ABSTRACT
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7. GENERAL STATEMENT OF TRAFFIC CONTROL PROBLEM
(SIGNAL PLAN CHOICE)
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objective function
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8. THE SET OF FEASIBLE CONTROLS (SIGNAL PLANS)
The formulation of the optimal control problem ),( cf JU implies GH¿QLQJWKHIHDVLEOHFRQWUROVHW fU , whose element fu U˜)( LVGH¿QHGE\ an ordered pair as follows:
Tu ),()( W ¸¸¹
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§ W
˜ u u
,
where
],,,,,[ K21 uuuuu !! k
LV WKH FRQWURO VWUXFWXUH VLJQDO SODQ VWUXFWXUH $ FRQWURO VWUXFWXUH component
)(],,,,,[ P21 K kuuuu Tkk
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),(}1,0{ PK pkukp ,
where P LV WKH VHW RI VLJQDO JURXS LQGLFHV LQ WKH FRPSOHWH VHW RI VLJQDO JURXSV7KHFRPSOHWHVHWRIVLJQDOJURXSVKDVWREHFKRVHQEHIRUHVWDUWRIWKH SUREOHPVROYLQJ 7KHVHFRQGHOHPHQWRIWKHSDLUWUHSUHVHQWVWKHF\FOHWLPHVSOLW
LH ],,,,,[ K21 WWWW W !! k
The terms control and signal planDUHXVHGDVV\QRQ\PV&RPSRQHQW ku is termed the control vector or phase
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p uuuuu ])()()(,)([)( P21 ˜˜˜˜ ˜ , where one vector component, )(˜pu UHSUHVHQWVWKHFRQWUROYDULDEOHDVVLJQHG WRVLJQDOJURXS pDc 9DOXHVRIWKLVYDULDEOHGXULQJDF\FOHWLPHDUHGH¿QHG E\WKHVHTXHQFH
LHE\ WKHYDOXHVRIYDULDEOHV kpu for K,,2,1 ! k 7KHGXUDWLRQRI HDFK component kpu RIWKDWVHTXHQFHLVGHWHUPLQHGE\WKHYDOXHRIFRPSRQHQW
kW of vector W with the same index, k+HQFHWKHFRQWUROYDULDEOH )(˜pu can be GH¿QHGE\WKHSDLU
)(,),()( PW ˜ pu Tpp u
7KHSUREOHPVRIRSWLPDOVLJQDOSODQFKRLFH ),( cf JU , can be formulated DVPDWKHPDWLFDORSWLPL]DWLRQSUREOHPV7KHVHWRIIHDVLEOHFRQWUROV fU , is GHWHUPLQHGE\ WKHFRQVWUDLQWV WKDWKDYH WREHVDWLV¿HGE\FKRVHQFRQWUROV In some problems, constraints on state variables of the process are also
SUHVHQW The set of feasible controls, fU GHSHQGV DOVR RQ WKH SUREOHPEHLQJ
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RIVLJQDOJURXSV This section presents mathematical expressions for control variable
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)(, ),( ),(
),()( K21
P» ¼
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ª WWWW
» ¼
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ª W
W ˜ p uuuu
u k
u u
Thus, control variable constraints are functions of vectors pu and W
8.2.1. The constraint of one interval of green indication
during the cycle
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T ppu ),()( W ˜ u , where ],,,,,[
k pppp uuuu !! u , }1,0{
k pu , ( Pp )
can be formulated as follows:
)(,2)2(mod)( K
puu k
where }P,,2,1{ ! P LVWKHVHWRIVLJQDOJURXSLQGLFHV
0 c t
0 c t
0 c t
u t1( )
u t2( )
u t3( )
W1 W W W W
» » »
¼
º
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],,,,[ 21 uuuuuu
2)2(mod)10()2(mod)00()2(mod)00(
)2(mod)00()2(mod)01()2(mod)(
kk uu
2)2(mod)01()2(mod)11()2mod()10(
)2(mod)00()2(mod)00()2(mod)(
kk uu
)2(mod)10()2(mod)00()2(mod)(
kk uu
8.2.2. Constraints of minimal duration of green indication intervals
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