CASE 2012 Aug. 20 -24 , 2012
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Transcript of 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
• 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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Background and Motivation
Two oxen and a single-log bridge(picture from Internet)
DEADLOCK
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Background and Motivation
(a) (b)
(c) (d)
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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?
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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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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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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;
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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Intrinsically Live Structure (ILS)
• 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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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;
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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Intrinsically Live Structure (ILS)
• 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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Intrinsically Live Structure (ILS)
• Main results - Restriction 1;
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Intrinsically Live Structure (ILS)
• Main results - Theorems;
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Intrinsically Live Structure (ILS)
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
• 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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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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Liveness and Ratio-enforcing Supervisor (LRS)
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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Liveness and Ratio-enforcing Supervisor (LRS)
• 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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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;
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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;
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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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;
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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Liveness and Ratio-enforcing Supervisor (LRS)
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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Liveness and Ratio-enforcing Supervisor (LRS)
• 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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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:(1) The existence is decided by the initial marking of a plant model;
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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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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;
2,663,888 statesIncluding 6 dead ones
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MIP & LRS
Iteration 2:Find the maximal insufficiently marked siphon by MIP;Control resource place p15
by v2;
2,613,824 statesIncluding 1 dead ones
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MIP & LRS
Iteration 3:Find the maximal insufficiently marked siphon by MIP;Control resource place p17
by v3;
2,500,037 statesNo dead onesLIVE
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MIP & LRS
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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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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.