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1492 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
V 1 1
2 V 2 h e
θ
≤− ρ
θ
+ ψ −
u λ u
≤− ρV
+ ψ
θ
2
(31)
where ρ = σγ
θ
Eq. (31) implies that for V
ψ
θ
> , V
0 ρ θ
<
and, therefore, θ is bounded. By integrating (31), we can
establish that:
From (32), we have
ψ
θ θ γ
ρ
2 2
− t
≤ (0) e ρ + 2
θ
(32)
(33)
−0.5ρt
θ ≤ θ(0) e + 2γθψ ρ
Using (33) and the fact that δ ( z)
and hu
λ
are bounded,
we can write
T
(
+ )
βξη ( , ) hu
φ ( z) θ δ( z)
λ
T
≤ βξη ( , ) hu
φ ( z) θ+ βξη ( , ) hu
δ( z)
λ
λ
T
≤ βξη ( , ) h φ ( z) θ +
uλ
+ βξη ( , ) h δ( z)
uλ
−0.
5ρt
T
≤ βξη ( , ) h φ ( z) θ(0)
e
uλ
T
+ βξη ( , ) h φ ( z) 2 γψ ρ+
uλ
+ βξη ( , ) h δ( z)
uλ
≤ ψ e + ψ
−0.5ρt
0 1
θ
(34)
+
Where ψ
0,
ψ
1
are some finite positive constants.
Lemma 1: The following inequality holds for all
Ξ> 0 and ς ∈ R with K = 0.2785 .
c
⎛ς
⎞
0≤ ς −ς
⋅tanh⎜
⎟≤ KcΞ
⎝Ξ
⎠
(35)
Theorem 1: Suppose that Assumption1-3 are satisfied
for the system (1), then the neural network controller and
adaptation law given by (24) guarantees the convergence
of the neural network parameters and to be uniformly
ultimately bounded of all the signal in the closed-loop
system.
Proof: Consider the Lyapunov function candidate:
V( e, η) = e T Pe +μV
( η ) (36)
Where μ > 0 is the design parameter. Considering (4),
(19), (20), (34) and lemma 1, differentiating V( e, η ) with
respect to time, we obtain
0
T
T T T bPe
Ve
⎛ ⎞
(,) η = e ( Ac
P+ PAc)
e− 2bPeλ
tanh⎜
⎟+
⎝ Ξ ⎠
T * dV0
() η
+2 bPehu
( u− u)
+ μ
λ
dt
T
T T
⎛bPe⎞
dV0
() η
=−eQe− 2bPeλtanh
⎜ ⎟+ μ +
⎝ Ξ ⎠ dt
T
( φ θ+
δ )
T
+2 bPeh ( z) ( z)
uλ
T
T T
⎛bPe⎞ dV0
() η
≤−eQe−2 bPeλ
tan h⎜
⎟ ++ μ +
⎝ Ξ ⎠ dt
T
−0.5
t
+ 2 bPe ψ e ρ + ψ
( 0 1)
(37)
If the design parameter λ is large enough to make
λ ≥ ψ 1
and considering assumption 4, we have
T T −0.5ρt
Ve ( , η) eQe 2 bPeψ
e +2ψ
K
0 1 c
≤− + Ξ+
∂V
( η)
[ q(0, η, u ) q( ξ, η, u) q(0, η, u )
η
η
]
0
+ μ
+ −
∂η
T
2
e Qe μσ
3
μσ
4Lξ
μσ
4Lq
≤ − − η + ξ η + η +
(38)
T
+ 2 b Pe ψ e + 2ψ
K Ξ
Then
−0.5ρt
0 1
Considering assumption 2 and
c
T −0.5ρt 2 T
2
2 −ρt
ψ0 ≤ + ψ0
2 b Pe e 0.5 e 2 b P e
ξ ≤ e + y y y ≤ e + b
(1) ( r−1)
T
(
d d
d
)
d
( λ )
Ve ( , η) ≤- ( Q) −0.5
e − +
Using the inequality
2 2
min
μσ3
η
+ μσ L e η + μσ L b η +
4 ξ
4
ξ d
T
2
2 −ρt
4 q
η
0 1 c
+ μσ L + 2 b P ψ e + 2ψ
K Ξ
4 ξ 4 ξ 1 4 ξ
2 2ε1
(39)
1 2 1
2
μσ L e η ≤ μσ L ε η + μσ L e (40)
μσ
( ) ( ( )) 2 2
Lb
ξ d
Lq μσ ε Lb
ξ d
Lq
1
+ η ≤ + η + (41)
4 4 2 2
4ε
2
Then (39) satisfies
V
( e, η)
⎛
1 ⎞ 2
≤- ⎜λmin ( Q) −0.5− μσ4Lξ
⎟ e +
⎝
2 ε1
⎠
⎡ 1
2
⎤ 2
−μ⎢σ3 − σ4Lξε1 − μ( σ4ε2( Lξbd
+ Lq)
)
2
⎥ η +
⎣
⎦
T
2
1
+ 2 bP ψ e + 2ψ
KΞ+
2 −ρt
0 1 c 2
4ε
2
(42)
where ε1,
ε
2
are suitable positive constants. We
adjust ε1,
ε
2
to
1
make σ ( ( )) 2
3
σ4Lξε1 μ σ4ε2
Lξbd
Lq
− − + > 0 .
2
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