The Design of Manufacturing Systems Part 15 pptx

© 2001 by CRC Press LLC Reducing Models The greater the number of D-structures in an , the larger the opportunity for deadlock and the more complex the design of the deadlock avoidance controller. To lower the complexity of controller synthesis, we first present the following method for reducing models. Definition 3 Let G ϭ (P ʜ R, T, F, m 0 ) be an and r be a resource place. Let H(r) ϭ {p ʦ P ͉ R(p) ϭ r}. The r-reduced G is a Petri net G(r) constructed from G by the following steps. 1. Remove the place r from G and all arcs which are incidental to or from r. 2. Repeat the following steps for each place p ʦ H(r). Select and remove p ʦ H(r) and all transitions in from G. For every pair of transitions (t 1 , t 2 ) of in G: • if in G, then add a new transition, denoted by t 1 ϩ t 2 , and some arcs which are incidental from or to t 1 ϩ t 2 so that ϭ گ{p, r} and ϭ گ{p, r}. •if ϭ ϭ {r 1 } in G, then add a new transition, denoted by t 1 ϩ t 2 , and some arcs which are incidental from or to t 1 ϩ t 2 so that ϭ گ{p, r, r 1 } and ϭ گ{p, r, r 1 }. We will call t 1 ϩ t 2 ␶ -transition, and let T ␶ denote the set of all ␶ -transitions in G(r). Notice that in the second situation above ϭ ϭ 0/ and ϭ ϭ r 1 . and so we can consider and as the same operation places requiring the same resource r 1 . In this manner, the r-reduced model may also be considered an model, and any model can be reduced by any resource place. Let G be an and r 1 and r 2 be two resource places. For the r 1 -reduced G(r 1 ), we can do an r 2 - reduced procedure for G(r 1 ) and obtain an {r 1 , r 2 }-reduced G(r 1 , r 2 ) in the same way. G(r 1 , r 2 ) is an model. In general, for any set of resource places RЈ, we can construct RЈ-reduced , denoted by G(RЈ). Example 4 Consider the G shown in Figure 9.1. The M 2 -reduced G(M 2 ) is given in Figure 9.4. The firing of ␶ -transition t 2 ϩ t 3 requires no resources; ϭ ϭ M 1 . The oper- ations and can be considered the same operation as in G(M 2 ), and G(M 2 ) is an . G contains at least ten D-structures and G(M 2 ) has only three D-structures. Let R K be the set of key resource places and G(R K ) ϭ (P Ј ʜ R Ј , T Ј , F Ј , m Ј 0 ) be the R K -reduced . Then G(R K ) is an model in which there are no key resources. The set of transitions in G(R K ) can be divided into two parts, T 0 and T 1 , where T 0 is a set of ␶ -transitions and every ␶ -transition t 1 ϩ t 2 ϩ … ϩ t k in T 0 corresponds to a maximal key path t 1 p 1 t 2 , p 2 … p kϪ1 t k , K Ն 2, in G. T 1 ϭ TЈگT 0 . Let T 2 denote the set of transitions of G which are in some key path. Then T ϭ T 1 ʜ T 2 . The complexity for reducing an by the set of key resources R K is linear with |{p ʦ P | R(p) ʦ R K }|. Since {p ʦ P | R(p) ʦ R K } is finite, the procedure for reducing an model is efficient, and any admits this reduction. Optimal Deadlock Avoidance Petri Net Controllers for a Class of s Let G be a marked and R K be the set of key resource places. Then G(R K ) is an which contains no key resources. For such a special class of s, we can first present the following deadlock avoidance Petri net controller: Definition 4 Let G ϭ (P ʜ R, T, F, m 0 ) be a marked , R K ϭ 0/. A controller for G is a marked Petri net defined by , R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN p • p • ʜ p • p • ϫ t 1 r() t r() 2  t 1 t 2 ϩ() • t • 1 t • 2 ʜ() t 1 t 2 ϩ() • t 1 • t 2 • ʜ t 1 r() t r() 2 t 1 t 2 ϩ() • t • 1 t 2 ʜ() t 1 t 2 ϩ() • t 1 • t 2 • ʜ( ) t 1 t 2 ϩ() r() t 1 t 2 ϩ() r() Rt 1 t 2 ϩ() p() () Rt 1 t 2 ϩ() p() () t 1 t 2 ϩ() p() t 1 t 2 ϩ() p( ) R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN Rt 2 t 3 ϩ() p() ()Rt 2 t 3 ϩ() p() () t 2 t 3 ϩ() p() t 2 t 3 ϩ() p() R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN CЈ P c TF c m 0 c ,, ,()ϭ . Models The greater the number of D-structures in an , the larger the opportunity for deadlock and the more complex the design of the deadlock avoidance controller. To lower the complexity of controller synthesis,. D-structures. Let R K be the set of key resource places and G(R K ) ϭ (P Ј ʜ R Ј , T Ј , F Ј , m Ј 0 ) be the R K -reduced . Then G(R K ) is an model in which there are no key resources. The set of transitions. in G. T 1 ϭ TЈگT 0 . Let T 2 denote the set of transitions of G which are in some key path. Then T ϭ T 1 ʜ T 2 . The complexity for reducing an by the set of key resources R K is linear with