Adaptive Neural Control for a Class of Stochastic Nonlinear Systems with Unknown
Parameters, Unknown Nonlinear Functions and Stochastic Disturbances
Chao-Yang Chena,b,∗, Wei-Hua Guib , Zhi-Hong Guanc , Ru-Liang Wangd , Shao-Wu Zhoua
a School

arXiv:1702.02072v1 [] 7 Feb 2017

of Information and Electrical Engineering, Hunan University of Science and Technology, Xiangtan, 411201, P.R. China
b School of Information Science and Engineering, Central South University, Changsha, 410012, P.R.China
c
College of Automation, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China
d Computer and Information Engineering College, Guangxi Teachers Education University, Nanning, 530001, P.R. China

Abstract
In this paper, adaptive neural control (ANC) is investigated for a class of strict-feedback nonlinear stochastic systems with unknown parameters, unknown nonlinear functions and stochastic disturbances. The new controller of adaptive neural network with
state feedback is presented by using a universal approximation of radial basis function neural network and backstepping. An adaptive neural network state-feedback controller is designed by constructing a suitable Lyapunov function. Adaptive bounding design
technique is used to deal with the unknown nonlinear functions and unknown parameters. It is shown that, the global asymptotically stable in probability can be achieved for the closed-loop system. The simulation results are presented to demonstrate the
effectiveness of the proposed control strategy in the presence of unknown parameters, unknown nonlinear functions and stochastic
disturbances.
Keywords: Unknown parameters, stochastic disturbances, unknown nonlinear functions, stochastic nonlinear, adaptive neural
control.
1. Introduction
In recent years, the study of the robust control system design
for nonlinear systems has attracted extensive attention. Lots
of significant developments have been obtained[1–12], and interesting results of adaptive nonlinear control have been everincreasing. Adaptive backstepping as a powerful method has a
large number of applied research about synthesizing controllers
for lower-triangular nonlinear systems. Backstepping design
technique is one of the methods to design nonlinear system by
structuring intermediate laws and Lyapunov functions step by
step. It has obtained a large number of successful applications
in nonlinear control. Such as [13–15].
Adaptive control is an important branch of robust control.
Noticeably, due to neural network control and fuzzy logic control have a good approximation ability over a compact domain,
they are very suitable for handle highly uncertain and nonlinear system, and they have become an important part of adaptive
control and a lot of research has been obtained, such as [16–
18, 20]. In the early development stage of neural network control schemes, the control schemes that derive parameter adaptive law in off-line environments was usually used [21], which
can perform well in some simple cases, but the stability, robustness and performance of nonlinear systems have few systematic
analytical methods. In order to avoid the above problems, the
reference [19] has obtain some adaptive neural control schemes
based on Lyapunov’s stability theory.
∗ Corresponding

author.
Email address: cychen@hnust.edu.cn (Chao-Yang Chen)

Neurocomputing

Stochastic nonlinear modelling has come to play an important role in many areas of industry, science and technology.
After the success of systematic control design for deterministic nonlinear systems, how to extend this technique to the case
of stochastic nonlinear systems has been an open research area
[22]-[30]. Therefore, it is a challenging and meaningful issue
fo the stability analysis and control design of nonlinear stochastic systems, and have attracted more and more scholars’ attention in recent years. The main technical obstacle in the Lyapunov function design for stochastic nonlinear systems is that
the gradient and the higher order Hessian term are involved Itô
stochastic differentiation. In [22, 23], strict-feedback stochastic
system is studied for the first time using a backstepping design. [24] tried to extend the results in [22], a class stochastic nonlinear systems with time-delay is investigated. By using
the quadratic Lyapunov function, [25, 26] studied the stabilization problem of stochastic nonlinear systems. In [25], a class
of stochastic nonholonomic system has investigated, and adaptive stabilization by state-feedback is resolved. For linear LTI
SISO plants, [27] proposed a modified adaptive backstepping
control. In [24], a class of stochastic nonlinear systems were
investigated via output feedback, linearly bounded unmeasurable states are involved in nonlinear function. By an adaptive neural control (ANC) scheme, [38] studied a class of nonaffine pure feedback stochastic nonlinear system. It is shown
that all the signals involved are semi-globally uniformly ultimately bounded under the action of the developed controller. A
simplified adaptive backstepping neural control (ABNC) strategies are proposed for a class of uncertain strict-feedback nonFebruary 8, 2017

linear systems in [42]. In [44], a class of stochastic non-strictfeedback stochastic nonlinear system is studied. However, the
system functions must are the monotonously increasing and it
should be bounded, which is a stricter assumption because it is
the difficulty to confirm a bounded and monotonously increasing unknown function. In [45], this constraints that bounded
and monotonously increasing for nonlinear functions are relaxed, but it is unavailable when the stochastic system with
unknown parameters and unknown disturbances. And then, a
series of studies on stochastic systems are carried out [28–30].
At the same time, the ANNC method has been many successful applications for some unknown nonlinear systems, such
as adaptive output-feedback control [31–33], pure-feedback [34–
37] and so on. In ANNC, the neural network is often used to online approximate unknown nonlinearity owning to their inherent approximation capabilities. Motivated by the above observations, this work focuses on the feedback nonlinear stochastic
systems with unknown parameters, unknown nonlinear functions and stochastic disturbances, using the adaptive neural control method. By using backstepping method, an adaptive neural
controller is designed. The proposed adaptive neural controller
guarantees that all the signals in the closed-loop are bounded.
Partial results of this paper have been presented in [39].
In this paper, the main contributions in this paper are as
follows: (i) The new controller of adaptive neural network
with state feedback is presented by using a universal approximation of radial basis function neural network and backstepping for a class of stochastic nonlinear systems. Compared
with the results in [46, 47], the gain function for virtual control signal xi+1 is constant 1. In this work, it is extended to a
function of the partial state variables. Therefore, the systems
in [46, 47] are special situations of the system considered for
this work. (ii) Corresponding model is more universal than
references [38, 42, 45]. The unknown disturbances and unknown nonlinear functions are considered. Adaptive bounding
design technique is used to deal with the unknown parameters
and unknown nonlinear functions. The upper bound of the disturbances is not necessary to know in the design of the adaptive controller. When the derivatives of the state are available
for feedback, the designed adaptive controller can be guarantee that all the signals of the closed-loop system are bounded;
(iii) A universal type adaptive feedback controller is designed,
which globally regulates all the states of the uncertain system
while keeping boundedness of all the states to guarantee global
asymptotically stable in probability. The main advantage of the
proposed control scheme is that the all closed-loop signals are
guaranteed to be globally bounded. However, many existing
adaptive backstepping neural control approaches (e.g., see [16–
19, 27, 32, 33, 36–38, 40, 41]) can only guarantee semi-globally
bounded for all the closed-loop signals.

where ϕ∗i = [ϕ∗i,1 , ϕ∗i,2 , · · · , ϕ∗i,nq ]T , ϕ∗i (0) = 0. And b∗i j ≥ 0, is unknown parameter values and ϕ∗i j ( x̄i ) is known smooth functions,
respectively. b∗i j is defined as the smallest nonnegative constant
such that (2) is satisfied.

2. Problem formulation and preliminaries

Remark 1. Consider the system (3) can not satisfy either the
linear parametrization conditions, such as following system

Consider the following stochastic nonlinear system
dxi
dxn

=

(gi ( x̄i )xi+1 + θi∗ T ψ∗i ( x̄i ) + fi ( x̄i ) + ∆i (x, t))dt

=

+φTi ( x̄i )dω 1 ≤ i ≤ n − 1,
(gn ( x̄n )u + θn∗ T ψ∗n ( x̄n ) + fn ( x̄n ) + ∆n (x, t))dt

y(t) =

+φTn ( x̄n )dω,
x1 (t),
T

(1)
n

where x = [x1 , x2 , · · · , xn ] ∈ R is the state vector with initial value x(0), x̄i = [x1 , x2 , . . . , xi ]; u ∈ R is the control input and y ∈ R is the system output, respectively; fi (·) ∈ R
and ψ∗i (·) ∈ Rqi , ψi (0) = 0 are unknown smooth functions with
fi (0) = 0, ψ∗i (0) = 0; gi (·) ∈ R, gi (·) , 0 and φi (·) ∈ Rr are
known smooth functions, θi∗ ∈ Rqi are unknown bounded parameters, i = 1, . . . , n; ∆i (x, t) is unknown disturbances. ω is an
independent r-dimensional standard Wiener process and ∆i (x, t)
satisfy the following assumption.
Assumption 1. [40] For unknown disturbances ∆i (x, t) and
∀(t, x) ∈ R+ × Rn , we have
|∆i (x, t)| ≤ p∗i Φ∗i ( x̄i )

(2)

where p∗i ≥ 0 and Φi ( x̄i ) are some unknown parameter values
and known smooth functions, respectively. And, Φ∗i (0) = 0, we
define p∗i is the smallest nonnegative constant such that (2) is
satisfied.
To simplify the notation, we let θ = [θ1∗ T , · · · , θn∗ T ]T ∈ Rq ,
P
where q := i qi . Therefore, we can rewrite the system (1) as
in following form:
dxi = (gi ( x̄i )xi+1 + θT Ψi ( x̄i ) + fi ( x̄i ) + ∆i (x, t))dt
+ φTi ( x̄i )dω 1 ≤ i ≤ n − 1,
dxn = (gn ( x̄n )u + θT Ψn ( x̄n ) + fn ( x̄n ) + ∆n (x, t))dt
+ φTn ( x̄n )dω,
y(t) = x1 (t),

(3)

where each Ψi : Ri 7→ Rq is given by
Ψi =[Ψi1 , Ψi2 , · · · , Ψin ]
=[0T1 , · · · , 0Ti−1 , ψ∗i , 0Ti+1 , · · · , 0Tn ]T
q

with 0i := [0, · · · , 0]T ∈ Ri .
Assumption 2. For unknown functions Ψi j ( x̄i ) and ∀x ∈ Rn , we
have
(4)
|Ψi j ( x̄i )| ≤ b∗i j ϕ∗i j ( x̄i )

dx1 = ((x21 + 1)x1 x2 + θ1 x21 + θ3 sin (tθ4 x2 ))dt + x1 dω,

2.1. Plant dynamics
In this section, we recall the basic background knowledge
concerning the stability properties of stochastic systems.

dx2 = (e x2 u + θ1 x22 + θ2 e x1 + (θ4 + θ3 sin x1 )x22 ) + e−x2 )dt,
y(t) = x1 (t).
2

It can be put in the form (2) by letting

are the input vector and weight vector, respectively. Denote the
number of nodes in the neural network is l > 1; and

g1 = x21 + 1, g2 = e−x2 , θ = [θ1 θ2 ]T ,
f1 = 0, f2 = e−x2 ,

φ1 = x1 , φ2 = 0,

Ψ1 = [x21 0]T ,

Ψ2 = [x22 e x1 ]T ,

∆1 = θ3 sin(tθ4 x2 ),

∆2 = (θ4 + θ3 sin

S (Z) = [s1 (Z), . . . , sl (Z)]T ,
with si (Z) being chosen as the following form Gaussian functions


 −(Z − µi )T (Z − µi ) 
 , (i = 1, 2, . . . , l.)
si (Z) = exp 
η2i

x1 )x22 .

The bounds on |∆1 | ≤ p∗1 , |∆1 | ≤ p∗1 x22 , where
p1 := |θ3 |, p2 := |θ3 | + |θ4 |.

where µi = [µ1 , µ2 , . . . , µq ]T and η represent the center of the
receptive field and the width of the Gaussian function, respectively. Is well known that neural network (7) can approximate
any continuous function defined on a compact set ΩZ ⊂ Rq ,
furthermore, the neural network can is given by

Definition 1. [48, 49] If for any ε > 0,
(
)
lim P sup k x(t)|| ≥ ε = 0,
x(0)→0

t≥0

Q(Z) = Qnn (Z, W ∗ ) + ε(Z) ∀ ∈ ΩZ ,

n
o
and for any initial condition x(0), we have P lim x(t) = 0 =
t→∞
1, then we claim that the solution {x(t) = 0} of system (3) is
asymptotically stable in the large.

where W ∗ is the ideal neural network weights and |ε(Z)| ≤ ε∗
is the neural network approximation error. W ∗ is ideal constant
weights, and for all Z ∈ ΩZ , there exist constant ε∗ > 0 such
that |ε| ≤ ε∗ . Moreover, W ∗ is bounded by kW ∗ k ≤ wm on the
compact set ΩZ , where wm is a positive constant.
W ∗ is usually unknown and need to be estimated in function
approximation. W ∗ is defined as follows:
#
"
∗
W = arg min sup |Qnn (Z, W) − Q(Z)| ,

Definition 2. [49] If
n
o
lim sup P kx(t)k ≥ ε = 0.

ε→∞ t≥0

The solution process {x(t), t ≥ 0} of system (3) is said to be
bounded in probability.
Denote L with

(W)

LV(x) =

∂V
1 n ∂2 V o
f + T r gT 2 g ,
∂x
2
∂x

3. Adaptive control design and stability analysis

Lemma 1. [41] For system (3), if there exists a positive definite
radial unbounded Lyapunov functions V1 (x) ∈ C 2 (Rn ; R) and
V2 (θ) ∈ C 2 (Rm ; R), and V(x, θ) = V1 (x) + V2 (θ) satisfying

In this section, adaptive control design is given by backstepping design, and there contain n steps in backstepping design procedure. At each step, an intermediate control function
αi (t) shall be developed using an appropriate Lyapunov function Vi (t). The design of both the control laws and the adaptive
laws are based on the following change of coordinates:

(6)

where constants λ > 0, K ≥ 0, then, the stochastic system (3)
is globe bounded stable in probability.

z1 = x1 , zi = xi − αi−1 , i = 2, . . . , n,

Lemma 2. [43] For any ǫ > 0 and u ∈ R, the following inequality holds

(9)

where αi (t) is an intermediate control. And, the system controller u(t) will is designed in the last step to stabilize the entire
closed-loop system.
Step 1: The Lyapunov function candidate is consider as

0 ≤ |u| − u tanh(u/ǫ) ≤ δǫ,
where δ = 0.2785.

z41 ϑ̃T1 Γ−1
p̃T Γ p1 p̃1 W̃1T Γ−1
ε̃2
w1 W̃1
ϑ1 ϑ̃1
V1 =
+
+ 1
+
+ 1 ,
4
2
2
2
2γε1

2.2. RBFNN Approximation
In this work, the RBFNN is is used to approximate a nonlinear continuous function. The continuous function Q(Z) : Rq →
R is given by
Qnn (Z, W) = W T S (Z),

Z∈ΩZ

which is unknown and needs to be estimated in control design.
Let Ŵ be the estimate of W ∗ , and let W̃ = Ŵ − W ∗ be the weight
estimation error.

(5)

the infinitesimal generator of the solution of stochastic system
(3) for any V(x) ∈ C 2 (Rn ; R).

LV(x) ≤ −λV(x) + K,

(8)

(10)

where Γ−1
ϑ1 and Γw1 are symmetric positive definite matrices,
Γ p1 ≤ 0, Γε1 ≤ 0. Let ϑ̂i is the ith times estimate of ϑ, for clarify
in every step, denote ϑi = ϑ for the ith step. In order to facilitate
of analysis, we denote gi ( x̄i (t)) = gi , others may be deduced by
analogy, Then, we have

(7)

where Z ∈ Ω ⊂ Rq and,
W = [w1 , w2 , . . . , wl ]T ∈ Rl ,

dz1 = (g1 (z2 + α1 ) + θ1T Ψ1 + f1 + ∆1 (x, t))dt + φT1 dω, (11)
3

Let Λ1 = ∆1 , p1 = p∗1 , Φ1 = Φ∗1 , ϕ1 = ϕ∗1 . By using (9) and (11),
we have

Then, we have
1
˙
3
LV1 ≤ − c1 z41 + z22 − ϑ̃T1 (Γ−1
ϑ1 ϑ̂1 − z1 ϕ1 )
4
3
˙
− ε̃1 (γε−11 ε̂˙ 1 + z31 ̟10 ) − p̃T1 (Γ−1
p1 p̂1 + z1 ̟11 )
˙ + z3 S (Z )) + |z |3 ε∗ − z3 ε∗ ̟
− W̃ T (Γ−1 Ŵ

dz1 = (g1 (z2 + α1 ) + θ1T Ψ1 + f1 + Λ1 (x, t))dt + φT1 dω. (12)
Note (5)(10) and (12), we can obtain

1

LV1 = z31 (g1 (z2 + α1 ) + θ1T Ψ1 + f1 + Λ1 (x, t))
˙
− θ̃T Γ−1 θ̂˙ − p̃T Γ−1 p̂˙ − γ−1 ε̃T ε̂˙ − W̃ T Γ−1 Ŵ
θ1

1

p1

1

ε1

1 1

1

w1

1

+ 3z21 φT1 φ1 /2.

(13)

3z4
3ǫ11
+ 1 k φ1 k4 ,
4
4ǫ11
1
3
4/3
g1 z31 z2 ≤ g1 z41 + z42 ,
4
4

≤
≤
≤
=

q
X

k=1
q
X

1

1

1

1

1 1

10

ϑ̂˙ 1 = − Γϑ1 (z31 ϕ1 − σϑ1 ϑ̂1 ), (σϑ1 > 0)
ε̂˙ 1 = − γε1 (z31 ̟10 − σε1 ε̂1 ), (σε1 > 0)
p̂˙ 1 = − Γ p1 (z3 ̟11 − σ p1 p̂1 ), (σ p1 > 0)

(14)
(15)

(19)

1

˙ = − Γ (z3 S (Z ) − σ Ŵ ), (σ > 0)
Ŵ
w1
1
w1 1
1
w1 1

and applying Assumption (2), we have
θ1T Ψ1

1

The adaptive laws are chosen as

Let Q1 (Z1 ) = f1 . Applying Yong’s inequalities, we have
3z21 φT1 φ1 /2 ≤

w1

+ |z1 |3 pT1 Φ1 (x1 ) − z31 pT1 ̟11 + |z1 |3 ϑT1 ϕ1 (x1 ) − z31 ϑT1 ̟12
3
(18)
+ ǫ11 .
4

Note (18) and Lemma (2), we have
|θ1k |ϕ∗1k

1
LV1 ≤ − c1 z41 + z22 + σϑ1 ϑ̃T1 ϑ̂1 + σε1 ε̃1 ε̂1 + σ p1 p˜1 T p̂1
4
3
T
+ σW1 W̃1 Ŵ1 + 0.2785(ε10 + ε11 + ε12 ) + ǫ11 . (20)
4

|θ1k |b1k ϕ∗1k

k=1
ϑT1 ϕ∗1
ϑT1 ϕ1

Applying following inequalities
(16)

1
1
σϑ1 ϑ̃T1 ϑ̂1 ≤ − σϑ1 k ϑ̃1 k2 + σϑ1 k ϑ1 k2 ,
2
2
1
1
2
σε1 ε̃1 ε̂1 ≤ − σε1 k ε̃1 k + σεε1 kε1 k2 ,
2
2
1
1
2
σ p1 p̃1 p̂1 ≤ − σ p1 k p̃1 k + σ p1 k p1 k2 ,
2
2
1
1
T
2
σw1 W̃1 Ŵ1 ≤ − σw1 k W̃1 k + σw1 k W1∗ k2 ,
2
2

where
ϑ1 = [|θ11 |b11 , · · · , |θ1q |b1q ]T , ϕ∗1 = [ϕ∗11 , · · · , ϕ∗1q ]T .
From (13)-(16) and applying Assumption 1, we have
3
1
LV1 ≤ g4/3
z41 + z42 + z31 g1 α1 + |z1 |3 ϑT1 ϕ1 + z31 W1∗ T S (Z1 )
1
4
4
˙
T −1 ˙
+ |z1 |3 ε∗1 + |z1 |3 p1 Φ1 (x1 ) − θ̃T Γ−1
θ1 θ̂ − p̃1 Γ p1 p̂1
3
˙
− γε−11 ε̃T1 ε̂˙ 1 − W̃1T Γ−1
w1 Ŵ1 + ǫ11
4
3z41
k φ1 k4 .
(17)
+
4ǫ11

(21)

we can obtain
1
1
1
LV1 ≤ − c1 z41 + z42 − σϑ1 k ϑ̃1 k2 − σε1 k ε̃1 k2
4
2
2
1
1
1
− σ p1 k p̃1 k2 − σw1 k W̃1 k2 + σ p1 k p1 k2
2
2
2
1
3
1
+ σϑ1 k ϑ1 k2 + σw1 k W1∗ k2 + ǫ11
2
2
4
+ 0.2785(ε10 + ε11 + ε12 )
1
(22)
≤ − λ1 V1 + K1 + z42 ,
4

Let intermediate law as
3 4/3
α1 = g−1
1 (−c1 z1 − g1 z1 − β10 − β11 − β12
4
3z1
T
k φ1 k4 ),
− Ŵ1 S (Z1 ) −
4ǫ11
z3
β10 = ε̂1 ̟10 , ̟10 := tanh[ 1 ],
ε10
z3 Φ1 ( x̄1 )
];
β11 = p̂T1 ̟11 , ̟11 := Φ1 ( x̄1 ) tanh[ 1
ε11
z3 ϕ1 ( x̄1 )
β12 = ϑ̂T1 ̟12 , ̟12 := ϕ1 ( x̄1 ) tanh[ 1
];
ε12

where





σ p1
σϑ1
σw1 


,
4c , γ
,
λ1 = min 
,
,



−1
 1 ε1 σε1 max(Γ−1 ) max(Γ−1

p1 ) max(Γw1 )
ϑ1

1
1
1
K1 = σ p1 k p1 k2 + σϑ1 k ϑ1 k2 + σw1 k W1∗ k2
2
2
2
3
+ ǫ11 + 0.2785(ε10 + ε11 + ε12 ).
4

4

Step i: (2 ≤ i < n). Similarly, we can obtain
h
i
dzi = gi (zi+1 + αi ) + θiT Ψi + fi + ∆i (x, t) dt + φi dω
 i−1
X ∂αi−1
− 
(g j x j+1 + θTj Ψ j + f j + ∆ j )
∂x j
j=1
+

i−1
X
∂αi−1

ϑ̂˙ j +

i−1
X
∂αi−1

˙ +
Ŵ
j

i−1
X
∂αi−1

∂ε̂ j

i−1
i−1
X

∂αi−1 ˙
1 X ∂2 αi−1
+
gk gTj  dt
p̂ j +
∂ p̂ j
2 k, j=1 ∂xk ∂x j
j=1
∂ϑ̂ j

j=1

−

i−1
X
∂αi−1

∂x j

j=1

j=1

∂Ŵ j

with 2 ≤ i ≤ n.
By using (23)–(30), we have
1
3
+ z4i+1 + z3i gi αi
LVi ≤ LVi−1 + g4/3
i
4
4
+ |zi |3 ϑTi ϕ̄i + z3i Wi∗ T S (Zi )
+ |zi |3 ε∗i + |z|3i pTi Φ̄i − z3i

ε̂˙ j

j=1

φ j dω.

∂αi−1 ˙
∂αi−1 ˙
∂αi−1 ˙
∂αi−1 ˙
ε̂ j +
p̂ j )
Ŵ j +
ϑ̂ j +
∂ε̂ j
∂ p̂ j
∂Ŵ j
∂ϑ̂ j
i−1
z3 X
∂2 αi−1
3ǫi1
− i
gk gTj +
2 k, j=1 ∂xk ∂x j
4

+

(23)

Applying Yongs inequalities, we have

+

i−1
i−1
X
X
3z2i
∂αi−1 T
∂αi−1
φ j )(φi −
φ j)
(φi −
2
∂x
∂x j
j
j=1
j=1
i−1
X
3ǫi1 3z4i
∂αi−1
≤
k φi −
φ j k4 ,
+
4
4ǫi1
∂x
j
j=1

3
1
gi z3i zi+1 ≤ g4/3
z4 + z4 .
4 1 i 4 i+1

Qi (Zi ) = fi −

∂x j

j=1

|Λi | = |∆i −

i−1
X
∂αi−1
j=1

|Υi | = |θiT Ψi −

∂x j

∆ j |, (2 ≤ i ≤ n)

i−1
X
∂αi−1
j=1

Vi =Vi−1 +

f j ( x̄ j ), (2 ≤ i ≤ n)

∂x j

θiT Ψ j |, (2 ≤ i ≤ n)

(31)

We select intermediate law αi , 2 ≤ i ≤ n − 1 as
"
1
3
− ci zi − zi − g4/3
zi − βi0 − βi1 − βi2
αi = g−1
i
4
4 1
i−1
1 X ∂2 αi−1
gk gTj
− ŴiT S (Zi ) +
2 k, j=1 ∂xk ∂x j

(24)
(25)

i−1
X
∂αi−1 ˙
∂αi−1
∂αi−1 ˙
∂αi−1 ˙
ϑ̂ j +
(
Ŵ j +
x j+1 +
ε̂ j
∂x
∂ε̂ j
∂
Ŵ
∂
ϑ̂
j
j
j
j=1

i−1
X

∂αi−1
3zi
∂αi−1 ˙
4
k φi −
φ j k  .
p̂ j ) −
+
(32)
∂ p̂ j
4ǫi1
∂x j
j=1

+

(26)

(27)
Let
(28)

βi0 = ε̂i ̟i0 ,

ϑ̃T Γ−1
p̃T Γ pi p̃i W̃iT Γ−1
ε̃2
z4i
ϑi ϑ̃
wi W̃i
+
+ i
+
+ i ,
4
2
2
2
2γεi
(29)

̟i0 := tanh[

z3i
],
εi0

z3i Φi
],
εi1
z3 ϕi
βi2 = ϑ̂Ti ̟i2 , ̟i2 := ϕi ⊙ tanh[ i ],
εi1

βi1 = p̂Ti ̟i1 , ̟i1 := Φi ⊙ tanh[

Define operator |θi |1 = [|θi1 |, |θi2 |, · · · , |θiq |]T , then using assumption (1) and (2), we have
|Λi | ≤ pi T Φ̄i , |Υi | ≤ ϑTi ϕ̄i ,

i−1
X
3z4i
∂αi−1
˙
k φi −
φ j k4 −ϑ̃Ti Γ−1
ϑi ϑ̂i
4ǫi1
∂x
j
j=1

˙
−1 ˙
T −1 ˙
− W̃iT Γ−1
Wi Ŵi − p̃i Γ pi p̂i − γεi ε̃i ε̂i .

Let
i−1
X
∂αi−1

i−1
X
∂αi−1
x j+1
(
∂x j
j=1

(33)

where

(30)

z3 Φi
Φi ⊙ tanh[ i ]
εi1

∗
∗
i−1
i−1
 ∂αi−1
z3i ∂α
z3i ∂α
∂αi−1 ∗
∂xi−1 Φi−1
∂x1 Φ1
∗

Φ1 tanh[
], · · · ,
Φi−1 tanh[
],
= 
∂x1
εi1
∂xi−1
εi1
T
z3 Φ∗ 
Φ∗i tanh[ i i ] ,
εi1

where
pi = [p∗1 , p∗2 , · · · , p∗i ]T ,
iT
h ∂αi−1
∂αi−1 ∗
∂αi−1 ∗
|Φ∗1 , |
|Φ2 , · · · , |
|Φi−1 , Φ∗i ,
Φ̄i = |
∂x1
∂x2
∂xi−1
iT
h
∗ T
∗ T
ϑi = (|θ1 |1 •b1 ) , (|θ2 |1 •b2 ) , · · · , (|θi |1 •b∗i )T ,
h ∂αi−1 T ∂αi−1 T
∂αi−1 ∗ T ∗ T iT
ϕ̄i = |
.
|ϕ∗1 , |
|ϕ∗2 , · · · , |
|ϕ
, ϕi
∂x1
∂x2
∂xi−1 i−1
Let
∂αi−1 ∗
∂αi−1 ∗ ∂αi−1 ∗
Φ1 ,
Φ2 , · · · ,
Φ , Φ∗ ]T ,
∂x1
∂x2
∂xi−1 i−1 i
h ∂αi−1 T ∂αi−1 T
∂αi−1 ∗ T ∗ T iT
ϕi =
,
ϕ∗1 ,
ϕ∗2 , · · · ,
ϕ
, ϕi
∂x1
∂x2
∂xi−1 i−1

Φi = [

5

z3 ϕi
ϕi ⊙ tanh[ i ]
εi2

i−1 ∗
i−1 ∗
 ∂αi−1
z3i ∂α
z3i ∂α
∂αi−1 ∗
∂x1 ϕ1
∂xi−1 ϕi−1
∗


= 
ϕ tanh[
], · · · ,
ϕ tanh[
],
∂x1 1
εi2
∂xi−1 i−1
εi2
T
z3i ϕ∗i 
∗
] .
ϕi tanh[
εi2

and using Lemma (2) we can obtain

A similar procedure is employed, by using (24 − 30, 33, 35), we
have
n
n
X
X
LVn ≤ −
λi Vi +
Ki ,

1
˙
3
LVi ≤LVi−1 − z4i − ci z4i − W̃iT (Γ−1
wi Ŵi + zi S (Zi ))
4
3
˙
− ε̃i (γε−1i ε̂˙ i + z3i ̟i0 ) − p̃Ti (Γ−1
pi p̂i + zi ̟i1 )


˙
3
− ϑ̃Ti Γ−1
ϑi ϑ̂i + zi ̟i2
+ 0.2785(εi0 + εi1 + εi2 ) +

3ǫi1 1 4
+ zi+1 .
4
4

i=1

where





σ pi
σwi 
σϑi


,
,
,
4ci , γεi σεi ,
λi = min 


−1
−1
−1


max(Γ ) max(Γ pi ) max(Γwi ) 

(34)

ϑi

1
1
1
Ki = σ pi k pi k2 + σϑi k ϑi k2 + σwi k Wi∗ k2
2
2
2
3
+ ǫi1 + 0.2785(εi0 + εi1 + εi2 ),
4

The adaptive laws are chosen as
ε̂˙ i = − γεi (z3i ̟i0 − σεi ε̂i ), (σεi > 0),
p̂˙ i = − Γ pi (z3i ̟i1 − σ pi p̂i ), (σ pi > 0),
ϑ̂˙ = − Γ (z3 (̟ − σ ϑ̂ ), (σ > 0),
i

ϑi

i

ϑi i

i2

(35)

with 1 ≤ i ≤ n.
At the present stage, ANC design has been completed based
on backstepping technique. The main result can be summarized
by the following theorem.

ϑi

˙ = − Γ (z3 S (Z ) − σ Ŵ ), (σ > 0).
Ŵ
i
wi i
i
wi i
wi
Note (21), (34), (35) can be written as
1
LVi ≤ LVi−1 − z4i − ci z4i + σϑi ϑ̃Ti ϑ̂i + σεi ε̃i ε̂i
4
T
+ σ pi p̃i p̂i + σWi W̃iT Ŵi + 0.2785(εi0 + εi1 + εi2 )
3
1
+ ǫi1 + z4i+1
(36)
4
4
1
1
1
≤ LVi−1 − ci z4i + z4i+1 − σϑi k ϑ̃i k2 − σεi k ε̃1 k2
4
2
2
1
1
1
2
2
− σ pi k p̃i k − σwi k W̃i k + σ pi k pi k2
2
2
2
3
1
1
2
∗ 2
+ σϑi k ϑi k + σwi k Wi k + ǫi1
2
2
4
1 4
1 4
≤ (LVi−1 − zi ) − λ1 Vi + Ki + zi+1
4
4
i
i
X
X
1
≤ −
λ jV j +
K j + z4i+1 ,
(37)
4
j=1
j=1

Theorem 1. Consider stochastic nonlinear systems (1), assumptions (1) and (2). We can design state-feedback adaptive controller (38) and adaptation laws (19) and (35) by using backstepping approach. For bounded initial conditions, the following prosperities hold.
(I) The closed-loop system’s the equilibrium at the origin is
asymptotically stable in probability.
(II) The closed-loop system has an almost surely unique solution on [0, ∞) for each x(0), ϑ̂(0), Ŵ(0), ε̂(0), p̂(0), system
state x(t) and the parameter estimates ϑ̂(t), Ŵ(t), ε̂(t), p̂(t)
satisfy
n
o
P lim x(t) = 0 = 1;
t→∞
n
P lim ϑ̂(t), lim Ŵ(t), lim ε̂(t) and lim p̂(t)
t→∞
t→∞
t→∞
t→∞
o
exist and are f inite = 1.

where





σ pi
σwi 
σϑi


,
,
λi = min 
,
,
4c , γ


−1 ) 

 i εi σεi max(Γ−1 ) max(Γ−1
)
max(Γ
pi
wi

4. Simulation studies

ϑi

The following example is given to show the effectiveness of
the proposed adaptive neural control algorithms for stochastic
nonlinear system (39). we consider the following second-order
system.

1
1
1
Ki = σ pi k pi k2 + σϑi k ϑi k2 + σwi k Wi∗ k2
2
2
2
3
+ ǫi1 + 0.2785(εi0 + εi1 + εi2 ).
4
Step n: We select the control
"
1
−1
u =gn − cn zn − zn − βn0 − βn1 − βn2
4
n−1
1 X ∂2 αn−1
− ŴnT S (Zn ) +
φk φTj
2 k, j=1 ∂xk ∂x j
+

dx1 = (x2 + f1 (x1 ) + ∆1 (x, t))dt + x1 cos x1 dω,
dx2 = ((1 + 0.5 sin x1 )u + θ2∗ ψ∗2 + f2 (x))dt
+ sin x2 dω,

(39)

y = x1 ,
where

n−1
X
∂αn−1 ˙
∂αn−1 ˙
∂αn−1 ˙
∂αn−1
x j+1 +
ε̂ j
Ŵ j +
ϑ̂ j +
(
∂x
∂ε̂ j
∂Ŵ j
j
∂ϑ̂ j
j=1


n−1
X

3zn
∂αn−1 ˙
∂αn−1
4
p̂ j ) −
+
k φn −
φ j k  .
∂ p̂ j
4ǫn1
∂x j
j=1

i=1

(38)
6

g1 = 1, θ1∗ ϕ∗1 = 0,

f1 = x1 sin x1 ,

∆1 = 0.5x1 sin x2 t,

φ1 = x1 cos x1 ,

g2 = (1 + 0.5 sin x1 ),

θ2∗ ψ∗2 = 0.02x2,

f2 = x2 cos x2 ,

∆2 = 0,

φ2 = sin x2 .

shows that boundedness of weights kŴ1 k and kŴ2 k, Fig.4 displays the boundedness of approximation error ε̂1 and ε̂2 , and
Fig.5 and Fig.6 show the response curve of the adaptive parameter k p̂1 k, k p̂2 k, kϑ̂1 k and kϑ̂2 k.

The adaptive laws can be designed in the following:
ε̂˙ 1 = γε1 (z31 ̟10 − σε1 ε̂1 ),
p̂˙ 1 = Γ p1 (z31 ̟11 − σ p1 p̂1 ),
˙ = Γ (z3 S (Z ) − σ Ŵ ),
Ŵ
1
w1 1
1
w1 1
˙ϑ̂ = Γ (z3 ϕ − σ ϑ̂ ),
1

ϑ1

1 1

ϑ1 1

ε̂˙ 2 = γε2 (z32 ̟20 − σε2 ε̂2 ),
p̂˙ 2 = Γ p2 (z32 ̟21 − σ p2 p̂2 ),
˙ = Γ (z3 S (Z ) − σ Ŵ ),
Ŵ
2
w2 2
2
w2 2
˙ϑ̂ = Γ (z3 ̟ − σ ϑ̂ ).
2

ϑ2

2

22

ϑ2 2

Finally, the system control law can be designed as
"
∂α1
1
T
ϕ1 )
u = g−1
2 −c2 z2 − zn − β20 − β21 − ϑ̂2 (ϕ2 −
4
∂x1
∂α1
∂α1 ˙
1 ∂2 α1 2
∂α1 ˙
φ +(
ϑ̂1 +
− Ŵ2T S (Z2 ) +
Ŵ1
x2 +
2 ∂x21 1
∂x1
∂
Ŵ1
∂ϑ̂1
#
∂α1 4
∂α1 ˙
3z2
∂α1 ˙
(φ2 −
φ1 ) ,
ε̂1 +
p̂1 ) −
+
∂ε̂1
∂ p̂1
4ǫ21
∂x1

Figure 1: System states x1 :“solid line.” x2 :“dash line”.

where
i
h
ω20 = tanh z32 /ε20 ,
i
h
 ∂α
 1 φ∗ tanh z3 ∂α1 φ∗ /ε21
1
1
∂x
∂x
2
1
ω21 =  1
0



 .


The initial conditions and design parameters are given as following.
θ̂1 (0) = [0 0.1]T ,

θ̂2 (0) = [0 0.8]T ,

b̂1 = [0 1]T ,

b̂2 = [0 0.1]T ,

ε̂1 (0) = 0.1e-3,

ε̂2 (0) = 0, p̂1 (0) = 0.1,

p̂2 (0) = [0 0.15]T ,

Ŵ1 (0) = Ŵ1 (0) = 0,

Γϑ1 = diag(0.3 0.3),

Γϑ2 = diag(0.25 0.25),

γε1 = 0.3, γε2 = 0.4,

Γ p1 = 0.3, Γ p2 = [0.4 0],

σϑ1 = 0.3, σϑ2 = 0.25,

σε1 = 0.3, σε1 = 0.4,

σ p1 = 0.3, σ p2 = 0.4,

σw1 = 1.5, σw2 = 0.3,

Figure 2: Control input u

ε10 = ε20 = ε11 = ε21 = 0.3, c1 = c2 = 0.3.
Specifically, neural network Ŵ1T S (Z1 ) contains 27 nodes (i.e.
l1 = 27) with centers µl (l = 1, . . . , l1 ) evenly spaced in [−1.5, 1.5]
and widths ηl = 0.8(l = 1, . . . , l1 ). Neural networks Ŵ2T S (Z2 )
contains 64 nodes(i.e. l2 = 64) with centers µl (l = 1, . . . , l2 )
evenly spaced in [−1.5, 1.5]×[−1.5, 1.5]×[−1.5, 1.5]×[−1.5, 1.5],
and widths ηl = 1.5(l = 1, . . . , l1 ).
The simulation results are shown in Fig.1–Fig.6, from which
we can see that the controller renders the resulting closed-loop
system asymptotically stable and the limits of estimated parameters exist and are finite. Fig.1 shows that x1 and x2 converge
to zero rapidly. Fig.2 displays the control input signal u, Fig.3

Figure 3: Boundedness of weights k Ŵ1 k:“solid line.”k Ŵ2 k:“dash line”.

7

technique is used to deal with the unknown parameters and unknown functions. we have designed a universal type adaptive
feedback controller, which keep boundedness of all the states
to guarantee global asymptotically stable in probability. Simulation has been conducted to show the performance of the proposed approach.
Acknowledgements
This work is partially supported by the National Natural
Science Foundation of China (61503133, 51374107, 51577057,
61304162), by Major State Basic Research Development Program (973) sub-project (61325309), and by Postdoctoral Science Foundation of China under Grant (2016M592449), and by
Natural Science Foundation of Hunan Province (2016JJ6043,
14JJ3110), and by Research Foundation of Education Bureau
of Hunan Province (15C0548).

Figure 4: Boundedness of approximation error ε̂1 :“solid line.”ε̂2 :“dash line”.

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