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1.4 Non Iterative Approach
The restriction of TS identification method for the case presented in the previous
section does not mean the non-existence of solutions rather than an incentive for
their search. As it has been seen, the problem comes from the fact that the solution
should fulfil:
X
t
Y
X
t
XP
∇
J
=
−
=
0
(1.31)
But as it was shown above, the columns of the matrix X are linearly dependent and
consequently
X
t
X
is not an invertible matrix, therefore it is impossible to calculate
P through:
X
t
X
)
−
1
X
t
Y
P
=
(
(1.32)
Nevertheless, as the rows of
X
t
are linearly dependent, the independent term in
equation (
1.31
)
X
t
Y
will have the same dependence among its rows and thereupon
the rank of the system matrix will be the same as the rank of the extended matrix
with the independent term.
X
t
X
X
t
X
X
t
Y
rank
(
)
=
rank
(
|
)
(1.33)
And so the system has solution. In other words, the system is a compatible inde-
terminate one, that is, if P is a solution of (
1.31
) and K belongs to the Kernel of
X
t
X
X
t
X
K
∈
Ker
(
)
(1.34)
then
P
∗
=
K
will also be a solution. Therefore, the problem is not the lack of
a solution rather the existence of infinite solutions and the key idea is the ability to
find one of them. Several proposals can be made to select a solution. In our case, the
aim is to find solutions with lower norm.
P
+
1.4.1 Parameters' Weighting Method
An effective approach with few computational effort, based on the well known para-
meters' weighting method, is proposed. The main target is to improve the choice of
the performance index and minimize it. It is characterized by extending the objective
function by including a weighting
γ
of the norm of P vector.
m
k
=
1
(
2
2
p
j
2
2
2
J
=
y
k
−ˆ
y
k
)
+
γ
=
Y
−
XP
+
γ
P
(1.35)
j
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