tailieunhanh - Petri nets applications Part 14

Tham khảo tài liệu 'petri nets applications part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 512 Petri Nets Applications Sequence-detectability implies the knowledge of all firing sequences of an IPN. In others words there is a function As Ak Q Mo oAb Q Mo X Q Mo L Q Mo where As w Q Mo ơw. The problem of determining whether or not a system is sequence-detectable has a high computational complexity. However the following definition provides conditions that reduce the computational complexity. Definition 12. An IPN given by Q Mo is event detectable if the firing of any transition tk e T at a marking Mk e R Q Mo can be uniquely determined through the information provided by the input symbol a X tk and the output signals q Mk where q C k is the column of q C corresponding to transition tk . Note that this definition implies that all events can be detected and distinguisable . their firing can be detected and distinguisable from each other after its occurrence and before another event occurs. Thus an event-detectable IPN system is also sequence-detectable. Event detectable has a structural characterization captured in the next lemma that can be tested in a polynomial time. Lemma 1 An IPN given by Q M o is event detectable if only if Vi e 1 2 3 . m C i 0 and Vj k e 1 2 3 . m such q C j q C k then X tj L tk . Proof. You can find the proof in Ramirez-Trevino et al. 2003 . Observe that an IPN given by Q Mo where it has transitions tk tj e T with L tk X tj and q C k ọC j 0 the firing of tk tj cannot be distinguisable in this case tk tj are called indistinguishable. Definition 13. An IPN given by Q Mo is marking-detectable if there is an integer k X such that Vw e A Q Mo it holds that the information provided by w and Q Mo suffices to uniquely determine the marking Mi reached by firing ơw. In other works there is a function Am A Q Mo AB Q Mo X Q Mo R Q Mo where Am w Q Mo Mi As w Q Mo ơw and M0 Mi. Example 4 Consider the IPN shown in figure where Pm pi P5 P6 and Tc ti ts . Its incidence matrix shown in the figure and its ouput function is the matrix of .