Synthesis of the Supervising Agent in MAS

Transcription

1 Synhesis of he Supervising Agen in MAS Franišek Čapkovič Insiue of Informaics, Slovak Academy of Sciences Dúbravská cesa 9, Braislava, Slovak Republic hp:// Absrac. The sysemaic approach o he synhesis of he agen-supervisor supervising a group of agens is presened. I is based on resuls of he DES (discree even sysems) conrol heory. The agens as well as he agen-supervisor are undersood o be DES and hey are modelled by Peri nes (PN). The mehod based on PN place invarians is used in order o synheise he supervisor. The illusraive example is inroduced o explain he elemenary seps of he proposed echnique. A counerexample illusraing he limis of he approach is inroduced oo. Keywords: Agen, invarians, MAS, modelling, Peri nes, supervisor. 1 Inroducion Agens are usually undersood o be persisen (no only sofware, bu also maerial, personal, ec.) eniies ha can perceive, reason, and ac in heir environmen, and communicae wih oher agens. Hence, muliagen sysems (MAS) can be apprehended as a composiion of collaboraive agens working in shared environmen. The agens ogeher perform a more complex funcionaliy. Communicaion enables he agens in MAS o exchange informaion. Thus, he agens can coordinae heir acions and cooperae wih each oher. However, an imporan quesion arises here: Wha communicaion mechanisms enhance he cooperaion beween communicaing agens? In his paper he synhesis of he supervising agen is presened. I uilizes some resuls of he DES (discree even sysems) conrol heory, especially hose concerning he heory of place invarians [8,6] and he supervisor [9] and/or conroller synhesis based on his heory. Mahemaical and/or graphical descripion of he agens as well as he global MAS will be del on he background of place/ransiion Peri nes (P/T PN) described e.g. in [8]. Consider wo simple agens A 1, represening, respecively, wo processes P 1, P 2 given in Fig. 1a). While he resources used by he agens are sufficien, he agens can work auonomously. However, when he resources are limied a coordinaed acion is necessary. In such a case a supervisor or a conroller is necessary. The procedure of he muex (muual exclusion) frequenly occurs in Parially suppored by he Slovak Gran Agency for Science (VEGA) under gran # 2/6102/26. M. Bubak e al. (Eds.): ICCS 2008, Par III, LNCS 5103, pp , c Springer-Verlag Berlin Heidelberg 2008

2 546 F. Čapkovič differen sysems when i is necessary o solve conflics among parallelly operaing subsysems. I is very frequen also in DES like flexible manufacuring sysems (e.g. a robo serves wo ranspor bels), communicaion sysems (wo users wan o use he same channel), ranspor sysems (wo rains wan o ener he same segmen of he railway), ec. The PN model of he simples form of he muex - he muex of wo processes - is given in Fig. 1b). I is no very complicaed o synhesize such a supervisor in his simple case. Conrariwise, in case of more agens and more complicaed condiions for co- p 1 p 2 p 3 3 p 4 4 Process P 1 Process P 2 a) p 1 p 3 5 p 2 p 3 p 4 4 Process P 1 Process P 2 b) Fig. 1. The PN-based model of he a) wo simple processes and b) heir muex operaion among hem, founding he supervisor is by far more inricaed. Consider he group of 5 auonomous agens Gr A = {A 1,,A 3,A 4,A 5 } given in Fig.2. Consider e.g he siuaion when i is necessary o ensure ha only one agen from he subgroup Sgr 1 = {A 1,A 4,A 5 } and only one agen from he subgroup Sgr 2 = {,A 4,A 5 } and only one agen from he subgroup Sgr 3 = {A 3,A 4,A 5 } can simulaneously cooperae wih oher agens from Gr A. In oher words, he agens inside he designaed subgroups mus no work simulaneously. Even, he agens A 4 and A 5 can work only individually (any cooperaion wih oher agens is excluded). However, he agens A 1,, A 3 can work simulaneously. In such a siuaion he synhesis of he supervisor realizing such a muex (i.e. deciding which agens can simulaneously work and when) is no so simple. Consequenly, a sysemaic (orderly, no sray) approach has o p 1 p 3 p 5 p 7 p p 2 p 4 p 6 p 8 4 Agen A 1 Agen Agen A 3 Agen A 4 Agen A p 10 0 Fig. 2. The PN-based model of he group of 5 simple agens be found. There exiss he DES conrol synhesis mehod based on he PN place invarians (P-invarians) in DES conrol heory [11,7,2,4]. Is principle can be uilised for our purposes.

3 2 The Supervisor Synhesis Synhesis of he Supervising Agen in MAS 547 Before saring he supervisor synhesis i is necessary o inroduce he mahemaical model of P/T PN. Namely, he abiliy o express he saes of agens, he abiliy o es properies by means of he PN invarians and he reachabiliy graph (RG) as well as he abiliy o synhesize he supervisory conrollers by means of he invarians and RG [3,5] uilizing linear algebra predesinae PN o be frequenly used. The model has he form as follows x k+1 = x k + B.u k, k =0,N (1) B = G T F (2) F.u k x k (3) where k is he discree sep of he dynamics developmen; x k =(σp k 1,..., σp k n ) T is he n-dimensional sae vecor of DEDS in he sep k; σp k i {0,c pi }, i =1,..., n express he saes of he DEDS elemenary subprocesses or operaions - 0 (passiviy) or 0 <σ pi c pi (aciviy); c pi is he capaciy of he DEDS subprocess p i as o is aciviies; u k =(γ k 1,..., γ k m ) T is he m-dimensional conrol vecor of he sysem in he sep k; is componens γ k j {0, 1}, j =1,..., m represen occurring of he DEDS elemenary discree evens (e.g. saring or ending he elemenary subprocesses or heir aciviies, failures, ec. - 1 (presence) or 0 (absence) of he corresponding discree even; B, F, G are srucural marices of consan elemens; F = {f ij } n m,f ij {0,M fij },i = 1,..., n, j = 1,..., m expresses he causal relaions among he saes of he DEDS (in he role of causes) and he discree evens occuring during he DEDS operaion (in he role of consequences) - 0 (nonexisence), M fij > 0 (exisence and mulipliciy) of he corresponding causal relaions; G = {g ij } m n,g ij {0,M gij },i=1,..., m, j =1,..., n expresses very analogically he causal relaions among he discree evens (causes) and he DEDS saes (consequences); because F and G are he arcs incidence marices he marix B is given by means of hem according o (2); (.) T symbolizes he marix or vecor ransposiion. The main idea of he approach o he supervisor synhesis consiss in imbedding of addiional PN places (slacks) and finding he srucural inerconnecions beween hem and he original PN places. Consequenly, a new PN subne represening he supervisor and is inerface wih he original sysem are found and added o he PN model. Thus, he desired behaviour of agens is forced. The definiion of he P-invarian v of PN in general is he following B T.v = 0/ (4) wih v being n-dimensional vecor and 0/ is m-dimensional zero vecor. However, usually here are several invarians in PN models. Hence, he se of he P-invarians of PN is creaed by he columns of he n n x -dimensional (n x expresses he number of invarians) marix V being he soluion of he equaion V T.B = 0/ (5)

4 548 F. Čapkovič In PN wih he slacks menioned above we have o use he following srucure wih augmened marices. Consequenly, he equaion (5) has o be wrien in he form [ ] B [L, I s ]. = 0/ Bs where I s is (n x n x )-dimensional ideniy marix and L describes condiions (as o marking of some places in he PN models of agens) of he desired MAS behaviour. While B is he known srucural marix of he agens (wihou he supervisor), B s is he supervisor srucure which is searched by he synhesis process. Thus, L.B + B s = 0/; B s = L.B; B s = G T s F s (6) where he acual srucure of he marix L has o be respeced. The augmened sae vecor and he augmened srucural marices (of he original sysem ogeher wih he supervisor) are he following x a = [ ] ( ) ( ) x F ; F a = ; G T G xs Fs a = T G T s where he submarices F s and G T s correspond o he inerconnecions of he incorporaed slacks wih he acual (original) PN srucure. 2.1 The Illusraive Example I can be seen ha in he above menioned case of he five agens he form of L follows from he imposed condiions prescribing he limied cooperaion of agens expressed mahemaically as σ p2 + σ p8 + σ p10 1 (7) σ p4 + σ p8 + σ p10 1 (8) σ p6 + σ p8 + σ p10 1 (9) The form of he augmened marix [L, I s ] follows from he modified condiions (inroducing he slacks) which are given mahemaical erms as σ p2 + σ p8 + σ p10 + σ p11 = 1 (10) σ p4 + σ p8 + σ p10 + σ p12 = 1 (11) σ p6 + σ p8 + σ p10 + σ p13 = 1 (12) where p 11,p 12,p 13 are he addiional places - i.e. he slacks s 1,s 2,s 3 (in general, he number i of such slacks s i can also be differen from i =3,ofcourse)-ha ensure he desired eliminaions of he agens cooperaion aciviies. The number of he slacks depends on he number of condiions and on heir form. As i is clear from he equaions (10)-(12), only one place from he se of four places

5 Synhesis of he Supervising Agen in MAS 549 p 2,p 8,p 10,p 11 can be acive while he res ones have o be passive. Analogically, he same is valid for he ses p 4,p 8,p 10,p 12 and p 6,p 8,p 10,p 13. F A G A F A F = 0 0 F A F A4 0 ; G = 0 G A G A G A4 0 (13) F A G A5 F Ai = ( ) 10 ; G 01 Ai = ( ) 01 ; B 10 Ai = ( ) 1 1 ; i =1,..., 5 (14) L = ; B = (15) B s = L.B = (16) Hence, by wirue of (6), we have F s = ; G T s = (17) Thus, we obain he srucure of MAS given in Fig. 3 where he supervisor is represened by he srucure consising of he places p 11, p 12, p 13 ogeher wih he inerconnecions joining hem wih he elemenary agens. Such MAS exacly fulfils he prescribed condiions. The reachabiliy graph (RG) [8] of he sysem is given in Fig. 4. Is nodes are he saes reachable from he iniial sae x 0 and is edges are weighed by corresponding PN ransiions enabling he sae-o-sae ransi. The conrol synhesis based on RG is described in he auhor s works [3,5]. Consequenly, his problem is no discussed here. In he ligh of he above inroduced facs we can illusrae also he sysemaic synhesis of he simples muex presened in Fig. 1. Here we have σ p2 + σ p4 1 (18) σ p2 + σ p4 + σ p5 = 1 (19)

6 550 F. Čapkovič L =(0, 1, 0, 1); B = ; L.B =(1, 1, 1, 1) (20) B s = L.B =( 1, 1, 1, 1); F s =(1, 0, 1, 0); G T s =(0, 1, 0, 1) (21) Of course, he resul is he same like ha given in he Fig. 1.b) - compare (21) wih Fig. 1b). p 11 p 12 p 13 p 1 p 3 p 5 p p 2 p 4 p 6 p p 9 9 p 10 0 Fig. 3. The PN-based model of he conrolled cooperaion of he agens x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 a) x x 7 9 x 1 1 x 3 x 4 x x 6 x 7 x b) 1 x x 0 Fig. 4. The RG of he five cooperaing agens a) in he form where he edges are oriened bidirecionally, bu weighs of he elemenary edges are no inroduced because hey are differen depending on he orienaion; b) in he form where boh he orienaion of edges and heir weighs are inroduced implicily 3 Applicabiliy of he Approach The mehod described above is suiable for he so called pure PN. P/T PN is pure when here is no loop beween a place and a ransiion - i.e. i is he ne wihou he loops of he form given in Fig. 5a). Namely, in such a case

7 Synhesis of he Supervising Agen in MAS 551 he marices F, G T have nonzero elemen equal o 1 in he same place - in general, when here exiss he loop beween p i and j he elemen f(i, j) ofhe marix F is equal o he elemen gt(i, j) ofhemarixg T. The effec is, ha he difference G T - F = B gives us a deformed informaion abou he sysem srucure, because he elemens f(i, j), gt(i, j) eleminae each oher (namely, gt(i, j) -f(i, j) = 0). Forunaely, in PN heory a simple soluion of such a problem is possible. I is sufficien o replace he ransiion in quesion by he ficive srucure inroduced in Fig. 5b). The auxiliary place p ogeher wih is inpu and oupu ransiions and replace he original ransiion. Imeans ha he impureness in he form of a loop can be replaced (wihou changing any dynamic propery of he ne) very simply by he replacemen of he ransiion in quesion by wo ficive ransiions and a ficive place se beween hem. Such a replacemen is invarian. Afer removing he loop problems he original mehod of he supervisor synhesis can be uilized also for he sysems modelled by he impure P/T PN. p p p a) b) Fig. 5. The replacemen of he loop. The original loop is on he lef. The srucure replacing he loop is given on he righ. 3.1 Dealing wih Problems of Muual Ineracions Among Agens Many imes, anoher form of demands on he cooperaion among agens occurs. Especially, he siuaion - see e.g. [1] - when here exis muual ineracions among he agens o be supervised. In such a case a paricular marking (i.e. he sae of a concree place) of one agen affecs (e.g. sops) firing a ransiion in anoher agen. The siuaion is illusraed in Fig. 6a), where he ransiion A 1 of he agen A 1 is prevened from firing under marking he place A2 p of he agen. As we can see in ha picure, a figmen ha he place p represens he supervising agen A 3 (or, beer said, is par) can be acceped and consecuively, he above described approach o he supervisor synhesis can be uilized. Namely, he ransiion A1 can be subsiued by means of he auxilary ficive place p ogeher wih is ficive inpu and oupu ransiions,and analogically o ha displayed in Fig. 5b). Hence, he new siuaion - i.e. new srucure - is displayed in Fig.6b). I can be seen ha i is he classical muex. Such a subsiuion can be uilized in synhesis of he supervisor in case of agens wih more complicaed srucure. For example, he well known kind of he readerwrier cooperaion named as he Owicki/Lampors muex (defined e.g. in [10]) can be auomaically synheised using his subsiuion. However, i is necessary o say, ha here exis cases where he synhesis of he supervisor based on invarians is no so simple (i.e. fully auomaized) or even, i is impossible. Here, he experience and invenion of he creaor of he

8 552 F. Čapkovič p Agen A 1 A 3 Agen A 1 Agen a) p A 1 p p Agen A 3 Agen A 1 Agen b) p Fig. 6. The prevenion of he ransiion A 1 (he par of he Agen A 1) from firing under he marking p (he par of he Agen )bymeansofp - he par of he supervisor, i.e. he Agen A 3. a) The original schema; b) The subsiuive schema. agens supervisor has o be used. In he nex illusraive example such a sysem is presened and described. 3.2 The Couner-Example In order o illusrae such a couner-example le us consider he so called Peerson s muex algorihm [10] given in Fig. 7. This sysem consiss of wo agens A 1 (wrier) and (reader). The PN places describe he following aciviies: p 1 - pending 1 of A 1 ; p 2 - pending 2 of A 1 ; p 3 -criicala 1 ; p 4 - finished A 1 ; p 5 - pending 0 of A 1 ; p 6 -quiea 1 ; p 7 - pending 1 of ; p 8 - pending 2 of ; p 9 - criical ; p 10 - finished ; p 11 - pending 0 of ; p 12 -quie ; p 13 -aa 1 (lef); p 14 -a (righ). The sae p 4 signals o ha A 1 is presenly no sriving o become criical. This allows o easily access is criical region, by he acion represened by he ransiion 2. Likewise, he sae p 10 allows A 1 o access is criical sae, by he acion represened by he ransiion 5. The shared oken alernaes beween he saes p 13 and p 14 The sep from p 1 o p 2 resuls in he oken in p 13 :by acion expressed by he ransiion in case A 1 obains he oken from p 13,or by acion expressed by he ransiion 3 in case a A 1 held he oken anyway. The sep from p 7 o p 8 likewise resuls in he oken in p 14. Hence, he oken is always a he sie ha execued he sep from pending 1 o pending 2 mos recenly. Afer leaving p 6 along he quiescen acion represened by he ransiion 7, he A 1 akes hree seps o reach is criical sae p 3. In he firs sep, he fair acion represened by he ransiion 6 brings A 1 from p 5 o p 1 and removes he sae p 4. Fairness of he ransiion 6 is local, because he ransiion 6 is local o A 1,wihp 4 he only forward branching he place in he ransiion 6,whichis conneced o he by he loop (p 4, 2 ). The second sep, from p 1 o p 2,resuls in he shared oken in he place p 13, as described above. The hird sep brings A 1 o p 3, wih acion expressed by he ransiion 5 in case signals wih p 10,

9 Synhesis of he Supervising Agen in MAS 553 p 1 p 7 p 13 p p 2 p p p 5 3 p 9 p 11 p p 12 4 p 4 p 10 Agen A 1 Agen Fig. 7. The PN-based model of he wo agens communicaion - he wrier (agen A1) and he reader (agen A2) - in he form of he Peerson s muex ha i is presenly no ineresed in going criical, or wih acion represened by he ransiion 4 in case more recenly execued he sep from p 7 o p 8. The algorihm s overall srucure guaranees ha one of p 10 or p 14 will evenually carry a oken ha remains here unil evenually eiher he even represened by 5 or 4 occurs. The wo agens are srucurally symmerical, bu he iniial sae favors (if he place p 14 conains he oken) or A 1 (if he place p 13 conains he oken). In his example he invarians-based approach o he supervisor synhesis fails. Namely, any saisfying marix L canno be found. Searching for reasons i was deeced ha he problem consiss in he specific srucure of he P-invarians of he global sysem presened in Fig. 7. As we can see from he following marix of he global sysem P-invarians he invarians of he agens A 1, are muually disjoin V T = Conclusion The mehod of he DES conrol heory based on PN place invarians was uilized for he synhesis of he agen (wih prescribed properies) supervising a group of agens in MAS. The illusraive example documening he soundness of he mehod was inroduced. I was shown ha he mehod is simply applicable for auonomous agens modelled by pure PN. Addiionally, i was described how he applicabiliy can be exended for he agens modelled by impure PN. Namely,

10 554 F. Čapkovič a simple subsiuion of he ransiion in he loops of he PN model is sufficien o remove he impureness. As a resul of he research i can be said ha he mehod is very useful for he synhesis of he agens supervising a group of auomous agens. In case of a group of cooperaing agens some problems can occur. Therefore, he couner-example was presened. I was shown ha e.g. for he reader-wrier in he form of he so called Peerson s muex he supervising agen canno be synheised by he proposed mehod. Hence, here exiss a challenge for he fuure invesigaion on his way, especially, o define general condiions for he validiy of he approach. References 1. Bordbar, B., Giacomini, L., Holding, D.J.: UML and Peri Nes for Design and Analysis of Disribued Sysems. In: IEEE Inernaional Conference on Conrol Applicaions (CC000) and IEEE Inernaional Symposium on Compuer-Aided Conrol Sysem Design (CACSD 2000), pp IEEE Press, Piscaaway (2000) 2. Buy, U., Darabi, H.: Sidesepping Verificaion Complexiy wih Supervisory Conrol. In: Workshop on Sofware Engineering for Embedded Sysems: From Requiremens o Implemenaion, 8 pages (2003), hp:// shaz/sees/schedule.hm 3. Čapkovič, F.: An Applicaion of he DEDS Conrol Synhesis Mehod. Journal of Universal Compuer Science 11, (2005) 4. Čapkovič, F.: DES Modelling and Conrol vs. Problem Solving Mehods. In. J. Inelligen Informaion and Daabase Sysems 1, (2007) 5. Čapkovič, F.: Modelling, Analysing and Conrol of Ineracions among Agens in MAS. Compuing and Informaics 26, (2007) 6. Marinez, J., Silva, M.: A simple and fas algorihm o obain all invarians of generalized Peri ne. In: Applicaion and Theory of Peri Nes, vol. 52, pp Springer, New York (1982) 7. Moody, J.O., Ansaklis, P.J.: Supervisory Conrol of Discree Even Sysems Using Peri Nes. Kluwer Academic Publishers, Boson (1998) 8. Muraa, T.: Peri Nes: Properies, Analysis and Applicaions. Proceedings IEEE 77, (1989) 9. Ramadge, P.J., Wonham, W.M.: Supervisory Conrol of a Class of Discree Even Processes. SIAM Journal on Conrol and Opimizaion 25, (1987) 10. Reisig, W.: Elemens of Disribued Algorihms. Modeling and Analysis wih Peri Nes. Springer, Berlin (1998) 11. Yamalidou, K., Moody, J.O., Ansaklis, P.J., Lemmon, M.D.: Feedback Conrol of Peri Nes Based on Place Invarians. Auomaica 32, (1996)