Discussion on Supervisory Control by Solving Automata Equation

Discussion on Supervisory Contr ol by Solving A utomata Equat ion V icto r Bushkov Department of EECS T omsk State Univer sity T omsk, 634050, Russia v .bushk ov@gmail.com Nina Y e vtushenko Department of EECS T omsk Sta te University T omsk, 634 050, Russia ninayevtushenko@yahoo.com T iziano V illa Department of CS University of V er on a 37134 V er ona, I taly tiziano.vil la@univr .it Abstract In this paper we consider the supervisory contr ol pr oblem thr ough lan guage equation solving. The equa- tion solving appr oach allows to deal with mor e gen eral topologies and to find a lar gest supervisor w hich can be used as a reservoir for deriving an optimal contr oller . W e intr oduce the notions of solutio ns under partial con- tr ollability and partial o bservability , and we s how how supervisory co ntr ol pr oblems with partial contr ollabil- ity and partial observability can be solved by employing equation solving methods. 1. Intr oduction The pro blem of superv isory con trol is well known [1, 2]. A discrete e vent system P , called the plant, should be co ntrolled by a superv isor C in o rder to mee t the specification S . In other word s, we ar e r equired to construct a supervisor ( also c alled a controller) C that combined with P satisfies S . I n this p aper, we assume that all the behaviors are described by regular languages and thus, can be represen ted by finite auto mata. Sometimes more superv isor restriction s are im- posed. When co nsidering pa rtial con trollability some actions of the plant cann ot be disabled by a supervi- sor , while under partial observability some plant actio ns cannot be observed by a supervisor . According to the pro blem statement, the problem of constru cting a supervisor is very close to the pr oblem of solving a language (o r an autom ata) eq uation and it is known [3] how to d eriv e a largest solu tion to the au- tomata equ ation P ⋄ X ∼ = S , where S is th e behavior o f the overall system, P is the behavior of th e kn own part of the system, X is the unknown component, and ∼ = is a parallel com position operator . Howe ver, these methods cannot be directly used to solve the supervisory control problem due to the presen ce of u ncontr ollable and un- observable events (wh ich are usu ally defined in a differ- ent way f or langua ge and auto mata eq uations). In this paper, we describe particular solutio ns of an auto mata equation under such limitations. 2. Pr elimi naries An automa ton is a quintup le P = ( P , Σ , p 0 , T P , F P ) , where P is a finite non- empty set of states with the in i- tial state p 0 and the subset F P of final (acceptin g) states, Σ is an alph abet, and T P ⊆ P × Σ × P is a transition relation which is extended to words in a usual w ay . The lang uage accepted by P is the set L ( P ) = { ( α ∈ Σ ∗ : ∃ p ∈ F P ( p 0 , α , p ) ∈ T p ) } . An automaton is tr im if from each state a final state c an be re ached. An au- tomaton with a prefix-clo sed lang uage is a prefix -closed automaton . Moreover, automa ton I ni t ( P ) is a trim au - tomaton with th e langu age that is the p refix-closur e of the langu age of P . An au tomaton R i s a r edu ction of an au tomaton P if L ( R ) ⊆ L ( P ) (written, R ≤ P ). If L ( R ) = L ( P ) then automata R and P are equiv a lent (written, R ∼ = P ). Given two au tomata P and C with lan- guages L ( P ) ⊆ Σ ∗ 1 and L ( C ) ⊆ Σ ∗ 1 , let E be a non- empty subset of Σ 1 ∪ Σ 2 . The parallel composition P ⋄ E C is the auto maton ( P ⇑ Σ 2 ∩ C ⇑ Σ 1 ) ⇓ E . When clear fro m the context, instead o f P ⋄ E C we simply write P ⋄ C . If E = Σ 1 = Σ 2 , then P ⋄ E C ∼ = P ∩ C with th e language L ( P ) ∩ L ( C ) . Correspond ingly , given the auto maton S with the langu age L ( S ) ⊆ E ∗ we consider an automata equation P ⋄ E X ∼ = S , where X is a n un known automa - ton with the langu age over alp habet Σ 2 . An a utomaton C with th e lang uage over alph abet Σ 2 is a s olution to the equation if P ⋄ E C ∼ = S . It is known that a solv able equa- tion P ⋄ E X ∼ = S has a largest solution M = P ⋄ E S [3]: the langu age of e ach so lution is contained in the lan- guage o f a largest solu tion. As usual, a number of par- ticular solution s can be considered when solving au- tomata equation [3]. In this p aper, all automata in an automata equation are assumed to be trim. 3. Super visor synthesis by solving au- tomata equations 3.1. Desc ribing the set of supe rvisors Let P = ( P , Σ , p 0 , T P , F P ) and S = ( S , Σ , s 0 , T S , F S ) be trim automata which descr ibe the plant and the spec- ification be havior , co rrespond ingly . The pro blem is to derive a superv isor C = ( C , Σ , c 0 , T C , F C ) with a prefix - closed language su ch that P ⋄ C ∼ = S . Since P , S and C are d efined over the same alphabet, we are req uired to solve th e equation P ∩ X ∼ = S . T hen the equation is known to have a largest solution P ∩ S ∼ = P ∪ S and we denote by ( P ∪ S ) pre f the largest subautom aton of P ∪ S with a prefix -closed lang uage. Th us, ther e exists a su- pervisor C such that P ∩ C ∼ = S iff P ∩ ( P ∪ S ) pre f ∼ = S . On the other hand, f or each C such that P ∩ C ∼ = S it holds that L ( C ) ⊇ L ( S ) and thus, th e following state- ment holds. Proposition 1. Given the plan t P and the spec ification S , ther e exists a supervisor C such that P ∩ C ∼ = S iff P ∩ ( P ∪ S ) pre f ∼ = S . Mor eover , when a supervisor ex- ists an auto maton C with a prefix-closed la nguage is a supervisor iff I nit ( S ) ≤ C ≤ ( P ∪ S ) pre f . Howe ver, not e very supe rvisor is of practical use. If the languages of the plan t and th e specification ar e not prefix-clo sed then the intersection P ∩ C is not ne c- essary a trim auto maton and th us, a deadlo ck or a live- lock can occur during the joint work of th e pla nt and the superv isor . T o escape such drawback s the notion of a progre ssi ve (non -block ing) superv isor is u sed. A su- pervisor C is progressiv e if the autom aton P ∩ C is trim. If the equ ation P ∩ X ∼ = S is solvable th en a superv isor with lan guage I nit ( L ( S )) is pro gressiv e. Howe ver, i t is not always the case for the supervisor ( P ∪ S ) pre f . Example 1. Consider P and S with the languages { a , ab c } a nd { a } defined ov er the alphab et { a , b , c } , correspo ndingly . The lan guage o f a largest superv isor C h as each word except of abc and all continu ations of this word ; howev er , C is not p rogressive, since the au- tomaton P ∩ C is not trim. The notio n of a pr ogressive superv isor coincides with the notion of a progressiv e solution of an automata equation [ 4] and thus, a largest prog ressi ve su pervisor exists if the eq uation P ∩ X ∼ = S is solvable. A largest progr essi ve super visor can be d erived in the same way as a largest p rogre ssi ve solutio n is derived, i.e., by delet- ing ’b ad’ sequen ces from the language of the automa- ton ( P ∪ S ) pre f . A sequen ce is ’bad ’ if it is in the lan- guage I ni t ( L ( P )) while ha v ing no continuation in L ( S ) . For this reason, d ifferently from the gen eral case of the largest progre ssi ve solution to automata equatio ns the following pro position holds. Proposition 2. Each automato n C with a p r efi x-closed language is a pr ogres sive supervisor iff I ni t ( S ) ≤ C ≤ ( I ni t ( P ) ∪ I ni t ( S )) pre f , wher e ( I nit ( P ) ∪ I nit ( S ) ) pre f is the lar gest pr ogr essive supervisor . 3.2. D escribing the set of supervisors under partial controllability When talking abo ut partial controllab ility one as- sumes that a sup ervisor cann ot prevent the occurre nce of unc ontr olla ble actions, i.e., alphab et Σ is partitio ned into two sub sets Σ c and Σ uc , where Σ c and Σ uc are the sets of co ntrollab le and unc ontrollab le action s, respec- ti vely . Given an autom aton C o ver alph abet Σ , we ob - tain the Σ uc -extension C ⇑ Σ uc of C by adding at each state of C a self-loop lab eled with each action a ∈ Σ uc such that there is no transition from this state under action a . A solution C o f the equatio n P ∩ X ∼ = S is a solu- tion un der pa rtial co ntr ollab ility if C ⇑ Σ uc is a solution of the equation P ∩ X ∼ = S . The following statemen t establishes nec essary and suf ficient cond itions for the equation solvability und er partial controllability . Proposition 3. Given solvable equa tion P ∩ C ∼ = S . (i) The equ ation is solvable und er partial contr o lla- bility iff I nit ( L ( S ))( Σ uc ) ∗ ⊆ L ( P ) ∪ L ( S ) . (ii) If the eq uation P ∩ X ∼ = S is solvab le under partial contr olla bility , then it has a la r gest solution un der partial contr ollability . Howe ver, it may occu r that neither I ni t ( S ) nor ( P ∪ S ) pre f are solutions under partial controllability . Example 2. Consider P and S with the lan guages { ε , ba } an d { ε } over Σ = { a , b } , corre sponding ly . Let Σ uc = { a } . Th e language of ( P ∪ S ) pre f contains all words over Σ , except those that h ave ba as a p refix. Then the langu age of (( P ∪ S ) pre f ) ⇑ Σ uc contains the word ba . As a re sult, (( P ∪ S ) pre f ) ⇑ Σ uc is not a s olu- tion of the equation P ∩ X ∼ = S . Example 3. Consider P and S with the lan guages { ε , b , ab } and { ε } over Σ = { a , b } , correspo ndingly . Let Σ uc = { a } . The automato n I nit ( S ) ⇑ Σ uc is not a solution , since its lan guage conta ins the word a b . Bu t the equa- tion P ∩ X ∼ = S is solvable under partial contro llability , for example, an automato n with the lang uage { ε , b , a } is a solution under partial controllability . A largest solutio n un der partial co ntrollability can be o btained b y iter ativ ely eliminating each state s t of the automaton ( P ∪ S ) pre f , such th at f rom st there are no transitions under some uncon trollable action, un til ev ery state has a tran sition f or every uncon trollable ac- tion; if the resulting automaton is no t a solution, then the equation has no solutions and the intersection of the resulting automato n with the plan t giv es the largest con- trollable behavior we could achie ve. Howev er , as th e following propo sition states, if th e languag es of P and S are prefix-clo sed, th en the re is no n eed for trimming of the automaton ( P ∪ S ) pre f . Proposition 4. If the lan guages of P an d S ar e prefix- closed an d the equ ation P ∩ X ∼ = S is solvable under p ar- tial contr ollab ility then an au tomaton C with a pr efix- closed la nguage is a supe rvisor iff I nit ( S ) ≤ C ≤ ( P ∪ S ) pre f . A solution C o f the e quation P ∩ X ∼ = S is a pr o - gr essive solu tion under partial contr ollability if C ⇑ Σ uc is a progressive solution of the equation P ∩ X ∼ = S . Un- like the case wh en all e vents are contro llable, a progres- si ve solution o f the equation is n ot always progr essi ve under partial contro llability . Example 4. Let Σ uc = { a } , L ( P ) = { ε , ab } , L ( S ) = { ε } . Then automato n C with the lan guage L ( C ) = { ε } is a p rogr essi ve solution o f the equ ation and is a solu- tion under par tial con trollability; however , C is not a progr essi ve solution under partial contro llability . Nev ertheless, it turns o ut that if the equatio n P ∩ X ∼ = S has a pro gressive solution, then a pr ogressive solution under partial controllability is eq uiv alen t to a correspo nding progressi ve solution . Proposition 5 . If the equa tion P ∩ X ∼ = S has a pr o- gr essive solutio n un der p artial con tr ollability a nd C is a pr efi x-closed solution then: (i) C is a pr ogr e ssive solutio n un der partial contr ol- lability iff C is a pr ogr essive solution. (ii) C is a pr ogr essive solution u nder p artial con tr ol- lability iff I nit ( S ) ≤ C ≤ ( P ∪ S ) pre f . 3.3. Desc ribing the set of supervisors und er partial observability When talking abo ut p artial observability on e as- sumes that the supe rvisor cannot ‘see’ the occurrence of unobser vable actions, i.e., the Σ is partition ed into two subsets Σ o and Σ uo , where Σ o and Σ uo are the sets of ob- servable and uno bservable actions, r espectively . How- ev er , the plant can observe eac h action of the supervisor and cor respond ingly under comp lete contro llability the plant can ex ecute an action if f both, the p lant and the supervisor, ar e read y to execute the action at their cu r- rent states. After executing an a ction u nobservable by a supe rvisor the plan t moves to the n ext state wh ile the supervisor remains at its current state. If an actio n is observable by a sup ervisor then both, the plant and the supervisor, execute a co rrespond ing tr ansition. Here we notice that in gener al case, p artial contr ollability and observability are considered inde penden tly . Uncontro l- lable actions can b e observable while controllable ac- tions can be unobservable and vice versa. Since a su- pervisor cann ot ‘see’ u nobservable action s, it is neces- sary to impose ad ditional co nditions in order to have a solution of the equation P ∩ X ∼ = S un der partial obser v- ability . Giv en an autom aton C over alphab et Σ , we obtain the Σ uo -folding C ⇓ Σ uo of C by replacing eac h transition ( c 1 , a , c 2 ) o f C , such that a ∈ Σ uo , with a self-loop at state c 1 . Let Σ = Σ o ∪ Σ uo . A solution C of the equation P ∩ X ∼ = S is a solution und er partial observability if C ⇓ Σ uo is a solution of the equation P ∩ X ∼ = S . Giv en an automaton C over alphabet Σ = Σ o ∪ Σ uo , we obtain the a utomaton C real by addin g a self-loo p at each state { c 1 , . . . , c n } o f the determin istic restriction C ⇓ Σ o labeled with each action a ∈ Σ uo such that fr om some state c i ∈ { c 1 , . . . , c n } the re is a transition und er a in the automaton C . Proposition 6. The e quation P ∩ C ∼ = S is solvable un- der partial observab ility iff ( I ni t ( L ( S ))) real ⊆ L ( P ) ∪ L ( S ) . Unfortu nately , the un ion of two solutions under partial o bservability is no t necessary a solution under partial observability and thus, a largest solution does not exist under pa rtial ob servability . W e demon strate this by a simple example. Example 5. L et Σ o = { b } , L ( P ) = { ε , ab } , and L ( S ) = { ε } . Consider auto mata C 1 and C 2 with the langu ages L ( C 1 ) = { ε , a } and L ( C 2 ) = { ε , b } wh ich are solutions of the eq uation P ∩ X ∼ = S . The auto maton C 1 ∪ C 2 has the lan guage { ε , a , b } and thus, the languag e of ( C 1 ∪ C 2 ) ⇓ Σ uo equals ε , a ∗ , a ∗ b . The intersection of this languag e with L ( P ) has the word a b whic h is not co n- tained in L ( S ) , i.e. , C 1 ∪ C 2 is not a supervisor u nder partial observability . A solution C of the equ ation P ∩ X ∼ = S is a pr o - gr essive so lution under partial observability if C ⇓ Σ uo is a progressiv e solution of th e equation P ∩ X ∼ = S . A solution u nder p artial ob servability that is a prog res- si ve solution of the equ ation is not necessary a progres- si ve solution u nder p artial observability , even when the equation has progressive solutions under partial observ- ability . Mo reover , a progr essi ve solution under par tial observability is not alw ays a progressive so lution of the equation. A solution C of the eq uation P ∩ X ∼ = S is a so- lution un der pa rtial contr o llability and ob servability if ( C ⇓ Σ uo ) ⇑ Σ uc is a solution of the equation. A solution C of the equ ation P ∩ X ∼ = S is a pr ogress ive solu- tion u nder pa rtial con tr ollability a nd observability if ( C ⇓ Σ uo ) ⇑ Σ uc is a pro gressive solution of the equ ation. It can be shown that ( C ⇓ Σ uo ) ⇑ Σ uc ∼ = ( C ⇑ Σ uc ) ⇓ Σ uo . Some- times a special case o f partial controllability and ob- servability is considered wh en each un observable actio n cannot b e co ntrolled, i.e., Σ uo ⊆ Σ uc . In this case, there exists a lar gest supervisor . Example 6. Let Σ uo = Σ uc = { b } , L ( P ) = { b , baa } , and L ( S ) = { b } . Then a utomaton C with the languag e L ( C ) = { ε , b , ba } is n ot a pro gressiv e solu tion of the equation, wh ile ( C ⇓ Σ uo ) ⇑ Σ uc is a progr essi ve solution of the eq uation. Therefo re C is a pro gressive solutio n u n- der par tial controllability and obser vability in spite o f the fact that it is not pro gressive without the partial con- trollability and observability limitation . Example 7. Let Σ uo = Σ uc = { b } , L ( P ) = { b , baa } , and L ( S ) = { b } . Then a utomaton C with the languag e L ( C ) = { ε , b , a } is a progr essi ve solu tion o f the equa- tion, howe ver, ( C ⇓ Σ uo ) ⇑ Σ uc is not a pro gressive solution . Proposition 7 . Let Σ uo ⊆ Σ uc and let Z b e au toma- ton with the langu age L ( I ni t ( S ))( Σ uc ) ∗ . Th e e quation P ∩ X ∼ = S is solvable under partial co ntr ollab ility an d observability iff L ( Z real ) ⊆ L ( P ) ∪ L ( S ) . Proposition 8. I f Σ uo ⊆ Σ uc and the equatio n P ∩ X ∼ = S is solva ble und er partial contr ollability and observabil- ity then there exists a larg est solution under partial con - tr ollability and ob servability Howe ver, similar to th e partial contro llability the automaton ( P ∪ S ) pre f is not always a largest solu- tion u nder pa rtial con trollability and observability , and in order to get a largest sup ervisor we n eed to trim ( P ∪ S ) pre f . 4. Conclusion In this pap er , we h ave co nsidered the pro blem of synthesizing a sup ervisor throu gh automata eq uation solving. W e have discussed prog ressiv e (n on-blo cking) supervisor s as well as supervisors under partial control- lability and observability and ha ve shown that most spe- cial kinds of sup ervisors can be derived as p roper so- lutions of a corresponding autom ata equ ation. More- over , the co mplexity of solv ing a cor respond ing au- tomata eq uation is not exponential as in genera l case b ut rather poly nomial w .r .t. to th e n umber of states of the plant and the specification . A largest pr oper superv isor (if exists) can be deriv ed by trimming a largest solution to the automata equation . M oreover , differently from the gener al case each redu ction of such trim au tomaton is also a supervisor . Each largest superv isor can be used as a reservoir for deri ving an optimal superviso r that can be simpler than a traditio nal s uperv isor . Also, since the ap proach based on langu age equation solving can deal wi th more gener al topolog ies, this approach can be used for deriving s uperv isors when the plant, the speci- fication and the supervisor have different sets of actions [5, 6]. Acknowledgmen ts The first au thor g ratefully acknowledges supp ort from the Bortnik Fund (contract 6 360 /885 8). Th e sec- ond auth or gratefully a cknowledges sup port of RFBR- NSC (grant 06-08- 8950 0). Refer ences [1] P . J. Ramadge and W . M. W onham, “The Co ntrol of Dis- crete Even t Systems, ” Pr oceedings of the IE EE , V ol. 77, No. 1, pp. 81–98, 1989. [2] C. C. Cassandras and S. Lafortune, Intr oduction to Dis- cr ete Event Systems , 2nd ed., Springer, 200 7. [3] N. Y evtu shenko , T . Villa, R. Brayton, A. Petrenko , and A. Sangiov anni-V incentelli, “S olution of Parallel Lan- guage Equations for Logic Synthesis, ” in ICCAD , 2001, pp. 103–110. [4] K. El-Fakih, N. Y e vtushenk o, S . Buffa lov , and G. v . Bochmann, “Progressiv e S olutions to a Paral- lel Automata Equation, ” Theore tical Computer Science , V ol. 362, No. 1, pp. 17–32, 2006. [5] A. Aziz, F . Balarin, R. K. Brayton, M. D. DiBenedetto, A. Saldanha, and A. L . Sangiov anni-V incentelli, “Super - visory Control of F inite State Machines, ” P r oceedings of Confer ence on Computer -Aided V erification , Li ` ege, Bel- gium, 1995, pp. 279–292 . [6] R. Kum ar , S. Nelv agal, and S. I. Marcus, “ A Discrete Event Systems App roach for Protocol Con version , ” Dis- cr ete Event Dynamical Systems: Theory and Applica- tions , V ol. 7, No. 3, pp. 295–345, Ju. 1997.