Information Technology Reference
In-Depth Information
region. A stronger concept is
asymptotic stability
where the system tends to
an equilibrium point as time grows beyond any bounds. However, there might
be many ways of choosing a
u
term such that the system is stable. Thus one
can consider additional constraints on the control input and the plant state. A
general formulation is to introduce a non-negative cost function
K
and consider
the accumulated cost
R
0
u
(
t
))
dt
where
T
is a xed bound, or the amortized cost
R
0
K
(
x
(
t
)
;
u
(
t
))
e
−at
dt
where
a
is a positive discount factor. One can then try to nd an optimal control which
minimizes the cost.
In summary, we would expect to nd stability and some sort of optimality
as the generic top level requirements also for an embedded system.
K
(
x
(
t
)
;
Hybrid Systems.
In practical engineering, it has long been recognized that a
plant might have several operating modes with dierent characteristics. When
mode changes happen infrequently, one can analyze each mode in isolation and
use the conventional theory outlined above. However, if modes change rapidly,
for instance when controlled by computers, it is uncertain whether the transients
can be ignored. Thus there is increasing interest in theories combining the eects
of discrete transitions and continuous evolutions - theories for
hybrid systems
.
However, the classical questions of stability and optimal control are still central
to any extended theory. We shall not detour into hybrid systems theory, but
refer the interested reader to the presentation by Branicky in [1].
2.2
Duration Calculus
The usual language for control engineering is the conventional notation of math-
ematical analysis, and it serves its purpose very well, when used in in traditional
mathematical argumentation. However, formal reasoning is rather cumbersome
when all formulas contain several quantiers over time points, state values, etc.
This was the rationale for Duration Calculus which we summarize in the follow-
ing.
Syntax.
The syntax of Duration Calculus distinguishes (
duration
)
terms
,each
one associated with a certain type, and (
duration
)
formulas
. Terms are built from
names of elementary states like
x
or
Gas
,and
rigid variables
representing time
independent logical variables and are closed under arithmetic and propositional
operators. Examples of terms are
:
Gas
and
x
=
0
(of Boolean type) and
x
−
c
(of type vector of real).
Terms of type (vector of) real are also called
state expressions
and terms of
Boolean type are called
state assertions
.Weuse
f
,
g
for typical state expressions
and
P
,
Q
for typical state assertions.
Duration terms are built from
b.
f
and
e.
f
denoting the
initial
and
nal value
of
f
in a given interval, and
R
f
denoting the
integral
of
f
in a given interval. For
Search WWH ::
Custom Search