Observer-based fault detection for networked control systems with network Quality of Services

Applied Mathematical Modelling 34 (2010) 1653–1661

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Observer-based fault detection for networked control systems with network Quality of Services Chen Peng a, Dong Yue a,*, Engang Tian a, Zhou Gu b a b

School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing, Jiangsu 210042, PR China College of Power Engineering, Nanjing Normal University, Nanjing, Jiangsu 210042, PR China

a r t i c l e

i n f o

Article history: Received 27 July 2008 Received in revised form 3 September 2009 Accepted 7 September 2009 Available online 11 September 2009 Keywords: Fault detection (FD) Networked control systems (NCSs) Markov jumping systems (MJSs) Linear matrix inequalities (LMIs)

a b s t r a c t This paper addresses the problem of the fault detection for linear time-invariant systems over data networks with limited network Quality of Services (QoS). An integrated index gk , which related with data dropout, network-induced delay and error sequence, is presented to described the non-ideal QoS, the probabilistic switching between different gk is assumed to obey a homogeneous Markovian chain. Then by view of the augmented matrices approach, the fault detection error dynamic systems are transferred to Markov jumping systems (MJSs). With the developed model and using the bounded real lemma (BRL) for MJSs, an H1 observer-based fault detection filter is established in terms of linear matrix inequalities (LMIs) to guarantee that the error between the residual and the weighted faults is made as small as possible. A simulation example is provided to show the effectiveness of the present methods. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Over the last decade a lot of interest has been devoted to Fault detection and isolation (FDI) technology due to an increasing demand for higher performance, higher safety and reliability standards. Fruitful theoretical results and increasing applications in industrial practice, such as in power plant coal mills and robotic systems can be found in [1–9]. Furthermore, with the rapid development of communication networks, recently, a great amount of effort has been devoting to the problems of FDI for networked control systems [10–15]. Data communication networks are used in control systems to interconnect various system devices and components. Such control systems are commonly referred to as Networked control systems (NCSs) [16,17], in which measurement and control signals are transmitted over data networks. NCSs have many advantages over conventional control systems. However, there are also challenges in NCS analysis and design, for instance, time-varying network-induced delay and packet dropout [18– 20]. The study on some kind of NCSs has received much attentions recently, the representative works in this field can be seen in [21–26]. Considering a networked control system, the existence of unknown input and network-induced delay may seriously affect the performance of observer-based FDI systems. Recent advances in fault detection over data network include He et al. [12], Mao et al. [11], and Fang et al. [14], and so on. Such as, Mao et al. utilized the multi-rate sampling method together with the augmented state matrix method to model the long random delay networked control system as MJSs, then based on the Riccati equation to have the robustness analysis and H1 filter design [11]. Considering the free multiple state delays system, He et al. studied the fault detection problem for a class of discrete-time networked system with both the random

* Corresponding author. Tel.: +86 25 85481180; fax: +86 25 85481197. E-mail addresses: [email protected] (C. Peng), [email protected] (D. Yue). 0307-904X/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2009.09.014

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C. Peng et al. / Applied Mathematical Modelling 34 (2010) 1653–1661

communication delays and the stochastic date missing, with the help of bounded real lemma (BRL) of MJSs [27,28], a sufficient condition for the existence of the desired fault detection filter is established in terms of a set of LMIs [12]; Fang et al. employed a Takagi–Sugeno (T–S) model to represent an NCS with different network-induced delays, then developed parityequation and fuzzy-observer-based approaches for fault detection of an NCS [14], respectively. Regardless of these recent developments, to the best of authors’ knowledge, the FD problems for NCSs with non-ideal QoS, such as network-induced delay and data dropout described by a unified framework have not been fully investigated yet, which are not only theoretically interesting and challenging, but also very important in practical engineering applications, and motivates our present study. This paper addresses the problem of the fault detection for linear time-invariant systems over data networks with limited network QoS. The systems under consideration are assumed to be stabilizable. For theoretical development of the fault detection over data networks, a new Markovian jumping system (MJS) model will be developed that combines Souza’s idea for MJSs [28] and our recent idea for network modeling [24,26], which allows us to describe both fault detection and NCSs in a unified framework. With the developed model and using the BRL for MJSs [27,28], an H1 fault detection filter is established in terms of linear matrix inequalities (LMIs) to guarantee a convergent error dynamics if there is no fault, and to make the effect of disturbances on the residual satisfy a prescribed H1 performance. A simulation example is provided to show the effectiveness of the present methods. The paper is organized as follows. Section 2 proposes a mathematical model to describe both fault detection and NCSs in a unified framework. In Section 3, an H1 fault detection filter is designed with consideration of limited network QoS. Numerical example is given in Section 4 to demonstrate the effectiveness of the proposed method. Section 5 concludes the paper. Notation: Throughout the paper, N stands for positive integers, Rn denotes the n-dimensional Euclidean space, Rnm is the set of n  m real matrices, I is the identity matrix of appropriate dimensions. The notation X > 0 (respectively, X P 0), for X 2 Rnn means that the matrix X is a real symmetric positive definite (respectively, positive semi-definite). 2. Problem statement Consider the controlled process is a linear time-invariant system



_ xðtÞ ¼ AxðtÞ þ BuðtÞ þ Ef ðtÞ þ D1 xðtÞ yðtÞ ¼ CxðtÞ þ D2 xðtÞ

;

ð1Þ

where xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input vector, yðtÞ 2 Rr is the measurement output, the unknown input xðtÞ 2 Rd includes model uncertainties and external plant disturbances, and xðtÞ belongs to L2 2 ½0; 1Þ. f ðtÞ 2 Rq is the fault to be detected. A; B; C; E; D1 and D2 are constant matrices with appropriate dimensions, C is of full row rank. The following assumptions, which are common in NCS research reported in the open literature [26,29,30], are also made in this work: (1) The sensors are clock-driven, the controller and actuators are event-driven; (2) Data, either from measurement or for control, is transmitted with a single-packet, and full state variables are available for measurements; (3) The real input uðtÞ, realized through a zero-order hold, is a piecewise constant function. For example, the uðtÞ is keep constant in zero-order holding period in Fig. 1. (4) In case of non-ordered sequences, the time stamping technique is applied to choose the latest message. Therefore, the control signal is chosen as the solid line in Fig. 1.

Fig. 1. Time diagram for data transmission.

C. Peng et al. / Applied Mathematical Modelling 34 (2010) 1653–1661

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It is worth mentioning that the sampling period of a sensor is prescribed for control algorithm design, and thus the sensor can be assumed to be clock-driven. However, an actuator does not change its output to the plant under control until an updated control signal is received, implying that the actuator is event-driven. From the above assumptions, using a similar modeling technique in [26,29,30], we model the real system (1) controlled over network as

8 _ ¼ AxðtÞ þ BuðtÞ þ Ef ðtÞ þ D1 xðtÞ; t 2 ½ik T þ sik ; ikþ1 T þ sikþ1 Þ > < xðtÞ yðtÞ ¼ CxðtÞ þ D2 xðtÞ ; >   : þ uðt Þ ¼ uðt  sik Þ; t 2 ik T þ sik ; k ¼ 1; 2; . . .

ð2Þ

where uðt þ Þ ¼ lim^t!tþ0 uð^tÞ, T denotes the sampling period, ik ðk ¼ 1; 2; 3; . . .Þ are some integers such that fi1 ; i2 ; i3 ; . . .g  f0; 1; 2; 3; . . .g. sik is the time from the instant ik T when sensors sample from the plant to the instant when actuators send control actions to the plant. Here, we have assumed that the control computation and other overhead delays are included in sik . Simultaneously, we choose the present arrive actuator instant ik T þ sik and next arrive actuator instant 1 Þ; t0 P 0. Then the system (2) ikþ1 T þ sikþ1 as duration of zero-order hold, Obviously, [1 k¼1 ½ik T þ sik ; ikþ1 T þ sikþ1  ¼ ½ t 0 ; can be rewritten as



_ xðtÞ ¼ AxðtÞ þ Buðik TÞ þ Ef ðtÞ þ D1 xðtÞ;

t 2 ½ik T þ sik ; ikþ1 T þ sikþ1 Þ

yðtÞ ¼ CxðtÞ þ D2 xðtÞ

ð3Þ

:

Since uðik TÞ ¼ u½t  ðt  ik TÞ, define gðtÞ ¼ t  ik T; k ¼ 1; 2; 3; . . . in every internal ½ik T þ sik , ikþ1 T þ sikþ1 Þ. Then Eq. (3) becomes



_ xðtÞ ¼ AxðtÞ þ Buðt  gðtÞÞ þ Ef ðtÞ þ D1 xðtÞ; yðtÞ ¼ CxðtÞ þ D2 xðtÞ

t 2 ½ik T þ sik ; ikþ1 T þ sikþ1 Þ

:

ð4Þ

From the definition of gðtÞ, it can be seen that gðtÞ is discontinuous at the points ik T þ sik ; k ¼ 1; 2; 3; . . .. If ikþ1  ik and sik have upper bound, then gðtÞ is a bounded piecewise continuous function. Therefore, in some sense, the NCS (3) is equivalent to a linear system with a bounded time-varying delay. Remark 1. In (2), fi1 ; i2 ; i3 ; . . .g is a subset of f0; 1; 2; 3; . . .g. Moreover, it is not required that ikþ1 > ik . If jðikþ1  ik Þj ¼ 1 it means no data packet dropout in the transmission. If ikþ1 > ik þ 1, it means that some data packet dropout and no nonordered sequence occurs. If ikþ1 < ik þ 1, it means that non-ordered sequence occurs, which includes sk ¼ s0 and sk < h as the special cases. It is shown in Fig. 1. Therefore, (2) can be viewed as a general form of NCS, where the effect of the networked-induced delay, data packet dropout and non-ordered sequence are simultaneously considered [26,30]. From the above definition of gðtÞ, we have gðtÞ 6 sup½ðikþ1  ik ÞT þ sikþ1  6 hT (h is a positive integer). Then similar to the discrete operation in Hu and Zhu [23], integration of (4) over a sampling interval [kT; ðk þ 1ÞT) yields

xkþ1 ¼ As xk þ

h X

Bki uki þ Efk þ D1 xk ;

ð5Þ

i¼0

R tk RT RT where As ¼ eAT , Bki ¼ tki eAðTsÞ dsB, tk1 ¼ 0; tk0 ¼ T, E ¼ 0 eAðTtÞ Edt, D1 ¼ 0 eAðTtÞ D1 dt. i1 Assume that the filter is embedded in the system through the communication network. For simplify, which is said as network-based filter. In NCS, since the signal transmitted over communication network, there have the number of i, i 2 f0; 1; . . . ; hg possible network-induced delay. Therefore, the system output yik obtained in network-based filter can be expressed as

yik ¼

h X

dðgk ; iÞCxki þ D2 xk ;

ð6Þ

i¼0

where dð; Þ stands for the Kronecker delta, i.e.

dðj; iÞ ¼



0

if j – i

1 if j ¼ i

:

ð7Þ

gk is a stochastic variable whose role is to determine the size of the occurred gðtÞ at time k. Similar to [11,12,23], we assume that ½gk  is a discrete-time homogeneous Markov chain taking values in the finite state space N :¼ f0; 1; . . . ; hg and the staP tionary transition probability matrix K ¼ ½kij , where kij :¼ Probfgkþ1 ¼ j j gk ¼ i:g and 0 6 kij 6 1, m j¼0 kij ¼ 1. T For the convenient development, setting the augmented state vector of the plant be zk ¼ ½xk ; . . . ; xTkh ; uTk1 ; . . . ; uTkh T , and combining Eqs. (5) and (6) into a unified framework, we obtain (

e k þ Bu e kþe e 1 xk zkþ1 ¼ Az Efk þ D ; e i zk þ D 2 xk yi ¼ C k

ð8Þ

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C. Peng et al. / Applied Mathematical Modelling 34 (2010) 1653–1661

where

2

½As ; 0nn ; . . . ; 0nn  |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} 6 6 h 62 3 6 Inn    0 0nn 6 66 7 . . .. 66 7 . . 0nn 5 640 6 6 0    Inn 0nn hðhþ1Þ e ¼6 A 6 6 ½0mn ; . . . ; 0mn  6 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 6 hþ1 62 3 6 6 0mn    0 66 7 66 . . .. 7 640 . . 5 4 0    0mn ðh1Þðhþ1Þ 2

Bk 6 0 6 0hnm e¼6 B 60 4 mm

0ðh1Þmm

3

3

2

E 6 6 0hnm e E¼6 60 4 mm

7 7 7; 7 5

0ðh1Þmm

7 7 7; 7 5

3

½Bk1 ; . . . ; Bkh  2

3

0nm 6 6 40 0

 .. . 

½ 0mm

   0 hh

2

Imm

6 6 40 0

0 7 .. 7 . 5 0nm hh

3

 0 . . .. . .

7 7 5

   Imm

7 7 7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 7 5

ðh1Þh

3

2

D1 6 6 0hnm e1 ¼ 6 D 60 4 mm

0ðh1Þmm

7 7 7; 7 5

  mm ; . . . ; 0mm  e i ¼ ½dðgk ; 0ÞC; . . . ; dðgk ; iÞC; . . . ; dðgk ; hÞC; ½0 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} C : h

The main purpose of this paper is to investigate the problem of fault detection for NCSs with non-ideal network QoS, which includes the residual generation and a residual evaluation stage. Now, we are in a position to achieve this objective coupled with the above NCS augmented model (8). The network-based filter of the following Luenberger observer form is sought:

(

 e ^zk þ Bu e k þ K i yi  y ^zkþ1 ¼ A ^k k ; e i ^zk ^k ¼ C y

ð9Þ

where ^zk is the augmented state estimation vector, K i is the filter’s gain to be designed. ^k Þ, where S is a suitable weighting matrix designed Set the filter error be ek ¼ zk  ^zk , and the residual vector ek be Sðyik  y to assure isolability properties. Then the overall fault detection dynamics is governed by the following Markov jumping systems (MJSs):

(

ekþ1 ¼ Ai ek þ Bi v k

ek ¼ C i ek þ Dv k

ð10Þ

;

where

ei þ Ki C e i Þ; Ai ¼ ð A e i; Ci ¼ SC

h i e 1; e Bi ¼ K i D2 þ D E ;

D ¼ ½ SD2

0 ;

v k ¼ ½ xk

T

fk  :

In order to reduce the false alarm rate, we also use threshold selector to evaluate the residual for the residual generator. The following 2-norm of residual signal is chosen as residual evaluation function as that in [3].

8( )1=2 9 L < X = T ; J th ¼ sup E es es ; x2l2 ;f ¼0 : s¼0

ð11Þ

where L denotes the maximum time step of the evaluation function. Then, the decision is made based on the following rule:



JðkÞ > J th ) A fault is detected JðkÞ 6 Jth ) No faults

:

ð12Þ

Remark 2. In this paper, we only consider the observer-based network fault detection filter design problem, and assume that the control input signal uk in (9) can be directly obtained from the control input in (8). However, if the control signal uk Ph is transmitted over network, then the uk should be replaced by i¼0 dðgk ; iÞuki in (9) due to the negative effect of the integrated delay gk .

C. Peng et al. / Applied Mathematical Modelling 34 (2010) 1653–1661

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After the above manipulations, we are now attempting to formulate some practically computable criteria to obtain the filter gain K i described by (9). The following definition is useful in deriving the criteria. Definition 1. For the given filter gain matrix K i , system (10) is said to be mean-square stable with an H1 norm bound c, if the following hold: (1) For system (10) with

( 1 X

E

v k  0, is said to be mean-square stable if the following holds [28]:

) eTk ek

! 0;

as k ! 1

ð13Þ

k¼0

for any initial condition e0 and initial distribution g0 2 N. (2) For a given scalar c > 0, the following inequality holds for any non-zero

vk

( ) ( ) 1 1 X X T 2 T E ek ek 6 c E vk vk : k¼0

ð14Þ

k¼0

Remark 3. Based on performance index (14), the design of fault detection filter (9) is formulated as an H1 -filtering problem, which can be solved by using optimization technique, and will be shown in the following section.

3. Main results To finish the filter design based on the MJSs model (10), we first introduce a lemma concerning the norm of discrete-time Markovian jump linear system [27,28]. Lemma 1. Given a scalar c > 0 and a stochastic system



xkþ1 ¼ Ai xk þ Bi xk ; zkþ1 ¼ C i xk þ Di xk

ð15Þ

where

Ai ¼ AðhðkÞÞjhðkÞ¼i ; Bi ¼ BðhðkÞÞjhðkÞ¼i ;

C i ¼ CðhðkÞÞjhðkÞ¼i ; Di ¼ DðhðkÞÞjhðkÞ¼i ; and hðkÞ is a discrete-time homogeneous markov chain with finite state space N ¼ f0; 1; . . . ; hg and stationary transition probability matrix K ¼ ½kij , where kij :¼ Probfkðk þ 1Þ ¼ j j kðkÞ ¼ i:g. Then the following conditions are equivalent: (1) System (15) is internally mean-square stable and its H1 norm defined by

sup xl2 ;x–0;xð0Þ¼0

E½kzk2  kxk2

ð16Þ

is less than c. (2) There exist matrices Pi and Gi satisfying the following LMIs:

2

Pi

6 6 G i Ai 6 6 40 Ci

3

ATi GTi

0

Gi þ GTi  P i

Gi Bi

BTi GTi

cI

7 0 7 7 > 0; 7 DTi 5

0

Di

cI

C Ti

where

Pi ¼

h X

kij Pj ; i ¼ 0; . . . ; h:

j¼0

Based on above Lemma 1 and the MJSs model (10), we have the following result to determine the filter gain K i .

ð17Þ

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C. Peng et al. / Applied Mathematical Modelling 34 (2010) 1653–1661

Theorem 1. Given scalars S, c > 0, if there exist symmetric positive definite matrices P i , Gi > 0 with appropriate dimensions, matrices Li with appropriate dimensions, such that the following LMIs hold

2

e i þ Li C e i ÞT ðGi A

i P

0

0

6 e i þ Li C e i Þ Gi þ GT  P i e 1 Þ Gi e 6 ðGi A ðLi D2 þ Gi D E i 6 6 T T T T e 60 0 ðD2 Li þ D 1 Gi Þ cI 6 6 e 40 E T GTi 0 cI Ci

0

SD2

3

C Ti

7 7 7 T T 7 > 0; D2 S 7 7 7 5 0 0

ð18Þ

cI

0

where

Pi ¼

h X

kij Pj ;

i ¼ 0; . . . ; h:

ð19Þ

j¼0

ei, C e i, e e 1 , D2 and C i are defined in (8) and (10), then system (10) is mean-square stable with its H1 norm being less than c and A E, D L K i ¼ G1 i. i Proof 1. Define Li ¼ Gi K i , then based on Lemma 1 and MJS model (10), it is direct to derive the Eq. (18), the detailed process is omitted here. This completes the proof. h So far the filter has been designed to satisfy the Definition 1. Also, the results in Theorem 1 suggest some other types of optimization problems to obtain the minimum noise attenuation level bound c. Therefore, an optimal fault detection filter can be readily found by solving the following optimization problem: (P1) The optimal H1 filter design problem

Minimize : c;

s:t: ð18Þ and ð19Þ:

ð20Þ

Remark 4. Theorem 1 is derived based on Lemma 1, the main advantage of Theorem 1 is that it transfers the non-ideal network QoS to an integrated delay bound and utilizes the MJSs model to couple the fault detection of NCSs and non-ideal network QoS into a unified framework. Otherwise, the computational complexity of Theorem 1 mainly depends on the range of the finite state space N :¼ f0; 1; . . . ; hg of the Markov chain, which can be solved using the efficient LMI toolbox/Matlab. 4. Numerical example Consider the system (1) with the following parameters

 A¼

0

1

3 4

C ¼ ½ 1 1 ;

 ;

B¼

  2

D2 ¼ 0:1;

1

 ;

D1 ¼

0:1 0:05

 ð21Þ

;

E ¼ B:

ð22Þ

Assume that the sampling period of the NCS is 0:01 s, the state space of the Markov chain is N :¼ f 0 culation, the Markovian jump system model is obtained as (10) with the following parameters.

2

0:9999

6 0:0294 6 6 61 6 60 6 e A¼6 60 6 60 6 6 40 0

3

0:0098 0

0

0 0 0:0199 0:0198

0:9606 0

0

0 0 0:0105

0 1

0 0

0 0

0 0 0 0 0 0

0

1 0

0 0 0

0

0

1 0 0 0

0

0

0

0 0 0

0:0115 7 7 7 7 0 7 7 0 7 7; 7 0 7 7 0 7 7 5 0

0

0

0

0 0 1

0

e ¼ ½ 0:0201 0:0095 0 0 0 0 1 0 T ; B e E ¼ ½ 0:02 0:0095 0 0 0 0 0 0 T ; e 1 ¼ ½ 0:001 0:0005 0 0 0 0 0 0 T ; D e 0 ¼ ½ 1 1 0 0 0 0 0 0 ; C

1

2 g. By some cal-

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C. Peng et al. / Applied Mathematical Modelling 34 (2010) 1653–1661

e 1 ¼ ½ 0 0 1 1 0 0 0 0 ; C e 2 ¼ ½ 0 0 0 0 1 1 0 0 : C The transition probability matrix is given by [12]

2

k0;0

6 K :¼ 4 k1;0 k2;0

k0;1 k1;1 k2;1

k0;2

3



0:5 0:4 0:1

3

7 6 7 k1;2 5 ¼ 4 0:2 0:6 0:2 5; k2;2 0:2 0:3 0:5

the initial model is set to be

fk ¼

2

ð23Þ

s0 ¼ 0. The fault signal fk is given as

1 þ 0:5 cosð4pkÞ for k ¼ 100; 101; . . . ; 200 ; 0 others

ð24Þ

and the unknown input xk is supposed to be a random uniformly distributed over ½ 0:5

0:5 .

3 2.5 2

τ(k)

1.5 1 0.5 0 −0.5 −1

0

100

200

300

400

500

Time step k Fig. 2. Measurement mode over network.

0.15

x 1 x 2

0.1

Filter error e(k)

0.05

0

−0.05

−0.1

−0.15

−0.2

0

50

100

150

200

250

300

Time step k Fig. 3. Filter error signal eðkÞ.

350

400

450

500

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C. Peng et al. / Applied Mathematical Modelling 34 (2010) 1653–1661

With the given parameters and based on the optimal H1 filter design problem (P1), we obtain the minimum noise attenuation level bound copt ¼ 1:177 of the fault detection dynamics (10), and the parameters of the fault detection filter in different modes are given by

2

0:2105

3

6 0:0626 7 7 6 7 6 6 0:2026 7 7 6 6 0:0676 7 7 6 K0 ¼ 6 7; 6 0:1601 7 7 6 6 0:0546 7 7 6 7 6 4 0:0000 5

2

0:2081

3

6 0:0602 7 7 6 7 6 6 0:2001 7 7 6 6 0:0649 7 7 6 K1 ¼ 6 7; 6 0:1975 7 7 6 6 0:0728 7 7 6 7 6 4 0:0000 5

0:0000

2

0:1648

3

6 0:0418 7 7 6 7 6 6 0:1566 7 7 6 6 0:0443 7 7 6 K2 ¼ 6 7: 6 0:1605 7 7 6 6 0:0532 7 7 6 7 6 4 0:0000 5

0:0000

ð25Þ

0:0000

Now, we consider the time-domain simulation using the obtained fault detection filter. Fig. 2 shows the measurement mode with random delays satisfying Markov transition probability matrix described by (23). gk ¼ 0; 1; 2 means that the measurement transmitted over the network ideally and with one-step and two-step delay, respectively. Fig. 3 shows the filter error described by (10). From Fig. 3, it can be seen that the error dynamic system (10) 1

0.5

Residual vector ε(k)

0

−0.5

−1

−1.5

−2

−2.5

0

50

100

150

200

250

300

350

400

450

500

Time step k Fig. 4. Residual signal

ek .

18 16 14 12

Jk

10 8 6 4 2 0

0

100

200

300

400

Time step k Fig. 5. Evolution of JðkÞ.

500

600

C. Peng et al. / Applied Mathematical Modelling 34 (2010) 1653–1661

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trends to stable under the external impact of fault signal fk and disturbance signal xk . Fig. 4 shows the generated residual nP o1=2 k T is presented in Fig. 5. We select a threshold as signal ek , and the evolution of JðkÞ ¼ s¼0 e ðsÞeðsÞ nP o1=2 300 T , after 500 times simulations, we obtain an average value J th ¼ 3:9304. From Fig. 5, it can be J th ¼ supf ¼0 E s¼0 e ðsÞeðsÞ shown that 3:7901 ¼ Jð108Þ < J th < Jð109Þ ¼ 4:2241. Therefore, the fault can be detected in nine time steps after its occurrence. 5. Conclusion This paper is concerned with the problem of the fault detection for linear time-invariant systems over data networks with limited network QoS. A new Markovian jumping system (MJS) model has been developed to couple observer-based fault detection filter with our recent idea for network modeling [24,26] in a unified framework. With the developed model and using the bounded real lemma (BRL) for MJSs [27,28], an H1 fault detection filter is established in terms of linear matrix inequalities (LMIs) to guarantee the convergent error dynamics if there is no fault in the system, and to make the effect of disturbances on the residual satisfying a pre-described H1 performance. Numerical example has been given to demonstrate the effectiveness of the proposed method. Acknowledgment The authors acknowledge Natural Science Foundation of China (NSFC) for its support under Grant Nos. 60704024, 60774060 and 60834002. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

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