¹` ^a` bghfºlnmpag
v
`ln~Bg
ve®
oq`ojm9jkmao»Xg
v
gmtb¼~Ijomtb`jk¨
j|p`g
ve®
rsboqjkm[¯³rkmp½ijkmp` b
v
rsoqmngZ<|_^¹boqfg¡rk`~IgZi\be`h£w¾ng¼~
v
jk|nqgf¿oq`Gbej¹ijkf~nlabeg¼jkma«bngLÀI^Ábng
no»Xg
v
gmtb³~pr
v
bo{rsaj
v
ng
v
no{`cbej
v
ogZ`¯
§
no{ir
v
gPbeng#~Bj`e` oq|nqg!g·a~nqrkmpr3beojmp`:js¨Ibng#j|p`g
ve®
rk|nqg!g
®
ghmb`£
´OgÂaj¹mnjsboqf~Ijt` gbprsbboqfg o{`jk|I` g
v®
rs|pgµbeng¡rsoqf.js¨` ln~Bg
ve®
oq`oqjkm¦o{`Gbj¹oqma¨©g
v
beng¼~pr
v
bo{rs
j v ng v omnzjs¨Ibngg ® gmtbe`:rkmpbengov ~Bj`e` oq|nqgö voqmnnr3begh`h£s¾noq`:o{`:rkinoqg ® gh|t^%ijkmI` o{ag v oqmnzr±fjaag
jk¨:bngG` ^a` bgfÄlnmpng v `ln~Bg ve® oq`ojm[¯no® gmÂrk`zrboqfg u gb v o)mngbZ¯Brkmp bngjkma«bngLÀI^ ijkmp` b v lpi\beojm
jk¨zrsm½lnma¨©jqaoqmnI¯³klno{agh¦|_^¹bng¼jk|I` g v® r3beojmp`h£)Å`oqmnOrO`^tf%|Ijo{i v gh~ v gZ` ghmtber3beojm[¯)bno{`G~prs~Bg v
~ vgZ` ghmb`rmng
§
ag¶pmpobeojm jk¨)benglnma¨©jqnomnjk¨)beofg u gb vo(mngbe`
§
obÂaghmp`gzbeofg£
ÆÂÇtÈXÉeÊË
3Ìk ijkmIil
vev
ghmtb`^a`cbegf`¯nnoqrkkmnjt` o{`¯tbeofg
u
gb
v
o(mngbe`h¯nlnma¨©jqnomn
Í{ÎwÏÐ ÑsÒϱÓBÔÕÐcÖ¥×3Ø
Ù:ÚtÛÝÜ#Þ:ßàcá"Þ:âhÜÜ¥ãtätäaßà¥åcæ\çè3éêåcÚtæzëqàcæìíÚîÃïÃî)Ù6äkàcßð«æíeåcܱñsÞ:âhì¼âì_ç¼òXæàcÜ¥ÛÝëqßhàcóôIë{ãtìtçæ\çêè3éêåcÚtæ
õGÛìÛÝÜRåcöeàcæçæ÷âGîPæí Útæeàcí Útæ(øpä_âà¥åcìtæeàcÜ#ßë:åcÚtæñsÞ³âì äkàcßð«æíeåzâàcæùLìàcÛâkôBúà âhìíæIÙBæ÷ÝæíßhóîÃûü±ôIýÃ÷Ýí\âåcæ÷Õô
þ ßZÜ õGæeå¥àcÛÝíÜôhâhìtç#ÿ!ìtÛ æàcÜ¥ÛÝåLé#ßhëßà çæ\âãøätâà¥åcìtæàcÜXßhëkåcÚtæ)òXæeàcÜ¥Ûë{ßàcó6äàcßðRæíåâàcæùLìàcÛâkôúkà âhìtíeæBÙBæe÷æeíßZó
î!ûü±ôkùLïÙ!ôÃàcätÚæãtÜôâhìtçGÿ!ìÛ æeàcÜ¥Ûå«éßëÃàcæìtßhèt÷Ýæ
ù«îPù¥ñký[ï#ñ:âhí Ú_âì)àcæeå âZìæZô:âóätãtÜçkææ\âãt÷ÝÛÝæãIô¹úÄîPæìtìæÜíæ\çæ pôúà âìtíæhô
!"#%$&'()"*&+&-,/.*0,1*,2'&3(421
ù«îPù¥ñký[ï#ñ:âhí Ú_âì)àcæeå âZìæZô:âóätãtÜçkææ\âãt÷ÝÛÝæãIô¹úÄîPæìtìæÜíæ\çæ pôúà âìtíæhô
)5&6-7289(;:&170<-18+&=.*83(8.'->?&-?/"*&@.A(421
Unité de recherche INRIA Rennes
-#1 /# 4¼/+#5['ê8-O/ #
4= #) 4 879!$:¹$)* #/
r%` lp~Ig v® o{` oqjkm agh``^_` bdhfgh`!aoq` b v oq|nlh`Ãbegf~p` v gXgZ`cblnmng±~ v jk|nqdfgri\belprsqobk£
y±rsmI`Ãigz~prk~noqg v mnjklp`rk|Ij v ajmp`w{r`ln~Bg ve® oq`ojm"`jklp`P¨©j v fg±ag vgZijmp`cb v lIi\boqjkmpoq` bjov gh`~pr v
beov j|p` g v® rsboqjkm ® mpgfgmtbe`ª²~pr v g·_ghf~ngrs{r v fgh`¬\£*jlp`±~ v jk~Bj`jkmp`±lnmngrk~n~ v j_ipgGag
®_v rsoqgijmpil vv gmpiglabeoqo{`rkmb±gZ` v Z` gZrsla·Âag u gb v obgf~Bj v g{`µXqg%fjaadhgG¶pmpobg·a~nqo{iobgfghmb
qgh` v ghqrsboqjkmp`Ãagzihrslp`ersqob*gb!ag±ijkmIil vev ghmpiggmtb v g*qgh` ® hmngfghmb`wj|p` g v® rk|ngZ` fo{`P~Ir v qgz`^a`c
bedfg"!tlnogh` bzj|p`g ve® ~pr v ~nqlp`oghl v `zirk~abghl v `gb#!lpo)gZ`cbijkmtb v rkomtb*~pr v agZ`zr` ~Bghibe`#begf~Bj v g{`£
g%~ v j|ndhfgGijkmp`oq` bg$ irs{ilpg v rkl ® jk:gZ`ao»% v gmtbegh`±no{`cbejkov gZ`GªL` jlp`z¨©j v fgGj v v g~pr v beogh{¬
!tlno~Bgl
®
gmtbzg·_~po&!tlng
v
gZ`'
®
mpgfgmtbe`jk|p`g
ve®
Z`£(*jklI`*mof~Bj`jkmp`z~pr`'!tlngqgbegf~p`±` job±jk|a
`g
ve®
rs|nqgµ_qgz|plab*agz{r`ln~Bg
ve®
oq`ojmêgh` b#oqma¨)
v g v
*j v v gz~pr
v
boqgXagZ`
®
hmngfgmtb`Pgb!qgh`nrsbgh`ag
beo
v
~Ijt``o|pgZ`£
u
jkl
v
ighqrGmnjlp`*ijmp`oq
v
jmp`!lnmÂfjaadqgalÂ` ^a` bdhfGg+`ln~Bg
ve®
o{`g
v
¯najkmnm,` jlp`qr
¨©j
v
fglnm
v
h`ghrklÂag
u
gb
v
obghfG~Bj
v
g«¯pgbzmnjlp`*ijkmp` b
v
lno{` jmp`*rsl
®
jk)lnm¡~po{rsgklno{~Ir
v
qgh`
j|p`g
ve®
rsboqjkmp`h£-Ãgbr
v
bo{iqg%~
v
Z` ghmbeg%lpmngmnjkl
®
gqgG¶ImnobeojmÁalO~nqoqrkkgGlpm
v
h`ghrklWag
u
gb
v o
begf~Bj v g«¯nnrsmp`qg!tlngh[qgzbegf~p`#gZ`cb v g~ v Z` ghmb±ngfrkmnoqd v g`^_f|Bjkqo.!tlng£
/Ë Z É 10
`^a`cbedfgh`
v
~Ir
v
bo{`¯aao{rsmnj` bo{is¯
v
h`ghrkla·"ag
u
gb
v
oXbegf~Ij
v
ghq`h¯nh~no{rsg
, -/.10!235476980:;3%.=<>.74?2@A<B0C@D4FE32G
jkmnobj
v
oqmn
v
ghrk«boqfgijkmIil
vev
ghmtb±`^_` bghf`zo{`r¼iprsqqgmnomn"ber` ¢X£(¤¥mbno{`~prs~Bg
v § gG¨©j
v
fln{r3beg
fjaaghR|prk`ghW` ln~Bg
ve®
oq`oqjkm¡|_^¼fghrsmI`jk¨Pno{nagm¹` ber3begno{` bj
v ^ v gZijmp`cb
v
lIi\boqjkm¯n¨
v
jkfg
®
gmtbGª²gk£p£
rkqr v fê¬:j|p`g ve® rsboqjkmp`h£k´Og*¨©jqj
§
r`js¥irkqghGb v lpgzijmpil vv gmpi^%rk~n~ v jrkilI` oqmnbeofg u gb v oXmngb`µ
bengfj_ng[ag¶pmngh`g·a~no{iobq^bengirklp`rkobc^rsmp ijmpil vv gmpi^ v ghqrsboqjkmp`!|Bgb
§
ggmbpg±j|p`g ve® rk|nqg
g ® ghmb`¯~ v jaalIigh¹|_^Obpg"` ^a` bghf=lnmIag v ` lp~Ig v® o{` oqjkm¹jkm<no»Xg v gmtb%~Bjkoqmtbe`js¨#jk|p`g ve® r3beojm[¯rsmI
ijkmp` b v rsoqmngZ½|_^Dboqfgr` ~Bghibe`h£Ã¾ng~ v j|nqgf oq`"bj¦ijkf~nlabegWjmaLbeng«Àp^½bngOao»Bg v ghmb ~pr v bo{rs
j v ng v no{`cbej v ogZ`¯
§
poqiÂr v g±bng~Ijt``o|pgg·_~pqrkmpr3beojmp`js¨bngj|p`g ve® rk|nqg±g ® ghmb`£'H*m¼of~Bj v berkmb
rk~n~nqoqihr3beojmo{`!bng`ln~Bg ve® oq`ojmjk¨begqghijfflnmno{irsboqjkmp`mpgb
§ j v ¢_`h¯
§
no{ifjsbeo® rsbghbenoq`
§ j v ¢X£
´¸obnjlabijmp`oqag v oqmnboqfgk¯r mprsbl v rswihrsmpao{nrsbgGbj ¨©j v frsqoJIhgbeng~ v jk|nqgf=r v gG`er3¨©g u gb v o
mpgbe`
§
ob½| v rkmpinoqmnW~ v jaigZ``gh`%rsmp¦lnma¨©jqaoqmnt`£³¾ng ~ v g ® ojlp`
§ j v ¢¹js¨jl v v jkln~<lI` gZÁbpoq`
¨v rsfg
§ j v ¢bej"ag¶Imngbpgno{`cbej v ogZ`#rsmp¼rêao{` b v o|nlnbghÂrkj v obenf bej|nlnoqq bengfrk`#r"ijkqqghi\beojm
jk¨pijmp` o{` bgmtbqjairk ® og
§
`%K+(L£Z¾pgÃrs~p~ v jtrkiag¶pmngh`bpgP~Bj`e` oq|nqgPg·a~n{rsmIr3boqjkmI`[rk`[bng!lnmpag v q^_omp
g ® ghmbw` b v lpi\bel v g#js¨Bbng*lpma¨©jk{aoqmnjs¨Bbng~ v jaalIi\bwjs¨Bbpg u gbv opmngbwfj_ngIrkmpGjs¨(rsmrki^aiqoqi u gb v o
mpgb
v
g~
v
gh`gmtboqmn%bng~Ir
v
bo{rsXj
v
ag
v
jk¨)bengjk|p`g
ve®
gZ"rkqr
v
f`h£
¤¥mÂbeno{`~prk~Ig
v §
gg·tbegmp¡jkl
v
fGgbnja¡bjbeofg
u
gb
v
o)mpgbe`h¯Xrsqj
§
oqmnbpg%agZ` oqkmng
v
bej"fjaag
beofgijmp` b
v
rsoqmb`¯
v
gh` b
v
oqiboqmn%|_^benoq`
§ r ^Gbng`gb#js¨)~Ijt``o|pgg·a~n{rsmIr3boqjkmI`¯t´gajGmnjkboqf~Ijt` g
bepr3bboqfGgoq`#j|p`g
ve®
rk|nqgkµabengrsoqf jk¨w` ln~Bg
ve®
oq`oqjkm¼oq`#bjêoqma¨©g
v
bng~Ir
v
bo{rs[j
v
ag
v
oqmnjs¨:bengg
®
gmtbe`
rkmp¼bengo
v
~Bj`e`o|nqg¶
v
oqmnpr3bgZ`£XÅ`oqmnr` ^_f|Bjkqo{i
v
g~
v
gh`gmtbersboqjkm[¯pbno{`±~prs~Bg
v ~ v
gh`gmtbe`±rêmng
§
ng¶pmnoboqjkm¼jk¨bpglnma¨©jk{aoqmnjs¨)boqfg u gbv o[mngbe`
§
obÂaghmp` g±beofgk£
jaaghR|pr` gZ½ao{rsmnj`o{`%lI` oqmn¹boqfg u gb v o*mngbe`rkmpD~pr v bo{rs#j v ag v `Gpr`rsv gZrka^Á|Bgghm9rk_
vgZ``ghomMKqZ L£3¤¥mbeno{`
§ j v ¢X¯Zbghf~Ij v rk v ghr` jmnoqmnzo{`:|pr` gZjkm¼ª©qoqmnghr v ¬)qjkko{is£s¾pgö v ` b v g¨©g v gmpig
bejboqfg u gb v o[mngb*lnma¨©jk{aoqmnê`ggf`!bej|IgoqmÁhkn¯_|_^NH£B]_ghfGghmnj ® ¯H£!OPrk¢kj ® qg ® rsmIPH£$Qzjtgh
frkmp`NKq +L oqm<bengijkmtbg·tbGjs¨*pr v § r v g ® g v o¶Iihr3beojm[£)¾ngh^ÁaghrkÃjmn^
§
ob¦r !tlnobeg v gh` b v o{i\bgZ
iqr``jk¨mpgbe`h¯pihrsqgZ #RSUTVWT/X'#5 Y Z#[]\^X#R#¯nom
§
no{iÂrsm_^"injoqigoq` v gZ` j® gh
oqmpagh~Ighmpaghmbe^Âjk¨wboqfgk£(¤¥m_K L ¯[¾£`H*l v r"rkmpbaI£ oqoqlp`±ko® gGr"~pr v beoqrk)j v ng v ` ghfrkmtbo{i`bjêbeofg
u gbv oBmpgbe`h¯_|prk`ghêjkmêbeng±mnjkmp`g!tlnghmtbo{rsB~ vjaigh`e` gZ`P`gfrsmtbo{i`P¨©j v lnmtboqfghmngb#`^_` bghf`£CH9beofg
~ v
jaigZ``:jk¨[rboqfg
u
gb
v
opmpgbPo{`Pag¶pmngZrk`Pr±b
v
raoboqjkmprkq^%ijkmp` b
v
lpi\beghirklp`ersn~
v
jaigZ``:bepr3bÃprk`wr
®
rko{boqfompp£CH*m"rkj
v
obenf¨©j
v
ingZi¢_omn
®
rko{amngZ``Ãjs¨[ro
®
gmbeofoqmno{`P~
v
gZ` ghmbegh(£¤RbÃo{`P~
v j ® gh
bepr3bbengoqmbeg
v
gZr
®
omnt`jk¨Ãbengboqfg~
v
j_igh`e` gZ`r
v
g%oqm¹|noxcgZi\beojm
§
obObeng¶
v
omnÂ`einghnlngZ`£dc!lab
lpma¨©j
v
blnmIr3bgh^¯ bngh^ajmnjkb:~
v j ® o{ag!r
§ r ^bj
v
g~
v
gh`gmtb³rsq_bng
®
rsqoq%~
v
jaigh`e`gh`lp`omn±bengmnjsbeojm
jk¨#lnma¨©jk{aoqmn¡jk¨!beofg
u
gb
v
oPmngbh¯:rk`lp`lprkPoqm¹bnglnmtboqfGgZÁir` g£DH ¨©g
§
^ghr
v
`{r3beg
v
ª²omDsktk¬¯
e
£/fqgo{`inIrki¢rkmp -z£]tbghnmnjoqmMKZ
L
o
®
g!bng#¶
v
` bwmnjsbeojmjs¨[r±¶pmnobg*~
v
g¶n·%jk¨Ibeng*lnma¨©jqnomnjk¨
rboqfg u gb v o(mngbh£p¾nghov fgbenja v ghoqgh`!jkm¼r%b v rsmp`{r3boqjkm"bj
§ r v n`!rsm"j v nompr v ^~pqrig(g b v rsmp`oboqjkm
mpgbh£(¾no{` v g !tlnov gh`bj¼ijmp`oqag v jmnq^¡aoq`ei v gbegbeofgrsmp¡bj gm_lnfg v r3beg%rkbpgG`obelpr3beojmp`h£X¾poq`
rkq`j v gqoqgh`jkm"bngoqmtb v j_nlpi\beojmjk¨mng
§ b v rsmp`oboqjkmp`h¯
§
noqi v g~ v gh`gmtbPbengiqjai¢bo{i¢a`£!H*bpjkln
v gqg ® rkmb¨©j v fjaagh¥inghi¢_oqmnp¯ obwoq`:mpjsbwiqghr v bepr3bwobPrsqqj
§
`)lI`bj v gZij ® g v irslI`rkoboqgh`:rsmI%ijkmpil v
v gmIioqgh`h¯Br` v g!tlnov gh¡oqmbngao{rskmpj`oq`±rs~n~po{irsboqjkm[£`fnl vbeng v fj v gk¯
§
gr v g%ijkm ® omIigh¡bprsbzbeofg
ijkmp` b v rsoqmtbe`Ãflp` b#|Ig±b v gZr3bgZêoqm r`^_f|Bjkqoqi
§ r ^¯_lp` oqmn%bngrkmprsqjk%js¨` bersbgi{rk`e`ijkmI`cbv lpiboqjkmp`
jk¨Bc±£$c!g v bpjkfoghlhKin¯ L£
¾ng v gZ`cbzjs¨:bng%~prs~Bg v oq`zj v rkmnojIgh rk`*¨©jkqj
§
`h£B]aghi\beojmWag¶pmngh`*bngGao»Bg v ghmb±omp v ghaoqgmtbe`
jk¨wjkl v fjaag«|Irk`ghÂ`ln~Ig v® o{` oqjkm¯pmprkfGgh^ bengGnoqrkkmnjt` o{``gbeln~[¯Ibng%boqfg u gb v omngb±fjaag:rsmI
obe`G~pr
v
beoqrkÃj
v
ng
v
` ghfrkmtbo{i`h£:]_gZi\beojm
+
agh`ei
v
o|Bgh`%beng `^_f|Bjkqoqilnma¨©jqnomnbgZinmno&!tlnglI` gZÁbj
ijkf~nlabegbng` ^_f|Bjkqo{i~
v
j_igh`e` gZ`¯
§
poqiÁ`g
ve®
grk`g·a~nqrkmpr3beojmp`h£c!g¨©j
v
ggmtbeg
v
omn bengêkghmng
v
rs
ihrk`gk¯
§
gijkmI` o{ag
v
beng` oqf~nqgh` birk`gjs¨g·_bghmpagh¨
v
ggRinjoqigbeofg
u
gb
v
o(mngb`K L
£n´Ogijmpiqlpag
oqm¡]_ghiboqjkmPip£
:<1.^3C:90: @M@`0!2$:Z.^@`0 <>.749< 2!0:;<>AZ32$4^@2 @ <>.10!:"8
!" Ë
k$#) Ë 0Ç
gblI`ijmp`oqag v r v ghrkPao{`cb v oq|nlabegh<`^a`cbegf¼¯
§
no{iÁ~ v jaalpigZ`jm¹r¡ko® gm<`gb%js¨*` ghmp`j v `` jfg
g ® ghmb`ª©j v rkqr vf`¬nl v oqmn¼obe`qo¨©g£[´Ogijkmp`o{ag v bengG¨©jkqj
§
oqmnÂ` gbln~¨©j v noqrkkmnjt` o{`¯(r``lnfomp
bepr3b±rs{r v f`r v gmnjsbqj` bh£&%Prki¡`gmI` j v#v gZij v n`#obe`jaihrs)rkqr v f`oqmW`g!tlngmIigk¯
§
noqg v gh`~Bghi\beomp
ihrslp`ersqobc^<ª©o«£g£[bpgjk|p`g ve® gZ`g!tlnghmpigZ`irsmpmnjsbijmtb v rkao{i\bbpg"irklp`rk³rsmpbghf~Ij v rkwj v ag v oqmn
ng¶pmngZomGbeng*fjaagÕ¬\£t¾ngao»Bg v ghmtbÃ` ghmp`j v `:~Ig v ¨©j v f ompng~Bgmpaghmtbq^Grkmprk`^tmIi v jmnjklI` q^k¯3rsmI
r"` oqmnkqg%`ln~Bg ve® oq`j v ijkqgZi\b`*beng v ghij v n`¨v jkfÄbngGao»Bg v ghmb` ghmp`j v `£B¾tlI`zrkmt^ oqmbeg v gZr ® omnjk¨
beng v ghij v n`èv jf ao»Bg v ghmtbz` ghmp`j v `oq`#~Bj`e`o|nqgk¯IrsmpÂihrslp`ersqobeogZ`#rsmpbegf~Ij v rk[j v ag voqmnrsfjkmp
rkqr v f`èv jkf ao»Xg v gmtb` ghmp`j v `!r v g±qj` bh£p¾no{`#r v inobgZi\bl vg±oq`*oqlI`cb v rsbghoqmMfoqkl v gk£
fpj v bengWag ® ghj~nfgmtbêjs¨bpgWg·nrkfG~pg¯
§
gijkmI` o{ag v bprsbêbng` ^a` bgf lnmpng v `ln~Bg ve® oq`ojm
~ vjaalpigh`zbpg`g!tlnghmpigZ`(')')<jmO`gmI` j v H¯rsmp+**Djkm¹` ghmp`j v cz£,±o® ghmWbengGboqfg u gb v o³mngb
fjaaghnjs¨`fol
v
gª²g¨Õb¬¯sbpg*kjtrsajs¨Bbpg*aoqrkkmnjt` g
v
oq`wbjijf~nlabg*rknbng#~Bj`e` oq|nqg#g·a~n{rsmprsboqjkmp`
`nj
§
m9oqm foqkl
v
g¦£-%³·a~nqrkmpr3beojmp` r
v
gW{rs|BgqgZ9~pr
v
beoqrkj
v
ag
v
`h£.%Ãrki9mnjaagOo{`{rs|Bgqgh |_^6r
b v
rkmp`obeojmjs¨[beng
u
gb
v
oXmngbÃfjaag(rkmpêr~Bj`e`o|nqgzpr3bgzko
®
ghm|t^r%` ^_f|Bjkqo{i*g·a~
v
gh`e`ojm[£ jsbo{ig
bepr3bbpgnoqrkkmnjt` g
v
oqma¨©g
v
`)bng!~Ijt``oq|ngirklp`ersqobeogZ`|Bgb
§
gghm%rs{r
v
f`¯Zrk`
§
gq_r`[beng~Ijt``oq|ng!nr3begh`
¨©j
v
ghri¹js¨!bnghf £:¾ng¶
v
` b%rs{r
v
f`/')' rkmp0**¸oqfG~p^bprsbb
v
rsmI` oboqjkmp`2143rsmp0165"r
v
g%¶
v
gZ
b §
o{ig%rsmpÂijkmpil vv gmtbe^£p¾ng{rk` b7)OihrsmÂ|Bgg·_~pqrkomngZ¼|_^"b
§
j"ao»Bg v ghmtb*b v rsmp`oboqjkmp`#oqmWijkmaÀpo{i\b
ª168rsmp 1:93¬\£
; <2
Ç>=zÇ
" Ç k³Ì Ç@? " Ë "
AÂË hn
Ë "
´gaghmnjsbegCB$D
3
bengom ® g v ` g±js¨³r|noxcghi\beojmEB£a´gaghmnjsbegCBFGÁbeng v gh` b v oqiboqjkm js¨r
frk~n~noqmn BWbj"r"` gbIH£X¾ng v gh` b v oqiboqjkm¼pr`*noqkng v ~ v oqj v obc^"bprkm¼bng%om ® g v `gkµJB D 3
FGLKNM BFGPOQD
3 £
´OgaghmnjsbegSR%benglp`lprk)ijf~Ijt` oboqjkmÂjk¨¨©lpmpi\beojmp`£T aghmnjsbegh`#beng%`gbzjs¨³mnjkmpmngtr3bo® g v r3beojmprs
m_lnf%|Ig v `h£
¾oqfg u gb v o[mngbe`
§ g v g±oqmtb v jaalpigZ"oqmK L£
H #[^\5#[D# ª²¾ u z¬Poq`!rblp~ngVU
KXWY-Z4[7Z:\^]4_@Z:\J`badceZb_fegZihfeg&j
§
ng v g Y rkmp
[ r v g¶pmnobg
`gb`jk¨
kl/
rkmp
#R4$#R$(v gZ` ~Bghibo
®
gq^k¯
\]4_
rkmp
\J`badc
frs~ghrkib
v
rsmp`obeojm/1.m
[
bejobe`
"XX#
jk¨Õbghm<agmnjkbgZond1/p:q;r
K\]4_sM 1OSt
YuM ne1>v
Kxw
Orkmp¹obe` $(#[X# js¨Õbghm<agmpjsbgZ014nyp:q;r
K\`zadciM 1O{t
YS|
_}feg+~[&
T rsmI
hfeg~[
T %r``jaio{r3begbng QdkX#s TVikl&
_fegM
1Ozrsmp kl#W#
%@ T/ekl&
hfegM 1O
§
obOghrki¡b v rsmp`obeojmE1\£`H ¾ u o{` vgh~ v gh`gmtbgZÂrk`±r" v rk~n
§
ob¡b
§
jêbc^_~Bgh`
jk¨mnjaagZ`µ_~n{rkigh`±ª²iov iqgh`¬Prkmpb v rkmp` oboqjkmI`zª²|pr v `e¬£t¾pg±ijt` gZêoqmbeg ve® rsP
_}fegM 1O
ZihfegM 1O:[oq`
§ v obbghm
mpghr v ghri¹bv rsmp`obeojm[£ fnj v bng ~nl v ~Bj`gêjs¨`ln~Bg ve® o{`ojm[¯
§
gijkmp`oqng vkekkj T #[\5X#R]#
Diagnoser
System under super- vision
αΥαΥ
ββ
Sensor B Sensor
A
P1 ● P2 ●
P3 P4
t1
t3
t2 t4
[0,∞[
α
[2,2]
γ
[1,2]
[0,0] β γ
t1 t2
t4
t1 t2
t3
e1(α)
e2(β)
e3(γ)
e4(α)
e5(β) e6(γ)
t1 t2
t4
t1 t2
t4
Explanations
P5
e1(α)
e2(β)
e3(γ)
e4(α)
e5(β) e7(γ)
"! #$%&('*)+,-.'/10 ('234- #,! 5'#,3687"9'9):;!9'<
WU Z9=IZ*>j
§
ng v g =
oq`!r`gb!js¨g ® ghmbÃbc^_~IgZ`zª²j v rs{r v f`¬\¯trsmp
>
bengzbc^t~pomn%js¨(b v rkmp` oboqjkmI`ª'
Z * Z)
oqm fol v g ¬\£
H ##) js¨!rboqfg u gb v ompgboq`zko® ghm¡|_^¡rb v oq~ng
W1? Zbg`"@Z9A j ¯ §
pg v g ? t Y
o{`r y9B@
ngmnjkbgh
§
ob¼bejk¢gmp`ª©bno{i¢¼ajsb`e¬¯
A
m+T o{`*obe`znr3begrsmp
g `@ ~;? &
T°r``jaio{r3bgZ`#r T#)
#Y g `@@M,C OED
A §
obghribjk¢gmWª²fr v ¢kgh~pqrigZ¬
C m ?
£CH¸b v rkmp`obeojmy1.m
[
o{` ekJT om"bng
` bersbg
W(? Z g`"@ Z#A j
o¨wrsq(js¨:ob`*oqmn~nlabz~nqrigZ`#r v g±fr v¢gh(µ&ne1-t
?
£p¤Rbe` T #WZ iks
g` _sM 1O#oq`
bengnr3begzjk¨|novbe"js¨bng±^kjlnmnkgZ`cbÃbj¢kgm"om ob`!omn~plab!~pqrigh`hµ
g` _sM 1O p:q;r
KGFIHJLKMLNOg`"@sM,C
O\£!H*qXbng
beofg u gb v oXmngb`
§
gzijkmp`o{ag v om"bno{`r v bo{iqgzr v g iX ¯ao«£g£_omghrki v ghriprs|pg` bersbg
W1? Zbg`@ Z9A j
¯_o¨
r%b v rkmp` oboqjkm>1Po{`#ghmprs|nqghoqm
W(? Z g `@@Z#A@j
¯abengm 14nQP
M1?SR ni1O
K w £
H ~ v jaigZ``Ãjs¨:rG¾ u ` ber v b`!om¼rsm $#Rzk`#:#W
W1?UT Z g `@T Z9AT j ¯ §
poqi oq`o® gm|t^beng $#Rzk
y9B@
? T
rsmp%bpg#omnobo{rsnpr3bg
A T
£3¤¥mnobeoqrkq^k¯rsq_bngbejk¢kghmp`³ir
vv
^±bng*pr3bg
A T
rk`³pr3bgjk¨B|no
v
b[µ
V C m ? T g `@ T MWC O p6q;r
KGA T £
¾ngb v rkmp`obeojm 1!irsm ¶ v gr3b*nrsbg
AXZY[A
¨v jf ` ber3beg
W1? Z g `@@Z9A j
¯no¨cµ
\
1Ão{`gmprk|nqgh(µ&ni1 t
?|
\
bngfoqmnof%lnf ng{r ^êoq` v ghringh[µ
AXZY g` _sM 1O&]
_fegM 1O
|
\
bngghmprs|nqgh"b v rsmI` oboqjkmp`#njGmpjsb*j ® g v brs¢g*bpg±frs·_oqflpf aghqr ^a`hµ
V 1 X m [ ne1
X t ? K_^ AX
D
g` _sM 1 X
O&]
hfegM 1X
O£
¾ng¶
v
oqmn±js¨^1)r3bwnrsbg
AX
qghrn`bj±beng*`cbr3beg
W4M1?`R nd1Oz 14n
Z g`"@
X
Z9AX;j
¯ §
ng
vg g `@
X
MWC O p6q;r
K g`"@sM,C
O
o¨
C m
?SR nd1rsmI
g `@
X
M,C O p6q;r
KGAX o¨
C
m>14ns£
foqmprsq^
§
grk`e` lnfgbepr3b#beofg
T(}
µ §
ngmÂoqma¶pmnobgh^ frsm_^êb
v
rkmp`obeojmp`!¶
v
gk¯nbeofgmpghigZ`c
`er
v
oq^êao
® g v
gh`Ãbjoqma¶pmnobc^k£
¤¥mÂbengoqmnobo{rs`cbr3bg%js¨:bpgmngbzjs¨foqkl
v
gêk¯
C 3 rsmp
C 5 r v gfr
v
¢kgZ rkmp¼bengo
v
nrsbgjk¨w|no
v
be
o{`*p£143rsmI 165r v ggmprk|ngZÂrsmpbengov nrsbgjk¨:ghmprs|poqmno{`bngomnobo{rsnr3begn£165irsm¼¶ vgoqm¼bng
oqmnobo{rs!`cbr3bgr3bGrkm_^¡boqfg"|Bgb
§
gghm¸"rsmp<n£ -Ãnj_jt` gbeofgk£DH*¨Õbg v bpoq`¶ v omn
C
3rkmp
C 9 r v g
fr v ¢gh(¯z143!o{`³beng*jmn^%gmprk|nqghGbv rsmI` oboqjkmrsmI%obÃIrk`wrkv ghra^
§
rsobgh #beofg*lpmnobZ£143ihrsm¶ v g*r3b
rkm_^boqfg
A
¯k~ v j ® o{agh%obÃoq`³ v gZr3beg v bIrsm¼k£ -Ãjmp`oqag v 143P¶ v gh`wrsb³boqfg + £ C 8 rsmI
C 9 r v g#fr v ¢kgh%oqm
beng±mng
§
` bersbg¯_rkmpb v rkmp`obeojmp`1 8 rsmI{19 r v gghmprs|nqgh[¯_rkmpbengov nr3begzjk¨gmprk|nqomnGo{` + |IgZirklp` g
beng^pr ® g!xclp` b|BgghmÂgmprk|ngZ |t^"bng¶ voqmnjk¨1 3 £I¾)j¶ v g¯168
§
jkln{pr ® gbej
§
rkobz%beofglnmpob`£
c!labPb v rsmp`obeojmS1:9zirkmnmnjkb
§
rkobÃrsbÃrsq«£t]_j/1:9
§
oqpmngZigZ``er v oq^¶ v gªLr3bwbeofg + ¬\¯trsmp 168ihrsmnmpjsb³¶ v gk£
9Xy9B ¾ng`gfrsmtbeoqih`#js¨:boqfg u gb v o)mngb`r v gjk¨Õbgmag¶pmpgh¼oqm¡rê` qoqktbq^ no»Xg v gmtb
§ r ^Xµabng
` bersbgzjs¨(bpgzmngbÃo{`!ko® gmêr`Ãr~prsov
W(? Zsj
¯ §
ng v g ?
oq`Pbngzfr v ¢toqmnI¯rsmI
frk~p`Ãghrkighmprs|pgZ
b v rkmp`obeojmE1#bj"bngaghqr ^ bIr3bprk`zg{rs~p`gh` oqmpigob
§
rk`zgmprk|nqgh(¯Ibprsb±o{`
AC g ` _sM 1O
§
objkl v
mpjsbersboqjkmp`h£w¤Rb"o{`êfj v g¡ijkm ® ghmnoghmtb¨©j v lI`bj<rsb brki6boqfg¡oqma¨©j v fr3boqjkm6jm6bngWbjk¢gmp`êjs¨bng
fr v ¢_oqmn bprkmÁjkm¹bpgêgmprk|nqghb v rsmp`oboqjkmp`h£[´Ogêpr ® gêipj`gmObngnr3begjs¨#|nov beÁjs¨bngêbjk¢gmp`
v r3beng v beprsmObengov rsgk¯[|Bghihrslp`g
§ g §
rsmtbbejÂfrk¢kgGbpgêof~pri\bjk¨Ãbeng¶ v omp jk¨Ãb v rkmp`obeojmp`rk`
qjairk!r`%~Bj`e`o|nqgk£>HmpÁbengÂrskgjs¨gZrkiÁbejk¢gm¦oqm<beng¼fr v ¢toqmnflp` bG|Bg ln~Bpr3bgZ¦gZrkiÁbeofg
rWb v rsmI` oboqjkm 1¶ v gh`h¯
§
ng v ghr`beng¡nr3begjk¨z|nov bDprk`%bj¹|BgÂ` gbGjmn^<¨©j v bng bejk¢kghmp`%bprsbêr v g
i v gZr3begh¼|_^ 1\£fpl v bng v fj v gk¯Blp`lprs:`gfrsmtbo{i`js¨Õbegmnghrs
§
ob¡bengag{r ^¼|Igb
§
ghgmÂbeng¶ v oqmn"jk¨
b §
jêijmp`ghilnbo® gb v rsmI` oboqjkmp`h£n¤¥m¼benoq`~prs~Bg v §
glI` gbengrk|p` jlnbg¶ v oqmnênr3begjs¨:bengbv rsmI` oboqjkmp`
oqmp` bgZrk(£t¾noq`P¶nb`P|Bgbbg v bjjkl v rs~n~ v jriom
§
poqi
§
gzr v g#mnjkbÃoqmtbg v gh` bgZomêbngbjsbrsBj v ag v oqmn
jk¨bengg ® gmtbe`h£
= a
0 Ë 3Ì Ç " Ç <"X
k
= Ë Ç h Ç ´Og
§
oqq:ng¶pmng%bpgG`gb js¨ª©¶pmnobgZ¬z~ v jaigZ``gh`js¨Pr"`er3¨©g%boqfg u gb v o)mngb` ber v beomp
rsbznrsbg
AT
oqmÂbengoqmnobo{rsfr
v
¢_oqmn
?UT
£B¾ngZ` g~
v
jaigh`e` gZ`*r
v
gbnjt` g%agh`ei
v
o|Bgh¡om Kq
L
£I´Og%ng¶pmng
bengf omIalpi\beo
®
gq^%rkmplI` g*rirkmnjkmpoqihrsaijaaoqmnoq¢kg#oqmMK
L
£k¾pg#~
v
jaigh`e` gZ`)~
vj ®
o{ag!r~Ir
v
bo{rs_j
v
ag
v
v g~ v gZ` ghmbr3beojm"js¨)bngg·aghilnboqjkmp`h£
%Pri%~ v jaigZ``
§
oqqa|Ig*r±~prkov p:q;r
K WyZ/j
¯ §
pg v g
o{`wr` gb:jk¨ ($#[ ¯srsmp
X~
T frs~p`
gZrki¹g ® gmtbbjWob`¶ v ompWnr3begk£
o{``jkfgboqfGgZ` v g~ v gh`gmtbeghOrk`rÂ`gb%js¨#~prsov `
M@ZSM
OO\£%Pri
g ® ghmb
o{`#r~Irsov Mn ZM
OOÃbepr3b*ij_ngh`rsm jaiil vvgmpigjk¨bpgb v rsmp`obeojm
M
O!oqmbng~ v jaigh`e`h£n
o{`r"`gbjs¨P~Irsov ` p:q;r
K M n
Z6\hQ_M eO4O(m
EY
£[]_lpiOrê~prkov o{`zihrsqgZWr T#[ rkmp v g¨©g v `bj
beng*bejk¢gmbepr3b!prk`Ã|Ighgm i v ghr3begh|_^Gbngzg ® gmtb noqmêbpg~n{rkig
\hQ_sM
eO£_´g±`r ^%bprsbÃbengg ® ghmb
p:q;r
KuM
n ZM
O4O
!
bengijkmIaobeojmp`±om n
£[]_^_ffgb
v
o{irkq^¼bng`gbS
M@ZC O
C m M
O n
Gjk¨
ijkmpnobeojmp`!bprsbr v g "Q#WT |_^
o{`#agmpjsbgZ
ns£
fpj v rsqÃ`gb! 2js¨ijkmpaoboqjkmI`¯
§
g agmnjkbg#"
hQ_sM O p:q;r
K
\hQ_sM
eO$ m% yt¯rkmp
§
nghmÁbng
v
gh` b
v
o{i\boqjkmGjs¨
\hQ_
bj& o{`³omsxcghi\beo
®
gk¯
§
g*aghmnjsbeg
\h _ D 3 F'
obe`³oqm
® g v
`gk¯srsmI¨©j
v
rsq
Y
t("
hQ_sM O\¯
"
hQ_
D 3
F' MY O p6q;r
K
\^hQ_
D 3
F' M,C O)*Im
Y
t£
¾ng½` gbWjs¨êijmpaoboqjkmp`Âbprsb v gfrsoqm r3b¡beng<gmI js¨bng¦~ v jaigh`e`
WyZ/j
ª²fGgZrsmnoqmn¸bepr3b
beng^pr ® g¹|Bgghmi v ghr3begh|_^rsm g ® gmtb¡jk¨
¯zrsmp mnj¸g ® gmtb¡jk¨
pr`Âijmp` lpfGgZbnghf¬Âoq`