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y
ˆ( )
k
=
H
( )
k
¸
.
(10)
tion can be easily extended into a weighted least
squares estimator
j
−
1
=
T
e
T
e
where
H
(
k
is a set of independent specified
basis and
¸
j
is the matrices of parameters to be
estimated. On the other hand, the true model can
be thought as a contaminated estimation model
θ
j
H
( )
k w
H
( )
k
H
( )
k w k
y
( ),
(16)
j
j
where
w
e
is the appropriate weighting parameter
and
y
( )
k
=
H
( )
k
¸
+
e
( ).
k
(11)
j
T
(17)
T
T
T
H
( )
k
=
[ (
y
k
−
1
) , ...,
y
(
k
−
n
) ,
u
(
k
−
1
) , ..., (
u
k m
−
) ]
From Eq. (11), the error dynamics is given by
e
( )
k
=
y
( )
k
−
H
( )
k
¸
.
(12)
j
and
Using the fact that a scalar equals its transpose,
substituting Eq. (12) into Eq. (9) leads to the fol-
lowing objective function
θ
j
=
[
a
, ...,
a
,
b
, ...,
b
].
(18)
1
,
j
n j
,
1
,
j
m j
,
The nonlinear HRC MIMO ARX-TS fuzzy
modeling approach proposed in this chapter
can be summarized: 1) nonlinear behavior of a
building-MR damper system are represented by
a family of multiple MIMO ARX input models
that are integrated into a nonlinear time-varying
model through fuzzy rules; 2) the premise part
of the multiple MIMO ARX-TS fuzzy model is
determined by using the HRC algorithm; 3) the
consequent part parameters are optimized by a
family of weighted linear least squares. Finally,
the effectiveness of the HRC MIMO ARX-TS
fuzzy model is demonstrated from a benchmark
building structure in the following section. Note
that the proposed HRC MIMO ARX-TS fuzzy
model was used for a semiactive nonlinear fuzzy
control system design (Kim et al. 2009).
1
2
T
T
T
T
J
=
J
(
¸
)
=
y
( )
k
y
( )
k
−
2
y
( )
k
H
( )
k
¸
+
¸
H
( )
k
H
( )
k
¸
.
j
j
j
j
(13)
In this problem, the goal is to find
¸
j
such that
the objective function
J
is minimized. For mini-
mization of the quadratic function of Eq. (13), the
following necessary condition can be derived
T
T
∇ =
j
J
H
( )
k
H
( )
k
¸
−
H
( )
k
y
( )
k
=
0
,
¸
j
(14)
where
∇
¸
j
J
is a Jacobian matrix, the first partial
derivative of J with respect to
¸
j
. For solution of
this equation, the following analytical least squares
estimator is available
5. CASE STUDY
−
1
=
T
T
θ
j
H
( )
k
H
( )
k
H
( )
k
y
( ).
k
(15)
5.1. MR Damper
By simply adding an appropriate weighting
parameter of
w
e
, the linear least squares estima-
In recent years, smart control systems have been
considered for large civil structures because it
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