A Structural Approach to the Enforcement of Language and Disjunctive Constraints

Transcription

1 A Srucurl Aroch o he Enforcemen of Lnguge nd Disjuncive Consrins Mrin V. Iordche School of Engineering nd Eng. Tech. LeTourneu Universiy Longview, TX Pnos J. Ansklis Dermen of Elecricl Engineering Universiy of Nore Dme Nore Dme, IN 66 June 0, 00

2 Ouline There re lredy numerous resuls on he enforcemen of liner consrins Lµ. Cn we del wih leled PNs? Cn we del wih more generl secificions? We ddress he following oics: Bckground: Generlized Liner Consrins (Lµ + Hq + Cv ) Lnguge Consrins Disjuncive Consrins ( W L i µ i ) i

3 Bckground Generlized Liner Consrins Noion: µ he mrking, µ 0 he iniil mrking, D he incidence mrix, q he firing vecor, nd v he Prikh vecor. Le µ i denoe µ( i ) nd v j denoe v( j ). The se equion: µ = µ 0 + Dv. TOKEN TRANSITION ARCS ( fires) ( fires) TRANSITIONS PLACES µ 0 = [ 0 0 ] T µ = [ 0 ] T µ = [ 0 0 ] T v = [ ] T v = [ 0 0 ] T v = [ 0 ] T q = [ 0 0 ] T q = [ 0 0 ] T

4 Bckground Generlized Liner Consrins The generlized liner consrins cn descrie lces rirrily conneced o PN. They hve he form: They require he iniil se (µ 0, v 0 ) o sisfy Lµ + Hq + Cv () Lµ 0 + Cv 0 Furher, rnsiion i my fire from curren se (µ, v) iff () Lµ + Hq + Cv for q(i) = nd q(j) = 0 j i. () Lµ + Cv, where v = v + q nd µ i µ. The generlized liner consrins descrie he P-ye lnguges of free-leled PNs.

5 Bckground Generlized Liner Consrins The generlized liner consrins descrie he P-ye lnguges of free-leled PNs. () () (c) unconsrined oerion ( ) v ( ) v v 0 ( ) v + v q + v v v 0 v v + v

6 Bckground Generlized Liner Consrins The enforcemen of he generlized liner consrins hs een sudied y Iordche nd Ansklis [ACC 00, TAC 8()]. This roch hs een followed: Enforce dmissile Lµ + Hq + Cv y simle mrix oerions. If Lµ + Hq + Cv re no dmissile, find L µ + H q + C v such h. L µ + H q + C v Lµ + Hq + Cv. L µ + H q + C v is dmissile. The roch used o find L µ+h q+c v reduces he rolem o he enforcemen of consrins L µ in rnsformed PN, for which numerous mehods re ville.

7 Lnguge Consrins Generlized liner consrins corresond o P-ye lnguges of free-leled PNs. We cn del lso wih he enforcemen of P-ye lnguges of generl leled PNs. In he following exmle, he even is unconrollle. d c 6 c PLANT SPECIFICATION

8 Lnguge Consrins Mehod Se The firs se is o comose he ln nd secificion models. d c c

9 Lnguge Consrins Mehod Se The firs se is o comose he ln nd secificion models. 9 7 ( ) ( ) d c ( ) ( ) ( ) ( ) c Becuse he suervisor cn disinguish eween is own rnsiions, we cn relel he ne o ke in ccoun his fc.

10 Lnguge Consrins Mehod Se Nex, we cn idenify he consrins ssocied wih he suervisor lces: 9 7 ( 6 ) v + v v 0 ( 7 ) v + v v v 0 ( 8 ) v v v ( 9 ) v v ( ) ( ) c ( ) ( ) ( ) ( ) d c 6 8 6

11 Lnguge Consrins Mehod Se Then, he consrins cn e rnsformed o n dmissile form 9 7 ( 6 ) v + v v 0 ( 7 ) v + v v v 0 ( 8 ) v v v + µ c d c ( 9 ) v v + µ 6 8 6

12 Lnguge Consrins This rocess hs chnged he secificion such h i is dmissile: d ORIGINAL ADMISSIBLE The closed-loo generes sulnguge of he originl secificion.

13 Lnguge Consrins Admissiiliy: A lnguge secificion is dmissile if he comosiion ln-suervisor is such h he suervisor never ems o inhii ln-enled unconrollle rnsiions, or deec closed-loo-enled unoservle rnsiions, or disinguish eween ln rnsiions wih he sme lel. The generlized consrin roch is exended o leled PNs o: Reduce he enforcemen of Lµ + Hq + Cv in leled PN o he enforcemen of L e µ e e in differen leled PN. Find n dmissile soluion L e, µ e e,, nd use i o derive n dmissile secificion L µ + H q + C v of he originl rolem. Srucurl mehods could e used wih he following sufficien condiions for dmissiiliy, T, ρ( ) = ρ( ) L e D(, ) = L e D(, ) () T, ρ() Σ uc {λ} L e D(, ) 0 () T, ρ() Σ uo {λ} L e D(, ) = 0 ()

14 Disjuncive Consrins Disjuncions hve he form where L i Z m i n nd i Z m i W L i µ i i, or equivlenly V W j i A j l i µ c i where l i Z n, c i Z nd A j is se of inegers. We cn ly here lierure resuls h reduce roosiionl logic o inequliies, y mens of uxiliry vriles.

15 Disjuncive Consrins Le δ i e uxiliry vriles: [δ i = ] [l i µ c i ] () Thus, X i A j δ i _ i A j l i µ c i (6) Assuming l i µ is ounded, m i l i µ M i, [δ i = ] [l i µ c i ] ecomes l i µ + (M i c i )δ i M i (7) l i µ + (c i + m i )δ i c i + (8)

16 Disjuncive Consrins Mehod Se Given re PN nd secificion 6 [µ 0] [µ 0] (9) The firs se is o idenify nd cree coies of he rnsiions h increse/decrese l i µ For µ 0: T = { } nd T + = {+ }. For µ 0: T = { } nd T + = {+ }.

17 Disjuncive Consrins Mehod Se The second se is o dd lces d i s inu (ouu) lces o T i (T + i ): d d We inend o chieve δ i = µ(d i ).

18 Disjuncive Consrins Mehod Se The nex se is o enforce he consrins involving he uxiliry vriles. Assume m = m = 0 nd M = nd M =. d d + + µ + δ [ ] µ + δ [e ] 6 µ + δ [ ] µ + δ [e ] e e Now, [µ(d i ) = ] [l i µ c i ] is enforced. We only need o enforce µ(d ) + µ(d ) o finish our rolem.

19 Disjuncive Consrins Mehod Se Since [µ(d i ) = ] [l i µ c i ] is lredy enforced, we only need o enforce µ(d ) + µ(d ) + h + e e d d 6 The lce h is oined.

20 Disjuncive Consrins Resul Our reviuos develomens descrie closed-loo sysem corresonding o he comosiion of he ln α α α α α α 6 6 h wih he following suervisor: d d + + α α α α α α α α e e Noe h he suervisor is no free-leled (hough he ln is).

21 Finl Remrks Srucurl mehods cn design PN suervisors for wide clss of secificions: conjuncive liner consrins disjuncive liner consrins (under cerin oundedness ssumions) P-lnguge secificions Srucurl mehods romise comuionl gins. The cos is h mehods my e suoiml, mening: he suervisor my e overly resricive soluions my exis even when no soluions re found. The closed-loo my well dedlock. Addiionl mehods need o e lied o ensure he closed-loo is live.