DSpace at VNU: Control problem on timed place transition nets

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 an g Chi T h a n h College o f Science, V N Ư A b s t r a c t: D esign, a n a ly s is and control of large sy ste m s are com plicated because of their state space explosion and behaviours In th is paper we solve the control problem on Tim ed P lace/T ransition nets by piopoòing a concurrent composition, which is a good solution for syste m s design We also show th at the safety and aliveness of Tim ed P lace/T ransition nets arc preserved by the controller Increase of concurrency in com posed Timed Place/Transition nets is also considered K e y w o r d s : P e tr i net, control, composition, concurrency, safety, aliveness and concurren t step In tr o d u c tio n Control on c o n c u rre n t sy ste m s and its a p plication a re a problem co ncentrating much of in te re s t T he g e n eral control problem on sy s te m s w as form alized by L Alfaro, T A H e n z in g e r a n d F Y c M ang in [ ] T h is probl d ll is m eaningful to c o n cu rren t sy ste m s in th e com positional level T he m a in foundation of design, analy sis and control problem is a composition o peratio n T he o peratio n depends so much on the k in d of sy ste m s T herefore, sy ste m s p ro p e rtie s preserv ed by controller may be d if fe re n t1 To date we h a v e some good m a th e m a tic a l models for re p re s e n tin g concurrent system s One of th e m , w hich w as proposed e a rlie s t an d vigorously investig ated is the net model in tro d u c e d by C.A P etri The inode], we will use for the control problem is Tim ed P la c e /T n s itio n net We recall some n o ta tio n s concerning Petri nets, which w ere defined in [2,3,6] A Petri net is a trip le N = (P, T, F), w here p, T a re disjo in t sets and F c (PiT) u (TIP) is a rela tio n , so-called the flow relation of th e n e t N Let N = (P, T, F) be a P e tri net N For a n e le m e n t X g Xn, we denote: *x = { y G XN I y F X* = { y e XN I X X } F y } xw = P u T , is the se t of all ele m e n ts of and it is called the p re-set ot‘ X, and it is called the p ost-set of X, a n d they a re s im ila r for a s u b s e t of XN A n e t is sim p le i f a n d o n ly i f i t s t w o d if f e r e n t e l e m e n t s h a v e no c o m m o n p r e ­ s e t a n d p o s t- s e t This paper is supported by the N a tio n a l N a tm a l Science C o u ncil o f V ie tn a m u n d er the project nr 3 48 the ne 49 Control Problem on T im ed Pla c e/T ran sition nets A sim ple n e t h a s been being used to re p re s e n t s ta tis tic a l s tru c tu re of a system From a sim ple n e t one can c o n stru ct d ifferen t n e t m odels by adding some aspects The T im ed P la c e /T n s itio n n e t is such a net I t is th e P la ce /T ran sitio n net introduced in [2 ], ad d ed a d u r a tio n function a n d is defined a s follows: D e f i n i t i o n 1.1 T he -tuple I = (P, T, F, K, M°, w , D) is called a Timea Place I T n sitio n net (Tim ed T/P net, for short) iff: ) N = (P T F) is a sim ple n et, w h e re a s a n e le m e n t of p is called a place and an e l e m e n t o f T IS C3.116CÌ a t r a n s i t i o n ) K ' p -> N u {oc} is a function show ing a capacity on each place ) w ■ F -> N\ {*} is a function a ssig n in g a w eight on each arc of 4) M° : p the flow N u ỊocỊ is an in itia l m a r k in g , w hich is n o t g r e a te r t h a n capacity on places, i.e.: V p e p , M°(p) < K(p) 5) D • T {[a b] I < a < b} is a function p o in tin g o u t a d ura tio n , in which the tr a n s itio n will be p erfo rm ed only The in itia l m a r k in g r e p r e s e n ts given to k en 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 la c e I f t o k e n s o n e a c h placf belonging to th e p re -s e t of some tra n s itio n a re g r e a te r t h a n or e q u al to weight 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 th f initial m a r k in g can a c tiv a te th e c orrespo nding tra n s itio n A fter perform ing tht tra n s itio n , to k en s on each place belonging to th e p re -s e t of th is tra n s itio n art decreased by w e ig h t of th e arc connecting th e c o rre sp o n d in g place to th is transition and to kens on each place b elonging to th e p o st-se t of th is tr a n s it i o n a re increase* by w eight of th e arc c o n necting th is tr a n s itio n to th e c o rre sp o n d in g place It m u a be e n su re d t h a t new to k e n s a re not g re a te r t h a n th e c ap a city of t h a t place W h en th e in itia l m a r k in g a ctiv a te s some tr a n s it i o n in s u ita b le tim e, thỉ tra n s itio n is p erfo rm ed a n d th e n we get a new m a rk in g , th e new m ark in g cai activ ate a n o th e r tr a n s itio n a n d th e process re p e a te d ly c o n tin u e s in such a w a \ Therefore 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 as follows: T he m a rk in g M : p —> N u {oc} can a c tiv a te a tra n s itio n t 1) Vpe*t , iff: 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 rk in g M is so-called t-a ctiva tin g the tr a n s itio n t, we get th e following new m ark in g: M ( p ) -W ( p ,t) i f p e 't A t * V M(p) + W(t,p) if pet* V t M (p ) = i M(p) - w (p, t) + W(t, p) if pet* \ t |M(p) otherw ise and we often w rite th a t: M[ t > M A fter perform ance >f H o a n g Chi T h a n h T he m a rk in g M ’ can activate some oth er tra n s itio n and th en we get a n o th e r ,avking M ” The set of all m arking s reachable from the m a rk in g M is denoted by FM] In [5] we applied th e control problem on C ondition/E vent system s In this jiper, we solve the control problem on Timed P lace/T ransition nets, which a re one , no dels u su a lly used to re p re se n t real system s The problem is defined as follows: Given a T im ed Place I Transition net 1A (a plant) Ịace/ 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 th e c o m p o s e d n e t ■•ịộr d e f i n e d p r o p e r t i e s F in d a Tim ed I A # Ĩ.ỊỊ m e e t s the We show t h a t the safety and aliveness of Tim ed Place/T ransition nets are reserved by the controller Increase 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 lace/T ran sitio n n e ts and show th a t the safety a n d aliveness a re preserved y the composition Section proposes the notation of a cu rrent step and jrsiders in c re a sin g of concurrency in composed Tim ed Place/T ransition nets, ira'.ly, some conclusions and directions for future research are given in Section C om p osition o f tim ed P/T n ets Given two Tim ed P/T nets I, = (Pi, T„ F„ K„ M,°, w „ Dj) , i = , , we compose vv> these n e ts by the following way D e f i n i t i o n 2.1 The Timed P/T net I = (P, T, F, K, M°, w, D) , where: 1) p = P , u P , 2) T = T , u T , 3) F = F , u F, 4) olovs The w eight function w on arcs of th e flow rela tio n F is determ ined as w ,(e) ,if eeFj \ F W(e) =