Decentralized Control of Petri Nets

Transcription

1 Decenralized Conrol of Peri Nes Marian V. Iordache and Panos J. Ansaklis Dearmen of Elecrical Engineering Universiy of Nore Dame Nore Dame, IN 6556 June, 00 M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

2 Ouline The goal of he aer is o exend he suervision based on lace invarians (SBPI) o a decenralized seing >. Overview of he SBPI. The Decenralized Seing. Decenralized Admissibiliy. Enforcing D-Admissible Consrains 5. Enforcing D-Inadmissible Consrains (a) Enforcemen Wih Communicaion (b) Enforcemen Wihou Communicaion (c) Enforcemen Wih Resriced Communicaion M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

3 Overview of he Suervision Based on Place Invarians Suervision Based on Place Invarians: inroduced by several researchers (Giua, Yamalidou, Moody, and ohers). The secificaion of he SBPI is Lµ b. Case I: All ransiions are conrollable and observable. Le D be he incidence marix of he lan Peri ne. The suervisor can be designed as a Peri ne of incidence marix D s = LD If µ 0 is he iniial marking of he lan, he iniial marking of he suervisor is µ s0 = b Lµ 0 The laces of he suervisor are called conrol laces. The closed-loo is a Peri ne of incidence marix [ ] D D c = LD M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

4 Overview of he Suervision Based on Place Invarians Examle The se of consrains µ( )+µ( ) Targe Peri ne µ( )+µ( ) is described by Lµ b wih: [ 0 L = 0 ] b = [ ] The incidence marix is: D = 0 0 Suervised Peri ne The suervisor has wo conrol laces (as L has wo rows): D s = LD = [ 0 0 ] C C M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

5 Overview of he Suervision Based on Place Invarians Examle The iniial marking of he suervisor is µ s0 = b Lµ 0 = Noe ha for all reachable markings µ s = b Lµ [ ] Targe Peri ne This aroach is called suervision based on lace invarians, as i creaes for each row of L a lace invarian. In aricular: Suervised Peri ne µ( )+µ( ) µ(c )= µ( )+µ( ) µ(c )= C C M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 5

6 Overview of he Suervision Based on Place Invarians Case II: No all ransiions are conrollable and observable. A suervisor should no inhibi unconrollable ransiions or observe firings of unobservable ransiions. Then, he suervisory aroach of Case I can sill be used if (bu no only if) LD uo =0and LD uc 0 () where D uc and D uo are he resricions of he incidence marix D o he columns of he unconrollable and unobservable ransiions, resecively. To enforce Lµ b we can roceed as follows:. If L saisfies (), find he suervisor as in Case I. Oherwise:. Transform Lµ b o L a µ b a such ha L a µ b a Lµ b and L a saisfies (). Then he suervised PN is obained as in Case I by enforcing L a µ b a insead of Lµ b. M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 6

7 Overview of he Suervision Based on Place Invarians Examle Assume unobservable and he same secificaion: µ( )+µ( ) µ( )+µ( ) 0 As D = 0 [ ] Noe ha LD uo = ( L = [ 0 0, D uo = 0. ] b = [ and D uc is emy. ]) Therefore, he consrains are ransformed o Targe Peri ne Suervised Peri ne µ( )+µ( ) µ( )+µ( ) and enforced by he conrol laces C and C. is unobservable C C M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 7

8 Ouline The goal of he aer is o exend he suervision based on lace invarians (SBPI) o a decenralized seing. Overview of he SBPI >. The Decenralized Seing. Decenralized Admissibiliy. Enforcing D-Admissible Consrains 5. Enforcing D-Inadmissible Consrains (a) Enforcemen Wih Communicaion (b) Enforcemen Wihou Communicaion (c) Enforcemen Wih Resriced Communicaion M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 8

9 Cenralized vs Decenralized SUPERVISOR Nework Connecion PLANT PLANT... PLANT Nework Connecion SUPERVISOR SUPERVISOR... SUPERVISOR PLANT PLANT PLANT M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 9

10 Decenralized Suervision Examle Pars bin Assembly area Pars bin Comuer Comuer Nework Connecion T c, = T o, = {, } T c, = T o, = {, } Secificaion: µ + µ M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 0

11 Decenralized Suervision Examle Pars bin Assembly area Pars bin T c, = T o, = {, } T c, = T o, = {, } Comuer Comuer Comuer T c, = { 5 } Nework Connecion T o, = {,,,, 5 } 5 6 Secificaion: 5 µ + µ µ 5 µ 6 M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

12 Decenralized Suervision Given: he Peri ne model of he sysem he ses of conrollable and observable T c,i and T o,i, i =... he secificaion Lµ b. Problem : Find he suervisors S... S such ha. The join oeraion of S... S ensures he lan saisfies Lµ b.. Each S i conrols only ransiions in T c,i and observes only ransiions in T o,i. M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

13 Decenralized Suervision wih Communicaion Problem : Solve Problem when communicaion is allowed. Communicaion can be used o enable S i o conrol T c,j, / T c,i. j i observe T o,j, / T o,i. j i Remark: Cenralized suervision assumes: T c = T c,j and T o = ha is, full (maximum) communicaion! Oimaliy crieria: minimum communicaion. maximally ermissive design. j=... j=... T o,j M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

14 Ouline The goal of he aer is o exend he suervision based on lace invarians (SBPI) o a decenralized seing. Overview of he SBPI. The Decenralized Seing >. Decenralized Admissibiliy. Enforcing D-Admissible Consrains 5. Enforcing D-Inadmissible Consrains (a) Enforcemen Wih Communicaion (b) Enforcemen Wihou Communicaion (c) Enforcemen Wih Resriced Communicaion M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

15 Decenralized Admissibiliy In cenralized suervision: i is (comuaionally) easy o enforce consrains Lµ b on fully conrollable and observable PNs. in arially conrollable and observable PNs, we say ha Lµ b is c-admissible if i can be enforced as if he PN were fully conrollable and observable. consrains Lµ b ha are no c-admissible are ransformed o a c-admissible form L a µ b a such ha L a µ b a Lµ b. In decenralized suervision: we exend c-admissibiliy o d-admissibiliy, such ha d-admissible consrains Lµ b are (comuaionally) easy o enforce. checking wheher a se of consrains is d-admissible is (comuaionally) racable. he definiion we roose allows us o ransform consrains Lµ b ha are no d-admissible o d-admissible consrains L a µ b a such ha L a µ b a Lµ b. enforce consrains ha are no d-admissible by enabling communicaion M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 5

16 Decenralized Admissibiliy Definiion Le Lµ b, L Z m P and b Z m be a se of consrains. A consrain of Lµ b is denoed by lµ c, l Z P and c Z. lµ c is d-admissible wih resec o (N,µ 0,T c,...t c,n,t o,...t o,n ),ifhere is C {,,...n}, C =, such ha lµ c is c-admissible wih resec o (N,µ 0,T c,t o ),wheret c = T c,i and T o = T o,i. i C i C Lµ b is d-admissible if each of is consrains lµ c is d-admissible. c-admissibiliy is a secial case of d-admissibiliy, in he sense ha if lµ c is c-admissible w.r.. (N,T c,i,t o,i ), lµ c is d-admissible (se C = {i}). lµ c d-admissible imlies If firing a lan-enabled ransiion violaes lµ c hen i C: T c,i. All suervisors S i wih i Care able o know he value of c lµ. an algorihm checking wheher a se of consrains is d-admissible is in he aer. M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 6

17 Ouline The goal of he aer is o exend he suervision based on lace invarians (SBPI) o a decenralized seing. Overview of he SBPI. The Decenralized Seing. Decenralized Admissibiliy >. Enforcing D-Admissible Consrains 5. Enforcing D-Inadmissible Consrains (a) Enforcemen Wih Communicaion (b) Enforcemen Wihou Communicaion (c) Enforcemen Wih Resriced Communicaion M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 7

18 Enforcemen of D-admissible Consrains Le D and µ 0 be he incidence marix and he iniial marking of a PN N. Recall he cenralized enforcemen of a c-admissible consrain lµ c on (N,µ 0 ): A conrol lace C is generaed such ha for all :. If ld(,) > 0, henc and he weigh is W (C, ) =ld(,).. If ld(,) < 0, henc and he weigh is W (, C) = ld(,). The iniial marking of C is c lµ 0. In he decenralized enforcemen of a d-admissible consrain lµ c, for all i C: Define x i N, as he sae variable of S i. Iniialize x i o c lµ 0. S i disables a ransiion if T c,i and x i <ld(,). If T o,i fires and ld(,) 0,henx i = x i ld(,). I can be roved ha he decenralized suervisor i C S i enforces lµ c andhaiis equally ermissive o he cenralized suervisor S enforcing lµ c in he fully conrollable and observable version of N. M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 8

19 Enforcemen of D-Admissible Consrains Examle Desired consrain: µ + µ. Iniial marking µ 0 =[0,,, 0] T. Decenralized seing: T c, = {, }, T c, = {, }, T o, = T o, = {,,, }. The suervisor S : iniializes x o 0. disables if x =0 incremens x if or fires. decremens x if or fires. The suervisor S : iniializes x o 0. disables if x =0 incremens x if or fires. decremens x if or fires. A grahical reresenaion is ossible, however i may be boh helful and misleading. C Cenralized conrol C Subsysem Decenralized conrol Subsysem C M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 9

20 Ouline The goal of he aer is o exend he suervision based on lace invarians (SBPI) o a decenralized seing. Overview of he SBPI. The Decenralized Seing. Decenralized Admissibiliy. Enforcing D-Admissible Consrains 5. Enforcing D-Inadmissible Consrains > (a) Enforcemen Wih Communicaion (b) Enforcemen Wihou Communicaion (c) Enforcemen Wih Resriced Communicaion M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 0

21 Enforcemen of D-Inadmissible Consrains via Communicaion µ + µ is d-inadmissible for T c, = T o, = {, } and T c, = T o, = {, }. Global sysem Subsysem Subsysem The consrain becomes d-admissible if he ransiions and are communicaed o subsysem and he ransiions and o subsysem. Then T o, = T o, = {,,, }, T c, = {, } and T c, = {, }. Global sysem Subsysem Subsysem M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

22 Enforcemen of D-Inadmissible Consrains via Communicaion D-inadmissible consrains can be made admissible by communicaion:. Le T c,l = i=...n T c,i and T o,l = i=...n T o,i.. Is he secificaion c-admissible wih resec o (N,T c,l,t o,l )? If no, ransform i o be c-admissible.. Le S be he cenralized SBPI suervisor enforcing he secificaion. Le T c be he se of ransiions conrolled by S and T o he se of ransiions deeced by S.. Find a se C such ha T c,i T c. i C 5. The communicaion can be designed as follows: for all T o \( T o,i ), a subsysem j such ha T o,j ransmis he firings of o all suervisors S k wih / T o,k and k C. 6. Design he decenralized suervisor according o he algorihm for d-admissible consrains. i C M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

23 Enforcemen of D-Inadmissible Consrains via Communicaion In he algorihm No communicaion resricions considered. These are considered laer. The suervisor is equally ermissive o he cenralized suervisor. In he communicaion olicy roosed in he algorihm: The conrol decisions are aken locally (no conrol decisions are communicaed). Assuming broadcas, here is less communicaion raffic han in he cenralized soluion, which remoely observes and conrols he ransiions in T o and T c, resecively. Beer communicaion olicies may be ossible. (The oimal olicy can be obained by solving an ineger rogram.) M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

24 Enforcemen of D-Inadmissible Consrains via Communicaion µ + µ is d-inadmissible for T c, = T o, = {, } and T c, = T o, = {, }. Global sysem Subsysem Subsysem T c,l = T o,l = {,,, }; µ + µ is c-admissible w.r.. (N,T c,l,t o,l ). T c and T o found from he cenralized SBPI: C T c = {, } T o = {,,, } C = {, } M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

25 Enforcemen of D-Inadmissible Consrains via Communicaion C SUPERVISOR Nework Connecion PLANT PLANT CENTRALIZED Broadcas:,,,and. Remoely conrol: and. SUPERVISOR PLANT DECENTRALIZED Nework Connecion SUPERVISOR PLANT C Broadcas: and. Remoely conrol: C Broadcas: and. Remoely conrol: M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 5

26 Enforcemen of D-Inadmissible Consrains via Communicaion C C Sill anoher soluion... Broadcas: Remoely conrol:. Broadcas: and. Remoely conrol: In general, several equally ermissive and decenralized soluions are ossible. The oimal soluion deends on he relaive cos of broadcas/remoe conrol. M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 6

27 Ouline The goal of he aer is o exend he suervision based on lace invarians (SBPI) o a decenralized seing. Overview of he SBPI. The Decenralized Seing. Decenralized Admissibiliy. Enforcing D-Admissible Consrains 5. Enforcing D-Inadmissible Consrains (a) Enforcemen Wih Communicaion > (b) Enforcemen Wihou Communicaion (c) Enforcemen Wih Resriced Communicaion M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 7

28 Enforcemen of D-Inadmissible Consrains via Transformaions Secificaion: Lµ b (d-inadmissible) Goal: Find L µ b... L m µ b m ha are d-admissible such ha Remarks: (L µ b L µ b...l m µ b m ) Lµ b () - Each L i µ b i has a differen se C i. - The ses C i are given. - Any soluion can be found if all ses C i are given. If so, m = n. - However, we could discard he ses C i wih T (i) o = T o,i =. i C i - In racice, we exec mos ses C i o have T (i) o =. We roose o simlify () o: [(L + L +...L m )µ (b + b +...b m )] Lµ b () M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 8

29 Enforcemen of D-Inadmissible Consrains via Transformaions The following aramerizaion is used: L + L +...L m = R + R L () b + b +...b m = R (b + ) (5) for R N m P, R N m m such ha R > 0 and R is diagonal. Admissibiliy consrains where T (i) uc = T uc,i and T (i) uo i C i L i D(,T (i) uc ) 0 (6) L i D(,T (i) uo ) = 0 (7) = T uo,i. i C i Then he roblem is o find a feasibile soluion of ( 7). The unknowns are R, R, L i, and b i, and ineger rogramming can be used o find hem. Drawbacks: The comuaional comlexiy of ILP and he fac ha a ermissiviy requiremen seems raher hard o be encoded as linear consrains. M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 9

30 Examle Secificaion: µ + µ ; T c, = T o, = {, } and T c, = T o, = {, }. Subsysem Subsysem Global sysem Take m =, C = {} and C = {}. Decenralized soluion: µ (as L µ b )andµ (as L µ b ). C C M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 0

31 Ouline The goal of he aer is o exend he suervision based on lace invarians (SBPI) o a decenralized seing. Overview of he SBPI. The Decenralized Seing. Decenralized Admissibiliy. Enforcing D-Admissible Consrains 5. Enforcing D-Inadmissible Consrains (a) Enforcemen Wih Communicaion (b) Enforcemen Wihou Communicaion > (c) Enforcemen Wih Resriced Communicaion M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

32 Resriced Communicaion The revious ILP aroach can be used wih communicaion exensions. Noe ha: - Communicaion allows imroving ermissiviy. - Some consrains are no enforcible wihou communicaion. Exensions: The binary variables α ij and ε ij are inroduced. α ij =iff he firing of j is announced o he suervisors of C i. ε ij =iff a suervisor from C i remoely conrols j. In aricular, in he broadcas case α ij = α j i =...m (α j = iff each firing of j is broadcas, i.e., all suervisors are announced when j fires). ε ij = ε j i =...m (ε j =iff all suervisors are allowed o remoely conrol j ). Communicaion consrains can be incororaed as exressions of α ij and ε ij. M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

33 Resriced Communicaion Define B i U and Bi L as uer and lower bounds of L id. Le A =[α ij ] and E =[ε ij ]. The admissibiliy consrains L i D(,T (i) uc ) 0 and L id(,t (i) uo )=0are relaced by: L i D(,T (i) uo ) [Bi U diag(a(i, ))] T (i) uo L i D(,T (i) uo ) [Bi L diag(a(i, ))] T (i) uo L i D(,T (i) uc ) [Bi U diag(e(i, ))] T (i) uc (8) (9) (0) Given he weigh marices C and F, he objecive of he ILP can be se o o minimize communicaion. min A,E,L i,b i,r,r Trace(CA + FE) () C/F may reflec saisics on how ofen he ransiions j are fired/require conrol. M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

34 Manufacuring Examle (Adaed from [Lin, 990]) Machines: M and M. Buffers: B... B. Robos: H... H. Two ossible manufacuring sequences: - γ τ π α τ π α η - γ τ π α τ π α η B and B share common buffer sace. B and B share also common sace. γ η τ π α H B τ π α H B M M [ η ] [ M ] [ α ] [ B ] [ π ] [ H ] [ τ ] [ M ] [ α ] [ B ] [ π ] [ H ] [ τ ] [ M ] [ γ ] [ M ] B H α π τ B H α π τ γ η 5 [ M ] [ γ ] [ M ] [ τ ] [ H ] [ π ] [ B ] [ α ] [ M ] [ τ ] [ H ] [ π ] [ B ] [ α ] [ M ] [ η ] M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes.

35 Decenralized Suervision T c, = { } T o, = {,, } T c, = { 5 } T o, = { 5, 6, 7, 8 } T c, = { 0 } T o, = { 0,, } T c, = {, 6 } T o, = {,, 5, 6 } Avoid buffer overflow: µ + µ and µ 6 + µ 0. C [ η ] [ α ] [ π ] [ τ ] [ α ] [ π ] [ τ ] [ γ ] Take m =and C i = {i}, i =... C [ γ ] [ τ ] [ π ] [ α ] [ τ ] [ π ] [ α ] [ η ] Soluion wihou communicaion: µ + µ (sub-) µ 5 + µ 6 (sub-) µ 9 + µ 0 (sub-) µ + µ (sub-) C C M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 5

36 Decenralized Suervision T c, = { } T o, = {,, } T c, = { 5 } T o, = { 5, 6, 7, 8 } T c, = { 0 } T o, = { 0,, } T c, = {, 6 } T o, = {,, 5, 6 } Fairness: v 8 v 6 and v 6 v 8. (v i : he number of firings of i.) [ η ] [ α ] [ π ] [ τ ] [ α ] [ π ] [ τ ] [ γ ] No acceable soluion wihou communicaion! 6 C 5 C [ γ ] [ τ ] [ ] [ ] [ ] [ ] [ ] [ ] π α τ η π α Resul: subsysem : broadcas 8 and enforce µ 5 +µ 6 +µ 7 +v 8 v 6 subsysem : broadcas 6 and enforce v 6 v 8 M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 6

37 Conclusion This aer exends he SBPI o he decenralized seing. The suervisors can be designed by consrain ransformaion for: - no communicaion - resriced communicaion - minimal communicaion This work shows ha he decenralized suervision of PNs can be racable. On he negaive side: - Our ILP aroach is suboimal. - Difficul o include ermissiviy requiremens in he ILP. M.V. Iordache and P.J. Ansaklis, Decenralized Conrol of Peri Nes. 7