- Email: [email protected]
CSP GENERATION FROM PETRI-NETS MODELS
Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain
www.elsevier.com/locate/ifac
%52 )'0'4#6+10 (41/ 2'64+0'65 /1&'.5 &CPKGN 4KGTC /KSWGN # 2KGTC #PVQPK )WCUEJ 'PIKP[GTKC FG 5KUVGOGU
K #WVQO CB VKEC 7PKXGTUKVCV #WVBQPQOC FG $CTEGNQPC $CTEGNQPC 5RCKP f&CPKGN4KGTC/KIWGN#PIGN2KGTCg"WCDGU +PUVKVWVQ FG 4QD Q y VKEC G +PHQTOCy VKEC +PFWUVTKCN 72%%5+% $CTEGNQPC 5RCKP )WCUEJ"GUCKKWREGU
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
-G[YQTFU /GVJQFQNQI[ 2GVTKPGVU %QPUVTCKPV 5CVKUHCEVKQP 2TQDNGOU 2NCPPKPI 2TQFWEVKQP U[UVGOU
+0641&7%6+10
U[PEJTQPK\CVKQP CPF TGUQWTEGU UJCTKPI YJKEJ CTG VJG OCKP EJCTCEVGTKUVKEU QH C (/5
+P VJG NCUV HGY [GCTU OCP[ OGVJQFU CPF VQQNU JCXG DGGP FGXGNQRGF VQ KORTQXG RTQFWEVKQP RGTHQTOCPEG KP VJG OCPWHCEVWTKPI KPFWUVT[ 6JGUG CRRTQCEJGU VT[ VQ VCEMNG EJCPIGU KP RTQFWEVKQP QDLGEVKXGU UWEJ CU JKIJ RTQFWEVKQP FKXGTUKV[ KPUVGCF QH JKIJ RTQFWEVKQP XQNWOG OCMG VQ QTFGT KPUVGCF QH OCMG VQ UVQEM CPF \GTQ UVQEM KPUVGCF QH LWUV KP VKOG RQNKEKGU #NVJQWIJ 14 1RGTCVKQPU 4GUGCTEJ OGVJQFQNQIKGU JCXG DGGP RTQXGF VQ RGTHQTO YGNN KP HTQPV QH UQOG V[RGU QH RTQDNGOU VJG[ HCNN UJQTV VQ VCEMNG RTGUGPV ƀGZKDNG OCPWHCEVWTKPI UEJGFWNKPI RTQFWEVKQP FGOCPFU
6JWU 20 HQTOCNKUO KU WUGF VQ URGEKH[ CNN VJG GXGPVU VJCV NGCF VQ C U[UVGO UVCVG EJCPIG GI DGIKPPKPI QT GPFKPI QH C RTQFWEVKQP CEVKXKV[ VQIGVJGT YKVJ CNN VJG RTGEQPFKVKQPU VJCV UJQWNF DG UCVKUſGF KP QTFGT CP GXGPV EQWNF QEEWT GI TCY OCVGTKCN CPF TGUQWTEGU TGSWKTG OGPVU GVE 0QVG VJCV C 20 OQFGN QH C RTQFWEVKQP U[U VGO KU GPQWIJ VQ UKOWNCVG KVU DGJCXKQWT YJKEJ OGCPU VJCV VJG OQFGN HQTOCNK\GU DQVJ
#TEJKVGEVWTG 2TQFWEVKQP %QPUVTCKPVU #WVQOCVGF VTCPURQTV WPKVU KG EQPXG[QTU TQDQVU OCPKRWNC VQTU GVE DKPF VJG OCVGTKCN ƀQY KP VJG RTQFWEVKQP U[UVGO .QIKUVKE 2TQFWEVKQP %QPUVTCKPVU &GURKVG ƀGZKDNG RTQFWEVKQP OGEJCPKUOU CNNQY FKHHGTGPV UEJGFWNKPI RQNKEKGU RTQFWEV TGEKRGU TGFWEG UWD U[UVGO KPVGTCEVKQP FWG VQ UGXGTCN HCEVQTU UWEJ CU VJG QRGTCVKQP UGSWGPEG KP YJKEJ VJG TCY OCVGTKCN OWUV DG RTQEGUUGF
2GVTKPGVU 20 JCXG UJQYP VQ DG UWEEGUUHWN VQQNU HQT OQFGNNKPI (NGZKDNG /CPWHCEVWTKPI 5[UVGOU (/5 FWG VQ UGXGTCN CFXCPVCIGU UWEJ CU VJG EQPEKUGPGUU QH GODQF[KPI DQVJ VJG UVCVKE UVTWEVWTG CPF VJG F[PCO KEU VJG CXCKNCDKNKV[ QH VJG OCVJGOCVKECN CPCN[UKU VGEJ PKSWGU CPF KVU ITCRJKECN PCVWTG ,GPUGP 5KNXC CPF 8CNGVVG
241
: (P T ) ! N KU C HWPEVKQP VJCV FGſPGU VJG YGKIJV QH FKTGEVGF CTEU HTQO RNCEGU VQ VTCPUKVKQPU O : (T P ) ! N KU C HWPEVKQP VJCV FGſPGU VJG YGKIJV QH FKTGEVGF CTEU HTQO VTCPUKVKQPU VQ RNCEGU M0 KU VJG KPKVKCN OCTMKPI
%QPUVTCKPV 2TQITCOOKPI %2 JCU DGGP OCKPN[ EJQUGP DGECWUG QH KVU IQQF QRVKOKUCVKQP RGTHQTOCPEG 5KPEG %2 KU WUWCNN[ GODGFFGF KP FGENCTCVKXG RTQITCOOKPI VJG WUGT FQGU PQV PGGF VQ YTKVG CP CNIQTKVJO VQ UQNXG VJG RTQDNGO DWV QPN[ VQ OQFGN VJG RTQDNGO VQ DG UQNXGF 6JGTGHQTG QPEG VJG OQFGN KU IGPGTCVGF %2 ECP QRVKOKUG KV YKVJQWV TGSWKTKPI CP GZRGTV VQ TWNG KV
I
# OCTMKPI KU CP CTTC[ VJCV CUUKIPU VQ GCEJ RNCEG C PQP PGICVKXG KPVGIGT +H C OCTMKPI CUUKIPU VQ RNCEG p C XCNWG + k k 2 Z p KU OCTMGF YKVJ k VQMGPU TGRTGUGPVGF ITCRJKECNN[ D[ DNCEM FQVU
+P VJKU RCRGT VJG OCKP CURGEVU QH C 20 CPCN[UKU OGVJQFQNQI[ VQ FGCN CWVQOCVKECNN[ YKVJ CNN VJG EQP UVTCKPVU QH C RTQFWEVKQP U[UVGO CTG RTGUGPVGF 6JG CKO QH VJKU OGVJQFQNQI[ KU VQ CWVQOCVK\G VJG EQPUVTCKPVU OQFGNNKPI RJCUG QH C RTQFWEVKQP U[UVGO ō FGUETKDGF RTGXKQWUN[ KP 20 HQTOCNKUO ō KP QTFGT VQ IGPGTCVG C SWCUKQRVKOCN UEJGFWNKPI HQT C RCTVKEWNCT U[UVGO UVCVG CPF RTQFWEVKQP IQCN OCMKPI VJG QRVKOKUCVKQP RJCUG VTCPURCTGPV VQ VJG WUGT 5GG (KI
# VTCPUKVKQP ti KU UCKF VQ DG GPCDNGF D[ C OCTMKPI M KH 8p 2 P : M (p) I (p t i ) 6JG ſTKPI VTCPUKVKQP IGPGTCVGU C PGY OCTMKPI M 0 YJKEJ ECP DG EQORWVGF D[ YKVJFTCYKPI I (p k ti ) VQMGPU HTQO GCEJ p k KPRWV RNCEG QH ti CPF D[ CFFKPI O(p j ti ) VQMGPU VQ GCEJ p j QWVRWV RNCEG QH t i +P OCPWHCEVWTKPI VGTOU VTCPUKVKQPU CTG WUGF VQ OQFGN QRGTCVKQPU ſTKPI C VTCPUKVKQP ECP TGRTGUGPV C VCUM QT RTQEGUU KPKVKCVKQP QT CP GPFKPI QH C VCUM RNCEGU CTG WUGF VQ OQFGN DWHHGTU CPF TGUQWTEGU UVCVWU EQPPGEVKPI CTEU URGEKH[ NQIKECN TGNCVKQPUJKRU CPF TGUQWTEG EQP UVTCKPVU COQPI QRGTCVKQPU CPF VQMGPU TGRTGUGPV OC VGTKCN CPF TGUQWTEGU EQPFKVKQPU
(KI 2TGUGPVGF OGVJQFQNQI[ +V UJQWNF DG PQVGF VJCV FGURKVG VJGTG CTG UGXGTCN 20 CPCN[UKU VQQNU VQ KORTQXG U[UVGO RGTHQTOCPEG VJQUG ECP DG WUGF QPN[ YJGP VJG 20 OQFGN UJQYU EGTVCKP EJCTCEVGTKUVKEU YJKEJ KU PQV CNYC[U VJG ECUG QH 20 OQFGNU QH RTQFWEVKQP U[UVGOU 1P VJG QVJGT JCPF FGURKVG %2 QHHGTU C RQYGTHWN VGEJPQNQI[ VQ FGCN YKVJ VJG DGUV RNCPPKPI RQNKE[ HQT C RTQFWEVKQP U[UVGO KV TGSWKTGU C FGGR MPQYNGFIG KP DQVJ VJG RTQFWEVKQP U[UVGO CPF VJG %2 VGEJPQNQI[ CPF VKOG VQ FGXGNQR VJG %2 OQFGN 6JWU VJG RTGUGPV YQTM VTKGU VQ EQODKPG VJG OCKP CFXCPVCIGU QH DQVJ OGVJQFQNQIKGU KP QTFGT VQ QHHGT IQQF UEJGFWNKPI RQNKEKGU OKPKOK\KPI VJG VKOG URGPV KP VJG OQFGNNKPI CPF QRVKOK\CVKQP VCUMU
%QPUVTCKPVU CTKUG KP OQUV CTGCU QH JWOCP GPFGCXQWT # EQPUVTCKPV KU UKORN[ C NQIKECN TGNCVKQP COQPI UGX GTCN WPMPQYPU QT XCTKCDNGU GCEJ VCMKPI C XCNWG KP C IKXGP FQOCKP 6JG EQPUVTCKPV VJWU TGUVTKEVU VJG RQU UKDNG XCNWGU VJCV XCTKCDNGU ECP VCMG %2 KU VJG UVWF[ QH EQORWVCVKQPCN U[UVGOU DCUGF QP EQPUVTCKPVU 6JG OCKP KFGC KU VQ UQNXG RTQDNGOU D[ UVCVKPI EQPUVTCKPVU TG SWKTGOGPVU CDQWV VJG RTQDNGO CTGC CPF EQPUGSWGPVN[ ſPFKPI C UQNWVKQP UCVKUH[KPI CNN VJG EQPUVTCKPVU
6JG HQWPFCVKQPU HQT VJG CPCN[UKU QP C 20 OQFGN CPF VJG IGPGTCVKQP QH VJG %2 OQFGN CTG RTGUGPVGF 6JKU CPCN[UKU KU OCFG D[ FGVGEVKPI UVTWEVWTGU KP VJG 20 YJKEJ CTG VTCPUNCVGF FKTGEVN[ KPVQ VJG %52 EQPUVTCKPVU
6JG GCTNKGUV KFGCU NGCFKPI VQ %2 OC[ DG HQWPF KP VJG #TVKſEKCN +PVGNNKIGPEG #+ YKVJ VJG UEGPG NCDGNNKPI RTQDNGO 9CNV\ CPF VJG KPVGTCEVKXG ITCRJKEU
5WVJGTNCPF
+P UGEVKQPU CPF 20 CPF %2 DCEMITQWPFU CTG RTG UGPVGF 5GEVKQP KPVTQFWEGU VJG RWTRQUGF OGVJQFQN QI[ 5GEVKQP KNNWUVTCVGU VJKU OGVJQFQNQI[ D[ OGCPU QH CP GZCORNG (KPCNN[ 5GEVKQPU CPF FKUEWUU VJG DGPGſVU CPF RTGUGPV VJG EQPENWUKQPU CPF HWVWTG YQTM
)CNNCKTG CPF ,CHHCT CPF .CUUG\ PQVGF VJCV NQIKE RTQITCOOKPI YCU LWUV C RCTVKEWNCT MKPF QH %2 6JG DCUKE KFGC DGJKPF .QIKE 2TQITCOOKPI .2 CPF FGENCTCVKXG RTQITCOOKPI KP IGPGTCN KU VJCV VJG WUGT UVCVGU YJCV JCU VQ DG UQNXGF KPUVGCF QH JQY VQ UQNXG KV YJKEJ KU XGT[ ENQUG VQ VJG KFGC QH EQPUVTCKPVU
%10564#+06 241)4#//+0) $#%-)4170&
4GEGPV CFXCPEGU RTQOKUG VJCV %2 CPF 1RGTCVKQPU 4GUGCTEJ 12 ECP GZRNQKV GCEJ QVJGT KP RCTVKEWNCT VJG %2 ECP UGTXG CU C TQQH RNCVHQTO HQT KPVGITCVKPI XCTKQWU EQPUVTCKPV UQNXKPI CNIQTKVJOU KPENWFKPI VJQUG FGXGNQRGF CPF EJGEMGF VQ DG UWEEGUUHWN KP 14
2'64+0'65 $#%-)4170& # 2GVTKPGV 20 KU C RCTVKEWNCT MKPF QH FKTGEVGF ITCRJ VQIGVJGT YKVJ CP KPKVKCN UVCVG ECNNGF VJG KPKVKCN OCTMKPI #P QTFKPCT[ 20 KU C VWRNC N = (P TIO M 0 )
6JGP %2 EQODKPGU KFGCU HTQO C PWODGT QH ſGNFU KPENWFKPI #TVKſEKCN +PVGNNKIGPEG %QODKPCVQTKCN #NIQ TKVJOU %QORWVCVKQPCN .QIKE &KUETGVG /CVJGOCVKEU 0GWTCN 0GVYQTMU 1RGTCVKQPU 4GUGCTEJ 2TQITCOOKPI .CPIWCIGU CPF 5[ODQNKE %QORWVCVKQP
=
fp1 : : : pn g KU VJG UGV QH RNCEGU TGRTG UGPVGF ITCRJKECNN[ D[ EKTENGU T = ft1 : : : tm g KU VJG UGV QH VTCPUKVKQPU TGRTG UGPVGF ITCRJKECNN[ D[ DQNF NKPGU QT TGEVCPINGU
P
242
6JG RTQDNGOU UQNXGF WUKPI %2 CTG ECNNGF %QPUVTCKPV 5CVKUHCEVKQP 2TQDNGOU %52 # %52 KU FGſPGF CU
#NIQTKVJO 6TCPUKVKQPU (KTKPIU $QWPFKPI 5VGR +PKVKCNKUCVKQP QH 0 D[ CRRN[KPI TWNGU 6!2 CPF 2!6 5GG DGNQY HQT GCEJ RNCEG CPF VTCP UKVKQP TGURGEVKXGN[ (KTUV 6!2 KU CRRNKGF VQ GXGT[ RNCEG KP VJG 20 CPF NCVGT 2!6 ECNEWNCVGU VJG KPKVKCN ŌXCNWGU HQT GXGT[ VTCPUKVKQP
C UGV QH XCTKCDNGU X
=f
x1 : : : x
sg
HQT GCEJ XCTKCDNG x i C ſPKVG UGV Di QH RQUUKDNG XCNWGU KVU FQOCKP CPF C UGV QH EQPUVTCKPVU TGUVTKEVKPI VJG XCNWGU VJCV VJG XCTKCDNGU ECP UKOWNVCPGQWUN[ VCMG 5WEJ CU $CTVyCM UC[U ő%QPUVTCKPV RTQITCOOKPI KU CP GOGTIGPV UQHVYCTG VGEJPQNQI[ HQT FGENCTCVKXG FGUETKRVKQP CPF GHHGEVKXG UQNXKPI QH JCTF TGCN NKHG RTQD NGOU GURGEKCNN[ KP CTGCU QH RNCPPKPI CPF UEJGFWNKPIŒ
!2
!6
C 6
D 2
2'64+0'6 /1&'.5 61 %2 /1&'.5 (KI )GPGTCN ECUGU HQT RNCEG CPF VTCPUKVKQP HGGFKPI %52 %QORQPGPVU +FGPVKſECVKQP
6!2 ſPFU VJG OCZKOWO PWODGT QH VQMGPU GCEJ RNCEG p 2 P ECP EQPVCKP FWTKPI CNN VJG U[UVGO TWP 6JKU KU ECNEWNCVGF CRRN[KPI VQ VJG IGPGTCN ECUG UJQYP KP (KI C VJG HQTOWNC RTGUGPVGF KP 'S
+P QTFGT VQ CPCN[UG VJG 20 OQFGN CPF IGPGTCVG VJG %52 RTQDNGO VJG ſTUV UVGR KU VQ KFGPVKH[ VJG GNGOGPVU EQORQUKPI VJG %52 6JGUG GNGOGPVU CTG
8CTKCDNGU 6JGTG KU C XCTKCDNG HQT GCEJ ſTKPI QH C 20 VTCPUKVKQP *GPEG GXGT[ VTCPUKVKQP KU CUUQEKCVGF VQ C NKUV QH VKOGU 6JG NGPIVJ QH VJKU NKUV KU IKXGP D[ VJG PWODGT QH VKOGU VJG VTCPUKVKQP KU ſTGF QT C JKIJGT DQWPF KH WPMPQYP 6JGUG CTG ECNNGF VTCPUKVKQP XCTKCDNGU 6JG UGEQPF UGV QH XCTKCDNGU CTG VJQUG TGRTG UGPVKPI VJG PWODGT QH ſTKPIU QH VJG VTCPUKVKQPU 6JGUG XCTKCDNGU CTG ECNNGF ſTKPI XCTKCDNGU #RCTV QH VJGUG VJGTG CTG CNUQ $QQNGCP XCTK CDNGU YJKEJ KPFKECVG VJG RCVJU HQNNQYGF D[ VQ MGPU KP DKHWTECVKQPU &QOCKPU 6JG FQOCKPU QP VJG VTCPUKVKQP XCTKCDNGU CTG FGſPGF WUKPI VJG MPQYNGFIG CDQWV VJG RTQDNGO 7UWCNN[ VJG HCEV QH TGFWEKPI VJG FQOCKPU QP VJG XCTKCDNGU OCMGU VJG UGCTEJ HCUVGT (KTKPI XCTK CDNGU FGRGPF QP VJG DQWPFU HQWPF D[ VJG 6TCPUK VKQPU (KTKPIU $QWPFKPI CNIQTKVJO 5GG #NI %QPUVTCKPVU 6JG[ CTG IGPGTCVGF KP VJG 20 UVTWEVWTGU FGVGE VKQP RJCUG 5GG 5GEVKQP CPF TGUVTKEV VJG RCVJU YJKEJ ECP DG HQNNQYGF D[ GXGT[ VQMGP CPF VJG VKOGU YJGP VTCPUKVKQPU ECP DG ſTGF
( )=
tok p
m X ( )+ i
tokini p
i=1
Oti p
YJGTG tokini (p) KU VJG KPKVKCN PWODGT QH VQMGPU QH RNCEG p KP VJG KPKVKCN OCTMKPI M 0 2!6 KPKVKCNKUGU VJG PWODGT QH VKOGU GCEJ VTCPUK VKQP t 2 T OKIJV DG ſTGF FWTKPI VJG U[UVGO TWP 6JG PWODGT QT GZRTGUUKQP KH VJGTG CTG WPMPQYP XCNWGU KU IKXGP D[ VJG CRRNKECVKQP QH VJG HQTOWNC KP 'S VQ VJG IGPGTCN ECUG UJQYP KP (KI D
( ) = i=1min :::n
t
j
tok(pi )
p
I it
k
j Ipit 6= 0
5VGR 2TQRCICVKQP QH VJG ŌXCNWGU KP i HTQO KVU XCNWGU KP i1 +P VJKU UVGR DQVJ PWODGTU CPF GZ RTGUUKQPU CTG RTQRCICVGF 5VGR 4GUQNWVKQP QH VJG GZRTGUUKQPU KP i +H C GZ RTGUUKQP TGRTGUGPVKPI j EQPVCKPU KVUGNH j KV KU PQV EQPUKFGTGF CPF KU UWDUVKVWVGF D[ KPſPKV[ 1 #HVGT VJKU UKORNKſECVKQP min HWPEVKQPU CTG CRRNKGF KH RQUUKDNG + then go to 5VGR = Z 5VGR if 9j 2 i j j 2 else 'PF
2GVTKPGV 5VTWEVWTGU &GVGEVKQP +P VJKU UVGR VJG 20 UVTWEVWTGU CTG GZVTCEVGF KP QTFGT VQ IGPGTCVG VJG EQPUVTCKPVU CPF TGFWEG VJG UGCTEJ URCEG 'CEJ UVTWEVWTG EQTTGURQPFU VQ C UGV QH EQPUVTCKPVU YJKEJ CTG CWVQOCVKECNN[ IGPGTCVGF
%CNEWNCVKQP QH VJG $QWPFU QH VJG 6TCPUKVKQPU (KTKPIU 5KPEG VJG XCTKCDNGU CTG VJG ſTKPIU QH VJG VTCPUKVKQPU KP VJG 20 KV KU PGEGUUCT[ VQ DQWPF VJG PWODGT QH VKOGU VJG[ JCRRGP 1VJGTYKUG VJG PWODGT QH XCTKCDNGU YQWNF DGEQOG WPOCPCIGCDNG
5VTWEVWTG 4 4 EQPUVTCKPVU 5GG 'S CTG IGPGTCVGF HQT GXGT[ UKPING VTCPUKVKQP KP VJG 20 6JKU KU PQV C UVTWEVWTG DWV C YC[ QH TGOQXKPI U[OOGVTKGU HTQO VJG RTQDNGO 5KPEG VJGTG KU C XCTKCDNG HQT GCEJ ſTKPI QH
#P CNIQTKVJO YJKEJ FGRGPFU QP VJG 20 CPF VJG KPKVKCN UVCVG M0 QH VJG RTQDNGO KU RTQRQUGF
243
C VTCPUKVKQP D[ WUKPI 4 VJG XCNWGU QH VJGUG XCTKCDNGU
CPF JGPEG VJG ſTKPIU CTG IKXGP C WPKSWG QTFGT
t t 1 8j = 2 : : : l ij
ij
length(Ti ) = l
5VTWEVWTG 4 6JKU UVTWEVWTG KPENWFGU VYQ EQPUGEWVKXG VTCPUKVKQPU YKVJ C FKTGEV RCVJ DGVYGGP VJGO 'CEJ ſTKPI VKOG QH VJG GZKV VTCPUKVKQP KU TGNCVGF YKVJ QPG QH VJG UQWTEG VTCPUKVKQP 5GG 'S
(KI 5VTWEVWTG ( VJTQWIJ C WPKSWG RNCEG 5GG (KI $QQNGCP XCTKCDNGU CTG CNUQ PGEGUUCT[ KP VJKU ECUG 5GG 'S
tout (tin + timein) Bin Bdi M 8j = 1 : : : n length(Tin) = n 8i = 1 : : : m length(Tout ) = m
t t + time 8j = 1 : : : l length(t ) = length(t ) = l ij
i
k
kj
i
k
Bdi = diff
YJGTG timek KU VJG RTQEGUUKPI VKOG CUUQEKCVGF VQ VTCPUKVKQP k
i
j
X (j B i
k
ink
=1
)
6JGUG EQPUVTCKPVU CTG PQV IGPGTCVGF KH DQVJ VTCPUKVKQPU
HQTOKPI 4 DGNQPI VQ C OQTG EQORNGZ UVTWEVWTG KG ( , QT (, 5VTWEVWTG +5 6JKU TGRTGUGPVU CP KPKVKCN UVQEM
KG CP KUQNCVGF RNCEG HGGFKPI VYQ QT OQTG VTCPUKVKQPU +5 UVTWEVWTG FQGU PQV CFF VGORQTCN EQPUVTCKPVU DWV EQPUVTCKPVU TGNCVGF YKVJ VJG PWODGT QH ſTKPIU QH VJG GZKV VTCPUKVKQPU 5GG 'S
X tok
( ) 8i j feeds(pin t ) i
ini pin
(KI 5VTWEVWTG , 5VTWEVWTG ,( 6JKU KU VJG OQUV EQORNGZ UVTWE VWTG YJGTG C NKUV QH KPRWV VTCPUKVKQPU HGGFU CP QWVRWV NKUV QH VTCPUKVKQPU 5GG (KI 6JKU UVTWEVWTG CNUQ TG SWKTGU VJG WUG QH $QQNGCP XCTKCDNGU 5GG 'S
i
YJGTG pin KU VJG RNCEG TGRTGUGPVKPI VJG KPKVKCN UVQEM
Tout (Tin + timein ) Bin out Bdi M 8j = 1 : : : n length(Tin) = n
8i = 1 : : : m length(Tout) = m i
5VTWEVWTG ( 6JKU UVTWEVWTG KU HQTOGF D[ C UKP ING VTCPUKVKQP YJKEJ HGGFU C UGV QH VTCPUKVKQPU VJTQWIJ C UKPING RNCEG 5GG (KI # NKUV QH $QQNGCP XCTKCDNGU CTG WUGF VQ UGNGEV VJG RCVJ HQNNQYGF D[ GCEJ VQMGP KP VJG UVTWEVWTG 6JGUG XCTKCDNGU GPCDNG QT FKUCDNG EQPUVTCKPVU FGRGPFKPI QP VJGKT EQTTGURQPFGPEG VQ VJG UGNGEVGF RCVJ 5GG 'S
tout (tin + timein ) Bout Bdi M 8j = 1 : : : n length(Tin ) = n 8i = 1 : : : m length(Tout ) = m i
j
Bdi = diff(i
XXB j
in
k
=1
ink
j
)
j
j
X (i B j
Bdi = diff
k
=1
outk
)
(KI 5VTWEVWTG ,(
YJGTG HWPEVKQP diff TGVWTPU \GTQ 0 KH VJG RCTCOG VGTU CTG GSWCN CPF QPG 1 KH VJG[ CTG FKHHGTGPV CPF M KU CP KPVGIGT VJCV FQOKPCVGU VJG GZRTGUUKQP C ŎDKI/ŏ VGTO
%52 .CDGNNKPI &GſPKVKQP 5KPEG VJG CKO QH VJKU OGVJQFQNQI[ KU VQ ſPF VJG QRVKOWO PQV QPN[ C UQNWVKQP JCU VQ DG HQWPF DWV QRVKOCNKV[ OWUV DG RTQXGF #NVJQWIJ C RQUUKDKNKV[ KU VQ IGPGTCVG VJG EQORNGVG UQNWVKQPU VTGG VJKU YQWNF OGCP C JKIJ EQORWVCVKQPCN EQUV
5VTWEVWTG , 6JKU KU EQORNGOGPVCT[ VQ UVTWE VWTG ( # UGV QH VTCPUKVKQPU HGGF C UKPING VTCPUKVKQP
244
6 4GVKTG ſPCN RKGEG HTQO / JQWTU +PKVKCN OCTMKPI
1P VJG QVJGT JCPF QRVKOCNKV[ ECP DG RTQXGF D[ NC DGNNKPI VJG XCTKCDNG VQ QRVKOKUG ſTUV CPF UGNGEVKPI KVU XCNWGU HTQO DGVVGT VQ YQTUG UQNWVKQPU 6JKU OGCPU VJCV KP VJG OQOGPV C UQNWVKQP KU HQWPF CNN VJG RQUUKDNG DGVVGT UQNWVKQPU JCXG DGGP CNTGCF[ TGLGEVGF 1RVKOCNKV[ KU RTQXGF CPF HWTVJGT UGCTEJ KU PQV PGEGUUCT[
M0
= 01
5 0 0 1 0 0 1 0 0]
6TCPUKVKQP CPF $QQNGCP XCTKCDNGU CTG NCDGNNGF QPN[ QPEG HQT GCEJ KPUVCPEG QH VJG XCTKCDNG VQ QRVKOKUG CP[ QH VJGKT HGCUKDNG XCNWGU IKXGU CP QRVKOCN UQNWVKQP 2CVJU CTG UGNGEVGF D[ NCDGNNKPI $QQNGCP XCTKCDNGU ſTUV 6TCPUKVKQP XCTKCDNGU CTG NCDGNNGF NCVGT
':#/2.' 'ZCORNG 2TQEGUU CPF #UUGODNG (CEVQT[ 6JG U[UVGO KU EQORQUGF D[ VYQ OCEJKPGU YJKEJ RGT HQTO VJG PGZV QRGTCVKQPU / OCMGU VYQ FKHHGTGPV QR GTCVKQPU FGRGPFKPI QP VJG V[RG QH RKGEG KV KU YQTMKPI YKVJ 6JKU KU C UJCTGF TGUQWTEG *GPEG KV KU PGEGUUCT[ VQ RNCP VJG QTFGT VJG RKGEGU CTG RWV KPVQ KV 1P VJG QVJGT JCPF / KU WUGF VQ LQKP VYQ RTQEGUUGF RKGEGU # CPF $ 6JG U[UVGO ECP DG UGGP KP (KI
(KI 6JG 2GVTKPGV QH VJG U[UVGO
6JG CKO KU IKXGP RKGEGU QH V[RG # CPF RKGEGU QH V[RG $ VQ RGTHQTO 5 RTQEGUUGF CPF CUUGODNGF RKGEGU
6TCPUKVKQPU (KTKPIU $QWPFKPI +P QTFGT VQ ECNEWNCVG VJG JKIJGT DQWPFU QH VJG VTCPUK VKQPU ſTKPIU #NIQTKVJO KU CRRNKGF VQ VJG 20
0 = min(10 1 + 3 ) min(5 4 ) 1 2 min(3 4 1 + 6 ) 5 ] 1 = min(10 1 + 1 ) min(5 2 ) min(10 1 + 3 ) min(5 4 ) min(1 2 1 + 5 ) min(3 4 1 + 6 )] = = min(10 ) min(5 ) min(10 ) min(5 ) min(1 2 ) min(3 4 )] = = 01 5 10 5 min(1 2 ) min(3 4 )] = 2 = 01 5 10 5 min(10 5) min(10 5)] = 01 5 10 5 5 5] =
(KI 6JG UVWFKGF U[UVGO
1 1
2GVTKPGV QH VJG 5[UVGO 6JG 20 QH VJG UVWFKGF U[UVGO 5GG (KI JCU VJG PGZV EQORQPGPVU 2NCEGU 2 5VQEM QH RKGEGU # 2 5VQEM QH RKGEGU $ 2 2KGEG # DGKPI RTQEGUUGF KP / 2 2KGEG $ DGKPI RTQEGUUGF KP / 2 / HTGG 2 5VQEM QH RKGEGU # RTQEGUUGF 2 5VQEM QH RKGEGU $ RTQEGUUGF 2 / HTGG 2 / CUUGODNKPI C ſPCN RKGEG 2 5VQEM QH CUUGODNGF RKGEGU 6TCPUKVKQPU 6 /QXG RKGEG # VQ / JQWT 6 /QXG RKGEG $ VQ / JQWT 6 4GVKTG RKGEG # HTQO / JQWTU 6 4GVKTG RKGEG $ HTQO / JQWTU 6 /QXG RTQEGUUGF RKGEGU # CPF $ VQ / JQWT
1
1
1
1
5VTWEVWTGU (QWPF 6JG UVTWEVWTGU HQWPF CPF CWVQOCVKECNN[ VTCPUNCVGF KPVQ EQPUVTCKPVU CTG CU HQNNQYU 4 =6? =6? =6? =6? =6? CPF =6? 4 =6 6? =6 6? CPF =6 6? +5 ( , =6 6 6 6? ,( =6 6 6 6?
!
!
!
! !
4GUWNVU 6JG ſTKPI VKOGU HQWPF CHVGT VJG QRVKOKUCVKQP RJCUG CTG VJG PGZV
245
5KNXC / CPF 4 8CNGVVG 2GVTK PGVU CPF ƀGZKDNG OCPWHCEVWTKPI .GEVWTG 0QVGU KP %QORWVGT 5EK GPEG XQN #FXCPEGU KP 2GVTK 0GVU RR Ō 5WVJGTNCPF + 5MGVEJRCF C OCPOCEJKPG ITCRJKECN EQOOWPKECVKQP U[UVGO 2TQE +(+2 5RTKPI ,QKPV %QORWVGT %QPHGTGPEG 9CNV\ &. 7PFGTUVCPFKPI NKPG FTCYKPIU QH UEGPGU YKVJ UJCFQYU KP 2U[EQNQI[ QH %QORWVGT 8KUKQP /E)TCY*KNN 0GY ;QTM 9CPI . CPF 5 9W /QFGNKPI YKVJ EQNQTGF VKOGF QDLGEVQTKGPVGF RGVTK PGVU HQT CWVQOCVGF OCPWHCEVWTKPI U[UVGOU %QORWVGTU CPF +PFWU VTKCN 'PIKPGGTKPI Ō
# ECUG UVWF[ KP OQFGNKPI CPF RGTHQT OCPEG GXCNWCVKQP QH OCPWHCEVWTKPI U[UVGOU WU KPI EQNQTGF RGVTK PGVU 2TQEGGFKPIU QH VJG VJ 'W TQRGCP 5KOWNCVKQP 5[ORQUKWO '55 ŏ
6 = ? 6 = ? 6 = ? 6 = ? 6 = ? 6 = ? 6JGP VJG QRVKOCN VKOG HQT VJG RGTHQTOCPEG QH RKGEGU
RTQEGUUGF CPF CUUGODNGF KU 44 JQWTU T 6 5 +time6 $'0'(+65 1( 6*' 274215'& /'6*1&1.1); 6JG RTQRQUGF CRRTQCEJ KORTQXGU UGXGTCN CURGEVU QH CEVWCN UEJGFWNKPI VQQNU UQOG QH YJKEJ CTG
6JG URGEKſ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ſTUV CPF URGEKCNK\KPI NCVGT 4'('4'0%'5 $CTVyCM 4 %QPUVTCKPV RTQITCOOKPI +P RWTUWKV QH VJG JQN[ ITCKN 2TQEGGFKPIU QH 9&5 KPXKVGF VCNM )CNNCKTG * .QIKE RTQITCOOKPI (WTVJGT FG XGNQROGPVU +''' 5[ORQUKWO QP .QIKE 2TQ ITCOOKPI ,CHHCT , CPF ,. .CUUG\ %QPUVTCKPV NQIKE RTQ ITCOOKPI 6JG #%/ 5[ORQUKWO QP 2TKPEKRNGU QH 2TQITCOOKPI .CPIWCIGU ,GPUGP - %QNQWTGF 2GVTK 0GVU $CUKEU %QP EGRVU #PCN[UKU /GVJQFU CPF 2TCEVKECN 7UG 8QN 5RTKPIGT8GTNCI $GTNKP
246
www.elsevier.com/locate/ifac
%52 )'0'4#6+10 (41/ 2'64+0'65 /1&'.5 &CPKGN 4KGTC /KSWGN # 2KGTC #PVQPK )WCUEJ 'PIKP[GTKC FG 5KUVGOGU
K #WVQO CB VKEC 7PKXGTUKVCV #WVBQPQOC FG $CTEGNQPC $CTEGNQPC 5RCKP f&CPKGN4KGTC/KIWGN#PIGN2KGTCg"WCDGU +PUVKVWVQ FG 4QD Q y VKEC G +PHQTOCy VKEC +PFWUVTKCN 72%%5+% $CTEGNQPC 5RCKP )WCUEJ"GUCKKWREGU
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
-G[YQTFU /GVJQFQNQI[ 2GVTKPGVU %QPUVTCKPV 5CVKUHCEVKQP 2TQDNGOU 2NCPPKPI 2TQFWEVKQP U[UVGOU
+0641&7%6+10
U[PEJTQPK\CVKQP CPF TGUQWTEGU UJCTKPI YJKEJ CTG VJG OCKP EJCTCEVGTKUVKEU QH C (/5
+P VJG NCUV HGY [GCTU OCP[ OGVJQFU CPF VQQNU JCXG DGGP FGXGNQRGF VQ KORTQXG RTQFWEVKQP RGTHQTOCPEG KP VJG OCPWHCEVWTKPI KPFWUVT[ 6JGUG CRRTQCEJGU VT[ VQ VCEMNG EJCPIGU KP RTQFWEVKQP QDLGEVKXGU UWEJ CU JKIJ RTQFWEVKQP FKXGTUKV[ KPUVGCF QH JKIJ RTQFWEVKQP XQNWOG OCMG VQ QTFGT KPUVGCF QH OCMG VQ UVQEM CPF \GTQ UVQEM KPUVGCF QH LWUV KP VKOG RQNKEKGU #NVJQWIJ 14 1RGTCVKQPU 4GUGCTEJ OGVJQFQNQIKGU JCXG DGGP RTQXGF VQ RGTHQTO YGNN KP HTQPV QH UQOG V[RGU QH RTQDNGOU VJG[ HCNN UJQTV VQ VCEMNG RTGUGPV ƀGZKDNG OCPWHCEVWTKPI UEJGFWNKPI RTQFWEVKQP FGOCPFU
6JWU 20 HQTOCNKUO KU WUGF VQ URGEKH[ CNN VJG GXGPVU VJCV NGCF VQ C U[UVGO UVCVG EJCPIG GI DGIKPPKPI QT GPFKPI QH C RTQFWEVKQP CEVKXKV[ VQIGVJGT YKVJ CNN VJG RTGEQPFKVKQPU VJCV UJQWNF DG UCVKUſGF KP QTFGT CP GXGPV EQWNF QEEWT GI TCY OCVGTKCN CPF TGUQWTEGU TGSWKTG OGPVU GVE 0QVG VJCV C 20 OQFGN QH C RTQFWEVKQP U[U VGO KU GPQWIJ VQ UKOWNCVG KVU DGJCXKQWT YJKEJ OGCPU VJCV VJG OQFGN HQTOCNK\GU DQVJ
#TEJKVGEVWTG 2TQFWEVKQP %QPUVTCKPVU #WVQOCVGF VTCPURQTV WPKVU KG EQPXG[QTU TQDQVU OCPKRWNC VQTU GVE DKPF VJG OCVGTKCN ƀQY KP VJG RTQFWEVKQP U[UVGO .QIKUVKE 2TQFWEVKQP %QPUVTCKPVU &GURKVG ƀGZKDNG RTQFWEVKQP OGEJCPKUOU CNNQY FKHHGTGPV UEJGFWNKPI RQNKEKGU RTQFWEV TGEKRGU TGFWEG UWD U[UVGO KPVGTCEVKQP FWG VQ UGXGTCN HCEVQTU UWEJ CU VJG QRGTCVKQP UGSWGPEG KP YJKEJ VJG TCY OCVGTKCN OWUV DG RTQEGUUGF
2GVTKPGVU 20 JCXG UJQYP VQ DG UWEEGUUHWN VQQNU HQT OQFGNNKPI (NGZKDNG /CPWHCEVWTKPI 5[UVGOU (/5 FWG VQ UGXGTCN CFXCPVCIGU UWEJ CU VJG EQPEKUGPGUU QH GODQF[KPI DQVJ VJG UVCVKE UVTWEVWTG CPF VJG F[PCO KEU VJG CXCKNCDKNKV[ QH VJG OCVJGOCVKECN CPCN[UKU VGEJ PKSWGU CPF KVU ITCRJKECN PCVWTG ,GPUGP 5KNXC CPF 8CNGVVG
241
: (P T ) ! N KU C HWPEVKQP VJCV FGſPGU VJG YGKIJV QH FKTGEVGF CTEU HTQO RNCEGU VQ VTCPUKVKQPU O : (T P ) ! N KU C HWPEVKQP VJCV FGſPGU VJG YGKIJV QH FKTGEVGF CTEU HTQO VTCPUKVKQPU VQ RNCEGU M0 KU VJG KPKVKCN OCTMKPI
%QPUVTCKPV 2TQITCOOKPI %2 JCU DGGP OCKPN[ EJQUGP DGECWUG QH KVU IQQF QRVKOKUCVKQP RGTHQTOCPEG 5KPEG %2 KU WUWCNN[ GODGFFGF KP FGENCTCVKXG RTQITCOOKPI VJG WUGT FQGU PQV PGGF VQ YTKVG CP CNIQTKVJO VQ UQNXG VJG RTQDNGO DWV QPN[ VQ OQFGN VJG RTQDNGO VQ DG UQNXGF 6JGTGHQTG QPEG VJG OQFGN KU IGPGTCVGF %2 ECP QRVKOKUG KV YKVJQWV TGSWKTKPI CP GZRGTV VQ TWNG KV
I
# OCTMKPI KU CP CTTC[ VJCV CUUKIPU VQ GCEJ RNCEG C PQP PGICVKXG KPVGIGT +H C OCTMKPI CUUKIPU VQ RNCEG p C XCNWG + k k 2 Z p KU OCTMGF YKVJ k VQMGPU TGRTGUGPVGF ITCRJKECNN[ D[ DNCEM FQVU
+P VJKU RCRGT VJG OCKP CURGEVU QH C 20 CPCN[UKU OGVJQFQNQI[ VQ FGCN CWVQOCVKECNN[ YKVJ CNN VJG EQP UVTCKPVU QH C RTQFWEVKQP U[UVGO CTG RTGUGPVGF 6JG CKO QH VJKU OGVJQFQNQI[ KU VQ CWVQOCVK\G VJG EQPUVTCKPVU OQFGNNKPI RJCUG QH C RTQFWEVKQP U[UVGO ō FGUETKDGF RTGXKQWUN[ KP 20 HQTOCNKUO ō KP QTFGT VQ IGPGTCVG C SWCUKQRVKOCN UEJGFWNKPI HQT C RCTVKEWNCT U[UVGO UVCVG CPF RTQFWEVKQP IQCN OCMKPI VJG QRVKOKUCVKQP RJCUG VTCPURCTGPV VQ VJG WUGT 5GG (KI
# VTCPUKVKQP ti KU UCKF VQ DG GPCDNGF D[ C OCTMKPI M KH 8p 2 P : M (p) I (p t i ) 6JG ſTKPI VTCPUKVKQP IGPGTCVGU C PGY OCTMKPI M 0 YJKEJ ECP DG EQORWVGF D[ YKVJFTCYKPI I (p k ti ) VQMGPU HTQO GCEJ p k KPRWV RNCEG QH ti CPF D[ CFFKPI O(p j ti ) VQMGPU VQ GCEJ p j QWVRWV RNCEG QH t i +P OCPWHCEVWTKPI VGTOU VTCPUKVKQPU CTG WUGF VQ OQFGN QRGTCVKQPU ſTKPI C VTCPUKVKQP ECP TGRTGUGPV C VCUM QT RTQEGUU KPKVKCVKQP QT CP GPFKPI QH C VCUM RNCEGU CTG WUGF VQ OQFGN DWHHGTU CPF TGUQWTEGU UVCVWU EQPPGEVKPI CTEU URGEKH[ NQIKECN TGNCVKQPUJKRU CPF TGUQWTEG EQP UVTCKPVU COQPI QRGTCVKQPU CPF VQMGPU TGRTGUGPV OC VGTKCN CPF TGUQWTEGU EQPFKVKQPU
(KI 2TGUGPVGF OGVJQFQNQI[ +V UJQWNF DG PQVGF VJCV FGURKVG VJGTG CTG UGXGTCN 20 CPCN[UKU VQQNU VQ KORTQXG U[UVGO RGTHQTOCPEG VJQUG ECP DG WUGF QPN[ YJGP VJG 20 OQFGN UJQYU EGTVCKP EJCTCEVGTKUVKEU YJKEJ KU PQV CNYC[U VJG ECUG QH 20 OQFGNU QH RTQFWEVKQP U[UVGOU 1P VJG QVJGT JCPF FGURKVG %2 QHHGTU C RQYGTHWN VGEJPQNQI[ VQ FGCN YKVJ VJG DGUV RNCPPKPI RQNKE[ HQT C RTQFWEVKQP U[UVGO KV TGSWKTGU C FGGR MPQYNGFIG KP DQVJ VJG RTQFWEVKQP U[UVGO CPF VJG %2 VGEJPQNQI[ CPF VKOG VQ FGXGNQR VJG %2 OQFGN 6JWU VJG RTGUGPV YQTM VTKGU VQ EQODKPG VJG OCKP CFXCPVCIGU QH DQVJ OGVJQFQNQIKGU KP QTFGT VQ QHHGT IQQF UEJGFWNKPI RQNKEKGU OKPKOK\KPI VJG VKOG URGPV KP VJG OQFGNNKPI CPF QRVKOK\CVKQP VCUMU
%QPUVTCKPVU CTKUG KP OQUV CTGCU QH JWOCP GPFGCXQWT # EQPUVTCKPV KU UKORN[ C NQIKECN TGNCVKQP COQPI UGX GTCN WPMPQYPU QT XCTKCDNGU GCEJ VCMKPI C XCNWG KP C IKXGP FQOCKP 6JG EQPUVTCKPV VJWU TGUVTKEVU VJG RQU UKDNG XCNWGU VJCV XCTKCDNGU ECP VCMG %2 KU VJG UVWF[ QH EQORWVCVKQPCN U[UVGOU DCUGF QP EQPUVTCKPVU 6JG OCKP KFGC KU VQ UQNXG RTQDNGOU D[ UVCVKPI EQPUVTCKPVU TG SWKTGOGPVU CDQWV VJG RTQDNGO CTGC CPF EQPUGSWGPVN[ ſPFKPI C UQNWVKQP UCVKUH[KPI CNN VJG EQPUVTCKPVU
6JG HQWPFCVKQPU HQT VJG CPCN[UKU QP C 20 OQFGN CPF VJG IGPGTCVKQP QH VJG %2 OQFGN CTG RTGUGPVGF 6JKU CPCN[UKU KU OCFG D[ FGVGEVKPI UVTWEVWTGU KP VJG 20 YJKEJ CTG VTCPUNCVGF FKTGEVN[ KPVQ VJG %52 EQPUVTCKPVU
6JG GCTNKGUV KFGCU NGCFKPI VQ %2 OC[ DG HQWPF KP VJG #TVKſEKCN +PVGNNKIGPEG #+ YKVJ VJG UEGPG NCDGNNKPI RTQDNGO 9CNV\ CPF VJG KPVGTCEVKXG ITCRJKEU
5WVJGTNCPF
+P UGEVKQPU CPF 20 CPF %2 DCEMITQWPFU CTG RTG UGPVGF 5GEVKQP KPVTQFWEGU VJG RWTRQUGF OGVJQFQN QI[ 5GEVKQP KNNWUVTCVGU VJKU OGVJQFQNQI[ D[ OGCPU QH CP GZCORNG (KPCNN[ 5GEVKQPU CPF FKUEWUU VJG DGPGſVU CPF RTGUGPV VJG EQPENWUKQPU CPF HWVWTG YQTM
)CNNCKTG CPF ,CHHCT CPF .CUUG\ PQVGF VJCV NQIKE RTQITCOOKPI YCU LWUV C RCTVKEWNCT MKPF QH %2 6JG DCUKE KFGC DGJKPF .QIKE 2TQITCOOKPI .2 CPF FGENCTCVKXG RTQITCOOKPI KP IGPGTCN KU VJCV VJG WUGT UVCVGU YJCV JCU VQ DG UQNXGF KPUVGCF QH JQY VQ UQNXG KV YJKEJ KU XGT[ ENQUG VQ VJG KFGC QH EQPUVTCKPVU
%10564#+06 241)4#//+0) $#%-)4170&
4GEGPV CFXCPEGU RTQOKUG VJCV %2 CPF 1RGTCVKQPU 4GUGCTEJ 12 ECP GZRNQKV GCEJ QVJGT KP RCTVKEWNCT VJG %2 ECP UGTXG CU C TQQH RNCVHQTO HQT KPVGITCVKPI XCTKQWU EQPUVTCKPV UQNXKPI CNIQTKVJOU KPENWFKPI VJQUG FGXGNQRGF CPF EJGEMGF VQ DG UWEEGUUHWN KP 14
2'64+0'65 $#%-)4170& # 2GVTKPGV 20 KU C RCTVKEWNCT MKPF QH FKTGEVGF ITCRJ VQIGVJGT YKVJ CP KPKVKCN UVCVG ECNNGF VJG KPKVKCN OCTMKPI #P QTFKPCT[ 20 KU C VWRNC N = (P TIO M 0 )
6JGP %2 EQODKPGU KFGCU HTQO C PWODGT QH ſGNFU KPENWFKPI #TVKſEKCN +PVGNNKIGPEG %QODKPCVQTKCN #NIQ TKVJOU %QORWVCVKQPCN .QIKE &KUETGVG /CVJGOCVKEU 0GWTCN 0GVYQTMU 1RGTCVKQPU 4GUGCTEJ 2TQITCOOKPI .CPIWCIGU CPF 5[ODQNKE %QORWVCVKQP
=
fp1 : : : pn g KU VJG UGV QH RNCEGU TGRTG UGPVGF ITCRJKECNN[ D[ EKTENGU T = ft1 : : : tm g KU VJG UGV QH VTCPUKVKQPU TGRTG UGPVGF ITCRJKECNN[ D[ DQNF NKPGU QT TGEVCPINGU
P
242
6JG RTQDNGOU UQNXGF WUKPI %2 CTG ECNNGF %QPUVTCKPV 5CVKUHCEVKQP 2TQDNGOU %52 # %52 KU FGſPGF CU
#NIQTKVJO 6TCPUKVKQPU (KTKPIU $QWPFKPI 5VGR +PKVKCNKUCVKQP QH 0 D[ CRRN[KPI TWNGU 6!2 CPF 2!6 5GG DGNQY HQT GCEJ RNCEG CPF VTCP UKVKQP TGURGEVKXGN[ (KTUV 6!2 KU CRRNKGF VQ GXGT[ RNCEG KP VJG 20 CPF NCVGT 2!6 ECNEWNCVGU VJG KPKVKCN ŌXCNWGU HQT GXGT[ VTCPUKVKQP
C UGV QH XCTKCDNGU X
=f
x1 : : : x
sg
HQT GCEJ XCTKCDNG x i C ſPKVG UGV Di QH RQUUKDNG XCNWGU KVU FQOCKP CPF C UGV QH EQPUVTCKPVU TGUVTKEVKPI VJG XCNWGU VJCV VJG XCTKCDNGU ECP UKOWNVCPGQWUN[ VCMG 5WEJ CU $CTVyCM UC[U ő%QPUVTCKPV RTQITCOOKPI KU CP GOGTIGPV UQHVYCTG VGEJPQNQI[ HQT FGENCTCVKXG FGUETKRVKQP CPF GHHGEVKXG UQNXKPI QH JCTF TGCN NKHG RTQD NGOU GURGEKCNN[ KP CTGCU QH RNCPPKPI CPF UEJGFWNKPIŒ
!2
!6
C 6
D 2
2'64+0'6 /1&'.5 61 %2 /1&'.5 (KI )GPGTCN ECUGU HQT RNCEG CPF VTCPUKVKQP HGGFKPI %52 %QORQPGPVU +FGPVKſECVKQP
6!2 ſPFU VJG OCZKOWO PWODGT QH VQMGPU GCEJ RNCEG p 2 P ECP EQPVCKP FWTKPI CNN VJG U[UVGO TWP 6JKU KU ECNEWNCVGF CRRN[KPI VQ VJG IGPGTCN ECUG UJQYP KP (KI C VJG HQTOWNC RTGUGPVGF KP 'S
+P QTFGT VQ CPCN[UG VJG 20 OQFGN CPF IGPGTCVG VJG %52 RTQDNGO VJG ſTUV UVGR KU VQ KFGPVKH[ VJG GNGOGPVU EQORQUKPI VJG %52 6JGUG GNGOGPVU CTG
8CTKCDNGU 6JGTG KU C XCTKCDNG HQT GCEJ ſTKPI QH C 20 VTCPUKVKQP *GPEG GXGT[ VTCPUKVKQP KU CUUQEKCVGF VQ C NKUV QH VKOGU 6JG NGPIVJ QH VJKU NKUV KU IKXGP D[ VJG PWODGT QH VKOGU VJG VTCPUKVKQP KU ſTGF QT C JKIJGT DQWPF KH WPMPQYP 6JGUG CTG ECNNGF VTCPUKVKQP XCTKCDNGU 6JG UGEQPF UGV QH XCTKCDNGU CTG VJQUG TGRTG UGPVKPI VJG PWODGT QH ſTKPIU QH VJG VTCPUKVKQPU 6JGUG XCTKCDNGU CTG ECNNGF ſTKPI XCTKCDNGU #RCTV QH VJGUG VJGTG CTG CNUQ $QQNGCP XCTK CDNGU YJKEJ KPFKECVG VJG RCVJU HQNNQYGF D[ VQ MGPU KP DKHWTECVKQPU &QOCKPU 6JG FQOCKPU QP VJG VTCPUKVKQP XCTKCDNGU CTG FGſPGF WUKPI VJG MPQYNGFIG CDQWV VJG RTQDNGO 7UWCNN[ VJG HCEV QH TGFWEKPI VJG FQOCKPU QP VJG XCTKCDNGU OCMGU VJG UGCTEJ HCUVGT (KTKPI XCTK CDNGU FGRGPF QP VJG DQWPFU HQWPF D[ VJG 6TCPUK VKQPU (KTKPIU $QWPFKPI CNIQTKVJO 5GG #NI %QPUVTCKPVU 6JG[ CTG IGPGTCVGF KP VJG 20 UVTWEVWTGU FGVGE VKQP RJCUG 5GG 5GEVKQP CPF TGUVTKEV VJG RCVJU YJKEJ ECP DG HQNNQYGF D[ GXGT[ VQMGP CPF VJG VKOGU YJGP VTCPUKVKQPU ECP DG ſTGF
( )=
tok p
m X ( )+ i
tokini p
i=1
Oti p
YJGTG tokini (p) KU VJG KPKVKCN PWODGT QH VQMGPU QH RNCEG p KP VJG KPKVKCN OCTMKPI M 0 2!6 KPKVKCNKUGU VJG PWODGT QH VKOGU GCEJ VTCPUK VKQP t 2 T OKIJV DG ſTGF FWTKPI VJG U[UVGO TWP 6JG PWODGT QT GZRTGUUKQP KH VJGTG CTG WPMPQYP XCNWGU KU IKXGP D[ VJG CRRNKECVKQP QH VJG HQTOWNC KP 'S VQ VJG IGPGTCN ECUG UJQYP KP (KI D
( ) = i=1min :::n
t
j
tok(pi )
p
I it
k
j Ipit 6= 0
5VGR 2TQRCICVKQP QH VJG ŌXCNWGU KP i HTQO KVU XCNWGU KP i1 +P VJKU UVGR DQVJ PWODGTU CPF GZ RTGUUKQPU CTG RTQRCICVGF 5VGR 4GUQNWVKQP QH VJG GZRTGUUKQPU KP i +H C GZ RTGUUKQP TGRTGUGPVKPI j EQPVCKPU KVUGNH j KV KU PQV EQPUKFGTGF CPF KU UWDUVKVWVGF D[ KPſPKV[ 1 #HVGT VJKU UKORNKſECVKQP min HWPEVKQPU CTG CRRNKGF KH RQUUKDNG + then go to 5VGR = Z 5VGR if 9j 2 i j j 2 else 'PF
2GVTKPGV 5VTWEVWTGU &GVGEVKQP +P VJKU UVGR VJG 20 UVTWEVWTGU CTG GZVTCEVGF KP QTFGT VQ IGPGTCVG VJG EQPUVTCKPVU CPF TGFWEG VJG UGCTEJ URCEG 'CEJ UVTWEVWTG EQTTGURQPFU VQ C UGV QH EQPUVTCKPVU YJKEJ CTG CWVQOCVKECNN[ IGPGTCVGF
%CNEWNCVKQP QH VJG $QWPFU QH VJG 6TCPUKVKQPU (KTKPIU 5KPEG VJG XCTKCDNGU CTG VJG ſTKPIU QH VJG VTCPUKVKQPU KP VJG 20 KV KU PGEGUUCT[ VQ DQWPF VJG PWODGT QH VKOGU VJG[ JCRRGP 1VJGTYKUG VJG PWODGT QH XCTKCDNGU YQWNF DGEQOG WPOCPCIGCDNG
5VTWEVWTG 4 4 EQPUVTCKPVU 5GG 'S CTG IGPGTCVGF HQT GXGT[ UKPING VTCPUKVKQP KP VJG 20 6JKU KU PQV C UVTWEVWTG DWV C YC[ QH TGOQXKPI U[OOGVTKGU HTQO VJG RTQDNGO 5KPEG VJGTG KU C XCTKCDNG HQT GCEJ ſTKPI QH
#P CNIQTKVJO YJKEJ FGRGPFU QP VJG 20 CPF VJG KPKVKCN UVCVG M0 QH VJG RTQDNGO KU RTQRQUGF
243
C VTCPUKVKQP D[ WUKPI 4 VJG XCNWGU QH VJGUG XCTKCDNGU
CPF JGPEG VJG ſTKPIU CTG IKXGP C WPKSWG QTFGT
t t 1 8j = 2 : : : l ij
ij
length(Ti ) = l
5VTWEVWTG 4 6JKU UVTWEVWTG KPENWFGU VYQ EQPUGEWVKXG VTCPUKVKQPU YKVJ C FKTGEV RCVJ DGVYGGP VJGO 'CEJ ſTKPI VKOG QH VJG GZKV VTCPUKVKQP KU TGNCVGF YKVJ QPG QH VJG UQWTEG VTCPUKVKQP 5GG 'S
(KI 5VTWEVWTG ( VJTQWIJ C WPKSWG RNCEG 5GG (KI $QQNGCP XCTKCDNGU CTG CNUQ PGEGUUCT[ KP VJKU ECUG 5GG 'S
tout (tin + timein) Bin Bdi M 8j = 1 : : : n length(Tin) = n 8i = 1 : : : m length(Tout ) = m
t t + time 8j = 1 : : : l length(t ) = length(t ) = l ij
i
k
kj
i
k
Bdi = diff
YJGTG timek KU VJG RTQEGUUKPI VKOG CUUQEKCVGF VQ VTCPUKVKQP k
i
j
X (j B i
k
ink
=1
)
6JGUG EQPUVTCKPVU CTG PQV IGPGTCVGF KH DQVJ VTCPUKVKQPU
HQTOKPI 4 DGNQPI VQ C OQTG EQORNGZ UVTWEVWTG KG ( , QT (, 5VTWEVWTG +5 6JKU TGRTGUGPVU CP KPKVKCN UVQEM
KG CP KUQNCVGF RNCEG HGGFKPI VYQ QT OQTG VTCPUKVKQPU +5 UVTWEVWTG FQGU PQV CFF VGORQTCN EQPUVTCKPVU DWV EQPUVTCKPVU TGNCVGF YKVJ VJG PWODGT QH ſTKPIU QH VJG GZKV VTCPUKVKQPU 5GG 'S
X tok
( ) 8i j feeds(pin t ) i
ini pin
(KI 5VTWEVWTG , 5VTWEVWTG ,( 6JKU KU VJG OQUV EQORNGZ UVTWE VWTG YJGTG C NKUV QH KPRWV VTCPUKVKQPU HGGFU CP QWVRWV NKUV QH VTCPUKVKQPU 5GG (KI 6JKU UVTWEVWTG CNUQ TG SWKTGU VJG WUG QH $QQNGCP XCTKCDNGU 5GG 'S
i
YJGTG pin KU VJG RNCEG TGRTGUGPVKPI VJG KPKVKCN UVQEM
Tout (Tin + timein ) Bin out Bdi M 8j = 1 : : : n length(Tin) = n
8i = 1 : : : m length(Tout) = m i
5VTWEVWTG ( 6JKU UVTWEVWTG KU HQTOGF D[ C UKP ING VTCPUKVKQP YJKEJ HGGFU C UGV QH VTCPUKVKQPU VJTQWIJ C UKPING RNCEG 5GG (KI # NKUV QH $QQNGCP XCTKCDNGU CTG WUGF VQ UGNGEV VJG RCVJ HQNNQYGF D[ GCEJ VQMGP KP VJG UVTWEVWTG 6JGUG XCTKCDNGU GPCDNG QT FKUCDNG EQPUVTCKPVU FGRGPFKPI QP VJGKT EQTTGURQPFGPEG VQ VJG UGNGEVGF RCVJ 5GG 'S
tout (tin + timein ) Bout Bdi M 8j = 1 : : : n length(Tin ) = n 8i = 1 : : : m length(Tout ) = m i
j
Bdi = diff(i
XXB j
in
k
=1
ink
j
)
j
j
X (i B j
Bdi = diff
k
=1
outk
)
(KI 5VTWEVWTG ,(
YJGTG HWPEVKQP diff TGVWTPU \GTQ 0 KH VJG RCTCOG VGTU CTG GSWCN CPF QPG 1 KH VJG[ CTG FKHHGTGPV CPF M KU CP KPVGIGT VJCV FQOKPCVGU VJG GZRTGUUKQP C ŎDKI/ŏ VGTO
%52 .CDGNNKPI &GſPKVKQP 5KPEG VJG CKO QH VJKU OGVJQFQNQI[ KU VQ ſPF VJG QRVKOWO PQV QPN[ C UQNWVKQP JCU VQ DG HQWPF DWV QRVKOCNKV[ OWUV DG RTQXGF #NVJQWIJ C RQUUKDKNKV[ KU VQ IGPGTCVG VJG EQORNGVG UQNWVKQPU VTGG VJKU YQWNF OGCP C JKIJ EQORWVCVKQPCN EQUV
5VTWEVWTG , 6JKU KU EQORNGOGPVCT[ VQ UVTWE VWTG ( # UGV QH VTCPUKVKQPU HGGF C UKPING VTCPUKVKQP
244
6 4GVKTG ſPCN RKGEG HTQO / JQWTU +PKVKCN OCTMKPI
1P VJG QVJGT JCPF QRVKOCNKV[ ECP DG RTQXGF D[ NC DGNNKPI VJG XCTKCDNG VQ QRVKOKUG ſTUV CPF UGNGEVKPI KVU XCNWGU HTQO DGVVGT VQ YQTUG UQNWVKQPU 6JKU OGCPU VJCV KP VJG OQOGPV C UQNWVKQP KU HQWPF CNN VJG RQUUKDNG DGVVGT UQNWVKQPU JCXG DGGP CNTGCF[ TGLGEVGF 1RVKOCNKV[ KU RTQXGF CPF HWTVJGT UGCTEJ KU PQV PGEGUUCT[
M0
= 01
5 0 0 1 0 0 1 0 0]
6TCPUKVKQP CPF $QQNGCP XCTKCDNGU CTG NCDGNNGF QPN[ QPEG HQT GCEJ KPUVCPEG QH VJG XCTKCDNG VQ QRVKOKUG CP[ QH VJGKT HGCUKDNG XCNWGU IKXGU CP QRVKOCN UQNWVKQP 2CVJU CTG UGNGEVGF D[ NCDGNNKPI $QQNGCP XCTKCDNGU ſTUV 6TCPUKVKQP XCTKCDNGU CTG NCDGNNGF NCVGT
':#/2.' 'ZCORNG 2TQEGUU CPF #UUGODNG (CEVQT[ 6JG U[UVGO KU EQORQUGF D[ VYQ OCEJKPGU YJKEJ RGT HQTO VJG PGZV QRGTCVKQPU / OCMGU VYQ FKHHGTGPV QR GTCVKQPU FGRGPFKPI QP VJG V[RG QH RKGEG KV KU YQTMKPI YKVJ 6JKU KU C UJCTGF TGUQWTEG *GPEG KV KU PGEGUUCT[ VQ RNCP VJG QTFGT VJG RKGEGU CTG RWV KPVQ KV 1P VJG QVJGT JCPF / KU WUGF VQ LQKP VYQ RTQEGUUGF RKGEGU # CPF $ 6JG U[UVGO ECP DG UGGP KP (KI
(KI 6JG 2GVTKPGV QH VJG U[UVGO
6JG CKO KU IKXGP RKGEGU QH V[RG # CPF RKGEGU QH V[RG $ VQ RGTHQTO 5 RTQEGUUGF CPF CUUGODNGF RKGEGU
6TCPUKVKQPU (KTKPIU $QWPFKPI +P QTFGT VQ ECNEWNCVG VJG JKIJGT DQWPFU QH VJG VTCPUK VKQPU ſTKPIU #NIQTKVJO KU CRRNKGF VQ VJG 20
0 = min(10 1 + 3 ) min(5 4 ) 1 2 min(3 4 1 + 6 ) 5 ] 1 = min(10 1 + 1 ) min(5 2 ) min(10 1 + 3 ) min(5 4 ) min(1 2 1 + 5 ) min(3 4 1 + 6 )] = = min(10 ) min(5 ) min(10 ) min(5 ) min(1 2 ) min(3 4 )] = = 01 5 10 5 min(1 2 ) min(3 4 )] = 2 = 01 5 10 5 min(10 5) min(10 5)] = 01 5 10 5 5 5] =
(KI 6JG UVWFKGF U[UVGO
1 1
2GVTKPGV QH VJG 5[UVGO 6JG 20 QH VJG UVWFKGF U[UVGO 5GG (KI JCU VJG PGZV EQORQPGPVU 2NCEGU 2 5VQEM QH RKGEGU # 2 5VQEM QH RKGEGU $ 2 2KGEG # DGKPI RTQEGUUGF KP / 2 2KGEG $ DGKPI RTQEGUUGF KP / 2 / HTGG 2 5VQEM QH RKGEGU # RTQEGUUGF 2 5VQEM QH RKGEGU $ RTQEGUUGF 2 / HTGG 2 / CUUGODNKPI C ſPCN RKGEG 2 5VQEM QH CUUGODNGF RKGEGU 6TCPUKVKQPU 6 /QXG RKGEG # VQ / JQWT 6 /QXG RKGEG $ VQ / JQWT 6 4GVKTG RKGEG # HTQO / JQWTU 6 4GVKTG RKGEG $ HTQO / JQWTU 6 /QXG RTQEGUUGF RKGEGU # CPF $ VQ / JQWT
1
1
1
1
5VTWEVWTGU (QWPF 6JG UVTWEVWTGU HQWPF CPF CWVQOCVKECNN[ VTCPUNCVGF KPVQ EQPUVTCKPVU CTG CU HQNNQYU 4 =6? =6? =6? =6? =6? CPF =6? 4 =6 6? =6 6? CPF =6 6? +5 ( , =6 6 6 6? ,( =6 6 6 6?
!
!
!
! !
4GUWNVU 6JG ſTKPI VKOGU HQWPF CHVGT VJG QRVKOKUCVKQP RJCUG CTG VJG PGZV
245
5KNXC / CPF 4 8CNGVVG 2GVTK PGVU CPF ƀGZKDNG OCPWHCEVWTKPI .GEVWTG 0QVGU KP %QORWVGT 5EK GPEG XQN #FXCPEGU KP 2GVTK 0GVU RR Ō 5WVJGTNCPF + 5MGVEJRCF C OCPOCEJKPG ITCRJKECN EQOOWPKECVKQP U[UVGO 2TQE +(+2 5RTKPI ,QKPV %QORWVGT %QPHGTGPEG 9CNV\ &. 7PFGTUVCPFKPI NKPG FTCYKPIU QH UEGPGU YKVJ UJCFQYU KP 2U[EQNQI[ QH %QORWVGT 8KUKQP /E)TCY*KNN 0GY ;QTM 9CPI . CPF 5 9W /QFGNKPI YKVJ EQNQTGF VKOGF QDLGEVQTKGPVGF RGVTK PGVU HQT CWVQOCVGF OCPWHCEVWTKPI U[UVGOU %QORWVGTU CPF +PFWU VTKCN 'PIKPGGTKPI Ō
# ECUG UVWF[ KP OQFGNKPI CPF RGTHQT OCPEG GXCNWCVKQP QH OCPWHCEVWTKPI U[UVGOU WU KPI EQNQTGF RGVTK PGVU 2TQEGGFKPIU QH VJG VJ 'W TQRGCP 5KOWNCVKQP 5[ORQUKWO '55 ŏ
6 = ? 6 = ? 6 = ? 6 = ? 6 = ? 6 = ? 6JGP VJG QRVKOCN VKOG HQT VJG RGTHQTOCPEG QH RKGEGU
RTQEGUUGF CPF CUUGODNGF KU 44 JQWTU T 6 5 +time6 $'0'(+65 1( 6*' 274215'& /'6*1&1.1); 6JG RTQRQUGF CRRTQCEJ KORTQXGU UGXGTCN CURGEVU QH CEVWCN UEJGFWNKPI VQQNU UQOG QH YJKEJ CTG
6JG URGEKſ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ſTUV CPF URGEKCNK\KPI NCVGT 4'('4'0%'5 $CTVyCM 4 %QPUVTCKPV RTQITCOOKPI +P RWTUWKV QH VJG JQN[ ITCKN 2TQEGGFKPIU QH 9&5 KPXKVGF VCNM )CNNCKTG * .QIKE RTQITCOOKPI (WTVJGT FG XGNQROGPVU +''' 5[ORQUKWO QP .QIKE 2TQ ITCOOKPI ,CHHCT , CPF ,. .CUUG\ %QPUVTCKPV NQIKE RTQ ITCOOKPI 6JG #%/ 5[ORQUKWO QP 2TKPEKRNGU QH 2TQITCOOKPI .CPIWCIGU ,GPUGP - %QNQWTGF 2GVTK 0GVU $CUKEU %QP EGRVU #PCN[UKU /GVJQFU CPF 2TCEVKECN 7UG 8QN 5RTKPIGT8GTNCI $GTNKP
246