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with vanishing perturbations. The nominal system
V
V
is exponentially stable.
Thus, it can be proved that the system is asymptotically stable [26]
i.e. V
(
e
)
≤−
λ
→
0
as
t
→
∞
. Thus, the definition of
V
(
e
) can be used to conclude that
e
(
t
)
→
0as
t
→
∞
. Since the system is asymptotically stable,
V
(
e
)
∈
L
∞
; hence
e
(
t
)
∈
L
∞
.Since
e
(
t
)
∈
L
∞
, and using Assumptions 12.1 and 12.7,
y
(
t
)
,
u
(
t
)
∈
L
∞
, thus
y
(
t
)
,
u
(
t
)
∈
L
∞
, a linear analysis proves that
·
e
(
t
)
L
∞
.Since
e
(
t
),
y
(
t
),
u
(
t
),
ω
(
t
),
b
(
t
)
∈
∈
L
∞
.
Also, the gains
L
∞
. Hence, all the signal are bounded and the
proposed estimator identifies/estimates the states asymptotically. As
y
3
(
t
),
u
1
(
t
),
and
u
2
(
t
) can be estimated, it is possible to partially recover the motion parameters,
i.e. b
1
(
t
) and
b
2
(
t
).
ρ
3
(
t
),
ρ
4
(
t
),
ρ
5
(
t
)
∈
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