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For a run
π
=
M
,
(
X
)
X
∈
Var
of H and t
∈
Time let
π
(
t
)
denote the state
M
(
t
)
X
(
t
)
π
(
t
)=
{
M
→
}∪{
X
→
|
X
∈
Var
}
.
assigning to the mode observable M and the all variables X
∈
Var the values in
the run
π
at time t.
Thetimesequence(
τ
i
)
i
∈
N
identifies the points in time, at which mode-switches
may occur, which is expressed in Clause (2). Only at those points discrete transi-
tions (having a noticeable effect on the state) may be taken. On the other hand, it
is not required that any transition fires at some point
τ
i
,whichpermitstocoverbe-
haviors with a finite number of discrete switches within the framework above. Our
simple plant models with only one mode provide examples. As usual, we exclude
zeno behavior (in Clause (1)). As a consequence of the requirement of transition
separation, after each discrete transition some time must elapse before the next
one can fire. Clause (3) forces all local and output variables (whose dynamics is
constrained by the set of differential equations associated with this mode) to actu-
ally obey their respective equation. Clause (4) requires, for each mode, the valua-
tion of continuous variables to meet the local invariant while staying in this mode.
Clause (5) forces a discrete transition to fire when its trigger condition becomes
true. The effect of a discrete transition is described by Clause (6). Whenever a dis-
crete transition is taken, local and output variables may be assigned new values,
obtained by evaluating the right-hand side of the respective assignment using the
previous value of locals and outputs and the current values of the input. If there is
no such assignment, the variable maintains its previous value, which is determined
by taking the limit of the trajectory of the variable as
t
converges to the switching
time
τ
i
+1
. Values of inputs may change arbitrarily. They are not restricted by the
clauses, other that they obey mode invariants and contribute to the satisfaction
of discrete transitions when those fire.
A.2 Parallel Composition
The parallel composition of two such hybrid automata
H
1
and
H
2
presupposes
the typical disjointness criteria for modes, local variables, and output variables.
Output variables of
H
1
which are at the same time input variables of
H
2
,and
vice versa, establish communication channels with instantaneous communica-
tion. Those variables establishing communication channels become local vari-
ables of
H
1
H
2
(in addition to the local variables of
H
1
and
H
2
), for other
variable sets we simply take the union of those not involved in communication.
Modes of
H
1
H
2
are the pairs of modes of the component automata. One may
define the set of runs of
H
as those tuples of trajectories which project to runs
of
H
1
and
H
2
, respectively. It is not always possible to give a hybrid automaton
for
H
1
H
2
, because of problems with cycles of instantaneous communications.
Therefore, we impose the following additional condition on the composability of
hybrid automata.
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