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4 ClosedLoopPerformance
The key of this work is to determine the learning rate
η
(
k
) for every time index
k
. Substitute the control effort
u
(
k
) given by (3) into the system formulation
(2), we have
T
(
k
)
x
(
k
+1)=
F
N
(
k
)+
G
N
(
k
)
β
φ
(
k
)
.
(9)
Thus, the next time step error can be rewritten as
T
(
k
)
e
(
k
+1)=
x
d
(
k
+1)
− F
N
(
k
)
− G
N
(
k
)
β
φ
(
k
)
,
(10)
Substitute (10) into (7), the adaptation law can be obtained as
T
(
k
)
(
k
+1)=[
I − η
(
k
)
G
2
N
(
k
)
β
φ
φ
(
k
)]
β
(
k
)+
η
(
k
)[
x
d
(
k
+1)
,
−F
N
(
k
)]
y
p
(
k
)
φ
(
k
)
.
(11)
For the convenient presentation, let us select learning be
γ
η
(
k
)=
(
k
)
,
(12)
T
(
k
)
G
u
φ
φ
μ
k
,u
k
)
∂u
k
∂f
N
(
when
G
u
is the positive upper bound of
as given by assumption 2b
and 0
<γ<
2 is the designed parameter.
Next, we introduce the proof of system performance based on the proposed
controller with the following theorem.
Theorem 2.
System stability(Closed loop system convergence)
Let the desired trajectory
x
d
(
k
) be bounded and the upper bound of
G
N
(
k
)be
known as
G
u
. Determine the control effort
u
(
k
) by (3) and tune parameters by
(7) with the varying learning rate given by (12) when 0
<γ<
2. Then the
tracking error
e
(
k
) is bounded for the nonlinear system given by (1).
Proof.
Let define the Lyapunov function candidate as
V
(
k
)=
1
2
e
2
(
k
)
,
(13)
thus the change of Lyapunov function can be given as
ΔV
(
k
)=
V
(
k
+1)
− V
(
k
)
,
=
Δe
(
k
)[
e
(
k
)+
Δe
(
k
)
2
]
.
(14)
Let consider the next time index error can be written by
e
(
k
+1) =
e
(
k
)+
Δe
(
k
)
,
where
Δe
(
k
) can be estimated by
Δe
(
k
)
≈ Δ
e,β
(
k
)
Δ
β
(
k
), when
Δ
e,β
(
k
)=
[
∂e
(
k
+1)
∂β
1
(
k
)
∂e
(
k
+1)
∂β
2
(
k
)
∂e
(
k
+1)
∂β
l
(
k
)
]
T
. By using the chain rule, we have
···
∂e
(
k
+1)
∂β
i
(
k
)
∂e
(
k
+1)
∂x
(
k
+1)
∂x
(
k
+1)
∂u
(
k
)
∂u
(
k
)
∂β
i
(
k
)
,
=
=
−G
N
(
k
)
φ
i
(
k
)
.
(15)
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