VNU. JO U R N A L O F S C I E N C E , Mathematics - Physics, T.xx, N04, 2004
CONTROL PROBLEM ON TIMED PLACE/TRANSITION NETS
H o a n g C h i T h a n h College o f Science, V N Ư
A b s t r a c t : D esig n , a n a ly s is and control of large s y s te m s are com plicated because of their s ta te space ex plo sio n and behaviours. In th is paper we solve the control problem on T im ed P la ce/T ra n sitio n nets by piopoòing a concurrent composition, which is a good so lu tio n for s y s te m s design. We also sh o w t h a t the safety and a liv e n e ss of T im ed P la ce/T ra n sitio n nets arc preserved by the controller. Increase of concurrency in com posed T im ed P lace/T ransition n e ts is also considered.
K e y w o r d s : P e t r i n e t, control, composition, co n cu rren c y , safety, a liv en ess and c o n cu rren t step.
1. I n t r o d u c t io n
Control on c o n c u r r e n t s y s te m s a n d its a p p lic a tio n a r e a problem c o n ce n tra tin g much of in te r e s t. T h e g e n e ra l control problem on s y s te m s w a s form alized by L.
Alfaro, T. A. H e n z in g e r a n d F. Y.
c.
M a n g in [1]. T h is probl d ll is m eaningful to c o n c u rre n t sy s te m s in th e com positional level. T h e m a in fo u n d atio n of design, a n aly sis a n d control pro b lem is a com position o p e ra tio n . T h e o p e ra tio n d ep en d s so much on th e k in d of s y s te m s. T h erefo re, sy s te m s p r o p e r tie s p re se rv e d by controller may be d i f f e r e n t1.To d a te we h a v e som e good m a th e m a tic a l m odels for r e p r e s e n tin g c o n cu rren t system s. One of th e m , w hich w as proposed e a r lie s t a n d vigorously in v e s tig a te d is th e n et model in tro d u c e d by C.A. P e tri. T he inode], we will u se for th e control problem is T im ed P la c e /T r a n s itio n net. We recall some n o ta tio n s co ncerning Petri nets, which w ere d efin ed in [2,3,6].
A Petri n et is a tr ip le N = (P, T, F), w h ere p, T a r e d isjo in t sets a n d F c (PiT) u (TIP) is a re la tio n , so-called the flo w relation of th e n e t N.
L et N = (P, T, F) be a P e tr i net. x w = P u T , is th e s e t of all e le m e n ts of th e n et N. For a n e le m e n t X gXn, we denote:
*x = { y G XN I y F X } a n d it is called th e p re -s e t ot‘ X, X* = { y e XN I X F y } a n d it is called the p o st-se t of X,
a n d th ey a re s im ila r for a s u b s e t of XN.
A n e t i s s im p le i f a n d o n l y i f i t s t w o d i f f e r e n t e l e m e n t s h a v e n o c o m m o n p r e s e t a n d p o s t - s e t .
T his paper is supported by the N a tio n a l N a tm a l Science C o u n c il o f V ie tn a m u n d er the project nr. 2 3 0 2 0 3
48
A sim p le n e t h a s b e en b ein g u sed to r e p r e s e n t s t a t i s t i c a l s tr u c tu r e of a system . F rom a sim p le n e t one can c o n s tru c t d iffe re n t n e t m o d els by adding some asp ects T he T im e d P la c e /T r a n s itio n n e t is such a net. I t is th e P la c e /T ra n s itio n n et in tro d u c ed in [2], a d d e d a d u r a t i o n fu n ctio n a n d is d efin ed a s follows:
D e f i n i t i o n 1.1. T h e 7-tu p le I = (P, T, F, K, M°, w , D) is called a Timea Place I T r a n sitio n net (T im ed T/P n e t, for sh o rt) iff:
1) N = (P T F) is a sim p le n e t, w h e r e a s a n e le m e n t of p is called a place and
a n e l e m e n t o f T IS C3.116CÌ a t r a n s i t i o n .
2) K ' p -> N u {oc} is a fu n ctio n sh o w ing a cap a city on e a c h place.
3) w ■ F -> N \ {*} is a fu n ctio n a s s ig n in g a w e ig h t on e ach arc of the flow 4) M° : p N u ỊocỊ is an in itia l m a r k in g , w hich is n o t g r e a t e r t h a n capacity on places, i.e.: V p e p , M°(p) < K(p).
5) D • T {[a b] I 0 < a < b} is a fu n ctio n p o in tin g o u t a d u ra tio n , in which th e t r a n s it i o n will be p e rfo rm e d only.
T he in itia l m a r k i n g r e p r e s e n ts given to k e n s on each place of a net. The tokens
a r e n o t g r e a t e r t h a n t h e c a p a c i t y o f t h e c o r r e s p o n d i n g p l a c e . I f t o k e n s o n e a c h placf
belonging to th e p r e - s e t of som e t r a n s i t i o n a re g r e a t e r t h a n or e q u a l to w eight o'
t h e a r c c o n n e c t i n g t h i s p l a c e t o t h e t r a n s i t i o n , i . e. i t i s e n o u g h f o r “p a y i n g ” , t h e n t h f
in itial m a r k i n g c a n a c tiv a te th e c o rre sp o n d in g tr a n s it i o n . A fte r p e rfo rm in g tht t r a n s itio n , to k e n s on each place belo n g in g to th e p r e - s e t of t h is tr a n s itio n art decreased by w e ig h t of t h e arc c o n n ectin g th e c o rre sp o n d in g p lace to t h is transition and to k en s on each place b e lo n g in g to t h e p o st-se t of t h is t r a n s i t i o n a r e increase*
by w eig h t of th e arc c o n n e c tin g t h is t r a n s it i o n to th e c o rre s p o n d in g place. I t m u a be e n s u r e d t h a t new to k e n s a r e not g r e a t e r t h a n th e c a p a c ity of t h a t place.
W h e n th e i n it i a l m a r k i n g a c tiv a te s som e t r a n s i t i o n in s u ita b le tim e, thỉ t r a n s itio n is p e rfo rm e d a n d t h e n we get a new m a r k in g , t h e new m a rk in g cai a ctiv a te a n o th e r t r a n s i t i o n a n d th e process r e p e a te d ly c o n tin u e s in such a w a \ T h erefo re th e a c tiv itie s h a p p e n e d on a T im ed P/T n e t will be m athem atically form alized a s follows:
T h e m a r k in g M : p —> N u {oc} can a c tiv a te a t r a n s i t i o n t iff:
1) V pe*t , M(p) > W(p, t) a n d 2) Vpet* , M(p) < K(p) - W(t, p).
In such a case, th e m a r k i n g M is so-called t-a ctiv a tin g . A fter p erfo rm an ce >f th e t r a n s i t i o n t, we get th e following new m a rk in g :
M ( p ) - W ( p ,t) i f p e 't A t * V M(p) + W (t,p) if p e t* V t M ( p ) = i
M(p) - w (p, t) + W(t, p) if pet* \ t
|M (p) otherw ise
Control P ro b lem on T i m e d P la c e / T r a n s i t i o n net s 49
an d we often w rite t h a t: M[ t > M
9 H o a n g Chi T h a n h T h e m a r k in g M ’ can a ctiv a te some o th e r tr a n s itio n a n d th e n we get a n o th e r ,avking M ”... The set of all m a rk in g s reachable from th e m a rk in g M is d enoted by
F M].
In [5] we ap p lied th e control problem on C o n d itio n /E v en t system s. In this jiper, we solve th e control problem on Timed P la ce /T ran sitio n nets, which a re one , n o d e ls u s u a lly used to r e p r e s e n t real system s. The problem is defined as follows:
Given a T im e d Place I Transition net 1A (a plant). F in d a Tim ed
Ịa ce / T r a n s i t i o n n e t Ĩ.ỊỊ (a c o n t r o l l e r ) s u c h t h a t t h e c o m p o s e d n e t I A # Ĩ.ỊỊ m e e t s t h e
■•ịộr d e f i n e d p r o p e r t i e s .
We show t h a t th e safety a n d aliveness of Tim ed P lace /T ran sitio n n ets are reserved by th e controller. In crease of concurrency in composed Timed lace/Transition n e ts is also considered.
This p a p e r is organized as follows, In section 2, we define a composition on iried P la ce /T ra n s itio n n e ts and show th a t the safety a n d aliv en ess a re preserved
y the com position. Section 3 proposes the n o tatio n of a c o n cu rren t step a n d jrsiders in c re a s in g of concurrency in composed T im ed P lace /T ran sitio n nets, ira'.ly, some conclusions and directions for fu tu re re se a rc h are given in Section 4. . C o m p o sitio n o f tim e d P/T n e ts
4) The w e ig h t function w on arcs of th e flow re la tio n F is determ in ed as olovs
Given two Tim ed P/T n ets I, = (Pi, T„ F„ K„ M,°, w „ Dj) , i = 1, 2, we compose vv> these n e ts by the following way.
D e f i n i t i o n 2.1. The Tim ed P/T n et I = (P, T, F, K, M°, w , D) , where:
1) p = P , u P , 2) T = T , u T , 3) F = F , u F,
w ,(e) , if eeFj \ F 2
W(e) = < min(W1(e),W2(e)) ,ife e F , n F2 w2(e) ,if e e F2 \ F j 5) And th e cap acity function on places is defined as follows
K t (p)
K(p) = ■ max(Kj (p),Kjj(p)) K2(e)
, if p 6 Pj \ P2 ,ifp € Pj n P2 , i f p e P * \ P , 6) The in itia l m a rk in g is combined like t h a t
M?(p)
M °(p)= mcx(M ?(p),M“(p)) M"(p)
,i f p e p, \ P 2 ,if p e p, n P2 ,if pe P2 \ p ,
Control Problem on Tim ed Place/Transition nets 5
7) At the end, th e d u ra tio n function for perform ance of e ach tran sit,i(n i d e te rm in e d in th e following way
Dj(t) jifteT^NTg
D(t) =
ị
D | ( t ) n D 2(t) ,ifteT\ n T 2 D2(t) ,if tGT2 \ Tịis called a composition of two n e ts Zj and
z2.-
And th e n we denoteI = I , # 12-
K1 M l 0
Pl 3 3
P: 2 0
P; 8 0
P: 4 1
D1
n [0.1CO
tj [2,8]
tj [2,6]
K2 M°2
p* 5 5
Pi 7 1
P' 5 0
D2 ti [1,12]
t* [5,10]
tj m
Note t h a t th e w eight function w , the capacity function K, th e in itial m a i ũ n M° and the d u ra tio n function D of th e composed n e t h ave been b e in g const]ricte<
from corresponding asp ects of two com ponent n ets in th e following way: I r th p a rtia l dom ains we choose values of th e corresponding fun ctio n on all pa-tia
e l e m e n t s , w h i l s t in t h e c o m m o n d o m a in w e c h o o s e o n ly m i n i m u m ( m a x i m U.1 o
intersection) of v alu es of th e corresponding function on comm on e le m e n t s . Phi.
illu s tr a te s th e g en eral principle of composition proposed by th e a u t h o r in [4].
The control problem on Tim ed P/T n e ts is solvable a n d a T im ed pi,rỉ ne alw ays can control a n o th e r Timed P/T net.
E x a m p l e 2.2. C onsider two Timed P/T*nets given in F ig u re 1. a) a n d b).
a) b)
Figure 1: Two Timed P/T nets
In s u c h a case, t r a n s it i o n s in th e step
u
can be p e rfo rm e d c o n c u rre n tly and f.fter t h e i r p e rfo rm an c e we get th e following m ark in g :54 H o a n g Ch i T h a n h
M ’(p)
M ( p ) - X W(p,t) , i f p e #u \ i r
teU
M(p)+ X W(t,p)
,if
peU*V u
t € u
M ( p ) - X W (p ,t)+ X W (t,p) ,if p e * ư n u *
teư teư
M(p) ,otherwise.
We also denote th a t: M[
u
> M ’ a n d th e m a r k in g M is called U -activating.Such as above, we can find ste p s se q u en c es on th e net. As big a re th e ste p s as high :oncurrency is.
L et M°[ Uj > M 