Synthesis of Supervisors Enforcing General Linear Vector Constraints in Petri Nets

Transcription

1 The 00 Aerican onrol onference, May -0, Anchorage, Alaska. Synhesis of Suervisors Enforcing General Linear Vecor onsrains in Peri Nes Marian V. Iordache and Panos J. Ansaklis Absrac This aer considers he roble of enforcing linear consrains conaining arking ers, firing vecor ers, and Parikh vecor ers. Such consrains increase he exressiviy ower of he linear arking consrains. We show how his new ye of consrains can be enforced in Peri nes. In he case of fully conrollable and observable Peri nes, we give he consrucion of a suervisor enforcing such consrains. In he case of Peri nes wih unconrollable and/or unobservable ransiions, we reduce he suervisor synhesis roble o enforcing linear arking consrains on a ransfored Peri ne. Inroducion In his aer we consider a suervisory conrol roble for discree even syses odeled as Peri nes, in which we desire o enforce a cerain ye of secificaions. Thus we have a lan which is absraced as a Peri ne (PN), and a secificaion on he behavior of he PN lan. We desire o find a suervisor such ha he closed-loo of he lan and he suervisor saisfies he secificaion. We resric our aenion o suervisors which can be reresened as PNs, and o secificaions in he for of conjuncions of linear inequaliies involving he arking, he firing vecor and he Parikh vecor of he lan PN. We describe such secificaions nex. Efficien ehods have been roosed in [,,, 7] for he synhesis of suervisors enforcing ha he arking µ of a PN saisfies consrains of he for Lµ b () The ehods address boh he fully conrollable and observable PNs and he PNs which ay have unconrollable and unobservable ransiions. The consrains () have been exended in [, 7] o he for Lµ + Hq b () Dearen of Elecrical Engineering, Universiy of Nore Dae, IN, USA. E-ail: iordache., ansaklis.@nd.edu. The arial suor of he Naional Science Foundaion (NSF ES99-), of DARPA/ITO-NEST Progra (AF- F ) and of he Lockheed Marin ororaion is graefully acknowledged. which adds a firing vecor er. In such consrains an eleen q i of he firing vecor q is se o if he ransiion i is o be fired nex; else q i = 0. Wihou loss of generaliy, H has been assued o have nonnegaive eleens. In his aer we consider consrains which add o () a Parikh vecor er: Lµ + Hq + v b () In () v is he Parikh vecor, ha is v i couns how ofen he ransiion i has fired since he iniial arking µ 0. As an exale, Parikh vecor consrains can be used o describe fairness requireens, such as he consrain ha he difference beween he nuber of firings of wo ransiions is liied by one. Adding he Parikh vecor er in () increases he exressiviy ower of linear consrains. In fac, any suervisor ileened as addiional laces conneced o he ransiions of a lan PN can be reresened by consrains of he for Hq + v b () The conribuion of his aer is as follows. In secions and we show ha any lace of a PN can be seen as a suervisor lace enforcing a consrain of he for (). Previously his roery was known for consrains of he for v b and PNs wihou self-loos []. Then we show how o obain suervisors enforcing consrains () in PNs. We firs give he soluion for he case of fully conrollable and observable PNs in secion. Then, in secion we urn our aenion o PNs which ay have unconrollable and unobservable ransiions. There we firs define adissible consrains as he consrains for which he ehod for fully conrollable and observable PNs can sill be used. Then, by using ne ransforaions, we reduce our roble o he suervisory synhesis roble for consrains of he for (), for which effecive ehods exis. Our aroach also exends he indirec ehod of [] on enforcing consrains (), as boh couled and uncouled consrains can be considered. Finally, an exale is given in secion. In he lieraure, Parikh vecor consrains and arking consrains have been searaely considered for vecor DES (VDES) in []. The VDES considered in [] corresond o PNs wihou self-loos. I has been shown here how o consruc he oial conroller via ineger rograing. A less couaionally burdensoe aroach, however no always oial, has been

2 Figure : Peri nes for Exale. given in [, ], which considers arking consrains and firing vecor consrains. This aer exends soe of he aroaches of [, ] by including he Parikh vecor consrains of []. Algebraic Reresenaions of PNs We denoe a PN srucure by N =(P, T, F, W), where P is he se of laces, T he se of ransiions, F he se of ransiion arcs, and W he weigh funcion. We also denoe by D he incidence arix, and by D + and D is coonens corresonding o weighs of arcs fro ransiions o laces, and weighs of arcs fro laces o ransiions, resecively. The coon algebraic PN reresenaion is via he following sae equaion: µ = µ 0 + Dv () where µ 0 is he iniial arking. The oeraion of a PN can also be described hrough inequaliies of he for (). Indeed, fro () we derive: ( D)v µ 0 () Le = D. The inequaliy v µ 0 deerines he oeraion of a PN only if he ne has no self-loos. To deal wih self-loos, an addiional er is inroduced: Hq + v µ 0 (7) where { D + H i,j = i,j if D i,j 0 () 0 oherwise Noe ha H i,j 0 for all i and j. The vecor q has he following eaning. Afer we fire fro µ 0 a sequence σ of Parikh vecor v, he ransiion i is enabled iff Hq (i) + (v + q (i) ) µ 0,whereq (i) is a vecor q wih zero eleens exce for he i h one, which is one. Exale. onsider he PNs of Figure. The PN is no resriced: he firings of, and are free. Therefore H and are ey arices. However, by adding he laces, and as in he PN, he following inequaliies aear in (7): v (9) v v 0 (0) v + v () (c) Figure : Illusraive exale. where he inequaliies are generaed by,,and, resecively. The inequaliies of he PN (c) are: q + v () v v 0 () v v + v () Noe ha boh µ and v can describe he sae of a PN. We choose o denoe by R(N,µ 0 ) all airs (µ, v) σ such ha µ 0 µ, and he Parikh vecor of he firing sequence σ is v. Enforcing Generalized Linear onsrains In his aer, a suervisor of a PN N =(P, T, F, W) is he PN ileenaion of a a Ξ : M T for soe M N P N T. For siliciy, he suervisor is also denoed by Ξ. A suervisor Ξ resrics he oeraion of a Peri ne N by forbidding all ransiions / Ξ(µ, v) o fire, where (µ, v) is he PN sae. APN(N,µ 0 ) and a suervisor Ξ are in closed-loo if Ξ suervises (N,µ 0 ); he closed-loo is denoed by (N,µ 0, Ξ). Given (N,µ 0, Ξ), we denoe he se of all reachable saes (µ, v) byr(n,µ 0, Ξ). We desire o enforce consrains of he general for (). For () is ore exressive han for (). Indeed, consider he closed-loo PN of Figure. There is no lace invarian involving he conrol lace, so canno be obained by enforcing () []. However he following consrain of he for () describes : v + v + v In fac, as shown in he revious secion, every lace of a PN can be seen as a conrol lace resricing he firings of he ne ransiions. We say ha a suervisor Ξ enforces () on a PN (N,µ 0 ) if (µ, v) R(N,µ 0, Ξ): () is saisfied. We say ha Ξ oially enforces () if (µ, v) R(N,µ 0, Ξ): Ξ is defined a (µ, v), and a ransiion i enabled in he lan (N,µ) is disabled by Ξ a (µ, v) (i.e. i / Ξ(µ, v)) iff firing i leads o a sae (µ,v ) such ha Lµ +v b or Lµ+Hq (i) +v b, where q (i) is he vecor q corresonding o firing i. X denoes he nuber of eleens of X.

3 Suervisor synhesis in he fully conrollable and observable case This secion describes he synhesis of he oial suervisor enforcing consrains () in PNs in which all ransiions are conrollable and observable. The oial suervisor is obained by exending he forulas given in [] for consrains of he for (). Le D + lc = ax(0, LD ) () D lc = ax(0,ld+ ) () The suervisor is given by he incidence arices: D c + = D + lc +ax(0,h D lc ) (7) Dc = ax(d lc,h) () The iniial arking of he suervisor is: µ c0 = b Lµ 0 (9) where µ 0 is he iniial arking of he lan. Noe ha in equaions ( ) he oeraor ax is defined as follows. If A is a arix, B =ax(0,a) is he arix of eleens B ij =0forA ij < 0, and B ij = A ij for A ij 0. Furherore, for wo arices A and B of hesaesize, =ax(a, B) is he arix of eleens ij =ax(a ij,b ij ). Noe ha equaions (7), () and (9) define a suervisor which can be reresened as a PN of incidence arices D + c and D c, and wih iniial arking µ c0. We call he laces of he suervisor conrol laces. Theore. The suervisor defined by he incidence arices D + c and D c of (7) and () and of iniial arking given by (9), oially enforces (). The heore can be roved by verifying ha in he closed-loo ne (which has he incidence arices [D +T,D c + T ] T,[D T,Dc T ] T and he iniial arking [µ T 0,µT c0 ]T ), a conrol lace revens a ransiion o fire iff firing violaes (). To his end i can be roven by inducion ha µ c = b v Lµ (0) Noe ha he suervisors we build for () ay no creae a lace invarian in he closed-loo ne. Suervisor synhesis in he case of PNs wih unconrollable and/or unobservable ransiions. Adissibiliy A ransiion is unconrollable if he suervisors are no given he abiliy o direcly inhibi i. A ransiion is unobservable if he suervisors are no given he abiliy Figure : Unconrollabiliy/unobservabiliy illusraion. o direcly deec is firing. In our aradig he suervisors observe ransiion firings, no arkings. For insance, consider he PN of Figure. Assue firs ha is conrollable and is unconrollable. Then, in case canno be direcly inhibied; i will evenually fire. However, in case can be indirecly revened o fire by inhibiing. Now assue ha is unobservable and is observable. This eans ha we canno deec when fires. In oher words, he sae of a suervisor is no changed by firing. However, we can indirecly deec ha has fired by deecing he firing of. We are ineresed in adissible consrains, ha is consrains which can be oially enforced as in secion, in sie of our inabiliy o deec or conrol cerain ransiions. We forally define adissibiliy as follows. Definiion. Le (N,µ 0 ) be a PN. Assue ha we desire o enforce a se of consrains (). onsider he suervisor defined by (7), (), and (9). We say ha he consrains () are adissible if for all reachable saes (µ, v) of he closed-loo ne i is rue ha:. If is unconrollable and is enabled by µ N in N, hen is enabled by µ in he closed-loo ne.. If is unobservable and is enabled by µ, firing does no change he arking of he conrol laces. Noe ha condiion in he definiion corresonds o he requireen ha he unobservable ransiions which are no dead a he iniial arking of he closedloo ne, have null coluns in D c = D c + D c (where D c + and Dc are defined in (7) and ()). For general PNs i ay no be easy o check wheher a consrain is adissible. A couaionally sile es is given in he following sufficien condiion. Le Dc,uc be he resricion of Dc o he coluns of he unconrollable ransiions. Le D c,uo be he resricion of D c o he coluns of he unobservable ransiions. Proosiion. The consrains () are adissible a all iniial arkings if D c,uo and Dc,uc are null arices. The condiion D c,uo = 0 ensures ha for any unconrollable ransiion, a conrol lace is eiher no conneced o i, or is conneced o i wih inu and ouu arcs of equal weigh. The condiion D c,uc =0ensuresha no conrol lace is in he rese of an unconrollable ransiion. We denoe by µ N he resricion of µ o he laces of N.

4 Figure : Illusraion of he -ransforaion.. Transforaions o adissible consrains When a consrain is adissible, i can be enforced as in secion. However, when a consrain is no adissible or we canno discern wheher i is adissible, we are ineresed o ransfor i o a for which we know is adissible. Thus we have he following roble. Given a se of consrains () on a PN (N,µ 0 ), find a se of adissible consrains L a µ + H a q + a v b a () so ha if Ξ is a suervisor oially enforcing () on (N,µ 0 ), hen (µ, v) R(N,µ 0, Ξ): () is saisfied. In secion. we consider a ransforaion aroach in which we ransfor he PN such ha he consrains () are aed ino arking consrains. Then he arking consrains can be ransfored o adissible consrains by using any of he aroaches in []. Firs we define he PN ransforaions we use.. The -Transforaion We illusrae he idea of he ransforaion on an exale. Assue ha we desire o enforce he consrain below on he PN of Figure µ + q + v v () By ransforing he ne as in Figure, () can be wrien wihou referring o v: µ + q + µ µ () We say ha he PN of Figure and he consrain () are he -ransforaion of he PN of Figure and of (). The inverse -ransforaion is also ossible. Given he consrain µ µ +µ + q () on he PN of Figure, we can a i o µ + q v +v () in he original PN. We roceed nex o forally define he direc and inverse ransforaions. The -Transforaion Inu: The PN N = (P, T, F, W), he consrains Lµ + Hq + v b, and oionally he iniial arking µ 0. Figure : Exale for he H-ransforaion. Ouu: The -ransfored PN N = (P,T,F, W ), he -ransfored consrain L µ + Hq b, and he iniial arking µ 0 of N.. Iniialize N o equal N, L o L, and le k = P.. For i =o T.a. If i,hei h colun of, is no zero.a.i. Se k = k +.a.ii. Add a new lace k o N such ha k = and k = { i }..a.iii. Se L =[L, i ]andµ 0 =[µ T 0, 0]T. The -Transforaion Inu: The PN N =(P, T, F, W), he -ransfored ne N = (P,T,F,W ), and a se of consrains L µ + Hq b on N. Ouu: The consrains Lµ + Hq + v b.. Se L o L resriced o he firs P coluns and o be a null arix.. For i = P +o P.a. Le j be he ransiion index such ha i = { j }..b. Se j = L,i.. The H-ransforaion This ransforaion is a odificaion of he indirec ehod for enforcing firing vecor consrains in []. We illusrae i on an exale. onsider he PN of Figure. Assue ha we desire o enforce µ + µ +µ + q () Then we ransfor he PN as shown in Figure. The ransforaion adds a lace and a ransiion which corresond o he facor q. The ransfored consrain is µ + µ +µ +µ (7) where he er µ is obained as follows. onsider firing in he ransfored ne. If µ µ and a is he coefficien of µ,wedesire a + µ + µ +µ =+µ + µ +µ where he facor is he coefficien of q in (). Thus we obain a =. Nex we forally define he H-ransforaion. j /L,i is he colun j/i of /L.

5 i i k j Figure : Illusraion of he ransiion sli oeraion. The H-Transforaion Inu: The PN N = (P, T, F, W), he consrains Lµ + Hq b, and oionally he iniial arking µ 0. Ouu: The H-ransfored PN N H =(P H,T H,F H, W H ), he H-ransfored consrain L H µ H b, and he iniial arking µ 0H of N H.. Iniialize N H o equal N, L H o L, and le j = T and k = P.. For i =o T.a. If H i,hei h colun of H, is no zero.a.i. Se j = j +andk = k +..a.ii. Add a new lace k and a new ransiion j o N H as in Figure, where j has he sae conrollabiliy and observabiliy aribues as i..a.iii. Se L H =[L H,H i +LD i ]andµ 0H =[µ T 0H, 0]T, where D i is he i h colun of D,andD corresonds o N. The H -Transforaion Inu: The PN N =(P, T, F, W), he H-ransfored ne N H =(P H,T H,F H,W H ), and a se of consrains L H µ H b on N H. Ouu: The consrains Lµ + Hq b.. Se L o L H resriced o he firs P coluns and H o be a null arix.. For k = P +o P H.a. Le i be he ransiion index such ha k = { i }..b. Se H i = L H,k L H D H,i. 7 Resriced Access Area Figure 7: Plan Peri ne in he exale. and he iniial arking, resecively.. Aly he H-ransforaion o N, L µ + Hq b, andµ 0. Le N H, L H µ H b, andµ H0 be he H-ransfored ne, he consrains, and he iniial arking, resecively.. Tes wheher L H µ H b is adissible. If so, exi, and declare Lµ + Hq + v b adissible.. Transfor L H µ H b o adissible consrains L Ha µ H b a, such ha a suervisor oially enforcing L Ha µ H b a also enforces L H µ H b. In case of failure, exi and declare failure o find adissible consrains.. Aly he H -ransforaion o L Ha µ H b a. Le L a µ + H a q b a be he ransfored consrain. 7. Aly he -ransforaion o L a µ + H a q b a.sel a µ + H a q + a v b a o he -ransfored consrains. We rove he following resul in []. Theore. Assue ha he algorih does no fail a se. Then L a µ + H a q + a v b a is adissible, and a suervisor oially enforcing i enforces also Lµ + Hq + v b.. Algorih for he ransforaion o adissible consrains We can use he - and H-ransforaions o obain adissible consrains as follows. Inu: APNN, consrains Lµ + Hq + v b, and oionally an iniial arking µ 0. Ouu: Adissible consrains L a µ+h a q+ a v b a. Iniialize L a o L, H a o H, and a o.. Aly he -ransforaion. Le N, L µ +Hq b, andµ 0 be he -ransfored ne, he consrains, H i /L H,k /D H,i is he colun i/k/i of H/L H/D H,andD H corresonds o N H. I is ossible o carry ou he algorih indeendenly of he iniial arking. Exale onsider he lan PN of Figure 7. I corresonds o a region of a facory cell in which auonoous vehicles (AV) access a resriced area (RA). The nuber of AVs which ay be a he sae ie in he RA is liied. The AVs ener he RA fro wo direcions: lef and righ; AVs coing on he lef side ener via or, and AVs coing on he righ side via or. The AVs exi he resriced area via 9 or 0. The oal arking of, and 7 corresonds o he nuber of lef AVs waiing in line o ener he RA; only one AV should be in he saes and 7,haisµ + µ 7. Any of he aroaches in [, ] can be used. Aroaches generaing disjuncive consrains can also be used by alying he ses and 7 o each coonen of he disjuncion.

6 n n q + µ + µ + µ + µ 7 + v +v + v + v v 9 v 0 + (7) q + µ + µ + µ + µ + v +v + v + v v 9 v 0 + () q µ µ µ µ (v +v + v + v v 9 v 0 ) 0 (9) q µ µ 7 µ µ (v +v + v + v v 9 v 0 ) 0 (0) v + v + v + v v 9 v 0 +µ + µ () The closed-loo PN is shown nex o he lan in Figure, where he conrol laces 9 corresond o he consrains (7), (), (9), (0), (), (), (), (), and (), in his order. Figure : losed-loo Peri ne. The arking of,,and has a siilar eaning. Le be he axiu nuber of AVs which can be a he sae ie in he RA; noe ha he nuber of AVs in he RA is v + v + v + v v 9 v 0.When he nuber of vehicles in he resriced area is and boh a lef and a righ AV ae o ener he resriced area (i.e. boh µ + µ 7 =andµ + µ =), arbiraion is required. When an AV is in and no arbiraion is required, i can ener he RA wihou soing. When arbiraion is required, i sos (eners he sae 7 ) and wais he arbiraion resul. The sae aly o and. We desire he following. When an AV eners he RA, if an arbiraion was required o decide ha i ay ener, he AV should ener via or ; if no arbiraion was required, i should ener via or. These consrains can be wrien as follows: q + µ + µ 7 (v + v + v + v v 9 v 0) + () q + µ + µ (v + v + v + v v 9 v 0) + (9) q µ + µ + v + v + v + v v 9 v 0 (0) q µ + µ 7 + v + v + v + v v 9 v 0 () In addiion we have he requireens ha µ + µ 7 () µ + µ () The requireen on he axiu nuber of AVs in he RA is v + v + v + v v 9 v 0 () We add he fairness consrains v v n () v + v n () As,, 9, 0 are unconrollable and 9, 0 unobservable, he consrains ( ) and () are inadissible. They are ransfored o 7 7 The consrains (0) and () canno be ransfored o (ore resricive) adissible consrains; (9) and (0) reresen relaxed (and adissible) fors of (0) and (). 7 onclusion Enforcing linear arking and firing vecor consrains can be done effecively in Peri nes. This aer has exended his class of consrains o include Parikh vecor consrains. Then, we have shown how hese ore exressive consrains can be enforced as effecively as linear arking consrains. We have also enhanced he revious echnique for enforcing firing vecor consrains in he resence of unconrollable and unobservable ransiions. Our algorihs are sofware ileened. References [] A. Giua, F. Diesare, and M. Silva. Generalized uual exclusion consrains on nes wih unconrollable ransiions. In Proceedings of he IEEE Inernaional onference on Syses, Man and yberneics, ages , Ocober 99. [] M. V. Iordache and P. J. Ansaklis. Synhesis of suervisors enforcing general linear vecor consrains in Peri nes. Technical reor isis-00-00, Universiy of Nore Dae, February 00. [] Y. Li and W. Wonha. onrol of Vecor Discree- Even Syses II - onroller Synhesis. IEEE Transacions on Auoaic onrol, 9(): 0, March 99. [] J. O. Moody and P. J. Ansaklis. Suervisory onrol of Discree Even Syses Using Peri Nes. Kluwer Acadeic Publishers, 99. [] J. O. Moody and P. J. Ansaklis. Peri ne suervisors for DES wih unconrollable and unobservable ransiions. IEEE Transacions on Auoaic onrol, (): 7, March 000. [] J. O. Moody, M. V. Iordache, and P. J. Ansaklis. Enforceen of even-based suervisory consrains using sae-based ehods. In Proceedings of he h onference on Decision and onrol, ages 7 7, 999. [7] E. Yaalidou, J. O. Moody, P. J. Ansaklis, and M. D. Leon. Feedback conrol of Peri nes based on lace invarians. Auoaica, ():, January 99.