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## Timed event graphs with multipliers and homogeneous min-plus systems.

Cohen, G.;
Gaubert, S.;
Quadrat, J.P.
In:
*IEEE Trans. on Automatic Control, Vol. 43, No. 9*, pages 1296-1302.
1998.

Abstract:
The authors study fluid analogues of a subclass of Petri nets, called
fluid timed-event graphs with multipliers, which are a tinted extension of
weighted T-systems studied in the Petri net literature. These event graphs
can be studied naturally, with a new algebra, analogous to the min-plus
algebra, but defined on piecewise linear concave Increasing functions,
endowed with the pointwise minimum as addition and the composition of
functions as multiplication. A subclass of dynamical systems in this
algebra, which have a property of homogeneity, can be reduced to standard
min-plus linear systems after a change of counting units. The authors give
a necessary and sufficient condition under which a fluid timed-event graph
with multipliers can be reduced to a fluid timed-event graph without
multipliers. In the fluid case, this class corresponds to the so-called
expansible timed-event graphs with multipliers of Munier, or to
conservative weighted T-systems. The change of variable is called here a
potential. Its restriction to the transitions nodes of the event graph is
a T-semiflow.

Keywords:
discrete-event systems, dynamic programming, max-plus algebra, robotics,
timed Petri nets, timed event graphs, weighted T-systems.

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