Reduction of the Supervisor Design Problem with Firing Vector Constraints

Transcription

1 wih Firing Vecor Consrains Marian V. Iordache School of Engineering and Eng. Tech. LeTourneau Universiy Longview, TX Panos J. Ansaklis Dearmen of Elecrical Engineering Universiy of Nore Dame Nore Dame, IN June 0, 005

2 Conex PNs develoed in Comuer Science o model concurren rocesses (CPs). Muual exclusion: a common consrain ye imosed on CPs. Exressed in PNs by: P i µ i c. Used also in he conex of AGV coordinaion for manufacuring sysems [Krogh and Holloway 99] and bach chemical rocesses [Tius and Lenarson, 999]. More comlex consrains Lµ b are necessary (L, b ineger marices) - due o unconrollabiliy and unobervabiliy - o exress more comlex secificaions The consrains Lµ b have been used o describe liveness enforcing suervisors [Iordache and Ansaklis, 000; Park and Reveliois, 00].

3 Conex In he conex of chemical rocess conrol [Yamalidou and Kanor 99], consrains describing how valves should be oened/closed resul in: Lµ + Hq b. The same arise also in railway neworks [Giua and Seazu 00], describing safey consrains (rains should no collide). The more general form Lµ + Hq + Cv b used in a raher comlex AGV coordinaion roblem [Iordache and Ansaklis, 00]. Fairness consrains of he form Cv b were used for a communicaion roocol [Genrich e al, 980] and a manufacuring alicaion [Li and Wonham, 99].

4 Conex The SBPI The suervision based on lace invarians develoed for consrains Lµ b. (All reachable markings µ consrained o saisfy Lµ b.) Le D be he incidence marix and µ 0 he iniial marking. The suervisor consiss of monior laces conneced according o D s = LD and of iniial marking µ s0 = b Lµ 0. µ + µ µ + µ PLANT Moniors 4 5 CLOSED LOOP

5 Conex Moniors Conversely, given a se of moniors, wha secificaion do hey enforce? Lµ + Hq + Cv b, in general. - The Cv erm no needed if hey ariciae in lace invarians. - The Hq erm needed if hey have selfloos. µ + µ µ + µ PLANT Moniors 4 5 CLOSED LOOP

6 Conex Beyond Moniors I is desirable o be able o deal wih even more general secificaions. For wha kind of secificaions is he closed-loo sill a Peri ne? - Prefix ye language secificaions. - In aricular, disjuncions W i L i µ b i, under some boundedness assumions. b 7 b 4 a c b b d c a 9 a PLANT SPECIFICATION

7 Conex Beyond Moniors As an examle, his is he lan. The secificaion is [µ 0] [µ 4 0]. α α α α 4 α 5 α Assuming he bounds µ and µ 4, his suervisor resuls: a h a d d α α α 5 α 4 α α α 4 α e e Noe ha he suervisor is no free-labeled (hough he lan is).

8 Conex Difficulies Difficulies arise due o arial unconrollabiliy and unobservabiliy. We have focused on srucural and suboimal mehods of suervisor design. - No necessarily leas resricive soluions; a soluion may no be found even when one exiss. - The deadlock revenion roblem needs o be deal wih searaely. + The suervisor is aramerized by he iniial marking µ 0 : + a numerical value of µ 0 is no necessary o design a suervisor; + changes in he numerical value of µ 0 do no require recalculaing he suervisor. + Poenial comuaional benefis (reachabiliy analysis avoided). We have addressed: Cenralized and decenralized conrol. Labeled PN lans wih or wihou concurrency. Disjuncive, P-ye language, generalized linear consrains. Also liveness secificaions.

9 BOOK

10 TOPIC This resenaion considers he suervision roblem for secificaions n d _ [L i µ + H i q + C i v b i ] () i= By adding sink laces, he Cv erm can be incororaed in he Lµ erm. Therefore, i is enough o focus on consrains n d _ [L i µ + H i q b i ] () i= The aer shows ha he roblem of enforcing () can be reduced o he roblem of enforcing () on a ransformed Peri ne, wihou loss of ermissiveness. n d _ [L H,i µ H b i ] () i=

11 Background Generalized Linear Consrains Noaion: µ he marking, µ 0 he iniial marking, D he incidence marix, q he firing vecor, and v he Parikh vecor. Le µ i = µ( i ), q j = q( j ) and v j = v( j ). q j : how many imes j is o be fired; v j : how many imes j has been fired. The sae equaion: µ = µ 0 + Dv. ( fires) ( fires) 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

12 Background Generalized Linear Consrains The generalized linear consrains can describe laces arbirarily conneced o a PN. They have he form: They require he iniial sae (µ 0, v 0 ) o saisfy Lµ + Hq + Cv b (4) Lµ 0 + Cv 0 b Furher, a ransiion i may fire from a curren sae (µ, v) iff (a) Lµ + Hq + Cv b for q(i) = and q(j) = 0 j i. (b) Lµ + Cv b, where v = v + q and µ i µ. The generalized linear consrains describe he P-ye languages of free-labeled PNs. The enforcemen of he generalized linear consrains has been sudied by Iordache and Ansaklis [ACC 00, TAC 48()].

13 Seing The lan is assumed o be a double-labeled PN. ρ : T K (o : T O) associaes conrol (observaion) evens o ransiions. K c K (O o O) denoe he ses of conrollable (observable) evens. A double-labeled PN can model boh he unconrollabiliy of ClPNs (Krogh and Holloway) and he unobservabiliy of labeled PNs. The resuls are obained under he concurrency seing (ransiion-bag assumion, q N). 4 5 e (α) (β) (β) (γ) (γ) (γ) e e e e 4 e 4 Above, K = {e, e, e, e 4 } and O = {α, β, γ}.

14 GLCs and Concurrency Inerreing W i L iµ + H i q + C i v b i is a nonrivial issue. Le s begin wih Lµ + Hq + Cv b, under concurrency. Le H d = max(0, LD + C, H). Consider he moniors enforcing Lµ + Hq + Cv b ha are consruced under he no concurrency assumion (Iordache and Ansaklis, 00). The moniors enable q when H d q µ c = b Lµ Cv. This will be he concurrency inerreaion of Lµ + Hq + Cv b: a any ime, µ, q and v are ermissible if hey saisfy H d q b Lµ Cv. Inuiively, his corresonds o q enabled iff a all inermediary sages of he firing of q, he secificaion is saisfied (Lemma.). This will be he concurrency inerreaion of W i L iµ + H i q + C i v b i : a any ime, µ, q and v are ermissible if hey saisfy H d,i q b i L i µ C i v for some i.

15 Aroach Given W n i= L iµ + H i q b i, he aroach is o find W m i= La i µ + Ha i q ba i ha is feasible such ha W m i= La i µ + Ha i q ba i W n i= L iµ + H i q b i. N i L µ + H q i i b i H TRANSF N H L i H,i µ H,i b i PLANT AND SPECS TRANSFORMED PLANT AND SPECS (DESIGN) i a a a i i b i L µ + H q FEASIBLE SPECS INVERSE H TRANSF i L a H,i µ H,i a b i FEASIBLE CONSTRAINTS

16 The H-Transformaion Illusraion The ransformaion slis ransiions o subsiue a q i erm by a marking erm This ransformaion mas ino µ + µ + µ + q 5 (5) µ + µ + µ + 4µ 5 5 (6) The erm 4µ 5 is obained as follows. Consider firing in he ransformed ne: µ µ. The coefficien a of is o saisfy ha a + µ + µ + µ = + µ + µ + µ

17 The H-Transformaion Definiion Inu: N = (P, T, D, D + ), Lµ + Hq b, and oionally µ 0 and a se T s,h T. Ouu: N H = (P H, T H, D H, D+ H ), L Hµ H b, and µ H0.. Le T s = { T : ρ() = ρ( ) for some T s.. T s,h or H d (, ) 0}, where H d = max(ld, H, 0).. Le N H = N, L H = L, and µ H0 = µ 0.. For all T s : k j (a) Sli in, k, and j. (b) Le L H (, k ) = H d (, i ) + LD (, i ) and µ H0 ( k ) = For all T s : (a) o( ) = o(). (b) Exend he se of conrol evens s.. ρ( ) / {ρ() : T }. (c) ρ( ) is conrollable iff ρ() is conrollable. (d) For all T s, ρ( ) = ρ( ) iff ρ() = ρ( )....

18 The H-Transformaion Definiion The H-ransformaion for a se of consrains W i L iµ + H i q b i :. Le H d,i = max(l i D, H i, 0) and T s,h T s,h S i { T : H d,i(, ) 0}.. For all i, aly he H-ransformaion o he consrains L i µ + H i q b i wih he argumen T s,h above. Le L H,i µ H b i, N H, and µ H0 be he resul.. Final resul: W i L H,iµ H b i, N H, and µ H0.

19 The H-Transformaion Definiion The H -Transformaion of L H µ H b Inu: N = (P, T, D, D + ), N H = (P H, T H, D H, D+ H ), and L Hµ H b. Ouu: The H -ransformed consrains Lµ + Hq b.. Se L(, ) = L H (, ) P and H o he null marix.. For all k P H \ P (a) Le i be he ransiion such ha { i } = k. (b) Se H(, i ) = L H (, k ) L H D H (, i). The H -Transformaion of W i L H,iµ H b i. For all i, aly he H -ransformaion o he consrains L H,i µ H b i. Le L i µ + H i q b i be he ransformed consrains.. Final resul: W i L iµ + H i q b i.

20 Noaion S W i L iµ + H i q b i, S H W i L H,iµ H b i, S a H W i La H,i µ H b a i, and S a W i La i µ + Ha i q ba i. N i L µ + H q i i b i H TRANSF N H L i H,i µ H,i b i PLANT AND SPECS TRANSFORMED PLANT AND SPECS (DESIGN) i a a a i i b i L µ + H q FEASIBLE SPECS INVERSE H TRANSF i L a H,i µ H,i a b i FEASIBLE CONSTRAINTS

21 Noaion Fac: S H H S H and S H S S = S. Bu is i rue ha S H H S and S H S H S H = S H? Theorem. (a) The H-ransformaion of any S saisfies P H \ P : j LH (, ) L H D + H (, ) L H (, ) L H D H (, ) (7) T \ (P H \ P) : L H D H (, ) 0. (8) H (b) Given N H and S H, assume S H S and S H (S H, N H ). If L H saisfies (7 8) and he H-ransformaion has T s,h = (P H \ P), hen N H = N H and S H = S H.

22 Feasibiliy Le Ξ be he oimal suervisor for S (i.e. wihou resricion), Ξ H for S H, Ξ a H for Sa H, and Ξ a for S a. Ξ is feasible if i resecs he conrollabiliy and observabiliy consrains of (N, µ 0 ). S is feasible when Ξ is feasible. H-feasibiliy: weaker han feasibiliy. Defined for S H such ha he following resul holds rue. Theorem. Assume S H S H. Then S is feasible iff S H is h-feasible.

23 Permissiveness The join H-ransformaion of S and S selecs T s,h s.. S H (S H, N H ) and S H (S H, N H ) N H = N H. Noaion: Ξ Ξ : Ξ is a leas as resricive as Ξ. Ξ Ξ : Ξ is more resricive han Ξ. Theorem. Ξ Ξ (Ξ Ξ ) iff Ξ H Ξ H (Ξ H Ξ H ).

24 Procedure. (S, N) H (S H, N H ).. Find h-feasible S a H ha saisfy (7 8) and Ξa H Ξ H.. S a H H S a. S a is he soluion. The oal H-ransformaion slis all ransiions: T s,h = T. X: he se of suervisors oimally enforcing feasible consrains S. X H : he se of suervisors oimally enforcing h-feasible consrains S H ha saisfy (7 8). Theorem.4 a) S a is feasible and Ξ a Ξ. Assume ha he oal H-ransformaion is alied a se one. b) Ξ a is leas resricive among he suervisors of X enforcing S iff Ξ a H resricive among he suervisors of X H enforcing S H. is leas c) There is no suervisor Ξ Ξ a of X ha enforces S if here is no suervisor Ξ H Ξa H of X H ha enforces S H.

25 Remarks The roblem of enforcing W i L iµ + H i q b i can be solved in erms of he simler W i L H,iµ H b i in a ransformed PN, wihou loss of ermissiveness. The resuls were derived under he ransiion-bag concurrency seing. A loss of ermissiveness is ossible when his aroach is used for oher concurrency seings. The resuls obained under a very general unconrollabiliy and unobservabiliy seing. The roblem of enforcing W i L H,iµ H b i is sill comlex. - Under cerain assumions, including no concurrency, a soluion is available (Sremerssch and Boel, 00). - More work has been done on he aricular form L H µ H b. - A srucural and suboimal soluion o he enforcemen of W i L H,iµ H b i aears in he book of Iordache and Anasklis, 006. I alies o double-labeled PNs and he mos common concurrency seings.