Geology Reference
In-Depth Information
Box 1.
{
}
.
(
)
(
)
(
)
(
)
(
)
(
)
z
FZ
∈
y k
−
1
, ..,
y k
−
n
, ..,
y
k
−
n
,
u k
−
1
, ..,
u k m
−
, ..,
u
k m
−
1
1
p
1
1
p
n
i
∏
i
µ
(
z
)
1
2
i
R
:
If
z
is
p
and
z
is
p
and
and
z
is
p
i j
,
FZ
j
FZ
1
,
j
FZ
2
,
j
FZ
i j
,
i
w z
(
)
=
i
=
1
,
n
m
∑
∑
( )
=
(
)
+
(
)
j
FZ
(6)
Then
y
k
a
y
k
−
i
b u
k
−
i
,
N
n
r
i
i j
,
i j
,
∑
∏
i
µ
(
z
)
i
=
1
i
=
1
i j
,
FZ
j
=
1
i
=
1
(4)
where n is the number of delay steps in the output
signals; m is the number of delay steps in the
input signals;
y
( )
is the output;
u
( )
is the input;
a
i j
,
and
b
i j
,
are the consequent parameters (see
Box 1).
Note that the number of the fuzzy rules cor-
responds to the number of local linear MIMO
ARX models, i.e., the mth local linear MIMO
ARX dynamic model represents the mth fuzzy
rule that describes behavior of a nonlinear dy-
namic system in a local linear operating region.
However, a question would arise on how to blend
the multiple local linear MIMO ARX dynamic
models into an integrated nonlinear dynamic
system model, i.e., how to construct a bridge
across the multiple MIMO ARX models. One
solution is found in the fuzzy logic-based inter-
polation (Yen & Langari 1998). The multiple
local linear MIMO ARX models at the specific
operating point
z
i
FZ
can be blended
i
where
µ
i j
FZ
is the grades of membership of
z
i
FZ
;
N
r
is the number of local linear dynamic
models; and
n
i
is the number of premise variables.
Once the MIMO ARX-TS fuzzy model is set up,
the premise parameters
P
i,j
and the consequent
parameters
a
i,j
and
b
i,j
are determined such that
the MIMO ARX-TS fuzzy model describes be-
havior of a nonlinear dynamic system. In this
study, the premise parameters are determined via
clustering techniques, including the hierarchical,
the fuzzy C-means, and the subtractive clustering
algorithms, and the consequent part is optimized
using the weighted linear least squares estimation
algorithm.
,
(
z
)
4. OPTIMIZATION OF MIMO
ARX-TS FUZZY MODEL
In the MIMO ARX-TS fuzzy identification model,
the parameters of the premise and consequent parts
are optimized such that the MIMO ARX-TS fuzzy
model effectively represents nonlinear behavior of
the physical system. In particular, it is desirable
to group a large data set into subsets of data with
similar patterns for efficient determination of the
premise part, i.e., the small number of MFs but
reasonable pattern recognition. Appropriate meth-
N
N
,
n
n
r
r
=
1
∑
(
)
∑
∑
(
)
y
ˆ( )
k
=
w z
i
a y
(
k
− +
i
)
w z
i
b u
(
k
−
i
)
j
FZ
i j
,
j
FZ
i j
,
i
j
=
1
i
=
1
j
=
1
(5)
≤
(
)
≤
i
FZ
where
0
w z
j
1
is the normalized true
value of the
j
th
rule,
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