Geert Stremersch

Structuring Acyclic Petri Nets for Reachability Analysis and Control

The incidence matrices—from places to transitions and vice versa—of an acyclic Petri net can obtain a block-triangular structure by reordering their rows and columns. This allows the efficient solution of some reachability problems for acyclic Petri nets. This result is further used in supervisory control of Petri nets; supervisors for Petri nets with uncontrollable transitions are constructed by extending the method of Yamalidou et al. (1996) to Petri nets where transitions can be executed simultaneously. A large class of Petri nets with uncontrollable transitions is given for which the maximally permissive supervisor can be realized by a Petri net. The original specification is algorithmically transformed—by using the results for acyclic Petri nets—into a new specification to take the presence of uncontrollable transitions into account. The supervisor is obtained by simple matrix multiplications and no linear integer programs need to be solved. Furthermore, a class of Petri nets is given for which the supervisor can be realized by extending the enabling rule with OR-logic.

Decomposition of the supervisory control problem for Petri nets under preservation of maximal permissiveness

Decomposing the design of supervisory control laws for Petri nets is an efficient way to tackle its complexity. We consider legal sets which are the union of two sets. In general, this can make the control law obtained via decomposition too restrictive, i.e., it disables more transitions than necessary. Generalizing existing results, we give structural conditions under which the control law via decomposition is maximally permissive.

Enforcing k-safeness in controlled state machines

We design a supervisor which enforces k-safeness in state machines, i.e. the marking of every place is not allowed to be greater than k /spl epsiv/ N, and which is itself a Petri net. This is done by extending the control design method based on invariants to Petri nets which contain uncontrollable transitions. We show that this supervisor is maximally permissive-disables as few transitions as possible-and is minimal-contains as few control places as possible. Finally, we show that the design of the supervisor can be done using min-plus algebra.

Reduction of the supervisory control problem for Petri nets

The authors prove a reduction theorem for the supervisory control problem for general Petri nets with general legal sets. To design control laws guaranteeing that the marking stays within the legal set, it suffices to consider a sub-Petri net of the full model. This extends existing design algorithms, allows to prove an important property of maximally permissive control laws and limits the number of events which need to be observed.

Linear Algebraic Design of Supervisors for Partially Observed Petri Nets

Abstract The design of supervisors for Petri nets containing both uncontrollable and unobservable transitions is studied. The control goal is that the marking always satisfies a linear inequality, defining a so-called legal set. Supervisors are designed by using linear programming techniques. One only has to find the minimal —w.r.t. the componentwise partial order— vertices of a polyhedron. Moreover, the presented method allows to make an approximation of the worst-case uncontrollable behaviour of the original Petri net without doing any reachability analysis.

Controlled Petri nets and general legal sets

Proves a reduction theorem for the supervisory control of general controlled Petri nets, with general legal sets. The reduction theorem shows that in order to design a maximally permissive control law guaranteeing that the marking always remains in the legal set, it is sufficient to consider a sub-Petri net of the full model. This extends the design algorithms which were previously known for special classes of Petri nets, and for special classes of legal sets. The reduction theorem allows us to prove a useful property of maximally permissive control laws, and to limit the number of events which must be observed.

On the influencing net and forbidden state control of timed Petri nets with forced transitions

We discuss forbidden state feedback control design for real-time discrete event systems modeled by timed Petri nets. The firing time of a state-enabled transition lies in an interval which can be modified for controllable transitions. Once the upper bound is reached, the transition, controllable or uncontrollable, is forced to fire. We construct necessary and sufficient conditions for the control to satisfy forbidden state control objectives for the case of a timed marked graph. From these equations we derive what the influencing net looks like.

Discrete Event Systems

a) Sometimes, even simple grammars can produce tricky languages. We can interpret the 1s and 2s of the second production rule as opening and closing brackets. Hence, L(G) consists of all correct bracket terms where at least one 0 must be in each bracket. Choose w = 102 ∈ L(G). Let w = xyz with |xy| ≤ p and |y| ≥ 1 (pumping lemma). Because of |xy| ≤ p, xy can only consist of 1s. According to the pumping lemma, we should have xyz ∈ L for all i ≥ 0. However, by choosing i = 0 we delete at least one 1 and get a word w′ = 1p−|y|02p with |y| ≥ 1. w′ is not in L since it has fewer 1s than 2s. This means that w is not pumpable and hence, L(G) is not regular.

Forbidden state control synthesis for timed Petri net models

Complex, computer controlled plants can be analyzed efficiently by reducing the very large state space to a finite state space, using abstraction. At the same time the model can be decomposed in smaller components. Automata can be used as models for — components of — such discrete event systems. Proper behaviour of the system means that the global state of the system never reaches forbidden subsets, or equivalently, that certain forbidden sequences of transitions between states never occur. Ramadge and Wonham [6] developed a framework for control of discrete event systems. The state evolution can be constrained by blocking some controllable transitions. For a class of untimed Petri nets Holloway and Krogh [4] developed an efficient algorithm for synthesizing maximally permissive control laws, guaranteeing that the state never reaches some forbidden set. It turns out that this maximally permissive control law only depends on the marking of places in a subnet of the Petri net, and that control action is only required at transitions at the boundary of this same subnet. This subnet is called the influencing net [1, 4, 7].

Continuous Versus Discrete Events

When a legal set A is described by a finite disjunction of linear inequalities and its influencing net A A uc is acyclic, the problem of deciding whether a given state belongs to A can be written as an integer linear programming problem (see (5.32) on page 126). In this chapter we construct a closed-form solution to the corresponding linear programming problem (take δ ∈ ℚ + #ū and z ∈ ℚ in (5.32)). We do this by introducing continuous Petri nets in Section 1. The difference with the Petri net definition of Chapter 1 is that in the enabling condition (1.3) the transition vectors δ can belong to ℚ + m \ {0} and are not restricted to Δ = ℕ m \ {0}. In Section 2, this approach allows us, by using reachability results for acyclic Petri nets, to construct a subset of A. With this subset as the control goal, permissive control laws can be constructed.