1. Control and Deadlock Recovery of Timed Petri Nets Using Observers Alessandro Giua DIEE – Department of Electrical and Electronic Engineering University of Cagliari, Italy Joint work with: - Carla Seatzu (U. of Cagliari) - Francesco Basile (U. del Sannio)

2. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 2 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

3. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 3 0 – PETRI NETS A place/transition net is a 4-ple : N =( P,T,Pre, Post ) P={ p 1, p 2, …, p m } set of places (circles); T={ t 1, t 2, …, t n } set of transitions (bars); Pre : matrix denoting # of arcs from places to transitions Post : matrix denoting # of arcs from transitions to places p1p1 p2p2 p 3 p5 p5 t6t6 t 1 p8p8 t2t2 p4p4 p5p5 p 6 p7p7 t3t3 t4t4 t5t5

4. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 4 Net system (a net N with initial marking M 0 ): Set of firable sequences: Set of reachable markings: Siphon: a set of places S such that if a transition inputs into S then it also outputs from S (Ex: S = {p 1, p 2 } ) p 1 p2p2 p3p3 t 1 t 3 t 2 An empty siphon will always remain empty  all its output transitions are deadlocked PETRI NETS (cont’d)

5. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 5 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

6. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 6 1 - MOTIVATION FOR DISCRETE EVENT OBSERVERS Two approaches to design of observers for discrete event Systems Computer science approach (CSA): the state is unknown because the system structure is nondeterministic Control theory approach (CTA): the system structure is deterministic but the initial state is unknown

7. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 7 CTA - Supervisory control theory is based on language specifications (a set of legal words): language specification event-feedback MOTIVATION (cont’d)

8. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 8 When dealing with Petri nets it is natural to use state specifications (a set of legal markings): state specification state-feedback MOTIVATION (cont’d)

9. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 9 A mixed structure is often used: State specification= marking of the net Output events = transitions firing When the net structure and the initial marking is known (and the net labeling is deterministic) event observation is sufficient to reconstruct the net marking. MOTIVATION (cont’d)

10. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 10  If the initial marking is not completely known: use “observers” to estimate the marking after the word of events w has been observed  In our approach the observer determines two parameters Estimate: Bound: MOTIVATION (cont’d) legal markings

11. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 11 C (w) M w MOTIVATION (cont’d) Unlike other approaches based on automata, the PN structure allows one to “describe” the set of consistent markings in terms of these two parameters that are recursively updated. Linear constraint set

12. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 12 In this talk we present:  Algorithms for computing estimate and consistent set  Algorithms for control using observers  Algorithms for deadlock recovery  Deadlock analysis of the closed loop system PROBLEM: incomplete information due to the presence of an observer in the control loop may lead to deadlock. MOTIVATION (cont’d) All these problems are solved using the same approach based on integer programming

13. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 13 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

14. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 14 2 - RELEVANT LITERATURE State-feedback control with partial observability - Li & Wonham [CDC88] [T-AC93](state observ.) - Takai, Ushio & Kodama [T-AC95](state observ.) - Zhang & Holloway[Allerton95](event observ.) Derived nec & suff condition for optimality given a mask.

15. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 15 DES state estimation for FSM / Predicate Transformers - Ramadge [CDC86](FMS) - Caines, Greiner, Wang [CDC88] [CDC89](FMS) - Özveren, Willsky [T-AC90](FMS) - Kumar, Garg & Markus [T-AC93](PT) DRAWBACK These approaches enumerate at each step the set of consistent states (high complexity). No notion of “estimate error”. LITERATURE (cont’d)

16. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 16 Diagnosis - Wang, Schwartz [T-net 93] (state estimation) - Ushio, Onishi & Okuda[SMC98] (place observation) Petri net observability - Meda, Ramirez[SMC98] (interpreted nets) - Ramirez, Riveda, Lopez [ICRA2000] Partial knowledge of the marking - Cardoso, Valette & Dubois[ICATPN90] Concept of macromarking and “membership function”. LITERATURE (cont’d)

17. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 17 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

18. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 18 3 - MAIN IDEA Initially observed sequence: Initial marking Estimate: Set of consistent markings:

19. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 19 firing is detected MAIN IDEA (cont’d)

20. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 20 After fires Observed sequence: Actual marking Estimate: Set of consistent markings: MAIN IDEA (cont’d)

21. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 21 firing is detected MAIN IDEA (cont’d)

22. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 22 Observed sequence: Actual marking Estimate: Set of consistent markings: After fires MAIN IDEA (cont’d)

23. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 23 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

24. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 24 4 - MARKING ESTIMATION Hypothesis: - The net structure is known - The transition firing can be observed - The initial marking is not known Algorithm 1 - Initial estimate: Let 2 - Wait until fires 3 - Update previous estimate: 4 - New estimate: 5 - ; goto 2.

25. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 25 ESTIMATION (cont’d) Can define place error: Estimate is a lower bound: The set of markings consistent with observation w is: and estimation error: Error functions are non-increasing: Properties

26. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 26 Properties An observed word is marking complete if A net system is: - Marking Observable (MO) if there exists a complete word - Strongly Marking Observable (SMO) in k steps if: a) all with are complete b) all with that are not complete can be continued in a word ESTIMATION (cont’d)

27. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 27 Observer reachability graph Each node of the graph is labeled with: The real marking M w The estimation error u w = M w -  w ESTIMATION (cont’d)

28. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 28 Observer coverability graph If the net is unbounded, is it possible to construct an observer coverability graph (OCG). The error vector u is now only an upper bound. ESTIMATION (cont’d)

29. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 29 Analysis of properties Theorem 1 A net system is: marking observable iff marking observable if there exist a node in the OCG with strongly marking observable iff in the OCG for each dead node and for each node in a cycle ESTIMATION (cont’d)

30. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 30 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

31. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 31 5 – MARKING ESTIMATION WITH INITIAL MACROMARKING Sometimes partial information on the initial marking is available Example: assume the net starts from marking (known) evolving unobserved until it reaches ; at this point we start observations. Then we may use the information that This characterization in terms of PN reachability is hard to use but we can approximate it using a matrix of invariants :

32. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 32 Generalizing, we define an initial macromarking. The set of places is written as: For each, the token content of is known to be Nothing is known about the marking in Let be the char vector of and define Macromarking: MACROMARKING (cont’d)

33. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 33 A MANUFACTURING EXAMPLE p1 p1 p 10 p3 p3 p 4 p5 p5 p6 p6 p 7 p 8 p 9 p 2 t 6 t7 t7 t 1 t4 t4 t2 t2 t 3 t 5 p 11 p 12 M 0 (p 11 )+M 0 (p 12 ) = 1 M 0 (p 1 )+M 0 (p 3 )+M 0 (p 4 ) = 5 M 0 (p 1 )+M 0 (p 5 )+M 0 (p 6 ) = 5 M 0 (p 1 )+M 0 (p 3 )+M 0 (p 6 ) +M 0 (p 11 ) = 6 M 0 (p 1 )+M 0 (p 4 )+M 0 (p 5 ) +M 0 (p 12 ) = 5 Initial macromarking: we know the token content in each cycle M 0 (p 2 )+M 0 (p 8 ) +M 0 (p 9 ) = 6 M 0 (p 2 )+M 0 (p 3 )+M 0 (p 4 )+M 0 (p 7 )+M 0 (p 10 ) = 6 M 0 (p 2 )+M 0 (p 5 )+M 0 (p 6 )+M 0 (p 7 )+M 0 (p 10 ) = 6 M 0 (p 2 )+M 0 (p 3 )+M 0 (p 6 )+M 0 (p 7 )+M 0 (p 10 )+M 0 (p 11 ) = 7 M 0 (p 2 )+M 0 (p 4 )+M 0 (p 5 )+M 0 (p 7 )+M 0 (p 10 )+M 0 (p 12 ) = 6 MACROMARKING (cont’d)

34. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 34 Algorithm (estimation with macromarking) 1 – Initial estimate with 2 - Initial bound 3 - Let the current observed word be w=w 0. 4 - Wait until t fires. 5 - Update the estimate to 6 - New estimate: 7 - New bound: 8 - Goto 4. MACROMARKING (cont’d)

35. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 35 Elementary results The estimate is a lower bound: The error functions are non-increasing The set of markings consistent with the observation w is: MACROMARKING (cont’d) This set can also be characterized as:

36. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 36 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

37. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 37 6 - CONTROL USING OBSERVERS GMEC specifications: a set of linear constraints for j = 1, …, q. Example: The set of legal markings is:

38. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 38 Control with observer Prevent the firing of t after w has been observed iff there exists a legal consistent marking M such that the firing of t from M leads to a forbidden marking i.e., if exists j such that  where Control pattern: CONTROL (cont’d)

39. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 39 EXAMPLE The firing of t 1 is legal from but t 1 is disabled

40. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 40 EXAMPLE Actual marking M w Estimation error u w = M w -  w Bound

41. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 41 Usually, the control law using observers is not optimal since it can disable the firing of transitions that do not yield illegal markings. Such a control law may easily cause the controlled plant to block. We want to add to the observer the possibility of recovering from deadlocks caused by the incomplete information. CONTROL (cont’d)

42. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 42 A MANUFACTURING EXAMPLE p1 p1 p 10 p3 p3 p 4 p5 p5 p6 p6 p 7 p 8 p 9 p 2 t 6 t7 t7 t 1 t4 t4 t2 t2 t 3 t 5 p 11 p 12 M(p 9 )  3 M(p 3 )+M(p 5 )  3

43. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 43 ( 4 5 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 1 5 5 6 5 6 6 6 7 6 ) ( 4 5 0 1 0 1 0 1 0 0 1 0 / 0 0 0 1 0 0 0 0 0 0 1 0 / 0 4 5 5 4 6 5 6 6 5 ) ( 5 5 0 0 0 0 1 1 0 0 1 0 / 1 0 0 0 0 0 1 0 0 0 1 0 / 0 4 4 4 4 6 5 5 5 5 ) ( 5 6 0 0 0 0 0 0 0 0 1 0 / 1 1 0 0 0 0 0 0 0 0 1 0 / 0 4 4 4 4 5 5 5 5 5 ) t1t1 t4t4 t3t3 Deadlock Actual marking M w Estimate  w Bound  w

44. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 44 p1 p1 p 10 p3 p3 p 4 p5 p5 p6 p6 p 7 p 8 p 9 p 2 t 6 t7 t7 t 1 t4 t4 t2 t2 t 3 t 5 p 11 p 12 M(p 9 )  3 M(p 3 )+M(p 5 )  3 Only the green tokens have been detected t 6 and t 7 are disabled by the controller ? ? ?

45. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 45 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

46. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 46 IDEA: use the info that the net is deadlocked to improve the estimate (reducing the set of consistent markings) Theorem: In an ordinary net a marking M is dead iff: is a siphon for all MwMw 7 - DEADLOCK RECOVERY AND ESTIMATE UPDATE AFTER NET TIME-OUT Set of blocking markings

47. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 47 Given a structurally bounded net N, a marking M is dead iff  a vector (  {0,1} m such that: DEADLOCK RECOVERY (cont’d) The set of blocking markings of N: is the characteristic vector of a siphon contains only empty places contains all empty places each transitions has at least a pre arc coming from

48. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 48 DEADLOCK RECOVERY (cont’d) Algorithm (Control pattern updating after net time-out) Let C = C ( ,B ). Assume f(., C ) has led the net to a time-out. 1. Let i=0 and f 0 = f(., C ). 2. Let T i ={t  T | f i (t)=1} and let N i the net obtained by N removing all transitions not in T i. 3. Update the control pattern to f i+1 =f(., C  M b (N i )) 4. If f i+1 = f i THEN exit: (the deadlock procedure has failed) 5. Wait until (a) a transition fires (net has recovered from deadlock) (b) a new net time-out occurs: let i=i+1 and go to 2.

49. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 49 A unique linear algebraic formalism for: –state estimation –control –deadlock recovery DEADLOCK RECOVERY (cont’d) Main advantages of the approach: This procedure is denoted NTO procedure (net time-out procedure).

50. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 50 A MANUFACTURING EXAMPLE (cont’d) p1 p1 p 10 p3 p3 p 4 p5 p5 p6 p6 p 7 p 8 p 9 p 2 t 6 t7 t7 t 1 t4 t4 t2 t2 t 3 t 5 p 11 p 12 M 0 (p 11 )+M 0 (p 12 ) = 1 M 0 (p 1 )+M 0 (p 3 )+M 0 (p 4 ) = 5 M 0 (p 1 )+M 0 (p 5 )+M 0 (p 6 ) = 5 M 0 (p 1 )+M 0 (p 3 )+M 0 (p 6 ) +M 0 (p 11 ) = 6 M 0 (p 1 )+M 0 (p 4 )+M 0 (p 5 ) +M 0 (p 12 ) = 5 M(p 9 )  3 Initial macromarking: we know the token content in each cycle M(p 3 )+M(p 5 )  3 M 0 (p 2 )+M 0 (p 8 ) +M 0 (p 9 ) = 6 M 0 (p 2 )+M 0 (p 3 )+M 0 (p 4 )+M 0 (p 7 )+M 0 (p 10 ) = 6 M 0 (p 2 )+M 0 (p 5 )+M 0 (p 6 )+M 0 (p 7 )+M 0 (p 10 ) = 6 M 0 (p 2 )+M 0 (p 3 )+M 0 (p 6 )+M 0 (p 7 )+M 0 (p 10 )+M 0 (p 11 ) = 7 M 0 (p 2 )+M 0 (p 4 )+M 0 (p 5 )+M 0 (p 7 )+M 0 (p 10 )+M 0 (p 12 ) = 6

51. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 51 ( 4 5 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 1 5 5 6 5 6 6 6 7 6 ) ( 4 5 0 1 0 1 0 1 0 0 1 0 / 0 0 0 1 0 0 0 0 0 0 1 0 / 0 4 5 5 4 6 5 6 6 5 ) ( 5 5 0 0 0 0 1 1 0 0 1 0 / 1 0 0 0 0 0 1 0 0 0 1 0 / 0 4 4 4 4 6 5 5 5 5 ) ( 5 6 0 0 0 0 0 0 0 0 1 0 / 1 1 0 0 0 0 0 0 0 0 1 0 / 0 4 4 4 4 5 5 5 5 5 ) t1t1 t4t4 t3t3 NTO

52. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 52 ( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) t6t6 ( 4 5 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 1 5 5 6 5 6 6 6 7 6 ) ( 4 5 0 1 0 1 0 1 0 0 1 0 / 0 0 0 1 0 0 0 0 0 0 1 0 / 0 4 5 5 4 6 5 6 6 5 ) ( 5 5 0 0 0 0 1 1 0 0 1 0 / 1 0 0 0 0 0 1 0 0 0 1 0 / 0 4 4 4 4 6 5 5 5 5 ) ( 5 6 0 0 0 0 0 0 0 0 1 0 / 1 1 0 0 0 0 0 0 0 0 1 0 / 0 4 4 4 4 5 5 5 5 5 ) t1t1 t4t4 t3t3 NTO

53. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 53 NTO ( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) t6t6 ( 5 5 0 0 0 0 0 0 1 1 1 0 / 5 0 0 0 0 0 0 0 1 1 1 0 / 0 0 0 0 0 5 5 5 5 5 ) t7t7 ( 4 5 1 0 1 0 0 0 1 0 1 0 / 4 0 1 0 1 0 0 0 1 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) t2t2 ( 4 5 1 0 0 1 0 0 1 0 0 1 / 4 0 1 0 0 1 0 0 1 0 0 1 / 0 0 0 0 0 5 5 5 5 5 ) t5t5 ( 4 5 1 0 0 1 0 1 0 0 1 0 / 4 0 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 5 5 5 5 5 ) t1t1 ( 4 5 0 1 0 1 0 1 0 0 1 0 / 4 0 0 1 0 1 0 1 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) ( 5 5 0 0 0 0 1 1 0 0 1 0 / 5 0 0 0 0 0 1 1 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) ( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) t4t4 t3t3

54. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 54 NTO ( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) t6t6 ( 5 5 0 0 0 0 0 0 1 1 1 0 / 5 0 0 0 0 0 0 0 1 1 1 0 / 0 0 0 0 0 5 5 5 5 5 ) t7t7 ( 4 5 1 0 1 0 0 0 1 0 1 0 / 4 0 1 0 1 0 0 0 1 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) t2t2 ( 4 5 1 0 0 1 0 0 1 0 0 1 / 4 0 1 0 0 1 0 0 1 0 0 1 / 0 0 0 0 0 5 5 5 5 5 ) t5t5 ( 4 5 1 0 0 1 0 1 0 0 1 0 / 4 0 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 5 5 5 5 5 ) t1t1 ( 4 5 0 1 0 1 0 1 0 0 1 0 / 4 0 0 1 0 1 0 1 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) ( 5 5 0 0 0 0 1 1 0 0 1 0 / 5 0 0 0 0 0 1 1 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) ( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 ) t4t4 t3t3 ( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 6 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 0 0 0 0 0 )

55. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 55 Proposition: if the initial macromarking is such that (i.e., each column of V is a P-invariant) then, for all observed words w, DEADLOCK RECOVERY (cont’d) If a marking is consistent with the observation w then it is also consistent with the initial observation

56. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 56 Theorem 1: if the initial macromarking is such that then the closed loop system will never time out if the following constraint set does not admit feasible solutions DEADLOCK RECOVERY (cont’d) where

57. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 57 Definition: the maximal control pattern for a set C is: where and DEADLOCK RECOVERY (cont’d) When a controlled system times out, if it is deadlocked eventually a control pattern is reached

58. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 58 Theorem 2: if the initial macromarking is such that then the closed loop system will always recover from a time-out if the following constraint set does not admit feasible solutions DEADLOCK RECOVERY (cont’d) where

59. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 59 p1 p1 p 10 p3 p3 p 4 p5 p5 p6 p6 p 7 p 8 p 9 p 2 t 6 t7 t7 t 1 t4 t4 t2 t2 t 3 t 5 p 11 p 12 M(p 9 )  3 M(p 3 )+M(p 5 )  3 A MANUFACTURING EXAMPLE (cont’d) Initial macromarking: the net is a marked graph each cycle corresponds to a P-invariant the initial macromarking is such that

60. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 60 A MANUFACTURING EXAMPLE (cont’d) Theorem 1 does not apply: the following constraint set admits feasible solutions the net might time out (it actually does) Theorem 2 does apply: the following constraint set does not admit feasible solutions the closed loop system with net time-out recovery is deadlock-free

61. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 61 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

62. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 62 8 - USING TIMING INFORMATION TO IMPROVE THE PROCEDURE We extend the previous approach to exploit available information on the timing structure so as to obtain a better estimate of the set of consistent markings.  A known delay time  (t) is associated to each transition.  We say that a transition t has timed-out at time now if it has been control enabled without firing during [ now -  (t), now ].  We can be sure that at time now the actual marking M w is such that  M w |t .

63. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 63 IDEA: If T to is the set of transitions that have timed out at time now, we know for sure that the actual marking is such that We compute a (possibly) less restrictive control pattern using as set of consistent markings i.e., for all we compute USING TIMING INFORMATION (cont’d)

64. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 64 USING TIMING INFORMATION (cont’d) Main Advantages: Accelerates the state estimation Accelerates the deadlock recovery procedure Enables to recover from partial deadlocks The new approach is denoted TTO procedure (transition time-out procedure).

65. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 65 A MANUFACTURING EXAMPLE (cont’d) p1 p1 p 10 p3 p3 p 4 p5 p5 p6 p6 p 7 p 8 p 9 p 2 t 6 t7 t7 t 1 t4 t4 t2 t2 t 3 t 5 p 11 p 12 M(p 9 )  3 M(p 3 )+M(p 5 )  3  (t 1 ) = 2  (t 2 ) = 5  (t 3 ) = 3  (t 4 ) = 1  (t 5 ) = 2  (t 6 ) = 6  (t 7 ) = 3 Transition time-out Transition firing Delays:

66. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 66 ( 3 4 1 0 1 0 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 6 5 5 5 5 5 4 4 4 1 ) ( 3 4 0 1 1 0 0 1 0 0 1 0 / 0 0 0 1 1 0 0 1 0 0 1 0 / 4 4 4 4 4 3 3 3 3 0 ) ( 3 4 1 0 1 0 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 6 5 5 5 5 5 4 4 4 1 ) ( 3 4 0 1 1 0 0 1 0 0 1 0 / 0 0 0 1 0 0 0 0 0 0 1 0 / 5 4 5 4 5 4 3 4 3 0 ) t1t1 ( 3 4 0 1 1 0 0 1 0 0 1 0 / 0 0 0 1 1 0 0 1 0 0 1 0 / 4 4 4 4 4 3 3 3 3 0 ) t2t2 ( 3 4 0 1 0 1 0 1 0 0 0 1 / 0 0 0 1 0 1 0 1 0 0 0 1 / 4 4 4 4 4 3 3 3 3 0 ) t4t4 ( 4 4 0 0 0 0 1 1 0 0 0 1 / 1 0 0 0 0 0 1 1 0 0 0 1 / 4 4 4 4 4 3 3 3 3 0 ) now = 1 now = 2 now = 3 now = 4 now = 7 now = 8 { t 4 } { t 5 } { t 3, t 4, t 5 } { t 1, t 3, t 4, t 5 }

67. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 67 t4t4 ( 4 4 0 0 0 0 1 1 0 0 0 1 / 1 0 0 0 0 0 1 1 0 0 0 1 / 4 4 4 4 4 3 3 3 3 0 ) ( 4 4 0 0 0 0 1 1 0 0 0 1 / 4 0 0 0 0 0 1 1 0 0 0 1 / 4 4 4 4 4 0 0 0 0 0 ) t3t3 ( 4 5 0 0 0 0 0 0 0 0 0 1 / 4 1 0 0 0 0 0 0 0 0 0 1 / 4 4 4 4 4 0 0 0 0 0 ) ( 4 5 0 0 0 0 0 0 0 0 0 1 / 4 5 0 0 0 0 0 0 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 ) now = 8 now = 9 now = 11 now = 12 now = 14 { t 1, t 4, t 5 } { t 1, t 2, t 4, t 5, t 7 } { t 1, t 2, t 3, t 4, t 5, t 7 }

68. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 68 OUTLINE 0)Petri nets 1)Motivation for discrete event observers 2)Relevant literature 3)Main idea 4)Marking estimation 5)Marking estimation with initial macromarking 6)Control using observers 7)Deadlock recovery and estimate after net time out 8)Using timing information to improve the procedure 9)Conclusions and future work

69. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 69 We provided a unique linear algebraic formalism for: state estimation, control, deadlock recovery. We showed how timing information can be used to accelerate the state estimation and to detect the observer induced deadlock. Some sufficient conditions for deadlock recovery have been derived. 9 – CONCLUSIONS

70. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 70 FUTURE WORK Language completeness: A word is language complete if This may allow to use observers in event feedback. Partial event observability: assume some events are unobservable or undistinguishable. This may destroy the linear algebraic formalism in the general case. Look for restricted cases.

71. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 71 Associate a probabilistic structure to the transition firing and define, where is the probability of having a complete word after k firings. Under which conditions ? FUTURE WORK (cont’d)

72. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 72 A. Giua, C. Seatzu, “Observability of place/transition nets,” IEEE Trans. on Automatic Control, Vol. 47, No. 9, pp. 1424-1437, September, 2002. F. Basile, A. Giua, C. Seatzu, “Observer based state-feedback control of timed Petri nets with deadlock recovery,” IEEE Trans. on Automatic Control, Vol. 49, No. 1, pp. 17-29, Jan 2004. F. Basile, A. Giua, C. Seatzu, "Observer-based state-feedback control of timed Petri nets with deadlock recovery: theory and implementation," Proc CESA'2003 Multiconference (Lille, France), Jul 2003. A. Giua, C. Seatzu, J. Júlvez, "Marking estimation of Petri nets with pairs of nondeterministic transitions," Asian Journal of Control, June 2004. To appear. A. Giua, D. Corona, C. Seatzu, “State estimation and control of nondeterministic -free labeled Petri nets”, Proc. WODES’04. To appear. REFERENCES

73. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 73 - Structurally Strongly Marking Observable (sSMO) if the system is SMO for all - Structurally Marking Observable (sMO) if the system is MO for all A net system is: - Uniformly Marking Observable (uMO) if the system is Marking Observable for all - Uniformly Strongly Marking Observable (uSMO) if the system is SMO for all ESTIMATION (cont’d)

74. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 74 uniformly MO iff the semi-linear set is a home-space for all Theorem 2 A net system is: uniformly SMO only if it is bounded Similar results hold for structural MO and SMO. This is a finite union of linear sets with the same period and the home space property is decidable (Johnen & Frutos Escrig; 89) ESTIMATION (cont’d)

75. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 75 DEADLOCK RECOVERY (cont’d) MwMw After a deadlock recovery procedure is invoked we should remember the set of consistent markings is The linear characterization of this set is rather complex (it involves also a vector ). We propose to use a simpler approximation.

76. A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 76 DEADLOCK RECOVERY (cont’d) MwMw Compute for all places: Define the new set: Compute: Define