Optimal control policy for a well-known S³PR

Abstract

It has been a hot research topic to compare the effectiveness of new control policies by testing against a wellknown S 3 PR model. So far, only the control policy by Piroddi et al. may reach the optimal number of states among all approaches for a well-known benchmark using a siphon-based approach. The resulting model is a generalized Petri net since some control arcs are weighted, which complicates the system. The only improvement that can be made is to reduce the number of control arcs (by 3), and the number of weighted control arcs (by 9) as we report in this paper. Also the token count is reduced. This is achieved by replacing two monitors with weighted arcs by two new monitors without weighted arcs. INA (Integrated Net Analyzer) analysis indicates that the resulting controlled model is live and reaches the same 21581 states by Piroddi et al. We develop a formal theory for explaining the cause of state losses and providing the foundation for the above improvement model.

An elementary siphon-based deadlock control algorithm with maximally reachable number to cope with deadlock problems in ordinary Petri nets

“…From the existing literature, we know that those policies to make ESs explicitly controlled and DSs implicitly controlled, respectively, can reduce the SC of finally live controlled Petri net systems to some degree. However, this method of adding a CP and CT to each solved SMS 40 without considering different controllability between ESs and DSs may result in a relatively complex structural (N Ã , M Ã ) with MRN. Partially motivated by the advantages of ESs to design deadlock control policies, 14,[22][23][24] an IPP of P-invariants of an extended net system (N 0 , M 0 ) is constructed to test controllability of all DSs in (N 0 , M 0 ) formed by all ESs which are controlled by the addition of CPs and CTs.…”

“…As stated in Introduction, Chao 40 proposes a novel idea to add a CP and CT but not a CP to each solved SMS in an S 3 PR system with deadlocks, resulting in a live controlled Petri net system with MRN. For each S i 2 P and the corresponding ½S i , the method of adding corresponding CP (denoted as V S i ) with M 0 (V S i ) = 0 and CT (denoted as t S i ) for S i is briefly described as follows:…”

“…From the above mentioned, adding a CP and CT to each solved SMS 40 can obtain a live controlled Petri net system (N Ã , M Ã ) with MRN. On the basis of Definitions 1-6, SMSs are classified into ESs and DSs.…”

“…Li and Zhou-policy only adds the CPs to all ESs and some DSs that cannot meet the controllability and then achieves a liveness-enforcing supervisor, whose SC is enormously depressed compared with that of E-policy. Subsequently, Li and Zhou 9 and Chao and Chen 14 and Chao 40 propose the further research results including the novel methods to compute ESs, controllability of DSs (or compound siphons), and the liveness conditions associated with the controlled ESs and DSs for S 3 PR and S 3 PGR (systems of simple sequential processes with general resources requirement), which enrich the theory of elementary siphon and its applications. But BP and CC are not ideal for the livenessenforcing supervisors synthesized from the controlled ESs and DSs.…”

“…The method of adding CPs and CTs (resp. CP and CT) 40 is first reviewed in section ''A two-stage deadlock control policy.'' Accordingly, an IPP to test the controllability for all DSs in N 0 and a two-stage DCA with MRN are developed in the same section.…”