ON HYBRID PETRI NETS 15
T = {T
1
, T
2
,..., T
m
} is a finite, not empty, set of transitions; P ∩
T = ∅, i.e. The sets P and T are disjointed;
h: P ∩ T → {D,C}, called “hybrid function,” indicates for every node whether it is
a discrete node (sets P
D
and T
D
) or a continuous node (sets P
C
and T
C
);
Pre: P × T → R
+
or N, is the input incidence mapping;
4
Post:
P × T → R
+
or N, is the output incidence mapping; m
0
: P →
R
+
or N is the initial marking.
In the definition of Pre, Post and m
0
, N corresponds to the case where
P
i
∈ P
D
, and R
+
corresponds to the case where P
i
∈ P
C
.
Pre and Post functions must meet the following criterion: if P
i
and T
j
are a place and
a transition such that P
i
∈ P
D
and T
j
∈ T
C
then Pre(P
i
, T
j
) = Post(P
i
, T
j
) must be verified.
As usual,
◦
P
i
and P
i
◦
denote respectively the sets of input transitions and the set of
output transitions of place P
i
;
◦
T
j
and T
j
◦
denote respectively the sets of input places and
the set of output places of transition T
j
. To abbreviate, D-place, D-transition, C-place, and
C-transition, may stand for discrete place, discrete transition, continuous place, and
continuous transition, respectively.
In a hybrid PN, the characteristic vector s of a sequence S is a vector for which each
component is either an integer corresponding to the number of firings of a D-transition or
a non-negative real number corresponding to a firing quantity of a C-transition. A
marking m can be deduced from a marking m
0
due to a sequence S, using the
fundamental relation:
m = m
0
+ W · s. (1
0
)
InEquation(1),allthecomponentsofthevectorsareintegers,whilein(1
0
)thecomponents of s
are either integer or non-negative real numbers: this is the only difference between both
equations. Then, we have the same properties as stated in Remark 2, i.e., the results for P-
invariants and T-invariants are similar for a hybrid PN and for a discrete PN.
2.2.2. A hybrid Petri net allows modeling of ...
Informally, there are two parts in a hybrid PN, a discrete part (containing P
D
and T
D
) and
a continuous part (containing P
C
and T
C
), and these parts are interconnected thanks to
arcs linking a discrete node (place or transition) to a continuous node (transition or
place). In some cases, one part can influence the behavior of the other part without
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