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where
ρ
i
∈
R
,∀
i
=
{
1
,...,
6
}
are observer gains,
c
1
∈
R
is the known, nonzero
constant introduced in (12.38) and
ˆ
i
(
t
) are given by (12.48). The estimate
y
3
(
t
) is
generated based on the locally Lipschitz projection defined in (12.15) where
ω
·
φ
(
) is
now defined as
ρ
3
(
·
e
6
+
ρ
6
e
6
)
c
1
b
3
y
3
b
3
+
y
2
y
3
ˆ
y
1
y
3
ˆ
φ
ω
1
−
ω
2
−
y
1
b
3
e
1
−
y
2
b
3
e
2
+
c
1
b
3
e
6
+
.
(12.55)
In (12.50)-(12.55) the observer gains
ρ
1
,
ρ
2
and
ρ
6
∈
R
are strictly positive con-
stants, and the gains
ρ
3
(
t
),
ρ
4
(
t
),and
ρ
5
(
t
) are defined as
ρ
3
(
t
)
> |
(2
y
3
+
δ
)
|
b
3
|
+
|
b
3
|
(
u
1
+
u
2
)+
y
2
ω
1
−
y
1
ω
2
|,
(12.56)
J
4
+
q
(
u
1
)+
y
2
ω
1
−
ρ
4
(
t
)
> |
(
y
3
+
δ
)
|
b
3
|
+
|
b
3
|
u
1
+
u
1
y
1
ω
2
|,
(12.57)
J
5
+
q
(
u
1
)+
y
2
ω
1
−
ρ
5
(
t
)
> |
(
y
3
+
δ
)
|
b
3
|
+
|
b
3
|
u
2
+
u
2
y
1
ω
2
|.
(12.58)
12.4.2.3
Error Dynamics
Differentiating (12.12) and (12.37), and using (12.50)-(12.54) yields the following
closed-loop error dynamics
·
e
1
=
e
4
−
ω
1
+(1 +
y
1
)
˜
y
1
y
2
˜
y
2
˜
y
1
b
3
e
3
−
ρ
1
e
1
−
ω
2
−
ω
3
,
(12.59)
·
e
2
=
e
5
−
(1 +
y
2
) ˜
y
2
b
3
e
3
−
ρ
2
e
2
−
ω
1
+
y
1
y
2
˜
ω
2
+
y
1
˜
ω
3
,
(12.60)
·
e
4
=
b
3
u
1
e
3
+
y
3
b
3
e
4
+(
y
2
ω
1
−
y
1
ω
2
)
e
4
+
y
2
u
1
˜
y
1
u
1
˜
ω
1
−
ω
2
1
e
1
+
y
1
b
3
(
·
e
6
+
ρ
6
e
6
c
1
b
3
4
(
·
e
1
+
+
u
1
J
4
e
4
+
q
(
u
1
)
e
4
−
ρ
ρ
))
−
e
1
,
·
e
5
=
b
3
u
2
e
3
+
y
3
b
3
e
5
+(
y
2
ω
1
−
y
1
ω
2
)
e
5
+
y
2
u
2
˜
y
1
u
2
˜
ω
1
−
ω
2
2
e
2
+
y
2
b
3
(
·
e
6
+
ρ
6
e
6
c
1
b
3
5
(
·
e
2
+
−
ρ
−
,
+
u
2
J
5
e
5
+
q
(
u
2
)
e
5
ρ
))
e
2
·
e
6
=
c
1
b
3
e
3
−
ρ
6
e
6
.
(12.61)
By defining auxiliary signals
Ψ
1
(
t
)
∈
R
and
Ψ
2
(
t
)
∈
R
as
ω
1
+(1 +
y
1
)
˜
y
1
y
2
˜
y
2
˜
Ψ
1
−
ω
2
−
ω
3
,
(12.62)
(1 +
y
2
)
˜
ω
1
+
y
1
y
2
˜
ω
2
+
y
1
˜
Ψ
2
−
ω
3
,
(12.63)
(12.59) and (12.60) can be re-expressed as
·
e
1
=
e
4
−
y
1
b
3
e
3
−
ρ
1
e
1
+
Ψ
1
,
(12.64)
·
e
2
=
e
5
−
y
2
b
3
e
3
−
ρ
2
e
2
+
Ψ
2
,
(12.65)
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