Embedded Systems 5 BF - ES - 1 -

Transcription

1 Embedded Sysems 5 - -

2 REVIEW: Peri Nes Def.: N=C,E,F) is called a Peri ne, iff he following holds. C and E are disjoin ses 2. F C E) E C); is binary relaion, flow relaion ) Def.: Le N be a ne and le x C E). x := {y y F x} is called he se of recondiions. x := {y x F y} is called he se of oscondiions. Examle: x x x - 2 -

3 Comeing Trains Examle: Conflic for resource rack - 3 -

4 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M M2 M3 Sli. M4 M5 M6 Sli 4. Sli

5 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M6 M Sli

6 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M6 M7 M2s Sli

7 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M5 M6s Sli

8 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M4 M8 M6 Sli

9 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M M2s Sli.2-9 -

10 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M8 M M2 M3 M4 M5 M6 M7 M2s M3s M4s M5s M6s M7s M8s - -

11 Realisic scenarios need more general definiions More han one oken er condiion, caaciies of laces weighs of edges sae sace of Peri nes may become infinie! ready k= k=2 acceed roduce 2 Sorage acce consume 2 send k=5 5 idle k= k=2 ready Producer Consumers - -

12 From condiions o resources c/e nes model he flow of informaion a a fundamenal level rue/false) here are naural alicaion areas for which he flow/ransor of resources and he number of available resources is imoran daa flow, documen-/workflow, roducion lines, communicaion neworks, www,..) lace/ransiion nes are a generalizaion of c/e nes - 2 -

13 From condiions o resources lace/ransiion nes are a generalizaion of c/e nes: sae elemens reresen laces where resources okens) can be sored ransiion elemens reresen local ransiions or ransor of resources a ransiion is enabled if and only if sufficien resources are available on all is inu laces sufficien caaciies are available on all is ouu laces a ransiion occurrence consumes okens from each inu lace and roduces okens on each ouu lace - 3 -

14 Place/ransiion nes mulile okens er lace Def.: P, T, F, K, W, M ) is called a lace/ransiion ne P/T ne) iff. N=P,T,F) is a ne wih laces P and ransiions T 2. K: P N {}) \{} denoes he caaciy of laces symbolizes infinie caaciy) 3. W: F N \{}) denoes he weigh of grah edges 4. M : P N {} reresens he iniial marking of laces M W Segmen of some ne) defaul: K = W = - 4 -

15 Examle P =, 2, 3) T = {, 2} F = {, ), 2, 2), 3, ),, 2), 2, ), 2, 3)} W = {, ) 2, 2, 2), 3, ),, 2), 2, ) 2, 2, 3) } m = 2,, ) - 5 -

16 Reachabiliy Reachabiliy grah: - 6 -

17 Reachabiliy - 7 -

18 Comuing changes of markings Firing ransiions generae new markings on each of he laces according o he following rules: - 8 -

19 Acivaed ransiions Transiion is acivaed iff Acivaed ransiions can ake lace or fire, bu don have o. The order in which acivaed ransiions fire is no fixed i is non-deerminisic)

20 Boundedness A lace is called k-bounded or k-safe if i conains in all reachable markings a mos k okens A ne is bounded if each lace is bounded Alicaion: laces reresen buffers and regisers avoid buffer overflow

21 Liveness A ransiion is live if in every reachable marking here exiss a firing sequence such ha he ransiion becomes enabled A ne is live if all is ransiions are live - 2 -

22 Deadlock A dead marking deadlock) is a marking where no ransiion can fire A ne is deadlock-free if no dead marking is reachable

23 Comuaion of Invarians We are ineresed in subses consising of laces whose number of okens remain invarian under ransiions, e.g. he number of rains commuing beween Amserdam and Paris Cologne and Paris) remains consan Imoran for correcness roofs, e.g. he roof of liveness

24 Shorhand for changes of markings )if, ), \ )if, \ )if, ) W W W W Le P: M ) = M)+ ) Firing ransiion: +: vecor add M = M+

25 Marix N describing all changes of markings Def.: Marix N of ne N is a maing N: P T Z inegers) such ha T: N,)=) Comonen in column and row indicaes he change of he marking of lace if ransiion akes lace. )if, ), \ )if, \ )if, ) W W W W

26 Examle: N = s

27 Place invarians

28 Characerisic Vecor R R c R if if ) Le: ) ) ) c c R P j R j R j ) R j Scalar roduc

29 Condiion for lace invarians Accumulaed marking consan for all ransiions if R n R c c Equivalen o N T c R = where N T is he ransosed of N ) ) ) c c R P j R j R j

30 - 3 - Sysem of linear equaions Sysem of linear equaions. Soluion vecors mus consis of zeros and ones. )... ) ) ) ) ) )... ) ) n R R R n m m n n c c c

31 Comeing rains examle

32 Alicaion o Thalys examle N T c R =, wih N T = c R,

33 Inerreaion of he s invarian c R, Characerisic vecor describes laces for Cologne rain. We roved ha: he number of rains along he ah remains consan. C R, s

34 Alicaion o Thalys examle N T c R =, wih N T = c R, 2,,,,,,,,,,,,)

35 Inerreaion of he 2 nd invarian c R, 2,,,,,,,,,,,,) C R,2 We roved ha: None of he Amserdam rains ges los. s

36 Alicaion o Thalys examle N T c R =, wih N T = c R,

37 Soluion vecors for Thalys examle c R, c R, 2 c R, 3 C R,4 C R,3 C R, cr, 4 s We roved ha: he number of rains serving Amserdam, Cologne and Paris remains consan. he number of rain drivers remains consan. C R,2-37 -

38 Soluion vecors for Thalys examle I follows: each lace invarian mus have a leas one label a he beginning, oherwise dead a leas hree labels are necessary in he examle C R,4 C R,3 C R, s C R,2-38 -

39 Invarians & boundedness A ne is covered by lace invarians iff every lace is conained in some invarian. Theorem : a) If R is a lace invarian and R, hen is bounded. b) If a ne is covered by lace invarians hen i is bounded

40 Peri ne lan coordinaion for robocu eams G. Kones and M.G. Lagoudakis - 4 -

41 Passing Maneuver Teamwork Design Based on Peri Ne Plan P. F. Palamara, V. A. Ziaro, L. Iocchi, D. Nardi, and P. Lima - 4 -

42 Team sraegy Peri ne lan coordinaion for robocu eams G. Kones and M.G. Lagoudakis

43 Aacker role in he ressing defense acic BallOnO BallOnOwn BallNoGrabbed BallGrabbed sar.acswichtacic sar.acgotoball sar.acgobeweenballandogoall sar.acsayonmidline end.acbrabball end.acswichtacic end.acgotoball end.acgobeweenballandogoall end.acsayonmidline GOAL NearBall sar.acbrabball Peri ne lan coordinaion for robocu eams G. Kones and M.G. Lagoudakis

44 Midfielder role in he ressing defense acic BallOnOwn BallOnO BallNoGrabbed BallGrabbed sar.acswichtacic sar.acgotoball sar.acgobeweenballandowngoal sar.acsayonmidline end.acbrabball end.acswichtacic end.acgotoball end.acgobeweenballandowngoal end.acsayonmidline GOAL NearBall sar.acbrabball Peri ne lan coordinaion for robocu eams G. Kones and M.G. Lagoudakis

45 Defender role in he ressing defense acic BallGrabbed BallOnO BallOnOwn end.acbrabball sar.acswichtacic ar.acgobeweenballandowngoall sar.acgobeweenballandowngoall sar.acsayonmidline end.acsayonmidline end.acgobeweenballandowngoal end.acgobeweenballandowngoal sar.acbrabball end.acswichtacic NearBall GOAL Peri ne lan coordinaion for robocu eams G. Kones and M.G. Lagoudakis