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International Journal of Control, Automation, and Systems (2012) 10(6):1182-1192
DOI 10.1007/s12555-012-0613-0
ISSN:1598-6446 eISSN:2005-4092
http://www.springer.com/12555
Stability of a Class of Switched Stochastic Nonlinear Systems under
Asynchronous Switching
Dihua Zhai, Yu Kang*, Ping Zhao, and Yun-Bo Zhao
Abstract: The stability of a class of switched stochastic nonlinear retarded systems with asynchronous
switching controller is investigated. By constructing a virtual switching signal and using the average
dwell time approach incorporated with Razumikhin-type theorem, the sufficient criteria for pth mo-
ment exponential stability and global asymptotic stability in probability are given. It is shown that the
stability of the asynchronous stochastic systems can be guaranteed provided that the average dwell
time is sufficiently large and the mismatched time between the controller and the systems is sufficient-
ly small. This result is then applied to a class of switched stochastic nonlinear delay systems where the
controller is designed with both state and switching delays. A numerical example illustrates the effec-
tiveness of the obtained results.
Keywords: Asynchronous switching, average dwell time, Razumikhin-type theorem, stochastic
stability, switched stochastic nonlinear systems, time-delays.
1. INTRODUCTION
Switched systems are a special class of hybrid systems,
in the sense that the former is described by a family of
finite subsystems whose active modes are governed by a
switching rule. Switched systems are reasonable models
for various practical systems, such as networked control
systems, communication systems, flight control, etc. [1-
3]. One is usually interested in the stability, control-
lability, robustness, passivity, optimal control, sliding
mode control, etc., of such systems [4-14], among which
the stability is the most concerned and a number of tools,
e.g., multiple Lyapunov functions, average dwell time
approach, have been proposed from various perspectives
[11,15-17]. On the other hand, time delays are often seen
in practical engineering systems, which can be a severe
factor that deteriorates the system performance. A large
volume of studies can be seen from the literature [18-20];
some are undertaken within the switched system
framework [21-27].
Two types of controllers, the mode-dependent and the
mode-independent, are seen for switched systems. It is
believed that the mode-dependent controller is less con-
servative as it takes advantage of more information of the
system. The mode-dependent controller is often assumed
to be ideally synchronous with the switching of systems
[25] which, however, may not be true in reality due to
the presence of time delays. Specifically, on the one hand,
time-delay often appears in switched systems either in
input control or in output measurements. The former is
mainly due to actuator dynamics, the calculation of the
control gains, the communication delay between the
controller and the actuator, etc. while the latter can be
caused by the communication delay between the sensor
and the controller, etc. In some cases delay can indirectly
be induced by a phase lag in filtering out the noise from
the measurements. On the other hand, in the practical
implementation, due to unknown abrupt phenomena such
as component and interconnection failures, detecting the
switching rules also takes time. Those thus present a
great challenge at the boundary of switched systems and
time delay systems. Then the so-called asynchronous
switching is proposed, and a number of efforts have been
made, for example, the admissible delay of asynchronous
switching are given in [28,29]; state feedback stabili-
zation, input-to-state stabilization and output feedback
stabilization are studied in [30-36]; results have also
been reported for Markov jump linear systems [37-39],
just to name a few.
Switched stochastic systems have recently been pop-
ular due to the significant role played by the stochastic
modelling in many branches of engineering disciplines.
Consequently, the study of control synthesis for such sy-
stems with asynchronous switching, e.g., robust stabil-
ization, H∞ filtering, have been widely seen [40-43].
© ICROS, KIEE and Springer 2012
__________
Manuscript received May 31, 2012; revised August 24, 2012;
accepted September 6, 2012. Recommended by Editor Ju Hyun
Park.
This journal was supported in part by the National Natural
Science Foundation of P. R. China under grant No.60935001,
61174061, 60904024, 60934006, the Anhui Provincial Natural
Science Foundation (11040606M143), the Doctoral Foundation of
University of Jinan (XBS1012), the Fundamental Research Funds
for the Central Universities, the Program for New Century
Excellent Talents in University, and the Program for Youth
Innovation Promotion Association, CAS.
Dihua Zhai and Yu Kang are with the Department of Automa-
tion, University of Science and Technology of China, Hefei,
China (e-mails: [email protected], [email protected]).
Ping Zhao is with the School of Electrical Engineering,
University of Jinan, Jinan, China (e-mail: [email protected]).
Yun-Bo Zhao is with the Department of Chemical Engineering,
Imperial College London, London, United Kingdom and also with
CTGT centre, Harbin Institute of Technology, Harbin, China (e-
mail: [email protected]).
* Corresponding author.
Stability of a Class of Switched Stochastic Nonlinear Systems under Asynchronous Switching
1183
Using the boundedness assumption of nonlinear term, a
linear matrix inequality (LMI) approach incorporated wi-
th Lyapunov method is developed to meet the goal. In
another line switched stochastic retarded systems have
received much attention in recent years. Such systems
consist of a set of stochastic retarded subsystems that are
described by stochastic functional differential equations.
Works in related areas can be found in, for example, [44]
for the pth moment input-to-state stability of stochastic
nonlinear retarded systems under Markovian switching,
[24] for the stability under average dwell time switching
signal, and so forth. Despite all these results, to date
switched stochastic nonlinear retarded systems under as-
ynchronous switching have received little attention, whi-
ch motivates this study for us.
In this paper, we investigate the stability of a class of
switched stochastic nonlinear retarded systems under
asynchronous switching. The main challenge for a mode
dependent controller design is to deal with the mis-
matched period due to the existence of detection delays.
Our efforts are made towards the stability criteria for
such systems with respect to the mismatched interval.
For a more realistic situation, we consider the controller
with both state and switching delays. To describe and
deal with the asynchronous switching phenomenon, a
virtual switching signal is constructed and applied. Fin-
ally, a sufficient Razumikhin-type stability criterion is
derived to guarantee the stability of the closed-loop sys-
tem under average dwell time approach.
The remainder of the paper is organized as follows.
The problem is formulated and necessary definitions are
given in Section 2. The main results are then discussed in
Section 3, with an application given in Section 4. Section
5 concludes the paper.
Notions: +
! and +
" denote the set of positive
integer and nonnegative real numbers, respectively. Let
0.+
∪=! ! If x and y are real numbers, then x y∧
denotes the minimum of x and y. For vector ,
n
x∈! | |x
denotes the Euclidean norm. Let τ≥ 0 and ([ ,0]; )n
C τ− !
denote the family of all continuous n
! -valued functions
φ on [–τ, 0] with the norm || || | ( ) |:supϕ ϕ θ τ θ= − ≤
.0≤ Let 0
([ ,0]; )b n
C τ− !F be the family of all 0-F
measurable bounded ([ ,0]; )-n
C τ− ! valued random vari-
ables ( ) : 0.ξ ξ θ τ θ= − ≤ ≤ For t≥ 0, let ([ ,0];t
p
L τ−
F
)n
! denote the family of all -
tF measurable ([ ,C τ−
0]; )-n
! valued random variables ( ) :φ φ θ τ θ= − ≤ ≤ 0
such that 0
sup | ( ) |
p
τ θ
φ θ− ≤ ≤
E .< ∞ The transpose of
vectors and matrices is denoted by superscript T. Ci
denotes all the ith continuous differential functions; Ci,k
denotes all the functions with ith continuously differenti-
able first component and kth continuously differentiable
second component. Finally, the composition of two
functions : A Bα → and : B Cβ → is denoted by
: .A Cα β →!
2. PRELIMINARIES
Consider the following switched stochastic nonlinear
retarded systems
( ) ( )( , , , ( )) g ( , , , ( )) ,
t t t tdx f t x x u t dt t x x u t dw
σ σ
= + (1)
where ( )n
x x t= ∈! is the state vector, ( ) :tx x t θ= +
0τ θ− ≤ ≤ is ([ ,0]; )-n
C τ− ! valued random process,
( )l
u t∞
∈L is the control input. l
∞
L denotes denotes the
set of all the measurable and locally essentially bounded
input ( )l
u t ∈! on 0
[ , )t ∞ with the norm
0
0
[ , )
, ( ) 0[ , )
|| ( ) || sup inf sup ( , ) : \ t
s t
u s u w s w∞
∈Ω =∈ ∞
= ∈Ω
PA AA
(2)
w(t) is the m-dimensional Brownian motion defined on
the complete probability space 0
, ,( , ),t t t≥
Ω PF F with
Ω being a sample space, F being a σ -field, 0
t t t≥
F
being a filtration and P being a probability measure.
0: [ , )tσ ∞ → S (S is the index set, and may be infinite)
is the switching law and is right hand continuous and
piecewise constant on t. σ(t) discussed in this paper is
time dependent, and the corresponding switching times
are 1 2
.l
t t t< < < <! ! The il th subsystems will be
activated at time interval 1
[ , ).l lt t
+
Specially, when
0t t= (t
0 is the initial time), suppose
0 0 0( )t iσ σ= = ∈
.S For all ,i∈S : ([ ,0]; )
n n l
if C τ
+
× × − × →! ! ! !
n
! and : ([ ,0]; )
n n l n m
ig C τ
×
+
× × − × →! ! ! ! ! are
continuous with respect to t, x, u, and satisfy uniformly
locally Lipschitz condition with respect to x, u, and
( ,0,0,0) 0,if t ≡ ( ,0,0,0) 0.
ig t ≡
In practice, instantaneous switching signal detection is
impossible. In this paper, we are concerned the stability
of systems under the following mode-dependent state fe-
edback law: '( )
( ) ( , ),t t
u t h t xσ
= on 0
t t≥ with initial
data 0
0 0 ( ) : 0 ([ ,0]; )
b n
x x t Cθ τ θ ξ τ= + − ≤ ≤ = ∈ − !F
and 0 0 0 0
( ) ( ) .r r t r t iθ= + = = '( ) ( ( ))t t d tσ
σ σ= − is
utilized to denote the practical switching signal of
controller, where 0 ( ) .d t dσ
≤ ≤ Further, we assume
that, the detected switching signal σ'(t) is causal, i.e., the
ordering of the switching times of σ'(t) is the same as the
ordering of the corresponding switching times of σ(t).
Further, we also assume that 1
inf ,l l l
d t t∈ +
≤ −! which
guarantees that there always exists a period in which the
controller and the system operate synchronously. This
period is called matched period. Let '
1 l lt
≥ denotes the
switching times of σ'(t), '
( )( ),
ll l t lt t d t
σ
= + then '
'( )
ltσ =
( ) ,l lt iσ = and
' '
1 1 2 2t t t t< < <
'
1.
l l lt t t
+
< < < < <! !
We assume '
0 0.t t=
Due to the existence of the detection delay, there exist-
s a period in 1
[ , )l lt t
+
such that the mode-dependent fee-
dback control input u(t) and the li th subsystem operate
asynchronously. We call the time interval '
[ , )l lt t the
mismatched period, and '
1[ , )l lt t
+
the matched period.
For convenience, let '
1( , ) [ , ),
a l l l lT t t t t
+
=
'
1( , ) [ ,
s l l l
T t t t+
=
1).
lt+
For any0
s t≥ , let ( )a
T t s− denote the total time
of the mismatched time interval on [ , ].s t Then, for any
,l+
∈! we have
Dihua Zhai, Yu Kang, Ping Zhao, and Yun-Bo Zhao
1184
1
1 1
1 1 1 2
1
1 1 2
1
1 1
1 1
( , )
( ) ( , )
( ) ( , )
( ) ( , )
( )
( ) ( , )
( ) ( , ),
l a l l
a l l s l l
a l l l a l l
l
a i i s l l
i l
a l
n
a i i n a n n
i l
n
a i i s n n
i l
t t t T t t
T t t t T t t
T t t t t t T t t
T t t t T t t
T t t
T t t t t t T t t
T t t t T t t
+
+ +
+ + + +
+
+ + +
=
−
+ +
=
+ +
=
− ∈⎧
⎪− ∈
⎪
⎪ − + − ∈
⎪
⎪− ∈
⎪⎪
− = ⎨
⎪
⎪
− + − ∈⎪
⎪
⎪
− ∈
⎩
∑
∑
∑
! !
⎪
⎪
where 1
( )a l l
T t t+
− is the length of the 1
( , ).a l l
T t t+
For each i∈S , :
n l
ih
+
× →! ! ! is a smooth fun-
ction. Then, the closed-loop system can be transformed
into the following retarded-type systems
( ) ( )( , , ) ( , , ) ,
t t t tdx f t x x dt g t x x dw
σ σ
= + (3)
where ( ) ( ) '( )( , , ) ( , , , ( , )),
t t t t t tf t x x f t x x h t xσ σ σ
= ( )( ,
tg tσ
( ) '( ), ) ( , , , ( , )).
t t t t tx x g t x x h t x
σ σ
= ( )tσ is a virtual swit-
ching signal from 0
[ , )t ∞ to ×S S with ( ) ( ( ),t tσ σ=
'( ))tσ and switching times 0
l lt
≥ 0 0( ).t t= Clearly,
we have 2 1l lt t
−= and
'
2,
l lt t= for any 1.l ≥ Assume
that the composite functions f and g are sufficiently
smooth, such that system (3) has an unique solution on
0.t t τ≥ −
The following definitions are needed for the stability
of closed-loop system (3).
Definition 1: The equilibrium ( ) 0x t = of system (3)
is globally asymptotically stable in probability (GASiP),
if for any 0,ε > there exists a class KL function β
such that 0 0
| | ( || ||, ) 1 , .x t t t tβ ξ ε< − ≥ − ∀ ≥P E
Remark 1: Class ,K ,
∞
K KL functions are defined
in [45]. And in the sequel, class ∞
CK and ∞
VK
function are the two subsets of class ∞
K functions that
are convex and concave, respectively.
Definition 2: The equilibrium ( ) 0x t = of system (3)
is pth moment exponentially stable, if there exists a class
KL function β (where ( , )tβ ⋅ will converge to zero by
the way of exponential decay as ),t →∞ such that
0 0| | ( || || , ), .
p p
x t t t tβ ξ< − ∀ ≥E E (4)
Definition 3 [15]: For any given constants τ*
0> and
N0, let ( , )N t s
σ
denote the switch number of ( )tσ in
[ , ),s t for any 0,t s t> ≥ and let
0 0 0 ( ) : ( , ) , [ ,] )[ *, .
t s
N t s N s t tS Nσ
σ
τ
τ
∗
−
⋅ ≤ + ∀ ∈=
Then τ*
is called the average dwell-time of *
0[ , ],S Nτ
and
0 0 0
sup sup
( , )t t t s t
t s
N t s N
σ
σ
τ
≥ > ≥
−
−
! is called the average
dwell-time of ( ).σ ⋅
Lemma 1 [23]: If 0
( ,) ][ ,S Nσ τ
∗
⋅ ∈ then ( )σ ′ ⋅ ∈
0[ , ]
d
S Nτ
τ
∗
∗
+ and 0
( ) [ , 2 .]
2
d
S N
τ
σ
τ
∗
∗
⋅ ∈ +
Definition 4 [18]: For any given 2,1
(n
V C+
∈ ×! !
; ),+
× × !S S define a diffusion operator associated with
system (3), ,VL from ([ ,0]; )n
C τ+
− × × ×! ! S S to
,! by
( )
2
( ) ( )2
( , , ( ))
( , , ( )) ( , , ( ))
( , , )
1 ( , , ( ))
[ ( , , ) ( , , )],
2
t
t t
T
t t t t
V x t t
V x t t V x t t
f t x x
t x
V x t t
trace g t x x g t x x
x
σ
σ σ
σ
σ σ
σ
∂ ∂
= +
∂ ∂
∂
+
∂
L
(5)
where ( , , ( )) ( , , ( ( .), ))V t t V t t tσ σ σ ′⋅ ⋅!
In what follows let ( )( , )
tV tσ
⋅ denote ( , , .( ))V t tσ⋅
3. MAIN RESULTS
Based on the average dwell-time approach, we give th-
e sufficient criteria for GASiP and pth moment exponen-
tial stability for a class of switched stochastic nonlinear
retarded systems. We first consider the stability of ordin-
ary switched systems under asynchronous switching and
then the case with retarded-type state-feedback delay.
Let xt = x(t) in (3), then, it can be transformed into
( ) ( )d ( , )d ,( , )d
t tx f t x t g t x w
σ σ
= + (6)
where ( ) ( ) ( )( , ) ( , , ( , ) ,)
t t tf t x f t x h t xσ σ σ ′
!( )( , )
tg t xσ
!
( ) ( )( , , ( , )).
t tg t x h t xσ σ ′
Similarly, for any 2,1
(n
V C∈ ×!
; ),+ +
× ×! !S S we can define the infinitesimal operat-
or L from n
+
× × ×! ! S S to ,! associated with sy-
stem (6), by
( ) ( )
( ) ( )
2
( )
( ) ( )2
( , ) ( , )
( , ) ( , )
( , )1
[ ( , ) ( , )].
2
t t
t t
tT
t t
V x t V x t
LV x t f t x
t x
V x t
trace g t x g t x
x
σ σ
σ σ
σ
σ σ
∂ ∂
= +
∂ ∂
∂
+ ×
∂
Then, for closed-loop switched system (6), we have
the following result.
Lemma 2: If there exist functions 1,α
2,α
∞
∈K
class C2,1
Lyapunov function ( )( , )
tV x tσ
and some
positive constants λs, λ
a and µ≥ 1, such that
1 ( ) 2(| |) ( , ) (| |);
tx V x t x
σ
α α≤ ≤ (7)
and for any ,l∈!
( ) 1
( )
( ) 1
( , ), ( , ),
( , )
( , ), ( , ),
s t s l l
t
a t a l l
V x t t T t t
LV x t
V x t t T t t
σ
σ
σ
λ
λ
+
+
− ∈⎧⎪≤ ⎨
∈⎪⎩
(8)
hold almost surely. For any ,r+
∈!
Stability of a Class of Switched Stochastic Nonlinear Systems under Asynchronous Switching
1185
1( ) ( )
( ( ), ) ( ( ), ).r rt r r t r r
V x t t V x t tσ σ
µ−
≤E E (9)
Further, if there also exist some nonnegative constants
ρ ≥ 0 such that
,
s
s a
λ
ρ
λ λ
<
+
(10)
and for any t ≥ t0,
0 0( ) ( ).
a
T t t t tρ− ≤ − (11)
Then, system (6) is GASiP for all
ln( )
(1
.
)s a
µ
τ
λ ρ λ ρ
∗
>
− −
Proof: Denote ( )( , )
tW x t
σ
by ( )
.( , )st
te V x t
λ
σ Then,
in each time interval 1
[t , t ),l l+
according to inequality
(8), we can obtain that
1
( )
( ) 1
0, ( , );
( , )
( ) ( , ), ( , ).
s l l
t
s a t a l l
t T t t
LW x t
W x t t T t tσ
σ
λ λ
+
+
∈⎧⎪≤ ⎨
+ ∈⎪⎩
According to (9) and ( ) ( )
d
( , ) ( ,
dt t
LW x t W x
t
σ σ
=E E
),t when 1
( , ),a l l
t T t t+
∈ ( ( ), ) ( ( ),l
W x t t W x tσ σ
≤E E
( )( )
) ;s a l
t t
lt e
λ λ+ −
when 1 0 1
( , ),) [ ,s l l
t T t t t t+
∈ ∪ ( )
t
Wσ
E
( ) 1 1( ( ), ) ( ( ( )), ( )).
t l a l l l a l lx t t W x t T t t t T t t
σ + +
≤ + − + −E
Then, for 1
[ , ),l l
t t t+
∈ ,l∈! it holds that ( )
t
Wσ
E
2
( ) ( )
( )( ( ), ) ( ( ), ) .
s a a l
l
T t t
t l lx t t W x t t e
λ λ
σµ
+ −
≤ E Thus,
2
2 1
1
2( 1)
2
2( 2)
( ) ( )
( ) ( )
( ) ( )2
( )
( ) ( )3
( ) 1 1
( ) ( )5
( ) 2 2
2 (
( ( ), ) ( ( ), )
( ), )
( ( ), )
( ( ), )
s a a l
l
s a a l
l
s a a l
l
s a a l
l
T t t
t t l l
T t t
t l l
T t t
t l l
T t t
t l l
N
W x t t W x t t e
W x t t e
W x t t e
W x t t e
σ
λ λ
σ σ
λ λ
σ
λ λ
σ
λ λ
σ
µ
µ
µ
µ
µ
−
−
−
−
−
+ −
+ −
+ −
− −
+ −
− −
≤
≤
≤
≤
≤
≤
!
E E
E
E
E
0 0
0
0 0
0
, ) ( ) ( )
( ) 0 0
2 ln( )
2 [( ) ]( )
( ) 0 0
( ( ), )
( ( ), )
s a a
s a
t t T t t
t
d
N t t
t
W x t t e
W x t t e
λ λ
σ
µ
λ λ ρ
τ τ
σµ
∗ ∗
+ −
+ + + −
≤
E
E
for any 1
[ , ),l l
t t t+
∈ .l∈! Then,
0 0
0
0
( )
2 ln( )
2 [( ) ]( )
( ) 0 0
( ) 0 0 0
( ( ), )
( ( ), )
( ( ( ), ) ., )
s a s
t
d
N t t
t
t
V x t t
V x t t e
V x t t t t
σ
µ
λ λ ρ λ
τ τ
σ
σ
µ
β
∗ ∗
+ + + − −
≤
−!
E
When
ln( )
(1
,
)s a
µ
τ
λ ρ λ ρ
∗
>
− −
it’s easy to verify that
.β ∈KL For any 0,ε > take .
β
β
ε
= ∈
! KL By Che-
byshev’s inequality, we have
0
0
( ) ( ) 0 0 0
( )
1
( ) 0 0 0
( ( ), ) ( ( , ), )
( ( ), )
, [ , ).
( ( ( ), ), )
t t
t
l l
t
V x t t V x t t t
V x t t
t t t
V x t t t t
σ σ
σ
σ
β
ε
β+
≥ −
≤ < ∀ ∈
−
!
!
P
E
Let 1
1 2.( , ) ( ( ), )r s r sβ α β α
−
= °
!
β is a KL function if
the average dwell time is satisfied. Thus, we have
0 0 0| ( ) | (| |, ) 1 , .x t x t t t tβ ε< − ≥ − ∀ ≥P
This completes the proof.
Remark 2: The conditions (10) and (11) in Lemma 2
implies that the considered system can be stable provided
that the mismatched period is sufficiently small. The co-
ndition (8) indicates that the switched control systems
can start from unstable systems. Change the condition
(11) into the one that 0 0 0 0
( ): ,( )a
t t T t t t tτ ρ∀ ≥ − ≤ + −
where τ0 can be interpreted as an initial offset which
allows us to start with a subsystem with mismatched
controller. Clearly, we have 0 1 0
0 .t tτ≤ ≤ − Then, do
th-e similar analysis, we can also get the conclusion.
The following result can be obtained directly from the
proof of Lemma 2.
Corollary 1: System (6) is pth moment exponentially
stable for all
ln( )
(1
,
)s a
µ
τ
λ ρ λ ρ
∗
>
− −
if 1
α and 2
α in
Lemma 2 are such that 1 1( )
p
s c sα = and 2 2( )
p
s c sα =
where c1 and c
2 are positive constants.
The following lemma is useful for the stability of
retarded system (3) under asynchronous switching.
Lemma 3: For any C2,1
function , ,( )V x tσ
let ( )U t
( , )V x tσ
= for 0.t t≥ Then ( )U tE is continuous.
Proof: Based on ˆ’Ito s formula, we have
0
0
( ) ( ) 0 0 ( )( , ) ( ( ), ) ( , )d
t
t t s s
t
V x t V x t t V x s sσ σ σ
= + ∫ L (12)
0
( )
( )
( ( ), )
( , ( ), )d ( ).
ts
s s
t
V x s s
g s x s x w s
x
σ
σ
∂
+
∂∫
Since 0
([ , ,0]; )b n
Cξ τ∈ − !F we can find an integer k0
such that 0
|| || kξ < a.s. Then, for each integer 0,k k>
define a sequence of stopping time k
ρ as infk
tρ = ≥
0 0:| ( ) | , .t x t k k k> ! > Clearly,
kρ →∞ as .k →∞
If t is replaced by k k
tτ ρ= ∧ in (12), then the stoch-
astic integral (second integral) in (12) defines a marting-
ale (with k fixed and t varying), not just a local mar-
tingale. Thus, 0
( ) ( ) 0 0 ( ( ), ) ( ( ), )
kk k t
V x V x t tσ τ σ
τ τ =E E
0
( ) ( , )d .
k
s s
t
V x s s
τ
σ
+ ∫E L Letting k →∞ and using
Fubini’s theorem incorporated with Fatou’s lemma
yields
0
0
0 ( )
0 ( )
( ) ( ) ( , )d
( ) ( , ) d ,
t
s s
t
t
s s
t
U t U t V x s s
U t V x s s
σ
σ
= +
= +
∫
∫
E E E
E E
L
L
Dihua Zhai, Yu Kang, Ping Zhao, and Yun-Bo Zhao
1186
for all t ≥ t0, which implies ( )U tE is continuous.
Theorem 1: Suppose there exist functions 1
,α∞
∈K
2,α
∞
∈VK class C2,1
Lyapunov function ( )( , )
tV x tσ
and some constants 0,s
λ > 0,a
λ > 1,µ ≥ 0ρ ≥ and q
> 1, such that inequalities (7) and (9)-(11) hold, and
moreover, for any ,l∈!
( )
( ) 1
( ) 1
( ( ), )
( (0), ), ( , );
( (0), ), ( , ),
t
s t s l l
a t a l l
V t
V t t T t t
V t t T t t
σ
σ
σ
ϕ θ
λ ϕ
λ ϕ
+
+
− ∈⎧⎪< ⎨
∈⎪⎩
E
E
E
L
(13)
provided those ([ ,0]; )t
p n
Lϕ τ∈ − !F
satisfying that
, ( )
min ( ( ), ) ( (0), ),i j ij t
V t q V tσ
ϕ θ θ ϕ∈
+ ≤E ES (14)
for any [ ,0]θ τ∈ − . Further, if we also have
( (1 ) )s a
e qλ ρ λ ρ τ− −
≤ . (15)
Then, system (3) is GASiP for all
ln( )
(1
.
)s a
µ
τ
λ ρ λ ρ
∗
>
− −
Proof: According to (7) and Jensen’s inequality, we
obtain that
0( ) 0 0 2 0
[ ( 1) ]
2
( ( ), ) (| ( ) |)
( ) ,s a
tV x t t x t
Me
σ
λ ρ λ ρ θ
θ θ α θ
α ξ− +
+ + ≤ +
≤ ≤
E E
E
for any 0 ,[ , ]θ τ∈ − where 2( ).M α ξ! E Then, we
have 0
0
[ ( 1) ]( )
( ) ( ( ), ) ,
s at t
tV x t t Me
λ ρ λ ρ
σ
− + −
≤E0
[ ,t t τ∈ −
0.]t Now, for any
0 1[ , ,)t t t∈ we prove that
0
0
[ ( 1) ]( )
( ) ( ( ), ) .
s at t
tV x t t Me
λ ρ λ ρ
σ
− + −
≤E (16)
Suppose there exists some 0 1
( , )t t t∈ such that
0
0
[ ( 1) ]( )
( ) ( ( ), ) .
s at t
tV x t t Me
λ ρ λ ρ
σ
− + −
>E Let inft t
∗
=
0
0
[ ( 1) ]( )
0 1 ( )( , ) : ( ( ), ) .
s at t
tt t V x t t Me
λ ρ λ ρ
σ
− + −
∈ >E Con-
sidering the continuity of 0
( )tVσ
and x(t) on 0 1
[ , ),t t
without loss of generality, we have 0 1
( , )t t t
∗
∈ and
0
0
( ( 1) ( )
( )
)
( ( ), .)s a
t t
tV x t t Me
λ ρ λ ρ
σ
∗
− + −∗ ∗
=E Further, there
exists a sequence n
t!
1( ( , ),n
t t t
∗
∈! for any )n
+
∈!
with l ,imn
n
t t
∗
→∞
=! such that
0( )
( ( ), )t n n
V x t tσ
>! !E
0[ ( 1) ]( )
.
s a nt t
Me
λ ρ λ ρ− + −
!
From the definition of t*
, we have
0 0
0
0
( ) ( )
( ( 1) )
( )
( )
( ( ), ) ( ( ), )
( ( ), )
( ( ), ),
s a
t t
t
t
V x t t V x t t
e V x t t
q V x t t
σ σ
λ ρ λ ρ θ
σ
σ
θ θ∗ ∗ ∗ ∗
− + ∗ ∗
∗ ∗
+ + ≤
≤
≤
E E
E
E
and further
0, ( )
min ( ( ), ) ( ( ), ),i j ij t
V x t t q V x t tσ
θ θ∗ ∗
∈
+ + ≤E ES
for any 0 .[ , ]θ τ∈ − On the other hand, from Lemma 3,
we have ( , ) ( , ),t
D V x t V x tσ σ
+
=E E L where
( )
( ) ( )
0
( , )
( ( ), ) ( , )
lim sup .
t
t h t
h
D V x t
V x t h t h V x t
h
σ
σ σ
+
+
+
→
+ + −
=
E
E E
From inequality (13), we can obtain
0 0( ) ( )
( ( ), ) ( ( ), ).t s t
D V x t t V x t tσ σ
λ+ ∗ ∗ ∗ ∗
< −E E (17)
for any 0 1
[ , .)t t t∈ Clearly, there exists a positive const-
ants h > 0, which is small enough, such that (17) holds
on [ ].,t t h
∗ ∗
+ Then,
0 0
0
( ) ( )
[ ( 1) ]( )
( ( ), ) ( ( ), )
,
s
s a
h
t t
t t
V x t h t h V x t t e
Me
λ
σ σ
λ ρ λ ρ∗
−∗ ∗ ∗ ∗
− + −
+ + ≤
≤
E E
which is contradiction proves (16).
Considering the continuity, we further get that (16)
holds on all interval [t0,t1]. According to condition (9),
when 1 1
,t t t= = we have
1 0
1
[ ( 1) ]( )
( ) 1 1 ( ( ), )
s at t
tV x t t Me
λ ρ λ ρ
σµ
− + −
≤E (18)
Let ( ) ( )( ) ( ( ) ,, )
st
t tW t e V x t t
λ
σ σ= then,
1( ) 1
( )t
W tσ
µ≤E
0( ) 1
( ) .t
W tσ
E For any ,l∈! in time interval 1
[ , ),l lt t
+
we also have
1
( )
( ) 1
0, ( , );
( )
( ) ( ), ( , ).
s l l
t
s a t a l l
t T t t
W t
W t t T t tσ
σ
λ λ
+
+
∈⎧⎪< ⎨
+ ∈⎪⎩
L
Similarly, from Lemma 2 and Remark 2, we can obtain
( ) 0 1
0
( ) 0 0
0
2 ( , ) ( ) ( )
( ) ( ) 1
2 ( , ) ( ) ( )
( ) 1
( ) ( )
( ) ,
ts a a
ts a a
N t t T t t
t t
N t t T t t
t
W t W t e
W t e
σ
σ
λ λ
σ σ
λ λ
σ
µ
µ
+ −
+ −
≤
=
E E
E
which means
( ) 0 1 0 0 1
0 0
1 0
( )
2 ( , ) [ ( 1) ]( ) ( ) ( ) ( )
2 ln( )
2 [( ) ]( )
( )( )
( ( ), )
.
ts a s a a s
s a s
s a
t
N t t t t T t t t t
d
N t t
t t
V x t t
Me e e
e Me
σ
σ
λ ρ λ ρ λ λ λ
µ
λ λ ρ λ
ρ λ λτ τ
µ
µ
∗ ∗
− + − + − − −
+ + − + −
+ −
≤
≤
E
This completes the proof by Lemma 2.
Theorem 2: Let 1
.sup l l l
t tς+
∈ −
= − < ∞! Suppose
there exist functions 1
,α∞
∈K2
,α∞
∈VK class C2,1
Lyapunov function ( )( , )
tV x tσ
and some constants λs >
0, λa > 0, µ≥ 1, and q > 1, such that inequalities (7), (9),
(13) and (14) hold. Further, if there also exists
nonnegative constant ρ, such that for any 0,t tτ≥ ≥
s
s a
λ
ρ
λ λ
<
+
, (19)
( ) ( )a
T t tτ ρ τ− ≤ − . (20)
Stability of a Class of Switched Stochastic Nonlinear Systems under Asynchronous Switching
1187
Moreover, condition (15) is also satisfied. Then, system
(3) is GASiP for all
ln( )
(1
.
)s a
µ
τ
λ ρ λ ρ
∗
>
− −
Proof: Following the proof of Theorem 1, we have
0
0
[ ( 1) ]( )
( ) ( ( ), ) ,
s at t
tV x t t Me
λ ρ λ ρ
σ
− + −
≤E 0 1
[ , ,]t t t∀ ∈
and
1 0
1 0
( ) 1 1 ( ) 1 1
[ ( 1) ]( )
( ( ), ) ( ( ), )
.s a
t t
t t
V x t t V x t t
Me
σ σ
λ ρ λ ρ
µ
µ
− + −
≤
<
E E
Divide the time interval into 1 2
( , )a
T t t and 1 2
( , ).s
T t t
From condition (13), if (14) holds, we have
( )
( ) 1 2
( ) 1 2
( ( ), )
( ( ), ), ( , );
( ( ), ), ( , ).
t
s t s
a t a
D V x t t
V x t t t T t t
V x t t t T t t
σ
σ
σ
λ
λ
+
− ∈⎧⎪< ⎨
∈⎪⎩
E
E
E
Let ( .( ) , )st
W t e V x t
λ
σ σ= By the continuity of ( ,V x
σ
t) on any interval 1
[ , ),l lt t−
,l+
∈! it’s easy to verify
that ( )
( )t
W tσ
E is continuous on 1
[ , ).l lt t−
Moreover,
( ) ( ) ( )( ) ( ) ( ., )
st
t s t t tW t W t e V x t
λ
σ σ σλ= +L L Then
( )
1 2
( ) 1 2
( )
0, ( , );
( ) ( ), ( , )
.
t
s
s a t a
D W t
t T t t
W t t T t t
σ
σ
λ λ
+
∈⎧⎪< ⎨
+ ∈⎪⎩
E
E
Similar to Theorem 1, for any 1 2
[ , ),t t t∈ we have
2 1
1
( ) ( )
( ) ( ) 1 ( ) ) ,(
s a aT t t
t tW t W t e
λ λ
σ σµ
+ −
≤E E
i.e.,
1 2 1
1
0 2 1 1 0
( )
( ) ( ) ( )
( ) 1 1
( ) ( )( ( ) ( ))2
( , )
( ( ) )
.
, s s a a
s s a a
t
t t T t t
t
t t T t t t t
V x t
V x t t e
Me e
σ
λ λ λ
σ
λ λ λ ρ
µ
µ
− − + + −
− − + − + −
≤
=
E
E
By the continuity of ( )( , )
tV x tσ
and x, we have
1
0 2 1 1 0
( ) 2 2
( ) ( )( ( ) ( ))2
( ( ), )
.s s a a
t
t t T t t t t
V x t t
Me e
σ
λ λ λ ρ
µ
− − + − + −
≤
E
Now, suppose that for some 2,l ≥ we have
1 1 0
0 1
( )
( )( ( ) ( ))
( )2
( ( ), )
,
l
s a a i i
s i
t
T t t t t
t tl
V x t t
Me e
σ
λ λ ρ
λ
µ
+
=
+ − + −
− −
∑
≤
E
for any 1
[ , ).l l
t t t+
∈ If for any 1 2
[ , ],l l
t t t+ +
∈ we have
0
1
1 1 0
1
( )2 2
( )
( )( ( ) ( ))
( ( ), )
,
s
l
s a a i i
i
t tl
t
T t t t t
V x t t Me
e
λ
σ
λ λ ρ
µ
+
+
=
− −+
+ − + −
≤
∑
×
E
(21)
then by mathematical induction, the above assumption
holds. Actually, by the above induction, (21) holds
clearly. Thus, for all t ≥ t0,
( , )0
1
0 1
( ) ( )
2 ( , )
( ) ( ( ), )
N t t
s a a i i
i
T t t
N t t
tV x t t Me
σ
σ
λ λ
σµ
+
=
+ −∑
≤E
1 0 0( ) ( ) ( )
.
s a st t t t
e e
λ λ ρ λ+ − − −
× (22)
Combining (20) and (22), it follows that
( ) ( ( ), )
tV x t tσ
E
( , ) 10 0 0 0
0 0
0 0
( ) ( )2 ( , ) ( ) ( ) ( )
2 ( , ) [ ( 1) ]( ) ( )
2 ln( )
2 [ ( 1) ]( )
( )
s a N t ts s a
s a s a
s a
s a
t tN t t t t t t
N t t t t
d
N t t
Me e e
Me e
e Me
σ σ
σ
λ λ ρλ λ λ ρ
λ ρ λ ρ λ λ ρς
µ
λ ρ λ ρ
λ λ ρςτ τ
µ
µ
µ
+
∗ ∗
+ −− − + −
− + − +
+ − + + −
+
≤
≤
≤
0( , ).t tβ ξ −
!" E
If
ln( )
(1
,
)s a
µ
τ
λ ρ λ ρ
∗
>
− −
then ( , ) .β ⋅ ⋅ ∈
! KL The proof
is completed similarly to Lemma 2.
Remark 3: λa and λ
s are referred to the instability
margin and the stability margin, respectively. From (15),
it is seen that for any fixed q, τ and stability margin λs, a
large instability margin λa can be compensated by a small
0 < ρ < 1. That is to say, the switched system is stable
provided that the proportion of mismatched time is small
enough.
Remark 4: Condition (20) requires this upper bound
to hold uniformly over any interval [ , )tτ with arbitrary
starting point ,tτ ≤ which offers a simple strategy to
design the switching signal. Since all detection delays are
bounded, we can design the switching signal with
sufficiently large average dwell-time directly.
Similar to Corollary 1, the following useful corollary
is obtained.
Corollary 2: System (3) is pth moment exponentially
stable for all
ln( )
(1
,
)s a
µ
τ
λ ρ λ ρ
∗
>
− −
if α1 and α
2 in
Theorem 1 or in Theorem 2 are such that α1(s) = c
1sp
and
α2(s) = c
2sp
where c1 and c
2 are positive constants.
Remark 5: In reference [24], by using the Razumik-
hin method and the average dwell time approach, the
criterion of pth moment exponentially stability for a class
of switched stochastic nonlinear systems is developed.
However, the focus of our work is on stability analysis
under asynchronous switching, which is very different
from [24], and this is also the major contribution of our
work. In fact, if we let σ'(t) ≡ σ(t), i.e., we consider
synchronous switching, then the closed-loop dynamic in
(3) is the same as (2.1) in [24]. In this case, ρ ≡ 0 in
Corollary 2. Then, the result in Corollary 2 is the same
the result in [24].
Remark 6: In this paper, we consider only the
detection delay, assumed to satisfy some conditions, i.e.,
1( ) inf .
l l ld t d t tσ ∈ +
≤ ≤ −! However, if there exists a
detection error, then the assumption may not hold. (The
same problem also appears in the results in reference
[41-43], etc.) This difficulty leaves for our future endeavor.
Dihua Zhai, Yu Kang, Ping Zhao, and Yun-Bo Zhao
1188
4. APPLICATION AND EXAMPLE
We apply the general Razumikhin-type Corollary 2 to
deal with the pth moment exponentially stability for a
special type of switched stochastic nonlinear delay feed-
back systems, where the controller is designed with both
state and switching delays.
Consider the following switched stochastic nonlinear
control system
1 1
( ) 2
0 0 0 0
d ( , , , ( ))d ( , , , ( ))d ,
( ) ( , ),
( ) ( ), ( ) ( ) , ,
t
x F t x y u t t G t x y u t w
u t H t y
x s s s t i t s t
σ σ
σ
φ σ σ τ
′
⎧ = +
⎪=⎨
⎪= = = − ≤ ≤
⎩
(23)
on 0,t t≥ where
1 1( ) ( ( )),y t x t d t= −
2 2( ) ( ( )).y t x t d t= −
1 20 max ( ), ( ) .d t d t τ≤ ≤ Assume :
n n
iF
+
× × ×! ! !
,
l n
→! ! :
n n l n m
iG
×
+
× × × →! ! ! ! ! and :i
H
n l
+
× →! ! ! are continuous with ( ,0,0,0)itF ≡ 0 and
( ,0,0,0) 0i
G t ≡ for all ,i∈S and moreover system
(23) has the unique solution. For convenience, we trans-
fer the system (23) into the following type, i.e.,
( ) 1 2 ( ) 1 2d ( , , , )d ( , , , )d .
t tx F t x y y t G t x y y w
σ σ
= + (24)
For any given 2,1
( ),( ; )
n n n
tV Cσ + +
∈ × × ×! ! ! ! ! the
diffusion operator ( )t
Vσ
L in (5) becomes from n
×!
n n
+
× × × ×! ! ! S S to ! by [18]:
( ) 1 2
( ) ( )
( ) 1 2
2
( )
( ) 1 2 ( ) 1 22
( , , , )
( , ) ( , )
( , , , )
( , )1
[ ( , , , ) ( , , , )].
2
t
t t
t
t
t t
V x y y t
V x t V x t
F t x y y
t x
V x t
trace G t x y y G t x y y
x
σ
σ σ
σ
σ
σ σ
∂ ∂
= +
∂ ∂
∂
+
∂
L
Using Corollary 2, the following result is obtained.
Corollary 3: Let 1
.sup l l l
t tς+
∈ −
= − < ∞! For each
,
li ∈S ,l∈! suppose there exist a class C
2,1
Lyapunov
function ( )( ( ), )
tV x t tσ
and some positive constants c1, c
2,
,
lsi
λ ,
li
λ ,
lai
λ 1µ ≥ and 1,q > such that 1 ( ) 2| ( ) | ( ( ), ) ,| ( ) |
p p
tc x t V x t t c x t
σ
≤ ≤ (25)
and when 1
( , )s l l
t T t t+
∈ with 1 10 0
( , ) , ,[ )s
T t t t t=
( ) 1 2
2
( )
,1
( , , , )
( , ) min ( , ( ));l l
t
si t i k ij k k
i jk
V x y y t
V x t V y t d t
σ
σ
λ λ
∈
=
< − + −∑S
L
(26)
when 1
( , )a l l
t T t t+
∈ with 0 1
( ) ,,a
T t t =∅
( ) 1 2
2
( )
,1
( , , , )
( , ) min ( , ( )),l l
t
ai t i k ij k k
i jk
V x y y t
V x t V y t d t
σ
σ
λ λ
∈
=
< + −∑S
L
(27)
provided those ([ ,0]; )t
p n
Lϕ τ∈ − !F
satisfying that
, ( )
min ( ( ), ) ( (0), )i j ij t
V t q V tσ
ϕ θ θ ϕ∈
+ ≤E ES (28)
for any [ ,0].θ τ∈ − And for all ,r+
∈! we have
1( ) ( )
( ( ), ) ( ( ), ) .r rt r r t r r
V x t t V x t tσ σ
µ−
≤E E (29)
Let ,maxl l
k i i kλ λ
∈
= S
2
1
min l l
s i si kk
qλ λ λ∈
=
= −∑S
0> and 2
1
max .
l la i ai k
k
qλ λ λ∈
=
= +∑S If there also
exist some nonnegative constant ρ, such that (19), (20)
and (15) hold. Then, system (24) is pth moment
exponentially stable for all
ln( )
(1
.
)s a
µ
τ
λ ρ λ ρ
∗
>
− −
Proof: Taking the expectation on the both sides of (26)
and (27), by Fatou’s lemma and from (28), we have
( ) 1 2
( ) 1
( ) 1
( ( ), ( ), ( ), )
( ( ), ), ( , );
( ( ), ), ( , ).
t
s t s l l
a t a l l
V x t y t y t t
V x t t t T t t
V x t t t T t t
σ
σ
σ
λ
λ
+
+
− ∈⎧⎪< ⎨
∈⎪⎩
E
E
E
L
For any 0
t t≥ and ([ ,,0]; )t
p n
Lϕ τ∈ − !F
define
( ) ( ) 1 2( , (0), ) ( , (0), ( ( )), ( ( ))),
t tf t F t d t d tσ σ
ϕ ϕ ϕ ϕ ϕ= − −
and
( ) ( ) 1 2( , (0), ) ( , (0), ( ( )), ( ( ))).
t tg t G t d t d tσ σ
ϕ ϕ ϕ ϕ ϕ= − −
Thus, all the conditions in Corollary 2 are satisfied. This
completes the proof.
The following example is considered to demonstrate
the effectiveness of the proposed method.
Example 1: Consider the following switched stochas-
tic nonlinear systems.
( ) ( ) ( ) ( )d [ ( , )]d d ,
t t t tx A x B u f t x t C x w
σ σ σ σ
= + + +
where :
n n
if
+
× →! ! ! is an unknown nonlinear func-
tions satisfying 2
| ( , ( )) | | ,( ) |i if t x t U x t≤ for any i∈
,S where 2
|| ||⋅ denotes the 2-norm of the matrix. Set
( ) ( )( ) ) ( )),( (
t tu t K y t K x t d t
σ σ′ ′
= = − then
( ) ( ) ( ) ( )
( )
d [ ( , )]d
d .
t t t t
t
x A x B K y f t x t
C x w
σ σ σ σ
σ
′
= + +
+
(30)
Let ( ) ( , .( ) ( ))t t tσ σ σ ′= ∈ ×S S For system (30), take
( ) ( )( ) .
T
t tV x x P xσ σ
= Following reference [46], the con-
clusion that 1
,
T T T T T
HFE E F H HH E Eε ε
−
+ ≤ + holds
for any 0,ε > when ,
T
F F I≤ where I is an identity
matrix with appropriate dimension. Then
1 2
1 1
1 2
( ) [ ]
( , ) ( , ),
T T T
ii i ii ii i i ii i ii ii
T T T T
i i ii i i i ii i
V x x A P P A C P C P P x
y K B P B K y f t x P f t x
ε ε
ε ε
− −
≤ + + + +
+ +
L
and
3 4( ) [ ]
T T T
ij i ij ij i i ij i ij ijV x x A P P A C P C P P xε ε≤ + + + +L
1 1
3 4( , ) ( , ),
T T T T
j i ij i j i ij iy K B P B K y f t x P f t xε ε
− −
+ +
Stability of a Class of Switched Stochastic Nonlinear Systems under Asynchronous Switching
1189
for any ,i j∈S and .j i≠ Let 1
i iiX P
−
= and ii
K =
.
T
i iX K If there exist positive constants β
1 and β
2 such
that
1
1 2, .
i ijX I P Iβ β
−
> < (31)
Then,
1 2( )
T T T
ii i ii ii i i ii i ii iiV x x A P P A C P C P Pε ε≤ + + + +⎡
⎣L
1 1
2 1 1,
T T T T
i i i i ii i iU U x y K B P B K yε β ε
− −
⎤+ +⎦
and
3 4
1 1
4 2 3
( )
.
T T T
ij i ij ij i i ij i ij ij
T T T T
i i j i ij i j
V x x A P P A C P C P P
U U x y K B P B K y
ε ε
ε β ε− −
⎡⎣
≤ + + + +
⎤+ +⎦
L
According to Corollary 3, if there exist positive con-
stants λsi, λ
ai and λ
i such that
1 2
1 1
2 1 1
,
T T T
i ii ii i i ii i ii ii
T T T T
i i i i ii i i
T T
si ii i ii
x A P P A C P C P P
U U x y K B P B K y
x P x y P y
ε ε
ε β ε
λ λ
− −
⎡ + + + +⎣
⎤+ +⎦
< − +
and
3 4
1 1
4 2 3
,
T T T
i ij ij i i ij i ij ij
T T T T
i i j i ij i j
T T
ai ij i ij
x A P P A C P C P P
U U x y K B P B K y
x P x y P y
ε ε
ε β ε
λ λ
− −
⎡ + + + +⎣
⎤+ +⎦
< +
which means
11 22, 0,diagΩ = Ω Ω < (32)
11 22, 0,diagΣ = Σ Σ < (33)
where 1
11 1 2 2
T T
i ii ii i i ii i ii iiA P P A C P C P Pε ε ε
−
Ω = + + + + + ×
1,
T
i i si iiU U Pβ λ+
1
22 1,
T T
i ii i i ii i iP K B P B Kλ ε
−
Ω = − +11
Σ =
1
3 4 4 2,
T T T
i ij ij i i ij i ij ij i i ai ijA P P A C P C P P U U Pε ε ε β λ
−
+ + + + + −
1
22 3,
T T
i ij j i ij i jP K B P B Kλ ε
−
Σ = − + εk > 0, 1,2, .3,4k = Us-
ing 1 1
, ii ii
diag P P− −
to pre- and post- multiply the left
terms of matrix inequality (32); and using Schur’s comp-
lement lemma, then (32) is equivalent to
11
1
2 1
1
0 0
* 0 0 0
0,* * 0 0
* * *
* * * *
T
i i i i
i
T
i i ii i
i
X U X C
I
X
X K B
X
ε β
λ
ε
−
⎡ ⎤Ω
⎢ ⎥
−⎢ ⎥
⎢ ⎥<−
⎢ ⎥
⎢ ⎥−
⎢ ⎥
−⎢ ⎥⎣ ⎦
(34)
where 11 1 2
.T
i i i i i i si iX A A X X X Xε ε λΩ = + + + + More-
over, if there also exist q > 1 and µ≥ 1, such that, Pii, Pij,
λsi, λai, λi, q and µ satisfy the corresponding conditions in
Corollary 3, then system (30) is 2nd moment exponen-
tially stable.
Specify system (30) as follows
1
5 2
,
5 3
A
− −⎡ ⎤
=⎢ ⎥
−⎣ ⎦
1
1 0
,
0 1
B
−⎡ ⎤
=⎢ ⎥
⎣ ⎦
2
4 0
,
1 5
A
−⎡ ⎤
=⎢ ⎥
−⎣ ⎦
2
1 2
,
0 1
B
−⎡ ⎤
=⎢ ⎥
⎣ ⎦
1
0.2 0
,
0.3 0.5
C
⎡ ⎤
=⎢ ⎥−⎣ ⎦
2
0.3 0.2
,
0 0.5
C
−⎡ ⎤
=⎢ ⎥
⎣ ⎦
1
0.5cos( ) 0.1sin(| |)
( , ) ,
0 0.1sin( )
t x
f t x x
t
⎡ ⎤
=⎢ ⎥
−⎣ ⎦
1
0.5 0.1
,
0 0.1
U
⎡ ⎤
=⎢ ⎥
−⎣ ⎦
2
0.1cos( )sin(| |) 0
( , ) ,
0 0.5sin( )
t x
f t x x
t
⎡ ⎤
=⎢ ⎥
⎣ ⎦
2
0.1 0
.
0 0.5
U
⎡ ⎤
=⎢ ⎥
⎣ ⎦
( ) 0.8cos( ).d t t= Take 1
2.8,s
λ =
23,
s
λ =
10 0 ,. 3
a
λ =
20 0 ,. 1
a
λ =
17,0.0λ =
21,0.1λ =
11,β =
12,1.ε =
2ε =
1.5, 3
3ε = and 4
4.ε = Then, using the LMI toolbox
in the MATLAB, we get
11
1
0.5762 0.0354
,
0.0354 0.2707
0.2647 0.0399
,
0.0212 0.2050
P
K
⎡ ⎤
=⎢ ⎥
⎣ ⎦
⎡ ⎤
=⎢ ⎥
⎣ ⎦
12
2
22
21
189.9085 44.5369
,
44.5369 87.1685
0.4129 0.1255
,
0.1191 0.1606
0.3792 0.0493
,
0.0493 0.4606
128.9302 6.1472
,
6.1472 96.2041
P
K
P
P
⎡ ⎤=⎢ ⎥
⎣ ⎦
⎡ ⎤=⎢ ⎥
⎣ ⎦
−⎡ ⎤=⎢ ⎥−⎣ ⎦
−⎡ ⎤=⎢ ⎥−⎣ ⎦
and 1.7 ,737µ =
2268.2155.β = Take q = 3, then λ =
0.11, 47,2.s
λ = and 36.0.a
λ = Set 8,0.τ = ρ =
1
0.4 4,36
2
s
a s
λ
λ λ
=
+
then ( (1 ) )
3s a
q e
λ ρ λ ρ τ− −
= > = 2.6859.
Then, according to the above analysis, system (30) is 2nd
moment exponentially stable for all τ*
> 0.4640s. The
simulation results are shown in Figs. 1, 2, and 3. Figs. 1
and 2 show the Brownian motion w(t) and the switching
signal in system (30), respectively. More-over, the detec-
tion delay in Fig. 2 satisfies the conditions in Corollary 3.
Finally, Fig. 3 shows the trajectory of x(t) under the ini-
tial data x0 = ( 8, 6)± ∓ . Under the asynchronous switching
signal σ'(t) in Fig. 2, the state x(t) will converge to zero.
Dihua Zhai, Yu Kang, Ping Zhao, and Yun-Bo Zhao
1190
Fig. 1. Response curve of Brownian motion w(t).
Fig. 2. Switching signal σ(t) and the detected σ'(t).
Fig. 3. Response curve of x(t).
5. CONCLUSION
The stability of a class of stochastic nonlinear retarded
systems under asynchronous switching is investigated.
Based on the average dwell time approach, the correspo-
nding Razumikhin-type stability criteria on globally
asymptotically stable as well as pth moment exponential-
ly stable are given. It is shown that the switched system
can be stable when the mismatched interval is small
enough while the average dwell time is large enough.
Finally, we apply the results to a class of stochastic
nonlinear delay systems where the design of controller is
considered with both state and switching delays and
meaningful results are obtained, which are illustrated by
a numerical example.
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Dihua Zhai received his B.S. degree in
Automation from the Anhui University, P.
R. China, in 2010. He is currently a M.S.
student in the Department of Automation,
University of Science and Technology of
China, P. R. China. His research interests
are in the stability theory and control of
switched systems, networked control
systems and stochastic systems, etc.
Yu Kang received his Dr. Eng. degree in
Control Theory and Control Engineering,
University of Science and Technology of
China, P. R. China, in 2005. From 2005
to 2007, he was a postdoctoral fellow in
the Academy of Mathematics and Sys-
tems Science, Chinese Academy of
Sciences, P. R. China. Currently, he is an
Associate Professor in the Department of
Automation, University of Science and Technology of China,
China. Dr. Kang’s current research interests are in the adap-
tive/robust control, variable structure control, mobile manipula-
tor, Markovian jump systems, etc.
Ping Zhao received his B.S. degree from
the University of Jinan, P. R. China, in
2002, an M.S. degree from the Qufu
Normal University, P. R. China, in 2005,
and a Ph.D. from Academy of Mathemat-
ics and Systems Science, Chinese Acad-
emy of Sciences. He is currently a teach-
er of University of Jinan. His research
interests are in the stability theory and
control of stochastic and nonlinear systems.
Yun-Bo Zhao received his B.Sc. degree
in Mathematics from Shandong Universi-
ty, Shandong, P. R. China, in 2003, an
M.Sc. degree in Systems Theory from
the Institute of Systems Science, Chinese
Academy of Sciences, Beijing, P. R.
China, in 2007, and a Ph.D. degree from
the University of Glamorgan, Pontypridd,
U.K., in 2008. He is currently a Research
Associate with Imperial College London, London, U.K. His
research interests include systems biology, networked control
systems, and Boolean networks.
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1. Stability of a class of switched stochastic nonlinear systems under asynchronousswitchingAccession number: 20132816476064Authors: Zhai, Dihua (1); Kang, Yu (1); Zhao, Ping (2); Zhao, Yun-Bo (3, 4)Author affiliation: (1) Department of Automation, University of Science and Technology of China, Hefei, China; (2)School of Electrical Engineering, University of Jinan, Jinan, China; (3) Department of Chemical Engineering, ImperialCollege London, London, United Kingdom; (4) CTGT Centre, Harbin Institute of Technology, Harbin, ChinaCorresponding author: Kang, Y.([email protected])Source title: International Journal of Control, Automation and SystemsAbbreviated source title: Int. J. Control Autom. Syst.Volume: 10Issue: 6Issue date: December 2012Publication year: 2012Pages: 1182-1192Language: EnglishISSN: 15986446E-ISSN: 20054092Document type: Journal article (JA)Publisher: Institute of Control, Robotics and Systems, 193 Yakdae-dong Wonmi-gu, Bucheon, Gyeonggi-do, 420-734,Korea, Republic ofAbstract: The stability of a class of switched stochastic nonlinear retarded systems with asynchronous switchingcontroller is investigated. By constructing a virtual switching signal and using the average dwell time approachincorporated with Razumikhin-type theorem, the sufficient criteria for pth moment exponential stability and globalasymptotic stability in probability are given. It is shown that the stability of the asynchronous stochastic systems can beguaranteed provided that the average dwell time is sufficiently large and the mismatched time between the controllerand the systems is sufficiently small. This result is then applied to a class of switched stochastic nonlinear delaysystems where the controller is designed with both state and switching delays. A numerical example illustrates theeffectiveness of the obtained results. © 2012 Institute of Control, Robotics and Systems and The Korean Institute ofElectrical Engineers and Springer-Verlag Berlin Heidelberg.Number of references: 46Main heading: System stabilityControlled terms: Stochastic systems - SwitchingUncontrolled terms: Asynchronous switching - Average dwell time - Razumikhin-type theorem - Stochasticnonlinear systems - Stochastic stabilityClassification code: 721.1 Computer Theory, Includes Formal Logic, Automata Theory, Switching Theory,Programming Theory - 961 Systems ScienceDOI: 10.1007/s12555-012-0613-0Database: CompendexCompilation and indexing terms, Copyright 2016 Elsevier Inc.Data Provider: Engineering Village
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Record 1 of 1Title: Stability of a Class of Switched Stochastic Nonlinear Systems under Asynchronous SwitchingAuthor(s): Zhai, DH (Zhai, Dihua); Kang, Y (Kang, Yu); Zhao, P (Zhao, Ping); Zhao, YB (Zhao, Yun-Bo)Source: INTERNATIONAL JOURNAL OF CONTROL AUTOMATION AND SYSTEMS Volume: 10 Issue: 6 Pages: 1182-1192 DOI: 10.1007/s12555-012-0613-0 Published: DEC 2012 Times Cited in Web of Science Core Collection: 5Total Times Cited: 5Usage Count (Last 180 days): 1Usage Count (Since 2013): 15Cited Reference Count: 46Abstract: The stability of a class of switched stochastic nonlinear retarded systems with asynchronous switching controller is investigated. By constructing a virtualswitching signal and using the average dwell time approach incorporated with Razumikhin-type theorem, the sufficient criteria for pth moment exponential stability andglobal asymptotic stability in probability are given. It is shown that the stability of the asynchronous stochastic systems can be guaranteed provided that the average dwelltime is sufficiently large and the mismatched time between the controller and the systems is sufficiently small. This result is then applied to a class of switched stochasticnonlinear delay systems where the controller is designed with both state and switching delays. A numerical example illustrates the effectiveness of the obtained results.Accession Number: WOS:000311713500013Language: EnglishDocument Type: ArticleAuthor Keywords: Asynchronous switching; average dwell time; Razumikhin-type theorem; stochastic stability; switched stochastic nonlinear systems; time-delaysKeyWords Plus: H-INFINITY CONTROL; LINEAR-SYSTEMS; TIME-DELAY; HYBRID SYSTEMS; STABILIZATION; STATE; CONTROLLABILITY;REACHABILITY; CRITERIAAddresses: [Zhai, Dihua; Kang, Yu] Univ Sci & Technol China, Dept Automat, Hefei 230026, Peoples R China. [Zhao, Ping] Univ Jinan, Sch Elect Engn, Jinan, Peoples R China. [Zhao, Yun-Bo] Univ London Imperial Coll Sci Technol & Med, Dept Chem Engn, London SW7 2AZ, England. [Zhao, Yun-Bo] Harbin Inst Technol, CTGT Ctr, Harbin 150006, Peoples R China.Reprint Address: Kang, Y (reprint author), Univ Sci & Technol China, Dept Automat, Hefei 230026, Peoples R China.E-mail Addresses: [email protected]; [email protected]; [email protected]; [email protected] Identifiers:
Author ResearcherID Number ORCID Number
Zhao, Yun-Bo F-1699-2010 Publisher: INST CONTROL ROBOTICS & SYSTEMS, KOREAN INST ELECTRICAL ENGINEERSPublisher Address: BUCHEON TECHNO PARK 401-1506, 193 YAKDAE-DONG WONMI-GU, BUCHEON, GYEONGGI-DO 420-734, SOUTH KOREAWeb of Science Categories: Automation & Control SystemsResearch Areas: Automation & Control SystemsIDS Number: 045RCISSN: 1598-6446eISSN: 2005-409229-char Source Abbrev.: INT J CONTROL AUTOMISO Source Abbrev.: Int. J. Control Autom. Syst.Source Item Page Count: 11
Funding:
Funding Agency Grant Number
National Natural Science Foundation of P. R. China 60935001 61174061 60904024 60934006
Anhui Provincial Natural Science Foundation 11040606M143 Doctoral Foundation of University of Jinan XBS1012 Fundamental Research Funds for the Central Universities Program for New Century Excellent Talents in University Program for Youth Innovation Promotion Association, CAS
This journal was supported in part by the National Natural Science Foundation of P. R. China under grant No.60935001, 61174061, 60904024, 60934006,the Anhui Provincial Natural Science Foundation (11040606M143), the Doctoral Foundation of University of Jinan (XBS1012), the Fundamental ResearchFunds for the Central Universities, the Program for New Century Excellent Talents in University, and the Program for Youth Innovation PromotionAssociation, CAS.
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