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CASE 2012 Aug. 20 -24 , 2012

On Iterative Liveness-enforcement for a Class of Generalized Petri Nets YiFan Hou, Ding Liu, MengChu Zhou. CASE 2012 Aug. 20 -24 , 2012. Outline. Background and Motivation Intrinsically Live Structure (ILS) Liveness and Ratio-enforcing Supervisor (LRS) MIP & LRS

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CASE 2012 Aug. 20 -24 , 2012

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  1. On Iterative Liveness-enforcement for a Class of Generalized Petri NetsYiFan Hou, Ding Liu, MengChu Zhou CASE 2012 Aug. 20-24, 2012

  2. Outline • Background and Motivation • Intrinsically Live Structure (ILS) • Liveness and Ratio-enforcing Supervisor (LRS) • MIP & LRS • Conclusion and Future Work

  3. Outline • Background and Motivation • Intrinsically Live Structure (ILS) • Liveness and Ratio-enforcing Supervisor (LRS) • MIP & LRS • Conclusion and Future Work

  4. Background and Motivation DEADLOCK Two oxen and a single-log bridge (picture from Internet)

  5. Background and Motivation (b) (a) (d) (c)

  6. Background and Motivation • siphons do not carry any weight information; • the siphon-based method originally developed for ordinary Petri nets mostly cannot be directly used in generalized ones; • the siphon-based method originally developed for ordinary Petri nets yield a controlled system with very limited reachable states; • a new kind of structural objects tied with deadlock-freeness and liveness? • a new policy for deadlock-control / liveness-enforcement?

  7. Outline • Background and Motivation • Intrinsically Live Structure (ILS) • Liveness and Ratio-enforcing Supervisor (LRS) • MIP & LRS • Conclusion and Future Work

  8. Intrinsically Live Structure (ILS) • a structural object carrying weight information; • a structural intuitively reflecting circular waits; • a numerical relationship between initial marking and arc weights; (b) (a)

  9. Intrinsically Live Structure (ILS) • A WSDC is a subnet consisting of places, transitions, and their arcs that form a simple circuit of the digraph; • The competition path t2r2t3; • The upstream activity place pr2up and downstream one pr2down compete against each other; • The numerical relationship between the arc weights of and the initial number of tokens in the resource place;

  10. Intrinsically Live Structure (ILS) • A revised dining philosopher problem modeled by WS3PR; • A WSDC t2r1t14t5t11r4t8r3t5r2t2 expresses the circular wait relation among all resource places;

  11. Intrinsically Live Structure (ILS) • A competition path is a link of the whole chain of resource places; • Break the chain of circular wait by breaking a link of it; • The basic idea is to ensure that after a prioritized and maximal acquirement of tokens in the resource place by the upstream activity place, the remaining ones are still adequate for the downstream one to complete one operation; • Implemented by the numerical relationship between arc weights and initial markings;

  12. Intrinsically Live Structure (ILS) • A weight matrix is used to deal with the situation that multiple competition path with the same resource places;

  13. Intrinsically Live Structure (ILS) • Main results - Restriction 1;

  14. Intrinsically Live Structure (ILS) • Main results - Theorems;

  15. Intrinsically Live Structure (ILS) • A Live WS3PR with all WSDC satisfying Restriction 1;

  16. Outline • Background and Motivation • Intrinsically Live Structure (ILS) • Liveness and Ratio-enforcing Supervisor (LRS) • MIP & LRS • Conclusion and Future Work

  17. Liveness and Ratio-enforcing Supervisor (LRS) Basic idea: • Impose a well-designed supervisor with intrinsically live structures to break the chain of circular waits; • Consider the resource usage ratios of upstream and downstream activity places and the relation between them;

  18. Liveness and Ratio-enforcing Supervisor (LRS) Resource usage ratio (RU-ratio): an admissible range of RU-ratios

  19. Liveness and Ratio-enforcing Supervisor (LRS) • All RU-ratios

  20. Liveness and Ratio-enforcing Supervisor (LRS) • Rephrase Restriction 1 from the pespective of RU-ratio; • Make sure the structures of LRS monitors satisfy Restriction 2;

  21. Liveness and Ratio-enforcing Supervisor (LRS) • Design a control path satisfying Restriction 2; • Impose the control path to a competition one; • Make a competition path to be a puppet;

  22. Liveness and Ratio-enforcing Supervisor (LRS) • Designed a control path according to the control specification; • Impose the control path to the competition one virtually replacing its role in the chain; • Take over the token allocation of the resource place by the numerical relationship between arc weights and initial markings; • Design the control parameters of the competition path by setting a minimal RU-ratio of downstream activity place and solving the following mathematical programming problem;

  23. Liveness and Ratio-enforcing Supervisor (LRS)

  24. Liveness and Ratio-enforcing Supervisor (LRS) • The differences between LRS and siphon-monitor-based methods: • Basic idea; • Structural object; • Supervisor’s size; • RU-ratios and parameters;

  25. Liveness and Ratio-enforcing Supervisor (LRS) • The advantages of LRS: (1) The size of an LRS; (2) No new problematic structures; (3) Adjusting control parameters; (4) Intuitive and easy to understand; (5) A precise usage and robustness of resources; • The limitation of LRS: • The existence is decided by the initial marking of a plant model;

  26. Outline • Background and Motivation • Intrinsically Live Structure (ILS) • Liveness and Ratio-enforcing Supervisor (LRS) • MIP & LRS • Conclusion and Future Work

  27. MIP & LRS • Avoid enumerate all WSDCs in a plant net modeled with WS3PR; • Only find the problematic structure;

  28. MIP & LRS • Find a maximal insufficiently marked siphon by solving MIP problem 2; • Select a resource place from the maximal insufficiently marked siphon; • Design an LRS monitor for the resource place;

  29. MIP & LRS Process idle places: 2 Activity places: 11 Resource places: 6 Transitions: 14 3,334,653 states Including 30 dead ones

  30. MIP & LRS Iteration 1: Find the maximal insufficiently marked siphon by MIP; Control resource place p19 by v1; 2,663,888 states Including 6 dead ones

  31. MIP & LRS Iteration 2: Find the maximal insufficiently marked siphon by MIP; Control resource place p15 by v2; 2,613,824 states Including 1 dead ones

  32. MIP & LRS Iteration 3: Find the maximal insufficiently marked siphon by MIP; Control resource place p17 by v3; 2,500,037 states No dead ones LIVE

  33. MIP & LRS

  34. Outline • Background and Motivation • Intrinsically Live Structure (ILS) • Liveness and Ratio-enforcing Supervisor (LRS) • MIP & LRS • Conclusion and Future Work

  35. Conclusion and Future Work • Conclusion: (1) Avoid the enumeration of all WSDC; (2) All strict minimal siphons are minimally controlled; (3) The number of iterations is bounded by that of resource places; • Future work: • How to optimally select a shared resource place given a maximal insufficiently marked siphon; • How to extend this method to more general nets than WS3PR;

  36. Thanks for your attention!

  37. Related Publications [1] D. Liu, Z. W. Li, and M. C. Zhou, “Liveness of an Extended S3PR,” Automatica, vol. 46, no. 6, pp. 1008 –1018, 2010. [2] D. Liu, Z.W. Li, andM. C. Zhou, “Erratum to “Liveness of an Extended S3PR [Automatica 46 (2010) 1008-1018]”,” Automatica, vol. 48, no. 5, pp. 1003 – 1004, 2011. [3] D. Liu, Z. W. Li, and M. C. Zhou, “Hybrid Liveness-enforcing Policy for Generalized Petri Net Models of Flexible Manufacturing Systems,” accepted by IEEE Transactions on Systems, Man, and Cybernetics, Part A, 2012. [4] D. Liu, Z. W. Li, and M. C. Zhou, “A Parameterized Liveness and Ratio-Enforcing Supervisor for a Class of Generalized Petri Nets,” submitted to Automatica, 2012. [5] D. Liu, Z. W. Li, Y. F. Hou, and M. C. Zhou, “On Divide-and-Conquer Liveness enforcing strategy for Flexible Manufacturing Systems Modeled by a Class of Generalized Petri Nets,” Technical report, Xidian University, 2012. [6] Y. F. Hou, D. Liu, Z. W. Li, and M. Zhao, “Deadlock Prevention Using Divide-and-Conquer Strategy for WS3PR,“in Proceedings of IEEE ICMA 2010, pp. 1635 – 1640, 2010. [7] D. Liu, M. Zhao, H. S. Hu, and A. R. Wang, “Hybrid Liveness-enforcing Method for Petri Net Models of Flexible Manufacturing Systems,“ in Proceedings of IEEE ICMA 2010, pp. 1813 – 1818, 2010. [8] M. Zhao, Yifan Hou, and Ding Liu, “Liveness-enforcing Supervisors Synthesis for a class of Generalized Petri Nets based on Two-stage Deadlock Control and Mathematical Programming,“ International Journal of Control, vol. 83, no. 10, pp. 2053 – 2066, 2010.

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