Structure of weakly 2-dependent siphons

International Journal of Control

2014

Li and Zhou propose simpler Petri net controllers based on the concept of elementary siphons (generally much smaller than the set of all strict minimum siphons (SMSs) in large Petri nets) to minimise the addition of control places. SMSs can be divided into two groups: elementary and dependant; characteristic T-vectors of the latter are linear combinations of that of the former. A T-vector η is associated with each siphon S such that η(i) is the number of tokens gained in or lost from S by firing transition t i once. A dependent siphon S 0 strongly depends on elementary siphons S 1 , S 2 , . . . , S k if η 0 = a 1 η 1 + a 2 η 2 + · · · + a k η k with all a i (i = 1, 2, 3, . . . , k) positive. S 0 is a weakly dependent siphon if some a i is negative. The T-vectors (resp. number) for elementary siphons are mutually independent (linear to the size of the net). In an earlier paper, we show that there exists a third siphon S 3 such that η β = η 1 + η 2 -η 3 . This equation (called η relationship) plays an important role for optimal control of weakly dependent siphons. However, it assumes that all above S span between exactly two processes. For a well-known benchmark, however, most dependent siphons span more than two processes. This paper improves by removing this restriction and shows that η β = η 1 + η 2 -η 3 holds as long as S 1 ∩S 2 is another emptiable siphon.

Section: Resultsmentioning

confidence: 85%

Section: Resultsmentioning

confidence: 89%

“…We prove in Chao and Chen (2013) that η β = η 1 + η 2 -η 3 . In Figure 2, c 1 ∩c 2 = [p 15 t b p 14 ] is not a single resource place, where η 1 (t 15 ) = η 2 (t 15 ) = 1.…”

Section: Generalised η Relationshipmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

“…Thus, must be in a third basic circuit c 3 as stated in Chao and Chen (2013) from which to synthesise S 3 so that η β (t 15 ) = η 1 (t 15 ) + η 2 (t 15 ) − η 3 (t 15 ) = 1.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Improvement on ‘structure of weakly 2-dependent siphons’

International Journal of Control

2014

Controllability of weakly dependent siphons under elementary-siphon control

Transactions of the Institute of Measurement and Control

2015

Monitors are added to problematic siphons to avoid deadlocks. Li and Zhou (2004, 2006a,b, 2008a add monitors to elementary siphons only while controlling the rest-dependent siphons to save on costs. After failing a marking linear inequality (MLI) test, Li and Zhou perform a linear integer programming (LIP) test (NP-hard). We proposed a new MLI test earlier to avoid the LIP and extended it to systems of simple sequential processes with general resource requirements (S 3 PGR 2 ) for strongly dependent siphons (SDSs). The control policy for weakly dependent siphons (WDSs) is rather conservative due to some negative terms in the MLI. This paper shows that WDS and SDS have the same controllability (i.e. MLI). As a result, the control for WDS need no longer be that conservative. We also develop an optimization (by redundancy elimination) of the computation required for the LIP test to ensure deadlock prevention of systems of simple sequential processes with resources (S 3 PR). A favourable result for this policy is that any n-dependent (n . 2) WDS (similar to SDS) needs no monitor and hence the complexity for synthesizing a controller becomes polynomial. Application to a slight variant of a well-known benchmark is illustrated.

Controllability of Weakly Dependent Siphons under Elementary-Siphon Control

Chen

Huang

2013

AMM

Monitors are added upon problematic siphons to avoid deadlocks. Li and Zhou add monitors to elementary siphons only while controlling the rest dependent siphons to save cost. After failing a Marking Linear Inequality (MLI) test, they perform a Linear Integer Programming (LIP) test (NP-hard). We proposed a new MLI test earlier to avoid the LIP and extended it to S3PGR2 (systems of simple sequential processes with general resource requirements) for strongly dependent siphons (SDS). The control policy for weakly dependent siphons (WDS) is rather conservative due to some negative terms in the MLI. This paper shows that WDS and SDS have the same controllability (i.e., MLI). As a result, the control for WDS needs no longer be that conservative. We also develop an optimization (by redundancy elimination) of the computation required for LIP test to ensure deadlock prevention of S3PR (Systems of Simple Sequential Processes with Resources). A favorable result for this policy is that any n-dependent (n>2) WDS (similar to SDS) needs no monitor and hence the complexity for synthesizing a controller becomes polynomial.