Information Technology Reference
In-Depth Information
Bounded Critical Durations.
These formulas constrain the duration of some
critical controlled state. These states are typically intermediate states which are
unavoidable; nevertheless they should not endure. Another interpretation of a
critical duration constraint on a state
P
is to see it as approximate stability of
the non-critical state
P
.
A critical duration constraint denes the duration of a critical state
P
. This
state shall only have a duration of
c
within at most
t
time units:
(
R
P
:
>
c
^ `
t
)
−! d :
P
e
For nontrivial constraints, the parameters
c
and
t
satisfy 0
c
t
.
Another formulation is the following law.
((
R
P
)
R
P
>
c
^ `
t
)
−! d :
P
e
)
,
(
`
t
c
)
Only-if is trivial. The implies is by contradiction.
^
R
P
`
t
>
c
)f
Sum
g
^
R
P
c
);
R
P
`
t
^
((
`
t
>
>
0)
)f
Finite Variability
g
^
R
P
`
t
^
((
`
t
>
c
);
3
d
P
e
)
)f
Denition
g
^
R
P
`
t
^
((
`
t
>
c
);
true
;
d
P
e
;
true
)
)f
Sum
g
^
R
P
(
`
t
>
c
);
d
P
e
;
true
)f
Assumption
g
false
The rst step applies the Sum law and a simple property of reals: If
r
>
c
then there exists positive
r
1
;
c
and
r
1
+
r
2
=
r
. The second
step appeals to the nite variability underlying induction: If
P
has a positive
duration, then there is at least one subinterval where
P
holds. The fourth step
uses the global length constraint and the Sum law to combine the rst and
second subinterval. The nal step use that the assumption is violated on the
rst subintervals. The proof is completed using that
false
is a zero for chop.
r
2
such that
r
1
>
A critical duration constraint ensures that for a critical state
P
the propor-
tional upper bound is
c
=
t
in the limit.
((
R
P
(
R
P
>
c
)
^ `
t
)
−! d :
P
e)
(
c
=
t
)
`
+
c
)
The mean value of the critical duration (
R
P
t
. A critical
duration constraint is like an optimality constraint in dynamical systems; in a
design it is rened to more concrete constraints with instability of the critical
state and stability of its complement.
=`
) thus tends to
c
=
3 Design
Requirements are in the ProCoS approach the starting point for the design. How-
ever, another constraint on the design is a desire for a systematic decomposition
of the embedded computer system. A standard paradigm is given in Figure 3
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