The Relation between the Stabilization Problem for Discrete Event Systems Modeled with Timed Petri Nets via Lyapunov Methods and Max-Plus Algebra

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.


Introduction
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 [1] and the references quoted therein.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 [2].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.This paper proposes a methodology consisting in combining Lyapunov theory with max-plus algebra to give a precise solution to the stabilization problem for discrete event systems modeled with timed Petri nets.The presented methodology results to be innovative and it is not, in general, known.The main objective of the paper is to spread its results along large audiences.The paper is organized as follows.In Section 2, Lyapunov theory for discrete event systems modeled with Petri nets is given.Section 3 presents max-plus algebra and max-plus recurrence equations for timed event Petri nets.Section 4 considers the solution to the stabilization problem for discrete event systems modeled with timed Petri nets.Finally, the paper ends with some conclusions.

Definition 2
The system (1) is said to be practically stable, if given ( , ) and it is strictly increasing.
denote an n m × matrix of integers (the incidence matrix) where denote a firing vector where if j t T ∈ is fired then, its corresponding firing vector is = [0,..., 0,1, 0,..., 0] T k u with the one in the th j position in the vector and zeros everywhere else.The nonlinear difference matrix equation describing the dynamical behavior represented by a PN is: where if at step k , < ( )  : and consider the matrix difference equation which describes the dynamical behavior of the discrete event system modeled by a PN , see (7).Proposition 9 Let PN be a Petri net.PN is uniform practical stable if there exists a Φ strictly positive m vector such that Moreover, PN is uniform practical asymptotic stable if the following equation holds = ( ), 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 Stabilization
Definition 12 Let PN be a Petri net.PN is said to be stabilizable if there exists a firing transition sequence with transition count vector u such that system (7) remains bounded.
Proposition 13 Let PN be a Petri net.PN is stabilizable if there exists a firing transition sequence with transition count vector u such that the following equation holds Remark 14 By fixing a particular u , which satisfies (11), the state space is restricted to those markings that are finite.

Basic Definitions
⊕ ⊗   has the algebraic structure of a commutative and idempotent semiring.

Matrices and Graphs
Let n n max ×  be the set of n n × matrices with coefficients in max  with the following operations: The sum of matrices , The product of matrices , denote the matrix with all its elements equal to  and denote by n n max E × ∈  the matrix which has its diagonal elements equal to e and all the other elements equal to .
 Then, the following result can be stated.
Theorem 18 The 5-tuple max = ( , , , , )  ( , )  w i j ∈  is associated with any arc ( , )  i j ∈  .Let be any matrix, a graph ( ) A  , called the communication graph of A , can be associated as follows.Define ( ) = N A n and a pair ( , )  i j n n ∈ × will be a member of ( ) , where ( ) A  denotes the set of arcs of ( ) A  .Definition 22 A path from node i to node j is a sequence of arcs  , ; ) P i j m the set of all paths from node i to node j of length 1 m ≥ and for any arc ( , ) ( ) i j A ∈  let its weight be given by ij a then the weight of a path ( , ; ) p P i j m ∈ denoted by | | w p is defined to be the sum of the weights of all the arcs that belong to the path.The average weight of a path p is given by p .Given two paths, as for example, = (( , ), ( , )) p i i i i and = (( , ), (( , ) = (( , ), ( , ), ( , ), ( , )) p q i i i i i i i i  . The communication graph ( ) A  and powers of matrix A are closely related as it is shown in the next theorem.
 in the case when ( , ; ) P i j k is empty i.e., no path of length k from node i to node j exists in ( ) A  .

Definition 24
gives the maximal weight of any path from j to i .If in addition one wants to add the possibility of staying at a node then one must include matrix E in the definition of matrix A + giving rise to its Kleene star representation defined by: has average circuit weight less than or equal to  .Then it holds that:   be a graph and , i j ∈  , node j is reachable from node i , denoted as i j  , if there exists a path from i to j .A graph G is said to be strongly connected if is called irreducible if its communication graph is strongly connected, when this is not the case matrix A is called reducible.
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 [5].

Spectral Theory and Linear Equations
Definition 28 Let G a are negative then, the solution is unique.

Max-Plus Recurrence Equations for Timed Event Petri Nets
is called an Mth order recurrence equation. ) ⊕ can be expressed as: ˆ( 1) = ( ); 0 x k A x k k + ⊗ ≥ , which is known as the standard autonomous equation.

The Solution to the Stability Problem for Discrete Event Dynamical Systems Modeled with Timed Petri Nets
Definition 34 A TPN is said to be stable if all the transitions fire with the same proportion i.e., if there exists q ∈  such that This means that in order to obtain a stable TPN all the transitions have to be fired q times.It will be desirable to be more precise and know exactly how many times.The answer to this question is given next.Now starting with an unstable TPN , collecting the results given by: proposition ( 13), what has just been discussed about recurrence equations for TPN at the end of subsection (3.3) and the previous lemma (35) plus theorem (30), the solution to the problem is obtained.

Conclusion
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.

. Definition 1
The n vector valued function µ is called an eigenvalue and v an eigenvector.average circuit weight.Notice that since ( ) A  is a finite set, the maximum is attained (which is always the case when matrix A is irreducible).In case( ) = A ∅  define = λ  .Definition 29 A circuit ( ) p G A ∈is said to be critical if its average weight is maximal.The critical graph of A , denoted by ( graph consisting of those nodes and arcs that belong to critical circuits in ( ) irreducible, then there exists one and only one finite eigenvalue (with possible several eigenvectors).This eigenvalue is equal to the maximal average weight of circuits in ( ) .If the communication graph ( ) G A has maximal average circuit weight less than or equal to e , then * = x A b ⊗ solves the equation = ( ) x A x b ⊗ ⊕ .Moreover, if the circuit weights in ( ) k x k , called the state of the system, satisfies the Mth order recurrence equation: hypothesis of theorem (33) are satisfied, and setting ˆ( ) = ( ( ), ( 1),..., ( arbitrary.A an irreducible matrix and λ ∈  its eigenvalue then, Let v be an eigenvector of A such that 0 =x v then, It is assumed that at each time k there exists at least one transition to fire.If a transition is enabled then, it can fire.If an enabled transition W p t α (or ( , ) =W t p β ) then, this is often represented graphically by α , ( β ) arcs from p to t ( t to p ) each with no numeric label.
can be reached from some other marking M and, if we fire some sequence of d transitions with k u is utilized in the difference equation to generate the next step.Notice that if M