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Petri Net Part 16

Tham khảo tài liệu 'petri net part 16', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Using Transition Invariants for Reachability Analysis of Petri Nets 441 computed minimal singular T-invariants can be combined with non-complementary T-invariants of group 13 to produce new non-minimal singular T-invariants. Consider a linearly-combined T-invariant F fl f2 . fm f i x .i bF 16 with rational coefficients kj where Fj are minimal-support T-invariants of groups 13 14 and 15 and w is the number of elements in the three groups. In agreement with Corollary 1 we are looking only for those combined T-invariants F which yield fm 1 1. Thus the following constraint must hold for each linear combination F in 16 fm 1 EJ 1 kjfj m 1 1. 17 With kj 0 the product kjFj in 16 can be considered as a contribution of firings of transitions of T-invariant Fj to firings of transitions of the combined T-invariant F. On the other hand a negative coefficient kj in 16 may be interpreted as a reverse or backward firing of transitions corresponding to T-invariant Fj and this is not legal in the normal semantics of Petri nets. Thus for T-invariants of groups 14 and 15 taking into account 17 their coefficients kj must be in the following range 0 kj 1. 18 That is for groups 14 and 15 in which fj m 1 1 to satisfy 17 the following inequality must hold E kj - 1. 19 However coefficients kj for T-invariants of group 13 in 16 may have arbitrary nonnegative values without affecting the constraint 17 . As a particular case these T-invariants can be combined in 16 with coefficients kj 1. The case when T-invariants of group 13 can be included into combination 16 with arbitrary large coefficients is considered in Section 6. The linearly-combined T-invariants 16 with the constraints 17 18 and 19 are called minimal singular T-invariants of the complemented Petri net. As a subset they include all minimal-support T-invariants of group 14 . Minimal singular T-invariants of the complemented Petri net can be computed in the following way. Rewrite 16 as a system of linear algebraic equations K FT 20 .