Handbook of Industrial Automation - Richard L. Shell and Ernest L. Hall Part 8

Tham khảo tài liệu 'handbook of industrial automation - richard l. shell and ernest l. hall part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Figure 6 A Petri net model. A modified version of incidence matrix A is formed as -1 0 1 -1 0 0 I 1 1 0 I 0 0 0 0 0 0 1 0 0 0 0 J -1 0 0 1 -1 0 0 I 0 0 10 0 0 -1 0 0 1 -1 0 I 0 0 1 I 0 -1 I 0 0 0 10 0 0 0 0 1 0 0 0 0 0 1 We will add rows to eliminate a nonzero element in each row of A. Specifically 1 adding the third row to the first row 2 the fourth row to the second row 3 adding the fourth fifth and sixth rows to the third row. Determine if mf 3 0 I 7 is reachable from initial marking mữ 0 3 l r. Solution 1. Check the rank of A r rank J 1. 2. Partition A as An 1 J12 1 21 -IO P andA22 1Ơ T. 3. Use Eq. 16 to determine Bp. Bp 1 0 -1 1 0 0 1 1 1 0 0 0 1 4. Check where Am mf m0 3 -3 0 r. 1 1 0 3 o 0 0 1 -3 0 0 According to Theorem 1 mf can be reachable from mQ. The solution to Eq. 9 may become more complicated when one includes the constraint that the elements of X must be nonnegative integers. In this case an efficient algorithm 9 for computing the invariant of Petri net is proposed. The basic idea of the method can be illustrated by getting the minimal set of P-invar-iants through the following example. Example 7. The incidence matrix of the Petri net in Fig. 6 is -1 1 0 1 0 -1 0 1 -1 -1 -1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1 1 -1 0 0 1 0 -1 0 0 -1 0 1 -1 0 0 -1 0 0 1 1 0 -1 0 0 I 10 1 0 I 0 10 0 I 0 0 1 0 I 0 0 0 1 I 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 -1 I 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 -1 0 0 1 0 0 1 0 0 0 0 1 0 -1 0 1 0 0 0 1 0 0 -1 -1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 -1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 -1 0 1 0 0 0 1 0 0 -1 -1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 -1 1 0 0 0 0 0 1 The three P-invariants are Vi 10100 0 r V 2 0 1 0 1 0 0 7 . and x3 0 0 1 1 1 I 7. They are actually the first three rows of the modified identity matrix. This is because their associated three rows in the final version of modified A have all zero elements. Invariant Analysis of a Pure Petri Net The .