Information Technology Reference
In-Depth Information
2.1.1 Problem Formulation
Let n be the number of input variables and m the number of output variables
of a completely general system to model; a discrete Multiple Input Multiple
Output—MIMO—TS fuzzy model can be represented by the following set of rules
(Babuška 1995 ; Babuška et al. 1996 ; Nguyen et al. 1995 ; Takagi and Sugeno 1985 ):
R ( l , i ) :
is A l 1 i and
is A l ni
If x 1 (
k
)
...
and x n (
k
)
j = 1
n
(2.1)
Then y i (
a l 0 i +
a l ji x j (
k
) =
k
),
where l
M i is the index of the rule and M i the number of rules that model
the evolution of the i th system output, y i (
=
1
...
. a ij , j
)
=
...
n , represents the set of
adaptive parameters of the consequents of the rules, thus they must be determined
by the modeling process. k indicates the current sampling time.
Note that in using the above representation, there are multiple inputs and outputs,
and each output can be modeled by a different number of rules. This representation
facilitates the reduction of the total number of rules needed to model correctly a
complex system, and, therefore, facilitates the modeling process by reducing the
number of model parameters.
If input vector is extended in a coordinate (Andújar and Barragán 2005 ; Andújar
et al. 2009 )by
k
0
x 0 =
˜
1, extended vector
x takes the form:
˜
T
T
x
˜
= ( ˜
x 0 , ˜
x 1 ,..., ˜
x n )
= (
1
,
x 1 ,...,
x n )
(2.2)
Using the weighted average as a method of aggregation and the extension of the
state vector given in ( 2.2 ) , the output y i generated by the set of rules R ( l , i ) , can be
calculated by (Wang 1994 , 1997 ):
n
y i (
k
) =
h i (
x
(
k
)) =
a ji (
x
) ˜
x j (
k
),
(2.3)
j = 0
being a ji (
x
)
variables coefficients (Wong et al. 1997 ) defined by
M i
l
1 w i (
a l ji
x
)
=
a ji (
x
) =
,
(2.4)
M i
l
1 w i (
x
)
=
where w i (
is calculated by ( 2.5 ) and represents the degree of activation of the rules
of the fuzzy model:
x
)
n
j = 1 μ
w i (
l
l
x
) =
ji (
x j (
k
), σ
ji ).
(2.5)
Search WWH ::




Custom Search