MODELLING, ANALYSING AND CONTROL OF INTERACTIONS AMONG AGENTS IN MAS

Transcription

1 Compuing and Informaics, Vol. 26, 2007, MODELLING, ANALYSING AND CONTROL OF INTERACTIONS AMONG AGENTS IN MAS Franišek Čapkovič Insiue of Informaics Slovak Academy of Sciences Dúbravská cesa Braislava, Slovakia {Franisek.Capkovic, urrcapk}@savba.sk Revised manuscrip received 20 February 2007 Absrac. An alernaive approach o modelling and analysis of ineracions among agens in muliagen sysems (MAS) and o heir conrol is presened in analyical erms. The aenion is focused especially on he negoiaion process. However, he possibiliy of anoher form of he communicaion is menioned oo. The reachabiliy graph of he Peri ne (PN)-based model of MAS is found as well as he space of feasible saes. Trajecories represening he ineracion processes among agens in MAS are compued by means of he muual inersecion of boh he sraighlined reachabiliy ree (from a given iniial sae owards he erminal one) and he backracking reachabiliy ree (from he desired erminal sae owards he iniial one, however oriened owards he erminal sae). Conrol inerferences are obained on he basis of he mos suiable rajecory chosen from he se of feasible ones. Keywords: Agens, analysis, conrol, decision, discree-even sysems, modelling, peri nes 1 INTRODUCTION MAS are used in inelligen conrol, especially for a cooperaive problem-solving[27]. To analyse complicaed ineracions among agens modelling of hem is ofen used. The negoiaion belongs o he mos imporan ineracions. I is he process of mulilaeral bargaining for muual profi. In oher words [2, 25], he negoiaion is a de-

2 508 F. Čapkovič cision process where wo or more paricipans make individual decisions and inerac wih each oher in order o reach a compromise. In [13] Peri nes (PN) are used for e-negoiaions aciviies. The following five principle properies of e-negoiaion are defined here: 1. ineraciviy (i involves he agens o paricipae and communicae wih each oher); 2. informaiviy (i generaes, ransmis and sores informaion); 3. irregulariy(i behaves differenly according o he combinaion of agens, sraegies, evens, asks, issues, alernaives, preferences and crieria); 4. inegriy (i affords speed, consisency and absence of errors hrough efficien and effecive mechanisms); 5. inexpensiviy(i auomaes or semi-auomaes negoiaion aciviies o save ime and cos). Because PN can effecively help on his way (i.e. o express he properies o be saisfied, especially he firs hree ones ha are generic) PN were chosen o model MAS oo [17, 21]. On he base of previous experience [4, 5, 8] wih PN-based modelling and conrol synhesis of he discree even dynamic sysems (DEDS) and he agen cooperaion [6, 7] a new approach o modelling, analysis and conrol of he negoiaion process is proposed here. The negoiaion process is undersood o be DEDS. I seems o be naural, because he process is discree in naure and simulaneously i is causal. The approach consiss of: 1. creaing he PN-based mahemaical model of he negoiaion process; 2. generaing he space of feasible saes which are reachable from he given iniial sae; 3. uilizing he reachabiliy graph in order o find he feasible sae rajecories o a prescribed feasible erminal sae. Afer a horough analysing he se of possibiliies, he mos suiable sraegy (he conrol rajecory) can be chosen. In order o use DEDS-based approach i is necessary o menion DEDS operaion and causaliy firs. Likewise he PN-based model will be concisely menioned because i yields boh he user friendly analyical descripion of DEDS operaion (by means of he linear discree model consising of he sysem of linear difference equaions) and he basis for compuing he reachabiliy ree and/or he reachabiliy graph making possible simple esing he DEDS properies as well as assuring he sysem causaliy. As o he model creaion he approach is sufficienly general o be uilized no only for modelling he negoiaion process bu also for modelling he wider specrum of boh he agen behaviour and he forms of communicaion among he agens.

3 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS THE DEDS OPERATION AND CAUSALITY DEDS behaviour is driven by discree evens. Such a sysem persiss in a given sae as long as i is forced o change i owing o he occurrence of a discree even. As o he srucure DEDS can be very complicaed. Namely, he ypical represenaives of DEDS are flexible manufacuring sysems (FMS), communicaion sysems of differen kinds, ranspor sysems, ec. Taking ino accoun e.g. FMS as an example of DEDS, hey consis of many devices like robos, numerically conrolled machine ools, conveyers, auomaically guided vehicles, palles, buffers, ec., in order o produce he final produc(s) from raw maerial(s) and/or semiproduc(s). The muual cooperaion among he devices is driven by discree evens (saring or ending some echnological operaions and/or movemen of mobile devices, swiching he devices on and/or off, ec.). In PN-based modelling of DEDS he level of absracion is very imporan. From he global poin of view he FMS devices can be undersood o be aomic elemens of DEDS. However, a a deeper invesigaion hey can also be found o be DEDS, namely he subsysems of he global FMS wih parial aciviies and/or operaions being he aomic elemens. While in he global undersanding a robo is a device being only swiched on (o be acivaed) and/or off (o be deacivaed), going ino deails we can find ha i performs several aciviies i can be available (i.e. free and waiing for any acivaion), i can perform differen operaions like picking pars up, moving is wris wih one of he pars from one poin of is working space o anoher one, puing a par in a machine ool, on a palle or ino a buffer, aking a par off a machine ool or a buffer, ec. Analogically, he machine ools can perform several differen echnological operaions (e.g. drilling, milling, ec.). From he echnological poin of view FMSs realize a echnological processes while heir subsysems realize corresponding echnological subprocesses. The sae of FMS a a ime insan consiss of he saes all of he aomic aciviies in he sysem. Therefore, i is a vecor. Is dimensionaliy is equal o he global number of aciviies in he sysem. From he sysem heory poin of view he operaion of a simple example of DEDS e.g. one of he FMS devices can be imagined in such a way like i is illusraed in Figure 1. The course of a DEDS sae i.e. he sysem dynamics developmen represened here shows us how he sae changes is discree values (levels) x 1,x 2,...,x 6 owing o he occurrence of discree evens u 1,u 2,...,u 5. Here, x i represens he sae of he aomic aciviy a i, i = 1,...,6. Namely, if a i is performed x i is acive, oherwise x i is passive. In his very simple example only one of he six feasible aomic aciviies is performed in any sep of he DEDS dynamic developmen, e.g.: 1. x 3 he robo R moves from is waiing posiion o a conveyer; 2. x 1 R picks up a par from he conveyer; 3. x 2 R moves (wih he par in is wris) from he conveyer o a machine ool; 4. x 5 R pus he par ino he machine ool;

4 510 F. Čapkovič 5. x 4 R moves o is waiing posiion; 6. x 6 he machine ool sars a ooling operaion. The aciviies link o each oher hey pass sequenially. Namely, he sysem persissinadiscreesae(performsanoperaion)uniliisforcedochangeiowing o he occurrence of a discree even (ending he operaion and saring anoher one). Afer his change he sysem persiss in he new discree sae (performs he operaion) unil anoher discree even (ending/saring of an operaion) occurs and causes anoher change, ec. I is clear from Figure 1 ha here exis 6 differen discree levels of he sae, namely x 1,x 2,...,x 6. A any ime he sysem (in our simpleexample)canpersisonly inone ofhem, of course, because ihas acharacer of he sequenial process. Consequenly, he saes of he sysem can be expressed by he sae vecors wih he srucure x k = (x k 1,x k 2,...,x k 6) T, k = 0,1,...,K, where for each k only one of he componens x k i, i = 1,2,...,n (in our case n = 6) of he vecor x k is differen from zero. Thus, for K = 5 we have he following saes x 0 = (0,0,x 0 3,0,0,0) T, x 1 = (x 1 1,0,0,0,0,0) T, x 2 = (0,x 2 2,0,0,0,0) T, x 3 = (0,0,0,0,x 3 5,0)T, x 4 = (0,0,0,x 4 4,0,0)T, x 5 = (0,0,0,0,0,x 5 6 )T. Name k o be he sep of he sysem dynamics developmen. discree values of a sae x x 6 x 5 x 4 x 3 x 2 x 1 discree u 1 u 2 u 3 u 4 u 5 evens Fig. 1. The developmen of he sae in he simple example of DEDS I is necessary o say ha in large and complicaed FMS a number of devices can work simulaneously for example, wo producion lines of FMS. When boh lines conain a machine ool and he machine ools are served by he same robo R, he robo canno serve boh machines simulaneously. I is clear ha R is able o serve he machines eiher alernaively or in virue of a prescribed program. In such a case he course of he DEDS sae will be more complicaed han ha shown in Figure 1. Therefore, i is useful o esablish anoher graphical inerpreaion of he sysem dynamic developmen.

5 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS Feasible Saes and he Sae Space To avoid any misundersanding, i is necessary o disinguish boh he feasible sae vecors of he sysem regardless of he sep k in which hey occur (i.e. regardless of he causaliy relaions) and he sae vecors occurring consecuively (successively) during he sysem dynamics developmen in he seps k = 0,1,...,K of he sysem dynamics developmen, i.e causally. While he former vecors (hey are muually differen) represen he sysem sae space, he laer ones express he dynamics of he sysem wihin he framework of he given sae space. Le us denoe in our example of FMS he feasible saes as 1,..., 6, where 1 corresponds o x 0 (we will use he symbol, i.e. 1 x 0 ), 2 x 1, 3 x 2, 4 x 3, 5 x 4, 6 x 5. The se reach = { 1, 2,..., 6 } of he feasible saes is creaed by he iniial sae vecor x 0 and all of he muually differen saes reachable from x 0 regardless of he number of seps which are necessary for such a reachabiliy. The vecors of he se reach can be undersood o be he columns of he marix reach as follows. reach = ( 1, 2,..., 6 ) = I is he marix of feasible saes. As already said, by he erm feasible saes he iniial sae ogeher wih all of he saes reachable from i are mean. The dimensionaliy of he marix reach is (n N) wih n,n being, respecively, he number of he aomic aciviies being in he discree saes x k i,i = 1,...,n (he componens of he vecors x k, k = 0,1,...,K) of he sysem, and he number of all he feasible sae vecors i, i = 1,2,...,N. For he given iniial sae he marix reach represens he sysem sae space as a whole. However, when he iniial sae will be represened e.g. by he vecor x 0 = (0,x 0 2,0,0,0,0)T, hen x 1 = (0,0,0,0,x 1 5,0) T, x 2 = (0,0,0,x 2 4,0,0) T, x 3 = (0,0,0,0,0,x 3 6) T. Thus, 1 x 0, 2 x 1, 3 x 2 and 4 x 3. Consequenly, N = 4 and reach = ( 1, 2, 3, 4 ). However, in general, some of he feasible saes can occur repeaedly (inconsecuively) in a causal sequence of he saes {x 0,x 1,...,x K } in which he sysem finds iself during he dynamics developmen. Alhough no ime is explicily designaed on he horizonal axis in Figure 1 is implici presence is indispuable. Namely, he discree evens occur in concree ime insans bu he ime inervals beween he insans of occurrence of wo adjacen evens have differen lengh. However, aking no accoun of a ime we can inroduce a new simplified concepion of he course of DEDS sae as a formal represenaion of he sae rajecory. In such a case we will designae he seps of he DEDS dynamics developmen on he horizonal axis, namely wih he same geomeric disance beween pairs of adjacen seps. A he same ime we will designae he discree va- (1)

6 512 F. Čapkovič lues of he DEDS saeon he vericalaxis, namely likewisewih he same geomeric disance beween pairs of adjacen discree values. Consequenly, he course of he DEDS sae will be represened by a broken line consising of he elemenary linear segmens (he abscissae) connecing he poins wih he pairs of coordinaes (k i,x j1 ) and (k i+1,x j2 ), where k i is he i h sep of he DEDS dynamics developmen and x j1, x j2 (i is clear ha x j1 > x j2 or x j1 < x j2 ) are wo adjacen discree values of he sae. A he same ime, he discree even causing he change of he sae from he discree value o anoher one represens he parameer of he ransiion of he sae beween hese values. A such a concepion he course of he sae represened in Figure 1 will be ransformed ino he course represened in Figure 2a). To emphasize he sep of he sysem dynamics developmen, he lef upper index k is added o he symbol of i h discree even u i denoing he sep in which heevenoccurs. Thus, k u i meanshau i occursinhesepk. Becausehegraphical expression of saes is imbedded ino he laice, i is well-arranged, lucid, and easy o undersand. x 6 6 d x is u 4 c 4 u 5 r x 4 e 2 u 3 5 e x 3 1 v 0 u 1 a x 2 l 3 u x e 1 2 u 2 1 s sep k of he dynamics developmen 6 4 u f 5 e a 5 si 3 u 4 4 b le 2 u 3 3 s 0 1 u 2 2 a u 1 e 1 s sep k of he dynamics developmen a) Fig. 2. The simplified represenaion of: a) he course of he DEDS sae; b) he sae rajecory in he sae space As eviden from Figure 2 a), such a represenaion is unambiguous. Namely, he ransiion of he DEDS sae from he discree value in a sep k o he immediaely following one in he nex sep is realized jus by means of he corresponding discree even. Alhough he broken line is only ficive, i can be named as he sae rajecory, because i represens (by any linear par i.e. by any abscissa) he ransiion from he exising sae o anoher one being he consequence of he occurrence of he corresponding discree even. When he feasible saes are designaed on he verical axis, anoher simplified represenaion of he DEDS sae course can be inroduced. I is displayed in b)

7 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS 513 Figure 2b). As can be seen below, such a represenaion can be uilized also for displaying pahs of he PN reachabiliy ree and/or he reachabiliy graph. Namely, heir nodes can be designaed on he verical axis and heir lenghs can be designaed by he number of seps on he horizonal axis. The sysem can pass from a given discree sae o anoher sae by means of he occurrence of one discree even only. However, in general (in more complicaed DEDS), several discree evens can someimes be enabled in a given discree sae. Because only one of hem can be fired, here is a conflic among hem. In case of he forced evens a decision has o be made which of hem will be fired. Namely, each of hem can ransfer he sysem o anoher (however, again solely one) poenial nex sae. Moreover, hese saes are muually differen. Consequenly, he even ransferring he sysem ino he mos suiable sae is chosen, of course. The choice can be realized on he basis of a crierion or in virue of a prescripion. In case of FMS he choice is deermined by he echnological process. Namely, any discree even canno ransfer he sysem ino wo or more differen saes simulaneously. This fac is very imporan for he DEDS causaliy and i is uilized during he conrol synhesis. Chaining several successive discree evens yields he rajecory represening he ransiion of he sysem from he given iniial sae o a prescribed erminal one. Such a ransiion is causal. The DEDS causaliy can be (in an analogy wih he causaliy of coninuous sysems described in [1]) illusraed by Figure 3 where he sae in he sep k = j can be reached from he sae in he sep k = 0 (he iniial sae) afer j seps. However, he same sae can be reached also from he sae in he sep k = i, 0 < i < j, afer (j i) seps. k = 0 k = i k = j; j > i Fig. 3. The principle of causaliy in DEDS Le us undersand he rajecory in Figure 2b) o be he direced graph. Is adjacency marix is as follows: 0 0 u u A = u u 4 0 (2) 0 4 u The dimensionaliy of A depends on he number of he graph nodes. Because he nodes are represened by he feasible sae vecors he marix dimensionaliy is

8 514 F. Čapkovič N N. In our case N = 6. However, he nonzero elemens of A represening he discree evens depend on he sep k of he DEDS dynamics developmen. Namely, heelemensa i,j, i = 1,2,...,6, j = 1,2,...,6,ofheadjacencymarixArepresen he discree evens ransfering he sysem from i o he adjacen sae j. Thus a i,j = k u (i j), k u (i j) {0,1}, i = 1,...,6, j = 1,...,6 where k deermines he sep of he DEDS dynamics developmen when he even is enabled. Consequenly, he elemen a i,j depends on k. In oher words he elemen is k-varian (is a funcion of k). Therefore, he marix A can be named as he k-varian adjacency marix and/or he funcional adjacency marix and i will be denoed as A k. In order o avoid he complicaed lower index i j of he evens we will use only he ordinal number of he even like i was done in Figures 2a) and 2b) as well as in (2). Thus, for j = 0,1,... j u r = { 1 if j = k 0 oherwise ; r = 1,2,...m (3) where m is he oal number of discree evens. In our case m = 5. Consequenly, A 0 = A 1 = A 4 = Vicarious Sae Vecors In order o work wih he funcional adjacency marix A k in a mahemaical model of he DEDS dynamics, le us define he N-dimensional vicarious vecors k = ( k 1, k 2,..., k N ) T, k = 0,1,...,K which will ake he place of he real sae vecors x k, k = 0,1,2,...,K represening he sysem dynamics developmen. In any sep k he vecor k has only one nonzero componen, namely k i = { 1 if i = k +1 0 oherwise. ; i = 1,2,...,N (4) Thus, in our case 0 represening x 0 has he form 0 = (1,0,0,0,0,0) T, 1 represening x 1 has he form 1 = (0,1,0,0,0,0) T,..., 5 represening x 5 has he form 5 = (0,0,0,0,0,1) T. By means of he vicarious vecors we can develop he sysem dynamics as follows k+1 = A T k. k, k = 0,1,...,K. (5) The equaion (5) describes DEDS as a sae machine. In any sep k only one discree even can occur (can be fired). Such a discree even ransfer he sysem from k o k+1 by means of he discree even fired in he sep k.

9 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS The Example of he More Complicaed DEDS Consider he more complicaed sysem (as o he composiional srucure) wih he behaviour expressed by he direced graph in Figure 4. As we can see here are hree muually differen possibiliies of he ransiion o he nex sae already in he iniial sae x 0. The iniialsae vecor is he firs feasible vecor of he sysem i.e. 1 x 0. I can be seen from he same figure ha x 1 { 2, 3, 4 } i.e. eiher 2 x 1 or 3 x 1 or 4 x 1. Analogically, x 2 { 5, 6, 7 }, x 3 { 8, 9 }, x 4 { 10, 11 }, x 5 { 12, 13, 14 }. The marix reach = ( 1, 2,..., 14 ). 0 u u u u u u u u u u u u u u 2 4 u Fig. 4. The example of he more complicaed DEDS The sysem srucure is creaed by he digraph wih he vecors i, i = 1,2,..., 14 being is nodes. Consequenly, he adjacency marix of he graph is he following (14 14)-dimensional marix: 0 0 u u 2 1 k u u u u u u A= u u 1 4 (6) 0 3 u u u u u

10 516 F. Čapkovič In he k-varian adjacency marix A k only such discree evens appear which have is lef upper index equal o k. I means ha 0 u 1 1, 0 u 2 1, 0 u 3 1 appear in A 0, 1 u 1 2, 2 u 2 2, 2 u 3 2 appear in A 1, ec. Here i has o be repeaed again ha only one of he enabled discree evens can be fired in any sep of he sysem dynamics developmen u u u u u 2 2 u u u f 7 e a si 6 1 u u 1 3 b 5 le 1 u u u 1 2 s 3 a 0 u e 0 u 1 1 s sep of he dynamics developmen Fig. 5. The graphical expression all of he feasible rajecories Inroducing he 14-dimensional vicarious vecors k, k = 0,...K aking he places of he vecors x k, k = 0,1,...,K we can develop he sysem dynamics by means of (5). The sysem rajecories are given in Figure The Formal Expression of DEDS Causaliy The funcional (k-varian) marix A k is he ransiion marix beween wo causally adjacen saes k, k+1, k = 0,1,...,K. Consequenly, k k 1 k = A T k 1...A T 0. 0 = ( A T k i). 0 = ( A i ) T. 0 = Φ k,0. 0 (7) i=1 i=0

11 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS 517 because 1 =A T 0. 0, 2 =A T 1. 1=A T 1.AT 0. 0,..., k =A T k 1. k 1=A T k 1.AT k 2....A T The marix Φ k,0 is he ransiion marix of he sysem from he sae 0 o he sae k. 2.5 The DEDS Backward Causaliy Till now we considered he naural DEDS causaliy he causaliy in sraighforward direcion,i.e.inhedirecioncause even consequence. Insuchacasehesysembeing in he sae k passes o anoher sae k+1 in a naural way (i.e. owing o he even occurrence of a discree even). Thus he ransformaion k k+1 is realized. This kind of causaliy deermines he fuure on he basis of he presence. However, here are siuaions(he DEDS conrolor, beer said, he need of he DEDS conrol synhesis is one of hem), when i is imporan o answer he quesion wha even (symbolically?even ) caused ha he sysem finds iself in he presen sae k and/or from which previous sae k 1 (symbolically? k 1 ) he sysem passed?even (due o he even) ino he sae k i.e.? k 1 k. Thus, he quesion marks symbolize he quesions from which previous sae? and by means of wha discree even? he sysem passed o he sae k. Usually, any unambiguous answer does no exis in such a case. The sysem could pass ino he sae k from several (e.g. n cp ) differen previous saes i k 1 (causal predecessors), i = 1,2,...,n cp, namely owing o differen discree evens being he elemens of he marix A k 1. For example, in Figure 5 he feasible sae 9 corresponding o he real sae x 3 (represened by he vicarious sae 3 ) can be reached eiher from 5 (corresponding o 1 x 2 represened by 1 2 ), namely by means of he even 2 u 1 3, or from 6 (corresponding o 2 x 2 represened by 2 2 ), namely by means of he even 2 u 2 3. This kind of causaliy finds possible causes in he pas he consequence of which is jus he presence (he exising sae of he sysem). The backward causaliy is very imporan especially during he DEDS conrol synhesis. The mahemaical descripion of he backward causaliy is (a he above inroduced concepion of causaliy) very simple. I is sufficien o ranspose he ransiion marix A T k (i.e. o use he marixa k ) in order o obainsuch a procedure. For one sep he descripion is as follows: k 1 = A k 1. k. (8) In general (for i = 1,2,...), k i = A k i.a k i+1...a k 1. k. Thus, k 1 0 = A 0.A 1...A k 2.A k 1. k = ( A i ). k = Φ 0,k. k. (9) The marix Φ 0,k is he backward ransiion marix of he sysem from he sae k o he sae 0. I is eviden from comparing (7) and (9) ha Φ 0,k = Φ k,0 T. i=0

12 518 F. Čapkovič 3 SIMPLIFIED EPRESSING OF THE CAUSALITY Using he k-varian adjacency marix A k is no comforable as o he compuaions. The sandard (i.e. consan) adjacency marix A seems o be more suiable because he graph heory resuls can be uilized. Namely, he relaions beween powers of he digraph adjacency marix A and he number of pahs having a given lengh are defined and proved see e.g. [20, 22] and/or [10, 11, 14, 19, 24, 26]. The digraph reachabiliy marix is defined here as well. Therefore, he elemen a (k) i,j of he k h power A k of he adjacency marix A represens he number of he pahs having he lengh k from he node i o he node j. The reachabiliy marix R = N i=1 Ai = A 1 + A A N yields informaion abou he number of pahs having he lengh N or he lengh less han N. Replacing he ordinary arihmeic (in general e.g. c ij = n k=1 a ik b kj ) by he Boolean arihmeic(accordingly, c ij = n k=1 a ik b kj ), he elemen a (k) i,j of he marix A k is Booleanand expresses he nonexisence or exisence of he pah from he node i o he node j having he lengh k. The elemens of he logical reachabiliy marix R L = N i=1 Ai = A 1 A 2... A N yield informaion abou he reachabiliy in iself (decide he reachabiliy). Thenonzeroelemensr jk,j = 1,2,...,N,ofhek h column(r 1k,r 2k,...,r jk,..., r Nk ) T of he reachabiliy marix R conain he numbers of pahs from he node j o he node k, while he elemens of he same vecor in he logical reachabiliy marix R L decide wheher he node k is reachable from he node j or no. 3.1 Uilizing he Consan Adjacency Marix Consider formally ha all of he discree evens are enabled. Consequenly, he k-varian adjacency marix A k is urned o he consan adjacency marix A having he same srucure like in (6) however, all of he nonzero elemens are replaced by he ineger 1. Such an approach helps us find he space of feasible rajecories from a given iniial sae o a prescribed erminal sae. Using he marix A in he sraigh-lined developmen is as follows: { k } = A} T.A T...A {{ T.A T } k facors k 1. 0 = ( i=0 k 1 A T ). 0 = ( i=0 A) T. 0 = c Φ k,0. 0 (10) because { 1 } = A T. 0, { 2 } = A T.{ 1 } = A T.A T. 0,..., { k } = A T.{ k 1 } = A} T.A T.{{....A T }. 0. Here, he { i }, i = 1,2,...,k express aggregaed saes k facors because of he fac ha all of he discree evens are (formally) fired. Denoe hese vecors as sl { i }, i = 1,2,...,k (because hey represen he sraigh-lined developmen) and sore hem as he columns of he marix M 1 = ( sl 0, sl { 1 },..., sl { k }). (11)

13 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS 519 Analogically, he backward developmen is performed as follows k 1 { 0 } = } A.A...A.A {{}. k = ( A). 0 = c Φ 0,k. 0 = c Φ T k,0. 0 (12) k facors i=0 because { k 1 } =A. k, { k 2 } =A.{ k 1 } =A 2. k,..., { 0 } =A.{ k 1 } = A.A. }{{....A}. k. Here, he { i }, i = 0,1,...,k 1 also means aggregaed saes k facors because of he same reason. Denoe hese vecors as b { i }, i = 0,1,...,k 1 (because hey represen he backracking developmen) and sore hem as he columns of he marix M 2 = ( b { 0 }, b { 1 },..., b { k 1 }, b k ). (13) In boh cases such an approach has he real inerpreaion. The nonzero componens of he columns of M 1 express he number of pahs hrough he corresponding graph nodes in he sraigh-lined direcion, i.e. from he iniial sae 0 o he prescribed erminal sae k. The nonzero componens of he columns of M 2 conain he number of pahs hrough he corresponding graph nodes in he backward direcion, i.e. from he erminal sae k o he iniial sae 0, however, direced owards he erminal sae. 3.2 The Space of Feasible Trajecories Afer he column-o-column inersecion of he marices M 1, M 2 we have he final resul in he form of he marix M = M 1 M 2 = ( f 0, f { 1 },..., f { k 1 }, f k ) (14) M = ( sl 0 b { 0 }, sl { 1 } b { 1 },..., sl { k 1 } b { k 1 }, sl { k } b k ). (15) The column-o-column inersecion of wo corresponding columns is undersood o be finding minima of heir corresponding elemens. I is done as follows f { i } = sl { i } b { i } = min( sl { i }, b { i }), i = 0,1,...,k (16) wih sl { 0 } = 0, b { k } = k. The operaion of he inersecion ensures ha in he marix M only he pahs emerging from he given iniial sae and enering he prescribed erminal sae are sored. I can be said ha in he marix M he feasible rajecories are sored or, in oher words, ha M represens he space of he feasible rajecories. I is very imporan and ineresing ha he principle of causaliy allows us o find shorer rajecories when he longer ones have already been compued. Namely, having a disposal he marices M 1, M 2 (given, respecively, by (11), (13)) we can compue rajecories shorer for 1,2,...,j seps in such a way ha before he inersecionofhesemaricesweshifhemarixm 2 oheleffor1,2,...,j columns

14 520 F. Čapkovič as follows 1 M = M 1 1 M 2, where 1 M 2 = ( b { 1 },..., b { k 1 }, b k,0/) (17) 2 M = M 1 2 M 2, where 2 M 2 = ( b { 2 },..., b { k 1 }, b k,0/,0/) (18). j M = M 1 2 M 2, where j M 2 = ( b { j },..., b { k 1 }, b k,0/,...,0/ ). (19) }{{} j vecors Here, 0/ is he zero column vecor of he corresponding dimensionaliy. When an inersecion M 1 j M 2 does no exis, he marix j M is he zero marix. I means ha no rajecory shorer for j seps exiss in such a case The Illusraive Example To illusrae he procedure le us apply i o Figure 5. The real iniial sae vecor is x 0. I is he feasible sae and hus 1 x 0. The vicarious vecor 0 represens he iniial sae vecor in order o work wih he adjacency marix A. When he feasible sae 12 is prescribed o be he erminal sae, afer 5 seps we have M 1 = M 2 = M = However, i has o be said ha he number represening he sep k in which he erminal sae will be reached from he iniial sae is no predeermined. I is bounded only by he relaion k (N 1) which resuls from graph heory (where (N 1) is he maximal lengh of he pahs in he graph wih N nodes) as well as by he relaion sl { k } k (o be sure ha k is comprehended in sl { k }). When such a k is found, he number of he columns of M 1 is deermined and he backracking developmen saring from k can be performed in order o obain he marix M 2. Thus, in our case k = 5, 12 x 5, where x 5 is represened by 5. The firs column of he marix M is creaed by he iniial vicarious sae vecor 0

15 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS u u u u u f e a 6 si 1 u u b le 1 u u u s a 0 u e 0 u 1 1 s sep of he dynamics developmen Fig. 6. The graphical expression of he feasible rajecories from 1 o 12 represening he real iniial sae vecor x 0 ( 1 x 0 ). The las (k + 1)-h column is creaed by he erminal vicarious sae vecor 5 represening he real erminal sae vecor x 5, where 12 x 5. The columns placed beween hem poin ou (by heir nonzero elemens) he feasible ransiive saes as well as he number of pahs hrough hem. The marix M sores he feasible rajecories from he feasible sae 1 o he desired feasible erminal sae 12. We can disinguish he rajecoriesas wellasmakesure ofhecorrecnessofherajecoriesby meansofhe marix A displayed in (6). Namely, his marix guaranees causaliy. A he same ime we can assign he corresponding discree even realizing he ransiion beween any adjacen feasible saes ( i, j ) being a par of he rajecories in quesion. Consequenly, 0 u u u u u (20) 0 u u u u u (21) 0 u u u u u (22) The graphical expression of he rajecories is given in Figure 6.

16 522 F. Čapkovič 4 THE PN-BASED MATHEMATICAL MODEL OF DEDS Use he analogy beween he DEDS aomic aciviies a i {a 1,...,a n } and he PN places p i {p 1,...,p n } as well as beween he discree evens u j {u 1,...,u m } occurring in DEDS and he PN ransiions j { 1,..., m }. Consequenly, DEDS can be modelled by means of PN. The analyical model of DEDS based on PN has he form of he linear discree sysem as follows x k+1 = x k +B.u k, k = 0,...,K (23) B = G T F (24) F.u k x k (25) where k is he discree sep of he dynamics developmen; x k = (σ k p 1,...,σ k p n ) T is he n-dimensional sae vecor in he sep k; σ k p i {0,1,...,c pi }, i = 1,...,n express he saes of he DEDS aomic aciviies, namely he passiviy is expressed by σ pi = 0 and he aciviy is expressed by 0 < σ pi c pi ; c pi is he capaciy as o he aciviies e.g. he passiviy of a buffer means he empy buffer, he aciviy means a number of pars sored in he buffer and he capaciy is undersood o be he maximal number of pars which can be pu ino he buffer; 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 he occurrence of he DEDS discree evens (e.g. saring or ending he aomic aciviies, occurrence of failures, ec.) when he j h discree even is enabled γ k j = 1, when he even is disabled γ k j = 0; B, F, G are srucural marices of consan elemens; F = {f ij } n m, where f ij {0,M fij },i = 1,...,n,j = 1,...,m express he causal relaions beween he saes ofhededs(inheroleofcauses)andhediscreeevensoccuringduringhededs operaion (in he role of consequences) nonexisence of he corresponding relaion is expressed by M fij = 0, exisence and mulipliciy of he relaion are expressed by M fij > 0; G = {g ij } m n, where g ij {0,M gij },i = 1,...,m,j = 1,...,n express very analogically he causal relaions beween he discree evens (as he causes) and he DEDS saes (as he consequences); he srucural marix B is given by means of he arcs incidence maricesf and G accordingo (24); (.) T symbolizes he marix or vecor ransposiion. The PN marking which in PN heory is usually denoed as µ was denoed here by he leer x usually denoing he sae in sysem heory. The main reason is ha we work wih he erm sysem raher han wih he erm PN. Fuzzy PN [15] can be modelled analogically see [3]. Using he higher-level PN is no discussed in his paper. 4.1 The PN Reachabiliy Tree and he Reachabiliy Graph Exac definiions of he reachabiliy ree are inroduced in PN heory basic sources see e.g. [16, 18]. To have an idea abou he reachabiliy ree i is sufficien o inroduce here only a shor descripion of i. The PN reachabiiy ree G r = (V r,e r ) is

17 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS 523 he ree where he se of nodes V r = {v 0,v 1,...,v Nr } is represened by he se of PN saes i.e. he nodes v i, i = 0,...,N r represen he sae vecors x i, i = 0,...,N r and consequenly, V r = {x 0,x 1,...,x Nr }, wih he iniial sae x 0 being he roo of he reachabiliy ree. The se of edges E r = {e 1,e 2,...,e M } consiss of egdes marked by he PN ransiions j T,j = 1,...,m. Namely, wo nodes v i,v j V are conneced by he oriened arc e = e vi v j E direced from v i owards v j. The arc is marked by he PN ransiion = vi v j = xi x j T jus when i is enabled in he sae x i and by is firing he new sae x j will be reached. The reachabiliy ree has o involve a corresponding node for every PN sae and a corresponding edge for any PN ransiion enabled in he given sae. To avoid complicaions (especially he infinieness of he generaed reachabiliy ree) he so called dupliciy nodes are defined as he leaf nodes (he graph leaves). In such a way subrees which have been included ino he reachabiliy ree already are eliminaed. Namely, he node v i V r for which here is a node v j V r such ha v j v i is named he dupliciy node. Here he operaor represens he binary anireflexive ransiive anisymeric relaion expressing he ordering of nodes in he se V r. Consequenly, from he dupliciy node no edges emerge. Connecing all dupliciy nodes of he node v j V r ogeher and also wih he node v j iself we obain (afer doing his for j = 1,...,N r ) he reachabiliy graph from he reachabiliy ree. I is imporan ha boh he reachabiliy graph and he reachabiliy ree have he same adjacency marix. Anoher descripive definiion of he reachabiliy graph is presened in [12]. I is disincive on i ha i sars by defining wo funcions enabled(v): given a sae v, his funcion reurns he se of ransiions ha are enabled in v fire(v,): given a sae v, and a ransiion enabled(v), his funcion reurns he sae v reached from v by firing and consequenly, he reachabiliy graph is defined as G r = (V r,t,e r,v 0 ) being he graph wih he smalles ses of nodes V r, ransiions T, and edges E r such ha v 0 V r, where v 0 is he iniial sae of he sysem, and if v V r, hen for all enabled(v) i holds ha T, fire(v,) V r, and (v,,fire(v,)) E r. In PN heory he reachabiliy ree and/or graph are very imporan, especially for esing he PN properies. In his paper i will be uilized a he DEDS conrol synhesis. The Malab procedure for compuing he G r was inroduced in [9]. Is enries are he PN incidence marices F, G and he iniial sae x 0. The procedure compues he adjacency marix A r of G r in he quasi-funcional form A r (k) (wih inegers represening he indices of he ransiions as is nonzero elemens) and he se of he reachable saes given as columns of he marix reach. These marices fully characerize he PN reachabiliy graph. The funcional adjacency marix A k corresponding o he quasi-funcional marix A r (k) can be consruced when he

18 524 F. Čapkovič inegers in A r represening he indices of PN ransiions j T,j = 1,...,m are replaced by he ransiion funcions γ k j,j = 1,...,m,k = 0,1...,K. This marix represens he PN causaliy. I can be said ha when DEDS is modelled by PN, he sricness of he DEDS causaliy is rigorously adhered by he PN reachabiliy ree and/or he reachabiliy graph. Therefore, he reachabiliy graph canno be avoided a he DEDS conrol synhesis, of course. Applying he resuls of he previous secions (concerning he DEDS causaliy) o he PN reachabiliy graph we can uilize hem a modelling, analysing and conrol of ineracions among agens. However, i is necessary o keep in mind ha he number of feasible saes N = N r + 1 because also he iniial sae is one of hem. The aenion will be focused especially on he negoiaion process. 5 THE PRINCIPLE OF THE AGENTS NEGOTIATION PROCESS The negoiaion process iself consiss of several principle aciviies [13]. Especially, he following are mos imporan: defining he negoiaion environmen, iniial conac of agens, offer(s) and couner offer(s) among hem, evaluaion of proposals, and oucomes of he negoiaion process. The coordinaion plan of he negoiaion process can be formally described by DEDS modelled by PN as can be seen on he lef in Figure 7a). The PN places represen he aciviies and he PN ransiions represen he discree evens. The inerpreaion of he places and ransiions is as follows: p 1 = sar; p 2 = define negoiaion environmen; p 3 = iniial conac; p 4 = offer(s) and couner offer(s); p 5 = evaluaion; p 6 = oucomes; p 7 = end; 1 = saring negoiaion process; 2 = negoiaion plan(s); 3 = hand shake ; 4 = proposal(s); 5 = revised proposal(s); 6 = agreemen or qui; 7 = ending negoiaion process. However, he realiy is more complicaed. To illusrae i in deails, le us inroduce he PN-based model of he agen in general and he cooperaion of wo agens. The possible cooperaion of several agens will be poined ou oo. 6 THE PN-BASED MODEL OF AGENTS IN MAS In general, he composiional srucure of an agen from he collaboraion/negoiaion poin of view can looks like ha given in Figure 7b). The PN-based model represens he aomic aciviies of he agen as well as heir muual inerconnecion. The inerpreaion of he PN places is as follows: p 1 = he agen (A 1 ) is free; p 2 = a problem has o be solved by A 1 ; p 3 = A 1 is able o solve he problem (P A1 ); p 4 = A 1 is no able o solve P A1 ; p 5 = P A1 is solved; p 6 = P A1 canno be solved by A 1 and anoher agen(s) should be conaced; p 7 = A 1 asks anoher agen(s) o help him solve P A1 ; p 8 = A 1 is asked by anoher agen(s) o solve a problem P B ; p 9 = A 1 refuses he help; p 10 = A 1 acceps he reques of anoher agen(s) for help; p 11 = A 1 is no able o solve P B ; p 12 = A 1 is able o solve P B. However, i has o

19 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS 525 p 1 p 2 p 3 p p 6 p p 5 p 2 p p p p p 4 11 p p p p p 6 7 Agen p 7 a) b) Fig. 7. The PN-based model of: a) he coordinaion plan of he negoiaion process in general; b) a composiional srucure of he agen be said ha anoher concep of boh he agen srucure and he aciviies is no excluded. In our case parameers of he PN-based model are as follows: F = ; G = Analyse e.g. he siuaion when x 0 = (1,0,0,0,0,0,0,1,0,0,0,0) T, i.e. he siuaion when he agen A 1 is free and i is asked by he agen A 2 o solve he problem P B = P A2. Using he algorihm inroduced in [5] we have he following quasi-funcional adjacency marix A r (k) (is elemens are he indices of he PN ransiions) of he PN reachabiliy ree and he marix reach wih columns being he feasible saes (he iniial sae and all saes reachable from his iniial sae):

20 526 F. Čapkovič A r (k) = ; T reach = The reachabiliy graph (RG) of such a PN-based model is given in Figure Fig. 8. The reachabiliy graph of he agen in he siuaion given by he iniial sae Such a PN-based model of he agen is universal and i can be used for modelling oher agens of MAS oo. Namely, he same inerpreaion of places (however wih shifed numbering p i+12, i = 1,...,12) can be used e.g. for he agen A 2. The agen model given in Figure 7b) can be modified by addiional inernal connecions from 7 o p 2 and from 7 o p 4. Consequenly, he elemens g 72,g 74 are added as follows: G = Such a modificaion of he model does no influence he agen aciviies when he agen works auonomously and/or in pair. However, experimens wih he models showed ha he modified model has a favourable influence when he number of agens is greaer han 2. Namely, when he modified model of agens is used he dimensionaliy of he reachabiliy ree is less han he dimensionaliy of he reachabiliy ree in case of he original model. 6.1 Analysing he Model and he Conrol Synhesis HavinghePNmodeladisposalwecanusehewidespecrumofmehodsdeveloped for PN, especially he mehods based on PN invarians and on he PN reachabiliy graph which are mos imporan a esing he PN properies. When he graphical

21 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS 527 ool (suiable for drawing PN and esing he properies) is used on his way he model analysis is comforable. The DEDS conrol synhesis is based on he following simple idea: 1. developing he sraigh-lined reachabiliy ree (SLRT) from he given iniial saex 0 (represenedbyhevicariousvecor 0 )owardsheprescribederminal sae x (represened by he vicarious vecor ); 2. developinghebackrackingreachabiliyree(btrt)from owards 0, however, direced o ; 3. inersecing boh he SLRT and BTRT. All seps are performed numerically in Malab. To compue SLRT and/or BTRT as well as heir inersecion he approach presened in Secion 3 is uilized. Having he se of feasible rajecories, he mos suiable one (saisfying imposed conrol ask specificaions) can be chosen. More deails abou he procedure can be found in [6, 9]. The GraSim graphical ool was developed o suppor he DEDS conrol synhesis. I is differen from he graphical PN simulaor (he ool for PN drawing and esing). The inpu of GraSim is he RG of he PN model. I is creaed by mouse clicking on appropriae icons represening he RG nodes and edges as well as he marks for designaing he iniial and erminal saes. A is oupu he ool yields (in he graphical form) he feasible rajecories from a given iniial sae o a prescribed erminal one. The rajecories can be analysed one afer anoher. When a rajecory is chosen for analysing, he sequence of corresponding discree evens is displayed. A presen he choice of he mos suiable rajecory is no self-acing ye. Namely, he choice srongly depends on he concree kind of he sysem o be conrolled. Therefore, a generalizaion in his way is problemaic. Moreover, he crieria for he choice are usualy only verbal. To quanify hem an appropriae knowledge represenaion has o be developed. However, using logical and/or fuzzylogical PN seems o be hopeful on his way. 6.2 Cooperaion of Agens Having wo agens A 1, A 2, heir he collaboraion/negoiaion is given in Figure 9. Incaseofmoreagens e.g.n A heplacesofheagenk = 1,...,N A arenumbered p i+12.j, i = 1,...,12, j = k 1 = 0,...,N A 1. In case of several agens boh he PN model and he RG will be more inricae. While he model size depends on modules and heir inerface, he RG size depends on x 0 and on he srucure of he model blocks. As we can see we have n = 24 places and m = 20 ransiions in he PN model of he wo agens cooperaion. However, he number of ransiions is higher han a simple sum being m = 14. Namely, some ransiions have o be added as an inerface in order o connec boh of he agens. Consequenly, for wo agens A 1,

22 528 F. Čapkovič p 2 p p p p 4 p 11 p p 8 p 5 p 9 p 6 7 p 7 16 p p p 15 I 13 n p e p 23 p 22 p 13 r 9 f a c p p p e 20 p p 19 Fig. 9. The Peri ne-based model of he wo agens negoiaion A 2 he marices of he MAS parameers will have he following form, where he srucure of he conac inerface beween he agens is given by he (n 6)-dimensional marix F c and (6 n)-dimensional marix G c F = ( F1 0 F c1 0 F 2 F c2 ) G 1 0 ; G = 0 G 1 G c1 G c2 F c1 = F c2 = G T c 1 = G T c 2 =

23 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS 529 In general, for agens i = 1,2,...,N A he srucure of marices is as follows: G F F c1 0 G F F c2 F= G= G NA F NA 1 0 F cna G NA F NA F cna B B c1 0 B B c2 B= B NA 1 0 B cna B NA B cna G c1 G c2... G cna 1 G cna = ( ) blockdiag(b i ) i=1,na B c where B i = G T i F i ; B ci = G T c i F ci ; i = 1,2...,N A ; F c = (F T c 1,F T c 2,...,F T c NA ) T G c = (G c1,g c2,...,g cna ); B c = (B T c 1,B T c 2,...,B T c NA ) T wih F i,g i,b i, i = 1,2,...,N A, represening he parameers of he PN-based model of he agen A i, and wihf c,g c,b c represeninghe srucure ofhe inerfacebeween he agenscooperaing in MAS. The model properies can be esed by means of he graphical ool. The ool was developed in order o draw he Peri ne o be esed, o simulae is dynamics developmen for a chosen iniial marking, o compue is P-invarians and T-invarians and o compue and draw is reachabiliy ree. In case of wo agens cooperaion saring from he iniial sae x 0 = (σ p1,σ p2,...,σ pn ) T = ( A1 x T 0, A2 x T 0 )T wih A1 x 0 = (1,1,1,0,0,0,0,0,0,0,0,0) T, A2 x 0 = (1,1, 0,1,0,0,0,0,0,0,0,0) T, i.e. from he sae where only six places have nonzero marking, namely σ p1 = 1, σ p2 = 1, σ p3 = 1, σ p13 = 1, σ p14 = 1, σ p16 = 1, he reachabiliy graph given in Figure 10 is developed Fig. 10. The reachabiliy graph of he wo agens negoiaion

24 530 F. Čapkovič I has N = 13 nodes represening he feasible saes { 1, 2,..., 13 }. The edges express feasible pahs (rajecories) among he corresponding saes. Using boh graphical ools he ool for he PN model drawing and analysing he model properies as well as he ool GraSim deermined especially for finding and analysing he space of feasible rajecories and for he conrol synhesis is very comforable and user friendly. However, from he ergonomy (human facor engineering) poin of view, he observed area on he monior as well as he number of places, ransiions and heir inerconnecions in he former ool and he number of saes and he inricacy of connecions among hem in he laer one, which he human operaor is able o recognize, are limied. Hence, more formal approaches are found for DEDS analysis and conrol. Thus, large-scale mahemaical models can be handled numerically (e.g. by means of he Malab procedures). The marices A r (k) and reach characerizing he RG are as follows A r (k) = ; reach = ( 1 ) reach 2 reach 1 reach =

25 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS reach = The soluions of he conrol synhesis are illusraed in Figure γ γ 6 11 f e 10 a si 9 3 γ 3 b 8 le 7 2 γ 15 s 6 a 5 e 1 γ 14 4 v e 3 c 0 γ 9 2 o r 1 s sep of he dynamics developmen γ γ γ γ a) b) Fig. 11. The cooperaion of wo agens A1, A2 in MAS: a) he case when A1 solves (on he basis of he demand of A2) he problem o be solved by A2, however, A2 is no able o do his; b) he case when A1 refuses o help A2

26 532 F. Čapkovič The Problem of DEDS Conroller and Conrolled Objec The simple example inroduced above describes he problem of wo independen (auonomous) agens ha communicae each oher abou he possibiliy of a muual help in solving own problems. As o he auhoriies, he agens are equipollen. Namely, hey can help each oher; however, hey also can refuse he help. However, consider he conrolled objec (being e.g. a kind of DEDS or a sysem in general) o be he agen A sys and is conroller o be anoher agen A con. The role of he agen A sys is o work wih respec o prescribed rules. The role of he agen A con is o force A sys o work wih respec o he prescripions. I can be said ha a maser-slave relaion beween he agens occurs, where A con is he maseranda sys isheslave. Moreprecisely,heA sys hashepermanenproblemo behave wih respec o he rules, e.g. o pass from a given sae owards a prescribed sae when several alernaives are possible. The mos suiable possibiliy canno be chosen by he agen A sys iself bu on he conrary, i is deermined in he process of he conrol synhesis. The conrol synhesis yields he sequence of conrol inerferences which are obruded on A sys. The agen A con supervises wheher A sys respecs he inerferences or no. Such an agen A con is usually named supervisor. Onheoherhand,amoresophisicaedagenA con canpermanenlybeableosolve he problems concerning he A sys behaviour and obrudes he conrol inerferences upon A sys. Therefore, in case of conrol i canno be spoken abou a muual help beweena sys anda con. ThesiuaionisdisplayedinFigure12. Whileheprocessof he conrol synhesis is usually performed in he off-line way he muual cooperaion beween he conroller and he conrolled sysem is performed on-line. Conrol Synhesis off-line Conroller on-line Conrolled Sysem Fig. 12. Cooperaion of he agens A sys (he conrolled sysem) and A con (he conroller) in he process of conrolling A sys by A con Neverheless, he proposed PN-based approach o modelling, analysing and conrol synhesis is sufficienly general and i can be uilized in his case as well. Consider ha A 1 = A sys and A 2 = A con and uilize he above inroduced mahemaical model suiable for wo agens wih he same parameers, bu wih anoher iniial sae given as x 0 = ( A1 x T 0, A2 x T 0) T wih A1 x 0 = (1,1,0,1,0,0,0,0,0,0,0,0) T,

27 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS 533 A 2 x 0 = (1,0,0,0,0,0,0,0,0,0,0,0) T. Hence, A r (k) = The corresponding reachabiliy ree is on he lef in Figure Fig. 13. The reachabiliy graph of he cooperaion A sys and A con. The cooperaion wihou he feedback on he lef side; he cooperaion wih he feedback on he righ side Because he conrol problem needs no be solved in one cooperaive cycle he feedback connecion can be inroduced. I can be simply realized by adding he ransiion 19 ino he sysem inerface see Figure 14. I can be seen ha is inpu place is p 5 (namely, is firing realizes he process of he conrol problem solving) and is oupu places are he places deermining he iniial sae of he agens, i.e. he places p 1, p 2, p 4 and p 13. Of course, he PN incidence marices have o respec he new ransiion 19 oo. Consequenly, he addiional 19 h column of F is (0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) T and he 19h row of he marix G is (1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0). In he reachabiliy graph on he righ in Figure 13 such a feedback manifess iself by connecing 9 wih 1 by he arc marked by 19. Consequenly, he firs column of A r (k) is (0,0,0,0,0,0,0,0,19) T. The space of he feasible saes is he same, i.e. he marix reach does no change. I means ha when he problem is no solved during one cooperaive cycle of he agens heir cooperaion can coninue in anoher cycle. The sae rajecories are given in Figure 15.

28 534 F. Čapkovič p 2 16 p 14 p p 24 3 p p 6 15 I 13 5 n p 4 p p e 12 p 16 4 p p p p 13 9 r f p 8 p 5 p9 a p p p p 6 c p 18 e p p 19 Conrolled Sysem Conroller Fig. 14. The PN-based model of he feedback cooperaion of he A sys and A con Cooperaion of Three Agens Analyse he cooperaion of hree agens A 1, A 2, A 3 muually conneced as given in Figure 16. The srucural marices describing he agens ineracions can be wrien on he basis of his figure as follows F c1 = ; G T c 1 =

29 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS F c2 = ; G T c 2 = F c3 = ; G T c 3 = γ 5 17 γ f e 7 7 a si 4 γ 13 6 b 5 le 3 γ 10 4 s 2 γ 18 3 a 1 γ 7 2 e 0 γ γ 6 13 γ γ γ γ γ 2 s sep Fig. 15. The sae rajecories a he cooperaion of A sys and A con : on he lef a he cooperaion wihou feedback; on he righ a he feedback cooperaion

30 536 F. Čapkovič p 2 24 p 14 p p p p p 16 p 4 p p 4 p p p p 13 9 p 8 p 5 p 17 p9 p 21 p 20 p 6 p 18 Agen A 7 Agen A p 7 p p 26 p p p 28 p p p p p p Agen A 3 p p Fig. 16. The Peri ne-based model of he hree agens cooperaion Consider x 0 = ( A1 x T 0, A2 x T 0, A3 x T 0) T (26) where A1 x 0 = (1,1,1,0,0,0,0,0,0,0,0,0) T, A2 x 0 = (1,1,1,0,0,0,0,0,0,0,0,0) T, A 3 x 0 = (1,1,0,1,0,0,0,0,0,0,0,0) T. I is he siuaionwhere A 3 is no able o solve is problem P A3 and i has o ask eiher A 1 or A 2 for help. A 1, A 2 are able o solve heir problems. There are N = 40 differen feasible saes in his sysem. The reachabiliy graph is in Figure 17. The segmen of he reachabiliy graph demarcaed by he doed line indicaes he par of he graph which is acive during he sysem dynamics developmen in case when he iniial sae is given by (26). Of course, he ineracion wih oher

31 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS a 1 21 c i v e s e g m 13 en Fig. 17. The reachabiliy graph of he hree agens cooperaion developed from he given iniial sae. The acive segmen indicaes he par when A 1 or A 2 solves he problem P A3 pars of he graph should no be broken in order o preserve he model universaliy. Two soluions can be found for he given iniial sae. Namely, he soluion depends on he fac which of he agens A 1, A 2 will be asked by he agen A 3 for help. The soluions are as follows: where i = ( A1 T i, A2 T i, A3 T i ) T, i = 37,38, A1 37 = (1,1,1,0,0,0,0,0,0,0,0,0) T, A 2 37 = (0,1,1,0,0,0,0,0,0,0,0,0) T, A 3 37 = (0,0,0,0,1,0,0,0,0,0,0,0) T and A 1 38 =(0,1,1,0,0,0,0,0,0,0,0,0) T, A2 38 =(1,1,1,0,0,0,0,0,0,0,0,0) T, A3 38 = (0,0,0,0,1,0,0,0,0,0,0,0) T.

32 538 F. Čapkovič 7 GENERALIZATION The above described approach o modelling he agens in MAS is no rigid. I is suiable no only for modelling and analysing he negoiaion process bu also for he agen cooperaion in general. Namely, he PN subnes modelling he srucure of agens can be buil arbirarily (according o demands of he model creaor). Even, he agens can have he muually differen srucure. Also he inerface among he agens in MAS needs no be modelled only by he addiional PN ransiions. On he conrary, in general he inerface can be modelled by a PN subne consising of boh he addiional PN ransiions and/or he addiional PN places. The srucure of such an inerface can be creaed wihou resricions a pleasure of he model creaor. Le us inroduce e.g. he simple model of he agen communicaion in order o illusrae he possibiliies of he PN modelling he agens cooperaion in MAS. 7.1 The Example of he Agen Communicaion Consider he simple srucure of he agens defined in [23]. The inerpreaion of he places in he PN model in Figure 18 is as follows: p 1 A 1 does no wan o communicae; p 2 A 1 is available; p 3 A 1 wans o communicae; p 4 A 2 does no wan o communicae; p 5 A 2 is available; p 6 A 2 wans o communicae; p 7 communicaion; p 8 availabiliy of he communicaion channel(s) Ch (represening he inerface). The PN ransiion 9 fires he communicaion when A 1 is available and A 2 wans o communicae wih A 1, 10 fires he communicaion when A 2 is available and A 1 wans o communicae wih A 2, and 12 fires he communicaion when boh A 1 and A 2 wans o communicae each oher. I is clear ha he inerface (he communicaion channel) has a form of he PN module (subne) consising of boh he places and ransiions. In spie of his fac, he above inroduced approach o modelling, analysing and conrol synhesis can be uilized also in such a case, of course. I is sufficien o use he following model parameers and o choose an iniial sae. Afer doing his, here are no resricions as o using he approach. F = = F A1 0/ (3 4) F A1 A 2 0/ (3 4) F A2 F A2 A 1 0/ (2 4) 0/ (2 4) F Ch1,2

33 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS G G T = T A 1 0/ (3 4) G T A 1 A = 0/ (3 4) G T A 2 G T A 2 A / (2 4) 0/ (2 4) G T Ch 1, x T 0 = (0,1,0,0,1,0,0,1)T The number of agens cooperaing in such a kind of MAS can be greaer han wo, of course. p 3 p 6 p p 1 p 4 p p p 7 Agen A 1 Agen A 2 11 Inerface Fig. 18. The example of he PN-based model of he wo agens communicaion in general 8 CONCLUSIONS Firs of all, boh he sraighforward and backward causaliy of DEDS in general were described in his paper. The DEDS conrol synhesis mehod based on hem was inroduced. Then, he alernaive PN-based approach o modelling, analysis and conrol of he agen ineracions in MAS was presened. The approach yields he resuls in analyical erms (compued by means of Malab) as well as graphically (obained by means of he graphical ool he PN simulaor). Nex, he PN-based model of he ineracion process was creaed in analyical erms. The reachabiliy ree and/or he reachabiliy graph as well as he space of reachable saes were generaed on he basis of boh he model parameers and he given iniial sae of he sysem. The coherence beween he causaliy and he reachabiliy graph was poined ou. Finally, he feasible sae rajecories were found by means of

34 540 F. Čapkovič he muual inersecion of boh he SLRT represening he sraighforward causal developmen of he sysem and BTRT represening he backward causal developmen of he sysem. The graphical ool GraSim developed in order o auomae he process of finding he space of feasible rajecories a he conrol synhesis was menioned. Finally, he possibiliy of he wider uilizing he proposed approach was poined ou. Acknowledgemens The presened resuls were achieved in he projec # 2/6102/26 parially suppored by he Slovak gran agency VEGA. The auhor hanks VEGA for he parial suppor. REFERENCES [1] Bellman, R. Kalaba, R.: Dynamic Programming and Modern Conrol Theory. Academic Press, New York-London, [2] Brams, S. J. Kilgour, D. M.: Fallback Bargaining. Group Decision and Negoiaion, Vol. 10, 2001, pp [3] Čapkovič, F.: Modelling and Conrol of Discree Even Dynamic Sysems. BRICS Repor Series, RS-00-26, Universiy of Aarhus, Denmark, 2000, 58p. [4] Čapkovič, F.: Conrol Synhesis of a Class of DEDS. Kybernees, Vol. 31, 2002, No. 9/10, pp [5] Čapkovič, F.: The Generalised Mehod for Solving Problems of DEDS Conrol Synhesis. In: P.W.H. Chung, C. Hinde, M. Ali(Eds.): Developmens in Applied Arificial Inelligence, Lecure Noes in Arificial Inelligence, Vol. 2718, 2003, pp [6] Čapkovič, F.: An Applicaion of he DEDS Conrol Synhesis Mehod. In: M. Bubak, G.D. van Albada, P.M.A. Sloo, J.J. Dongarra (Eds): Compuaional Science ICCS Par III, Lecure Noes in Compuer Science, Vol. 3038, 2004, pp [7] Čapkovič, F.: DEDS Conrol Synhesis Problem Solving. Proc. 2 nd IEEE Conf. on Inelligen Sysems, Varna, Bulgaria. IEEE Press, Piscaaway, NJ, USA, 2004, pp [8] Čapkovič, F. Čapkovič, P.: Peri Ne-Based Auomaed Conrol Synhesis for a Class of DEDS. Proc IEEE Conf. on Emerging Technologies and Facory Auomaion, Lisbon, Porugal, IEEE Press, Piscaaway, NJ, USA, 2003, pp [9] Čapkovič, F.: An Applicaion of he DEDS Conrol Synhesis Mehod. Journal of Universal Compuer Science, Vol. 11, 2005, No. 2, pp [10] Diesel, R.: Graph Theory. Springer-Verlag, New York, [11] Fiedler, M.: Special Marices and heir Using in Numerical Mahemaics. SNTL Publishing House, Prague, Czechoslovakia, 1981 (in Czech). [12] Heljanko, K.: T Parallel and Disribued Sysems (4 ECTS). Lecure 5. Deparmen of Compuer Science, Universiy of Helsinki, Finland, pp hp://

35 Modelling, Analysing and Conrol of Ineracions Among Agens in MAS 541 [13] Hung, P. C. K. Mao, J. Y.: Modeling E-Negoiaion Aciviies wih Peri Nes. Proc. 35 h Hawaii In. Conf. on Sysem Sciences, Big Island, Hawaii, Vol. 1. IEEE Compuer Sociey Press (CD/ROM), 2002, 10 p. [14] Kirkland, S. Olesky, D. D. van den Driessche, P.: Digraphs wih Large Exponen. The Elecronics Journal of Linear Algebra, Vol. 7, 2000, pp [15] Looney, C. G.: Fuzzy Peri Nes for Rule-Based Decision-Making. IEEE Trans. Sys. Man Cybern., Vol. 18, 1998, No. 1, pp [16] Muraa, T.: Peri Nes: Properies, Analysis and Applicaions. Proceedings of he IEEE, Vol. 77, April 1989, No. 4, pp [17] Nowosawski, M. Purvis, M. Cranefield, S.: A Layered Approach for Modelling Agen Conversaions. Proc. 2 nd In. Workshop on Infrasrucure for Agens, MAS, and Scalable MAS, Monreal, Canada, 2001, pp [18] Peerson, J. L.: Peri Nes Theory and he Modelling of Sysems. Prenice-Hall Inc., Englewood Cliffs, New York, [19] Plesník, J.: Graph Algorihms. VEDA, Braislava, Czechoslovakia, 1983(in Slovak). [20] Preparaa, F. P. Yeh, R. T.: Inroducion o Discree Srucures. Addison- Wesley Publ. Comp., Reading, USA, [21] Purvis, M. Cranefield, S. Nowosawski, M. Ward, R. Carer, D. Oliveira, M. A.: Agenciies Ineracion Using he Opal Plaform. Proc. Workshop on Agenciies: Research in Large-Scale Open Agens Environmens, 1 s In. Join Conf. on Auonomous Agens and Muli-Agen, Bologna, Ialy, 2002, CD/ROM, 5 p. [22] Rosen, K.: Discree Mahemaics and is Applicaions. McGraw-Hill Inc., [23] Sain-Voirin, D. Lang, C. Zerhouni, N.: Disribued Cooperaion Modeling for Mainenance Using Peri Nes and Muli-Agens Sysems. In: In. Symp. on Compuaional Inelligence in Roboics and Auomaion, CIRA 03, Kobe, Japan, July 16 20, 2003, pp [24] Sedláček, J.: Inroducion o Graph Theory. Academia, Prague, 1977 (in Czech). [25] Thompson, L.: The Mind and Hear of he Negoiaor. Prenice-Hall Inc., [26] Wieland, H.: Indecomposable Non-Negaive Marices. Mah. Zeischrif, Vol. 52, 1950, pp (in German). [27] Yen, J. Yin, J. Ioerger, T. R. Miller, M. S. u, E. Volz, R. A.: Cas: Collaboraive Agens for Simulaing Teamwork. Proc. 17 h Inernaional Join Conference on Arificial Inelligence IJCAI 2001, Seale, USA, 2001, pp Franišek Ô ÓÚ received his maser degree in 1972 from he Slovak Technical Universiy, Braislava, Slovakia. Since 1972 he has been working wih he Slovak Academy of Sciences(SAS), Braislava, namely in a he Insiue of Technical Cyberneics, in a he Insiue of Conrol Theory and Roboics and in 2001 ill now a he Insiue of Informaics. In 1980 he received he Ph.D. degree from SAS. Since 1998 he has been he Associae Professor. He works in he area of modelling, analysing and conrol of Discree-Even Sysems (DES).