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f
=(
y
3
+
y
3
)
b
3
e
3
+(
y
2
ω
1
−
y
1
ω
2
)
e
3
−
h
1
e
1
−
h
2
e
2
−
k
3
e
3
.
>
>
Case 2:
y
3
(
t
)
y
3
and
φ
(
t
)
0: After using (12.11), (12.15), and the definition
of
¯
φ
(
t
) given by (12.17), the time derivative of
e
3
(
t
) in (12.12), can be determined as
y
3
−
y
3
y
3
−
y
3
·
e
3
=
·
y
3
−
φ
−
φ =
f
−
φ
.
(12.24)
δ
δ
Case 3:
y
3
(
t
)
<
y
3
and
φ
(
t
)
<
0: After using (12.11), (12.15), and the definition
˘
of
φ
(
t
) given by (12.17), the time derivative of
e
3
(
t
) in (12.12), can be determined as
y
3
−
y
3
y
3
−
y
3
·
e
3
=
·
y
3
−
φ
−
φ
=
f
−
φ
.
(12.25)
δ
δ
12.4.1.3
Stability Analysis
The stability of the proposed observer is analyzed in this section using a Lyapunov-
based approach. Since the proposed observer uses a projection law, the Lyapunov
analysis is examined for three possible cases of projection.
Theorem 12.1.
The observer in (12.13)-(12.15) is exponentially stable provided
Assumptions 12.1-12.4 are satisfied, and k
3
(
t
)
is selected according to (12.20).
Proof.
Consider a domain
¯
3
containing
e
(0) and a continuously differentiable
D ⊂
R
Lyapunov function,
V
(
e
t
) :
¯
+
,definedas
,
D×
[0
×
∞
)
→
R
1
2
e
T
e
V
(
e
)
.
(12.26)
Case 1:
y
3
≤
y
3
(
t
)
≤
y
3
or if
y
3
(
t
)
>
y
3
and
φ
(
t
)
≤
0orif
y
3
(
t
)
<
y
3
and
φ
(
t
)
≥
0:
Taking the derivative of
V
(
e
) and utilizing (12.21) and (12.23) yields
·
V
=
k
1
e
1
−
k
2
e
2
−
y
2
ω
1
+
y
1
ω
2
+
k
3
)
e
3
.
−
(
−
(
y
3
+
y
3
)
b
3
−
(12.27)
By choosing the gain
k
3
(
t
) according to (12.20), the expression for
·
V
(
t
) can be
upper bounded as
·
V
k
1
e
1
−
k
2
e
2
−
k
4
e
3
≤−
k
4
)
V
≤−
2min(
k
1
,
k
2
,
(12.28)
where
k
4
∈
R
is a strictly positive number. Using the Gronwall-Bellman lemma [7]
and (12.26) yields
e
(
t
)
≤
e
(0)
exp(
−
min(
k
1
,
k
2
,
k
4
)
t
)
.
(12.29)
>
>
Case 2:
y
3
(
t
)
y
3
and
φ
(
t
)
0: Taking the time derivative of
V
(
e
) and utilizing
(12.21) and (12.24) yields
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