_{1}

A discrete event system is a dynamical system whose state evolves in time by the occurrence of events at possibly irregular time intervals. Timed Petri nets are a graphical and mathematical modeling tool applicable to discrete event systems in order to represent its states evolution where the timing at which the state changes is taken into consideration. One of the most important performance issues to be considered in a discrete event system is its stability. Lyapunov theory provides the required tools needed to aboard the stability and stabilization problems for discrete event systems modeled with timed Petri nets whose mathematical model is given in terms of difference equations. By proving stability one guarantees a bound on the discrete event systems state dynamics. When the system is unstable, a sufficient condition to stabilize the system is given. It is shown that it is possible to restrict the discrete event systems state space in such a way that boundedness is achieved. However, the restriction is not numerically precisely known. This inconvenience is overcome by considering a specific recurrence equation, in the max-plus algebra, which is assigned to the timed Petri net graphical model.

A discrete event system, is a dynamical system whose state evolves in time by the occurrence of events at possibly irregular time intervals. Timed Petri nets are a graphical and mathematical modeling tool applicable to discrete event systems in order to represent its states evolution where the timing at which the state changes is taken into consideration Timed Petri nets are known to be useful for analyzing the systems properties in addition of being a paradigm for describing and studying information processing systems, where the timing at which the state changes is taken into consideration. For a detailed discussion of Petri net theory see [

NOTATION:

where

Definition 1 The

Definition 2 The system (1) is said to be practically stable, if given

Definition 3 A continuous function

Consider a vector Lyapunov function

Theorem 4 Let

for

that:

imply the practical stability properties of system (1).

Corollary 5 In Theorem (4): If

Definition 6 A Petri net is a 5-tuple,

Definition 7 The clock structure associated with a place

The positive number

Definition 8 A timed Petri net is a 6-tuple

Notice that if

Let

Let

corresponding firing vector is

where if at step

Let

Proposition 9 Let

Moreover,

Lemma 10 Let suppose that Proposition (9) holds then,

Remark 11 Notice that since the state space of a TPN is contained in the state space of the same now not timed PN, stability of PN implies stability of the TPN.

Lyapunov StabilizationDefinition 12 Let

Proposition 13 Let

Remark 14 By fixing a particular

NOTATION:

Definition 15 The set

Definition 16 A semiring is a nonempty set

Theorem 17 The max-plus algebra

Let

The product of matrices

and

Theorem 18 The 5-tuple

Definition 19 Let

Definition 20 A matrix

Definition 21 Let

Let

Definition 22 A path from node

Let us denote by

Theorem 23 Let

Definition 24 Let

Lemma 25 Let

Definition 26 Let

Remark 27 In this paper irreducible matrices are just considered. It is possible to treat the reducible case by transforming it into its normal form and computing its generalized eigenmode see [

Definition 28 Let

Let

average circuit weight. Notice that since

Definition 29 A circuit

Theorem 30 If

Theorem 31 Let

weight less than or equal to

Definition 32 Let

Theorem 33 The Mth order recurrence equation, given by equation

With any timed event Petri net, matrices

state of the system, satisfies the Mth order recurrence equation:

Definition 34 A TPN is said to be stable if all the transitions fire with the same proportion i.e., if there exists

This means that in order to obtain a stable

Lemma 35 Consider the recurrence relation

Proof. Let

Now starting with an unstable

The main objective of the proposal is to make it knowledgeable to large audiences. This paper gives a complete and precise solution to the stabilization problem for discrete event systems modeled with timed Petri nets combining Lyapunov theory with max-plus algebra. The presented methodology results to be innovative.

Zvi Retchkiman Konigsberg, (2015) The Relation between the Stabilization Problem for Discrete Event Systems Modeled with Timed Petri Nets via Lyapunov Methods and Max-Plus Algebra. Journal of Applied Mathematics and Physics,03,839-845. doi: 10.4236/jamp.2015.37104