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1
0.9
0.8
0.7
0.6
0.5
0.4
SG
0.3
0.2
0.1
FF−SG
0
0
500
1000
1500
2000
2500
3000
t
Fig. 2.
The parameter estimation errors
δ
versus
t
=[
α
1
,α
2
,α
3
,α
4
,α
5
,
]
T
=[0
.
2
,
0
.
24
,
0
.
9
,
1
.
08
,
−
0
.
1]
T
,
ϕ
(
t
)=[
h
(
u
(
t
−
1))
u
(
t
−
1)
,h
(
u
(
t
−
2))
u
(
t
−
2)
,
u
(
t
−
1)
,u
(
t
−
2)
,
−
y
(
t
−
1)]
T
.
Applying the proposed SG and FF-SG algorithms to estimate the parameters of
this system, the parameter estimates and their errors are shown in Tables 1-2
and the parameter estimation errors
δ
:=
ˆ
θ
−
θ
/
θ
versus
t
are shown in
Figure 2.
Tabl e 1.
The SG estimates and errors
t α
1
α
2
α
3
α
4
α
5
δ
(%)
100 -0.0422 0.0043 0.4938 0.5448 -0.2357 52.9384
200 -0.0291 0.0224 0.5536 0.6047 -0.2483 47.3742
300 -0.0180 0.0347 0.5844 0.6351 -0.2537 44.4015
500 -0.0108 0.0442 0.6168 0.6669 -0.2658 41.6292
1000 0.0009 0.0534 0.6589 0.7046 -0.2646 37.9839
1500 0.0055 0.0585 0.6771 0.7217 -0.2663 36.4216
2000 0.0102 0.0630 0.6906 0.7342 -0.2663 35.2145
2500 0.0135 0.0666 0.7024 0.7442 -0.2663 34.2374
3000 0.0160 0.0690 0.7093 0.7510 -0.2653 33.5802
True values 0.2000 0.2400 0.9000 1.0800 -0.1000
Let
α
i
be the
i
th element of the vector
ˆ
θ
. From the definition of
θ
,wehave:
a
1
=
α
5
,
b
2
=
α
2
α
1
. Furthermore, we can compute the estimates
m
1
=
α
3
+
α
2
,
α
2
.
m
2
=
α
3
−
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