![](bg8.png)
Finally, according to Corollary, the control law
which guarantees the meeting of both temporal
constraints is given by the following vector:
35
34 5
(1) (1)
() .
1. ( 1) ( 1) 1. ( 1)
xk xk
uk
xk xk xk
−⊕ −
⎛⎞
=
⎜⎟
−⊕ −⊕ −
⎝⎠
After comparison between the terms of each
component of the vector, the control law is
simplified to:
3
4
(1)
() .
(1)
xk
uk
xk
−
⎛⎞
=
⎜⎟
−
⎝⎠
This feedback can be interpreted by two places of
control connected to the timed event graph to
guarantee the respect of the temporal constraints.
The controlled graph is given in Figure 5.
Figure 5: Controlled timed event graph.
Remark 2:
The same example was treated with the method
developed in Min-Plus algebra (Amari and al.,
2005). It is worth noting that the synthesis of the
Max-Plus control is easier than that in the Min-Plus
algebra. In this case, there are not necessary
compute of power for the matrix.
7 CONCLUSION
We have developed a method for control synthesis
of timed event graphs subject to strict temporal
constraints. A generalization for timed event graphs
with multivariable control has been proposed in this
paper. This method is illustrated on an example. The
conditions (12) and (13) are shown here to be
sufficient conditions, we are investigating actually
the existence of necessary and sufficient conditions
for the synthesis of control laws which ensure the
meeting of the temporal constraints. We will
continue the comparison of this method with those
developed in (Lahaye et al., 2004) and (Atto and al.,
2006). We hope to apply this method for real
systems, notably for the verification and validation
of automated systems as well as telecommunication
processes and real-time software.
REFERENCES
Amari S., I. Demongodin and J.J. Loiseau, Sizing, cycle
time and plant control using dioid algebra,
Chapter 6
in Supply Chain Optimisation, Series Applied
Optimization, Springer. pp. 71-85, 2004.
Amari S., I. Demongodin and J.J. Loiseau, Control of
Linear Min-Plus Systems under Temporal
Constraints.
44
th
IEEE Conference on Decision and
Control and European Control Conference
, Seville,
2005.
Atto A.M, C. Martinez et S. Amari, Synthèse de
superviseur pour la commande des systèmes à
événements discrets soumis à des contraintes de temps
strictes. Soumis à la revue ARIMA, INRIA, ISSN
1638-5713, 2006.
Baccelli F., G. Cohen, G.J. Olsder, and J.-P. Quadrat,
Synchronization and Linearity:
An algebra for
Discrete Event Systems,
Wiley, 1992.
Berthomieu B. and M. Diaz, Modelling and verification of
time dependant systems using time Petri nets
, IEEE
Transactions on Software Engineering,
Vol. 17, No. 3,
pp. 259-273, 1991.
Bonhomme P., P Aygalin and S. Calvez, Systèmes à
contraintes de temps : Validation, Evaluation et
contrôle
, MSR, Toulouse, 2001.
Cofer D. D., Garg V.K., Control of event separation times
in discrete event systems.
In Proceedings 34th IEEE
Conference on Decision and Control
, pp. 2005 – 2010,
New Orleans, 1995.
Ghezzi C., D. Mandrioli, S. Morasca and M. Pezze, A
unified high-level Petri net formalism for time critical
systems,
IEEE Transactions on Software Engineering,
Vol. 17, No. 2, pp. 160-172, 1991.
Holloway L.E., B.H. Krogh and A. Giua, A survey of Petri
net methods for controlled discrete event systems,
Kluwer Academic Publishers pp.151-190, Boston,
1997.
Kim J.H, T.E Lee and H.Y Lee, Scheduling Analysis of
Time-Constrained Dual-Armed Cluster Tools,
IEEE
Transactions on Semiconductor Manufacturing, vol.
16, no. 3, august 2003.
Lahaye S., B. Cottenceau and A. Correïa, Commande des
graphes d’événements temporisés avec contraintes de
temps critique,
CIFA’04, Tunisie, 2004.
Moody J., K. Yamalidou, M. Lemmon and P. Antsaklis,
Feedback control of Petri nets based on place
invariants,
Automatica, vol 32, pp. 15-28, 1996.
t
3
1
3
t
4
tu
0
0
t
1
t
2
1
tu
0
0
ICINCO 2006 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
98