∞ filtering for sampled-data stochastic systems with limited capacity channel

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Signal Processing

Signal Processing 91 (2011) 1826–1837

0165-16

doi:10.1

$ Thi

2009CB

under G

Excellen

Fok Yin� Cor

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journal homepage: www.elsevier.com/locate/sigpro

HN filtering for sampled-data stochastic systems withlimited capacity channel$

Ming Liu �, Jia You, Xincheng Ma

School of Astronautics, Harbin Institute of Technology, Harbin 150001, China

a r t i c l e i n f o

Article history:

Received 3 November 2010

Received in revised form

3 January 2011

Accepted 12 February 2011Available online 22 February 2011

Keywords:

H1 filtering

Sampled-data filtering

Stochastic system

Quantization

External disturbance

84/$ - see front matter & 2011 Elsevier B.V. A

016/j.sigpro.2011.02.006

s work was supported in part by the 973 P

320600, by the National Natural Science Fo

rant 60825303, by the Foundation for the

t Doctoral Dissertation of China under Grant

g Tung Education Foundation (111064).

responding author.

ail addresses: [email protected] (M

[email protected] (J. You),

[email protected] (X. Ma).

a b s t r a c t

This paper investigates the H1 filtering problem for sampled-data stochastic systems

with limited capacity channel. The considered plant is described by a class of Ito

stochastic systems subject to external disturbance. The output measurements are

sampled and quantized, and then transmitted through a network medium. The aim of

this paper is focused on the design of full order filters by using the quantized sampled

outputs. In sampled-data systems, the value of the sampled signal increases abruptly at

sampling times, and traditional filter design results based on time-independent

Lyapunov–Krasovskii functionals (or Lyapunov–Razumikhin functions) may be con-

servative. The main contribution of this paper is to propose a new type of time-

dependent Lyapunov function for Ito stochastic systems which does not increase in

sampling times due to its special mathematical structure. Based on this approach,

sufficient conditions for the existence of the proposed filter are established such that

the filtering error system is stochastically stable and preserves a guaranteed H1performance. A numerical example is provided to illustrate the effectiveness of the

proposed filtering technique in this paper.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The filtering problem has long been a significant andactive research area due to its theoretical and practicalsignificance in the signal processing and the systems control(see e.g., [20,19,32,26–28,22] and the references therein). Inmodern industrial process, the output signal is alwaysrequired to be measured at discrete sampling time instantsfor a continuous-time plant. Naturally, the problem of H1

ll rights reserved.

roject under Grant

undation of China

Author of National

2007E4, and by the

. Liu),

filtering under sampled measurements has attracted greatattention over the past few decades, and a few significantresults have been reported in the literature (see e.g.,[17,14,13,18] and the references therein). On the other hand,recent years network-based control and filtering theory alsopromotes the development of filtering techniques forsampled-data systems [10]. However, in realistic systemswithin a networked environment, signals between the plantand the filter are transmitted over digital channels withlimited capacities, and thus they are required to be quan-tized before being sent to the filter side. The effect of outputquantization can be regarded as a class of nonlinear map-ping mathematically which is difficult to be dealt with. Inthis sense, traditional filtering design strategies must be re-evaluated and re-designed before being directly appliedto quantized sampled-data filtering systems. So far, somenovel and interesting results have been reported focused onfiltering problem of sampled-data and quantized systems inthe existing literature [3].

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dx.doi.org/10.1016/j.sigpro.2011.02.006

M. Liu et al. / Signal Processing 91 (2011) 1826–1837 1827

Stochasticity typically exhibit in many branches ofscience and engineering applications, which are arguablyone of the main resources in practice that have resulted inconsiderable system complexities [9,15,7,6,21]. Hence,filtering for stochastic system is an important researcharea that has attracted considerable interest [31,33], and alarge number of the results on filtering for linear ornonlinear systems with stochastic white noises have beenreported in the existing literature (see [11,25] and thereferences therein). In addition, recently another signifi-cant type of stochastic noises/disturbances described byBrownian motions (or Wiener processes) has also beenaddressed for the filtering problems in some referencesdue to its important application in practical industrialprocessing (see [24,8] and the references therein). Inparticular, recently the sampled-data filtering problemfor Ito stochastic systems has been addressed in [29]based on an impulsive system approach.

Generally speaking, there exist three main approachesfor the control and filtering problems of sampled-datasystems (namely lifting technique approach, impulsivemodel approach, and input delay approach). Input delayapproach is one of the most important approaches, wherethe system is modeled as a continuous-time system withthe delayed-control input. In this research area, a fewresults have been reported in the literature [1,2,4]. Inparticular, recently a time-independent Lyapunov functionapproach (including Lyapunov–Krasovskii functionals andLyapunov–Razumikhin functions) has been developed inFridman [2] for linear sampled-data systems. However, inpractical sampled-data systems, there exists inevitablephenomenon that the sampled signal occurs jumping

behavior at sampling times and the value of the sampledsignal increases abruptly at sampling times. In addition, ifthe effect of signal quantization is considered together, theincrease of the signal value will be larger. As mentioned inFridman [1], the main drawback of the approach of Frid-man [2] is that the value of the integral terms of theproposed Lyapunov function increases abruptly at sam-pling times due to the jumping behavior of the sampledsignal (this issue will be discussed in detail in Remark 2below).

It is fortunate that in recent years a new Lyapunovapproach has been developed in Fridman [1] and Naghsh-tabrizi et al. [12] where a class of time-dependent Lyapunovfunctions approaches are developed. The designed Lyapunovfunction of [1,12] do not increase in sampling times andthus it can reduce the conservativeness of the existing filterdesign results. It should be pointed out, however, only statefeedback control problems have been considered in Fridman[1] and Naghshtabrizi [12], while the filtering problem hasnot been adequately investigated. Besides, the time-depen-dent Lyapunov approaches of [1,12] cannot be applieddirectly to high complicated systems such as Ito stochasticsystems. It is because for Ito stochastic systems the deriva-tion of x(t) (or _xðtÞ) is available and cannot be used in thedesign of the Lyapunov functionals. It is desirable to extendthe time-dependent Lyapunov approach [1,12] to Ito stochas-tic systems to pursue less conservative filter design results.The objective of this paper is therefore to shorten such a gapby developing a new filtering technique.

In this paper, we investigate the problem of filterdesign for sampled-data stochastic systems with limitedcapacity channel. The output measurement is sampledwith non-uniform sampling intervals, and quantized in alogarithmic form before being transmitted over networks.A new interesting discontinuous Lyapunov function ispresented which does not increase its value abruptly insampling times due to its special structure. In addition,free weight matrices are introduced into the derivation ofthe Lyapuov function V(t) to provide more design free-dom. Based upon this proposed Lyapunov approach, a H1filter technique is developed to generate the state estima-tion of the considered plant by employing the quantizedsampled output measurements. Finally, a simulationexample is provided to illustrate the effectiveness andapplicability of the design filtering technique. It should bepointed out that our approach is not a trivial extension ofthe results in [1,12] since the Brownian motion stochasticnoises have been considered there, and our designedLyapunov function has a new structure different fromthose in [1,12].

The remainder of this paper is organized as follows: InSection 2, the problem of designing sampled H1 filters isformulated for Ito stochastic systems. Section 3 is topresent a new time-dependent Lyapunov function forthe stochastic systems. In Section 4, the sufficient condi-tions for stability analysis and filter design are estab-lished. Finally, an illustrative example is presented inSection 5 to demonstrate the feasibility of the proposedmethod.

Notation: Throughout the paper, J � J denotes the standardEuclidean vector norm. Given a symmetric matrix A, thenotation A40 ðo0Þ denotes a positive definite matrix(negative definite, respectively). In denotes an identity matrixwith dimension n. x(t�) (resp. x(t+)) denotes the left-limitoperator (resp. right-limit operator), i.e., xðt�Þ ¼ limt-t�xðtÞ(resp. xðtþ Þ ¼ limt-t þ xðtÞ), where t-t� (resp. t-tþ )denotes t which tends to t from left (resp. right).

2. Problem description

We consider the following Ito stochastic system:

dxðtÞ ¼ ½AxðtÞþBuðtÞ� dtþ½ExðtÞþGuðtÞ� dwðtÞ,

yðtÞ ¼ CxðtÞþDuðtÞ,

zðtÞ ¼ LxðtÞ,

8><>: ð1Þ

where xðtÞ 2 Rn is the state; yðtÞ 2 Rp is the output; zðtÞ 2

Rr is the state combination to be estimated; A 2 Rn�n,B 2 Rn�m, C 2 Rp�n, D 2 Rp�m, E 2 Rn�n, G 2 Rn�m and L 2

Rr�n are known constant matrices; oðtÞ is a standard one-dimensional Brownian motion on a probability spaceðO,F ,PÞ relative to an increasing family ðF tÞt40 ofs-algebra F t 2 F ; O is the sample space, F is thes-algebra of subsets of the sample space, and P is theprobability measure on F .

It is assumed that uðtÞ is an adapted and measurableprocess with respect to F t . Also, uðtÞ belongs toL2ð½0,1Þ;Rm

Þ, where L2ð½0,1Þ;RmÞ denotes the space of

nonanticipatory square-integrable process f ð�Þ ¼ ðf ðtÞÞt2½0,1Þ

M. Liu et al. / Signal Processing 91 (2011) 1826–18371828

on Rm with respect to ðF tÞt2½0,1Þ, and it satisfies

Jf J2¼

Z 10jf ðtÞj2 dto1: ð2Þ

In this paper, the output y(t) are sampled and quantizedbefore transmitted over networks. Define T9ft1,t2,t3, . . .gas a strictly increasing sequence of sampling times in ðt0,1Þfor initial time t0=0, and it is assumed that tk satisfies

tkþ1�tkrh, 8k40, ð3Þ

where h40 is a known constant. Then, the input of thefilter yq(tk) is assumed to be of the following form at thesampling instant tk with zero-order hold (ZOH):

yqðtÞ ¼ qðyðtkÞÞ, t 2 ½tk,tkþ1Þ: ð4Þ

In (4), the quantizer qð�Þ has the following form:

qð�Þ ¼ ½q1ð�Þ,q2ð�Þ, � � � qpð�Þ�T , ð5Þ

where qið�Þ is assumed to be symmetric, that is,

qiðyiðtkÞÞ ¼�qið�yiðtkÞÞ, i¼ 1, . . . ,p ð6Þ

and the set of quantized levels of qið�Þ is described by

Qi ¼ f7sðjÞi jsðjÞi ¼ ðriÞ

j� sð0Þi ,j¼ 71,72, . . .g [ f7sð0Þi g [ f0g,

0orio1, sð0Þi 40: ð7Þ

In (7), ri denotes the quantizer density of the sub-quantizerqið�Þ, and sð0Þi denotes the initial quantization values for theith sub-quantizer qið�Þ. In this work, qið�Þ is assumed to bethe following logarithmic quantization:

qiðyiðtkÞÞ ¼

sðjÞi if1

1þdisðjÞi oyiðtkÞr

1

1�disðjÞi ,

yiðtkÞ40,

0 if yiðtkÞ ¼ 0,

�qið�yiðtkÞÞ if yiðtkÞo0, i¼ 1,2, . . . ,p;

j¼ 71,72, . . . ,

8>>>>>>>><>>>>>>>>:

ð8Þ

where di ¼ ð1�riÞ=ð1þriÞ are the quantizer parameters.According to [3], the logarithmic quantizer (8) can be

characterized by the following form:

qðyðtkÞÞ ¼ ðIpþLðtkÞÞyðtkÞ, ð9Þ

where

LðtkÞ ¼ diagfL1ðtkÞ,L2ðtkÞ, . . . ,LpðtkÞg, LjðtkÞ 2 ½�sj,sj�,

j¼ 1, . . . ,p: ð10Þ

The objective of this paper is to design a H1 filter for thestochastic system (1). To this end, we consider thefollowing filter of full order n:

dxðtÞ ¼ ½AxðtÞþ ByqðtÞ� dt,

zðtÞ ¼ LxðtÞ,t 2 ½tk,tkþ1Þ, ð11Þ

where xðtÞ 2 Rn and zðtÞ 2 Rr are the estimation of x(t)and z(t) respectively; A 2 Rn�n and B 2 Rn�m are filtergains to be designed.

Remark 1. The Brownian motion oðtÞ and the externaldisturbance uðtÞ in system (1) are born from differentsources in practice, and they are independent of eachother. Hence, they can be dealt with separately in the

mathematical derivation. In fact, this issue has beendiscussed in [29,30]. As seen in the statement and analysisof [29,30], oðtÞ and uðtÞ are treated independently whenconsidering the mathematical expectation for the inves-tigated Lyapunov functions. On the other hand, the maindifficulty in this filtering problem is that: only quantizedsampled output q(y(tk)) are available in the filter side,which results in the addressed filter design is a quitecomplicated work to be addressed. In the followingdiscussion, we will present a time-dependent Lyapunovfunction approach to cope this filtering problem.

We define the following error variables:

exðtÞ9xðtÞ�xðtÞ, eðtÞ9½xT ðtÞ, eTx ðtÞ�

T , ezðtÞ9zðtÞ�zðtÞ:

Subtracting (11) from (1), we have

dexðtÞ ¼ ½ðA�AÞxðtÞþ AexðtÞ�ByqðtÞ

þBuðtÞ� dtþ½ExðtÞþGuðtÞ� doðtÞ: ð12Þ

Combining (12) with (1), we obtain the filtering errordynamics as follows:

deðtÞ ¼ ½AeðtÞþBuðtÞþB1yqðtÞ� dtþ½EeðtÞþGuðtÞ� doðtÞ,ezðtÞ ¼ LeðtÞ, t 2 ½tk,tkþ1Þ,

(

ð13Þ

where

A ¼A, 0

A�A, A

" #, B ¼

B

B

� �, B1 ¼

0

�B

� �, E ¼

E 0

E 0

� �,

G ¼G

G

� �, L ¼ ½0,L�: ð14Þ

For simplicity, we define

f1ðtÞ ¼ AxðtÞþBuðtÞ, f2ðtÞ ¼ ðA�AÞxðtÞþ AexðtÞ�ByqðtÞþBuðtÞ,

gðtÞ ¼ ExðtÞþGuðtÞ,

f ðtÞ ¼f1ðtÞ

f2ðtÞ

" #, g ðtÞ ¼

gðtÞ

gðtÞ

" #ð15Þ

and the filtering error dynamics (13) can be rewritten as

deðtÞ ¼ f ðtÞ dtþgðtÞ doðtÞ,ezðtÞ ¼ LeðtÞ, t 2 ½tk,tkþ1Þ:

(ð16Þ

Before formulating the problem to be investigated, weintroduce the following definition, proposition and lem-mas, which are useful for the subsequent analysis.

Definition 1 (Niu et al. [16]). Let g40 and a40 be givenpositive constants, system (16) is said to be stochasticallyexponentially stable with the decay rate a andg-disturbance attenuation if: (i) system (16) is stochasti-cally exponentially stable with the decay rate a foruðtÞ ¼ 0; and (ii) under zero initial conditions, it satisfiesJzðtÞJE2

ogJuðtÞJ2 for all nonzero uðtÞ 2 L2½0,1Þ.

Proposition 1 (Weak infinitesimal operator, Skorokhod

[23]). Let C2ðRn; ½Rþ Þ denote the family of all nonnegative

functions V(t, x(t)) on Rn which are continuously twice

differentiable in x. For V 2 C2ðRn; ½Rþ Þ, define an infinitesimal


operator LVðt,xðtÞÞ by

LVðt,xðtÞÞ ¼ limD-0þ

1

D½EfVðt,xðtþDÞÞjxðtÞg�Vðt,xðtÞÞ�:

Lemma 1. Let the function Vðt,xÞ : R�Rn-R is continu-

ous from the right along the trajectory of system (16),absolutely continuous for tatk and satisfies

(1)
given b 2 N, there exist Z140, Z240 such that
Z1jxjbrVðt,xÞrZ2jxj

b, 8x 2 R, ð17Þ

(2)
for tk 2 T , V(tk, x) satisfies
Vðtk,xÞr limt-t�

k

Vðt,xÞ, ð18Þ

(3)
given a40, V(t) satisfies
LVðt,xÞþ2aVðt,xÞr0, tatk, tk 2 T ð19Þ

qi (y (tk))

qi (y(tk-1))

qi (y (t))

tk t

Fig. 1. The jumping behavior of the quantized sampled signal.

then system (16) is stochastically exponentially stable with

the decay rate a.

The proof of Lemma 1 is similar to the proof of Lemma1 of [1], we omit it here for brevity.

Lemma 2. For any real vectors a, b and matrix M40 of

compatible dimensions, the following inequality holds:

aT bþbT araT MaþbT M�1b: ð20Þ

Lemma 3. Let E, H, and F(t) be real matrices of appropriate

dimensions with F(t) satisfying FT ðtÞFðtÞo I. Then, for any

scalar e40,

EFðtÞHþHT FT ðtÞET reEETþe�1HT H:

In the sequel, the main objective of this paper is asfollows:H1 filtering problem: Given a constant a40 and a

disturbance attenuation level g40, the parameters A andB of filter (11) are designed such that (i) the filtering errorsystem (16) is stochastically exponentially stable with thedecay rate a under uðtÞ ¼ 0, and (ii) the following H1performance index:

Jz�zJE2ogJuJE2

ð21Þ

holds under zero initial conditions for all uðtÞ 2 L2

ð½0,1Þ;RmÞ.

3. Time-dependent Lyapunov function

In this section, we propose a new time-dependentLyapunov approach for the filter design of the Ito stochas-tic systems (1). For the error system (16), we constructthe following discontinuous and time-dependent Lyapunovfunctional:

VðtÞ ¼ V1ðtÞþV2ðtÞþV3ðtÞþV4ðtÞþV5ðtÞþV6ðtÞ, ð22Þ

with

V1ðtÞ ¼ xT ðtÞP1xðtÞþeTx ðtÞP2exðtÞ, ð23Þ

V2ðtÞ ¼ ðtkþ1�tÞ

Z t

tk

e2aðs�tÞf T1 ðtÞR1f1ðtÞ ds, ð24Þ


Z t

tk

e2aðs�tÞf T2 ðtÞR2f2ðtÞ ds, ð25Þ


Z t

tk

e2aðs�tÞgT ðtÞR3gðtÞ ds, ð26Þ

V5ðtÞ ¼ ðtkþ1�tÞðxðtÞ�xðtkÞÞT R4ðxðtÞ�xðtkÞÞ, ð27Þ

V6ðtÞ ¼ ðtkþ1�tÞðexðtÞ�exðtkÞÞTR5ðexðtÞ�exðtkÞÞ,t 2 ½tk,tkþ1Þ,

ð28Þ

where P1, P2, R1, R2, R3, R4, R5 are positive and definematrices.

Remark 2. It is shown in Fig. 1 that the quantizedsampled signal yq(t)=q(y(t)) occurs jumping behavior attk 2 T , and we have yq(tk

�)= yq(tk�1), yq(tk

+)=yq(tk). Sub-

sequently, the value of f2(t) in error system (16) willincrease abruptly at sampling times tk. This unexpectedphenomenon will add the conservativeness for the stabi-lity analysis and filter design. On the other hand, inour proposed Lyapunov functions (23)–(28), it can beobserved that Vi(t) (i=2,3,4,5,6) decreases to zero abruptlyat sampling times tk due to their special structures. Thatmeans, our designed Lyapunov function V(t) will notincrease abruptly at sampling times. It is also noted thatthe introduction of V1(t) has guaranteed that V(t) ispositive and definite, and thus our designing V(t) is awell-defined Lyapunov function. Compared with the tra-ditional Lyapunov approaches, our proposed Lyapunovfunction V(t) in (23)–(28) is less conservative for thestability analysis and filter design. We also point out thatour design approach can be applied to both constantsampling period and uncertain sampling period cases.

Remark 3. It should be pointed out that V1(t)+V2(t)+V3(t)+V4(t) or V1(t)+V5(t)+V6(t) can also be employed toaddress the stability analysis and filter design of system (1),and they also satisfy the property of positive definite and donot increase at sampling times tk. However, to pursue a lessconservative results, we have chosen V(t) as the form of (22)since it is more feasible to obtain a LMIs solution as will beseen in Theorem 1.

Remark 4. We apply the time-independent Lyapunovfunctional of Fridman [2] to system (16)

V0ðtÞ ¼ eT ðtÞPeðtÞþ

Z 0

�h

Z t

tþ sf T2 ðsÞf2ðsÞ ds dy: ð29Þ

It is noted that the termR 0�h

R ttþ s f T

2 ðsÞf2ðsÞ ds dy (29) willincrease abruptly at sampling times tk, which will add the


conservativeness of the design results. In fact, comparedwith the Lyapunov function (29), our proposed Lyapunovfunction (23)–(28) has overcome this unexpected effectsince it does not increase abruptly in each sampling times.

Remark 5. We further consider the following time-depen-

dent Lyapunov approaches developed in [1]

~V ðtÞ ¼ eT ðtÞQ1eðtÞþðtkþ1�tÞ

Z t

tk

e2aðs�tÞ _eTðsÞQ2 _eðsÞ ds

þðtkþ1�tÞzTðtÞ ~XzðtÞ, ð30Þ

where Q140, Q240 and

~X ¼ðX1þXT

1 Þ=2 �X1þXT2

� �X2�XT2þðX1þXT

1 Þ=2

" #40,

zðtÞ ¼eðtÞ

eðtkÞ

" #: ð31Þ

Unfortunately, the Lyapunov function (30) cannot bedirectly applied to the stochastic system (16) since thereexists the Brownian motion oðtÞ in stochastic system (16),and _eðtÞ and _xðtÞ are not available in this case. To overcomethis difficulty, we have proposed the Lyapunov function(23)–(28) to deal with the Brownian motion oðtÞ. It can beseen that our Lyapunov function (23)–(28) has a totallydifferent structure compared with that of (30).

We define the following variables:

tðtÞ ¼ t�tk, t 2 ½tk,tkþ1Þ, tk 2 T , ð32Þ

ff1ðtÞ ¼

Z t

tk

f1ðsÞ=tðsÞ ds, ð33Þ

ff2ðtÞ ¼

Z t

tk

f2ðsÞ=tðsÞ ds, ð34Þ

fgðtÞ ¼

Z t

tk

gðsÞ=tðsÞ ds: ð35Þ

At the end of this section, we introduce the following zeroterms, which will be useful in the derivation as follows:

2½xT ðtÞS1þxT ðtkÞS2�

� xðtÞ�xðtkÞ�tðtÞff1ðtÞ�

Z t

t�tðtÞgðsÞ doðsÞ

� �¼ 0, ð36Þ

2½eTx ðtÞT1þeT

x ðtkÞT2�

� exðtÞ�exðtkÞ�tðtÞff2ðtÞ�

Z t

t�tðtÞgðsÞ doðsÞ

� �¼ 0, ð37Þ

2½xT ðtÞU1þxT ðtkÞU2þ f T1 ðtÞU3� � ðAxðtÞþBuðtÞ�f1ðtÞÞ ¼ 0, ð38Þ

2½eTx ðtÞW1þeT

x ðtkÞW2þ f T2 ðtÞW3� � ½ðA�AÞxðtÞþ AexðtÞ�ByqðtÞ�f2ðtÞ� ¼ 0,

ð39Þ

where S1, S2, T1, T2, U1, U2, U3, W1, W2, W3 are the freeweight matrices with appropriate dimensions to bedesigned.

4. Design of the H1 filter

In this section, we shall develop a robust filteringscheme for system (1) based on the analysis of Section 3.The following theorem provides a sufficient condition forthe stochastic exponential stability of the system (13) interms of the time-dependent Lyapunov method.

Theorem 1. Let the filter gains A and B are given, if there

exist positive and definite matrices R1, R2, R3, R4, R5, P1 and

P2, and matrices S1, S2, T1, T2, W1, W2, W3, U1, U2 and U3

with appropriate dimensions, such that the following

matrices inequalities hold:

G9

G11 G12 G13 G14 G15 G16 0 0 0 G1,10 0

� G22 0 G24 G25 G26 0 0 0 G2,10 0

� � G33 0 0 0 0 0 0 G3,10 0

� � � G44 0 0 0 0 0 0 0

� � � � G55 G56 G57 G58 0 0 G5,11

� � � � � G66 G67 G68 0 0 G6,11

� � � � � � G77 0 0 0 G7,11

� � � � � � � G88 0 0 0

� � � � � � � � G99 0 0

� � � � � � � � � G10,10 0

� � � � � � � � � � G11,11

26666666666666666666664

37777777777777777777775

o0,

ð40Þ

~G9

~G11~G12

~G13~G15

~G16 0 ~G1,10 0

� ~G22 0 ~G25~G26 0 ~G2,10 0

� � ~G33 0 0 0 ~G3,10 0

� � � ~G55~G56

~G57 0 ~G5,11

� � � � ~G66~G67 0 ~G6,11

� � � � � ~G77 0 ~G7,11

� � � � � � ~G10,10 0

� � � � � � � ~G11,11

266666666666666664

377777777777777775

o0,

ð41Þ

with

G11 ¼ ET P1Eþ2aP1þET P2E�R4þLT L

þS1þST1þS1R�1

3 ST1þU1AþAT UT

1þR4þR5,

G12 ¼ R4�S1þS1R�13 ST

2þAT UT2 ,

G13 ¼ P1þAT UT3 ,

G14 ¼�hS1,

G15 ¼ ðA�AÞT WT1 ,

G16 ¼ ðA�AÞT WT2 ,

G1,10 ¼U1BþET ½R4þR5þ2R3�G,

G22 ¼�R4�S2�ST2þS2R�1

3 ST2,

G24 ¼�hS2,

G25 ¼�CT ðIpþDðtkÞÞT B

TWT

1 ,

G26 ¼�CT ðIpþDðtkÞÞT B

TWT

2 ,


G33 ¼U3þUT3 ,

G44 ¼�he�2ahR1,

G55 ¼�R5þT1þTT1 þT1R�1

3 TT1 þW1Aþ A

TWT

1 ,

G56 ¼ R5�T1�TT1 þT1R�1

3 TT2 þ A

TWT

2 ,

G57 ¼�ATWT

3�W1,

G58 ¼�hT1,

G5,11 ¼�W1BðIpþDðtkÞÞD,

G66 ¼�R5�T2�TT2 þT2R�1

3 TT2 ,

G67 ¼�W2,

G68 ¼�hT2,


G7,7 ¼�W3�WT3 ,


G8,8 ¼�e�2ahhR2,

G9,9 ¼�e�2ahhR3,

G10,10 ¼�g2Iþ2R3þR4þR5,

G11,11 ¼�g2I

and

~G11 ¼ ET P1Eþ2aP1þET P2E�R4þ2ahR4þLT L

þS1þST1þS1R�1

3 ST1þU1AþAT UT

1

þðhþ2InÞR3þR4þR5,

~G12 ¼ R4�2ahR4�S1þS1R�13 ST

2þAT UT2 ,

~G1,10 ¼U1BþET ½hR3þR4þR5þ2R3�G,

~G22 ¼�R4þ2ahR4�S2�ST2þS2R�1

3 ST2 ,

~G33 ¼ hR1þU3þUT3 ,

~G55 ¼�R5þ2ahR5þT1þTT1 þT1R�1

3 TT1 þW1Aþ A

TWT

1 ,

~G56 ¼ R5�2ahR5�T1�TT1 þT1R�1

3 TT2 þ A

TWT

2 ,

~G57 ¼ hR5�ATWT

3�W1,

~G66 ¼�R5þ2ahR5�T2�TT2 þT2R�1

3 TT2 ,

~G67 ¼�hR5�W2,

~G7,7 ¼ hR2�W3�WT3 ,

~G10,10 ¼�g2InþhGT R3Gþ2R3þR4þR5,

~G11,11 ¼�g2I

then the error dynamics (16) is stochastically exponentially

stable with the decay rate a, and the performance in (21) is

achieved for all nonzero uðtÞ.

Proof. The proof is divided into the following two parts:(i) prove that JezðtÞJE2

ogJuðtÞJE2; and (ii) prove that the

filtering error system (13) is stochastic exponentiallystable with the decay rate a under uðtÞ ¼ 0. (i) Considerthe filtering error system (13) with uðtÞ ¼ 0. Then, by Itoformula [11], the infinitesimal operator LVðtÞ for V(t) in(23)–(28) along the trajectory of system (16) is derived asfollows:

LV1ðtÞþ2aV1ðtÞ

¼ 2xT1ðtÞP1f1ðtÞþTraceðgT ðtÞP1gðtÞÞþ2axT ðtÞP1xðtÞ

þ2eTx ðtÞP2f2ðtÞþTraceðgT ðtÞP2gðtÞÞþ2aeT

x ðtÞP2exðtÞ,

ð42Þ

LV2ðtÞ ¼ ½�e�2at�2aðtkþ1�tÞe�2at�

Z t

t�tðtÞe2asf T

1 ðsÞR1f1ðsÞ ds

þðtkþ1�tÞf T1 R1f1ðtÞ ð43Þ

and we have

LV2ðtÞþ2aV2ðtÞ

¼�e�2at

Z t

t�tðtÞe2asf T

1 ðsÞR1f1ðsÞ dsþðtkþ1�tÞf T1 R1f1ðtÞ

r�Z t

t�tðtÞe2aðs�tÞf T

1 ðsÞR1f1ðsÞ dsþðtkþ1�tÞf T1 ðtÞR1f1ðtÞ

r�e2ah

Z t

t�tðtÞf T1 ðsÞR1f1ðsÞ dsþðtkþ1�tÞf T

1 ðtÞR1f1ðtÞ: ð44Þ

By Jensen’s inequality, we haveZ t

t�tðtÞf T1 ðsÞR1f1ðsÞ dsZfT

f1ðtÞR1ff1

ðtÞ: ð45Þ

Then, (45) together with (44) yields that

LV2ðtÞþ2aV2ðtÞr�e2ahtðtÞfTf1ðtÞR1ff1

ðtÞþðtkþ1�tÞf T1 ðtÞR1f1ðtÞ:

ð46Þ

For V3(t) and V4(t), similar with the derivation for V2(t),we have

LV3ðtÞþ2aV3ðtÞr�e2ahtðtÞfTf2ðtÞR2ff2

ðtÞþðtkþ1�tÞf T2 ðtÞR2f2ðtÞ,

ð47Þ

LV4ðtÞþ2aV4ðtÞr�e2ahtðtÞfTg ðtÞR3fgðtÞþðtkþ1�tÞgT ðtÞR3gðtÞ:

ð48Þ

For V5(t), we have

LV5ðtÞþ2aV5ðtÞ

¼�ðxT ðtÞ�xT ðtkÞÞR4ðxðtÞ�xðtkÞÞ

þ2ðtkþ1�tÞðxT ðtÞ�xT ðtkÞÞR4f1ðtÞþgT ðtÞR4gðtÞ

þ2aðtkþ1�tÞðxT ðtÞ�xT ðtkÞÞR4ðxðtÞ�xðtkÞÞ: ð49Þ

For V6(t), we have

LV6ðtÞþ2aV6ðtÞ

¼�ðeTx ðtÞ�eT

x ðtkÞÞR5ðexðtÞ�exðtkÞÞ

þ2ðtkþ1�tÞðeTx ðtÞ�eT

x ðtkÞÞR5f2ðtÞþgT ðtÞR5gðtÞ

þ2aðtkþ1�tÞðeTx ðtÞ�eT

x ðtkÞÞR5ðexðtÞ�exðtkÞÞ: ð50Þ


We now consider the zero terms (37)–(39). In fact, it canbe derived from Lemma 2 that

�2½xT ðtÞS1þxT ðtkÞS2�

Z t

tk

gðsÞ doðsÞ

r ½xT ðtÞS1þxT ðtkÞS2�R�13 ½S

T1xðtÞþST

2xðtkÞ�

þ

Z t

tk

gT ðsÞ doðsÞ� �

R3

Z t

tk

gðsÞ doðsÞ� �

, ð51Þ

�2½eTx ðtÞT1þeT

x ðtkÞT2�

Z t

tk

gðsÞ doðsÞ

r ½eTx ðtÞT1þeT

x ðtkÞT2�R�13 ½T

T1 exðtÞþTT

2 exðtkÞ�

þ

Z t

tk


R3

Z t

tk


: ð52Þ

Also it can be shown thatZ t

tk


R3

Z t

tk


¼ TraceðgT ðtÞR3gðtÞÞ

¼ ½xT ðtÞETþuT ðtÞGT �R3½ExðtÞþGuðtÞ�: ð53Þ

We add the zero terms (37)–(39) into LVðtÞ, in the light of(23)–(28), (51), (52) and (53), one can obtain

LVðtÞþ2aVðtÞ

r2xT1ðtÞP1f1ðtÞþTraceðgT ðtÞP1gðtÞÞþ2axT ðtÞP1xðtÞ

PðhkÞ9

P11 P12 P13 P14 P15 P16 0 0 0 P1,10 0

� P22 0 P24 P25 P26 0 0 0 U2B 0

� � P33 0 0 0 0 0 0 U3B 0

� � � P44 0 0 0 0 0 0 0

� � � � P55 P56 P57 P58 0 0 P5,11

� � � � � P66 P67 P68 0 0 P6,11

� � � � � � P77 0 0 0 P7,11

� � � � � � � P88 0 0 0

� � � � � � � � P99 0 0

� � � � � � � � � P10,10 0

� � � � � � � � � � P11,11

26666666666666666666664

37777777777777777777775

o0 ð61Þ

þ2eTx ðtÞP2f2ðtÞþTraceðgT ðtÞP2gðtÞÞþ2aeT

x ðtÞP2exðtÞ

�e2ahtðtÞfTf1ðtÞR1ff1

ðtÞþðtkþ1�tÞf T1 ðtÞR1f1ðtÞ

�e2ahtðtÞfTf2ðtÞR2ff2

ðtÞþðtkþ1�tÞf T2 ðtÞR2f2ðtÞ

�e2ahtðtÞfTg ðtÞR3fgðtÞ�ðx

T ðtÞ�xT ðtkÞÞR4ðxðtÞ�xðtkÞÞ

þ2ðtkþ1�tÞðxT ðtÞ�xT ðtkÞÞR4f1ðtÞ

þ2aðtkþ1�tÞðxT ðtÞ�xT ðtkÞÞR4ðxðtÞ�xðtkÞÞ

�ðeTx ðtÞ�eT


þ2ðtkþ1�tÞðeTx ðtÞ�eT

x ðtkÞÞR5f2ðtÞ

þ2aðtkþ1�tÞðeTx ðtÞ�eT


þ2½xT ðtÞS1þxT ðtkÞS2� � ½xðtÞ�xðtkÞ�tðtÞff1ðtÞ�

þ2½eTx ðtÞT1þeT

x ðtkÞT2� � ½exðtÞ�exðtkÞ�tðtÞff2ðtÞ�

þ½xT ðtÞS1þxT ðtkÞS2�R�13 ½S

T1xðtÞþST

2xðtkÞ�

þ½eTx ðtÞT1þeT

x ðtkÞT2�R�13 ½T

T1 exðtÞþTT

2 exðtkÞ�

þ½ExðtÞþGuðtÞ�T ððtkþ1�tkþ2ÞR3þR4þR5Þ½ExðtÞþGuðtÞ�

þ2½xT ðtÞU1þxT ðtkÞU2þ f T1 ðtÞU3� � ðAxðtÞþBuðtÞ�f1ðtÞÞ

þ2½eTx ðtÞW1þeT

x ðtkÞW2þ f T2 ðtÞW3� � ½ðA�AÞxðtÞ

þ AexðtÞ�ByqðtÞ�f2ðtÞ�þxT ðtÞLT LxðtÞ�g2uT ðtkÞuðtkÞ�g2uT ðtÞuðtÞ:

ð54Þ

We define the following variable:

hk9tkþ1�tk ð55Þ

and it is shown that

tkþ1�t¼ tkþ1�tk�tðtÞ ¼ hk�tðtÞ: ð56Þ

In fact, by utilizing Ito’s formula, under zero initialconditions it holds that

EfVðtÞg ¼ EZ T

0LVðtÞ dt

� �: ð57Þ

If we define the following performance index function:

JðTÞ ¼ EZ T

0½eT

z ðtÞezðtÞ�g2uT ðtÞuðtÞ� dt

� �ð58Þ

for any T40, then it follows from (58) that

JðTÞrEZ T

0½eT

z ðtÞezðtÞ�g2uT ðtÞuðtÞþLVðtÞþ2aVðtÞ� dt

� ��EfVðTÞg

rEZ T

0xTðtÞPðhkÞxðtÞ dt

� �, ð59Þ

where

xðtÞ ¼ ½xT ðtÞ xT ðtkÞ f T1 ðtÞ f

Tf1ðtÞ eT

x ðtÞ eTx ðtkÞ f T

2 ðtÞ fTf2ðtÞ fT

g ðtÞ uT ðtÞ uT ðtkÞ�

T ,

ð60Þ

and

P11 ¼ ET P1Eþ2aP1þET P2E�R4þ2aðhk�tðtÞÞR4þLT L

þS1þST1þS1R�1

3 ST1þU1AþAT UT

1þðhk�tðtÞþ2ÞR3þR4þR5,

P12 ¼ R4�2aðhk�tðtÞÞR4�S1þS1R�13 ST

2þAT UT2 ,

P13 ¼ P1þAT UT3 ,

P14 ¼�tðtÞS1,

P15 ¼ ðA�AÞT WT1 ,

P16 ¼ ðA�AÞT WT2 ,

P1,10 ¼U1BþET ½ðhk�tðtÞÞR3þR4þR5þ2R3�G,

P22 ¼�R4þ2aðhk�tðtÞÞR4�S2�ST2þS2R�1

3 ST2 ,

P24 ¼�tðtÞS2,

P25 ¼�CT ðIpþDðtkÞÞT B

TWT

1 ,


P26 ¼�CT ðIpþDðtkÞÞT B

TWT

2 ,

P33 ¼ ðhk�tðtÞÞR1þU3þUT3 ,

P44 ¼�tðtÞe�2ahR1,

P55 ¼�R5þ2aðhk�tðtÞÞR5þT1þTT1 þT1R�1

3 TT1 þW1Aþ A

TWT

1 ,

P56 ¼ R5�2aðhk�tðtÞÞR5�T1�TT1 þT1R�1

3 TT2 þ A

TWT

2 ,

P57 ¼ ðhk�tðtÞÞR5�ATWT

3�W1,

P58 ¼�tðtÞT1,

P5,11 ¼�W1BðIpþDðtkÞÞD,

P66 ¼�R5þ2aðhk�tðtÞÞR5�T2�TT2 þT2R�1

3 TT2 ,

P67 ¼�ðhk�tðtÞÞR5�W2,

P68 ¼�tðtÞT2,


P7,7 ¼ ðhk�tðtÞÞR2�W3�WT3 ,


P8,8 ¼�e�2ahtðtÞR2,

P9,9 ¼�e�2ahtðtÞR3,

P10,10 ¼�g2Iþðhk�tðtÞÞGT R3Gþ2R3þR4þR5,

P11,11 ¼�g2I:

From (59), it can be shown that if Po0 holds, then JðTÞo0

for any T40, which implies that JezðtÞJE2rg2JuðtÞJE2

holds

for any nonzero uðtÞ 2 LE2ð½0,1Þ;Rm

Þ. On the other hand,

by the similar proof of Theorem 1 in Fridman [1], it can

be shown that PðhkÞo0 for any hk 2 ½0,h� if and only

Pðhk ¼ hÞo0 and Pðhk ¼ 0Þo0 hold. The detailed proof is

omitted here for brevity. On the other hand, by simple

derivation, it can be seen that Pðhk ¼ hÞo0 is just the matrix

condition Go0, while Pðhk ¼ 0Þ is the matrix condition

C9

C11 C12 C13 C14 C15 C16 0 0 0

� C22 0 C24 C25 C26 0 0 0

� � C33 0 0 0 0 0 0

� � � C44 0 0 0 0 0

� � � � C55 C56 C57 C58 0

� � � � � C66 C67 C68 0

� � � � � � C77 0 0

� � � � � � � C88 0

� � � � � � � � C99

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

26666666666666666666666666664

~Go0. That means, the matrices condition Go0 and ~Go0

can imply that JezðtÞJE2rg2JuðtÞJE2

holds for any nonzero

uðtÞ 2 LE2ð½0,1Þ;Rm

Þ.

(ii) Next, we establish the stochastic exponential stabi-

lity of the filtering error system (16) under the condition

of Theorem 1. In this situation, it is assumed that uðtÞ ¼ 0.

From the proof of part (i), we have

LVðtÞþ2aVðtÞrzTðtÞFðhkÞzðtÞ, ð62Þ

where

zðtÞ ¼ ½xT ðtÞ xT ðtkÞ f T1 ðtÞ f

Tf1ðtÞ eT

x ðtÞ eTx ðtkÞ f T

2 ðtÞ fTf2ðtÞ fT

g ðtÞ�T

ð63Þ

and

FðhkÞ9

P11 P12 P13 P14 P15 P16 0 0 0

� P22 0 P24 P25 P26 0 0 0

� � P33 0 0 0 0 0 0

� � � P44 0 0 0 0 0

� � � � P55 P56 P57 P58 0

� � � � � P66 P67 P68 0

� � � � � � P77 0 0

� � � � � � � P88 0

� � � � � � � � P99

266666666666666664

377777777777777775

o0:

ð64Þ

It easy to see that PðhkÞo0 yields FðhkÞo0. On the

other hand, we have proved that the matrix conditions

(40) and (41) yield PðhkÞo0. That means, the condition in

Theorem 1 can imply that FðhkÞo0, which yields that

LVðtÞo0. This completes the proof. &

The matrix conditions (40) and (41) proposed inTheorem 1 are not linear type, also there exists parameteruncertainty DðtkÞ in (40) and (41). Therefore, Theorem 1cannot be directly employed to check the stability of errorsystem (16) and filter design. Based on Theorem 1, in thefollowing theorem, we shall establish LMI conditions forthe existence of the proposed filter (11).

Theorem 2. Given parameters b240, b340, if there exist

positive and definite matrices R1, R2, R3, R4, R5, P1 and P2,and matrices W0, S1, S2, T1, T2, U1, U2 and U3 with

appropriate dimensions, such that the following LMIs hold:

C1,10 0 0 0

C2,10 0 0 0

C3,10 0 0 0

0 0 0 0

0 C5,11 W1B �W1B

0 C6,11 W2B �W2B

0 C7,11 0 �W3B

0 0 0 0

0 0 0 0

C10,10 0 0 0

� C11,11 0 0

� � �e1I 0

� � � �e2I

37777777777777777777777777775

o0, ð65Þ

~C9

~C11~C12 C13 C15 C16 0 ~C1,10 0 0 0

� ~C22 0 C25 C26 0 C2,10 0 0 0

� � ~C33 0 0 0 C3,10 0 0 0

� � � ~C55~C56

~C57 0 C5,11 0 0

� � � � ~C66~C67 0 C6,11 W1B �W1B

� � � � � ~C77 0 C7,11 W2B �W2B

� � � � � � ~C10,10 0 0 �W3B

� � � � � � � ~C11,11 0 0

� � � � � � � � �e1I 0

� � � � � � � � � �e2I

26666666666666666666664

37777777777777777777775

o0, ð66Þ


with

C11 ¼ ET P1Eþ2aP1þET P2E�R4þLT LþS1þST1þS1R�1

3 ST1

þU1AþAT UT1þR4þR5,

C12 ¼ R4�S1þS1R�13 ST

2þAT UT2 ,

C13 ¼ P1þAT UT3 ,

C14 ¼�hS1,

C15 ¼ AT WT1�XT ,

C16 ¼ AT WT2�b2XT ,

C1,10 ¼U1BþET ½R4þR5þ2R3�G,

C22 ¼�R4�S2�ST2þS2R�1

3 ST2þe1CT C,

C24 ¼�hS2,

C25 ¼�CT YT ,

C26 ¼�CTb2YT ,

C33 ¼U3þUT3 ,

C44 ¼�he�2ahR1,

C55 ¼�R5þT1þTT1 þT1R�1

3 TT1 þXþXT ,

C56 ¼ R5�T1�TT1 þT1R�1

3 TT2 þb2XT ,

C57 ¼�b2XT�W0,

C58 ¼�hT1,

C5,11 ¼�Y1D,

C66 ¼�R5�T2�TT2 þT2R�1

3 TT2 ,

C67 ¼�b2W0,

C68 ¼�hT2,

C6,11 ¼�b2YD,

C7,7 ¼�b3W0�b3WT0 ,

C7,11 ¼�b3YD,

C8,8 ¼�e�2ahhR2,

C9,9 ¼�e�2ahhR3,

C10,10 ¼�g2Iþ2R3þR4þR5,

C11,11 ¼�g2Iþe2DT D

and

~C11 ¼ ET P1Eþ2aP1þET P2E�R4þ2ahR4þLT L

þS1þST1þS1R�1

3 ST1þU1AþAT UT

1þðhþ2InÞR3þR4þR5,

~C12 ¼ R4�2ahR4�S1þS1R�13 ST

2þAT UT2 ,

~C1,10 ¼U1BþET ½hR3þR4þR5þ2R3�G,

~C22 ¼�R4þ2ahR4�S2�ST2þS2R�1

3 ST2þe1CT C,

~C33 ¼ hR1þU3þUT3 ,

~C55 ¼�R5þ2ahR5þT1þTT1 þT1R�1

3 TT1 þXþXT ,

~C56 ¼ R5�2ahR5�T1�TT1 þT1R�1

3 TT2 þb2XT ,

~C57 ¼ hR5�bT3XT�W0,

~C66 ¼�R5þ2ahR5�T2�TT2 þT2R�1

3 TT2 ,

~C67 ¼�hR5�b2W0,

~C7,7 ¼ hR2�b3W0�bT3W0,

~C10,10 ¼�g2InþhGT R3Gþ2R3þR4þR5,

~C11,11 ¼�g2Iþe2DT D

then the H1 filtering problem is solved by the filter (11).Furthermore, the filter gains are given by

A ¼W�10 X, B ¼W�1

0 Y : ð67Þ

0 1 2 3 4 5−3

−2

−1

0

1

2

3

Time (Sec.)

e1 (t)e2 (t)e3 (t)

Fig. 2. Trajectory of error variable e(t).

0 1 2 3 4 5−4

−3

−2

−1

0

1

2

3

Time (Sec.)

Fig. 3. Individual paths and the average of x(t).


Proof. Letting W1=W0, W2 ¼ b2W0, W3 ¼ b3W0, and wedefine

X9W0A, Y9W0B: ð68Þ

Then, in the light of Lemma 3, it is easy to prove that thematrix conditions (40) and (41) hold, if there exist scalarse140, e240, such that the LMIs (65) and (66) hold.Subsequently, the filter gains A and B can be solved from(68). This completes the proof. &

Remark 6. It should be pointed out that, the sampled-dataH1 filtering problem has been addressed for linear systemwith limited capacity channel in [3]. The difference betweenour work and that of [3] is as follows: (1) The plantinvestigated in [3] is linear system, while in this paper theaddressed plant is a class of Ito stochastic system withBrownian motion. (2) The Lyapunov function employed in[3] is time-independent type, while in this paper a newtime-dependent Lyapunov function approach is developed.(3) The effects of signal sampling, networked inducedcommunication delay, data packet loss and signal quantiza-tion are taken into simultaneous consideration in [3], whilein this paper we only focus on the issues of signal samplingand quantization. In this sense, we cannot perform anumerical comparison between our results and that of [3].

Remark 7. The sampled-data filtering problem for Itostochastic systems has been addressed in Xu and Chen[29]. However, the problem of the upper bound of thesampling period has not been considered there yet. Hence,we cannot perform a numerical comparison between ourresults and that in [29]. In addition, our design approach isan extension of the results of Fridman [1] and Naghshtabriziand Hespanha et al. [12]. However, the issues investigatedin [1,12] are state feedback control problems, and theBrownian motion has not been considered there. In thesequel, we cannot do the numerical comparison betweenour results and those in [1,12].

5. Simulation

We consider the system (1) with the following data: n=3,m=2, p=2, r=2. For the logarithmic quantizer (8), thequantizer densities are chosen as r1 ¼ 0:6667, r2 ¼ 0:7391.The initial quantizer points are chosen as sð0Þ1 ¼ 40, sð0Þ2 ¼ 40.It can be calculated that d1 ¼ 0:2, d2 ¼ 0:15. The samplinginstant tk is generated randomly with the upper bound h.That means, tkþ1�tkrh for k=1,2,y. In addition, the systemmatrices are selected as follows:

A¼

�2 0:1 0:1

�0:2 �5 0:1

�0:4 1:0 �10

264

375, B¼

0:2 0:2

0:5 0:2

0:4 0:3

264

375,

C ¼0:2 0:8 �0:3

1:0 0:3 0:5

� �,

D¼0:2 0:4

0:3 0:6

� �, E¼

0:3 1:0 0:6

1:3 0:5 �0:2

2:3 0:2 �0:4

264

375,

G¼

0:2 0:4

1:2 0:5

0:6 0:3

264

375, L¼

0:5 0:2 0:2

0:3 0:3 0:2

� �:

Furthermore, the decay rate is selected as a¼ 0:05, and theH1 performance index parameter is selected as g¼ 0:8.We choose b2 ¼ b3 ¼ 0:2, and solve the LMIs (65) and (66)in Matlab environment, one can obtain the following solu-tions:

A ¼

�82:0875 �32:0063 �61:3549

�117:2403 �50:9080 �94:6204

�125:4729 �43:2566 �116:9821

264

375,

B ¼

�49:6862 108:8155

�79:7732 162:8165

�89:4087 174:8570

264

375,

and it is checked numerically that the upper bound of thesampling period hk is h¼maxk2Nhk � 0:75 s. Without loss ofgenerality, we assume that the noises uðtÞ in system (1) hasthe following form:

uðtÞ ¼sinðtÞ

1þðt�1Þ2sinðtÞ

0:5þðt�1Þ2

" #ð69Þ

it can be checked that uðtÞ satisfies the constraint (2).For the stochastic noise oðtÞ in (69), the simulation is

performed by employing the approaches in Higham [5].

0 1 2 3 4 5−3

−2

−1

0

1

2

3

4

Time (Sec.)

Fig. 4. Individual paths and the average of ex(t).

0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

Time (Sec.)

y1 (t)q (y1(tk))

Fig. 5. Trajectories of y1(t) and q(y1(tk)).

0 1 2 3 4 5−1

0

1

2

3

4

5

6

Time (Sec.)

y2 (t)q (y2(tk))

Fig. 6. Trajectories of y2(t) and q(y2(tk)).


We select the simulation time t 2 ½0,T�, where T=5; thevariance is dt¼ T=N with N=104, and the step size isDt¼ R� dt with R=2; the number of discretized Brownianpaths is M=10; the initial state is chosen as x(0)=[2, 1,0.5]T, xð0Þ ¼ ½�1,1:5,2�T .

The simulation results are provided in Figs. 2–6,which show the effectiveness of the proposed methods.The trajectories of e(t) along an individual discretizedBrownian path is shown in Fig. 2. The trajectories of x(t)

and e(t) along 10 individual paths (dotted lines) and theaverage over 10 paths (solid line) are shown in Figs. 3 and4 respectively. The trajectories of output y1(t) and quan-tized sampled output q(y1(tk)) are shown in Fig. 5. Thetrajectories of output y2(t) and quantized sampled outputq(y2(tk)) are shown in Fig. 6. It can be seen that thefiltering dynamics (16) is stochastically stable.

6. Conclusion

In this paper, the sampled-data filtering problem hasbeen investigated for Ito stochastic systems with limitedcapacity channel. To overcome the unexpected effect ofabrupt increase of the quantized output measurement insampling times, a new time-dependent Lyapunov func-tion approach has been proposed for the investigated Itostochastic systems. A sufficient condition is further estab-lished such that the derived filtering error system isstochastically stable. Finally, a numerical example hasbeen provided to illustrate the effectiveness of the pro-posed filtering approach.

References

[1] E. Fridman, A refined input delay approach to sampled-data control,Automatica 46 (2) (2010) 421–427.

[2] E. Fridman, A. Seuret, J. Richard, Robust sampled-data stabilizationof linear systems: an input delay approach, Automatica 40 (8)(2004) 1441–1446.

[3] H. Gao, T. Chen, H1 estimation for uncertain systems with limitedcommunication capacity, IEEE Transactions on Automatic Control52 (11) (2007) 2070–2084.

[4] H. Gao, T. Chen, Stabilization of nonlinear systems under variablesampling: a fuzzy control approach, IEEE Transactions on FuzzySystems 15 (5) (2007) 972–983.

[5] D. Higham, An algorithmic introduction to numerical simulation ofstochastic differential equations, SIAM Review (2001) 525–546.

[6] H. Li, B. Chen, Q. Zhou, C. Lin, Robust exponential stability fordelayed uncertain Hopfield neural networks with Markovian jump-ing parameters, Physics Letters A 372 (30) (2008) 4996–5003.

[7] H. Li, B. Chen, Q. Zhou, W. Qian, Robust stability for uncertaindelayed fuzzy Hopfield neural networks with Markovian jumpingparameters, IEEE Transactions on Systems, Man, and Cybernetics,Part B 39 (1) (2009) 94–102.

[8] M. Liu, D.W.C. Ho, Y. Niu, Robust filtering design for stochasticsystem with mode-dependent quantized output, IEEE Transactionson Signal Processing 58 (12) (2010) 6410–6416.

[9] M. Liu, D.W.C. Ho, Y. Niu, Stabilization of Markovian jump linearsystem over networks with random communication delay, Auto-matica 45 (2) (2009) 416–421.

[10] C. Lu, W. Shyr, K. Yao, C. Liao, C. Huang, Delay-dependent H1control for discrete-time uncertain recurrent neural networkswith interval time-varying delay, International Journal of Innova-tive Computing Information and Control 5 (10(B)) (2008)3483–3494.

[11] X. Mao, Stochastic Differential Equations and their Applications,Horwood Publishing, 1997.

[12] P. Naghshtabrizi, J. Hespanha, A. Teel, Exponential stability ofimpulsive systems with application to uncertain sampled-datasystems, Systems & Control Letters 57 (5) (2008) 378–385.

[13] S. Nguang, P. Shi, On designing filters for uncertain sampled-datanonlinear systems, Systems & Control Letters 41 (5) (2000)305–316.

[14] S.K. Nguang, P. Shi, Nonlinear H1 filtering of sampled-data systems,Automatica 36 (2) (2000) 303–310.

[15] Y. Niu, D.W.C. Ho, J. Lam, Robust integral sliding mode control foruncertain stochastic systems with time-varying delay, Automatica41 (5) (2005) 873–880.

[16] Y. Niu, D.W.C. Ho, X. Wang, Sliding mode control for Ito stochasticsystems with Markovian switching, Automatica 43 (10) (2007)1784–1790.


[17] P. Shi, Robust filtering for uncertain delay systems under sampledmeasurements, Signal Processing 58 (2) (1997) 131–151.

[18] P. Shi, Filtering on sampled-data systems with parametric uncertainty,IEEE Transactions on Automatic Control 43 (7) (2002) 1022–1027.

[19] P. Shi, M. Karan, C. Kaya, Robust Kalman filter design for Markovianjump linear systems with norm-bounded unknown nonlinearities,Circuits, Systems, and Signal Processing 24 (2) (2005) 135–150.

[20] P. Shi, M. Mahmoud, S. Nguang, A. Ismail, Robust filtering forjumping systems with mode-dependent delays, Signal Processing86 (1) (2006) 140–152.

[21] P. Shi, Y. Xia, G. Liu, D. Rees, On designing of sliding-mode controlfor stochastic jump systems, IEEE Transactions on AutomaticControl 51 (1) (2006) 97–103.

[22] S. Shi, Z. Yuan, Q. Zhang, Fault-tolerant H1 filter design of a class ofswitched systems with sensor failures, International Journal ofInnovative Computing Information and Control 5 (11(A)) (2009)3827–3838.

[23] A. Skorokhod, Asymptotic Methods in the Theory of StochasticDifferential Equations, American Mathematical Society, 1989.

[24] Z. Wang, Y. Liu, X. Liu, H1 filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities, Automatica 44(5) (2008) 1268–1277.

[25] L. Wu, D.W.C. Ho, Reduced order L2=L1 filtering for a class ofnonlinear switched stochastic systems, IET Control Theory andApplications 3 (5) (2009) 493–508.

[26] L. Wu, P. Shi, H. Gao, C. Wang, A new approach to robust H1filtering for uncertain systems with both discrete and distri-buted delays, Circuits, Systems, and Signal Processing 26 (2) (2007)229–248.

[27] L. Wu, P. Shi, H. Gao, C. Wang, Filtering for 2D Markovian jumpsystems, Automatica 44 (7) (2008) 1849–1858.

[28] Z. Wu, H. Su, J. Chu, H1 filter design for singular time-delaysystems, Proceedings of the Institution of Mechanical Engineers,Part I: Journal of Systems and Control Engineering 223 (7) (2009)1001–1015.

[29] S. Xu, T. Chen, Robust H1 filtering for uncertain impulsive stochas-tic systems under sampled measurements, Automatica 39 (3)(2003) 509–516.

[30] S. Xu, T. Chen, H1 output feedback control for uncertain stochasticsystems with time-varying delays, Automatica 40 (12) (2004)2091–2098.

[31] L. Zhang, E. Boukas, Stability and stabilization of Markovian jumplinear systems with partly unknown transition probabilities, Auto-matica 45 (2) (2009) 463–468.

[32] L. Zhang, E. Boukas, Mode-dependent H1 filtering for discrete-timeMarkovian jump linear systems with partly unknown transitionprobabilities, Automatica 45 (6) (2009) 1462–1467.

[33] L. Zhang, P. Shi, E. Boukas, C. Wang, H1 control of switched lineardiscrete-time systems with polytopic uncertainties, Optimal Con-trol Applications and Methods 27 (5) (2006) 273–291.

∞ filtering for sampled-data stochastic systems with limited capacity channel

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