case 2012 aug. 20 -24 , 2012

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On Iterative Liveness-enforcement for a Class of On Iterative Liveness-enforcement for a Class of Generalized Petri NetsGeneralized Petri Nets

YiFan Hou, Ding Liu, MengChu ZhouYiFan Hou, Ding Liu, MengChu Zhou

CASE 2012Aug. 20-24, 2012

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Outline

• Background and MotivationBackground and Motivation

• Intrinsically Live Structure (ILS)Intrinsically Live Structure (ILS)

• Liveness and Ratio-enforcing Supervisor (LRS) Liveness and Ratio-enforcing Supervisor (LRS)

• MIP & LRSMIP & LRS

• Conclusion and Future WorkConclusion and Future Work

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Outline






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Background and Motivation

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

DEADLOCK

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(a) (b)

(c) (d)

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• 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?

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Outline






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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;

(a) (b)

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• 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;

nt

1t

2t

1r

2r

3r

nr

3t

4t

2 't 3 't

2( )in rw 2out( )rw

2( )in rw 2out( )rw

1out( )rw

nout( )rw

3out( )rw1( )in rw

3( )in rw

n( )in rw

2rupp

2rdownp

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• A revised dining philosopher problem modeled by WS3PR;

• A WSDC t2r1t14t5t11r4t8r3t5r2t2 expresses the circular wait relation among all resource places;

30

30 30

3030

1

2

7

4

2

1p2p 3p1t 2t 3t

4p

5p

6p

4t

5t

6t

7p

8p

9p

7t

8t

9t10p 11p

12p

10t

11t

12t

13p

14p

15p

13t

14t

15t

1r 2r

3r

4r

5r2

2

3

3 2

2

2

2

2

2

( ) ( )

( )

( )

( )

16p 17p

18p

19p

20p

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• 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;

tt htr

inw (r) out(r)w

(a)

rtt ht

t t'

upp downp

(b)

inw (r)

inw (r)

out(r)w

out(r)w

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• A weight matrix is used to deal with the situation that multiple competition path with the same resource places; r

t1t t2t tnt

h1t h2t hnt

1in(r)w

1out(r)w

2in (r)w

2out(r)w

in(r)nw

out(r)nw

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• Main results - Restriction 1;

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• Main results - Theorems;

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100

3

1

3

3

1

3

3

100100

2

2

2

2

2

2

2

2

2

22

2

2

2

2

1p

2p

3p 4p

5p 6p

7p 8p

9p

10p

11p

12p

13p

14p

15p

16p

17p

18p

19p

20p

21p

22p

23p

24p

25p

26p

1t

2t 3t

4t 5t

6t 7t

8t 9t

10t

11t

12t

13t

14t

15t

16t

17t

18t

19t

20t• A Live WS3PR with all WSDC

satisfying Restriction 1;

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Outline






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Liveness and Ratio-enforcing Supervisor (LRS)

Basic idea: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;

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Resource usage ratio (RU-ratio):Resource usage ratio (RU-ratio):

an admissible range of RU-ratios

tt htr

inw (r) out(r)w

(a)

rtt ht

t t'

upp downp

(b)

inw (r)

inw (r)

out(r)w

out(r)w

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• All RU-ratios 250

10

9

50

(a)

1p

2p

3p

4p

5p

6p

7p

8p1t

2t

3t

4t 5t

6t

7t

8t3

3

4

4

2

29p 1r

10p 2r

11p 3r

2

10

9

(b)

2t 7t

3t 6t4

3 29p 1r

10p 2r

11p 3r

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• Rephrase Restriction 1 from the pespective of RU-ratio;

• Make sure the structures of LRS monitors satisfy Restriction 2;

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• Design a control path satisfying Restriction 2;

• Impose the control path to a competition one;

• Make a competition path to be a puppet;

nt

1t

2t

1r

2r

3r

nr

3t

4t

2 't 3 'tv(

)inV

w

out( )Vw( )in Vwout(

)V

w

2rupp 2r

downp

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• 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;

tt ht

(a)

r

tt ht

t t'

upp downp

(b)

inw (r)

inw (r)

out(r)w

out(r)w

in(v)w out(v)w

v

vin(v)w

in(v)w

out(v)w

out(v)w

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tt ht

(a)

r

tt ht

t t'

upp downp

(b)

inw (r)

inw (r)

out(r)w

out(r)w

in(v)w out(v)w

v

vin(v)w

in(v)w

out(v)w

out(v)w

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• The differences between LRS and siphon-monitor-based methods:(1) Basic idea;(2) Structural object;(3) Supervisor’s size;(4) RU-ratios and parameters;

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• 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:(1) The existence is decided by the initial marking of a plant model;

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Outline






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MIP & LRS

• Avoid enumerate all WSDCs in a plant net modeled with WS3PR;

• Only find the problematic structure;

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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;

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MIP & LRS

Process idle places: 2Activity places: 11Resource places: 6Transitions: 14

3,334,653 statesIncluding 30 dead ones

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MIP & LRS

Iteration 1:Find the maximal insufficiently marked siphon by MIP;Control resource place p19

by v1;


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MIP & LRS


by v2;


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MIP & LRS


by v3;

2,500,037 statesNo dead onesLIVE

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MIP & LRS

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Outline






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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:

(1) How to optimally select a shared resource place given a maximal insufficiently marked siphon;

(2) How to extend this method to more general nets than WS3PR;

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Thanks for your attention!Thanks for your attention!

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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.

case 2012 aug. 20 -24 , 2012

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