Floating Traffic Control for Public Transportation System

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FLOATING TRAFFIC CONTROL FOR PUBLIC TRANSPORTATION SYSTEM H. Sasama and Y. Ohkawa Control Eng inee ring Laborat ory, Railway T echn ical R esea rch In st itut e ofJapan ese National Raz1ways, Tokyo , Japan

abstract . In this paper , I-Ie l>ropose a " Floating Traffic Control " without a fixed time table for a publlC transportation system with high traffic denSIty . ~enerally , passengers prefer uniformity of traffic interval to operation,1l j', unctuality in an urban transportation system where the traffic density 1S suff l cient l~ high . For a system with high traffic dencity , a traffic model has be0n developed and an optimal control is deriv e red a,;plying " Linear RC'C]ulator Theory ". Using the mode l, dynamics of the system and characteristIcs of the con trol have been investigated with seve r al examples . As the result wc can see that the " Floating Traffic Control " is effective not on ly to stabilize a disturbed traffic , but also to realize a For an actual traffic h i gh level service meeting the passengers demands . system , the traffic model is extended to a complex system with merging and branching . Furthermore , for practical use , sensitivity analysis to system disturbanc e and si mplification of calculating logic have been investigated . Finally as the results of these analysis , we can get a siymple and effect i ve control for the urban transportation system . Keywords . Public transportation , traffic model , urban ra il ways , Stability , optimal control , regulator theory

nJTRODUCTION Generally , mo s t of public transportation systems are controled according to fixed schedu l es . When the traffic is dis~~rbed , it must be restor ed to t he original schedule as a ru l e.This is necessary fo r passengers in l ong distance tran spo rtation with low traffic density On an intra - city railway , a bus line or a new urban transportation system with sufficiently high traffic density , the passengers arrive at the stations randomly regardless of the time tab l e , an - d rid e the first vehi cle available . In this type of transportation system , it i s possible and desirable to cont rol the traffic i n accordance with the number of passengers and operational state r egardless of the fixed schedule , and to improve the s'?rvice for passengers an~ the stabil itv of traffic operation . \'ie named it a " ,loating Traffic Control " to contro l the o~eratio~al schedule in accordan ce with the state of operation and passengers. Recently , system autonatization of intra - city railway has been developed , for example , CTC (Centralized Traffic Control) system such as on Yamanote lin e of JNR . It is possible to co ll ect real time data on passengers through

229

check i ng and issue i ng automatic ticket mach i nes . Now it comes to be necess ar y to i ntroduce a total control system, which is connected with those automatic sys t ems for a smooth t r af f ic contro l and better service to passengers . A model is deve l oped to represent the system dynamics , and the e f fect. of the contro l is investigated using the model . Then for practical use , sensitivity of the control to system disturbance is eva l uated , and simplificat i on of the control calculation is t r ied . TRAFFIC MODEL In o r d ','"

analize t.he effect of " Floating it is necessa r y to formulate the dynamic cha r acteristics of public transportation wi th high t r affic density .

Traffi,,

to

~'ont r ol ",

Delay of staying time Fu ndamental e qu ation . ef a tr ain at a station gives large effect on the t r affic dynam i cs . Consider i ng this mainly , the fol l owing equation can be derived . t~::::

t ;+ r ;+ s ~ i=1 , 2 , . . . . . . . . ,

(1)

H. Sasama and Y. Ohkawa

230

\ \ \ x '.

Putting

in the above equation , we have (5)

\

We call T:a stable schedule which represents an operation state with o ut con trol and disturbance at all .

\

\

I k - th station -

.

-T

\

\

'.

T~ \

j- \ uk" R~ \

\

fig .l

u ~= O

,-+ "

(k - l) - th s t a ti on -

Stab l e sche dul e and act ua l ope r ation

Al l the states can be given sequential l y from an initia l stat e , by eq . (4) , but it is not convenient to app l y a contro l theory . Then , introducing a new variable x ~ = t ~ - T ~ next equation can be derived . (6)

s (sec) inside line 100

outs id e lin e

~ o reove r , the following state vector x and control vector u a r e introduced .

100 s=0.07x + 23 . 7

50

. . ..

. ..... . . . ~ ~

."

, ~ .."

50

100

200

•

~,

~

I

X K

i

x~

i

~

(

100

tr ain inte r vnl Fig.2

...,. -

t'

+ 29 . 3

~ c .

~

,

.

o. 10x

s=

.

200

(s ec )

Stayi n g time at s tation to trai n int e rval

x •.

l

U"-

I

x!

t~:

r;.

Sk

departure time of i - th train f r om k - th station runni ng time of i - th train from (k - l) - th to k - th station Staying t ime of i - th train at k - th s tat l.on

Furthermo r e , follow s

r and s

can be r ep r esented as

"

R~ + u ~

(2)

s ; = c ~ (t ~ - t~') + O~

~

k= 1 , 2 , 3 ,

Cv.-Xt( ==

XV. _l

wh e r e

R~ : u~

Or.

A parameter c is de l ay rat e to represent the effect on a staying t i me of i - t h train at k - th sta tion according to a train int erval . Equation (3) means that as the interval t i me between the train s becomes longer , s taying time f o r passengers at the station becomes l onge r linearly because they come to sta t ion randomly . The value of cv. is about 0 . 05 0 . 2 , measured on Yamanote Line in Tokyo at ru sh hou r s on a winter morning . One of the resul ts is shown in F i g . l . Ne xt equation andeq . (3) .

is der ived from eq . (1) , eq . (2)

(4)

. ,K

o

l - c~ c~

c

l - c; c~

(8)

. c;;·

l - c ~"

c~

0

l -c~

or Ar<

K

standard running time of i - th train from (k - l) - th to k - th stat i on control with running time of i - th train from (k - l) - th to k - th station mi nimum staying time of i - th train at k - th station

k "

+ u ..

X"'I

+ B r;. uo<

A,= ( C< B = ( C'"

( 3)

(7 )

Using these vectors , eq . (6) can be repr esented as follows .

Xk

r '

I

'

: ut

('

wh e r e

Iu ~ I (u'

I

(9)

"

)

)"

All the state vectors x K can be ca l cu lat ed sequent ially in acco r dance with con tro l u from an i nit ial state vector x ~ using this equation . We ca ll this dynamic model a " Station Sequential Model ", t hat is most c onvenient f o r applying to the fl oat i ng traffic cont r ol . Otherwise , we can derive a " Train Sequential Model " introducing a train sequentia l vector for the s tate and control .

(l 0 )

Then the system dynamics ca n be r ep res ented as follows . ( 11)

Floatin g Tr af fi c Cont r ol

2

D f-

a:

fU1

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(15 )

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3

2

Z

,

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In th ese eq uati ons , p , q , r are weightin g coeff i c i en t s depend in g on th e p ur pose of co ntr o l, Equati o n (15) can be represe nt ed in th e fo llowing quad rati c f o r m,

\ \

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'\[\'

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1\\\ '\ ~ '\ ' '\ \\' "\ '\ ' \

TIME

jU ::> (MIN J

jj

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( 16 )

J

)U

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j

('

5

q': p;

An e xampl e o f unstable floating traffic

- P;

PK =

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-p: q't 2 r,;

- P:, 2

ce

d '/' ~ L :-'11 ,

l

X ~~I

-

a :..;

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3n

.tlv:'. ~<1C l

the t r affic 1:-: nor.:1 cont rn] u~~)

f:cfl c c'j 1!1 t h,~,

c] '_r:1(I t1t

0:

D 'I nClni .:- ~

( '~

sfqU I ' ! l l'

i : lt(' ~-' ;a l

<]l '·;cn

ql~ 2p! "

: ',1 <: '

:i x r"cj

~:r : ()'..·:li

(1 8)

If

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ql\r l< 'I~,l!! '/

as

(1 9)

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is

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Wh e re the feed-back coe ff icie nt matrix F i s given by calculating th e f o llowing Ricatti e quation , backward seque n tially from th e last e quation GK=Pj(.

..., F,,= (5 ,,+ B . G<. , B k ) B" G.",A " G.= P", + AK~ G" A,, - A: G~., B. F"

(2 0 )

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: 1."

•

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totil l trilvel inq int' crv':"11 , :Jun"" ~,.1! i +-.-; t. rJt:~! l 'c!!1rr,d r , ~ " ri ot i n

\·:i ·: " V.J.l" l l. '\ '; · i ~ ; t l 'C';: : :..l I :cd C.~ ( :

The con tro l that optimizes this quadrat i c e valuating f unction und e r equation ( 3) is known asa Linear Regu l ator P r ob l em in contr o l th eo ry , whe r e th e optimal contro l is given in th e following f eed - ba c k con trol

'J< ,": ,.1 ' 1"

I ll'

nC'cessary t o (~limjn ' I'( ~ riv' ·. I! )~· : , a.l. ~ I i ~ , 1Ylldm i c s and to mtl~:lt,l!!"l st ,lLI· I rd;- :'i (" J!1 ' 1 i ! i q ll r. VJIUl"lti!l~J

o

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t( )

' It'Jet'c),:

)

rJ

('( , ! ! t " n ' ]

t.l

_ p~

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" Lumpy 0pel".J.tlnll " I is Id~I'l'/ ~I" r;..- l il : :( ', : \-Jith () r:1jnol- di~.;~urb ~1J !C (' .. :" it; ~ 3 J~; ;In l' X cltn!'](· of the un: ; tcJbl(~· · :";,n.-;:.: i, ·s \vi t il 11') ( "UTlr n ) ] ( \1 \01. (J ) . It shows cvid( '!: i 1 :.' thd" i ll] i ! 1i I i ,j ] d,' ! .1Y o[ ..j ~ ; scco nd~ Clt ~_;t-<1r~ jnq St. .lt iflll qi '.' ( ' tl t (' t ~\ " ~ rll

and 7th trai" ,; 0 iy :-ru tion .

o

\; ( ,. : U!' .

·.> it:~\ I\ J!

n (_~l t :l '_

- P. '

~ ' Vd ;

v, !ri.Zl~- l. '

s t- ,"'!t",

(17)

,

, p l, !

l~ : ~ '

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f 1 ;11!l

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an unstClbl,-. c1yn('c'11 r:~

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to

p:

- P ,~,

o mod e l f;f '. i..3 in,) t;'"j(~

23 1

for Publi c Transp o rtati on Syster:1

" ," , .,

Co ntro l results, We can get a proper op timal control 'by se l ec ting the evaluatin g function in accordance with the p urpo se of an object system , Some examp l es of operational cont r ol under vari ous e va l uating functionsare investigated . Gene rally it is verified that the contro l minimizing t he ave r age waiting ti~ e and the ave r age congestion a the uniform interval ope rati o n al schedule , if the pa s sengers are generated uniforml y . Then we can say that th e unifo rm int e rva l ope r ation ca n maintain a s table traffi c and p r ope r se rvi ce to passenge rs . Th e control in Fig . 4 i s a n example o f min imizing the av e rag e square interval time o f all trains and stat i ons . Th e re we can see c learly that disturbed state of th e unstabl e traffi c has been contro l ed in t o a s tabl e one , sta rting f r om an initial distu rb ed state . Fo r commuters , it would be essent ial that t he dep arture time from their home town and t he arriving time at des ti n ations into thei r offices or schools be fix e d to a certain co nstant time . Then it wou l d be necessary to r e aliz e a p u nctual control corresponding to th e fix e d sch e dule with spec ial train s f o r

232

H. Sasama and Y. Ohk awa lin e A

UJ

2

o I Cl

3 '\

2

U

:z Cl

TRRIN

3 \

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4 5

_\

6

4

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r~

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TIME

\

1\

,jU

0---- - - - - - - - - -- - - - - __-----....,.... 1

;

1\ \

\ \

l'

\ ,\

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\

0 - - -0 - - - 0 - - - - - - - 0 - /

\

IU

(MINJ

l~

.

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.

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e-- - -

2 UJ3

;3lj u<;

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1: rCl ; l'1

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9

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10

k_,

"..

TIME

>-

ko .,.

k.

K- l

K

2'

./

k.,

/

- TYPE(B) -

/'

lin e B

\ 1\

\

\

\

\

line A

•

\

k...,.

1 2 • • • • k ~ " k... K- l K 0 - - -0 - - - 0 - - - - - - -0 - - - 0 - - -0 - - - - - - - 0 - - __ 0

\

or--+---j.....l..---1'\.,---l\'lrr\--->..d--'-<-\+'-<-\--f'-.-\---'kI\---',rt----J..rt-\--1 \ \ \ \ \ \ F'ig . 5

- TYPE(A) -

o---o---o-------o---p---o------_o____ o

\ \ \ \ \ \ , ~~~~~~~ \~\~\~~~~\~\r4I~\+--+~ .c:

>

,/

lin e A

2

\\\ \

\ 1\

\

C90E8

6

K

_ __ _ _ _ _ ____ .__ _

o ..• 0 - - - 0 - - - - - - - 0 2

K- J

,/

. \; ;

- - - -; >

(1' )

11

k ... ,. .

line B

\

1\

I~

2'

\

An e xample of the cont rol u nifo rming train interval

F'ig . 4

~: •

. . . , \; . ',

f\

\. \ \ 1\

\

2

0 - - - 0 - - - 0 - - - - - - - 0 - - -,0 - - - 0 - - - - - - - 0 - - - - 0

\

. .. _--- - - - - - ->... 0 ... 0 ... 0 ....... 0 ,

line B

• • . • (~_, )

(t)

(1' )

'

(MIN J

- TYPE(C) -

An e xamp le of the control punctual for a special train Fig "6

commuters , maintaining uniform stable traffic as much as possib l e . The control for such a case is shown i n Fig . 5 . In the figure broken lines represent the fixed stable schedule to which the train shou l d conv f orm . The fourth t r ain conforms to a fixed sc h edu l e in accordance with a l arge value of weighting coefficient g~ fo r the f;f th train in the evaluationg function (16) . Further , arranging the weighting coeff i cients according to the system purpose , it is possible to realize a cont r o l minimizing the average waiting time over the tota l l ine or a contro l matchig t he ar ival at a certain junction sta t ion . Of course it is possible to rea l ize the traditional operation by simply we i ghting to punctualityin the eva l uating function.

EXTENDED MODEL FOR A CmlPLEX SYSTEM WITH MERGIN G AND BRANCHING Actual t ra nsportation systems may often contain complex traffic lines such as merging and b r anching . We can mOdify this contro l model to make it applicable to these comple x systems . We exam ine A mode l for l ines wit h merging . a system with me r ging at k - th station as shown i n Fig . 6 . Suppose that trains Nos . 1, 3 , 5 , ,,'. a re operated on lin e A and Nos . 2 , 4 , 6 , .. , are operated on lin e B. In this case , th e train p re ceding t he i -t h t r ain is ( i - 2) -t h t r ain be for e merging . Consequent l y equation (3) is chan ged a s follows .

Traffic system with merging ( 21)

Then matri x C is modified as follows

o

l-c~

o

l-c~

o

c~

~

CI< =

l- c ! 0

(22)

c rkl 0

o

c~

l - c!c-' 0 l- c ~

This matrix C with suffix k represents the dynamics of both of trains for k - th station on l ine A and k - th stat i on on line B. The r eafte r, we must change th e inte rv al element in th e eva luating function to

~ ~P; \~'x~'1f for k=2 , 3 , 4 , .. , k , then change the matrlx ~ in equation ( 1 7) as follows .

p.=

I

t~p: <.~p; -~... -~

l

- p~

0 - p,;

0

r'j-2P: 0 - p~ 0 r;"2~ 0 ,.-p~ , (23)

"

, " 0

r~'
0

- P~

- rt' 0 rlo'+rt' 0 •

_p~"

' 0

'r~+EtJ

for k=2 , 3 , ... , k

Modifying the matrices CK and P " for k=2 , 3 , ",k as mentioned above , we c a n app l y the lin ea r r egulator to get the optima l cont r ol u through the same process as the p receding one .

233

System Floatin g Traffic Contro l for Public Tr anspor tation

1\ \11\ \ ~. \ \!1\ \ '{ \1 I~ z;~ ~ \ i~ \ '\:~ \ j \ '\ \,", !\

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20 25 30 TIME (MIN)

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---- -i -

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[MfN)

Fig . 7 shows an e x ample o f simu lation r esults of the floatin g cont rol for th e sys t e m with merging , wher e an initia l delay of 90 second s i s given to 4th trai n at 1s t s t ation . We can see that 3r d train is contro l ed to arrive la te before merging at 6th stat ion for interva l regula tion . Whe n th e numbe r of station s o n line B diffe rs from that on line A, we can ma k e them equa l by introdu cin g p r ope r dumm y s tati o n s . Then the coe ffi cients C KR co rr espond ing to th e dummy station s are set at zero and th e co ntrol u;' for th e dummy sta tion should be decided t o be zero by se tting the corr espond ing coe ffici e nts in matric es P K and QK ta t zero . We ca n apply this techniq ue t o s u c h cases as shown in Fig . 6 . Fo r A model for li nes wi th branch ing. extend ing the model t o a system with branch ing we must r edef in e the d e la y coeff i c i e nt s c as follows

..

c ' . d e l ay coe f . with use both of line A and c ' . de la y coef . with use on ly line A ( c ~' r O ck!.: delay coef . with use only li ne B ( c~ O ",,'

An exa mpl e of th e contro l for th e system with branch in g

Fig.9

An e xample of the cont r ol f o r t he system with merging

Fig . 7

\

G

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~o (f] 7

I

1 1

1

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5 6

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7

5

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-1 i'< .

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-

I

the passen gers who can lin e B ( c ~ =O fo r k) k ~ ) the passeng e r s who c an for train i on lin e B) the passeng e r s who can for train i on lin e a)

Thus , the ma tr ix r ep r esent ing the dynami cs comes become s as f o ll ows .

system

o c~

c'. 1

l - c~ c~\

c::.

( 25 )

l -~ ct

0

l

weight ing coeffi cient s in Therea fter the eva luating fun c tion a r e r e defined as follows weight for int e rval of all trains weight for int erva l of trains on li ne A weight for inte rv al of trains on line B The e valuati ng functio n fo r train inte rval is r ep r esen t ed wit h these coeffi cients . (r; (x~)' +p~ (x~ _ x ~L (r:;. (x~)' +p~ (x~ - x::'

r +p.~ (x~. -<' f)

'I

+Pk~(X~ - X~'z

(26)

t)

or

'1 1

( 27) o ws . Then equati on ( 2 1) is mOdi fi ed as foll - t~- ' ) + C';-A( t~ - t\;Z ) i for train on line t~- r ) + C ':-,1( t;: - t~l ) (t ~i for train on line

S ~=

c ~ (t~

S '= I(

c~

+ A +

D j;.

(24)

2

• k,.,. k b

,

-

-

_ ._ ;;;--

K-l

k h .,

K

---- o --- o

'o --- o ------- o --- o

~"

"..

k \.., • • • • K-l

Fi g . S

'.

'.

c. ,

o --- o ------- o --- ~ --- o -------

.--

o

D"k

lin e A 1

•

B

.. _----_._- ----- - - - - - --

•

p=

li ne B

Traffic system with branch ing

Qk , the matri ces After s etting above the with nce accorda in y ull ef r ca decide th e optima l c an defini tions , we lin ea r the app lying cont rol fl o ating rol cont f o xample e an is 9 . Fig r e gulato r . r es ults f o r the system with branch ing line.

;-

K

-=-

J

P~

Using thise t echn iqu e for the traffic sys t ems with merging and branch ing , we can extend the floatin g con tr o l me thod for a complex system with netwo rk l ines .

H. Sasama and Y. Ohkawa

234

SENSITIVITY TO

SYSTE~

that the disturbance gives no effect on the control re s ults , compa r ed with no disturban ce c a se .

DISTURBANCE

For applying this optimization method to actual traffic control , it is desirable that the control be ins e nsitiv e to any kind of system disturbance . If the optimal control is verified to be stable , this control me thod will be sufficiently applicable to actual traffic systems with many disturbance sources . In the fundamental traffic equation (9), three kinds of diturbances are possible . They are disturbances to the state valu e x, to the contro l value u and to the delay coefficient c . An examp l e of influence of the th r ee kinds of disturbances on the optimal c ontrol eva luati on is shown in Table 1 . Disturbance to the con trol value is thought to give the largest effect on the evaluation . In actual traffic contro l , the actual operation is dif f erent from the optimal contro l calculated theoretically , because a vehicle can not run in higher speed than a limit , and often it stops in accordance with th e signal conditition . In this case the difference acts as a disturbance to the control u for th e control system . The optimal control can not be completely realized , but we can get a sub - optimal control . For example as shown in Table 1 , we can get the sub - optimal control giving one - tenth of average waiting time (i . e . uniformity of train interval) and unpunctuality by controlling as optimally as possible under the given condition with disturbance . As th e second disturbance, the difference between the values in the model and an actual system , acts as a disturbance to the system parameter . Now it is proved theoretically that the feed - back cont rol by linear regulator always decreases th e sensitivity to its parameter and makes the system stable. Therefore , considering th e stability and sensitivity to the disturbance and change of delay rat e c~ this optimal control is found appropriate for this system . Table 1 shows the effect of e~ch kind of disturbance on the control re sults . It shows

disturbance ~(u, T )

o

2

4 5

no contr o l

~(c,

,e) ',ith x

disturbed with all of c , x , u no distu r bance

uniformi ty of train interval

consequently , the total influence of disturbance to the state vector x. is the accumulation of each sing l e step . In Table 1 , that the effect of a disturbance we can see given to the XK is less than that of a disturbance given to the control value u R In actual system, those three disturbances influence the contro l not separately but compositely . In this case the effects are not always accumulated but often counteract each other . Then the control result is improved as shown in the fourth case of Table 1 .

SIMPLIFICATION OF CONTROL CALCULATION For applying this method to actual traffic control, it is d es ira eble to make the control software and hardware as simple as possib le . Therefore it is desired that a mini - computer or micro computer be employed as the processor to calculate the control value u . From that point of view , th e calculation of large matrix for actual traffic contro l is not practical , but a subop timal control will suffice . Looking i nto the feed - back matrix F", which is adequate for the theoretically opti mal control , the e lem ents on th e diagonal and their neighbors are found to have significant values, but the others have negligibly small values , compared with those shown in Fig . 1 0 In the station sequence model for example , the precisely o9timal control u is represented by th e following equation , where f . j is the i - j e l ement of the fced - back matrix .

TABLE 1 Effects on the control result ~Ith each kind of disturbance ~o .

The third one i s a disturbance to state actual con trol system , variable x . In forecasted values are used for s ome parts of the difference between state vector x , and fore c asted values and realiz ed ones acts as a kind of the disturbance to x. In the linear resulator method , the feed - back matrix of each control stage is decided theoretically by dynami c programming method . If state vector x~ in a certain control stage is given an arbitary value , the optimal feed - back contro l matrices F, , Fe , .. , F t<. for the later contro l stage a r e not affected .

punctuality of train

100 . 0

100 .0

1.6

3.0

6.2

2.8

10.7

16.1

9.0

14 . 1

2.1

4.4

27 - 13 - 2 - 0 - 0 - 0 - 0 - 0 - 0 - 0 -13 27 - 13 - 2 - 0 - 0 - 0 - 0 - 0 - 0 - 2 - 13 26 - 12 - 2 - 0 - 0 - 0 - 0 - 0 - 0 - 2 - 1 2 29 - 12 - 2 - 0 - 0 - 0 - 0 - 0 - 0 - 2 - 12 29 - 12 - 2 - 0 - 0 - 0 F.= O. Ol - 0 - 0 - 0 - 2 - 12 29 - 12 - 2 - 0 - 0 - 0 - 0 - 0 - 0 - 2 - 12 29 - 12 - 2 - 0 - 0 - 0 - 0 - 0 - 0 - 2 - 12 29 - 12 - 2 - 0 - 0 - 0 - 0 - 0 - 0 - 2 - 12 28 - 14 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 2 - 14 17 Fig ,

10

Example of feed back matrix F

Float i ng T r af fi c Co ntr ol fo r Pu b l i c Tr a n spo r ta ti o n Svstem

',.(~ - (:, x ,

+ •• +:L\'· ' X ~ :: +:. \. x ~\, +f~ .. ,x~.·.', +

. . + f \. lx,,\ ) (28 )

I~ t~is e~~ation t~e coe f fi cien t values o t~cr L '.a:' f" ."f .. a:1d f ...., are aFicr o xi matel ', eC;:..la l to z e rc . T~e r efo r e , we can use the ~ecd - back control val:..le J calcula~ed ~ ith onlv c~ r ee elei.le r.~s x~ ~; x; .. and x~ ~: , neglecting t he others , in place of the optimal feed - back ~ ith all e l ements . Then J ~ is r e~ r esen t ed as follo,,'s .

(29) Or it i s possib l e to rep r esent th e f eed - b a ck mat r i x F in app r o x imation for mul a . f, , f'l f" f u f " f ,. f " f, . f .. , f~,

C O ~ CL~D I~ G

RE ~ AR K

',': e l_r o ,cosed an ope r ational control by "loatoing Traf : ic Co!~trol ' lethod for i:1tra - city t r ans~o r tation system with hi gh tra:fic density , and e xa mined t he t r af f ic d '/namics and the optimal cO:1t r ol using the ::1cthod .

Fr om t he e x amination it is con c luded t hat t he optimal cont r ol by li near r egu l ato r can r 0alize unifo rm inte r va l ope r a t ion , ma t ching a~ a junction , convent i ona l a nd pu n c t ua l ope r atio n , by adj u s tin g th e we i g htin g coeffic i ents o f t he e v a l uat i ng f u n ction i n accordance with t he t r anspo r t a tion demands .

o ~ o r eo v e r ,

f~L,

(30 )

F v. = f l l il

o

235

,

f i l. ll f. d f~ 1' 1 f i j

t he influ e n ce on th e co ntro l r es ult u nde r s y stem d i s tur ba n ce h as been i n v es ti g at ed and va r ious simp li ficat i o n s o f t he contro l calc ul a ti o n a r e t r ied . Th ey can ma ke t h i s method mo r e p r ac ti ca l a nd u se fu l.

.i J

I n ac tual tr a ffi c contro l , some p arts o f el e me n t s i n th e s t a te v ec to r x . mu st be f o r eca st ed t o ca l c ulat e t he con t r ol u a cco r d i ng t o e quati o n ( 9 ) . As th e f o r eca st in g l o gi c is comp l icat ed and tr o u b l e s o me , we c a n mak e a furth e r simplification of ne gl ec ting x~, Th e nwe h a v e

( 31 ) Simu l a ti ng a s impl i fi ed contr o l and comp ar i n g t he result wit h t ha t of th e t heo r e ti ca l op timal contr o l , we c an r e a c h t h e con c lu s i o n th a t a s imp l ifi ed c o ntrol s y s t em is s uffi c i e ntly p r a ctica l for actual t raffi c contr o l.

REFERE NCE I!. SA SAHA , Y. OHKA'.JA ; ( 1 97 9) Fl o a tin g Tr aff i c Co nt r o l f o r T r an s po r tati o n Sy st e m. Qua r te r y Re p o r t , R. T .R.I., J , K. R. , vol . 20 , No .3 'pp . 122 - 125 H .SASA}~ (1981 ) . Dive r s it i c a t io n in Tra n s portation Sy stem . J . So c . Inst ru m. & Cont r ol En g . Vo 1. 20 , ~o .l, pp . 82 - 86 S . ARAYA , S . SO~E ; Tr aff i c Dy n amics of Aut o ma t e d Gu i d e way Tr a n s it Sys t e m, IEEE Tr a n s . Sy st ., Ma n & Cy b e r n. . W. S. LEVINE , H . ATHANS ; ( 1966) On th e Op timal Err o r Reg ula t ion o f a St r ing of Moving Ve h ic l e s , IEE E Tr a n s . Aut o m. Co ntr o l, Vol.AC - ll , No . 3