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5.2.3 Analysis of the OLS EXIN ODE
From eq. (5.12), for
ζ
=
0(OLS),
dx
dt
=−
Rx
−
r
(5.19)
Its critical point (a minimum because the cost function is quadratic) is given by
R
−
1
r
x
=
(5.20)
(
R
is invertible because it is positive definite). From eq. (5.14),
R
=
A
T
A
/
m
and
r
A
T
b
=
/
m
. Replacing these expressions in eq. (5.20) gives
x
=
(
A
T
A
)
−
1
A
T
b
=
A
+
b
which is the closed-form OLS solution [see eq. (1.7)]. This confirms the validity
of eq. (5.12) for
ζ
=
0.
5.2.4 Convergence of DLS EXIN
In the data least squares problem,
ζ
=
1. Thus, using the formulas of this section,
the error function becomes
T
1
2
(
Ax
−
b
)
(
Ax
−
b
)
n
E
DLS
(
x
)
=
∀
x
∈
− {
}
0
(5.21)
x
T
x
The DLS EXIN learning law becomes
)
=
x
(
t
)
−
α (
t
) γ(
t
)
a
i
+
α (
t
) γ
(
t
)
x
(
t
)
2
x
(
t
+
1
(5.22)
where
δ(
t
)
γ(
t
)
=
(5.23)
x
T
(
t
)
x
(
t
)
and the corresponding
n
th-dimensional ODE is
x
T
Rx
x
−
2
x
T
r
x
+
x
x
T
x
−
Rx
r
+
dx
dt
=
1
x
T
x
−
(5.24)
Two questions arise:
1. Is the DLS error cost minimized by the DLS solution (1.55)?
2. Does the DLS EXIN learning law converge to the right solution?
The following theorem addresses these problems.
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