Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 796104, 19 pages
doi:10.1155/2012/796104
Research Article
pth Moment Exponential Stability of Stochastic
PWM Feedback Systems with Time-Varying Delays
Zhong Zhang and Lixia Ye
College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China
Correspondence should be addressed to Zhong Zhang, zhanguicqu@gmail.com
Received 22 September 2012; Accepted 12 November 2012
Academic Editor: Chuandong Li
Copyright q 2012 Z. Zhang and L. Ye. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper further studies the pth moment exponential stability of stochastic pulse-widthmodulated PWM feedback systems with distributed time-varying delays. We establish several
globally exponential stability criteria for such PWM feedback systems by using LyapunovKrasovskii functional and then present an upper bound of the parameter of PWM when the
system is stable and such system has stronger anti-interference performance than the system
without time-varying delays. Furthermore, we present two examples to show the effectiveness
and conservativeness of the theoretical results.
1. Introduction
Pulse-width modulation has extensively been used in attitude control systems, adaptive
control systems, signal processing, power control systems, modeling of neuron behavior, and
the like e.g., see 1–5. In many areas, especially engineering applications, how to keep
the scheduled operation or work of the state counts for much. Therefore, there has been a
growing research interest on the stability analysis for PWM feedback systems, and a set of
stability results have been established by a variety of methods 6–22. In the actual process,
however, it is always operated in all kinds of accidental or continued disturbances. Time delay
will inevitably occur in electronic neural networks owing to the unavoidable finite switching
speed of amplifiers. In recent years, the stability analysis of stochastic systems, especially the
systems with time delay, is interesting to many investigators, and many results of stability
criteria of these systems have been reported 15–22.
There are, however, only a few results concerning the qualitative properties of
stochastic impulsive systems with time-varying delays. In 15, the authors investigated
robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive
2
Abstract and Applied Analysis
stochastic systems with time-varying delays. Besides, Sun and Cao 17 give some definitions
on the pth moment exponential stability in mean and established several pth moment globally
stability criteria in mean. In 12, 13, Hou and Michel established new Lyapunov and
Lagrange stability results for pulse-width-modulation PWM feedback systems subjected
to random disturbance.
To the best of the authors’ knowledge, there are few if any results for the stability
analysis of stochastic PWM systems with time-varying delays. Based on the pulse-widthmodulation feedback system uniqueness, obviously, such system subjected to random
disturbance and time-varying delays is somewhat complex in comparison with most of
the systems in the literature. It is noted that the linear plant considered herein is Hurwitz
stable, that is, all the eigenvalues of the linear plant are in the left side of the complex
plane. In the present paper, we try to make the contribution on this issue. By choosing
reasonable Lyapunov-Krasovskii functional, combined with linear matrix inequalities and It
o
integration method, we will establish several Lyapunov and Lagrange criteria for pth moment
exponential stability in mean and then present an algorithm to compute the upper bound for
the parameters of PWM. We will characterize the relationship among the parameters of pulsewidth modulation, time-varying delays, and the coefficient of state vectors of the feedback
systems. It will be shown that when the random disturbance is sufficiently small such PWM
feedback system is pth moment exponentially stable in mean provided that the upper bounds
of parameters of pulse-width modulator are selected properly. We also demonstrate that such
system has the stronger anti-interference performance and tending to the equilibrium point
speed more quickly by means of two numerical examples.
2. Notations and Some Definitions
Let Ω, F, P denote the underlying probability space for all the systems that will be
considered, where Ω is the sample space, F is the σ-algebra of subsets of the sample space,
and P is the probability measure. An Rn -valued random variable x with domain X is a
measurable function from Ω to X ⊂ Rn . A family {xt, t ∈ I} of Rn valued random variables
with domain X defined on a probability space Ω, F, P is called a stochastic process with
index set I and state space X, F n .
Definition 2.1. Let X, d be a metric space, X ⊂ Rn , A ⊂ X, and let T ⊂ R . For any fixed a ∈ A
a is called the initial state, t0 ∈ T , a stochastic process {xt, ω, a, t0 , t ∈ Ta,t0 } with domain
X is called a stochastic motion if xt0 , ω, a, t0 a for all ω ∈ Ω, where Ta,t0 t0 , t1 ∩ T , t1 > t0 ,
and t1 is finite or infinite.
Definition 2.2. Let S be a family of stochastic motions with domain X given by
S ⊂ {x·, ·, a, t0 : xt0 , ω, a, t0 a, ω ∈ Ω, a ∈ A, t0 ∈ T }.
2.1
We call the four-tuple {T, X, A, S} a stochastic dynamical system.
Definition 2.3. Let {T, X, A, S} be a stochastic dynamical system. A set M ⊂ A is said to be
invariant with respect to system S or short, S, M is invariant if a ∈ M implies that P {ω :
xt, ω, a, t0 ∈ M for all t ∈ Ta,t0 } 1, for all t0 ∈ T and all x·, ·, a, t0 ∈ S.
Abstract and Applied Analysis
3
A
Input +
PWM
B
+
∫
Integrator
C1 , C2
G
White noise
X(t)
Figure 1: Block diagram of stochastic PWM feedback systems with time-varying delays.
Definition 2.4. x0 ∈ A is called an equilibrium point of a stochastic dynamical system {T, X, A, S}
if the set {x0 } is invariant with respect to S.
Definition 2.5. Let {T, X, A, S} be a stochastic dynamical system, and let d be the metric on X.
A set M ⊂ A is said to be the pth moment exponentially stable in mean i.e., S, M is said to
be the pth moment exponentially stable in mean if for any t0 ∈ T , there exists a δ δt0 , ε
and constants β > 0, k > 1 such that Edxt, ω, a, t0 , Mp < kap e−βt−t0 for any process
x·, ·, a, t0 ∈ S, whenever da, M < δ, where a is called the initial state and E· denotes
the expectation of a random process. If δ is independent of t0 , S, M is said to be the pth
moment uniformly exponentially stable in mean. S, M is said to be the pth moment uniformly
asymptotically exponentially stable in mean if it is uniformly stable in the pth mean and if there
exists δ > 0, τ > 0 and constants β > 0, k > 1 such that for any process x·, ·, a, t0 ∈ S,
whenever da, M < δ implies that Edxt, ω, a, t0 , Mp < kap e−βt−t0 for all t > t0 τ.
3. Main Results
The PWM feedback system to be considered in this paper is shown in Figure 1.
The pulse-width modulator is described by
M sgnekT ut met 0
t ∈ kT, kT Tk ,
t ∈ kT Tk , kT T ,
3.1
where et rt − yt with rt being the external input and yt the system output, and,
for k 0, 1, 2, . . ., the pulse-width Tk and the sign function sgn are given, respectively, by
⎧
T
⎪
⎪
⎪
⎨β|ekT |, |ekT | ≤ β ,
Tk ⎪
T
⎪
⎪
|ekT | > ,
⎩T,
β
⎧
⎪
σ > 0,
⎪
⎨1
sgnσ 0
σ 0,
⎪
⎪
⎩−1 σ < 0.
3.2
4
Abstract and Applied Analysis
The sampling period T , the amplitude of the pulse M, and β are all assumed to be constants.
And throughout this paper, we always assume that rt ≡ 0. Under these assumptions, the
PWM feedback system with the output function can be described by
dxt Axtdt Ad xt − ddt Butdt GxtdWt ,
yt C1 xt C2 xt − d,
3.3
where x ∈ Rn , y ∈ R, u ∈ R, is output of the pulse-width modulator, A, Ad , B, C1 , C2 , and
G are matrices of appropriate dimensions, d di ∈ Rn , 0 < di ≤ T − Tk , i 1, 2, . . . , n, and
Wt is a scalar wiener process.
T
Note that x1T , x2 0T , 0 is an equilibrium point of PWM feedback system 3.3.
S12
. Then S < 0 is equivalent to any
Lemma 3.1 Schur complement. Given the matrix S SS11
21 S22
one of the following conditions:
i S11 < 0, S22 − S21 S−1
11 S12 < 0,
ii S22 < 0, S11 − S12 S−1
22 S21 < 0.
Lemma 3.2 the It
o isometry. Assume Wt is a scalar wiener process. If φt, ω is bounded and
elementary, then
⎡
2 ⎤
T
T
2
⎣
⎦
E
φt, ωdWt
φt, ω dt .
E
S
3.4
S
Proof. Put ΔWj Wtj1 − Wtj , then
⎧
⎨0
E ei ej ΔWi ΔWj ⎩E e2 t − t j1
j
j
if i /
j,
if i j,
3.5
using that ei ej ΔWi and ΔWj are independent if i < j. Thus
⎡
2 ⎤
T
E⎣
φt, ωdWt ⎦ E ei ej ΔWi ΔWj
S
i,j
ej2 · ti − tj
3.6
j
E
T
2
φt, ω dt .
S
Theorem 3.3. Assume that the matrix A in 3.3 is Hurwitz stable. Then the equilibrium xe 0 of
PWM feedback system 3.3 is the pth p ∈ Z moment uniformly exponentially stable in mean in
the large provided that the following conditions are satisfied:
Abstract and Applied Analysis
5
i
Mβ <
inf
λm Γ1 τk ∈0,∞
λ2m Γ1 4λM Γ2 2λM Γ2 ,
3.7
where
Γ1 U1 τk C1 U1 τk C1 T ,
Γ2 C1T G1 τk G2 τk C1
3.8
with
Gi Tk WTk T Pi − IWTk Ui Tk Pi − IWTk ,
i 1, 2,
3.9
ii
T
ϕ22 − ϕ12 ϕ−1
11 ϕ12 < 0,
3.10
where
ϕ11 −I − MβΓ1 M2 β2 Γ2 ,
ϕ12 −Mβ U1 τk C2 U2 τk C1 T eAd
ϕ22
M2 β2 C1T G1 τk G2 τk C2 ,
T
T
T Ad
AT
AT
Ad
e
P2 e − P2 − Mβ U2 τk C2 e e
U2 τk C2
3.11
M2 β2 C2T G1 τk G2 τk C2 ,
iii there exists a constant δ > 0, such that, whenever G < δ,
2
2
max Ad 2 , G2 · eAd G T < μ,
3.12
where μ is scalar satisfying
c μ 1 − 2Φ μ − 4P μ > 0
3.13
6
Abstract and Applied Analysis
with
⎛
Φ
⎝
⎞
−MβP1 eAT Wτk C2
⎠,
P2 eAT −d eAd − MβWτk C2
P1 eAT I − MβWτk C1
−MβP2 eAT −d Wτk C1
3.14
where P1 P1T > 0, P2 P2T > 0 satisfying
e
eAT
AT −d
T
T
P1 eAT − P1 −I,
3.15
P2 e
AT −d
− P2 −I.
1/p
is nondecreasing in p, the equilibrium xe 0 is stable resp.,
Proof. Since EXp asymptotically stable, etc. implying that it is the qth moment stable resp., asymptotically
stable, etc. in mean for all q < p. Firstly, we will provide to prove the theorem for even
integers.
Integrating 3.3, we have
xt eAt−kT xkT t
eAt−s Ad xs − dds
kT
t
e
At−s
Busds 3.16
t
kT
e
At−s
GxsdWs .
kT
Therefore, when t kT T ,
xkT T e
AT
xkT kT T
eAkT T −s Ad xs − dds eAT xkT −
kT
e
AT
eAkT T −s Busds
kT
kT
kT T
kT T
kT Tk
kT T
eAkT T −s GxsdWs
kT
eAkT T −s BM sgnC1 xkT C2 xkT − dds
kT
eAkT T −s Ad xs − dds kT T
eAkT T −s GxsdWs
kT
I − MβWτk C1 xkT − MβWτk C2 xkT − d
kT T −d
kT −d
e
AkT T −d−s
Ad xsds kT T
kT
eAkT T −s GxsdWs .
3.17
Abstract and Applied Analysis
7
For t kT T − d, we have
xkT T − d eAT xkT − d kT T −d
e
kT −d
eAkT T −d−s Busds
Ad xs − dds kT Tk
e
AkT T −d−s
kT T −d
kT −d
eAkT T −d−s GxsdWs
eAkT T −d−s BM sgnC1 xkT C2 xkT − dds
kT
kT T −d
kT −d
kT −d
AkT T −d−s
eAT xkT − d −
kT T −d
Ad xs − dds kT T −d
kT −d
eAkT T −d−s GxsdWs
eAT −d −MβWτk C1 xkT eAd − MβWτk C2 xkT − d
kT T −d
kT −d
eAkT T −d−s Ad xs − dds kT T
eAkT T −s Gxs − ddWs
kT
3.18
with
τk β|C1 xkT C2 xkT − d|
Tk , Tk < T,
≥ T, Tk T,
⎧
⎪
0,
τk 0,
⎪
⎪
⎪
−ATk
⎪
⎪
I
−
e
⎨
A−1 B,
τk < T,
Wτk Tk
⎪
⎪
⎪
⎪
I − e−AT −1
T
⎪
⎪
⎩
A B WT , τk ≥ T.
τk
τk
To simplify our notations, let
H1 I − MβWτk C1 −MβWτk C2 ,
H2 −MβWτk C1 eAd − MβWτk C2 ,
hkT hkT − d kT T −d
kT −d
kT T −d
kT −d
eAkT T −d−s Ad xsds,
eAkT T −d−s Ad xs − dds,
3.19
8
Abstract and Applied Analysis
$
hkT
kT T
eAkT T −s GxsdWs ,
kT
$
hkT
− d kT T
eAkT T −s Gxs − ddWs ,
kT
T
XkT xT kT , xT kT − d .
3.20
Then, 3.17 and 3.18 are reduced to
$
xkT T eAT H1 XkT hkT hkT
,
$
xkT T − d eAT −d H2 XkT − d hkT − d hkT
− d.
3.21
Choosing the quadratic Lyapunov-Krasovskii functional V : Rn → R ,
3.22
V x xT tP1 xt xT t − dP2 xt − d.
Then,
∇EV x EV xkT T − EV xkT − EV xkT − EV xkT − d
E xT kT T P1 xkT T E xT kT T − dP2 xkT T − d
− E xT kT P1 xkT − ExkT − dP2 xkT − d
E X T kT H1T P1 − IH1 H2T P2 − IH2 − P XkT 2E
2E
E
E
T
$
hkT hkT
P1 eAT H1 XkT $
hkT − d hkT
− d
$
hkT hkT
T
T
P2 eAT −d H2 XkT $
P1 hkT hkT
T $
$
hkT − d hkT
− d P2 hkT − d hkT
− d
E X T kT H1T P1 − IH1 H2T P2 − IH2 − P XkT $ T kT ΦXkT 2E ZT kT ΦXkT 2E Z
$
$ T kT P ZkT
$
E ZT kT P ZkT 2E ZT kT P ZkT
E Z
,
3.23
Abstract and Applied Analysis
9
where
T
ZkT hT kT hT kT − d ,
T
$
$ T kT − d ,
$ T kT h
ZkT
h
%
Φ
⎛
⎞
−MβP1 eAT Wτk C2
3.24
⎝
⎠,
AT −d
AT −d Ad
−MβP2 e
Wτk C1 P2 e
e − MβWτk C2
⎞
⎛
ϕ11 ϕ12
T
T
⎠.
H H1 P1 − IH1 H2 P2 − IH2 − P ⎝
ϕT12 ϕ22
P1 eAT H1
P2 eAT −d H2
&
P1 eAT I − MβWτk C1
Then 3.23 is reduced to
$ T kT ΦXkT ∇EV x E X T kT HXkT 2E ZT kT ΦXkT 2E Z
$
$ T kT P ZkT
$
E ZT kT P ZkT 2E ZT kT P ZkT
E Z
.
3.25
Let λm · and λM · denote the minimum and maximum eigenvalues of a matrix, respectively.
By Lemma 3.1, it is obvious that H is a negative definite matrix if and only if ϕ11 < 0 and
T
ϕ22 − ϕ12 ϕ−1
11 ϕ12 < 0. Noting that
λM ϕ11 ≤ −1 − Mβλm U1 τk C1 U1 τk C1 T
M2 β2 λM C1T G1 τk G2 τk C1
3.26
ΘMβ .
From condition i it follows that ΘMβ < 0, and hence the claim that ϕ11 < 0 is
true. From condition ii we then conclude that H is a negative definite matrix based on
Lemma 3.1.
$
Now we establish an estimation on ZkT and ZkT
. Let μ > 0 be arbitrary. We will
show that there exists δ > 0 whenever G < δ, such that
E ZkT 2 < μE XkT 2 ,
'
'2 '
'$
E 'ZkT ' < μE XkT 2 .
3.27
10
Abstract and Applied Analysis
For t ∈ kT, kT T , we have
xt xkT t
Busds xkT Axsds kT
kT
t
t
Busds kT
xt − d xkT − d t−d
kT −d
t−d
kT −d
GxsdWs
Ad xsds
kT −d
GxsdWs ,
kT
t−d
kT −d
t
t−d
t
Axsds Busds xkT − d t
Axsds kT
Ad xs − dds
kT
kT
t
t
kT −d
t−d
kT −d
Busds t
Ad xs − dds
GxsdWs
Axs − dds kT
3.28
t−d
t−d
kT −d
Ad xs − dds
Gxs − ddWs .
kT
Then, we have
Xt XkT LkT t
$
AXsds
kT
t−d
kT −d
$ d Xsds A
t
3.29
$
GXsdW
s,
kT
where
$
A
%
&
A 0
,
0 A
$d A
( t
LkT kT
%
&
Ad 0
,
0 Ad
$
G
%
&
G 0
,
0 G
Busds
0
( t−d
.
Busds
0
kT −d
3.30
Abstract and Applied Analysis
11
Similarly, we have
T
ZkT hT kT hT kT − d
( kT T −d
AkT T −d−s
e
A
xsds
0
d
kT −d
( kT T −d AkT T −d−s
e
Ad xs − dds
0
kT −d
kT T −d
kT −d
$ d Xsds,
eAkT T −d−s A
3.31
T
$
$ T kT − d
$ T kT h
ZkT
h
( kT T
kT
kT T
eAkT T −s GxsdWs
0
( kT T AkT T −s
e
Gxs − ddWs
0
kT
$
eAkT T −s GXsdW
s.
kT
Noting that
E LkT 2 ≤ M2 β2 B2 C2 E XkT 2 ,
' '
' '
' '
' $'
'$ '
' $'
C C1 , C2 ,
'A' A,
'Ad ' Ad ,
'G' G,
E Xt2
5
⎡'
'2 ⎤
' t
'
'
' ⎦
$
≤ E XkT 2 E⎣'
AXsds
' E LkT 2
' kT
'
⎡
⎡'
'2 ⎤
T t
⎤
t
'
' t−d
'
'
$ Xsds' ⎦ E⎣
$
$
E⎣'
A
GxsdW
GxsdW
s
s ⎦.
'
' kT −d d
kT
kT
3.32
By Lemma 3.2, one observes that
⎡
T t
⎤
t
t
T
T
E⎣
GxsdWs
GxsdWs ⎦ E
x sG Gxsds .
kT
kT
kT
3.33
12
Abstract and Applied Analysis
Then the inequality 3.32 is reduced to
E Xt2
5
' '2 t−d '$ '
≤ K0 E XkT 2 'A
E Xs2 ds
d'
kT −d
%' '2 ' '2 & t ' $' ' $ '
'A
E Xs2 ds
' 'G'
kT
≤ K0 E XkT 2
%' ' ' ' ' ' & t
' $ '2 ' $ '2 ' $ '2
'A
' 'G' 'Ad '
K0 E XkT 2 A2 G2 Ad 2
t
kT −d
kT −d
E Xs2 ds
3.34
E Xs2 ds,
where K0 1 M2 β2 B2 C2 .
By the Gronwall inequality, we have
2
2
$ 2
E Xt2 ≤ 5K0 E XkT 2 eA G Ad t−kT ≤ 5K0 e
A2 G2 Ad 2 T
E XkT 2 .
3.35
Thus, we have
E ZkT 2
⎡
T kT T −d
⎤
kT T −d
$ d Xsds
$ d Xsds ⎦
E⎣
eAkT T −d−s A
eAkT T −d−s A
kT −d
kT −d
⎡
⎤
' '2 kT T −d T
'$ ' ⎣
≤ 'Ad ' E
eAkT T −d−s eAkT T −d−s Xs2 ds⎦
kT −d
2
2
2
≤ 5K0 K1 Ad 2 eA G Ad T E XkT 2
2
2
KAd 2 eG Ad T E XkT 2 ,
⎤
⎡
T kT T
kT T
'
'2 '
'$
$
$
E 'ZkT ' E⎣
eAkT T −s GXsdW
eAkT T −s GXsdW
s
s⎦
kT
E
kT T
kT
T
T
AkT T −s
AkT T −s $
$
X sG e
e
GXsds
T
kT
' '2 kT T T
' $'
2
AkT T −s
AkT T −s
e
e
≤ 'G' E
Xs ds
kT
2
2
≤ KG2 eG Ad T E XkT 2 ,
3.36
Abstract and Applied Analysis
13
where
' T
'
'
'
K1 max 'eA s eAs ',
s∈0,T 2
K 5K0 K1 eA T .
3.37
Thus, when condition iii is satisfied, we have
'
'2 '
'$
E 'ZkT
' < μE XkT 2 .
E ZkT 2 < μE XkT 2 ,
3.38
Then, we obtain
∇EV x EV xkT T − EV xkT − EV xkT EV xkT − d
c μ
≤−
EV XkT < 0.
λM P 3.39
Therefore,
c μ
EV XkT .
1−
λM P EV XkT T <
3.40
Noticing that V X0 xT 0P1 x0 ≤ P x02 P a2 , where a x0 is initial state.
For t ∈ kT, kT T , we have
EV xt ≤ EV XkT c μ
≤ 1−
EV XkT − T λM P ≤ ···
≤
k
c μ
1−
EV x0
λM P 3.41
≤
k
c μ
1−
P1 x02
λM P < et lnΘ/ρ P1 x02
≤ P a2 et lnΘ/ρ ,
where
c μ
Θ
1−
< 1,
λM P ρ minTk − Tk−1 .
k
3.42
14
Abstract and Applied Analysis
Noticing that EV xt ≥ λm P Ext2 , one obtains that
EV xt
E xt2 ≤
λm P 2
< a
P et lnΘ/ρ
λm P 3.43
ψa2 e−βt ,
where
ψ
P > 1,
λm P β − ln
Θ
> 0.
ρ
3.44
Thereforce, by virtue of Definition 2.5 with p 2, we know that the equilibrium point
of system 3.3 is uniformly exponentially stable in mean square.
Now we will proof the pth moment uniformly exponentially stable in mean of system
3.3.
For p 2q, q ≥ 1, we have
E V XkT T q − E V XkT q
EV XkT T − V XkT V XkT T q−1 · · · V XkT q−1
c μ
V XkT × V XkT T q−1 · · · V XkT q−1
≤E −
λM P 3.45
c μ ) *
≤−
E V XkT q · · · V XkT q × V XkT T λM P c μ
≤−
E V XkT q < 0.
λM P It is obvious that
c μ
q
E V XkT q .
E V XkT T ≤ 1 −
λM P 3.46
Abstract and Applied Analysis
15
For t ∈ kT, kT T , we have
E V xtq ≤ E V XkT q
c μ
E V XkT − T q
≤ 1−
λM P ≤ ···
k
c μ
≤ 1−
E V x0q
λM P 3.47
k
c μ
1−
P q x02q
λM P ≤
< et lnΘ/ρ P q x02q P q ap et lnΘ/ρ .
Similarly, we have
E V xtq
E xtp ≤
λm P q
P q t lnΘ/ρ
p
< a
e
λm P 3.48
p −βt
ψa
$
e ,
where ψ$ ψ p/2 > 1 is constant.
Therefore, we have shown that the trivial solution of system 3.3 is the pth moment
uniformly exponentially stable in mean for even integers. In the same way, the theorem is
satisfied for odd integers. Hence, we conclude the proof for all p > 0.
Remark 3.4. The upper of PWM is given by that can easily be computed and optimized. We
will employ a simple procedure in a specific example in Section 4.
Remark 3.5. The pth moment exponential stability considered, in this paper, the system
tending to equilibrium speeds more quickly than others. The change of the status vectors
as time increases will be showed in Figure 3.
Corollary 3.6. Assume that A is Hurwitz stable, C1 / 0, C2 0, namely, the output of PWM feedback
system yt linear dependences on current status vectors xt. If the parameter of PWM Mβ satisfies
+
λm Γ1 Mβ <
$2
λ2m Γ1 4λM Γ
$2
2λM Γ
3.49
whenever G is sufficiently small (an upper bound of G has been given in the proof of Theorem 3.3),
$ 2 τk U
$ T Tk U
$ 2 Tk C1 with
$ 2 CT G1 τk G
where Γ1 U1 Tk C1 U1 Tk C1 T , Γ
2
1
16
Abstract and Applied Analysis
$ 2 τk W T τk e−Ad T P2 − Ie−Ad Wτk , U
$ 2 τk P2 − Ie−Ad Wτk by choosing P satisfied
G
T
eAT P eAT − P −I.
4. Examples
Example 4.1. Consider the system 3.3 with one order Hurwitz stable plant described by
transfer function T s 1/s 1. The state space representation of this system is given by
A −1, Ad 0.1, B 1, C1 1, C2 −1, assuming the period T 1, the time delay d 0.1.
Hence WTk , Pi , Gi Tk , Ui Tk , i 1, 2, in Theorem 3.3 are calculated in this case as
1
1
e2Tk − 1
,
P
,
WT
,
2
k
Tk
1 − e−2
1 − e−1.8
2
2
1
1
e2Tk − 1
e2Tk − 1
,
G2 Tk ,
G1 Tk 2
1.8
Tk
Tk
e −1
e −1
P1 U1 Tk e2Tk − 1 1
,
Tk e 2 − 1
U2 Tk 4.1
e2Tk − 1 1
,
Tk e1.8 − 1
where the pulse width Tk satisfies 0 < Tk ≤ 0.9, and we obtain the estimation of the upper
bound of Mβ, that is, Mβ ≤ 0.9398. For Mβ ∈ 0, 0.9398, we compute μmax 0.7 such that
√
for all μ < μmax . cμ 1 − 2Φ μ − 4P μ > 0 is true. Next, we compute δmax 0.22 such
2
that K · fAd , G < μmax is true for all δ < δmax , where K 5K1 1 M2 β2 B2 C2 eA T
T
and K1 max eA s eAs . In Figure 2, we depict the estimates of the upper bound δmax of G
s∈0,T versus Mβ.
We observe that δmax decreases as Mβ increases. When the states are sufficiently far
away from the origin so that Tk T − d, the curvature of the curve reduces more slowly than
without time-varying delays. Namely, as Mβ increases, the anti-interference performance of
the system 3.3 is stronger than the stochastic PWM feedback system without time-varying
delays. Furthermore, as M increases for fixed β, the maximum G allowable to ensure the
pth moment uniform exponential stability in mean will decrease, besides, if disturbance of
the feedback system 3.3 is increased less than δmax , the trivial solution of system 3.3 is
the pth moment uniformly exponentially stable in mean by decreasing the value of Mβ, as
shown in Figure 2.
Example 4.2. Consider PWM feedback system 3.3 with transfer function T s 1/s 2
s 1.
The state space representation of this system is given by
%
A
&
−2 0
,
0 −1
T
B
1 1 ,
%
&
0.2 0
,
Ad 0 0.1
C1 1 1 ,
%
G
&
0.02 0
,
0 0.006
C2 0 0 .
4.2
Abstract and Applied Analysis
17
0.2
δmax
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Mβ
Figure 2: Upper bounds for G when Mβ ∈ 0, 0.9398.
2
Sample response
1.5
1
0.5
0
−0.5
0
1
2
3
4
5
6
7
8
9
10
t
x1
x2
Figure 3: Sample response of PWM feedback system 3.3 in Example 4.2.
In Figure 3, we plot the sample response of x x1 x2 T with M 0.2, T β 1, d 0.1, and
x0 2 0.5T . We observe the system 3.3 tending to the equilibrium point speed quickly.
5. Conclusions
We studied the stochastic PWM feedback systems with time-varying delays and established
several Lyapunov and Lagrange criteria for the pth moment exponential stability in mean,
then presented an algorithm to compute the upper bound for the parameters of PWM, and
finally given two numerical examples to verify the effectiveness of theoretical results. We
characterized the relationship among the parameters of pulse-width modulation, time delay,
18
Abstract and Applied Analysis
and the coefficient of state vectors of the feedback systems and showed that when the random
disturbance is sufficiently small such PWM feedback system is the pth moment uniformly
exponentially stable in mean provided that the upper bounds of parameters of pulse-width
modulator are selected properly.
Acknowledgment
The work was supported by the Fundamental Research Funds for the Central Universities of
China under Grants CDJZR10100015.
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