Liveness Enforcing Supervisory Policy Research Articles

On computing the supremal right-closed control invariant subset of a right-closed set of markings for an arbitrary petri net

A set of non-negative integral vectors is said to be right-closed if the presence of a vector in the set implies all term-wise larger vectors also belong to the set. A set of markings is control invariant with respect to a Petri Net (PN) structure if the firing of any uncontrollable transition at any marking in this set results in a new marking that is also in the set. Every right-closed set of markings has a unique supremal control invariant subset, which is the largest subset that is control invariant with respect to the PN structure. This subset is not necessarily right-closed. In this paper, we present an algorithm that computes the supremal right-closed control invariant subset of a right-closed of markings with respect to an arbitrary PN structure. This set plays a critical role in the synthesis of Liveness Enforcing Supervisory Policies (LESPs) for a class of PN structures, and consequently, the proposed algorithm plays a key role in the synthesis of LESPs for this class of PN structures.

Liveness enforcing supervisory policies tolerant to controllability failures for discrete-event systems modeled by Petri Nets

A Discrete Event System (DES) modeled by a Petri Net (PN) is live if it is possible to fire any transition, although not necessarily immediately, from any marking that is reachable from the initial marking. A Liveness Enforcing Supervisory Policy (LESP) for a PN enforces liveness by preventing the firing of a subset of transitions called the controllable transitions, which correspond to the preventable events in a DES.In this paper, we consider the existence and synthesis of LESPs for arbitrary PNs in the presence of faults, where a subset of controllable transitions become temporarily uncontrollable, for a finite number of event occurrences. Following the formal specification of the fault model, we present a necessary and sufficient condition for the existence of Fault-Tolerant LESPs (FT-LESPs) for arbitrary PNs. We show that, even when an LESP is given, the existence of an FT-LESP for an arbitrary PN is undecidable. We then identify a class of PNs for which the existence of FT-LESPs is decidable. We conclude with some suggestions for future research.

On Liveness Enforcing Supervisory Policies for Arbitrary Petri Nets

Neither the existence nor the nonexistence of a liveness enforcing supervisory policy (LESP) for an arbitrary Petri net (PN) is semidecidable. In an attempt to identify decidable instances, we explore the decidability of certain properties of the set of initial markings for which an LESP exists, and the decidability of the existence of a specific class of LESPs. We first prove that for an arbitrary PN structure, determining if there is an initial marking, or there are no initial markings, for which there is an LESP, is not semidecidable. Then, we characterize the class of PN structures for which the set of all initial markings for which an LESP exists is right-closed . We show that testing membership, or nonmembership, of an arbitrary PN in this class of PNs is not semidecidable. We then consider a restricted class of LESPs, called marking monotone LESPs (MM-LESPs). We show that the existence of an MM-LESP for an arbitrary PN is decidable.

On Invariant-Based Monitors That Enforce Liveness in a Class of Partially Controlled General Petri Nets

We consider a class of Petri Net structures where the existence of a liveness enforcing supervisory policy (LESP) for an initial marking implies there is a LESP for a larger initial marking. That is, the set of initial markings for which there is a LESP for any instance of this class is right-closed. If a transition is prevented from firing at a marking by a LESP, and all LESPs, irrespective of the implementation-paradigm that is chosen, prescribe the same control for the marking, then it is a minimally restrictive LESP. It is possible to synthesize the minimally restrictive LESP for any instance of this class that uses this right-closed set of markings. Alternately, one could consider invariant-based monitors for liveness enforcement in an instance of this class.

On a Sufficient Information Structure for Supervisory Policies That Enforce Liveness in a Class of General Petri Nets

A Petri net (PN) is said to be live if it is possible to fire any transition, although not immediately, from every reachable marking. A liveness enforcing supervisory policy (LESP) determines which controllable transition is to be prevented from firing at a marking, to ensure the supervised Petri net (PN) is live. In this paper, we restrict our attention to a class of general Petri nets (PN) structures, where the existence of an LESP for an instance initialized at a marking, implies the existence of an LESP when the same instance is initialized with a termwise-larger initial marking. We show that the minimally restrictive LESP for an instance N from this class is characterized by a collection of boolean formulae {Θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tc</sub> (N)} <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tc∈Tc</sub> , where Tc is the set of controllable transitions in the PN. The literals in Θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tc</sub> (N) are true if and only if the token-load of specific places meet a threshold. Consequently, appropriately placed threshold-sensors, which detect if the token-load of a place is greater than or equal to a predetermined threshold, provide sufficient information to implement the minimally restrictive LESP. The paper concludes with some directions for future research.

On Deciding the Existence of a Liveness Enforcing Supervisory Policy in a Class of Partially Controlled General Free-Choice Petri Nets

If there are transitions in a Petri net (PN) that cannot be prevented from firing by a supervisory policy, then we have a partially controlled PN. The existence of a liveness enforcing supervisory policy (LESP) in a partially controlled ordinary PN is undecidable. Consequently, there can be no algorithms that synthesize an LESP for an arbitrary ordinary (general) PN. In contrast, we identify a class of general free-choice PN (FCPN) structures, which strictly includes the class of ordinary FCPN structures, where the existence of an LESP in any marked member of the class is decidable.

On a Decidable Class of Partially Controlled Petri Nets With Liveness Enforcing Supervisory Policies

We identify a class of partially controlled Petri net (PN) structures, which is denoted by G, that strictly includes the class of partially controlled free-choice (FC) PN structures. We show that there is a supervisory policy that enforces liveness in an arbitrary instance N(m0), where N ∈ G, if and only if there is a similar policy for an FCPN that results when the construction procedure enunciated in this paper is executed with N and its controllable transition set as input. Since the existence of a supervisory policy in an arbitrary partially controlled FCPN is decidable, it follows that the existence of similar policies for any N(m0), where N ∈ G, is also decidable. Furthermore, when it exists, the minimally restrictive supervisory policy that enforces in a member of G is characterized by a right-closed set of markings.