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 Conclusion and Future Work

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

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

Background and Motivation

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?

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

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;

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;

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

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;

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

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

Intrinsically Live Structure (ILS) Main results - Theorems;

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

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

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;

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

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

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;

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;

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;

Liveness and Ratio-enforcing Supervisor (LRS)

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;

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;

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

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

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;

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

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

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

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

MIP & LRS

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

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;

Thanks for your attention!

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.