Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 852161, 16 pages
doi:10.1155/2012/852161
Research Article
Adaptive Neural Control for a Class of Outputs
Time-Delay Nonlinear Systems
Ruliang Wang1 and Jie Li2
1
Computer and Information Engineering College, Guangxi Teachers Education University,
Nanning 530023, China
2
School of Mathematical Sciences, Guangxi Teachers Education University, Nanning 530023, China
Correspondence should be addressed to Ruliang Wang, awangruliang@tom.com
Received 25 April 2012; Accepted 26 August 2012
Academic Editor: Yiu-ming Cheung
Copyright q 2012 R. Wang and J. Li. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper considers an adaptive neural control for a class of outputs time-delay nonlinear systems
with perturbed or no. Based on RBF neural networks, the radius basis function RBF neural
networks is employed to estimate the unknown continuous functions. The proposed control
guarantees that all closed-loop signals remain bounded. The simulation results demonstrate the
effectiveness of the proposed control scheme.
1. Introduction
The study of the time-delay systems has been one of the most active research topics in
recent years 1–15. The time-delay systems can be divided into four types: systems with
input delay 1–5, systems with state delay 6–9, 16–18, systems with both input and state
delays, and systems with both input and output delays 19. The effect of time delay on
stability and asymptotic performance has been investigated in 20. In 21, LyapunovKrasovskii functionals were used with backstepping to obtain a robust controller for a class
of single-input single-output SISO nonlinear time-delay systems with known bounds on
the functions of delayed states, but it was commented that results could not be constructively
obtained in 22. In 23, the problem of the adaptive neural-networks control for a class of
nonlinear state-delay systems with unknown virtual control coefficients is considered. In 24,
An adaptive control scheme combined with radius basis function RBF neural networks,
backstepping, and adaptive control is proposed for the output tracking control problem of
a class of MIMO nonlinear system with input delay and disturbances. Neural networks are
employed to estimate the unknown continuous functions; the control scheme ensures that
2
Mathematical Problems in Engineering
the closed-loop system is semiglobally uniformly ultimately bounded SGUUB. In 11 A
control scheme combined with backstepping, radius basis function RBF neural networks,
and adaptive control is proposed for the stabilization of nonlinear system with input and
state delay.
In this paper, we present an adaptive neural controller design procedure for a class
of output time-delay nonlinear systems with perturbed, based on backstepping, adaptive
control, and neural networks. RBF neural network is employed to the unknown continuous
function. A numerical example is provided to show the effectiveness of the control scheme.
2. Problem Formulation and Preliminaries
Consider the nonlinear time-delay system is described as follows:
ẋi xi1 gi y fi yt − τ νi t, 1 ≤ i ≤ n − 1,
ẋn ut gn y fn yt − τ νn t,
2.1
y x1 ,
where x x1 , x2 , . . . , xn T ∈ Rn is state, u ∈ R is control and y ∈ R is output vectors,
respectively. νi t i 1, 2, . . . , n is a time-varying disturbance. fi yt − τ, i 1, 2, . . . , n,
gi y, i 1, 2, . . . , n are unknown continuous functions.
Assumption 2.1. The unknown function fi y satisfies fi2 y ≤ ki , where ki i 1, 2, . . . , n is a
known constant.
Assumption 2.2. The time-varying disturbance νi t satisfies |νi t| ≤ di < 1, 1 ≤ i ≤ n, where
di i 1, 2, . . . , n is a known constant.
Lemma 2.3. xt ∈ ΩM , t ∈ −τmax , τ, where ΩM {x | x ≤ M}, M is an unknown constant.
3. RBF NN Approximation
In this paper, for a given δ > 0 and any continuous function Hi ηi defined on Ωi , there is a
perfect RBF neural network, which satisfies
Fi ζi WiT Si ζi δi ζi ,
3.1
where |δi ζi | ≤ δWi ∈ Rmi is the weight vector of the neural networks, mi is the number of
the NN nodes, ζi ∈ Ωi is the input vector, Si ζi si1 , . . . , simi T is defined by
⎡ T ⎤
ζi − μij
ζi − μij
⎦.
sij ζi exp⎣−
ξ2
3.2
Mathematical Problems in Engineering
3
According to the discussion in 21, 22, denote the best weight vector as follows:
Wi∗ arg minm sup WiT Si ζi − Hi ζi wi ∈R
i
ζi ∈Ωi
3.3
i be the estimate of W ∗ ,
which is unknown and needs to be estimated in control design. Let W
i
∗
i W
i − W .
and define W
i
4. Main Result
In this section, we will consider system 2.1.
I when νi t 0, i 1, 2, . . . , n,
Let us define error variables zi assistant functions and the virtual control αi ,
respectively, as follows:
z1 x1 ,
zi xi − αi−1 ,
4.1
2 ≤ i ≤ n.
Define the following sets:
Ωzi {zi ∈ R | |zi | < λi },
Ωozi {zi ∈ R | |zi | ≥ λi },
4.2
where λi is a small constant. Define assistant functions as
F1 g1 x Fi gi x −
i−1
i−1
∂αi−1 1 ˙
z
U
W
−
x
i
j
1
j
2
2λi j1
∂
W
j1
j
i−1
∂αi−1
j1
1
U1 x1 z1 ,
2λ21
∂xj
1 ∂αi−1
xj1 gj x1 −
zj
2 ∂xj
4.3
,
2 ≤ i ≤ n.
Define the virtual control as
T Si ξi ,
αi −ki0 zi − W
i
1 ≤ i ≤ n,
4.4
4
Mathematical Problems in Engineering
where
1 b1 3
2 λ22 τmax k1 ,
2 2 2λ1
⎛
⎞
i−1
1 ⎝ 2
bi
ki0 ci 2 2 λi1 τmax kj ⎠, 2 ≤ i ≤ n − 1,
2
2λi
j1
k10 c1 kn0
⎛
⎞
n−1
1 ⎝ 2
bn
1 2 λi1 τmax kj ⎠,
cn 2
2λn
j1
4.5
ξ1 x1 , x1 T ,
⎡
ξi T
⎣xTi , xi , αi−1 ,
∂i−1
∂xi−1
ψi−1 T j
∂W
⎤T
T
, ψi−1 ⎦ ,
∂xi−1
i−1
∂αi−1
j1
∂i−1
˙ ,
∂W
j
2 ≤ i ≤ n,
2 ≤ i ≤ n.
Theorem 4.1. System 2.1 with both input delay and state delay satisfies Assumptions 2.1 and 2.2.
The virtual control can be selected as 4.4. If the control law and the adaptive law are selected as
follows:
ut αn ,
4.6
˙ Γ z S ξ − σ W
W
i
i
i i i
i i ,
1 ≤ i ≤ n,
4.7
then the closed-loop system is semi-globally uniformly ultimately bounded.
Proof. Define the Lyapunov-Kresovskii functional V t as
Vzi t 1 2
z t,
2 i
VUi t i
1
2 j1
VWi t t
1 T −1 W Γ Wi ,
2 i i
Uj yσ dσ,
t−τh
Vi t Vzi t VUi t VWi t,
V t n
Vi t.
i1
1 ≤ i ≤ n,
4.8
4.9
4.10
4.11
Mathematical Problems in Engineering
5
Step 1. For the first differential equation of the the first subsystem, by 4.1, 4.3, we can get
ż1 ẋ1 x2 g1 x1 f1 x1 t − τh z2 α1 F1 ζ1 −
1
U1 x1 z1 g1 x1 t − τh .
2λ21
4.12
By
z1 f1 x1 t − τh ≤
1 2 1 2
z f x1 t − τh 2 1 2 1
4.13
differentiating 4.10 and using 4.12, the inequality below can be obtained easily.
˙ 1 U x − 1 U x t − τ T Γ−1 W
V̇1 z1 ż1 W
1
1 1
1 1
h
1 1
2
2
1
z1 z2 α1 F1 ζ1 − 2 U1 x1 z1 f1 x1 t − τh 2λ1
˙ 1 U x − 1 U x t − τ T Γ−1 W
W
1
1 1
1 1
h
1 1
2
2
z21
1 2 1
˙ .
T Γ−1 W
≤ z1 z2 z1 α1 z1 F1 ζ1 z1 U1 1 − 2 W
1
1 1
2
2
λ1
4.14
1 If z1 ∈ Ωoz1 , then |z1 | ≥ λ1 . Thus, substituting 4.4 and 4.7 into 4.14 results in
z21
1 2 1
˙
T
T Γ−1 W
V̇1 ≤ z1 z2 z1 −k10 z1 − W1 S1 ζ1 z1 F1 ζ1 z1 U1 1 − 2 W
1
1 1
2
2
λ1
z2 1
1
b1 3 2
z1 z1 F1 ζ1 − 1 τmax k1 λ22
z1 2 z2 2 − c1 2
2
2 2
2λ1
z1 S1 ζ1 − σ1 W
T S1 ζ1 1 z2 1 z1 2 W
1
− z1 W
1
2 1 2
1
b1 2 z21 T W
1
z1 − 2 τmax k1 λ22 z1 ε1 ξ1 − σ1 W
≤ z22 − c1 1
2
2
2λ1
1
b1 2 b1 2
1 ∗ 2 1 T W
1
z1 z1 τmax k1 λ22 − σ1 W
≤ z22 − c1 ε1 −
1
2
2
2
2b1
2
σ1 2 σ1 W ∗ 2 1 ε∗ 2 1 z2 − λ2
≤ − c1 z21 − VU1 T − W
1 2
2
1
1
2
2
2b1
2
≤
4.15
≤ − k1 V1 bv1 Θ1 ,
2
2
∗ 2
∗ 2
where k1 min{2c1 , σ1 /λmax Γ−1
1 , 1}, bv1 σ1 /2W1 1/2b1 ε1 . Θ1 1/2z2 − λ2 .
6
Mathematical Problems in Engineering
If there is no item Θ1 in 4.15, then
0 ≤ V1 t ≤ V1 0 − δv e−k1 t δv ,
4.16
where δv bv1 /k1 . Thus V1 is bounded.
2 If z1 ∈ Ωz1 , then |z1 | < λ1 , z1 is bounded. By the integral median theorem, we can
obtain
1
2
t
1
U1 yσ dσ τh U1 yθ ,
2
t−τh
θ ∈ t − τh , t.
4.17
By Assumption 2.1 and 4.9, it can be concluded that VU1 is bounded.
Differentiating Vw1 ,
T z1 S1 ξ1 − σ1 W
1
Vw1 W
1
2 1 1 2
1
2
∗ 2
z S1 ζ1 ≤ − σ1 − kw1 W1 σ1 W1 2
2
kw 1 1
4.18
≤ − k1 Vw1 b1 ,
where
k1 σ1 − kw1
,
λmax Γ−1
1
b1 2 m1 2
1
σ1 W1∗ λ1 .
2
kw 1
4.19
m1 is the number of neurons of the neural networks. Choose the parameter so that kw1 <
σ1 , k1 > 0. Therefore
0 ≤ Vw1 t ≤ Vw1 0 − δw1 e−k1 t δw1 ,
4.20
where δw1 b1 /k1 . Thus Vw1 t is bounded. Because Vz1 , VU1 , Vw1 are all bounded, V1 is
bounded when z1 ∈ Ωz1 .
Step i. For the ith 2 ≤ i ≤ n − 1 subsystem, by utilizing 4.14.3, we have
żi ẋi − α̇i−1
xi1 gi x1 fi x1 t − τh − α̇i−1
zi1 αi gi x1 fi x1 t − τh −
i−1
∂αi−1 j1
∂xj
i−1
∂αi−1 ˙ .
W
xj1 gi x1 fi x1 t − τh −
j
∂
w
j
j1
4.21
Mathematical Problems in Engineering
7
Differentiating 4.10 along track 4.21, we have
i
1
1
V̇i ≤ zi zi1 αi Fi ζi z2i f 2 x1 t − τh 2
2 j1 j
i−1
1
Uj x1 − Uj x1 t − τh 2 j1
2λ2i j1
1
≤ z1 z2 α1 F1 ζ1 − 2 U1 x1 z1
2λ1
˙
T Γ−1 W
f x t − τ W
−
i−1
z2i 1
Uj x1 1
h
1
1
1
1
1
˙ z v t
T Γ−1 W
U1 x1 − U1 x1 t − τh W
i
i i
i i
2
2
z2
1
1
≤ z1 z2 z1 α1 z1 F1 ζ1 z21 U1 1 − 12
2
2
λ1
˙
T Γ−1 W
z v t
W
i
i
i
4.22
i i
1
1
≤ z1 z2 z1 α1 z1 F1 ζ1 z21 z2i
2
2
2
i−1
z
˙ .
T Γ−1 W
Uj x1 1 − i2 W
i
i i
λ
j1
i
1 If zi ∈ Ωozi , then |zi | ≥ λi . Thus, substituting 4.4 and 4.7 into 4.22 results in
1
bi
V̇i ≤ zi zi1 zi Fi ζi z2i − ci 2 z2i
2
2
⎡
⎤
i−1
z2
T Si ζi − i 2 ⎣τmax kj λ2i1 ⎦ − zi W
i
2λi
j1
T zi Si ζi − σi W
i
W
i
⎤
⎡
i−1
bi
1
T W
i 1 d2
≤ zi zi1 − ci 1 z2i − ⎣τmax kj λ2i1 ⎦ zi εi ξi − σi W
i
2
2
2 i
j1
≤ − ci z2i −
4.23
i−1
τmax σ
2 σ W ∗ 2
kj − W
i i
2 j1
2
2
1 ∗ 2 1 2
−zi z2i1 − λ2i1
εi 2bi
2
≤ − ki Vi bvi Θi ,
2
∗ 2
∗ 2
where ki min{2ci , σi /λmax Γ−1
i , 1}, bvi σi /2Wi 1/2bi εi . Θi 1/2−zi 2
2
zi1 − λi1 .
8
Mathematical Problems in Engineering
If there is no item Θi in 4.23, then
0 ≤ Vi t ≤ Vi 0 − δv e−ki t δv ,
4.24
where δv bvi /ki . Thus Vi is bounded.
2 If zi ∈ Ωzi , similar to step 1, we have Vi is bounded.
Step n. This is the last step for the nth subsystem, similarly to the ith subsystem, if zn ∈ Ωozn ,
then |zn | ≥ λn . Thus we have
V̇n ≤ − cn z2n −
n−1
τmax σ
1 ∗ 2 1 2
2 σ
kj − W
ε − zn
Wn∗ 2 n 2 j1
2
2
2bn n
2
4.25
≤ − kn Vn bvn Θn ,
∗ 2
∗ 2
2
where kn min{2cn , σn /λmax Γ−1
n , 1}, bvn σn /2Wn 1/2bn εn . Θn −1/2zn .
By 4.25, it is easy to have
0 ≤ Vn t ≤ Vn 0 − δv e−kn t δv ,
4.26
where δv bvn /kn . Thus Vn is bounded.
1 If zn ∈ Ωzn , similar to step 1, we have Vn is bounded.
The Vi , 1 ≤ i ≤ n is bounded when zi ∈ Ωzi , 1 ≤ i ≤ n. In zi ∈ Ωozi , 1 ≤ i ≤ n:
V̇ t ≤
n
V̇i
i1
≤ −
n
n
n
ki Vi bvi Θi
i1
i1
4.27
i1
≤ − kv V t bv ,
where kv min{k1 , k2 , . . . , kn }, bv Then
n
i1
bvi .
0 ≤ V t ≤ V 0 − δv e−kv t δv ,
4.28
where δv bv /kv . Thus V t is bounded.
II When νi t /
0, i 1, 2, . . . , n.
Let us define error variables zi assistant functions and the virtual control αi ,
respectively, as follows:
z1 x1 ,
zi xi − αi−1 ,
2 ≤ i ≤ n.
4.29
Mathematical Problems in Engineering
9
Define the following sets:
Ωzi {zi ∈ R | |zi | < λi },
Ωozi {zi ∈ R | |zi | ≥ λi },
4.30
where λi is a small constant. Define assistant functions as
1
U1 x1 z1 ,
2λ21
i−1
i−1
i−1
∂αi−1 ∂αi−1
1 1 ∂αi−1
˙
xj1 gj x1 −
Wj −
zj ,
Fi gi x 2 zi Uj x1 −
∂xj
2 ∂xj
2λi j1
j1 ∂Wj
j1
F1 g1 x 2 ≤ i ≤ n.
4.31
Define the virtual control as
T Si ξi ,
αi −ki0 zi − W
i
4.32
1 ≤ i ≤ n,
where
1 b1 3
2 λ22 τmax k1 d12 ,
2 2 2λ1
⎛
⎞
i−1
1 ⎝ 2
bi
2⎠
ki0 ci 2 2 λi1 τmax kj di , 2 ≤ i ≤ n − 1
2
2λi
j1
k10 c1 kn0
4.33
⎛
⎞
n−1
1 ⎝ 2
bn
1 2 λi1 τmax kj dn2 ⎠,
cn 2
2λn
j1
ξ1 x1 , x1 T
⎡
ξi T
⎣xTi , xi , αi−1 ,
ψi−1 ∂i−1
∂xi−1
T i−1
∂αi−1
j1
j
∂W
∂i−1
T
∂xi−1
˙ ,
∂W
j
⎤T
, ψi−1 ⎦ ,
2 ≤ i ≤ n,
4.34
2 ≤ i ≤ n.
Theorem 4.2. System 2.1 with both input delay and state delay satisfies Assumptions 2.1 and 2.2.
The virtual control can be selected as 4.32. If the control law and the adaptive law are selected as
follow:
ut αn ,
˙
i Γi zi Si ξi − σi W
i ,
W
4.35
1 ≤ i ≤ n,
then the closed-loop system is semi-globally uniformly ultimately bounded.
4.36
10
Mathematical Problems in Engineering
Proof. Define the Lyapunov-Kresovskii functional V t as
Vzi t 1 2
z t,
2 i
i
1
VUi t 2 j1
VWi t t
1 T −1 W Γ Wi ,
2 i i
Uj yσ dσ,
t−τh
Vi t Vzi t VUi t VWi t,
V t 1 ≤ i ≤ n,
n
Vi t.
4.37
4.38
4.39
4.40
i1
Step 1. For the first differential equation of the first subsystem, by 4.29, 4.31, We can get
ż1 ẋ1 x2 g1 x1 f1 x1 t − τh ω1 t
z2 α1 F1 ζ1 −
1
U1 x1 z1 h1 x1 t − τh ω1 t.
2λ21
4.41
By
z1 f1 x1 t − τh ≤
1 2 1 2
z f x1 t − τh 2 1 2 1
4.42
differentiating 4.39 and using 4.41, the inequality below can be obtained easily.
˙ 1 U x − 1 U x t − τ T Γ−1 W
V̇1 z1 ż1 W
1
1 1
1 1
h
1 1
2
2
1
˙
T Γ−1 W
z1 z2 α1 F1 ζ1 − 2 U1 x1 z1 ω1 t f1 x1 t − τh W
1
1 1
2λ1
1
1
U1 x1 − U1 x1 t − τh 2
2
z21
1 2 1
˙ z ω t.
T Γ−1 W
≤ z1 z2 z1 α1 z1 F1 ζ1 z1 U1 1 − 2 W
1
1 1
1 1
2
2
λ1
4.43
Mathematical Problems in Engineering
11
1 If z1 ∈ Ωoz1 , then |z1 | ≥ λ1 . Thus, substituting 4.32 and 4.36 into 4.43 results in
z21
1 2 1
˙ z ν t
T
T Γ−1 W
V̇1 ≤ z1 z2 z1 −k10 z1 − W1 S1 ζ1 z1 F1 ζ1 z1 U1 1 − 2 W
1
1 1
1 1
2
2
λ1
z2 1
b1 3 2
1
T S1 ζ1 z1 z1 F1 ζ1 − 1 τmax k1 λ22 d12 − z1 W
z1 2 z2 2 − c1 1
2
2
2 2
2λ1
1
1
z1 S1 ζ1 − σ1 W
1 1 d2
z21 z1 2 W
2
2
2 1
1
b1 2 z21 T W
1
z1 − 2 τmax k1 λ22 z1 ε1 ξ1 − σ1 W
≤ z22 − c1 1
2
2
2λ1
1 2
b1 2 b1 2
1 ∗ 2 1 2 1
1 2
2
T W
1
≤ z2 − c1 d1 −
ε
z1 z1 τmax k1 λ2 d1 − σ1 W
1
2
2
2
2b1 1
2
2
2
σ1 2 σ1 W ∗ 2 1 ε∗ 2 1 z2 − λ2
≤ − c1 z21 − VU1 T − W
1 2
2
1
1
2
2
2b1
2
≤
≤ − k1 V1 bv1 Θ1 ,
4.44
2
2
∗ 2
∗ 2
where k1 min{2c1 , σ1 /λmax Γ−1
1 , 1}, bv1 σ1 /2W1 1/2b1 ε1 . Θ1 1/2z2 − λ2 .
If there is no item Θ1 in 4.44, then
0 ≤ V1 t ≤ V1 0 − δv e−k1 t δv ,
4.45
where δv bv1 /k1 . Thus V1 is bounded.
2 If z1 ∈ Ωz1 , then |z1 | < λ1 , z1 is bounded. By the integral median theorem, we can
obtain
1
2
t
1
U1 yσ dσ τh U1 yθ ,
2
t−τh
θ ∈ t − τh , t.
4.46
By Assumption 2.1 and 4.38, it can be concluded that VU1 is bounded.
Differentiating Vw1 ,
T z1 S1 ξ1 − σ1 W
1
Vw1 W
1
2 1 1 2
1
2
∗ 2
W
σ
z
≤ − σ1 − kw1 W
ζ
S
1
1
1 1
1
2
2
kw 1 1
≤ − k1 Vw1 b1 ,
4.47
12
Mathematical Problems in Engineering
where
k1 σ1 − kw1
,
λmax Γ−1
1
b1 2 m1 2
1
σ1 W1∗ λ1 .
2
kw 1
4.48
m1 is the number of neurons of the neural networks. Choose the parameter so that kw1 <
σ1 , k1 > 0. Therefore
0 ≤ Vw1 t ≤ Vw1 0 − δw1 e−k1 t δw1 ,
4.49
where δw1 b1 /k1 . Thus Vw1 t is bounded. Because Vz1 , VU1 , Vw1 are all bounded, V1 is
bounded when z1 ∈ Ωz1 .
Step i. For the ith 2 ≤ i ≤ n − 1 subsystem, by utilizing 4.29, 4.31, we have
żi ẋi − α̇i−1
xi1 gi x1 fi x1 t − τh − α̇i−1 νi t
zi1 αi gi x1 fi x1 t − τh νi t −
i−1
∂αi−1 j1
−
i−1
∂αi−1
j1
∂w
j
∂xj
xj1 gi x1 fi x1 t − τh w
˙ j .
4.50
Differentiating 4.39 along track 4.50,we have
i
1
1
V̇i ≤ zi zi1 αi Fi ζi z2i f 2 x1 t − τh 2
2 j1 j
−
i−1
z2i 2λ2i j1
Uj x1 i−1
1
Uj x1 − Uj x1 t − τh 2 j1
≤ z1
1
z2 α1 F1 ζ1 − 2 U1 x1 z1 νi t
2λ1
˙
T Γ−1 W
f1 x1 t − τh W
1
1 1
1
1
˙ z ν t
T Γ−1 W
U1 x1 − U1 x1 t − τh W
i
i i
i i
2
2
z21
1 2 1
≤ z1 z2 z1 α1 z1 F1 ζ1 z1 U1 1 − 2
2
2
λ1
˙ z ν t
T Γ−1 W
W
i
i i
i i
Mathematical Problems in Engineering
13
1
1
1
≤ z1 z2 z1 α1 z1 F1 ζ1 z21 z2i di2
2
2
2
2
i−1
z
˙ .
T Γ−1 W
Uj x1 1 − i2 W
i
i i
λ
j1
i
4.51
1 If zi ∈ Ωozi , then |zi | ≥ λi . Thus, substituting 4.32 and 4.36 into 4.51 results in
1
bi
V̇i ≤ zi zi1 zi Fi ζi z2i − ci 2 z2i
2
2
⎡
⎤
i−1
z2i
2
2
T Si ζi W
i
T zi Si ζi − σi W
− 2 ⎣di τmax kj λi1 ⎦ − zi W
i
i
2λi
j1
⎤
⎡
i−1
bi
1
1
T W
i 1 d2
≤ zi zi1 − ci 1 z2i − ⎣τmax kj di2 λ2i1 ⎦ di2 zi εi ξi − σi W
i
2
2
2
2 i
j1
≤ − ci z2i −
i−1
τmax σ
2 σ W ∗ 2 1 ε∗ 2 1 −z2 z2 − λ2
kj − W
i i
i
i1
i1
2 j1
2
2
2bi i
2
≤ − ki Vi bvi Θi ,
4.52
2
2
∗ 2
∗ 2
where ki min{2ci , σi /λmax Γ−1
i , 1}, bvi σi /2Wi 1/2bi εi . Θi 1/2−zi zi1 −
2
λi1 .
If there is no item Θi in 4.52, then
0 ≤ Vi t ≤ Vi 0 − δv e−ki t δv ,
4.53
where δv bvi /ki . Thus Vi is bounded.
2 If zi ∈ Ωzi , similar to step 1, we have Vi is bounded.
Step n. This is the last step for the nth subsystem, similarly to the ith subsystem, If zn ∈ Ωozn ,
then |zn | ≥ λn . Thus we have
V̇n ≤ − cn z2n −
n−1
τmax σ
1 ∗ 2 1 2
2 σ
kj − W
ε − zn
Wn∗ 2 n 2 j1
2
2
2bn n
2
4.54
≤ − kn Vn bvn Θn ,
∗ 2
∗ 2
2
where kn min{2cn , σn /λmax Γ−1
n , 1}, bvn σn /2Wn 1/2bn εn . Θn −1/2zn .
By 4.54, it is easy to have
0 ≤ Vn t ≤ Vn 0 − δv e−kn t δv ,
where δv bvn /kn . Thus Vn is bounded.
4.55
14
Mathematical Problems in Engineering
II If zn ∈ Ωzn , similar to step 1, we have Vn is bounded.
The Vi , 1 ≤ i ≤ n is bounded. When zi ∈ Ωzi , 1 ≤ i ≤ n. In zi ∈ Ωozi , 1 ≤ i ≤ n:
V̇ t ≤
n
V̇i
i1
≤ −
n
n
n
ki Vi bvi Θi
i1
i1
4.56
i1
≤ − kv V t bv ,
where kv min{k1 , k2 , . . . , kn }, bv Then
n
i1
bvi .
0 ≤ V t ≤ V 0 − δv e−kv t δv ,
4.57
where δv bv /kv . Thus V t is bounded.
Simulation Example
Consider the nonlinear system with input and state delays as follows:
ẋ1 x2 0.3y2 sin 2yt − 0.2 ω1 ,
ẋ2 ut − 0.20 0.2y2 − sin 2yt − 0.2 ω2 ,
4.58
y x1 .
Define virtual control as
1 b1 3
2
2
T S1 ξ1 ,
τmax k1 λ2 d1 z1 − W
α1 − c1 1
2 2 2λ21
1 b2
2
T S2 ξ2 ,
1 2 τmax k1 d2 z2 − W
α2 − c2 2
2
2λ2
4.59
where c1 c2 30, b1 b2 2, λ1 λ2 2, τmax 0.6, k1 2. d1 d2 0.6, Γ1 Γ2 600,
σ1 σ2 0.006, ω1 0.05 sin2πt, ω2 0.05 cos2πt.
˙ Γ z S ξ − σ W
W
1
1
1 1 1
1 1 ,
˙ Γ z S ξ − σ W
W
2
2
2 2 2
2 2 ,
z1 x1 ,
z2 x2 − α1 .
The result of control scheme is in Figures 1 and 2.
4.60
Mathematical Problems in Engineering
15
40
30
20
10
0
−10
−20
−30
−40
−50
−60
0
5
10
15
20
25
Time (s)
Figure 1: The control input ut.
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
20
25
Time (s)
Figure 2: System state x1 t “− and x2 t “ − .− .
5. Conclusion
For a class of outputs time-delay nonlinear systems with perturbed or not, a control scheme
combined with adaptive control, backstepping, and neural network is proposed. The radius
basis function RBF neural networks is employed to estimate the unknown continuous
functions. It is shown that the proposed method guarantees the semi-globally uniformly
ultimately boundedness of all signals in the adaptive closed-loop systems. Simulation results
are provided to illustrate the performance of the proposed approach.
Acknowledgments
This work was jointly supported by the Natural Science Foundation of China 60864001 and
Guangxi Natural Science Foundation 2011GXNSFA018161.
16
Mathematical Problems in Engineering
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