Information Technology Reference
In-Depth Information
The qualitative values of a quantity variable are not only the landmark values,
but also the open intervals between them. Hence, a quantity evaluates either to
a landmark value or to an interval between landmarks. For example, the water
level of tank
T
2
can evaluate to seven different symbolic values:
x
2
∈{
Zero,Zero..Empty,Empty,Empty..Reserve,
Reserve, Reserve..F ull, F ull
}
Please note, in this abstraction the syntax
Empty..Reserve
represents a sym-
bolic value expressing the imprecise knowledge that the concrete real value is
somewhere in between those landmarks. For modeling, it is sucient to define a
list of landmarks, the interval values are implicit.
In order to describe the dynamics of a continuous system on a qualitative
level, the symbolic values alone would be insucient. Therefore, the direction
of change
δ
=
df
{−
,
0
,
+
}
is also part of the abstract value space. This is an
abstraction of the first derivation of the continuous behavior. Hence, the type
of a qualitative variable is a pair consisting of a qualitative value and
δ
.For
example,
x
2
=(
Reserve,
) expresses the state of tank
T
2
when its water-level
reaches the reserve level and the water-level is still decreasing.
A typical evolution of the water level
x
2
when filling up the tank would be
−
(
Zero,
0)
,
(
Zero..Empty,
+)
,
(
Empty,
+)
,
(
Empty..Reserve,
+)
,
(
Reserve,
+)
,
(
Reserve..F ull,
+)
,
(
F ull,
0)
.
Note that jumps in the qualitative evolution are forbidden. The water level
cannot go from increasing to decreasing without first being steady. Furthermore,
the qualitative value cannot jump from one landmark value to the next without
the interval value in between.
Time abstraction.
Time intervals in which the qualitative behavior does not
change are abstracted away. As a consequence, we abstract from continuous time
to a temporal ordering of qualitative states. Figure 4 shows the relation between
a continuous function
v
(top-right) and the according qualitative behavior
q
(bottom-left). Here, four landmarks split the value space into three regions of
interest. The symbolic values show if a value is on a landmark or in between as
well as its direction of change. The symbols in the qualitative trace denote steady
behavior (circle), increasing behavior (arrow pointing upward), and decreasing
behavior (arrow pointing downward). In addition, the abstraction
s
of the time
intervals to steps in the evolution is also made explicit (bottom-right diagram).
The intended qualitative behavior is specified via Qualitative Differential
Equations (QDEs). Like Ordinary Differential Equations, a QDE defines the dy-
namics of a system via relating the symbolic variables and their first derivations.
Auxiliary variables may be used to link a set of QDEs. Our running example
serves to illustrate the modeling approach.
In order to model the controller environment of our example we have to define
the domains of the model quantities. The quantities
x
1
and
x
2
denoting the fill
