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

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)
(b)
a structural object carrying weight information;
a structural intuitively reflecting circular waits;
a numerical relationship between initial marking and arc weights;

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
(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 –1018, [2] D. Liu, Z.W. Li, andM. C. Zhou, “Erratum to “Liveness of an Extended S3PR [Automatica 46 (2010) ]”,” Automatica, vol. 48, no. 5, pp – 1004, [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, [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, [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, [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 – 1640, [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 – 1818, [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 – 2066, 2010.