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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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