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IEEE TRANSACTIONS ON CYBERNETICS 1
Robust Fuzzy Adaptive Tracking Control
for Nonaffine Stochastic Nonlinear
Switching Systems
Huanqing Wang, Peter Xiaoping Liu, Senior Member, IEEE,andBenNiu
Abstract—This paper is concerned with the trajectory tracking
control problem of a class of nonaffine stochastic nonlinear
switched systems with the nonlower triangular form under
arbitrary switching. Fuzzy systems are employed to tackle the
problem from packaged unknown nonlinearities, and the back-
stepping and robust adaptive control techniques are applied to
design the controller by adopting the structural characteristics of
fuzzy systems and the common Lyapunov function approach. By
using Lyapunov stability theory, the semiglobally uniformly ulti-
mate boundness in the fourth-moment of all closed-loop signals
is guaranteed, and the system output is ensured to converge to a
small neighborhood of the given trajectory. The main advantages
of this paper lie in the fact that both the completely nonaffine
form and nonlower triangular structure are taken into account
for the controlled systems, and the increasing property of whole
state functions is removed by using the structural characteris-
tics of fuzzy systems. The developed control method is verified
through a numerical example.
Index Terms—Fuzzy systems, robust control theory, stochastic
systems.
I. INTRODUCTION
SWITCHED systems, a typical class of hybrid systems,
provide a unified framework for describing a large amount
of practical systems with switching characteristics such as net-
worked control systems, aircraft systems, circuit and power
systems, and so on [1]–[4]. Given their diverse applications,
the investigation on stability analysis and control synthesis
for switched systems has kept attracting wide attentions and
many promising results have been reported in [4]–[14] and
the references therein. For example, in [4]–[9], many ele-
gant methods are developed to switched continuous-time or
Manuscript received May 26, 2017; accepted August 2, 2017. This
work was supported by the National Natural Science Foundation of China
under Grant 61773072, Grant 61773051, and Grant 61773073. This paper
was recommended by Associate Editor J. Qiu. (Corresponding authors:
Huanqing Wang; Peter Xiaoping Liu.)
H. Wang is with the Department of Mathematics, Bohai University, Jinzhou
121013, China, and also with the Department of Systems and Computer
Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada (e-mail:
ndwhq@163.com).
P. X. Liu is with the Department of Systems and Computer Engineering,
Carleton University, Ottawa, ON K1S 5B6, Canada, and also with the
School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong
University, Beijing 100044, China (e-mail: xpliu@sce.carleton.ca).
B. Niu is with the Department of Mathematics, Bohai University, Jinzhou
121000, China (e-mail: niubenzg@gmail.com).
Digital Object Identifier 10.1109/TCYB.2017.2740841
discrete-time linear systems. The authors in [10]–[14] studied
the control problems of switched nonlinear uncertain systems
as nonlinearity is inherent in practically industrial applica-
tions. Backstepping control which is first proposed in [15] has
been an effective approach to strict-feedback nonlinear sys-
tems. To make it applicable to switched nonlinear systems,
state-feedback controllers are constructed in [16] and [17]
under arbitrary switching signals. Chiang and Fu [18] con-
sidered the problem of adaptive stabilization including state-
feedback and output-feedback under arbitrary switching sig-
nals. However, the above mentioned works only for systems
with nonlinear functions being known or bounded by known
functions multiplying uncertain constants. These preconditions
make them unsuitable or infeasible for practical engineering
problems.
During the past two decades, adaptive backstepping con-
trol of nonlinear systems with unknown functions in lower
triangular form has received much attention by applying
fuzzy systems/neural networks to estimate uncertain nonlin-
ear functions [19]–[40]. With the development in both control
theory and engineering practice, there also have been some
researchers in the control field dedicated to the study of intel-
ligent control methodology of switched nonlinear systems.
Even the intelligent control for switched systems is gradually
becoming a hot research field, and many interesting researches
have emerged, for example, see [41]–[45] for determinis-
tic systems and [46] for stochastic cases. For deterministic
switched nonlinear systems, Long and Fei [41] considered
the problem of neural networks stabilization and distur-
bance attenuation of nonlinear switched impulsive systems.
In [42], an adaptive neural control strategy is developed for
nonlinear systems with switching jumps and uncertainties.
Long and Zhao [43] and Tong et al. [44] proposed the adap-
tive neural/fuzzy output feedback control methods for a class
of nonlinear systems with unmeasured states. For stochastic
switched nonlinear systems, adaptive fuzzy control is extend to
a class of switched stochastic nonlinear systems with unmod-
eled dynamics [46]. The above mentioned results [41]–[46]
are reached merely when the control systems are in strict-
feedback structure that is a special form of lower-triangular
systems.
The nonaffine systems, which commonly reside in practice
such as biochemical process, have a more general repre-
sentation compared with strict-feedback systems since there
are no appearance of the variables being used as virtual
2168-2267 c
2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
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2IEEE TRANSACTIONS ON CYBERNETICS
controls and real control. This means the control design for
nonaffine nonlinear systems is a difficult and meaningful
work, and many prominent research achievements are reported
in [47]–[53]. By applying adaptive neural control, a type of
pure-feedback systems is studied in [47] and [48] where the
actual input is in affine form. Then, in [49], an adaptive
neural control of a class of completely pure-feedback sys-
tem is developed using “ISS-modular” approach. At the same
time, the study of switched nonaffine nonlinear systems have
received increasing attention. In [54], a neural-based tracking
control is proposed for switched pure-feedback systems with-
out knowing the signs of gain is developed under arbitrary
switchings. Meanwhile, Li et al. [55] presented a fuzzy-based
backstepping design scheme without requiring the informa-
tion of fuzzy basis functions. Hereafter, a tracking control
method is proposed for switched stochastic pure-feedback sys-
tems [56]. Although significant progress has been achieved
in the control design of switched pure-feedback systems,
most of the control methods seem only well suitable for the
nonlinear systems satisfying a little strict restriction, that is,
the controlled systems are of lower-triangular systems. This
precondition limits the scope of applications of backstepping-
based control method since many practical systems, such as
the Ball and Beam system [57], cannot satisfy the idealized
structure. To relax the limitation on the systems structure,
Chen et al. [58] presented a fuzzy-based control approach
for nonlinear systems with nonstrict-feedback structure. Then,
Wang et al. [59] extended adaptive fuzzy backstepping con-
trol to nonstrict-feedback stochastic nonlinear systems. More
recently, approximation-based adaptive control are considered
in [60] and [61] for switched nonstrict-feedback nonlinear
systems. Up to date, to the best of our knowledge, there
have been no results dealing with uncertain nonaffine sys-
tems with both external stochastic disturbance and nonlower-
triangular form. Actually, it is still an open, but interesting
topic, which partly motivates us to carry out the present
study.
With the discussions above, in this paper, we consider
this interesting problem, i.e., fuzzy output tracking con-
trol for a class of switched nonlinear systems with com-
pletely nonaffine form and nonlower triangular structure.
By using the recursive backstepping technique, an effec-
tive adaptive fuzzy tracking control scheme is developed
and the stability of the closed-loop system is proved apply-
ing common Lyapunov function under arbitrary switchings.
Compared with the existing results, the main contributions
of this paper lie in the fact that: 1) both the completely
nonaffine form and nonlower triangular structure are taken
into account for the controlled systems; 2) only one adap-
tive law is require in the proposed control scheme; and
3) by adopting the structural characteristics of fuzzy sys-
tems, the increasing property of whole state functions is
removed and the process of controller design becomes
simpler.
This paper is organized as follows. In Section II, the system
description and preliminaries are stated. The main result is
presented in Section III. In Section IV, an illustrative example
is established. Finally, Section V concludes the work.
II. PROBLEM DESCRIPTION AND PRELIMINARIES
Consider the following nonaffine stochastic nonlinear
switching system:
⎧
⎪
⎨
⎪
⎩
dχi=fσ(t),i(¯χi,χ
i+1)dt +φT
σ(t),i(χ)dw
dχn=fσ(t),n¯χn,uσ(t)dt +φT
σ(t),n(χ)dw
y=χ1
(1)
where 1 ≤i≤n−1,χ =[χ1,χ
2,...,χ
n]T∈Rnis
the state, ¯χi=[χ1,χ
2,...,χ
i]T∈Ri; and y∈Rstands
for the system output; wstands for an r-dimensional stan-
dard Brownian motion defined on a complete probability
space (, F,{Ft}t≥0,P), where is a sample space, Fis a
σ−field, {Ft}t≥0is a filtration, and Pis a probability measure;
σ(t):[0,∞)→={1,...,m}denotes switching signal;
uσ(t)∈Rdenotes actuator. For ∀k∈, the drifting terms
fk,i(·):Ri+1→Rand diffusion terms φk,i(·):Rn→Rr,(1≤
i≤n)represent uncertain continuous nonlinear functions.
The purpose is to establish a fuzzy-based control approach
for switched system (1) so that the boundedness in 4th-moment
of all signals within the closed-loop system is ensured and a
desired reference yris tracked by the output y.
According to mean value theorem [63], one has
fk,i(¯χi,χ
i+1)=fk,i¯χi,χ0
i+1+gk,iχi+1−χ0
i+1
(1≤i≤n−1)(2)
fk,n(¯χn,uk)=fk,n¯χn,u0
k+gk,nuk−u0
k(3)
in which fk,i(·)is described between fk,i(¯χi,χ
i+1)
and fk,i(¯χi,χ0
i+1),gk,i:=gk,i(¯χi,χ
μi)=
[(∂fk,i(¯χi,χ
i+1))/(∂χi+1)]|χi+1=χμi,1≤i≤n,χ
n+1=
uk,χ
μi=μiχi+1+(1−μi)χ0
i+1,0<μ
i<1.Furthermore,
by substituting (3)into(1), and selecting χ0
i+1=0,u0
k=0,
we get
⎧
⎨
⎩
dχi=gk,iχi+1+fk,i(¯χi)dt +φT
k,i(χ)dw,1≤i≤n−1
dχn=gk,nuk+fk,n(¯χn)dt +φT
k,n(χ)dw
y=χ1.
(4)
For the ease of control design, the assumptions below are
imposed.
Assumption 1 [20]: The signs of gk,iare known, and there
are unknown scalars bland brso that, for 1 ≤i≤n
0<bl≤|gk,i|≤br<∞,∀¯χi+1∈Ri+1.(5)
Assumption 2 [20]: The reference signal yr(t)and its time
derivatives up to the nth order are continuous and bounded.
To recall some useful preliminaries, we first consider the
stochastic system as
dχ=f(χ)dt +h(χ )dw,∀χ∈Rn(6)
where χ∈Rnis state, wis specified as (1), and f(χ) and
h(χ) denote locally Lipschitz functions with f(0)=h(0)=0.
Definition 1 [62]: Let V(χ) ∈C2be a positive function
with respect of the system (6). A differential operator Lis
defined as
LV=∂V
∂χ f(χ) +1
2TrhT(χ ) ∂2V
∂χ2h(χ )(7)
where Tr() stands for the trace of .
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WANG et al.: ROBUST FUZZY ADAPTIVE TRACKING CONTROL FOR NONAFFINE STOCHASTIC NONLINEAR SWITCHING SYSTEMS 3
Lemma 1 [61]: Assume that there are a C2function
V(χ) :Rn→R+, two constants c1>0 and c2>0, class
K∞−functions ¯α1and ¯α2satisfying
¯α1(|χ|)≤V(χ)≤¯α2(|χ|)
LV(χ)≤−c1V(χ)+c2
for all χ∈Rnand t>t0. Then, there is an unique strong
solution of system (6) for each χ0∈Rnand it satisfies
E[V(χ)]≤V(χ0)e−c1t+c2
c1,∀t>t0.
Before start deriving the presented control algorithm, the
following lemma regarding the property of fuzzy approxima-
tion is stated.
Lemma 2 [64]: Suppose that F(χ) is a continuous func-
tion which is specified on a bounded closed set . Then with
∀>0, there is a fuzzy system TS(χ) satisfying
sup
χ∈
|F(χ)−TS(χ)|≤
in which =[ψ1,ψ
2,...,ψ
N]Trepresents the desired
weight vector and S(χ) =([s1(χ ), . . . , sN(χ)]T/N
i=1si(χ))
is a vector of fuzzy basis function with N>1 being the rule
number and si(χ) is selected as Gaussian function, that is
si(χ)=e−(χ−μi)T(χ−μi)
η2,i=1,...,N
where μi=[μi1,μ
i2,...,μ
iq]Tand ηdenote the center and
width of Gaussian function, respectively.
Lemma 3 [62]: For ∀(x,y)∈R2,the result below is true
xy ≤εl
l|x|l+1
mεm|y|m(8)
where ε>0,l>1,m>1,and (l−1)(m−1)=1.The result
of Lemma 3 is Young’s inequality.
III. MAIN RESULT
This section contains two parts. In the first part, a fuzzy
adaptive control approach is proposed via backstepping. Then,
the theory result, i.e., Theorem 1 will be presented after
finishing the controller design.
During the design procedure, the following coordinate
transformation is made:
ei=χi−αi−1¯χT
i−1,ˆ
θ,i=1,2,...,n(9)
where α0=yrand αidenotes a virtual input being designed
in step ias follows:
αi(Xi)=−λiei−e3
iˆ
θ
2a2
iSi(Xi)2(10)
where 1 ≤i≤n,λi>0 and ai>0 represent design con-
stants, Si(Xi)denotes the basis function vector with Xi=
[¯χT
i,yr,ˆ
θ]T∈Xi⊂Ri+1.When i=n,α
nstands for the
real controller uk(t), which is designed in step n.ˆ
θis used to
approximate the unknown parameter θexpressed by
θ=max1
bli2;i=1,2,...,n(11)
where blis defined in (5) and iis specified in the ith step.
The adaptive law satisfies the following differential
equation:
˙
ˆ
θ=
n
j=1
e6
jˆ
θ
2a2
jSj(Xj)2−k0ˆ
θ(12)
where k0and are positive design constants.
Now, we start the controller design procedure.
Step 1: Define e1=χ1−yr. For any k∈, one has
de1=gk,1χ2+fk,1(χ1)−˙yrdt +φT
k,1(χ)dw.(13)
Choose a Lyapunov functional candidate to be
V1=1
4e4
1+bl
2
˜
θ2(14)
where ˜
θ=θ−ˆ
θrepresents the estimate error, and denotes
a parameter.
Combining (7), (14), and the completion of squares, one
has
LV1≤e3
1gk,1χ2+fk,1(χ1)−˙yr
+3
2e2
1φT
k,1(χ)φk,1(χ ) −bl
˜
θ˙
ˆ
θ
≤e3
1gk,1χ2+Fk,1(χ)−3
2e4
1
+3
4l2
1−bl
˜
θ˙
ˆ
θ(15)
where Fk,1(Z1)=fk,1(χ1)−˙yr+(3/2)e1+
(3/4)l−2
1e1φk,1(χ)4with Z1=[χT,yr,˙yr]T.
It is easy to get that
e3
1Fk,1(Z1)≤|e3
1|¯
F1(Z1), ∀k∈(16)
where ¯
F1(Z1)=maxk∈{|Fk,1(Z1)|}.
Since the function Fk,1(Z1)contains the unknown functions
fk,1(χ1)and φk,1(χ), the prior knowledge of ¯
F1(Z1)cannot be
available as well. Then, fuzzy logic system T
1S1(Z1)is used
to estimate ¯
F1(Z1)satisfying
¯
F1(Z1)=T
1S1(Z1)+δ1(Z1)(17)
where δ1(Z1)is an error and meets |δ1(Z1)|≤ ε1,∀ε1>0.
Considering the fact that 0 <ST
1(·)S1(·)≤1 and Young’s
inequality (8) results in
e3
1¯
F1(Z1)=e3
1T
1S1(Z1)+|e3
1|δ1(Z1)
≤e6
112
2a2
1S1(X1)2+1
2a2
1+3
4e4
1+1
4ε4
1
≤ble6
1θ
2a2
1S1(X1)2+1
2a2
1+3
4e4
1+1
4ε4
1(18)
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4IEEE TRANSACTIONS ON CYBERNETICS
where θ=max{(1/bl)i2;i=1,2,...,n}and X1=
[χ1,yr]T.Further, by substituting (16)into(15) and using the
results (17) and (18), we can obtain
LV1≤e3
1gk,1χ2+ble3
1θ
2a2
1S1(X1)2−3
4e4
1+3
4l2
1
+1
2a2
1+1
4ε4
1−bl
˜
θ˙
ˆ
θ
≤e3
1gk,1α1+ble3
1θ
2a2
1S1(X1)2+e3
1gk,1e2
−3
4e4
1+d1−bl
˜
θ˙
ˆ
θ(19)
where e2=χ2−α1and d1=(3/4)l2
1+(1/2)a2
1+(1/4)ε4
1.
With the help of (8) and (5), the coupling term e3
1gk,1e2can
be split as
e3
1gk,1e2≤3
4e4
1+1
4b4
re4
2.(20)
By means of (20), we can rewrite (19)as
LV1≤e3
1gk,1α1+ble3
1θ
2a2
1S1(X1)2
+1
4b4
re4
2+d1−bl
˜
θ˙
ˆ
θ. (21)
Furthermore, by constructing the first intermediate control α1
in (10) when i=1, that is
α1=−λ1e1−e3
1ˆ
θ
2a2
1S1(X1)2
and by applying Assumption 1, one has
e3
1gk,1α1≤−λ1ble4
1−ble6
1ˆ
θ
2a2
1S1(X1)2.(22)
From (22), we rewrite (21)as
LV1≤−λ1ble4
1+bl
˜
θe6
1
2a2
1S1(X1)2−˙
ˆ
θ
+1
4b4
re4
2+d1(23)
where λ1,a1, and are positive constants.
Step 2: From (9) and Itˆo formula, we get
de2=gk,2χ3+fk,2(¯χ2)−α1dt
+φT
k,2(χ) −∂α1
∂χ1φT
k,1(χ)T
dw (24)
where
α1=∂α1
∂χ1fk,1(¯χ2)+
1
j=0
∂α1
∂y(j)
r
y(j+1)
r
+∂α1
∂ˆ
θ
˙
ˆ
θ+1
2
∂2α1
∂χ2
1
φT
k,1(χ)φk,1(χ ). (25)
Take a Lyapunov function as
V2=V1+1
4e4
2.(26)
According to (7), the following result holds:
LV2=LV1+e3
2gk,2χ3+fk,2(¯χ2)−α1
+3
2e2
2φT
k,2(χ) −∂α1
∂χ1φT
k,1(χ)T
×φT
k,2(χ) −∂α1
∂χ1φT
k,1(χ).(27)
By the utilization of (23) and the following inequality:
3
2e2
2φT
k,2(χ) −∂α1
∂χ1φT
k,1(χ)TφT
k,2(χ) −∂α1
∂χ1φT
k,1(χ)
≤3
4l−2
2e4
2φT
k,2(χ) −∂α1
∂χ1φT
k,1(χ)2+3
4l2
2.
Equation (27) can be written as
LV2=−λ1ble4
1+e3
2gk,2χ3+Fk,2(Z2)−3
2e4
2
+bl
˜
θe6
1
2a2
1S1(X1)2−˙
ˆ
θ+d1+3
4l2
2(28)
where
Fk,2(Z2)=fk,2(¯χ2)+3
2e2+1
4b4
re2−α1
+3
4l−2
2e2
φT
k,2(χ) −∂α1
∂χ1φT
k,1(χ)
2
(29)
where Z2=[χT,¯y(2)
r,ˆ
θ]Twith ¯y(2)
r=[yr,˙yr,¨yr]T.
Define ¯
F2(Z2)=maxk∈{|Fk,2(Z2)|}. Then, it is easily
verified that
LV2=−λ1ble4
1+e3
2gk,2e3+e3
2gk,2α2+e3
2¯
F2(Z2)
−3
2e4
2+bl
˜
θe6
1
2a2
1S1(X1)2−˙
ˆ
θ
+d1+3
4l2
2(30)
where e3=χ3−α2.
Since ¯
F2(Z2)is unknown, fuzzy system T
2S2(Z2)is applied
to approximate ¯
F2(Z2)in the following form:
¯
F2(Z2)=T
2S2(Z2)+δ2(Z2)(31)
in which δ2(Z2)is an estimation error and |δ2(Z2)|≤ ε2with
ε2>0.
It is follows from Young’s inequality that:
e3
2¯
F2(Z2)≤ble6
2θ
2a2
2S2(X2)2+1
2a2
2+3
4e4
2+1
4ε4
2(32)
e3
2gk,2e3≤3
4e4
2+1
4b4
re4
3.(33)
From (31)–(33), we have
LV2=−λ1ble4
1+e3
2gk,2α2+ble3
2θ
2a2
2S2(X2)2
+bl
˜
θe6
1
2a2
1S1(X1)2−˙
ˆ
θ
+
2
j=1
dj+1
4b4
re4
3(34)
where dj=(3/4)l2
j+(1/2)a2
j+(1/4)ε4
j.
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WANG et al.: ROBUST FUZZY ADAPTIVE TRACKING CONTROL FOR NONAFFINE STOCHASTIC NONLINEAR SWITCHING SYSTEMS 5
Design the intermediate control function α2in (10) with
i=2. Further, the following result is true:
e3
2gk,2α2≤−λ1ble4
2−ble6
1ˆ
θ
2a2
2S2(X2)2.(35)
By using (35), it can be easily obtained that
LV2=−
2
j=1
λjble4
j+bl
˜
θ⎛
⎝
2
j=1
e6
j
2a2
j
Sj(Xj)
2−˙
ˆ
θ⎞
⎠
+
2
j=1
dj+1
4b4
re4
3.(36)
Step i(3 ≤i≤n−1): According to (9) and Itˆoformula,we
have
dei=gk,iχi+1+fk,i(¯χi)−αi−1dt
+⎛
⎝φk,i(χ) −
i−1
j=1
∂αi−1
∂xjφk,j(χ)⎞
⎠
T
dw (37)
where
αi−1=
i−1
j=1
∂αi−1
∂χjfk,j(¯χj)+
i−1
j=0
∂αi−1
∂y(j)
r
y(j+1)
r
+1
2
i−1
p,q=1
∂2αi−1
∂χp∂χqφT
k,p(χ)φk,q(χ )
+∂αi−1
∂ˆ
θ
˙
ˆ
θ. (38)
Select a Lyapunov functional
Vi=Vi−1+1
4e4
i.(39)
Then, by using (7), it follows:
LVi=LVi−1+e3
i(gk,iχi+1+fk,i(¯χi)−αi−1)
+3
2e2
i⎛
⎝φk,i(χ) −
i−1
j=1
∂αi−1
∂χjφk,j(χ )⎞
⎠
T
×⎛
⎝φk,i(χ) −
i−1
j=1
∂αi−1
∂χ φk,j(χ)⎞
⎠(40)
where, by following similar process to step 2, LVi−1is
outlined as:
LVi−1=−
i−1
j=1
λjble4
j+bl
˜
θ⎛
⎝
i−1
j=1
e6
j
2a2
j
Sj(Xj)
2−˙
ˆ
θ⎞
⎠
+
i−1
j=1
dj+1
4b4
re4
i.(41)
Next, utilizing Young’s inequality to the final term of (40) and
taking (36) into account, we obtain
LVi=−
i−1
j=1
λjble4
j+bl
˜
θ⎛
⎝
i−1
j=1
e6
j
2a2
jSj(Xj)2−˙
ˆ
θ⎞
⎠
+
i−1
j=1
dj+e3
igk,iχi+1+Fk,i(Zi)−3
2e4
i+3
4l2
i(42)
where
Fk,i(Zi)=fk,i(¯χi)+3
2ei+1
4b4
rei−αi−1
+3
4eiφk,i(χ) −
i−1
j=1
∂αi−1
∂χjφk,j(χ )2(43)
where Zi=[χT,¯y(i)
r,ˆ
θ]Twith ¯y(i)
r=[yr,˙yr,¨yr,...,y(i)
r]T.
Set ¯
Fi(Zi)=maxk∈{|Fk,i(Zi)|}. Then, we can rewrite
(42)as
LVi=−
i−1
j=1
λjble4
j+e3
igk,iei+1+e3
igk,iαi
+e3
i¯
Fi(Zi)+bl
˜
θ⎛
⎝
i−1
j=1
e6
j
2a2
jSj(Xj)2−˙
ˆ
θ⎞
⎠
+
i−1
j=1
dj−3
2e4
i+3
4l2
i.(44)
Next, applying the fuzzy system T
iSi(Zi)to approximate the
unknown function ¯
Fi(Zi)such that
¯
Fi(Zi)=T
iSi(Zi)+δi(Zi)(45)
where δi(Zi)denotes the estimation error and satisfies
|δi(Zi)|≤εiwith εi>0 being a constant.
Similar to the derivations of (32) and (33), the following
results are true:
e3
i¯
Fi(Zi)≤ble6
iθ
2a2
iSi(Xi)2+1
2a2
i+3
4e4
i+1
4ε4
i(46)
e3
igk,iei+1≤3
4e4
i+1
4b4
re4
i+1.(47)
Then, designing the intermediate control function αiin (10)
and repeating the process from (35)to(36) produces
LVi=−
i
j=1
λjble4
j+bl
˜
θ⎛
⎝
i
j=1
e6
j
2a2
j
Sj(Xj)
2−˙
ˆ
θ⎞
⎠
+
i
j=1
dj+1
4b4
re4
i+1.(48)
Step n: From (9) and Itˆo expression, one get
den=gk,nuk+fk,n(¯χn)−αn−1dt
+⎛
⎝φk,n(χ) −
n−1
j=1
∂αn−1
∂χjφk,j(χ )⎞
⎠
T
dw (49)
where αn−1is defined in (38) with i=n.Choose a Lyapunov
function as
Vn=Vn−1+1
4e4
n.(50)
From (7), considering (41) with k=nand using the
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6IEEE TRANSACTIONS ON CYBERNETICS
completion of squares, we obtain
LVn≤−
n−1
j=1
λjble4
j+bl
˜
θ⎛
⎝
n−1
j=1
e6
j
2a2
j
Sj(Xj)
2−˙
ˆ
θ⎞
⎠
+e3
ngk,nuk+fk,n(¯χn)−αn−1+1
4b4
ren
+3
2e2
n⎛
⎝φk,n(χ) −
n−1
j=1
∂αn−1
∂χjφk,j(χ )⎞
⎠
T
×⎛
⎝φk,n(χ) −
n−1
j=1
∂αn−1
∂χjφk,j(χ )⎞
⎠+
n−1
j=1
dj
≤−
n−1
j=1
λjble4
j+bl
˜
θ⎛
⎝
n−1
j=1
e6
j
2a2
j
Sj(Xj)
2−˙
ˆ
θ⎞
⎠
+
n−1
j=1
dj+e3
ngk,nuk+Fk,n(Zn)
−3
4e4
n+3
4l2
n(51)
where
Fk,n(Zn)=fk,n(¯χn)+3
4en+1
4b4
ren−αn−1
+3
4en
φk,n(χ) −
n−1
j=1
∂αn−1
∂χjφk,j(χ )
2
(52)
with Zn=[χT,¯y(n)
r,ˆ
θ]Tand ¯y(n)
r=[yr,˙yr,¨yr,...,y(n)
r]T.
Let ¯
Fn(Zn)=maxk∈{|Fk,n(Zn)|}.Next,(51) is rewritten as
LVn=−
n−1
j=1
λjble4
j+e3
ngk,nuk+|e3
n|¯
Fn(Zn)
+
n−1
j=1
dj+bl
˜
θ⎛
⎝
n−1
j=1
e6
j
2a2
j
Sj(Xj)
2−˙
ˆ
θ⎞
⎠
−3
4e4
n+3
4l2
n.(53)
Furthermore, the term ¯
Fn(Zn)could be expressed by
¯
Fn(Zn)=T
nSn(Zn)+δn(Zn)(54)
where T
nSn(Zn)is a fuzzy system with estimation error δn(Zn)
satisfying |δn(Zn)|≤ εnfor ∀εn>0.
Following the estimation methods of (32), one has:
e3
n¯
Fn(Zn)≤ble6
nθ
2a2
nSn(Xn)2+1
2a2
n+3
4e4
n+1
4ε4
n(55)
where anis the design constant.
By combining (56) with (54) and (55), we have
LVn=−
n−1
j=1
λjble4
j+e3
ngk,nuk+ble3
nθ
2a2
nSn(Xn)2
+
n
j=1
dj+bl
˜
θ⎛
⎝
n−1
j=1
e6
j
2a2
jSj(Xj)2−˙
ˆ
θ⎞
⎠.(56)
Now, we are ready to design actual controller ukin the
following form:
uk=−λnen−e3
nˆ
θ
2a2
nSn(Xn)2.(57)
Further, with the consideration of (5), one has
e3
ngk,nuk≤−λnble4
n−ble6
nˆ
θ
2a2
nSn(Xn)2.(58)
Then, substituting (57)into(56) and applying (58) result in
LVn=−
n
j=1
λjble4
j+
n
j=1
dj
+bl
˜
θ⎛
⎝
n
j=1
e6
j
2a2
j
Sj(Xj)
2−˙
ˆ
θ⎞
⎠.(59)
At the present stage, we summarize the main result of this
note by the following theorem.
Theorem 1: For the stochastic nonlinear system (1) under
Assumptions 1 and 2, we design the virtual control sig-
nal αiin (10) and actual controller (57) with adaptive law
defined in (12). Then, under the bounded initial values,
all signals within the closed-loop system are semi-globally
uniformly ultimately bounded in fourth-moment and track-
ing error enters into a small neighborhood around zero by
appropriately choosing design parameters.
Proof: For provide the stability analysis, select a Lyapunov
functional V=Vn.From(59), by designing adaptive law
defined in (12), it follows that:
LVn=−
n
j=1
λjble4
j+
n
j=1
dj+blk0
˜
θˆ
θ. (60)
With the fact that (blk0/) ˜
θˆ
θ≤(blk0/2)θ2−(blk0/2) ˜
θ2,
we obtain
LV≤−
n
j=1
λjble4
j−blk0
2
˜
θ2+
n
j=1
dj+blk0
2θ2
≤−ν0V+μ0(61)
where ν0=min{4λjbl,k0,j=1,2,...,n}and μ0=
(bl/2)k0θ2+n
j=1dj.
From Lemma 1, we known that
E[V(t)]≤V(0)e−ν0t+μ0
ν0(62)
which implies that V(0)+(μ0/ν0)is the upper bound of
E[V(t)]. Therefore, it can be shown from the definition of V
that all signals within the closed-loop system are semi-globally
uniformly ultimately bounded in fourth-moment.
Next, taking the limits of both sides of (62) and noting
limt→∞ e−ν0t=0, one has
Ee4
1(t)=E(y(t)−yr(t))4≤4μ0
ν0,t→∞.(63)
Therefore, we conclude that the fourth-moment tracking error
enters into a small area around aero.
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WANG et al.: ROBUST FUZZY ADAPTIVE TRACKING CONTROL FOR NONAFFINE STOCHASTIC NONLINEAR SWITCHING SYSTEMS 7
Remark 1: Noted that several related researches on
approximation-based control algorithms were reported
in [54]–[56] for nonlinear switched systems with pure-
feedback form. The main differences between our result and
the ones in [54]–[56] can be summarized as follows: 1) the
results in [54] and [55] do not consider the effect of stochastic
disturbance, and the control system studied in this paper is of
pure-feedback stochastic nonlinear switched systems which
is in a more general form and 2) the nonlinear system is
reported in [56] for lower-triangular nonlinear systems, i.e.,
the function φσ(t),i(.) in ith subsystem is a function of the
previous states χj(j≤i).
IV. SIMULATION EXAMPLE
Example 1: To test the feasibility of our control scheme,
consider the following switched systems:
⎧
⎨
⎩
dχ1=f1,1(χ1,χ
2)dt +φT
1,1(χ1,χ
2)dw
dχ2=f1,2(¯χ2,u1)dt +φT
1,2(χ1,χ
2)dw
y=χ1
(64)
⎧
⎨
⎩
dχ1=f2,1(χ1,χ
2)dt +φT
2,1(χ1,χ
2)dw
dχ2=f2,2(¯χ2,u2)dt +φT
2,2(χ1,χ
2)dw
y=χ1
(65)
where f1,1(χ1,χ
2)=0.8(1+χ2
1)χ2+0.9χ3
2,φ
1,1(χ1,χ
2)=
χ1χ2
1,f1,2(¯χ2,u1)=u1+0.5u3
1+χ2e−0.5χ2
1,φ
1,2(χ1,χ
2)=
0.6χ2sin χ1;f2,1(χ1,χ
2)=(0.5+0.3sinχ1)χ2+
χ3
2,φ
2,1(χ1,χ
2)=0.1χ2sin χ1,f2,2(¯χ2,u2)=
u2+0.2sinu1+χ1χ2and φ2,2(χ1,χ
2)=0.5χ2cos χ1.
The reference is given as yr=0.5(sin(t)+sin(0.5t)).
Apparently, the switched system considered in this part
does not belong to lower triangular since the nonlinearities
φ1,1(χ1,χ
2)and φ2,1(χ1,χ
2)being functions of all states.
The currently available control approaches via backstepping
are not suitable in theory. To construct fuzzy controller,
eleven fuzzy sets are chosen over the interval [−10,10] for
states χ1,χ
2,yr, and ˆ
θ. The fuzzy membership functions are
constructed in the following form:
μFk
i(χi)=e−0.5(χi−ck)2,1≤i≤4,1≤k≤11
with χ3=yr,χ
4=ˆ
θ, and ck=10 −2(k−1). Following
the design procedure in Section III, the virtual control law,
adaptive law, and actual controller are designed in the forms:
α1=−λ1e1−e3
1ˆ
θ
2a2
1S1(X1)2
˙
ˆ
θ=
2
j=1
e6
jˆ
θ
2a2
jSjXj2−k0ˆ
θ
uk=−λ2e2−e3
2ˆ
θ
2a2
2S2(X2)2,k=1,2
where e1=χ1−yr,e2=χ2−α1,X1=[χ1,yr]T, and X2=
[χ1,χ
2,yr,ˆ
θ]T.The design parameters and initial conditions
are selected as λi=6,ai=1,i=1,2,k0=0.2 and =1,
[χ1(0), χ2(0)]T=[0.3,−0.2]Tand ˆ
θ(0)=0.01.
Fig. 1. System output yand target signal yr.
Fig. 2. State x2.
Fig. 3. Control signal uk.
Simulation results are displayed by Figs. 1–5.Fig.1
expresses the system output yand target signal yr.From Fig. 1,
it is shown that the output ytracks the target signal yrwell.
Fig. 2 displays the state x2. The boundedness of control signal
ukis shown in Fig. 3.Fig.4shows the trajectory of adap-
tive law ˆ
θ.Fig.5illustrates the evolution of switching signal.
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8IEEE TRANSACTIONS ON CYBERNETICS
Fig. 4. Adaptive parameter ˆ
θ.
Fig. 5. Trajectory of switched signal σ(t).
From the numerical simulation, it is clear that the ideal control
performance is obtained and all signals are bounded.
V. CONCLUSION
The output tracking control problem has been considered
for a class of switched nonlinear systems with completely non-
affine form and nonlower triangular structure under arbitrary
switchings in this paper. By using the common Lyapunov func-
tion method and backstepping, an effective tracking control
scheme has been developed and the stability of the closed-loop
system is proved. The primary contributions of this paper are
three folds: 1) the completely nonaffine form and nonlower
triangular structure are taken into account for the controlled
systems; 2) just one adaptive law is needed in the presented
control scheme; and 3) by adopting the structural charac-
teristics of fuzzy systems, the increasing property of whole
state functions is removed and the control design and sta-
bility analysis becomes much simpler simultaneously. In the
future works, we will pay attention to the control problems of
more complicated systems such as the output-feedback control
of nonaffine switched stochastic systems by applying fuzzy
adaptive control approach.
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Huanqing Wang received the B.Sc. degree in
mathematics from Bohai University, Jinzhou, China,
in 2003, the M.Sc. degree in mathematics from Inner
Mongolia University, Huhhot, China, in 2006, and
the Ph.D. degree from the Institute of Complexity
Science, Qingdao University, Qingdao, China, in
2013.
He was a Post-Doctoral Fellow with the
Department of Electrical Engineering, Lakehead
University, Thunder Bay, ON, Canada, in 2014.
He is currently a Post-Doctoral Fellow with the
Department of Systems and Computer Engineering, Carleton University,
Ottawa, ON, Canada. He has authored or co-authored over 30 papers in top
international journals. His current research interests include adaptive backstep-
ping control, fuzzy control, neural networks control, and stochastic nonlinear
systems.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
10 IEEE TRANSACTIONS ON CYBERNETICS
Peter Xiaoping Liu (SM’07) received the B.Sc. and
M.Sc. degrees from Northern Jiaotong University,
Beijing, China, in 1992 and 1995, respectively, and
the Ph.D. degree from the University of Alberta,
Edmonton, AB, Canada, in 2002.
He has been with the Department of Systems and
Computer Engineering, Carleton University, Ottawa,
ON, Canada, since 2002. He is currently a Professor
and the Canada Research Chair. His current research
interests include interactive networked systems and
teleoperation, haptics, surgical simulation, robotics,
intelligent systems, and context-aware systems.
Dr. Liu has published over 250 research articles. He serves as an Associate
Editor for several journals including the IEEE/ASME TRANSACTIONS ON
MECHATRONICS, the IEEE TRANSACTIONS ON CYBERNETICS, the IEEE
TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING,andthe
IEEE ACCESS. He is a Licensed Member of the Professional Engineers of
Ontario (P.Eng) and a fellow of Engineering Institute of Canada.
Ben Niu was born in Shandong, China, in 1982. He
received the B.S. degree in mathematics and applied
mathematics from Liaocheng University, Liaocheng,
China, in 2007, and the M.S. and Ph.D. degrees
in pure mathematics and control theory and appli-
cations from Northeastern University, Shenyang,
China, in 2009 and 2012, respectively.
He is currently an Associate Professor with
the College of Mathematics and Physics, Bohai
University, Jinzhou, China. His current research
interests include switched systems, stochastic sys-
tems, robust control, and intelligent control and their applications.
