P-invariant, T-invariant, support and T-condition

Definition of P-invariant: An integer y vector is called a P-invariant if Ay= 0.

Theorem: An m-vector y is an P-invariant iff M^Ty = MoT

for any Mo and any M∈R(Mo

Definition of T-invariant: An integer x vector. is called a T-invariant if A

)

Theorem: An n-vector x ≥ 0 is T-invariant iff there exists a initial marking Mo

σ

and a firing sequence starting from Mo back to Mo where its firing count vector equals to x.

Definition: The set of places p(transition t such that y(p)>0, x(t)>0) for P-invariant (T-invariant)is called the support of the P-invariant(T-invariant) and is defined as ||y || (|| x ||).

Definition: A minimum support is a support and none of its proper subset is a support.

Definition: An invariant y is said to be minimal if there is no other invariant y' such that y'(p) <y(p) for all Properties Necessary and Sufficient conditions

Structurally Bounded ∃ y>0, Ay ≤0 (or Not ∃ x > 0, A^Tx> 0) Conservative ∃ y>0, Ay=0 (or Not ∃ x > 0, A^Tx> 0)

Repetitive ∃ x > 0, A^Tx> 0

Consistent' ∃ x > 0, A^Tx = 0 (or Not ∃y,Ay>0)

p. Minimal support invariant is a unique minimal invariant corresponding to the minimalsupport.

Theorem: The condition is defined as a T-condition if for any ordinary Petri net(OP), Ay≤0 iff∀ t∈T, Y(• t) ≥Y(t •).

1.9 Summary

In this chapter, we provide a brief review of the fundamental knowledge for the Petri net theory. The structure of Petri nets are introduced and some examples of application are given for illustration. We also explain some important properties for Petri nets. These properties especially the structural properties, are very important concepts when we discuss the synthesis rules for Petri nets in later chapters. We also introduce the subclass of Petri nets. These subclasses enhance the analytical power but sacrifice some modeling capability. Best of all, readers will see that the knitting technique that will be introduced starting from chapter 3 through this book covers even more than asymmetric nets does and automatically reserves all those well behavior properties as well.

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書名: A New Approach to Petri Net Synthesis: The Knitting Technique

著（編、譯）者: 趙玉

出版者: 趙玉

電話: 02-29387694

地址: 台北市文山區指南路二段六四號政大資管系 出版年月: 民國九十九年五月

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