# Direct Adaptive Tracking Control for a Class of Pure-Feedback Stochastic Nonlinear Systems Based on Fuzzy-Approximation

Authors: Huanqing Wang; Xiaoping Liu; Qi Zhou; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The problem of fuzzy-based direct adaptive tracking control is considered for a class of pure-feedback stochastic nonlinear systems. During the controller design, fuzzy logic systems are used to approximate the packaged unknown nonlinearities, and then a novel direct adaptive controller is constructed via backstepping technique. It is shown that the proposed controller guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. The main advantages lie in that the proposed controller structure is simpler and only one adaptive parameter needs to be updated online. Simulation results are used to illustrate the effectiveness of the proposed approach.


---

## Body

## 1. Introduction

During the past decades, many control methods have been developed to control design of nonlinear systems, such as adaptive control [1–3], backstepping control [4], and fault tolerant control [5–9]. Particularly, backstepping-based adaptive control has been an effective tool to deal with the control problem of nonlinear strict-feedback systems without satisfying matching condition. So far, many interesting results have been obtained for deterministic nonlinear systems in [4, 10–21] and for stochastic cases in [22–35]. In the aforementioned papers, however, the existing results required that the nonlinear functions be in the affine forms; that is, systems are characterized by input appearing linearly in the system state equation.Pure-feedback nonlinear systems, which have no affine appearance of the state variables that can be used as virtual control and the actual control input, stand for a more representative form than strict-feedback systems. Many practical systems are in nonaffine structure, such as biochemical process [4] and mechanical systems [36]. Therefore, the study on stability analysis and controller synthesis for pure-feedback nonlinear system is important both in theory and in practice applications [37–42]. By combining adaptive neural control and backstepping, in [37, 38], a class of pure-feedback systems was investigated, which contained only partial nonaffine functions and at least a control variable or virtual control signal existed in affine form. In [39], an “ISS-modular” approach combined with the small-gain theorem was presented for adaptive neural control of completely nonaffine pure-feedback systems. Recently, in [41], an adaptive neural tracking control scheme was presented for a class of nonaffine pure-feedback systems with multiple unknown state time-varying delays. On the other hand, it is well known that stochastic disturbance often exists in practical systems and is usually a source of instability of control systems. So, the consideration of the control design for pure-feedback nonlinear systems with stochastic disturbance is a meaningful issue and has attracted increasing attention in the control community in recent years [43–45]. In [43], the problem of adaptive fuzzy control for pure-feedback stochastic nonlinear systems has been reported. Then, Yu et al. [44] presented an adaptive neural controller to guarantee the four-moment boundedness for a class of uncertain nonaffine stochastic nonlinear system with time-varying delays. However, the considered systems in [43–45] are of a special kind of pure-feedback stochastic nonlinear systems in which only the last equation was viewed as nonaffine equation.Motivated by the above observations, we will develop a novel fuzzy-based direct adaptive tracking control scheme for a class of pure-feedback stochastic nonlinear systems. The presented controller guarantees that all the signals in the closed-loop system remain bounded in probability and the tracking error converges to a small neighborhood around the origin in the sense of mean quartic value. The main contributions of this paper lie in that the structure of the proposed controller is simpler and only one adaptive parameter needs to be updated online. As a result, the computational burden is significantly alleviated and the control scheme may be more implemented in practice.The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section2. An adaptive fuzzy tracking control scheme is presented in Section 3. The simulation example is given in Section 4, and it is followed by Section 5 which concludes the work.


## 2. Problem Formulation and Preliminaries

To introduce some useful conceptions and lemmas, consider the following stochastic system:(1)
d
x
=
f
(
x
,
t
)
d
t
+
h
(
x
,
t
)
d
w
,
∀
x
∈
R
n
,
where x
∈
R
n is state vector, w is a r-dimensional independent standard Brownian motion defined on the complete probability space (
Ω
,
F
,
{
F
t
}
t
≥
0
,
P
) with Ω being a sample space, F being a σ-field, {
F
t
}
t
≥
0 being a filtration, and P being a probability measure, and f
:
R
n
×
R
+
→
R
n,  h
:
R
n
×
R
+
→
R
n
×
r are locally Lipschitz functions in x and satisfy f
(
0
,
t
)
=
0,  h
(
0
,
t
)
=
0.Definition 1 (see [25]).
For any givenV
(
x
,
t
)
∈
C
2,1, associated with the stochastic differential equation (1), define the differential operator L as follows:
(2)
L
V
=
∂
V
∂
t
+
∂
V
∂
x
f
+
1
2
Tr
⁡
{
h
T
∂
2
V
∂
x
2
h
}
,
where Tr
⁡
(
A
) is the trace of A.Remark 2.
The term(
1
/
2
)
Tr
⁡
{
h
T
(
∂
2
V
/
(
∂
x
2
)
h
} is called Itô correction term, in which the second-order differential ∂
2
V
/
∂
x
2 makes the controller design much more difficult than that of the deterministic system.Definition 3 (see [46]).
The trajectory{
x
(
t
),  t
≥
0
} of stochastic system (1) is said to be semiglobally uniformly ultimately bounded in pth moment, if, for some compact set Ω
∈
R
n and any initial state x
0
=
x
(
t
0
), there exist a constant ɛ
>
0 and a time constant T
=
T
(
ɛ
,
x
0
) such that E
(
|
x
(
t
)
|
p
)
<
ɛ, for all t
>
t
0
+
T. Particularly, when p
=
2, it is usually called semiglobally uniformly ultimately bounded in mean square.Lemma 4 (see [46]).
Suppose that there exist aC
2,1 function V
(
x
,
t
)
:
R
n
×
R
+
→
R
+, two constants c
1
>
0 and c
2
>
0, and class K
∞
-functions α
¯
1 and α
¯
2 such that
(3)
α
¯
1
(
|
x
|
)
≤
V
(
x
,
t
)
≤
α
¯
2
(
|
x
|
)
,
L
V
(
x
,
t
)
≤
-
c
1
V
(
x
,
t
)
+
c
2
,
for all x
∈
R
n and t
>
t
0. Then, there is a unique strong solution of system (1) for each x
0
∈
R
n and it satisfies
(4)
E
[
V
(
x
,
t
)
]
≤
V
(
x
0
)
e
-
c
1
t
+
c
2
c
1
,
∀
t
>
t
0
.In this paper, we consider a class of pure-feedback stochastic nonlinear systems described by(5)
d
x
i
=
f
i
(
x
¯
i
,
x
i
+
1
)
d
t
+
h
i
T
(
x
¯
i
)
d
w
,
1
≤
i
≤
n
-
1
,
d
x
n
=
f
n
(
x
¯
n
,
u
)
d
t
+
h
n
T
(
x
¯
n
)
d
w
,
y
=
x
1
,
where x and w are defined in (1), u
∈
R and y
∈
R represent the control input and the system output, and respectively, x
¯
i
=
[
x
1
,
x
2
,
…
,
x
i
]
T
∈
R
i, f
i
(
·
)
:
R
i
+
1
→
R,  h
i
(
·
)
:
R
i
→
R
r (i
=
1,2
,
…
,
n) are unknown smooth nonlinear functions.The control objective is to design a fuzzy-based adaptive tracking control lawu for system (5) such that the system output y follows a desired reference signal y
d in the sense of mean quartic value, and all signals in the closed-loop system remain bounded in probability. To this end, define y
¯
d
(
i
)
=
[
y
d
,
y
d
(
1
)
,
…
,
y
d
(
i
)
]
T,  i
=
1,2
,
…
,
n, where y
d
(
i
) denotes the ith time derivative of y
d.For the system (5), define
(6)
g
i
(
x
¯
i
,
x
i
+
1
)
=
∂
f
i
(
x
¯
i
,
x
i
+
1
)
∂
x
i
+
1
,
i
=
1,2
,
…
,
n
,
where x
n
+
1
=
u.Assumption 5 (see [42]).
The signs ofg
i
(
x
¯
i
,
x
i
+
1
),  i
=
1,2
,
…
,
n, are known, and there exist unknown constants b
m and b
M such that for 1
≤
i
≤
n,
(7)
0
<
b
m
≤
|
g
i
(
x
¯
i
,
x
i
+
1
)
|
≤
b
M
<
∞
,
∀
(
x
¯
i
,
x
i
+
1
)
∈
R
i
×
R
.Assumption 6 (see [17]).
The reference signaly
d
(
t
) and its nth order derivatives are continuous and bounded.In this note, fuzzy logic system will be used to approximate a continuous functionf
(
x
) defined on some compact sets. Adopt the singleton fuzzifier, the product inference, and the center-average defuzzifier to deduce the following fuzzy rules:R
i
: IFx
1 is F
1
i and x
2 is F
2
i and … and x
n is F
n
i.
Theny is B
i (i
=
1
,
2
,
…
,
N),wherex
=
[
x
1
,
…
,
x
n
]
T
∈
R
n and y
∈
R are the input and output of the fuzzy system, respectively. F
i
j and B
i are fuzzy sets in R. Since the strategy of singleton fuzzification, center-average defuzzification, and product inference is used, the output of the fuzzy system can be formulated as
(8)
y
(
x
)
=
∑
j
=
1
N
W
¯
j
∏
i
=
1
n
μ
F
i
j
(
x
i
)
∑
j
=
1
N
[
∏
i
=
1
n
μ
F
i
j
(
x
i
)
]
,
where W
¯
j is the point at which fuzzy membership function μ
B
j
(
W
¯
j
) achieves its maximum value, which is assumed to be 1. Let
(9)
s
j
(
x
)
=
∏
i
=
1
n
μ
F
i
j
(
x
i
)
∑
j
=
1
N
‍
[
∏
i
=
1
n
μ
F
i
j
(
x
i
)
]
,
S
(
x
)
=
[
s
1
(
x
)
,
…
,
s
N
(
x
)
]
T, and W
=
[
W
¯
1
,
…
,
W
¯
N
]
T. Then, the fuzzy logic system can be rewritten as
(10)
y
(
x
)
=
W
T
S
(
x
)
.
If all memberships are chosen as Gaussian functions, the lemma below holds.Lemma 7 (see [47]).
Letf
(
x
) be a continuous function defined on a compact set Ω. Then, for any given constant ɛ
>
0, there exists a fuzzy logic system (10) such that
(11)
sup
⁡
x
∈
Ω
|
f
(
x
)
-
W
T
S
(
x
)
|
≤
ɛ
.The following lemma will be used in this note.Lemma 8 (Young’s inequality [23]).
For∀
(
x
,
y
)
∈
R
2, the following inequality holds:
(12)
x
y
≤
ɛ
p
p
|
x
|
p
+
1
q
ɛ
q
|
y
|
q
,
where ɛ
>
0,  p
>
1,  q
>
1, and (
p
-
1
)
(
q
-
1
)
=
1.


## 3. Adaptive Fuzzy Control Design

In this section, a fuzzy-based adaptive tracking control scheme is proposed for the system (5). The backstepping design procedure contains n steps and is developed based on the coordinate transformation z
i
=
x
i
-
α
i
-
1,   i
=
1,2
,
…
,
n, where α
0
=
y
d. The virtual control signal and the adaption law will be constructed in the following forms:
(13)
α
i
=
-
1
2
a
i
2
z
i
3
θ
^
,
(14)
θ
^
˙
=
∑
i
=
1
n
σ
2
a
i
2
z
i
6
-
γ
θ
^
,
where a
i (i
=
1,2
,
…
,
n),  σ, and γ are positive design parameters and Z
1
=
[
x
1
,
y
d
,
y
˙
d
]
T
∈
Ω
Z
1
⊂
R
3,  Z
i
=
[
x
¯
i
T
,
θ
^
,
y
¯
d
(
i
)
T
]
T
∈
Ω
Z
i
⊂
R
2
i
+
2
(
i
=
2
,
…
,
n
)
.
θ
^ is the estimation of unknown constant θ which is specified as
(15)
θ
=
max
⁡
{
b
M
2
b
m
∥
W
i
∥
2
;
i
=
1,2
,
…
,
n
}
.
Specially, α
n is the actual control input u
(
t
).For simplicity, in the following, the time variablet and the state vector x
¯
i are omitted from the corresponding functions, and let S
i
(
Z
i
)
=
S
i.Step 1.
Sincez
1
=
x
1
-
y
d, the error dynamic satisfies
(16)
d
z
1
=
(
f
1
(
x
1
,
x
2
)
-
y
˙
d
)
d
t
+
h
1
T
d
w
.
Consider a Lyapunov function candidate as
(17)
V
1
=
1
4
z
1
4
+
b
m
2
σ
θ
~
2
,
where θ
~
=
θ
-
θ
^ is the parameter error. By (2) and the completion of squares, one has
(18)
L
V
1
≤
z
1
3
(
f
1
(
x
1
,
x
2
)
-
y
˙
d
)
+
3
z
1
2
2
h
1
T
h
1
-
b
m
σ
θ
~
θ
^
˙
≤
z
1
3
(
f
1
(
x
1
,
x
2
)
-
y
˙
d
+
3
4
l
1
-
2
z
1
∥
h
1
∥
4
)
+
3
4
l
1
2
-
b
m
σ
θ
~
θ
^
˙
,
where l
1 is a constant. Define a new function
(19)
w
1
=
-
y
˙
d
+
3
2
z
1
+
k
1
z
1
+
3
4
l
1
-
2
z
1
∥
h
1
∥
4
,
then (18) can be rewritten as
(20)
L
V
1
≤
z
1
3
(
f
1
(
x
1
,
x
2
)
+
w
1
)
-
(
k
1
+
3
2
)
z
1
4
+
3
4
l
1
2
-
b
m
σ
θ
~
θ
^
˙
.
Based on Assumption 5 and the fact of ∂
w
1
/
∂
x
2
=
0, one has
(21)
∂
[
f
1
(
x
1
,
x
2
)
+
w
1
]
∂
x
2
≥
b
m
>
0
.
According to Lemma 1 in [37], for each value of x
1 and w
1, there exists a smooth ideal control input x
2
=
α
¯
1
(
x
1
,
w
1
) such that
(22)
f
1
(
x
1
,
α
¯
1
)
+
w
1
=
0
.
Applying mean value theorem [48], there exists μ
1 (0
<
μ
1
<
1) which makes
(23)
f
1
(
x
1
,
x
2
)
=
f
1
(
x
1
,
α
¯
1
)
+
g
μ
1
(
x
2
-
α
¯
1
)
,
where g
μ
1
:
=
g
1
(
x
1
,
x
μ
1
)
=
(
∂
f
1
(
x
1
,
x
2
)
/
∂
x
2
)
|
x
2
=
x
μ
1,  x
μ
1
=
μ
1
x
2
+
(
1
-
μ
1
)
α
¯
1.
Apparently, Assumption5 on g
1
(
x
1
,
x
2
) is still valid for g
μ
1. Substituting (23) into (20) and applying the result (22) and z
2
=
x
2
-
α
1 results in
(24)
L
V
1
≤
z
1
3
g
μ
1
(
x
2
-
α
¯
1
)
-
(
k
1
+
3
2
)
z
1
4
+
3
4
l
1
2
-
b
m
σ
θ
~
θ
^
˙
≤
z
1
3
g
μ
1
z
2
+
z
1
3
g
μ
1
(
α
1
-
α
¯
1
)
-
(
k
1
+
3
2
)
z
1
4
+
3
4
l
1
2
-
b
m
σ
θ
~
θ
^
˙
.
Since α
¯
1 contains the unknown function h
1, α
¯
1 cannot be directly implemented in practice. According to Lemma 7, there exists a fuzzy logic system W
1
T
S
1
(
Z
1
) which can model the unknown function α
¯
1 over a compact set Ω
Z
1
⊂
R
3, such that, for any given ɛ
1
>
0,
(25)
α
¯
1
=
W
1
T
S
1
(
Z
1
)
+
δ
1
(
Z
1
)
,
|
δ
1
(
Z
1
)
|
≤
ɛ
1
,
where δ
1
(
Z
1
) is approximation error. Then, based on Young’s inequality, it follows that
(26)
-
z
1
3
g
μ
1
α
¯
1
≤
b
m
z
1
6
2
a
1
2
∥
W
1
∥
2
b
m
S
1
T
S
1
+
a
1
2
2
+
3
4
z
1
4
+
b
M
4
4
ɛ
1
4
≤
b
m
2
a
1
2
z
1
6
θ
+
1
2
a
1
2
+
3
4
z
1
4
+
1
4
b
M
4
ɛ
1
4
,
where the unknown constant θ is defined in (15) and the property of S
1
T
S
1
≤
1 is used in (26). By choosing virtual control signal α
1 in (13) with i
=
1, and using the property of θ
^
≥
0 and Assumption 5, the following result holds:
(27)
z
1
3
g
μ
1
α
1
≤
-
b
m
2
a
1
2
z
1
6
θ
^
.
Substituting (26) and (27) into (20) and using Young’s inequality to the term z
1
3
g
μ
1
z
2 produces
(28)
L
V
1
≤
-
k
1
z
1
4
+
1
4
b
M
4
z
2
4
+
ρ
1
+
b
m
σ
θ
~
(
σ
2
a
1
2
z
1
6
-
θ
^
˙
)
,
with ρ
1
=
(
1
/
2
)
a
1
2
+
(
1
/
4
)
b
M
4
ɛ
1
4
+
(
3
/
4
)
l
1
2.Step 2.
Based on the coordinate transformationz
2
=
x
2
-
α
1 and Itô formula, one has
(29)
d
z
2
=
(
f
2
(
x
¯
2
,
x
3
)
-
ℓ
α
1
)
d
t
+
(
h
2
-
∂
α
1
∂
x
1
h
1
)
T
d
w
with
(30)
ℓ
α
1
=
∂
α
1
∂
x
1
f
1
(
x
1
,
x
2
)
+
∑
j
=
0
1
∂
α
1
∂
y
d
(
j
)
y
d
(
j
+
1
)
+
∂
α
1
∂
θ
^
θ
^
˙
+
1
2
∂
2
α
1
∂
x
1
2
h
1
T
h
1
.
Take the following Lyapunov function:
(31)
V
2
=
V
1
+
1
4
z
2
4
.
Then, by using a similar procedure as Step 1, it follows that
(32)
L
V
2
=
L
V
1
+
z
2
3
(
f
2
(
x
¯
2
,
x
3
)
-
ℓ
α
1
)
+
3
2
z
2
2
(
h
2
-
∂
α
1
∂
x
1
h
1
)
T
(
h
2
-
∂
α
1
∂
x
1
h
1
)
.
It is noticed that
(33)
3
2
z
2
2
∥
h
2
-
∂
α
1
∂
x
1
h
1
∥
2
≤
3
4
l
2
2
z
2
4
∥
h
2
-
∂
α
1
∂
x
1
h
1
∥
4
+
3
l
2
2
4
,
where l
2 is a constant.Furthermore, it can be verified by substituting (28) and (33) into (32) that
(34)
L
V
2
≤
-
k
1
z
1
4
+
b
m
σ
θ
~
(
σ
2
a
1
2
z
1
6
-
θ
^
˙
)
+
z
2
3
(
f
2
(
x
¯
2
,
x
3
)
-
ℓ
α
1
+
3
z
2
4
l
2
2
∥
h
2
-
∂
α
1
∂
x
1
h
1
∥
4
+
1
4
b
M
4
z
2
∥
h
2
-
∂
α
1
∂
x
1
h
1
∥
4
)
+
ρ
1
+
3
4
l
2
2
.From the definition ofθ
^
˙ in (14), we have
(35)
∂
α
1
∂
θ
^
θ
^
˙
=
∂
α
1
∂
θ
^
(
∑
j
=
1
2
σ
2
a
j
2
z
j
6
-
γ
θ
^
)
+
∂
α
1
∂
θ
^
∑
j
=
3
n
σ
2
a
j
2
z
j
6
.
By combining (30), (34), and (35), the following result holds:
(36)
L
V
2
≤
-
k
1
z
1
4
+
ρ
1
+
z
2
3
(
f
2
(
x
¯
2
,
x
3
)
+
w
2
)
-
(
k
2
+
3
2
)
z
2
4
+
b
m
σ
θ
~
(
σ
2
a
1
2
z
1
6
-
θ
^
˙
)
-
∂
α
1
∂
θ
^
z
2
3
∑
j
=
3
n
σ
2
a
j
2
z
j
6
+
3
4
l
2
2
,
where
(37)
w
2
=
b
M
4
4
z
2
-
∂
α
1
∂
x
1
f
2
(
x
1
,
x
2
)
-
∑
j
=
0
1
∂
α
1
∂
y
d
(
j
)
y
d
(
j
+
1
)
-
1
2
∂
2
α
1
∂
x
1
2
h
1
T
h
1
+
3
4
l
2
-
2
z
2
∥
h
2
-
∂
α
1
∂
x
1
h
1
∥
4
-
∂
α
1
∂
θ
^
(
∑
j
=
1
2
σ
2
a
j
2
z
j
6
-
σ
θ
^
)
+
(
k
2
+
3
2
)
z
2
.
It can be seen from (37) that ∂
w
2
/
∂
x
3
=
0. Then, from Assumption 5, we have
(38)
∂
[
f
2
(
x
¯
2
,
x
3
)
+
w
2
]
∂
x
3
≥
b
m
>
0
.
According to Lemma 1 in [37], there exists a smooth ideal control input x
3
=
α
¯
2
(
x
¯
2
,
w
2
) such that
(39)
f
2
(
x
¯
2
,
α
¯
2
)
+
w
2
=
0
.
By applying mean value theorem [48], there exists μ
2 (0
<
μ
2
<
1) such that
(40)
f
2
(
x
¯
2
,
x
3
)
=
f
2
(
x
¯
2
,
α
¯
2
)
+
g
μ
2
(
x
3
-
α
¯
2
)
,
where g
μ
2
=
(
∂
f
2
(
x
¯
2
,
x
3
)
/
∂
x
3
)
|
x
3
=
x
μ
2,x
μ
2
=
μ
2
x
3
+
(
1
-
μ
2
)
α
¯
2. Assumption 5 on g
2
(
x
¯
2
,
x
3
) is still valid for g
μ
2.Substituting (39) and (40) into (36) produces
(41)
L
V
2
≤
-
k
1
z
1
4
+
ρ
1
+
z
2
3
g
μ
2
z
3
+
z
2
3
g
μ
2
(
α
2
-
α
¯
2
)
-
(
k
2
+
3
2
)
z
2
4
+
b
m
σ
θ
~
(
σ
2
a
1
2
z
1
6
-
θ
^
˙
)
-
∂
α
1
∂
θ
^
z
2
3
∑
j
=
3
n
σ
2
a
j
2
z
j
6
+
3
4
l
2
2
,
where z
3
=
x
3
-
α
2.Further, fuzzy logic systemW
2
T
S
(
Z
2
) is used to approximate the desired unknown virtual signal α
¯
i over a compact set Ω
Z
2
⊂
R
6 such that, for any given positive constant ɛ
i, α
¯
2 can be shown as
(42)
α
¯
2
=
W
2
T
S
2
(
Z
2
)
+
δ
2
(
Z
2
)
,
|
δ
2
(
Z
2
)
|
≤
ɛ
2
,
with δ
2
(
Z
2
) being the approximation error.Then, constructing the virtual control signalα
2 in (13) and repeating the same methods used in (26) and (27), one has
(43)
-
z
2
3
g
μ
2
α
¯
2
≤
b
m
2
a
2
2
z
2
6
θ
+
1
2
a
2
2
+
3
4
z
2
4
+
1
4
b
M
4
ɛ
2
4
,
z
2
3
g
μ
2
α
2
≤
-
b
m
2
a
2
2
z
2
6
θ
^
.
By taking (43) into account and using Young’s inequality to the term z
2
3
g
μ
2
z
3, (41) can be rewritten as
(44)
L
V
2
≤
∑
j
=
1
2
(
-
k
j
z
j
4
+
ρ
j
)
+
b
m
θ
~
σ
(
∑
j
=
1
2
σ
2
a
j
2
z
j
6
-
θ
^
˙
)
-
∂
α
1
∂
θ
^
z
2
3
∑
j
=
3
n
σ
2
a
j
2
z
j
6
+
1
4
b
M
4
z
3
4
,
where ρ
j
=
(
1
/
2
)
a
j
2
+
(
1
/
4
)
b
M
4
ɛ
j
4
+
(
3
/
4
)
l
j
2,
j
=
1,2.Remark 9.
The adaptive lawθ
^
˙ defined in (14) is a function of all the error variables. So, unlike the conventional approximation-based adaptive control schemes, (
∂
α
1
/
∂
θ
^
)
θ
^
˙ in (30) cannot be approximated directly by fuzzy logic system W
2
T
S
2
(
Z
2
). To solve this problem, in (35), (
∂
α
1
/
∂
θ
^
)
θ
^
˙ is divided into two parts. The first term in the right hand side of (35) is contained in α
¯
2 to be modeled by W
2
T
S
2
(
Z
2
), and the last term in (35) which is a function of the latter error variables, namely, z
j, j
=
3,4
,
…
,
n, will be dealt with in the later design steps. This design idea will be repeated at the following steps.
Step i(
3
≤
i
≤
n
-
1
). From z
i
=
x
i
-
α
i
-
1 and Itô formula, we have
(45)
d
z
i
=
(
f
i
(
x
¯
i
,
x
i
+
1
)
-
ℓ
α
i
-
1
)
d
t
+
(
h
i
-
∑
j
=
1
i
-
1
∂
α
i
-
1
∂
x
j
h
j
)
T
d
w
,
where
(46)
ℓ
α
i
-
1
=
∑
j
=
1
i
-
1
∂
α
i
-
1
∂
x
j
f
j
(
x
¯
j
,
x
j
+
1
)
+
∑
j
=
0
i
-
1
∂
α
i
-
1
∂
y
d
(
j
)
y
d
(
j
+
1
)
+
∂
α
i
-
1
∂
θ
^
θ
^
˙
+
1
2
∑
p
,
q
=
1
i
-
1
∂
2
α
i
-
1
∂
x
p
∂
x
q
h
q
T
h
p
.
Choose a Lyapunov function as
(47)
V
i
=
V
i
-
1
+
1
4
z
i
4
.
Then, it follows that
(48)
L
V
i
=
L
V
i
-
1
+
z
i
3
(
f
i
(
x
¯
i
,
x
i
+
1
)
-
ℓ
α
i
-
1
)
+
3
2
z
i
2
(
h
i
-
∑
j
=
1
i
-
1
∂
α
i
-
1
∂
x
j
h
j
)
T
×
(
h
i
-
∑
j
=
1
i
-
1
∂
α
i
-
1
∂
x
j
h
j
)
,
where the term L
V
i
-
1 in (48) can be obtained in the following form by straightforward derivations as those in former steps:
(49)
L
V
i
-
1
≤
∑
j
=
1
i
-
1
(
-
k
j
z
j
4
+
ρ
j
)
+
b
m
θ
~
σ
(
∑
j
=
1
i
-
1
σ
2
a
j
2
z
j
6
-
θ
^
˙
)
-
∑
m
=
1
i
-
2
∂
α
m
∂
θ
^
z
m
+
1
3
∑
j
=
i
n
σ
2
a
j
2
z
j
6
+
b
M
4
4
z
i
4
,
where ρ
j
=
(
1
/
2
)
a
j
2
+
(
1
/
4
)
b
M
4
ɛ
j
4
+
(
3
/
4
)
l
j
2.According to the definition ofθ
^
˙ in (14), one has
(50)
∂
α
i
-
1
∂
θ
^
θ
^
˙
=
∂
α
i
-
1
∂
θ
^
(
∑
j
=
1
i
σ
2
a
j
2
z
j
6
-
γ
θ
^
)
+
∂
α
i
-
1
∂
θ
^
∑
j
=
i
+
1
n
σ
2
a
j
2
z
j
6
.
Then, substituting (49) into (48) and taking (50) into account, (48) can be rewritten as
(51)
L
V
i
≤
∑
j
=
1
i
-
1
(
-
k
j
z
j
4
+
ρ
j
)
+
z
i
3
(
f
i
(
x
¯
i
,
x
i
+
1
)
+
w
i
)
-
(
k
i
+
3
2
)
z
i
4
+
b
m
θ
~
σ
(
∑
j
=
1
i
-
1
σ
2
a
j
2
z
j
6
-
θ
^
˙
)
-
∑
m
=
1
i
-
1
∂
α
m
∂
θ
^
z
m
+
1
3
∑
j
=
i
+
1
n
σ
2
a
j
2
z
j
6
+
3
4
l
i
2
,
where
(52)
w
i
=
b
M
4
4
z
i
-
∑
j
=
1
i
-
1
∂
α
i
-
1
∂
x
j
f
j
(
x
¯
j
,
x
j
+
1
)
-
∑
j
=
0
i
-
1
∂
α
i
-
1
∂
y
d
(
j
)
y
d
(
j
+
1
)
-
1
2
∑
p
,
q
=
1
i
-
1
∂
2
α
i
-
1
∂
x
p
∂
x
q
h
p
T
h
q
+
3
4
l
i
-
2
z
i
∥
h
i
-
∑
j
=
1
i
-
1
∂
α
i
-
1
∂
x
j
h
j
∥
4
-
∂
α
i
-
1
∂
θ
^
(
∑
j
=
1
i
σ
2
a
j
2
z
j
6
-
σ
θ
^
)
-
σ
2
a
i
2
z
i
3
∑
m
=
1
i
-
2
∂
α
m
∂
θ
^
z
m
+
1
3
+
(
k
i
+
3
2
)
z
i
.
Noting ∂
w
i
/
∂
x
i
+
1
=
0, it follows that
(53)
∂
[
f
i
(
x
¯
i
,
x
i
+
1
)
+
w
i
]
∂
x
i
+
1
≥
b
m
>
0
.
Based on Lemma 1 in [37], there exists a smooth ideal control input x
i
+
1
=
α
¯
i
(
x
¯
i
,
w
i
) which makes
(54)
f
i
(
x
¯
i
,
α
¯
i
)
+
w
i
=
0
.
In addition, applying mean value theorem [48], there exists μ
i (0
<
μ
i
<
1) such that
(55)
f
i
(
x
¯
i
,
x
i
+
1
)
=
f
i
(
x
¯
i
,
α
¯
i
)
+
g
μ
i
(
x
i
+
1
-
α
¯
i
)
,
where g
μ
i
=
(
∂
f
i
(
x
¯
i
,
x
i
+
1
)
/
∂
x
i
+
1
)
|
x
i
+
1
=
x
μ
i,x
μ
i
=
μ
i
x
i
+
1
+
(
1
-
μ
i
)
α
¯
i. Assumption 5 on g
i
(
x
¯
i
,
x
i
+
1
) is still valid for g
μ
i. Combining (51)–(55) yields
(56)
L
V
i
≤
∑
j
=
1
i
-
1
(
-
k
j
z
j
4
+
ρ
j
)
+
z
i
3
g
μ
i
z
i
+
1
+
z
i
3
g
μ
i
(
α
i
-
α
¯
i
)
-
(
k
i
+
3
2
)
z
i
4
+
b
m
θ
~
σ
(
∑
j
=
1
i
-
1
σ
2
a
j
2
z
j
6
-
θ
^
˙
)
-
∑
m
=
1
i
-
1
∂
α
m
∂
θ
^
z
m
+
1
3
∑
j
=
i
+
1
n
σ
2
a
j
2
z
j
6
+
3
4
l
i
2
,
where z
i
+
1
=
x
i
+
1
-
α
i. Subsequently, using fuzzy logic system W
i
T
S
(
Z
i
) to model α
¯
i over a compact set Ω
Z
i
⊂
R
2
i
+
2 such that for any given ɛ
i
>
0, α
¯
i can be expressed as
(57)
α
¯
i
=
W
i
T
S
i
(
Z
i
)
+
δ
i
(
Z
i
)
,
|
δ
i
(
Z
i
)
|
≤
ɛ
i
,
with δ
i
(
Z
i
) being the approximation error. Furthermore, constructing virtual control signal α
i in (13) and following the same line as the procedures used from (26) to (27), one has
(58)
-
z
i
3
g
μ
i
α
¯
i
≤
b
m
2
a
i
2
z
i
6
θ
+
1
2
a
i
2
+
3
4
z
i
4
+
1
4
b
M
4
ɛ
i
4
,
z
i
3
g
μ
i
α
i
≤
-
b
m
2
a
i
2
z
i
6
θ
^
.
By substituting (58) into (56) and using Young’s inequality, we have
(59)
L
V
i
≤
∑
j
=
1
i
(
-
k
j
z
j
4
+
ρ
j
)
+
b
m
θ
~
σ
(
∑
j
=
1
i
σ
2
a
j
2
z
j
6
-
θ
^
˙
)
-
∑
m
=
1
i
-
1
∂
α
m
∂
θ
^
z
m
+
1
3
∑
j
=
i
+
1
n
σ
2
a
j
2
z
j
6
+
1
4
b
M
4
z
i
+
1
4
,
where ρ
j
=
(
1
/
2
)
a
j
2
+
(
1
/
4
)
b
M
4
ɛ
j
4
+
(
3
/
4
)
l
j
2.Step n. In this step, the actual control input signal u will be obtained. By z
n
=
x
n
-
α
n
-
1 and Itô formula, we have
(60)
d
z
n
=
(
f
n
(
x
¯
n
,
u
)
-
ℓ
α
n
-
1
)
d
t
+
(
h
n
-
∑
j
=
1
n
-
1
∂
α
n
-
1
∂
x
j
h
j
)
T
d
w
,
where ℓ
α
n
-
1 is defined in (46) with i
=
n. Choosing the Lyapunov function
(61)
V
n
=
V
n
-
1
+
1
4
z
n
4
,
then, the following inequality can be easily verified by using (2), the completion of squares, and taking (59) with i
=
n
-
1 into account:
(62)
L
V
n
≤
∑
j
=
1
n
-
1
(
-
k
j
z
j
4
+
ρ
j
)
+
b
m
θ
~
σ
(
∑
j
=
1
n
-
1
σ
2
a
j
2
z
j
6
-
θ
^
˙
)
+
z
n
3
(
f
n
(
x
¯
n
,
u
)
+
w
n
)
-
(
k
n
+
3
4
)
z
n
4
+
3
4
l
n
2
,
where
(63)
w
n
=
ℓ
α
n
-
1
-
∑
m
=
1
n
-
2
‍
∂
α
m
∂
θ
^
z
m
+
1
3
σ
2
a
n
2
z
n
3
+
1
4
b
M
4
z
n
+
3
4
l
n
-
2
z
n
∥
h
n
-
∑
j
=
1
n
-
1
‍
∂
α
n
-
1
∂
x
j
h
j
∥
4
+
(
k
n
+
3
4
)
z
n
.
From the definition of w
n, Assumption 5 and Lemma 1 in [37], for every value x
¯
n and w
n, there exists a smooth ideal control input u
=
α
¯
n
(
x
¯
n
,
w
n
) such that f
n
(
x
¯
n
,
α
¯
n
)
+
w
n
=
0. By using mean value theorem [48], there exists μ
n (0
<
μ
n
<
1) such that
(64)
f
n
(
x
¯
n
,
u
)
=
f
i
(
x
¯
n
,
α
¯
n
)
+
g
μ
n
(
u
-
α
¯
n
)
,
where g
μ
n
=
(
∂
f
n
(
x
¯
n
,
u
)
/
∂
u
)
|
u
=
x
μ
n,x
μ
n
=
μ
n
u
+
(
1
-
μ
n
)
α
¯
n.Furthermore, (62) can be rewritten as
(65)
L
V
n
≤
∑
j
=
1
n
-
1
(
-
k
j
z
j
4
+
ρ
j
)
+
b
m
θ
~
σ
(
∑
j
=
1
n
-
1
σ
2
a
j
2
z
j
6
-
θ
^
˙
)
+
z
n
3
g
μ
n
(
u
-
α
¯
n
)
-
(
k
n
+
3
4
)
z
n
4
+
3
4
l
n
2
.
Next, using fuzzy logic system W
n
T
S
(
Z
n
) to approximate the unknown function α
¯
n on the compact set Ω
Z
n
⊂
R
2
n
+
2, constructing the actual control u in (13) with i
=
n, and repeating the same procedures as (58)-(59), one has
(66)
L
V
n
≤
∑
j
=
1
n
(
-
k
j
z
j
4
+
ρ
j
)
+
b
m
θ
~
σ
(
∑
j
=
1
n
σ
2
a
j
2
z
j
6
-
θ
^
˙
)
.
Further, choosing the adaptive law θ
^
˙ in (14) and using the inequality (
γ
b
m
/
σ
)
θ
~
θ
^
≤
-
(
γ
b
m
/
2
σ
)
θ
~
2
+
(
γ
b
m
/
2
σ
)
θ
2 result in
(67)
L
V
n
≤
-
∑
j
=
1
n
k
j
z
j
4
-
γ
b
m
2
σ
θ
~
2
+
∑
j
=
1
n
ρ
j
+
γ
b
m
2
σ
θ
2
,
where ρ
j
=
(
1
/
2
)
a
j
2
+
(
1
/
4
)
b
M
4
ɛ
j
4
+
(
3
/
4
)
l
j
2,j
=
1,2
,
…
,
n.Now, the actual control signalu is constructed. The main result will be summarized by the following theorem.Theorem 10.
Consider the pure-feedback stochastic nonlinear system (5), the controller (13), and adaptive law (14) under Assumptions 5 and 6. Assume that there exists sufficiently large compacts Ω
Z
i,i
=
1,2
,
…
,
n, such that Z
i
∈
Ω
Z
i, for all t
≥
0; then, for bounded initial conditions [
z
¯
n
(
0
)
T
,
θ
^
(
0
)
]
T
∈
Ω
0 (where Ω
0 is an appropriately chosen compact set),(i)
all the signals in the closed-loop system are bounded in probability;
(ii)
there exists a finite timeT
1 such that the quartic mean square tracking error enters inside the area for all t
>
T
1,
(68)
Ω
1
=
{
y
(
t
)
∈
R
∣
E
[
|
y
-
y
d
|
4
]
≤
8
ρ
,
∀
t
>
T
1
}
,
where the timeT
1 will be given later.Proof.
(i) For the stability analysis of the closed-loop system, choose the Lyapunov function asV
=
V
n. From (67), it follows that
(69)
L
V
≤
-
a
0
V
+
b
0
,
where a
0
=
min
⁡
{
4
k
j
,
γ,j
=
1,2
,
…
,
n
} and b
0
=
∑
j
=
1
n
‍
ρ
j
+
(
γ
b
m
/
2
σ
)
θ
2. Therefore, based on Lemma 1 in [37], z
j and θ
~ remain bounded in probability. Because θ is a constant, θ
^ is bounded in probability. It can be further seen that α
j is a function of z
j and θ
^. So, α
j is also bounded in probability. Hence, we conclude that all the signals x
j in the closed-loop system (5) remain bounded in the sense of probability.
(ii) From (69), the following inequality can be obtained directly by [24, Theorem 4.1]
(70)
d
E
[
V
(
t
)
]
d
t
≤
-
a
0
E
[
V
(
t
)
]
+
b
0
.
Let ρ
:
=
b
0
/
a
0
>
0, then (70) satisfies
(71)
0
≤
E
[
V
(
t
)
]
≤
V
(
0
)
e
-
a
0
t
+
ρ
.
Then, it can be easily verified that there exists a time T
1
=
max
⁡
{
0
,
(
1
/
a
0
)
ln
⁡
(
V
(
0
)
/
ρ
)
} such that
(72)
E
[
|
y
-
y
d
|
4
]
≤
4
E
[
V
(
t
)
]
≤
8
ρ
,
∀
t
>
T
1
.


## 4. Simulation Results

In this section, to illustrate the effectiveness of the proposed control scheme, we consider the following second-order pure-feedback stochastic nonlinear system:(73)
d
x
1
=
(
x
2
+
0.05
sin
(
x
2
)
)
d
t
+
0.1
cos
⁡
(
x
1
)
d
w
,
d
x
2
=
(
(
2
+
x
1
2
1
+
x
1
2
+
x
2
2
)
u
+
0.1
u
3
)
d
t
+
0.2
sin
(
x
1
x
2
)
d
w
,
y
=
x
1
.
The control objective is to design an adaptive controller for the system such that all the signals in the closed-loop system remain bounded and the system output y tracks the given reference signal y
d
=
0.5
(
sin
(
t
)
+
sin
(
0.5
t
)
). According to Theorem 10, choose the virtual control law α
1 in (13), actual control input u in (13) with i
=
2, and adaptive law in (14). The design parameters are taken as follows: a
1
=
a
2
=
0.06, γ
=
0.0015, and σ
=
25. The initial conditions are given by [
x
1
(
0
)
,
x
2
(
0
)
]
T
=
[
0.2,0.5
]
T and θ
^
(
0
)
=
0.The simulation results are shown in Figures1–4. Figure 1 shows the system output y and the reference signal y
d. Figure 2 and Figure 3 show that the state variable x
2 and adaptive parameter θ
^ are bounded. Figure 4 displays the control input signal u.Figure 1
System outputy and reference signal y
d.Figure 2
The state variablex
2.Figure 3
The adaptive parameterθ
^.Figure 4
The true control inputu.


## 5. Conclusions

In this paper, a novel fuzzy-based adaptive control scheme has been presented for pure-feedback stochastic nonlinear systems. The proposed controller guarantees that all the signals in the closed-loop systems are bounded in probability and tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. The main advantages of this control scheme are that the controller is simpler than the existing ones and only one adaptive parameter needs to be estimated online for ann-order system. Numerical results have been provided to show the effectiveness of the suggested approach. Our future research will mainly focus on the multi-input and multioutput (MIMO) pure-feedback stochastic nonlinear systems based on the result in this paper.



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*Source: 462468-2014-02-13.xml*