Information Technology Reference
In-Depth Information
This can be rewritten as follows:
P
Y
0
X
γ
2
2
2
2
2
J
=
Y
−
XP
+
γ
P
=
−
=
Y
a
−
X
a
P
I
(1.36)
Now the extended matrix
X
a
is of full rank, and the vector P can be computed as:
X
t
a
X
a
)
−
1
X
t
a
Y
a
P
=
(
(1.37)
Obviously the solution that minimizes this index is not optimum. However, for a
small value of
, it will be close to the optimum one and it will be unique as well.
The weighting of
γ
does not need to have a unique value, rather than some values
can be weighted more than others to choose the most suitable one.
γ
P
Y
0
X
m
2
2
2
p
j
2
2
J
=
1
(
y
k
−
y
k
)
+
γ
=
Y
−
XP
+
P
=
−
i
k
=
j
(1.38)
2
which should necessarily have
a value other than zero to guarantee that the extended matrix is invertible.
where
is a diagonal matrix formed by the values
γ
1.4.2 Parameters Tuning Using the Parameters'
Weighting Method
The parameters' weighting method can also be used for parameters tuning of TS
model from local parameters obtained through the identification of a system in an
operating region or from any physical input/output data.
We suppose that in this case we have a first estimation
p
0
p
1
p
2
...
p
n
]
T
P
0
=[
(1.39)
of the TS model parameters. In order to obtain such an estimation, the classical least
square method can be used around the equilibrium point. The objective is to obtain
a global approximation of the system.
p
0
+
p
1
x
1
+
p
2
x
2
+···+
p
n
x
n
y
ˆ
=
(1.40)
.The
parameters of the global approximation can be calculated by minimizing the follow-
ing quadratic performance index:
Let us analyze a set of input/output system samples
{
x
1
k
,
x
2
k
,...,
x
nk
,
y
k
}
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