Decentralized vs. Monolithic Control of Automata and Weighted Automata

Transcription

1 Decentralized vs. Monolithic Control of Automata and Weighted Automata Jan Komenda INSTITUTE of MATHEMATICS Academy of Sciences Czech Republic Institute of Mathematics, Brno branch, November 11, 2016

2 Outline Introduction to Supervisory Control Computational Difficulties Decentralized Control of Monolithical DES Weighted automata Residuation of (max,+) and (min,+) series

3 Supervisory control of automata Motivation for supervisory control Supervisory control as generalization of model checking (verification) Supervisory control solves logical problems of safety and liveness. Control problems in complex distributed plants: manufacturing, computer and communication networks Often encountered: partial observations, modular structure, specification is a property independent from the system Language based and state based framework (equivalent, bidirectional transformations)

4 Automata Generators Definition (Generator) A generator is a structure G = (Q,A,δ,q 0 ) Q is a (finite) set of states A is an alphabet (finite set of events) δ : Q A Q is a partial transition function q 0 is the initial state b a d c

5 Generators Behaviour 1 b a 2 3 e b d 4 5 The language generated by G = (Q,A,δ,q 0 ) is the set L(G) = {w A δ(q 0,w) Q} = {ε,b,bd,bde,a,ab,abe,...}

6 Control Objectives = properties to be imposed by control The main control objectives are: Safety language of the controlled system never exceeds the specification Required behavior language of the controlled system includes at least the minimal required behavior Noblockingness controlled system never deadlocks or livelocks (every reachable state is coreachable from a marked state) (for marked languages)

7 General idea Given a system (plant) G a generator Given a specification K a language or generator The aim is to synthesize a supervisor S such that S/G K supervisor as a map and automaton b c b a d a

8 General idea Given a system (plant) G a generator Given a specification K a language or generator The aim is to synthesize a supervisor S such that S/G K supervisor as a map and automaton b c b a d a

9 General idea Given a system (plant) G a generator Given a specification K a language or generator The aim is to synthesize a supervisor S such that S/G K supervisor as a map and automaton b c b a d a

10 Controlled Generator b a d c A controlled generator is a structure (G,A c,γ), where G is a generator over A, Ac A is the set of controllable events, A u = A \ A c is the set of uncontrollable events, and Γ = {γ A A u γ} is the set of control patterns. Example G = ({1,2,3},{a,b,c,d},δ,1,{1}) A = {a,b,c,d}, A c = {c}, A u = {a,b,d} Γ = {{a,b,d},{a,b,c,d}}

11 Supervisory control b a d c A supervisory control for (G,A c,γ) is a map S : L(G) Γ Example Γ = {{a,b,d},{a,b,c,d}} S(ε) = S(ab) = S((ab) ) = {a,b,d} S(a) = S(aba) = S(a(ba) ) = {a,b,c,d} for all other strings w, S(w) = {a,b,d}

12 Closed-loop system b c b a d a A closed-loop system associated with (G,A c,γ) and S is the minimal language L(S/G) A satisfying ε L(S/G) and if s L(S/G), a S(s), and sa L(G), then sa L(S/G). Example S(ε) = S(ab) = S((ab) ) = {a,b,d} S(a) = S(aba) = S(a(ba) ) = {a,b,c,d} for all other strings w, S(w) = {a,b,d} L(S/G) = {ε,...}

13 Closed-loop system b c b a d a A closed-loop system associated with (G,A c,γ) and S is the minimal language L(S/G) A satisfying ε L(S/G) and if s L(S/G), a S(s), and sa L(G), then sa L(S/G). Example S(ε) = S(ab) = S((ab) ) = {a,b,d} S(a) = S(aba) = S(a(ba) ) = {a,b,c,d} for all other strings w, S(w) = {a,b,d} L(S/G) = {ε,a,...}

14 Closed-loop system b c b a d a A closed-loop system associated with (G,A c,γ) and S is the minimal language L(S/G) A satisfying ε L(S/G) and if s L(S/G), a S(s), and sa L(G), then sa L(S/G). Example S(ε) = S(ab) = S((ab) ) = {a,b,d} S(a) = S(aba) = S(a(ba) ) = {a,b,c,d} for all other strings w, S(w) = {a,b,d} L(S/G) = {ε,a,ab,...}

15 Closed-loop system b c b a d a A closed-loop system associated with (G,A c,γ) and S is the minimal language L(S/G) A satisfying ε L(S/G) and if s L(S/G), a S(s), and sa L(G), then sa L(S/G). Example S(ε) = S(ab) = S((ab) ) = {a,b,d} S(a) = S(aba) = S(a(ba) ) = {a,b,c,d} for all other strings w, S(w) = {a,b,d} L(S/G) = {ε,a,ab,aba,...}

16 Closed-loop system b c b a d a A closed-loop system associated with (G,A c,γ) and S is the minimal language L(S/G) A satisfying ε L(S/G) and if s L(S/G), a S(s), and sa L(G), then sa L(S/G). Example S(ε) = S(ab) = S((ab) ) = {a,b,d} S(a) = S(aba) = S(a(ba) ) = {a,b,c,d} for all other strings w, S(w) = {a,b,d} L(S/G) = {ε,a,ab,aba,abab,...}

17 Closed-loop system b c b a d a A closed-loop system associated with (G,A c,γ) and S is the minimal language L(S/G) A satisfying ε L(S/G) and if s L(S/G), a S(s), and sa L(G), then sa L(S/G). Example S(ε) = S(ab) = S((ab) ) = {a,b,d} S(a) = S(aba) = S(a(ba) ) = {a,b,c,d} for all other strings w, S(w) = {a,b,d} L(S/G) = {ε,a,ab,aba,abab,...}

18 Supervisory control problem Problem Given a controlled generator (G,A c,γ) a specification K L(G) Is there a supervisor S : L(G) Γ such that L(S/G) = K? Example K = {ε,a,ab,aba,abab,...} S((ab) ) = {a,b,d}, S(a(ba) ) = {a,b,c,d}, S(w) = {a,b,d} L(S/G) = K b a c d b a

19 Known results Theorem (Ramadge & Wonham, 1987) Let (G,A c,γ) be a controlled generator, and K L(G) be a specification. There exists a supervisor S : L(G) Γ such that if and only if with respect to L(G). L(S/G) = K specification K is controllable

20 Controllability b a c d b a Definition Given a controlled generator (G,A c,γ) K L(G) Let A u = A \ A c, then K is controllable with respect to L(G) if KA u L(G) K Example {ε,a,ab,aba,...}{a,b,d} L(G) {ε,a,ab,aba,...}

21 Partial Observations Actions (events) partition A = A o A uo disjoint union observable events (A o ) and unobservable (A uo ) events Def. (natural projection) P : A A o : P(ε) = ε, { P(s)a, if a Ao, P(sa) = P(s), if a A uo, for L A : P(L) = w LP(w) Motivation: observations are impossible (e.g. fault) or too costly (sensors required) More generally, mask M : A B for B < A M is surjective and catenative and may have indistinguishable events: M(a 1 ) = M(a 2 ) = b B

22 Observer automaton continued Observer of G : replace events from A uo by ε and determinize (ε- removal). Remarkable similarity with the powerset automaton (over A) used in determinization! Unfortunately number of reachable states denoted Q obs can be exponential in Q. In special cases (P has observer property wrt L(G)): Q obs Q. Theorem. Obs(G) recognize P(L(G)), L(Obs(G)) = P(L(G)).

23 Example (Observer) 1 a τ 2 τ 3 a 4 Figure: An automaton. 1,2 a 2,3,4 a 4 Figure: Its observer.

24 Partial observations: Language properties P : A A o, where A o A... observable events Definition Given a controlled generator (G,A c,γ) K L(G) K is observable with respect to L(G) and P if for s prefix(k) and a A c : sa L(G), s a prefix(k), P(s) = P(s ) sa prefix(k).

25 Outline Introduction to Supervisory Control Computational Difficulties Decentralized Control of Monolithical DES Weighted automata Residuation of (max,+) and (min,+) series

26 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z 1x

27 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z a 2x 1x

28 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z 1x a c 2x 1y

29 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z b 3x 1x a c 2x 1y

30 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z 1x a 3x b c 2x 2y c 1y

31 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z 1x a c 2x b 1y c a 3x 2y

32 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z 3x b c 2x 2y a a c d 1x 1y 1z

33 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z c 3x b c 2x 2y a a c d 1x 1y 1z 3y

34 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z c 3x b b c 2x 2y a a c d 1x 1y 1z 3y

35 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z c 3x 3y b b c d 2x 2y 2z a a c d 1x 1y 1z

36 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z c 3x 3y b b c d 2x 2y 2z a a a c d 1x 1y 1z

37 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z c d 3x 3y 3z b b c d 2x 2y 2z a a a c d 1x 1y 1z

38 How are the large-scale systems constructed? Composed of a number of small subsystems Parallel composition (synchronous product) Example a b c d x y z c d 3x 3y 3z b b b c d 2x 2y 2z a a a c d 1x 1y 1z

39 Size of the composition G = G 1 G 2... G n L(G) = n i=1 P 1 i (L(G i )) Two systems with 3 states result in 3 2 = 9 states n systems with k states = k n 2 n states

40 Size of the composition G = G 1 G 2... G n L(G) = n i=1 P 1 i (L(G i )) Two systems with 3 states result in 3 2 = 9 states n systems with k states = k n 2 n states

41 Size of the composition G = G 1 G 2... G n L(G) = n i=1 P 1 i (L(G i )) Two systems with 3 states result in 3 2 = 9 states n systems with k states = k n 2 n states

42 Size of the composition G = G 1 G 2... G n L(G) = n i=1 P 1 i (L(G i )) Two systems with 3 states result in 3 2 = 9 states n systems with k states = k n 2 n states

43 Coordinator G = G 1 G 2... G n G 1 G 2 Subsystems have something in common shared events = communication Abstract the information coordinator G k The coordinator shares this information among the subsystems G 1 G 2... G n G k

44 Coordinated system main idea G k G 1 G 2... G n

45 Coordinated system main idea G k G 1 G 2... G n G k G 1 G k G 2 G k... G n G k

46 Coordinated system main idea G k G 1 G 2... G n G k G 1 G k G 2 G k... G n G k S 1 /(G 1 G k ) S 2 /(G 2 G k )... S n /(G n G k ) K

47 Coordination control problem Problem Given subsystems G 1 and G 2 over alphabets A 1 and A 2, resp. and a specification K L(G 1 G 2 ) Are there a coordinator G k over A k and supervisors S 1 and S 2 such that L(S 1 /[G 1 (G k )]) L(S 2 /[G 2 (G k )]) = K?

48 Simple example A c = {a 1,a 2,a 3 } p i = prepare a i = access (a unique resource) f i = finish p 1 a p 2 a p 3 a f 1 f 2 f 3 Specification: only one system can access the resource

49 Construction of a coordinator Given subsystems G 1 and G 2 over A 1 and A 2, respectively, and a specification K Construct a coordinator G k over A k as follows: 1. Init A k as the set of shared events 2. Extend A k so that K becomes conditionally decomposable

50 Conditional decomposability Definition A language K (A 1 A 2 ) is conditionally decomposable with respect to alphabets A 1, A 2, A k A 1 A if K = P 1+k (K) P 2+k (K) P 1+k : (A 1 A 2 ) (A 1 A k ) There always exists such a A k Example (Projection) P : {a,b,...,y,z} {e,h,l,o} P(abhdeadiladilasoaf ) = hello

51 Construction of a coordinator Given subsystems G 1 and G 2 over A 1 and A 2, respectively, and a specification K Construct a coordinator G k over A k as follows: 1. Init A k as the set of shared events 2. Extend A k so that K becomes conditionally decomposable 3. Set G k = P k (G 1 ) P k (G 2 )

52 Existential result Theorem There exist supervisors S 1, S 2, S k such that L(S 1 /[G 1 G k ]) L(S 2 /[G 2 G k ]) = K if and only if K is conditionally controllable for generators G 1,G 2,G k Definition: K A is conditionally controllable for G 1,G 2,G k if (i) P 1+k (K) is controllable w.r.t. L(G 1 ) L(G k ) (ii) P 2+k (K) is controllable w.r.t. L(G 2 ) L(G k )

53 LCC and Observer properties Definition (LCC) P : A A 0 is LCC wrt s L if σ u A o A u s.t.p 0 (s)σ u P 0 (L), there is no u (A \ A o ) s.t. suσ u L, or there is u (A u \ A o ) s.t. suσ u L. Definition (Observer) P : A A o, where A o A, isl-observer for L A if, t P(L) and s L, if P(s) t then there is u A s. t. su L and Psu) = t. P(s) t P(L) P P s u L Figure: Natural observer.

54 Supremal sublanguages procedure Theorem Given G 1, G 2, G k, and K conditionally decomposable Define supc 1+k = supc(p 1+k (K),L(G 1 ) L(G k )) supc 2+k = supc(p 2+k (K),L(G 2 ) L(G k )) Let P i+k k be (Pi i+k ) 1 (L i )-observer and LCC for (Pi i+k ) 1 (L i ) and P i+k be LCC for P 1 i+k (L i L k ). Alternatively, let L(G 1 ) L(G k ) and L(G 2 ) L(G k ) are mutually controllable wrt shared uncontrollable events A k,u. Then, supc 1+k supc 2+k = supc(k,l,a u ).

55 Simple example coordinator p 1 a p 2 a p 3 a f 1 f 2 f 3 A k = (A 1 A 2 ) (A 1 A 3 ) (A 2 A 3 ) = /0 Extended A k = {a 1,a 2,a 3 } Set G k = P k (G 1 ) P k (G 2 ) P k (G 3 )

56 Simple example coordinator p 1 a p 2 a p 3 a f 1 f 2 f 3 A k = (A 1 A 2 ) (A 1 A 3 ) (A 2 A 3 ) = /0 Extended A k = {a 1,a 2,a 3 } Set G k = P k (G 1 ) P k (G 2 ) P k (G 3 )

57 Simple example coordinator p 1 a p 2 a p 3 a f 1 f 2 f 3 A k = (A 1 A 2 ) (A 1 A 3 ) (A 2 A 3 ) = /0 Extended A k = {a 1,a 2,a 3 } Set G k = P k (G 1 ) P k (G 2 ) P k (G 3 ) a 1,a 2,a 3 Figure: Coordinator G k it says, communicate if you access the resource 1

58 Simple example supervisors We can compute the supervisors a 2,a 3 a 2,a 3 a 1,a 3 a 1,a 3 a 1,a 2 a 1,a 2 p 1 a p 2 a p 3 a f 1 f 2 f 3 Figure: Supervisors supc 1+k, supc 2+k, and supc 3+k The supervisors communicate via coordinator

59 Simple example supervisors We can compute the supervisors a 2,a 3 a 2,a 3 a 1,a 3 a 1,a 3 a 1,a 2 a 1,a 2 p 1 a p 2 a p 3 a f 1 f 2 f 3 Figure: Supervisors supc 1+k, supc 2+k, and supc 3+k The supervisors communicate via coordinator

60 Simple example supervisors We can compute the supervisors a 2,a 3 a 2,a 3 a 1,a 3 a 1,a 3 a 1,a 2 a 1,a 2 p 1 a p 2 a p 3 a f 1 f 2 f 3 Figure: Supervisors supc 1+k, supc 2+k, and supc 3+k The supervisors communicate via coordinator

61 Multi-level hierarchy Overall coord. group 1 coord. group 2 coord.... group m coord. G 1... G i1 G i G i2 G im G im Group 1 Group 2 Group m

62 Outline Introduction to Supervisory Control Computational Difficulties Decentralized Control of Monolithical DES Weighted automata Residuation of (max,+) and (min,+) series

63 Decentralized control problem Observation and Control tasks are distributed among several supervisors S i, i = 1,...,n Problem (Decentralized control) S i observes A o,i A S i control (can disable) A c,i A specification K L(G) When there exist supervisors S i, i = 1,...,n such that n i=1 L(S i/g i ) = K?

64 Decentralized control of DES Find supervisors S i over alphabets A o,i, i I = {1,...,n} s.t. Remarks L(S 1 S 2 S n G) = K Locals supervisors S i exist if and only if K is coobservable and controllable (K. Rudie, W.M. Wonham (1992). If K is not coobservable then find a controllable and coobservable sublanguage of K Definition.( Coobservability.) K L is coobservable wrt L & (A o,i ) i I if s prefix(k) and a A c, if sa L \ prefix(k), then i I s.t. s with P i (s) = P i (s ), s a prefix(k). Algorithm to check coobservability (decidable in a polynomial time) : Rudie, Willems, (1995).

65 Separability and Decomposability Definition. Separability R is separable wrt (A o,i ) i I if R = i I P i (R) = i I (P i ) 1 P i (R) Lemma. R is separable wrt (A o,i ) i I iff R i A o,i i I, s.t. R = i I R i. Definition. Decomposability A language K is decomposable with respect to alphabets (A i ) n i=1 and L if K = n i=1 P i(k) L.

66 Coobservability does not hold If specification K is not coobservable, computing a coobservable or decomposable sublanguage is a difficult problem We use communications: every K can be made coobservable by enlarging (communicating) local observations This is formalized by the concept of conditional decomposability Note that existence of a decomposable sublanguage is undecidable in general!

67 Separability implies Coobservability Lemma The property A o,i A c A c,i, for i = 1,2,...,n, is equivalent to A o,i A c,j A c,i, for i,j = 1,2,...,n. Proposition. Assume that K is decomposable with respect to (A o,i ) n i=1 and L, and that for i = 1,2,...,n, A o,i A c A c,i. Then K is coobservable with respect to L and (A o,i ) n i=1. Theorem. Assume that A o,i A c A c,i, for i = 1,2,...,n. If K is separable with respect to (A o,i ) n i=1, then K L is coobservable with respect to (A o,i ) n i=1 and L.

68 (over)approximation by modular system Idea: plunge decentralized control problem into coordination control problem by A i = A o,i and A c,i = A o,i A c,i. Note that conditional decomposability is just separability of K with respect to (A o,i A k ) n i=1. Theorem Let A o,i A c A c,i, for i = 1,2,...,n. If K = n i=1 P i(k) (separable) wrt (A o,i ) n i=1, then K L is coobservable wrt (A o,i ) n i=1 and L. Hence, separability implies coobservability

69 1. Constructive results of coordination control Conditional Decomposability A language K is conditionally decomposable with respect to alphabets (A i ) n i=1 and A k, where i j (A i A j ) A k if K = n i=1 P i+k(k), where P i+k : ( n i=1 A i) A i+k, for A i+k = A i A k. Idea If supc i+k are synchronously nonconflicting then n i=1 supc(p i+k(k),l(g i ) L(G k ),A i+k,u ) is controllable with respect to L and A u.

70 Application to decentralized control We over-approximate the plant language by n i=1 P i+k(l). Theorem. Let K = prefixk L = L(G) s.t. K is conditionally decomposable wrt (A i ) n i=1 and A k. If languages in the composition are synchronously nonconflicting then M = n i=1 supc(p i+k(k),p i+k (L),A i+k,u ) K is controllable wrt L and A u, and coobservable wrt L and (A i+k ) n i=1. Remark M is easy to compute (only supremal controllale sublanguages), it is coobsevable by construction.

71 3. Modular control with mutual controllability Definition (Lee, Wong 2002). L i A i, where i = 1,2,...,n, are mutually controllable if for all i,j = 1,2,...,n, L j (A j,u A i ) P j (P i ) 1 (L i ) L j. L P j P i L 1 L 2 Proposition If L i A i, for i = 1,2,...,n, are mutually controllable, then for any K = n i=1 K i L, holds. supc( n i=1 K i, n i=1 L i,a u ) = n i=1 supc(k i,l i,a i,u )

72 Mutual controllability in decentralized control Theorem Let K = prefix(k) L and K = n i=1 P i+k(k). If P i+k (L) and P j+k (L) are mutually controllable, for i,j = 1,2,...,n, then M = n i=1 supc(p i+k(k),p i+k (L),A i+k,u ) is a sublanguage of K controllable wrt L and A u, and coobservable wrt L and (A i+k ) n i=1. Remark Corollary: Optimality If L is CD, i.e. separable wrt (A o,i+k ), i I, then M is coobservable wrt L and (A o,i+k ), i I, and M = SupC(K,L,A uc ) (i.e. is the globally optimal solution)

73 Example K = {aa,ba,bbd,abc}, L = {aac,abc,bac,bbd}, A o,1 = A c,1 = {a,c}, and A o,2 = A c,2 = {b,d}. c 4 3 a 8 6 c c 7 5 b a 1 2 b a 0 b 9 d 10 6 c b a a 1 2 b a 0 b 9 d 10 Generator for the plant Generator for the specification

74 Example continued K is not coobservable wrt L and (A o,i ) 2 i=1 : none of the supervisors can distinguish s = ab and s = ba, and bac K while abc K. Find A k /0 s.t. K is cond. decomposable wrt A 1+k and A 2+k. Take A k = {b} : b is communicated As A 1+k A 2+k = {b}, P 1+k (L) and P 2+k (L) are mutually controllable. supc(p 1+k (K),P 1+k (L),A 1+k,u ) = {aa,ba,bb,ab}, supc(p 2+k (K),P 2+k (L),A 2+k,u ) = {bbd}. Conclusion: K = n i=1 supc(p i+k(k),p i+k (L),A i+k,u ) is coobservable wrt L and {a,b,c} and {b,d}.

75 Remarks Idea: Over-approximate plant in Decentralized Supervisory Control Problem and plunge it into modular control with coordinators Compute a by construction coobservable solution The proposed solution is computationally cheap Communications based on multi-level coordination and extension to conditional (inference) architectures of decentralized control Application to products of timed automata: zone abstractions are not compositional

76 Multilevel Hierarchy Subsystems are organize into groups starting from the lowest level: G k over A k G k1 over A k1 G k2 over A k G km over A km G 1... G i1 G i G i2 G im G im Group I 1 Group I 2 Group I m I j = {i j 1 + 1,i j 1 + 2,...,i j } k l k,l {1,...,m} (A I k A Il ) smaller than k l k,l {1,...,n} A k A l )!

77 Multilevel control motivation Centralized coordination suffers from several problems: For large n too many events must be included in A k! Too many events need to be communicated among all subsystems Coordinator as well as its supervisor are too large Our solution: divide subsystems into groups and associate each group with group coordinators that need much less events Top level coordination then may have much fewer events as well

78 Outline Introduction to Supervisory Control Computational Difficulties Decentralized Control of Monolithical DES Weighted automata Residuation of (max,+) and (min,+) series

79 Motivation for weighted automata Generalize logical automata (outputs are in arbitrary semiring, not just Boolean) Generalize linear systems in semirings (from discrete time to free monoids) Outputs semirings can represent e.g. time (max,+), price (min,+) or probability Infinite state systems: states are vectors over semirings But: decidability issues for even elementary problems

80 Control of (max,+) automata inspired by supervisory control (Max,+) automata: weights in (R { }, max, +). class of Timed Discrete Event (dynamical) Systems (TDES) with synchronization and resource sharing strong expressive power in terms of timed Petri nets: 1-safe TPNs can be viewed as heap models: special (max,+)-automata Supervisory control of (max,+) automata : based on formal power series Tensor products in terms of linear representations Hadamard product in terms of behaviors

81 Algebraic preliminaries: dioids Dioids are idempotent semirings, i.e. an idempotent semigroup (D,, ε), endowed with associative (unit element e), a M : a ε = ε a = ε and distributes over : (i) a,b,c D, (a b) c = (a c) (b c) (ii) a,b,c D, c (a b) = (c a) (c b) Idempotent semigroups have natural order a b iff a b = b. In complete structures : a = n=0 a n, with a 0 = e a D.

82 Dioid of formal power series Dioid (idempotent semiring) of formal power series : R max (A) = {s : A R max }. Notation : s = w A s(w)w R max (A) forms a dioid. For s,s R max (A): and pointwise addition: s s = w A (s(w) s (w))w Cauchy (convolution) multiplication: s s = ( s(u)s (v))w. w A Hadamard product: s s w A (s(w) s (w))w. uv=w

83 Residuation theory Residuation theory generalizes inversion An isotone f : D D, where D is a dioid, is said to be residuated if there exists an isotone map h : D D such that f h Id D and h f Id D. h is unique residual of f, denoted by f. If f is residuated then y D, sup{x D f (x) y} exists and belongs to this subset and is equal to f (y). Example: left and right multiplications are always residuated in complete dioids! Notation. a \y = max{x a x y} and y /a = max{x x a y}.

84 Residuation of Hadamard product Addition and Kleene star are not residuated! But, Hadamard product H y : R max (A) R max (A),s s y is residuated. Moreover, Proposition. H y is residuated and its residual is given by H y(s) = w A (s(w) /y(w))w. Proof. (H y H y)(s) = w A [(s(w) /y(w)) y(w)]w w A s(w)w = s (H y H y )(s) = w A [(s(w) y(w)) /y(w)]w w A s(w)w = s

85 Outline Introduction to Supervisory Control Computational Difficulties Decentralized Control of Monolithical DES Weighted automata Residuation of (max,+) and (min,+) series

86 Basic facts Definition. A rational D-series S is unambiguous if it is recognized by a finite D-automaton G = (α, µ,β) s.t. w supp(s), there exists at most one succesful path in G. Definition. A rational D-series is sequential if it is recognized by a deterministic D-automaton G = (α, µ,β), i.e. G has (i) a single initial state q 0 (α(q) = ε q q 0 ) and (ii) a deterministic transition rel. ( a and q, µ(a)(q,q ) ε for at most one state q ). Theorem. Lombardy and Mairesse (2006) A rational (max,+) series is a rational (min,+) series if and only if it is unambiguous. Consequence. A rational (max,+)series with inverse coefficients is rational (min,+) series if and only if it is unambiguous.

87 Residuation of Hadamard product of series For any S 1,S 2 D(A) and for w A, simply (S 1 /S 2 )(w) = S 1 (w)/s 2 (w). Major problems: (i) S 1 /S 2 need not be (max,+)-rational and (ii) Can we decide whether the controller series S 1 /S 2 is non-negative? max a + b a + b / max a + + b + b a Table: (max,+) product and the corresponding residuation.

88 Motivating example. Example. a/1, b/1 Figure: An automaton recognizing the length of a word. a/1, b/0 a/0, b/1 0 1 Figure: An automaton recognizing the maximum number of occcurences of a letter in a word.

89 Example text. First automaton recognizes S 1 = w A w w. The second S 2 = w A max( w a, w b )w. Note: w A, S 1 (w)/ max S 2 (w) = w max( w a, w b ) = min( w a, w b ). Therefore, S 1 / max S 2 = w A min( w a, w b )w is recognized by automaton (b) seen as a (min,+) automaton. S 1 /S 2 is an ambiguous (min,+) rational series. Hence, this series cannot be a (max,+) rational series.

90 Main theorem Theorem. Let S R min Rat(A) and T R max Rat(A). Then S / max T R min Rat(A); T / min S R max Rat(A). Sketch of the proof. The set of states of A is Q = Q S Q T and we set the following: (p,q) Q S Q T, α (p,q) = (α S ) p /(α T ) q ; a A, (p,q),(r,s) Q S Q T, µ(a) (p,q),(r,s) = µ S (a) (p,r) /µ T (a) (q,s) ; (p,q) Q S Q T, β (p,q) = (β S ) p /(β T ) q.

91 Deciding positivity Proposition. For WA A = (α, µ,β) be a weighted automaton, let S min and S max be the respective (min,+) and (max,+) series recognized by A. One can decide whether S min (w) < 0 (resp. S max (w) > 0) for some w in supp(s min ) (resp. in supp(s max )). Sketch of the proof. S min (w) < 0 for some w supp(s) iff there is a minimal path q 0 a 1 q 1...a n q n s.t. α q0 + n i=1 µ(a i) (qi 1,q i ) + β qn and this sum is negative or there exists a minimal cycle with finite negative weight through some state q j {q 0,...,q n }.

92 Discussion When specification S is (min,+) rational and system T is (max,+) rational, we can decide existence of a non-negative delay-controller, namely the (min,+) rational series S / max T Construction: any (min,+) automaton recognizing S / max T can be transformed to an equivalent (min,+) automaton (α, µ,β) in which all entries of α, µ, and β are either non-negative or equal to However, S / max T that fails to be sequential cannot be used for control Another soluton: determinization

93 Determinization Not all weighted automata can be determinized: no finite state deterministic WA recognizing the same series exist in general For weights in semirings it is worse than in rings Sufficient conditions for determinization exist: twin property (Choffrut), clones property (Kirsten) Sufficient conditions for determinization of series stemming from timed PN: bounded fairness Bounded fairness for automata: ω-languages Determinization can be used to build observers Extension to partial observations and decentralized control

94 Conclusions and Perspectives (Multilevel) coordination control It is based on the top-down decomposition, bottom-up approach also exists Application to decentralized supervisory control Timed systems: abstractions or determinization needed for control

95 Thank You