Information Technology Reference
In-Depth Information
value of f is a finite set, which at least contains f(a) and f(b), and the element in
set {t|t
∈
[a,b]
∧
f(t) is landmark value} is called differentiated point.lj
Definition 4.1
Assuming l
1
<l
2
<···<l
k
are the landmark values of f: [a,b]
ŗ
[-
¯
,
¯
], for any t
∈
[a,b], the qualitative state QS(f,t) of f in t is regulated as ordered
pare, which is defined as following:
l
,
f
=
l
Ê
Ë
j
t
j
qval
=
(
l
,
l
)
,
f
∈
(
l
,
l
)
Ì
j
j
+
1
t
j
j
+
1
Ê
'
inc
,
f
(
t
)
>
0
Ë
qdir
std
f
'
t
=
,
(
)
=
0
Í
Ì
'
dec
,
f
(
t
)
<
0
Definition 4.2
Assuming t
i
, t
i+1
are adjacent differentiating points, the
qualitative state QS(f, t
i
, t
i+1
) of f in (t
i
,t
i
+1) is regulated as
QS(
f,t
),
t
∈
(
t
i
,t
i+1
) (4.1)
Definition 4.3
Assuming the qualitative behaviors of f in [
a,b
] are qualitative
states sequence QS(
f,t
0
), QS(
f,t
0
,t
1
), QS(
f, t
1
), ··· ,QS(
f, t
n
), among them
t
i
(i=0,1,··· ,
n
) indicate all the differentiating points and
t
i
<
t
i+
1
, if F = {
f
1
, ···, f
n
},
the qualitative behaviors of F are
QS(
F,t
i
) = { QS(
f
1
,t
i
), ···, QS(
f
n
,t
i
)} (4.2)
QS(F
,t
i
,t
i+
1
)= { QS(
f
1
,t
i
,t
i
+1
), ···, QS(
f
n
,t
i
,t
i
+1
)} (4.3)
Among them ti are the elements of the union of
f
1
, ···, f
k
.
4.5.1
Qualitative state transformation
Qualitative state transformation often occurs in qualitative simulation. Assuming
f
is a differentiable function, it must obey the intermediate value theorem and
mean value theorem when transforming from one qualitative state to another
qualitative state. There are two kinds of qualitative state transformation: one is P
transformation, which transforms from time point to time interval, another is I
transformation, which transforms from time interval to time point. Following
gives the transformation table:
