A Dioid Linear Algebra Approach to Study a Class of Continuous Petri Nets

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1 A Dod Lnear Algebra Approach to Study a Class of Contnuous Petr Nets Duan Zhang, Huapng Da, Youxan Sun Natonal Laboratory of Industral Control Technology, Zhejang Unversty, Hangzhou, P.R.Chna, 327 {dzhang, hpda, yxsun}@pc.zju.edu.cn Abstract. Contnuous Event Graphs (CEGs), a subclass of Contnuous Petr Nets, are defned as the lmtng cases of tmed event graphs and Tmed Event Multgraphs. A set of dod algebrac lnear equatons wll be nferred as a novel method of analyzng a specal class of CEG, f treated the cumulated toen consumed by transtons as state-varables, endowed the monotone nondecreasng functons pontwse mnmum as addton, and endowed the lower-semcontnuous mappngs, from the collecton of monotone nondecreasng functons to tself, the pontwse mnmum as addton and composton of mappngs as multplcaton. As a new modelng approach, t clearly llustrate characterstc of contnuous events. Based on the algebrac model, an example of optmal Control s demonstrated. Introducton It s well now that max-plus algebra s powerful tools for Dscrete Event Dynamc Systems (DEDS) [-3]. Mn-plus algebra s the dual of max-plus algebra, and they are both dod, an dempotent semrng. Lnear model s so popular that t s adopted by most of classc control theory. Many evdences support that max-plus algebra, mn-plus algebra, dod and some other dempotent semrngs are approprate mathematc tools to descrbe the phenomena of DEDSs, especally synchronzaton. Wth the operatons of above algebrac systems, some logc nonlnear formulae come to lnear formulae. As a byproduct of the lnearzaton, the exstence of an eventual perodc regme s readly obtaned, the performance beng characterzed n terms of nvarants of the orgnal net [3]. An event graph s a Petr net such that each place has only one nput arc and one output arc. Tmed Event Graphs (TEG) are a subclass of event graphs satsfed that the toens have to stay a mnmum amount of tme n the places. The dynamc of TEGs can be represented as a max-plus lnear algebrac model []. Tmed Event Multgraphs (TEMGs), each arc has an ntegral weght, are extensons of TEGs. The max-plus lnear algebrac model of TEMGs was bult by G. Cohen [2] and HuaPng Da [3] n dfferent way respectvely. By those lnear formulae, the analyses of TEGs and TEMGs are analogous to tradtonal lnear system theory.

2 2 Duan Zhang, Huapng Da, Youxan Sun Comparng wth TEGs or TEMGs, Contnuous Event Graphs (CEGs), proposed by R. Davd and H. Alla [4-6], have contnuous places and contnuous transtons. As a lmtng case of TEMGs, CEGs are appled to descrbe the contnuous events system [7-], for example the flow control and congest control of TCP [2]. Unfortunately, the buldng of algebrac model for CEGs s more dffcult than TEMGs. G. Cohen [2] nduce a mn-plus algebrac model consdered only the case of contnuous places, but no the case of contnuous transtons. There s no unversal algebrac model s constructed for CEGs so far. In Secton 2, the defnton of CEG s revewed. Some propertes of CEGs wll be presented n Secton 3. A dod lnear algebrac representaton of a class of CEGs s gven n Secton 4 and an example wll tacled n Secton 5 wth the new model. 2 Defnton of CEG Let R s the set of real, ϖ =, R = R { ϖ} and R = [, ) { ϖ}. Defnton. A CEG s a 6-tuplet: < PT,, RWV,,, M > ; P s a set of contnuous places; T s a set of contnuous transtons; R = RPT RTP, where R PT P T and RTP T P, < n, n2 > R represent that n s a nput of n 2 and n 2 s a output of n ; W : R R s the weght of R ; V : T R s the maxmum frng veloctes of transtons; M : P R s the ntal marngs of places; every place has only one nput transton and one output transton; moreover we assume that every transton have at least one nput place and one output place p t 2 p t p 2 Fg.. A CEG CEGs can be llustrated by drected graphs (Fg. ) that denote contnuous places as double crcle, denote contnuous transtons as rectangle, and denote the elements of R as arrow. Defnton 2. Let t be a transton, t = { p P < p, t > R PT } s called the preset of t, t = { p P < t, p > R TP } s called the postset of t ; Let p be a place,

3 A Dod Lnear Algebra Approach to Study a Class of Contnuous Petr Nets 3 p = { t T < t, p > R TP } s called the preset of p, p = { t T < p, t > R PT } s called the postset of p. Defnton 3. Let Mar( p, ) denote the toen of place p at the epoch, V(, t ) denote the frng velocty of transton t, then: ) V(,) t = ; 2) f p t Mar( p, ) >, then V(, t ) = Vt (); 3) f p t Mar( p, ) =, denote T = { t p p t and Mar( p, ) = }, then V( t, ) = mn{mn{ V( p, ) W( p, p) / W( p, t)}, V( t)}. Note that f there s a p T loop satsfed that for any place p n the loop Mar( p, ) =, then for any transton t n the loop, V(, t ) =. Remar. The frng of transton t can lead change of the toen of some places: ) f p t, then W ( t, p) V ( t, ) d represents the mar nput p from t n 2 tme nterval [, 2 ]; 2) f p t, then W( p, t) V( t, ) d represents the mar nput t from p n 2 tme nterval [, 2 ]. 3 Propertes of CEGs p t W ( p, t) W ( t, q) q p n W ( p n, t) W t, q ) ( m q m Fg. 2. A contnuous transton Defnton 4. M () :[, ) R s a monotone nondecreasng functon: ) f t T, M ( ) s the total mar t consumed from the epoch to, namely Mt ( ) = V( t, u) du ; t

4 4 Duan Zhang, Huapng Da, Youxan Sun 2) f p P, M ( ) s the total mar entered p from the epoch to, namely M M p V t u du = ; p ( ) ( ) (, ) p Then we consder transton t wth t = { p, p2,..., p n } and t = { q, q2,..., q m } as Fg.2. Proposton. For the nput places of t, M ( ) = Mar( p, ) M ( ) W ( p, t) =, 2,..., n ; () p t for the output places of t, M ( ) = M () M ( ) W( t, q ) =,2,..., m. (2) q q t Proposton 2. Let t T, there exsts > satsfed M ( ) = M ( ) mn{ V( t), M ( ) M ( ) W( p, t), =,..., n}. (3) t t p t Proof. Assume that Mar( p, ) > =,2,..., n. let Mar( p, ) < < mn{ }, W ( p, t) V ( t) (4) then V ( t, ) = V (t) holds n tme nterval [, ]; and M p ( ) M ( ) W ( p, t) M ( ) M ( ) W ( p, t) > V ( t), (5) t p thus (3) satsfed. Now consder the case that p t Mar( p, ) = : ) f there exsts a > such that mn{ Mar( p, )} = holds for all, then Defnton 3 mples (3); 2) else there exsts no > such that mn{ Mar( p, )} = holds for all. As the lmtng case of Mar( p, ) > =,2,..., n, nequalty (3) stll holds n ths case. It s mpossble to represent the equatons n Proposton 2 as max-plus algebrac lnear form, snce there have some multplcaton operatons that are exponental operatons under max-plus algebra. t 4 Dod Representaton of a Class of CEGs M s a collecton of functons, λ M λ:r R, and λ s monotone nondecreasng. A mnmzaton operaton n R s defned by

5 A Dod Lnear Algebra Approach to Study a Class of Contnuous Petr Nets 5 r r R r r = mn( r, ). (6), 2 2 r2 Defne the operaton n M as pont wse mnmzaton ( λ ( ), λ ( )) λ, λ2 M λ ( ) λ2 ( ) = mn 2. (7) Obvously, λ M λ ( ) = λ ( ) and M. Proposton 3. M, s a monod, the dentty element s, s commutatve and dempotent. Proposton 4. { λ} M λ ( ) M, namely M s complete. A partal-order s nduced from as follow: Defnton 5. Mappng λ, λ M λ ( ) λ ( ) λ ( ) λ ( ) = λ ( ). (8) f: M M s sad to be lower-semcontnuous (l.s.c.) [3] f {λ f ( λ ( )) = f ( λ ( )) } M. Proposton 5. All the l.s.c. mappngs on M, denoted F, can be nduced a dod wth two operatons, and, defned as follow: f, f2 F λ M ( f f2)( λ) = f( λ) f2( λ) ( f f )( λ) = f ( f ( λ)). 2 2 Accordng to Proposton 4 and Proposton 5, the followng holds. Corollary. F s Complete. For brefness, may be omtted. Proposton 6. Let N and >, x M be a varable vector, v M be * constant one, and A F, the least soluton of equatons x = Ax v s x = A v, where A * = = A []. Defnton 6. λ M ) add mappng [ r ], r [, ) : [ r ] λ ( ) = λ( ) r ; 2) multplcaton mappng [ r ], r R : [ r ] λ( ) = r λ( ). The symbol [ ] may be omtted f no ambguty. Note that dentty element of < M, >. Let F = {[ r ] r [, )}, F2 = {[ r ] r R }, and Theorem. m n F = { f f F F, j, m, n N} F F.. j j 2 = j= and (9) () are both the ()

6 6 Duan Zhang, Huapng Da, Youxan Sun Proof. Because all the elements of F F2 s l.s.c.. Gven a class of CEGs satsfed Wt (, p) t T max{ Vt ( ) t ( t), p = t } Vt ( ). W( p, t) Theorem 2. Performances of the class of CEGs above mentoned could be descrbed as lnear algebrac equatons. Proof. Consderng a transton t TC, gven that ( t) = { t t ( t)} TC, as above assumed. Let v ( ) = V ( t ), p = t M p () W ( t, p ), m =, w =. W ( p, t ) W ( p, t ) The followng equaton holds. M p () M ( ) (, ) t W t p (3) Mt ( ) = mn{ V( t ), }. W( p, t ) In fact, accordng to (2), on the one hand once a transton t satsfed Mar (, ) =, st.. Mar( t, ) = ; On the other hand, let t = = mn{ t Mar( t, ) = }, then. (3) can be represented as follow: Let = t t2 t T M ( ) [ m ][ w ] M ( ) v t t (2) =. (4) x (,,..., ) T and v = ( v, v2,..., v ) T T. (4) mply x = Ax v (5) where A = ( ) s a mappng matrx, and a = [ m ][ w ]. a j Corollary 2. The least soluton to (5) s j j j * x = A v. 5 Example of optmal Control A manufacturng system represented by a CEG s shown n Fg. 5, where V ( t ) =, V ( t 2 ) =, V ( t 3 ) =, V ( t 4 ) =, M () =, M () =, M (), M () =, M () and M p () = m. p4 p 5 = 6 p p2 p 3 = Usng Theorem 2, the followng equatons holds Mt ( ) ( ) ϖ ϖ ϖ Mt (6) Mt ( ) ( ) 2 ϖ m ϖ Mt 2 M( ): = : ( ) Mt ( ) = ( ) 3 ϖ ϖ Mt = AM v. 3 Mt ( ) ( ) 4 ϖ ϖ ϖ M t 4

7 A Dod Lnear Algebra Approach to Study a Class of Contnuous Petr Nets 7 p 6 t p t 2 p 2 t 3 p 4 p 5 p 3 t 4 Fg. 3. Example of represented a CEG by dod Equatons Accordng Corollary 2, the least soluton to (6) s ( m ) ( m 2) m ( m ) Av=. 2 2 ( m ) The soluton ndcate that m are cannot affect the frng velocty of transton t 3, but that of t, t2 and t 4. (7) 6 Concluson In ths note, a lnear algebrac model F,, for performance evaluaton of a class of Contnuous event graph has been developed. And an nterestng and useful result of the example gves us a novel approach to compute the mnmum (optmal) ntal toens of places n a CEG by ts algebrac model. Acnowledgement Ths wor was supported by Natonal Natural Scence Foundaton of Chna (6348).

8 8 Duan Zhang, Huapng Da, Youxan Sun References. Baccell, F., Cohen, G., Olsder, G.J., Quadrat, J.P.: Synchronzaton and Lnearty: An Algebra for Dscrete Event Systems. Wley, New Yor (992) 2. Cohen, G., Gaubert, S., Quadrat, J.P.: Tmed-events graphs wth multplers and homogeneous Mn-plus systems. IEEE Trans. Automat.Contr. 43(998) Da, H., Sun, Y.: An Algebrac Model for Performance Evaluaton of Tmed Event Multgraphs. IEEE Trans. Automat. Contr. 48(23) Davd, R, Alla, H.: Petr Nets Grafcet Tools for Modelng Dscrete Event Systems. Hermes, Pars (992). 5. Davd, R., Alla, H.: Contnuous Petr Nets. European Worshop on Applcaton and Theory of Petr Nets. Saragosse (987) 6. Davd, R., Alla, H.: Autonomous and Tmed Contnuous Petr Nets. Internatonal Conference on Applcaton and Theory of Petr Nets. Pars (99) Davd, R., Xe, X., Alla, H.: Propertes of Contnuous Models of Transfer Lneswth Unrelable machnes and Fnte Buffers. IMA Journal of Math. Appled n Busness & Industry. 6 (99) Alla, H., Davd, R.: Modelng of Producton Systems by Contnuous Petr Nets. 3rd Internatonal Conference on CAD/CAM, CARS & FOF88. Detrot 3(988) 9. Le Bal, J., Alla, H., Davd, R.: Hybrd Petr Net. European Control Conference. Grenoble (99) Le Bal, J., Alla, H., Davd, R.: Asymptotc Contnuous Petr Nets: An effcent Approxmaton of Dscrete event Systems. IEEE Internatonal Conference on Robotcs and Automaton. Nce (992). Zerhoun, N., Alla, H.: Asymptotc Contnuous Petr Nets: An effcent Approxmaton of Dscrete event Systems. IEEE Internatonal Conference on Robotcs and Automaton. Cncnnat. Cncnnat 2(99) Baccell, F., Hong, D.: TCP s Max-Plus Lnear and what t tells us on ts throughput. INRIA Report. Pars (2)