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H∞ fault detection for nonlinear networked systems with multiple channels data transmission pattern
Information Sciences 221 (2013) 534–543
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H1 fault detection for nonlinear networked systems with multiple channels data transmission pattern Yong Zhang a,⇑, Zhenxing Liu a, Huajing Fang b, Huabin Chen c a
School of Information Science and Engineering, Wuhan University of Science and Technology, Wuhan 430081, China Department of Control Science and Engineering, Hashing University of Science and Technology, Wuhan 430074, China c Department of Mathematics, School of Science, Nanchang University, Nanchang 330031, China b
a r t i c l e
i n f o
Article history: Received 17 August 2011 Received in revised form 17 September 2012 Accepted 21 September 2012 Available online 29 September 2012 Keywords: Fault detection Networked control systems Markovian jump systems Partly unknown transition probability
a b s t r a c t This paper is concerned with fault detection for a class of complex networked control systems. By introducing transmission matrix, nonlinear delayed Markovian jump systems model with partially unknown transition probabilities are established by multiple channels data transmission framework. Based on the obtained model, mode-dependent fault detection filters are used for residual generator, the addressed fault detection problem is converted into nonlinear H1 attenuation problem. Then the desired mode-dependent fault detection filters are constructed in terms of linear matrix inequalities such that the fault detection systems are stochastically stable with H1 attenuation level. A numerical example is given to demonstrate the effectiveness of the proposed design approach. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The past decade have witnessed an ever increasing research interest in networked control systems (NCSs) due to their advantages in many aspects such as low cost, simple installation and maintenance, increased system agility, reduced system wiring, as well as high reliability [2,12,15,18,20,21,26,30]. For NCSs, it is well known that sensors or data packet are connected to fault detection station via communication medium with limited capacity. Therefore, network-induced delay, data missing (also called packet dropout or missing measurement) have inevitably emerged, which all might be potential sources to poor performance and instability, even fault. Consequently, it is not surprising that, in the past few years, fault detection problem of networked control systems with communication delays and/or missing measurements have been extensively considered by many researchers [3,7,8,11,27,28]. On the other hand, it should be pointed out that there are basically several approaches for the modeling of multiple packets transmission [6,16,23] or medium access constraint [5,10,13] about networked communication systems. An arguably popular approach is to model multiple packets transmission as switching systems [23]. The diagonal matrix approach in which every elements obey Bernoulli distributed white sequence taking on values of zero and one with certain probability [6]. Communication sequence approach [5,10,24] is the third one which models medium access constraint. Moreover, Markovian jump systems model has also been employed to represent multiple packets transmission networked control systems [16]. In almost all the literature mentioned about Markov jump systems (MJSs), the assumption of the transition probabilities has been made completely accessible [16,22,29]. Unfortunately, in many practical applications, this ideal assumption would inevitably limit the application of established results because of the difficulty and cost in obtaining precisely all the transition probabilities [7,19]. Very recently, some initial results have been obtained in [25] for Markovian ⇑ Corresponding author. E-mail address: [email protected] (Y. Zhang). 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.09.026
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Y. Zhang et al. / Information Sciences 221 (2013) 534–543
jump systems with partially unknown transition probabilities. However, so far, the fault detection for networked systems with multiple packets transmission or medium access constraint in a unified framework has not been fully investigated, which motivates us to shorten such a gap in the present investigation. In this paper, we aim to deal with the fault detection filter design problem for multiple channels data transmission NCSs. By adopting transmission matrix pattern, the interactivity of different channels in adjacent sampling period can be described by Markov chain, and nonlinear Markovian jump systems model with partly unknown transition probabilities is established. Based on the novel model, by utilizing mode-dependent fault detection filter as residual generator, fault detection of nonlinear multiple channels data transmission NCSs is formulated into nonlinear H1 attenuation problem. Furthermore, by applying stochastic Lyapunov–Krasovskii functional method, delay-interval-dependent and Markov-chain-elementdependent sufficient conditions are established in terms of certain linear matrix inequalities (LMIs)[1], and mode-dependent observer gain matrices and residual weighting matrices are characterized if these LMIs are feasible. Finally, simulation examples are presented to demonstrate the effectiveness of the proposed method. Notation: Throughout the paper, the superscript ‘1’ and ‘T’ stand for the inverse and transpose of a matrix, respectively; Rn denotes the n-dimensional Euclidean space and kk refers to the Euclidean norm for vectors. P > 0(P0) means P is a real symmetric positive definite (semi-definite) matrix. E{x} is the expectation of the stochastic variable x. Prob{} means the occurrence probability of event ‘‘’’. I and 0 represent identity matrix and zero matrix with appropriate dimensions in different place. In symmetric block matrices or complex matrix expressions, we utilize asterisk () to represent a term that is induced by symmetry and diag { } stands for a block diagonal matrix. 2. Problem formulation Consider a class of nonlinear networked systems described by:
xðk þ 1Þ ¼ AxðkÞ þ N1 gðxðkÞÞ þ B1 dðkÞ þ B2 f ðkÞ
ð1Þ
yðkÞ ¼ CxðkÞ
where xðkÞ 2 Rn is the state vector, yðkÞ 2 Rp is the output of the plant. dðkÞ 2 Rs and f ðkÞ 2 Rt are the disturbance input and fault signal, respectively, which belong to ‘2[0, 1). A, B1, B2, N1 and C are known real constant matrices with the appropriate dimensions. The nonlinear function g() satisfies g(0) = 0 and the following sector-bounded condition [17]:
½gðxÞ gðyÞ S1 ðx yÞT ½gðxÞ gðyÞ S2 ðx yÞ 6 0
ð2Þ
where S1, S2 2 Rnn are known real constant matrices, S = S1 S2 is symmetric positive definite matrix. Dinary-valued function rl ðkÞ : Z ! f0; 1g ðl ¼ 1; . . . ; jÞ is used to describe the lth channels transmission status in time k, where 1 means successful data transmission and 0 means data loss. Specifically, the lth channels output yl(k) is available to fault detection filter (FDF) only corresponding channels are accessible, i.e. rl(k) = 1, then, network-induced delay sl(k) inevitably exist and yl (k sl(k)) will be obtained by the FDF. Otherwise, when rl(k) = 0, the output of lth channels will be zero l ðkÞ as the lth channels signal received by the FDF, we and yl(k) will be ignored owing to its being unavailable. If we regard y can present the transmission dynamics of lth channels as:
l ðkÞ ¼ rl ðkÞyl ðk sl ðkÞÞ ðl ¼ 1; . . . ; pÞ y
ð3Þ
¼ max16l6p sl ðkÞ are the delay low and upper bound, respectively. where s = min16l6psl(k) and s Remark 1. For networked systems, at each sampling time k, due to the bandwidth constraint of the wireless/wire network, only some of these sensors or packets signal can be transmitted successfully. In this note, if we regard every sensors or packets as one information channel, then multi-input and multi-output [9], medium access constraint [5,10,13] and multiple packets transmission [6,16,23] can be described validly by multiple channels transmission framework. On the other hand, modeling is one of the most important issues in networked control systems which has received an ever increasing research attention [4–10,14,16,23,24]. For practical networked control systems, network-induced delay and packet-dropout often occurs simultaneously due to limited channel capacity and noise. Actually, pattern (3) describe the NCSs with data loss and time-varying delay simultaneously, so it is more comprehensive than separate consideration of delay pattern [9] or data loss pattern [6,16,23]. n o 1;1 1;p p1;1 To discuss validly above problem, we take transmission matrix as Mp , M 0;1 , ; . . . ; M p1;p ; M p;1 p ; Mp ; . . . ; Mp ; . . . ; Mp p p and matrix Ms;t p ¼ diagfr1 ðkÞ; . . . ; rp ðkÞg is expressed by the following form: 1;1 M0;1 0; . . . ; 0g; . . . ; M1;p . . . ; 0; 1g; . . . ; M p1;1 , diagf1; . . . ; 1; 0g; p , diagf0; . . . ; 0g; M p , diagf1; p , diagf0; p |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} p1
. . . ; Mp1;p p
,
diagf0; 1; . . . ; 1g; Mp;1 p |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} p1
p1
, diagf1; 1; . . . ; 1g |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
Then we represent the whole dynamics model as
p
p1
ð4Þ
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Y. Zhang et al. / Information Sciences 221 (2013) 534–543
ðkÞ ¼ M PlðkÞ yðk sðkÞÞ y
ð5Þ
h
iT h iT ðkÞ ¼ y T1 ðkÞ; . . . ; y Tp ðkÞ ; M plðkÞ 2 M p ; lðkÞ is a Markov chain taking valwhere yðk sðkÞÞ ¼ yT1 ðk s1 ðkÞÞ; . . . ; yTp ðk sp ðkÞÞ ; y ues in a finite state space R ¼ f1; 2; . . . ; 2p g with transition probability matrix K = [qij] given by
qij ¼ Probðlðk þ 1Þ ¼ jjlðkÞ ¼ iÞ ð8i; j 2 RÞ
ð6Þ
where qij P 0ði; j 2 RÞ is the transition probability from mode i to j and
P2p
j¼1
qij ¼ 1ð8i 2 RÞ.
Remark 2. Recently, stochastic or deterministic model has been introduced to describe data loss phenomena for networked control systems. Usually, Bernoulli distributed model is arguably the most popular stochastic pattern in which 0 stands for entire missing and 1 denotes the intactness. However, due to the communication constraint of multiple channels network, the transmission status of different channels in adjacent sampling period may interact. Actually, finite-state Markov chains can be used to model this correlated pattern validly. Therefore, Markovian jump model (5) reveal the characteristic of communication network more effective than [5,6,9,10,13,16,23]. Especially, if we regard Bernoulli random variable as special case of Markov chains, then, Bernoulli-distributed model of multiple packets transmission[6] is the special case of Markovian jump model (5). In this paper, observer-based fault detection filters are constructed as residual generator:
8 ðkÞ y ^ðkÞÞ > < ^xðk þ 1Þ ¼ A^xðkÞ þ LlðkÞ ðy ^ðkÞ ¼ C ^xðkÞ y > : ðkÞ y ^ðkÞÞ rðkÞ ¼ V lðkÞ ðy
ð7Þ
^ðkÞ 2 Rq represent the state and output estimation vectors, respectively, r(k) is the residual signal. Obwhere ^ xðkÞ 2 Rn and y server gain matrices Ll(k) and residual weighting matrices Vl(k) are filter parameters to be determined, which is assumed to jump synchronously with the modes in (5) and is hereby mode-dependent. Let the estimation error be e1 ðkÞ ¼ ½xðkÞ ^ xðkÞ, then the error systems can be obtained by substituting (5) and (7) into (1) lðkÞ
e1 ðk þ 1Þ ¼ ðA LlðkÞ CÞe1 ðkÞ LlðkÞ CxðkÞ LlðkÞ MP Cxðk sðkÞÞ þ N1 gðxðkÞÞ þ B1 dðkÞ þ B2 f ðkÞ
ð8Þ
T By setting re(k) = r(k) f(k), eðkÞ ¼ xT ðkÞ; eT1 ðkÞ , w(k) = [dT(k),fT(k)]T and combining (1) and (7), we have the following augmented error systems
8 > eðk þ 1Þ ¼ AlðkÞ eðkÞ þ A1lðkÞ Heðk sðkÞÞ þ NHgðeðkÞÞ þ BwðkÞ > > <
ð9Þ
r e ðkÞ ¼ C lðkÞ eðkÞ þ C 1lðkÞ Heðk sðkÞÞ þ DwðkÞ > > > : ; s þ 1; . . . ; 0 eðkÞ ¼ /ðkÞ; lðkÞ ¼ /ðkÞ; k ¼ s where
" AlðkÞ ¼
A
0
LlðkÞ C
A LlðkÞ C
#
" ;
A1lðkÞ ¼
"
#
0 lðkÞ
LlðkÞ M P C
;
B¼
B1
B2
B1
B2
# ;
" N¼
N1
#
N1
;
lðkÞ
C lðkÞ ¼ ½V lðkÞ C; V lðkÞ C, C 1lðkÞ ¼ V lðkÞ M P C, D ¼ ½0; I; H ¼ ½I; 0; /ðkÞ and u(k) are the given initial condition of e(k) and l(k), respectively, re (k) is the residual error which contains information on both the time and location of the occurrence of fault. Definition 1. [19]. Augmented error system (9) is said to be stochastically stable for w(k) = 0 and every initial condition / (k)(u(k)), such that
! 1 X 2 E keðkÞk j/ðkÞ; uðkÞ < 1 k¼0
The purpose of this paper is to design the mode-dependent observer-based fault detection parameters Ll(k) and Vl(k) such that the following requirements are satisfied simultaneously: (a) The zero-solution of augmented error system (9) with w(k) = 0 is stochastically stable. (b) Under the zero-initial condition, the residual estimation error re(k) satisfies 1 1 X X Efkr e ðkÞk2 g 6 c2 kwðkÞk2 k¼0
k¼0
for all nonzero w(k), where c > 0 is a given disturbance attenuation level.
ð10Þ
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Y. Zhang et al. / Information Sciences 221 (2013) 534–543
In this paper, threshold Jth and residual evolution function J(re) are selected as:
( Jðr e Þ ¼ E
)1=2
k¼k 0 þj X
r Te ðkÞr e ðkÞ
;
J th ¼ supdðkÞ2‘2 ;f ðkÞ¼0 Jðr e Þ:
k¼k0
where k0 denotes the initial evaluation time instant, j denotes the evaluation time steps. Once the evaluation function J(re) has been selected, we are able to determine the threshold Jth. Based on the threshold, the occurrence of fault can be detected by comparing Jth and J(re) according to the following test:
Jðre Þ P J th ) alarm for fault
ð11Þ
Jðre Þ < J th ) no fault 3. Main results
The following theorem provides a sufficient condition under which the augmented error system (9) is stochastically stable and the residual estimation error re(k) satisfies the H1 criterion (10) under zero-initial condition. In addition, the transition probabilities of jumping process {l(k), k P 0} in this paper are assumed to be partly accessed, i.e. some elements are unknown. For notation clarity, we denote R ¼ RiK þ RiUK ð8i 2 RÞ with RiK , fj : qij is knowng and RiUK , fj : qij is unknowng. and c > 0, augmented filtering error system (9) with partly unknown transition probabilities Theorem 1. For given scalars 0 6 s 6 s is stochastically stable with H1 attenuation level (10) if there exist matrices P i > 0ði 2 RÞ,Q > 0 and scalar e > 0 such that
2 6 6 6 4 2
qiK Ni1
Ni1
0 c2 I
0 c2 I
6 6 4
qiK Ri1 Ri2 PiK 0 q
i KI
Ri1 Ri2 Pj 0 I
3
7 0 7 7<0 0 5 P iK 3
0 7 7 < 0; 0 5 Pj
ð12Þ
8j 2 RiUK
ð13Þ
where
3 ^ þ 1ÞQ eS1 H Pi 0 eHT ST2 HT ½ðs ST1 S2 þ ST2 S1 5 ; S1 ¼ ; N ¼4 Q 0 2 e I X qiK , qij ; s s ¼ s^; Ri1 ¼ ½C i ; C 1i ; 0; DT ; Ri2 ¼ ½Ai ; A1i ; N; BT : 2
i 1
S2 ¼
ST1 þ ST2 ; 2
PiK ¼
X
rij Pj ;
j2RiK
j2RiK
Þ and construct Lyapunov functional candidate as Proof. Denote IðkÞ ¼ ½eðkÞ; eðk 1Þ; . . . ; eðk s
VðIðkÞ; kÞ ¼ eT ðkÞPlðkÞ eðkÞ þ
sþ1 X
k1 X
k1 X
eT ðmÞQeðmÞ þ
þ2m¼k1þh h¼s
eT ðmÞQeðmÞ:
m¼ksðkÞ
where PlðkÞ > 0ðlðkÞ 2 RÞ and Qi > 0 are symmetric positive definite matrices. Taking the difference of above functional along the solution of the augmented error system (9), we have
E½DVðIðkÞ; kÞ ¼ EfVðIðk þ 1Þ; k þ 1ÞjIðkÞg VðIðkÞ; kÞ X 6 eT ðk þ 1Þ qij Pj eðk þ 1Þ eT ðkÞPi eðkÞ
ð14Þ
j2R
s þ 1ÞeT ðkÞQ 3 eðkÞ þ ðs
ks X
eT ðmÞQ 3 eðmÞ
þ1 m¼ks
eT ðk sðkÞÞQ 3 eðk sðkÞÞ þ
ks X
eT ðmÞQ 3 eðmÞ
þ1 m¼ks
s þ 1ÞeT ðkÞQ 3 eðkÞ eT ðk sðkÞÞQ 3 eðk sðkÞÞ ¼ ðs
ð15Þ
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Y. Zhang et al. / Information Sciences 221 (2013) 534–543
For any scalar e > 0, it follows readily from (2) that
e
HeðkÞ HgðeðkÞÞ
T "
S1
S2
I
#
HeðkÞ 60 HgðeðkÞÞ
ð16Þ
Denoting f(k) = [eT(k), eT(k s(k))HT, gT(e(k))HT]T and combining (14–16) leads to T
E½DVðIðkÞ; kÞ ¼ e ðk þ 1Þ
2
X
qij Pj eðk þ 1Þ þ f ðkÞ4 T
j2R
2
X
¼ eT ðk þ 1Þ4PiK þ
T
¼ f ðkÞ
8
3
i 2
i
X
qij þ
j2RiK
3
qij 5Ni1 fðkÞ
j2RiUK
2
qij Pj 5eðk þ 1Þ þ fT ðkÞ4qiK þ
j2RiUK i 1
X
qiK N þ N þ
X
qij
9 = fðkÞ N þN ;
j2RiUK
i 1
X
3
qij 5Ni1 fðkÞ
j2RiUK
i 3
ð17Þ
where
2 6 Ni2 ¼ 6 4
ATi PiK Ai
ATi PiK A1
AT1 PiK A1
3
2 T Ai P j Ai 7 6 i and N ¼ 4 AT1 PiK N1 7 3 5 T i N1 PK N1 ATi PiK N1
ATi Pj A1 AT1 Pj A1
ATi Pj N1
3
7 AT1 Pj N1 5: NT1 Pj N1
By Schur complement, (12) and (13) implies qiK Ni1 þ Ni2 < 0 and Ni1 þ Ni3 < 0, then we know from (17) that
h i n h io X E½DVðIðkÞ; kÞ 6 kmin qiK Ni1 Ni2 fT ðkÞfðkÞ qij minj kmin Ni1 þ Ni3 fT ðkÞfðkÞ 6 ðb1 þ b2 ÞfT ðkÞfðkÞ
ð18Þ
j2RiUK
n h io n
n h ioo and b2 ¼ inf 1 qiK minj kmin Ni1 þ Ni3 . From (18), if we do mathematics where b1 ¼ inf kmin qiK Ni1 Ni2 expectations in both sides, we obtain that for any N P 0, N X E½VðIðN þ 1Þ; N þ 1Þ VðIð0Þ; 0Þ 6 ðb1 þ b2 Þ E½fT ðkÞfðkÞ k¼0
which yields 1 X E½fT ðkÞfðkÞ 6 k¼0
1 E½VðIð0Þ; 0Þ < 1 ðb1 þ b2 Þ
Thus, from Definition 1, the augmented error systems (9) with w(k) = 0 is stochastically stable. Note that b1 + b2 will reduce to only b1 (respectively, b2) if all the transition probabilities are known (respectively,unknown). Now, to establish the H1 performance, consider the following performance index:
JN ¼ E
N X
rTe ðkÞre ðkÞ c2 wT ðkÞwðkÞ
ð19Þ
k¼0
under zero initial condition, we have
JN ¼ E
N X
1 X T rTe ðkÞre ðkÞ c2 wT ðkÞwðkÞ þ DVðIðkÞÞ EVðN þ 1Þ 6 E r e ðkÞr e ðkÞ c2 wT ðkÞwðkÞ þ DVðIðkÞÞ
k¼0
k¼0
2 3 1 X
i i X i T i i ¼ w ðkÞ4 qK N1 þ N2 þ qij N1 þ N3 5wðkÞ
ð20Þ
j2RiUK
k¼0
where
" T
T
T
i 1
wðkÞ ¼ ½f ðkÞ; w ðkÞ ;
N ¼ "
Xi3 ¼ DT D c2 I; Ni2 ¼
Ni2 Xi2
Xi3
Ni1 þ Xi1 Xi2 # ;
i 3
X
# ;
h
n
o
Xi1 ¼ diag C Ti C i ; 0; 0 ; iT
Xi2 ¼ BT PiK Ai ; BT PiK A1i ; BT PiK N ;
Xi3 ¼ BT Pj B and Xi2 ¼ ½BT Pj Ai ; BT Pj A1i ; BT Pj NT :
Xi2 ¼ ½DT C i ; DT C 1i ; 0T ; "
Xi3 ¼ BT PiK B;
Ni3 ¼
Ni3 Xi2
Xi3
# ;
539
Y. Zhang et al. / Information Sciences 221 (2013) 534–543
P
Similarly, by Schur complement, (12) and (13) are equivalent to qiK Ni1 þ Ni2 þ j2Ri qij Ni1 þ Ni3 < 0, which means J1 < 0, P1 P1 UK 2 2 2 i.e. k¼0 Efkr e ðkÞk g 6 c h k¼0 kwðkÞk , this completes the proof. Theorem 1 provides delay-interval-dependent and Markov-chain-element-dependent sufficient conditions which guarantee the H1 performance as well as the stochastic stability of the augmented error system (9). The following theorem gives a sufficient conditions on the existence of mode-dependent observer-based fault detection filters, the slack matrix will be constructed with a special structure, which allows us to obtain the solution within strict linear matrix inequalities (LMIs) framework. and c > 0, if there exist matrices Li ; V i ; X i ; Y i ; U i ; P i ¼ P i1 P i2 > 0ði 2 RÞ; Q > 0 and Theorem 2. For given scalars 0 6 s 6 s P i3 scalar e > 0 such that
2
Hi11 Hi12 Hi13
6 6 4
3 7
Hi22 Hi23 7 5<0 H
j ,
ð21Þ
i 33
j1
j2
j3
,
1 X
qiK j2Ri
qij
Pj1
P2j
Pj3
; 8j 2 RiK
ð22Þ
K
j ,
P j1
Pj2
Pj3
; 8j 2 RiUK
ð23Þ
where
2 6
^ þ 1ÞQ P i1 eS1 ðs
Pi2
Pi3
Hi11 ¼ 4 2
i 12
H
eS2 6 ¼4 0
0 0
3
0
H
i 35
2
AT X Ti C T LTi
T T Hi13 ¼ 6 4 C Vi
AT Y Ti C T LTii
C T Mip V Ti
N T1 X Ti
NT1 U Ti
3
C T M ip LTi
0 6 ¼4 0
BT1 X Ti þ BT1 Y Ti
7 BT1 U Ti þ BT1 Y Ti 5;
I
BT2 X Ti þ BT2 Y Ti
BT2 U Ti þ BT2 Y Ti
0 0
i 22
C T V Ti
6
7 0 5; Q
2
7 0 0 5;
2
3
0
i 33
H
2
AT U Ti C T LTi
3
7 AT Y Ti C T LTi 7 5; C T Mip LTi
I 6 ¼4
0 1i X i
0 X Ti
2i Y i U Ti 3i Y i Y Ti
3 7 5
2
and H ¼ diagfeI; c I; c Ig. Then, there exists mode-dependent OBFDF such that the augmented error system (9) is stochastically stable (or stable for any switching sequence if RiK ¼ £, for all i 2 R) with H1 attenuation level (10). Moreover, if LMIs (21) have feasible solution, the desired OBFDF are given by
Li ¼ Y 1 i Li
and V i ¼ V i ;
i2R
ð24Þ
Proof. From Theorem 1, inequalities (12) and (13) can be rewritten as:
2
Ni1
6 6 6 6 4
0
Ri1 Ri2 j
c2 I
0
I
3
7 0 7 7<0 0 7 5 j
ð25Þ
where
j ,
8 <
1
qiK
PiK ; 8j 2 RiK
:P ;
8j 2 RiUK
:
8 9 < = 1 T Performing a congruence transformation to (25) by J ¼ diag I; . . . ; I; I; j Ri , one has :|fflfflffl{zfflfflffl} ; j
2
Ni1
6 6 6 6 4
0 2
c I
0
0
I
0
T Ri 1 j Ri
For an arbitrary matrix Ri ¼
1
q
i K
3
Ri2 RTi
Ri1
PiK Ri
1
q
i K
PiK
1
Xi Ui
1
q
i K
4
7 7 7<0 7 5
ð26Þ
Yi ; 8i 2 R, we have the following fact: Yi
PiK Ri
T
P 0;
T ðPj Ri ÞP1 j ðP j Ri Þ P 0:
540
Y. Zhang et al. / Information Sciences 221 (2013) 534–543 T Then, we obtain j RTi Ri P Ri 1 j Ri , so (26) can rewrite as
2
0
Ri1
Ri2 RTi
c2 I
0
0
I
0
RTi
Ni1
6 6 6 6 4
Now, partition Pi as
X
j ,
j2RiK
rij Pj ¼
j P i1
X
rij
3
Ri
7 7 7<0 7 5
ð27Þ
Pi2 , we acquire Pi3
Pj1
Pj2
Pj3
j2RiK
,
j1
j2
j3
:
j1 j2 Further define matrics variables Li ¼ Y i Li ,V i ¼ V i and j , . Replacing Li ¼ Y i Li , V i ¼ V i into (27), we readily j3 obtain (21). From Theorem 1, the augmented filtering error system (9) will be stochastically stable with H1 attenuation performance. Meanwhile, if the solution of (21) exists, the parameters of admissible OBFDF are given by (24). The proof is completed. h Remark 3. As the two extreme cases, i.e., when all the transition probabilities are known and unknown,the underlying sys tems are the traditional MJLS RiK ¼ R; RiUK ¼ U and the switched systems under arbitrary switching RiUK ¼ R; RiK ¼ U , respectively. Correspondingly, the fault detection results can be found in some existing references, [29] investigated linear discrete-time Markovian jump system (completely accessible), [18] studied linear discrete-time switched systems (completely inaccessible), [7] considered transition probabilities with polytopic uncertainties which require the knowledge of uncertainties structure and it can still be viewed as accessible. Therefore, observer-based fault detection with partly unknown transition probabilities of this paper is a more natural assumption to the Markovian jump systems and hence covers the existing ones.
4. Numerical example and simulation Consider a nonlinear networked system in (1) with following parameters:
0:2 0 0:4 ; B1 ¼ ; 0:3 0:4 0:2 0:2 0 : S2 ¼ 0:1 0:1
A¼
B2 ¼
0 ; 0:6
C¼
0:2 0:1 ; 0 0:4
N1 ¼
0:3 0:1 ; 0:2 0
S1 ¼
0:4 0 ; 0:1 0:3
Attention is focused on the design of mode-dependent OBFDF which stabilize stochastically the augmented error system (9) and satisfy the H1 attenuation level (10) In this paper, we consider two channels data transmission NCSs, according to the transmission pattern presented in Section 2, the transmission matrices are constructed:
M2;1 2 , diagf1; 1g;
M0;1 2 , diagf0; 0g;
M1;1 2 , diagf1; 0g;
M 1;2 2 , diagf0; 1g:
With above data transmission pattern, from Remark 3, we consider the case II with partly unknown transition probabilities in Table 1, where‘?’means that element is unknown. As the special cases of case II, corresponding results of traditional Markovian jump systems (Case I) and switched systems (Case III) can be included in our theorems. For nonlinear multiple ¼ 3, the channels networked systems with data loss and time-varying delay (9), if the delay bound are given as s = 1 and s H1 attenuation performance level is taken as c = 1.2, by applying Theorem 2, the OBFDF Lji and V ji ði ¼ 1; 2; 3; 4; j ¼ 1; 2; 3Þ can be designed as
Table 1 Different transition probabilities matrices. Completely known (Case I)
3 0:1 0:2 0:3 0:4 6 0:1 0:6 0:1 0:2 7 7 6 7 6 4 0:5 0:1 0:2 0:2 5 2
0:6 0:1 0:2 0:1
Partly unknown (Case II)
3 0:1 0:2 0:3 0:4 6 ? 0:6 ? 0:2 7 7 6 7 6 4 0:5 ? ? ? 5 2
0:6 0:1 0:2 0:1
Completely unknown (Case II)
2
? ? ? ?
3
6? ? ? ?7 6 7 6 7 4? ? ? ?5 ? ? ? ?
Y. Zhang et al. / Information Sciences 221 (2013) 534–543
L11 ¼
1:2753 0:3075 0:0838 0:0235 0:4806 0:0140 ; L12 ¼ ; L13 ¼ ; L14 ¼ ; 0:1355 1:3050 0:9045 0:0853 0:5998 1:1232 0:0439
0:4590 0:1112 0:1808
V 11 ¼ ½0:0006 0:0348; L21 ¼
0:2496
V 12 ¼ ½0:2186 0:2592;
V 13 ¼ ½0:0052 0:1871;
V 14 ¼ ½0:4506 0:0067;
1:1352 0:2784 0:0816 0:0011 0:6036 0:0238 ; L22 ¼ ; L23 ¼ ; L24 ¼ ; 0:1816 1:1620 0:9204 0:1095 0:6939 0:8905 0:0618
0:5036 0:1188
V 21 ¼ ½0:0060 0:0467; L31 ¼
541
V 22 ¼ ½0:1974 0:2293;
V 23 ¼ ½0:0144 0:1215;
V 24 ¼ ½0:4845 0:0122;
1:0355 0:2512 0:2071 0:0151 0:5683 0:0334 ; L32 ¼ ; L33 ¼ ; L34 ¼ ; 0:2444 1:0751 0:7929 0:1715 0:5619 0:8880 0:0559
0:4262 0:1014 0:3409
V 31 ¼ ½0:0721 0:0879;
V 32 ¼ ½0:2291 0:2569;
V 33 ¼ ½0:0418 0:1996;
V 34 ¼ ½0:3191 0:0317:
To demonstrate the effectiveness of designed OBFDF, an unknown disturbance d(k) is assumed to be band-limited white noise with power of 0.05. The fault signal f(k) is simulated as a square wave of 0.3 amplitude that occurred from 10 to 30 steps and the nonlinear function is given by g(x(k)) = sin (0.2 x(k)). Under cases I, II and III, the initial state of augmented error system (11) is assumed as e(0) = [0.1 0.2 0.2 0.1]T, corresponding evolution of residual estimation error signal re(k) and residual evaluation function J(re) are shown in Figs. 1–3, respectively. By using 300 Monte Carlo simulations for k0 = 0 and j = 100, we obtain the threshold in Table 2. From Table 2, we observe that when k = 19, 23 and 14, J(re) P Jth for the first time, which mean the fault f(k) can be detected 9, 13 and 4 time steps after its occurrence, respectively. Remark 4. It is clearly observed from the simulation that the augmented error systems (9) is stochastically stable with given H1 performance level, and the fault can also be detected effectively. Especially, it is easily verified that Markovian jump systems model partly unknown transition probabilities is more natural pattern which contain traditional Markovian jump systems (known transition probability) and switched systems (unknown transition probability) as its special case.
5. Conclusion In this paper, the fault detection problem has been addressed for a class of nonlinear networked systems with timevarying delay. Transmission matrix has been introduced to describe the multiple channels data transmission pattern, and Markovian jump system with partly unknown transition probabilities has been used to model the large-scale networked systems. Delay-interval-dependent and Markov-chain-element-dependent sufficient conditions are derived which contain traditional Markovian jump systems (known transition probability) and switched systems (unknown transition probability) as its special case. An illustrative example has been exploited to show the usefulness of the results obtained. The future
Fig. 1. Corresponding simulation of Case I.
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Y. Zhang et al. / Information Sciences 221 (2013) 534–543
Fig. 2. Corresponding simulation of Case II.
Fig. 3. Corresponding simulation of Case III. Table 2 Corresponding threshold and residual evolution function value for different cases.
Threshold jth Residual evolution function Jdr e ðkÞe
Completely known (Case I)
Partly unknown (Case II)
Completely unknown (Case III)
Jth = 1.0119 1.0086 = J(18) < Jth < J(19) = 1.0258
Jth = 1.0983 1.0981 = J(22) < Jth < J(23) = 1.1308
Jth = 0.6945 0.6269 = J(13) < Jth < J(14) = 0.7338
research topics would include the extension of the main results developed in this paper to more general complex systems such as networked systems with quantization, general fractional uncertainties, polynomial nonlinear systems and functional differential equations of the neutral type. Acknowledgments The authors would like to thank the editor and the reviewers for their helpful suggestions to improve the quality of this correspondence. This paper is partially supported by the National Natural Science Foundation of China (Grant Nos.
Y. Zhang et al. / Information Sciences 221 (2013) 534–543
543
61104027, 61174107, 61034006), China Postdoctoral Science Foundation (Grant No. 2011M500899), National Natural Science Foundation of Mathematical Tianyuan Fund (Grant No. 11126278). 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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Contents lists available at SciVerse ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
H1 fault detection for nonlinear networked systems with multiple channels data transmission pattern Yong Zhang a,⇑, Zhenxing Liu a, Huajing Fang b, Huabin Chen c a
School of Information Science and Engineering, Wuhan University of Science and Technology, Wuhan 430081, China Department of Control Science and Engineering, Hashing University of Science and Technology, Wuhan 430074, China c Department of Mathematics, School of Science, Nanchang University, Nanchang 330031, China b
a r t i c l e
i n f o
Article history: Received 17 August 2011 Received in revised form 17 September 2012 Accepted 21 September 2012 Available online 29 September 2012 Keywords: Fault detection Networked control systems Markovian jump systems Partly unknown transition probability
a b s t r a c t This paper is concerned with fault detection for a class of complex networked control systems. By introducing transmission matrix, nonlinear delayed Markovian jump systems model with partially unknown transition probabilities are established by multiple channels data transmission framework. Based on the obtained model, mode-dependent fault detection filters are used for residual generator, the addressed fault detection problem is converted into nonlinear H1 attenuation problem. Then the desired mode-dependent fault detection filters are constructed in terms of linear matrix inequalities such that the fault detection systems are stochastically stable with H1 attenuation level. A numerical example is given to demonstrate the effectiveness of the proposed design approach. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The past decade have witnessed an ever increasing research interest in networked control systems (NCSs) due to their advantages in many aspects such as low cost, simple installation and maintenance, increased system agility, reduced system wiring, as well as high reliability [2,12,15,18,20,21,26,30]. For NCSs, it is well known that sensors or data packet are connected to fault detection station via communication medium with limited capacity. Therefore, network-induced delay, data missing (also called packet dropout or missing measurement) have inevitably emerged, which all might be potential sources to poor performance and instability, even fault. Consequently, it is not surprising that, in the past few years, fault detection problem of networked control systems with communication delays and/or missing measurements have been extensively considered by many researchers [3,7,8,11,27,28]. On the other hand, it should be pointed out that there are basically several approaches for the modeling of multiple packets transmission [6,16,23] or medium access constraint [5,10,13] about networked communication systems. An arguably popular approach is to model multiple packets transmission as switching systems [23]. The diagonal matrix approach in which every elements obey Bernoulli distributed white sequence taking on values of zero and one with certain probability [6]. Communication sequence approach [5,10,24] is the third one which models medium access constraint. Moreover, Markovian jump systems model has also been employed to represent multiple packets transmission networked control systems [16]. In almost all the literature mentioned about Markov jump systems (MJSs), the assumption of the transition probabilities has been made completely accessible [16,22,29]. Unfortunately, in many practical applications, this ideal assumption would inevitably limit the application of established results because of the difficulty and cost in obtaining precisely all the transition probabilities [7,19]. Very recently, some initial results have been obtained in [25] for Markovian ⇑ Corresponding author. E-mail address: [email protected] (Y. Zhang). 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.09.026
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Y. Zhang et al. / Information Sciences 221 (2013) 534–543
jump systems with partially unknown transition probabilities. However, so far, the fault detection for networked systems with multiple packets transmission or medium access constraint in a unified framework has not been fully investigated, which motivates us to shorten such a gap in the present investigation. In this paper, we aim to deal with the fault detection filter design problem for multiple channels data transmission NCSs. By adopting transmission matrix pattern, the interactivity of different channels in adjacent sampling period can be described by Markov chain, and nonlinear Markovian jump systems model with partly unknown transition probabilities is established. Based on the novel model, by utilizing mode-dependent fault detection filter as residual generator, fault detection of nonlinear multiple channels data transmission NCSs is formulated into nonlinear H1 attenuation problem. Furthermore, by applying stochastic Lyapunov–Krasovskii functional method, delay-interval-dependent and Markov-chain-elementdependent sufficient conditions are established in terms of certain linear matrix inequalities (LMIs)[1], and mode-dependent observer gain matrices and residual weighting matrices are characterized if these LMIs are feasible. Finally, simulation examples are presented to demonstrate the effectiveness of the proposed method. Notation: Throughout the paper, the superscript ‘1’ and ‘T’ stand for the inverse and transpose of a matrix, respectively; Rn denotes the n-dimensional Euclidean space and kk refers to the Euclidean norm for vectors. P > 0(P0) means P is a real symmetric positive definite (semi-definite) matrix. E{x} is the expectation of the stochastic variable x. Prob{} means the occurrence probability of event ‘‘’’. I and 0 represent identity matrix and zero matrix with appropriate dimensions in different place. In symmetric block matrices or complex matrix expressions, we utilize asterisk () to represent a term that is induced by symmetry and diag { } stands for a block diagonal matrix. 2. Problem formulation Consider a class of nonlinear networked systems described by:
xðk þ 1Þ ¼ AxðkÞ þ N1 gðxðkÞÞ þ B1 dðkÞ þ B2 f ðkÞ
ð1Þ
yðkÞ ¼ CxðkÞ
where xðkÞ 2 Rn is the state vector, yðkÞ 2 Rp is the output of the plant. dðkÞ 2 Rs and f ðkÞ 2 Rt are the disturbance input and fault signal, respectively, which belong to ‘2[0, 1). A, B1, B2, N1 and C are known real constant matrices with the appropriate dimensions. The nonlinear function g() satisfies g(0) = 0 and the following sector-bounded condition [17]:
½gðxÞ gðyÞ S1 ðx yÞT ½gðxÞ gðyÞ S2 ðx yÞ 6 0
ð2Þ
where S1, S2 2 Rnn are known real constant matrices, S = S1 S2 is symmetric positive definite matrix. Dinary-valued function rl ðkÞ : Z ! f0; 1g ðl ¼ 1; . . . ; jÞ is used to describe the lth channels transmission status in time k, where 1 means successful data transmission and 0 means data loss. Specifically, the lth channels output yl(k) is available to fault detection filter (FDF) only corresponding channels are accessible, i.e. rl(k) = 1, then, network-induced delay sl(k) inevitably exist and yl (k sl(k)) will be obtained by the FDF. Otherwise, when rl(k) = 0, the output of lth channels will be zero l ðkÞ as the lth channels signal received by the FDF, we and yl(k) will be ignored owing to its being unavailable. If we regard y can present the transmission dynamics of lth channels as:
l ðkÞ ¼ rl ðkÞyl ðk sl ðkÞÞ ðl ¼ 1; . . . ; pÞ y
ð3Þ
¼ max16l6p sl ðkÞ are the delay low and upper bound, respectively. where s = min16l6psl(k) and s Remark 1. For networked systems, at each sampling time k, due to the bandwidth constraint of the wireless/wire network, only some of these sensors or packets signal can be transmitted successfully. In this note, if we regard every sensors or packets as one information channel, then multi-input and multi-output [9], medium access constraint [5,10,13] and multiple packets transmission [6,16,23] can be described validly by multiple channels transmission framework. On the other hand, modeling is one of the most important issues in networked control systems which has received an ever increasing research attention [4–10,14,16,23,24]. For practical networked control systems, network-induced delay and packet-dropout often occurs simultaneously due to limited channel capacity and noise. Actually, pattern (3) describe the NCSs with data loss and time-varying delay simultaneously, so it is more comprehensive than separate consideration of delay pattern [9] or data loss pattern [6,16,23]. n o 1;1 1;p p1;1 To discuss validly above problem, we take transmission matrix as Mp , M 0;1 , ; . . . ; M p1;p ; M p;1 p ; Mp ; . . . ; Mp ; . . . ; Mp p p and matrix Ms;t p ¼ diagfr1 ðkÞ; . . . ; rp ðkÞg is expressed by the following form: 1;1 M0;1 0; . . . ; 0g; . . . ; M1;p . . . ; 0; 1g; . . . ; M p1;1 , diagf1; . . . ; 1; 0g; p , diagf0; . . . ; 0g; M p , diagf1; p , diagf0; p |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} p1
. . . ; Mp1;p p
,
diagf0; 1; . . . ; 1g; Mp;1 p |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} p1
p1
, diagf1; 1; . . . ; 1g |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
Then we represent the whole dynamics model as
p
p1
ð4Þ
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Y. Zhang et al. / Information Sciences 221 (2013) 534–543
ðkÞ ¼ M PlðkÞ yðk sðkÞÞ y
ð5Þ
h
iT h iT ðkÞ ¼ y T1 ðkÞ; . . . ; y Tp ðkÞ ; M plðkÞ 2 M p ; lðkÞ is a Markov chain taking valwhere yðk sðkÞÞ ¼ yT1 ðk s1 ðkÞÞ; . . . ; yTp ðk sp ðkÞÞ ; y ues in a finite state space R ¼ f1; 2; . . . ; 2p g with transition probability matrix K = [qij] given by
qij ¼ Probðlðk þ 1Þ ¼ jjlðkÞ ¼ iÞ ð8i; j 2 RÞ
ð6Þ
where qij P 0ði; j 2 RÞ is the transition probability from mode i to j and
P2p
j¼1
qij ¼ 1ð8i 2 RÞ.
Remark 2. Recently, stochastic or deterministic model has been introduced to describe data loss phenomena for networked control systems. Usually, Bernoulli distributed model is arguably the most popular stochastic pattern in which 0 stands for entire missing and 1 denotes the intactness. However, due to the communication constraint of multiple channels network, the transmission status of different channels in adjacent sampling period may interact. Actually, finite-state Markov chains can be used to model this correlated pattern validly. Therefore, Markovian jump model (5) reveal the characteristic of communication network more effective than [5,6,9,10,13,16,23]. Especially, if we regard Bernoulli random variable as special case of Markov chains, then, Bernoulli-distributed model of multiple packets transmission[6] is the special case of Markovian jump model (5). In this paper, observer-based fault detection filters are constructed as residual generator:
8 ðkÞ y ^ðkÞÞ > < ^xðk þ 1Þ ¼ A^xðkÞ þ LlðkÞ ðy ^ðkÞ ¼ C ^xðkÞ y > : ðkÞ y ^ðkÞÞ rðkÞ ¼ V lðkÞ ðy
ð7Þ
^ðkÞ 2 Rq represent the state and output estimation vectors, respectively, r(k) is the residual signal. Obwhere ^ xðkÞ 2 Rn and y server gain matrices Ll(k) and residual weighting matrices Vl(k) are filter parameters to be determined, which is assumed to jump synchronously with the modes in (5) and is hereby mode-dependent. Let the estimation error be e1 ðkÞ ¼ ½xðkÞ ^ xðkÞ, then the error systems can be obtained by substituting (5) and (7) into (1) lðkÞ
e1 ðk þ 1Þ ¼ ðA LlðkÞ CÞe1 ðkÞ LlðkÞ CxðkÞ LlðkÞ MP Cxðk sðkÞÞ þ N1 gðxðkÞÞ þ B1 dðkÞ þ B2 f ðkÞ
ð8Þ
T By setting re(k) = r(k) f(k), eðkÞ ¼ xT ðkÞ; eT1 ðkÞ , w(k) = [dT(k),fT(k)]T and combining (1) and (7), we have the following augmented error systems
8 > eðk þ 1Þ ¼ AlðkÞ eðkÞ þ A1lðkÞ Heðk sðkÞÞ þ NHgðeðkÞÞ þ BwðkÞ > > <
ð9Þ
r e ðkÞ ¼ C lðkÞ eðkÞ þ C 1lðkÞ Heðk sðkÞÞ þ DwðkÞ > > > : ; s þ 1; . . . ; 0 eðkÞ ¼ /ðkÞ; lðkÞ ¼ /ðkÞ; k ¼ s where
" AlðkÞ ¼
A
0
LlðkÞ C
A LlðkÞ C
#
" ;
A1lðkÞ ¼
"
#
0 lðkÞ
LlðkÞ M P C
;
B¼
B1
B2
B1
B2
# ;
" N¼
N1
#
N1
;
lðkÞ
C lðkÞ ¼ ½V lðkÞ C; V lðkÞ C, C 1lðkÞ ¼ V lðkÞ M P C, D ¼ ½0; I; H ¼ ½I; 0; /ðkÞ and u(k) are the given initial condition of e(k) and l(k), respectively, re (k) is the residual error which contains information on both the time and location of the occurrence of fault. Definition 1. [19]. Augmented error system (9) is said to be stochastically stable for w(k) = 0 and every initial condition / (k)(u(k)), such that
! 1 X 2 E keðkÞk j/ðkÞ; uðkÞ < 1 k¼0
The purpose of this paper is to design the mode-dependent observer-based fault detection parameters Ll(k) and Vl(k) such that the following requirements are satisfied simultaneously: (a) The zero-solution of augmented error system (9) with w(k) = 0 is stochastically stable. (b) Under the zero-initial condition, the residual estimation error re(k) satisfies 1 1 X X Efkr e ðkÞk2 g 6 c2 kwðkÞk2 k¼0
k¼0
for all nonzero w(k), where c > 0 is a given disturbance attenuation level.
ð10Þ
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Y. Zhang et al. / Information Sciences 221 (2013) 534–543
In this paper, threshold Jth and residual evolution function J(re) are selected as:
( Jðr e Þ ¼ E
)1=2
k¼k 0 þj X
r Te ðkÞr e ðkÞ
;
J th ¼ supdðkÞ2‘2 ;f ðkÞ¼0 Jðr e Þ:
k¼k0
where k0 denotes the initial evaluation time instant, j denotes the evaluation time steps. Once the evaluation function J(re) has been selected, we are able to determine the threshold Jth. Based on the threshold, the occurrence of fault can be detected by comparing Jth and J(re) according to the following test:
Jðre Þ P J th ) alarm for fault
ð11Þ
Jðre Þ < J th ) no fault 3. Main results
The following theorem provides a sufficient condition under which the augmented error system (9) is stochastically stable and the residual estimation error re(k) satisfies the H1 criterion (10) under zero-initial condition. In addition, the transition probabilities of jumping process {l(k), k P 0} in this paper are assumed to be partly accessed, i.e. some elements are unknown. For notation clarity, we denote R ¼ RiK þ RiUK ð8i 2 RÞ with RiK , fj : qij is knowng and RiUK , fj : qij is unknowng. and c > 0, augmented filtering error system (9) with partly unknown transition probabilities Theorem 1. For given scalars 0 6 s 6 s is stochastically stable with H1 attenuation level (10) if there exist matrices P i > 0ði 2 RÞ,Q > 0 and scalar e > 0 such that
2 6 6 6 4 2
qiK Ni1
Ni1
0 c2 I
0 c2 I
6 6 4
qiK Ri1 Ri2 PiK 0 q
i KI
Ri1 Ri2 Pj 0 I
3
7 0 7 7<0 0 5 P iK 3
0 7 7 < 0; 0 5 Pj
ð12Þ
8j 2 RiUK
ð13Þ
where
3 ^ þ 1ÞQ eS1 H Pi 0 eHT ST2 HT ½ðs ST1 S2 þ ST2 S1 5 ; S1 ¼ ; N ¼4 Q 0 2 e I X qiK , qij ; s s ¼ s^; Ri1 ¼ ½C i ; C 1i ; 0; DT ; Ri2 ¼ ½Ai ; A1i ; N; BT : 2
i 1
S2 ¼
ST1 þ ST2 ; 2
PiK ¼
X
rij Pj ;
j2RiK
j2RiK
Þ and construct Lyapunov functional candidate as Proof. Denote IðkÞ ¼ ½eðkÞ; eðk 1Þ; . . . ; eðk s
VðIðkÞ; kÞ ¼ eT ðkÞPlðkÞ eðkÞ þ
sþ1 X
k1 X
k1 X
eT ðmÞQeðmÞ þ
þ2m¼k1þh h¼s
eT ðmÞQeðmÞ:
m¼ksðkÞ
where PlðkÞ > 0ðlðkÞ 2 RÞ and Qi > 0 are symmetric positive definite matrices. Taking the difference of above functional along the solution of the augmented error system (9), we have
E½DVðIðkÞ; kÞ ¼ EfVðIðk þ 1Þ; k þ 1ÞjIðkÞg VðIðkÞ; kÞ X 6 eT ðk þ 1Þ qij Pj eðk þ 1Þ eT ðkÞPi eðkÞ
ð14Þ
j2R
s þ 1ÞeT ðkÞQ 3 eðkÞ þ ðs
ks X
eT ðmÞQ 3 eðmÞ
þ1 m¼ks
eT ðk sðkÞÞQ 3 eðk sðkÞÞ þ
ks X
eT ðmÞQ 3 eðmÞ
þ1 m¼ks
s þ 1ÞeT ðkÞQ 3 eðkÞ eT ðk sðkÞÞQ 3 eðk sðkÞÞ ¼ ðs
ð15Þ
538
Y. Zhang et al. / Information Sciences 221 (2013) 534–543
For any scalar e > 0, it follows readily from (2) that
e
HeðkÞ HgðeðkÞÞ
T "
S1
S2
I
#
HeðkÞ 60 HgðeðkÞÞ
ð16Þ
Denoting f(k) = [eT(k), eT(k s(k))HT, gT(e(k))HT]T and combining (14–16) leads to T
E½DVðIðkÞ; kÞ ¼ e ðk þ 1Þ
2
X
qij Pj eðk þ 1Þ þ f ðkÞ4 T
j2R
2
X
¼ eT ðk þ 1Þ4PiK þ
T
¼ f ðkÞ
8
3
i 2
i
X
qij þ
j2RiK
3
qij 5Ni1 fðkÞ
j2RiUK
2
qij Pj 5eðk þ 1Þ þ fT ðkÞ4qiK þ
j2RiUK i 1
X
qiK N þ N þ
X
qij
9 = fðkÞ N þN ;
j2RiUK
i 1
X
3
qij 5Ni1 fðkÞ
j2RiUK
i 3
ð17Þ
where
2 6 Ni2 ¼ 6 4
ATi PiK Ai
ATi PiK A1
AT1 PiK A1
3
2 T Ai P j Ai 7 6 i and N ¼ 4 AT1 PiK N1 7 3 5 T i N1 PK N1 ATi PiK N1
ATi Pj A1 AT1 Pj A1
ATi Pj N1
3
7 AT1 Pj N1 5: NT1 Pj N1
By Schur complement, (12) and (13) implies qiK Ni1 þ Ni2 < 0 and Ni1 þ Ni3 < 0, then we know from (17) that
h i n h io X E½DVðIðkÞ; kÞ 6 kmin qiK Ni1 Ni2 fT ðkÞfðkÞ qij minj kmin Ni1 þ Ni3 fT ðkÞfðkÞ 6 ðb1 þ b2 ÞfT ðkÞfðkÞ
ð18Þ
j2RiUK
n h io n
n h ioo and b2 ¼ inf 1 qiK minj kmin Ni1 þ Ni3 . From (18), if we do mathematics where b1 ¼ inf kmin qiK Ni1 Ni2 expectations in both sides, we obtain that for any N P 0, N X E½VðIðN þ 1Þ; N þ 1Þ VðIð0Þ; 0Þ 6 ðb1 þ b2 Þ E½fT ðkÞfðkÞ k¼0
which yields 1 X E½fT ðkÞfðkÞ 6 k¼0
1 E½VðIð0Þ; 0Þ < 1 ðb1 þ b2 Þ
Thus, from Definition 1, the augmented error systems (9) with w(k) = 0 is stochastically stable. Note that b1 + b2 will reduce to only b1 (respectively, b2) if all the transition probabilities are known (respectively,unknown). Now, to establish the H1 performance, consider the following performance index:
JN ¼ E
N X
rTe ðkÞre ðkÞ c2 wT ðkÞwðkÞ
ð19Þ
k¼0
under zero initial condition, we have
JN ¼ E
N X
1 X T rTe ðkÞre ðkÞ c2 wT ðkÞwðkÞ þ DVðIðkÞÞ EVðN þ 1Þ 6 E r e ðkÞr e ðkÞ c2 wT ðkÞwðkÞ þ DVðIðkÞÞ
k¼0
k¼0
2 3 1 X
i i X i T i i ¼ w ðkÞ4 qK N1 þ N2 þ qij N1 þ N3 5wðkÞ
ð20Þ
j2RiUK
k¼0
where
" T
T
T
i 1
wðkÞ ¼ ½f ðkÞ; w ðkÞ ;
N ¼ "
Xi3 ¼ DT D c2 I; Ni2 ¼
Ni2 Xi2
Xi3
Ni1 þ Xi1 Xi2 # ;
i 3
X
# ;
h
n
o
Xi1 ¼ diag C Ti C i ; 0; 0 ; iT
Xi2 ¼ BT PiK Ai ; BT PiK A1i ; BT PiK N ;
Xi3 ¼ BT Pj B and Xi2 ¼ ½BT Pj Ai ; BT Pj A1i ; BT Pj NT :
Xi2 ¼ ½DT C i ; DT C 1i ; 0T ; "
Xi3 ¼ BT PiK B;
Ni3 ¼
Ni3 Xi2
Xi3
# ;
539
Y. Zhang et al. / Information Sciences 221 (2013) 534–543
P
Similarly, by Schur complement, (12) and (13) are equivalent to qiK Ni1 þ Ni2 þ j2Ri qij Ni1 þ Ni3 < 0, which means J1 < 0, P1 P1 UK 2 2 2 i.e. k¼0 Efkr e ðkÞk g 6 c h k¼0 kwðkÞk , this completes the proof. Theorem 1 provides delay-interval-dependent and Markov-chain-element-dependent sufficient conditions which guarantee the H1 performance as well as the stochastic stability of the augmented error system (9). The following theorem gives a sufficient conditions on the existence of mode-dependent observer-based fault detection filters, the slack matrix will be constructed with a special structure, which allows us to obtain the solution within strict linear matrix inequalities (LMIs) framework. and c > 0, if there exist matrices Li ; V i ; X i ; Y i ; U i ; P i ¼ P i1 P i2 > 0ði 2 RÞ; Q > 0 and Theorem 2. For given scalars 0 6 s 6 s P i3 scalar e > 0 such that
2
Hi11 Hi12 Hi13
6 6 4
3 7
Hi22 Hi23 7 5<0 H
j ,
ð21Þ
i 33
j1
j2
j3
,
1 X
qiK j2Ri
qij
Pj1
P2j
Pj3
; 8j 2 RiK
ð22Þ
K
j ,
P j1
Pj2
Pj3
; 8j 2 RiUK
ð23Þ
where
2 6
^ þ 1ÞQ P i1 eS1 ðs
Pi2
Pi3
Hi11 ¼ 4 2
i 12
H
eS2 6 ¼4 0
0 0
3
0
H
i 35
2
AT X Ti C T LTi
T T Hi13 ¼ 6 4 C Vi
AT Y Ti C T LTii
C T Mip V Ti
N T1 X Ti
NT1 U Ti
3
C T M ip LTi
0 6 ¼4 0
BT1 X Ti þ BT1 Y Ti
7 BT1 U Ti þ BT1 Y Ti 5;
I
BT2 X Ti þ BT2 Y Ti
BT2 U Ti þ BT2 Y Ti
0 0
i 22
C T V Ti
6
7 0 5; Q
2
7 0 0 5;
2
3
0
i 33
H
2
AT U Ti C T LTi
3
7 AT Y Ti C T LTi 7 5; C T Mip LTi
I 6 ¼4
0 1i X i
0 X Ti
2i Y i U Ti 3i Y i Y Ti
3 7 5
2
and H ¼ diagfeI; c I; c Ig. Then, there exists mode-dependent OBFDF such that the augmented error system (9) is stochastically stable (or stable for any switching sequence if RiK ¼ £, for all i 2 R) with H1 attenuation level (10). Moreover, if LMIs (21) have feasible solution, the desired OBFDF are given by
Li ¼ Y 1 i Li
and V i ¼ V i ;
i2R
ð24Þ
Proof. From Theorem 1, inequalities (12) and (13) can be rewritten as:
2
Ni1
6 6 6 6 4
0
Ri1 Ri2 j
c2 I
0
I
3
7 0 7 7<0 0 7 5 j
ð25Þ
where
j ,
8 <
1
qiK
PiK ; 8j 2 RiK
:P ;
8j 2 RiUK
:
8 9 < = 1 T Performing a congruence transformation to (25) by J ¼ diag I; . . . ; I; I; j Ri , one has :|fflfflffl{zfflfflffl} ; j
2
Ni1
6 6 6 6 4
0 2
c I
0
0
I
0
T Ri 1 j Ri
For an arbitrary matrix Ri ¼
1
q
i K
3
Ri2 RTi
Ri1
PiK Ri
1
q
i K
PiK
1
Xi Ui
1
q
i K
4
7 7 7<0 7 5
ð26Þ
Yi ; 8i 2 R, we have the following fact: Yi
PiK Ri
T
P 0;
T ðPj Ri ÞP1 j ðP j Ri Þ P 0:
540
Y. Zhang et al. / Information Sciences 221 (2013) 534–543 T Then, we obtain j RTi Ri P Ri 1 j Ri , so (26) can rewrite as
2
0
Ri1
Ri2 RTi
c2 I
0
0
I
0
RTi
Ni1
6 6 6 6 4
Now, partition Pi as
X
j ,
j2RiK
rij Pj ¼
j P i1
X
rij
3
Ri
7 7 7<0 7 5
ð27Þ
Pi2 , we acquire Pi3
Pj1
Pj2
Pj3
j2RiK
,
j1
j2
j3
:
j1 j2 Further define matrics variables Li ¼ Y i Li ,V i ¼ V i and j , . Replacing Li ¼ Y i Li , V i ¼ V i into (27), we readily j3 obtain (21). From Theorem 1, the augmented filtering error system (9) will be stochastically stable with H1 attenuation performance. Meanwhile, if the solution of (21) exists, the parameters of admissible OBFDF are given by (24). The proof is completed. h Remark 3. As the two extreme cases, i.e., when all the transition probabilities are known and unknown,the underlying sys tems are the traditional MJLS RiK ¼ R; RiUK ¼ U and the switched systems under arbitrary switching RiUK ¼ R; RiK ¼ U , respectively. Correspondingly, the fault detection results can be found in some existing references, [29] investigated linear discrete-time Markovian jump system (completely accessible), [18] studied linear discrete-time switched systems (completely inaccessible), [7] considered transition probabilities with polytopic uncertainties which require the knowledge of uncertainties structure and it can still be viewed as accessible. Therefore, observer-based fault detection with partly unknown transition probabilities of this paper is a more natural assumption to the Markovian jump systems and hence covers the existing ones.
4. Numerical example and simulation Consider a nonlinear networked system in (1) with following parameters:
0:2 0 0:4 ; B1 ¼ ; 0:3 0:4 0:2 0:2 0 : S2 ¼ 0:1 0:1
A¼
B2 ¼
0 ; 0:6
C¼
0:2 0:1 ; 0 0:4
N1 ¼
0:3 0:1 ; 0:2 0
S1 ¼
0:4 0 ; 0:1 0:3
Attention is focused on the design of mode-dependent OBFDF which stabilize stochastically the augmented error system (9) and satisfy the H1 attenuation level (10) In this paper, we consider two channels data transmission NCSs, according to the transmission pattern presented in Section 2, the transmission matrices are constructed:
M2;1 2 , diagf1; 1g;
M0;1 2 , diagf0; 0g;
M1;1 2 , diagf1; 0g;
M 1;2 2 , diagf0; 1g:
With above data transmission pattern, from Remark 3, we consider the case II with partly unknown transition probabilities in Table 1, where‘?’means that element is unknown. As the special cases of case II, corresponding results of traditional Markovian jump systems (Case I) and switched systems (Case III) can be included in our theorems. For nonlinear multiple ¼ 3, the channels networked systems with data loss and time-varying delay (9), if the delay bound are given as s = 1 and s H1 attenuation performance level is taken as c = 1.2, by applying Theorem 2, the OBFDF Lji and V ji ði ¼ 1; 2; 3; 4; j ¼ 1; 2; 3Þ can be designed as
Table 1 Different transition probabilities matrices. Completely known (Case I)
3 0:1 0:2 0:3 0:4 6 0:1 0:6 0:1 0:2 7 7 6 7 6 4 0:5 0:1 0:2 0:2 5 2
0:6 0:1 0:2 0:1
Partly unknown (Case II)
3 0:1 0:2 0:3 0:4 6 ? 0:6 ? 0:2 7 7 6 7 6 4 0:5 ? ? ? 5 2
0:6 0:1 0:2 0:1
Completely unknown (Case II)
2
? ? ? ?
3
6? ? ? ?7 6 7 6 7 4? ? ? ?5 ? ? ? ?
Y. Zhang et al. / Information Sciences 221 (2013) 534–543
L11 ¼
1:2753 0:3075 0:0838 0:0235 0:4806 0:0140 ; L12 ¼ ; L13 ¼ ; L14 ¼ ; 0:1355 1:3050 0:9045 0:0853 0:5998 1:1232 0:0439
0:4590 0:1112 0:1808
V 11 ¼ ½0:0006 0:0348; L21 ¼
0:2496
V 12 ¼ ½0:2186 0:2592;
V 13 ¼ ½0:0052 0:1871;
V 14 ¼ ½0:4506 0:0067;
1:1352 0:2784 0:0816 0:0011 0:6036 0:0238 ; L22 ¼ ; L23 ¼ ; L24 ¼ ; 0:1816 1:1620 0:9204 0:1095 0:6939 0:8905 0:0618
0:5036 0:1188
V 21 ¼ ½0:0060 0:0467; L31 ¼
541
V 22 ¼ ½0:1974 0:2293;
V 23 ¼ ½0:0144 0:1215;
V 24 ¼ ½0:4845 0:0122;
1:0355 0:2512 0:2071 0:0151 0:5683 0:0334 ; L32 ¼ ; L33 ¼ ; L34 ¼ ; 0:2444 1:0751 0:7929 0:1715 0:5619 0:8880 0:0559
0:4262 0:1014 0:3409
V 31 ¼ ½0:0721 0:0879;
V 32 ¼ ½0:2291 0:2569;
V 33 ¼ ½0:0418 0:1996;
V 34 ¼ ½0:3191 0:0317:
To demonstrate the effectiveness of designed OBFDF, an unknown disturbance d(k) is assumed to be band-limited white noise with power of 0.05. The fault signal f(k) is simulated as a square wave of 0.3 amplitude that occurred from 10 to 30 steps and the nonlinear function is given by g(x(k)) = sin (0.2 x(k)). Under cases I, II and III, the initial state of augmented error system (11) is assumed as e(0) = [0.1 0.2 0.2 0.1]T, corresponding evolution of residual estimation error signal re(k) and residual evaluation function J(re) are shown in Figs. 1–3, respectively. By using 300 Monte Carlo simulations for k0 = 0 and j = 100, we obtain the threshold in Table 2. From Table 2, we observe that when k = 19, 23 and 14, J(re) P Jth for the first time, which mean the fault f(k) can be detected 9, 13 and 4 time steps after its occurrence, respectively. Remark 4. It is clearly observed from the simulation that the augmented error systems (9) is stochastically stable with given H1 performance level, and the fault can also be detected effectively. Especially, it is easily verified that Markovian jump systems model partly unknown transition probabilities is more natural pattern which contain traditional Markovian jump systems (known transition probability) and switched systems (unknown transition probability) as its special case.
5. Conclusion In this paper, the fault detection problem has been addressed for a class of nonlinear networked systems with timevarying delay. Transmission matrix has been introduced to describe the multiple channels data transmission pattern, and Markovian jump system with partly unknown transition probabilities has been used to model the large-scale networked systems. Delay-interval-dependent and Markov-chain-element-dependent sufficient conditions are derived which contain traditional Markovian jump systems (known transition probability) and switched systems (unknown transition probability) as its special case. An illustrative example has been exploited to show the usefulness of the results obtained. The future
Fig. 1. Corresponding simulation of Case I.
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Y. Zhang et al. / Information Sciences 221 (2013) 534–543
Fig. 2. Corresponding simulation of Case II.
Fig. 3. Corresponding simulation of Case III. Table 2 Corresponding threshold and residual evolution function value for different cases.
Threshold jth Residual evolution function Jdr e ðkÞe
Completely known (Case I)
Partly unknown (Case II)
Completely unknown (Case III)
Jth = 1.0119 1.0086 = J(18) < Jth < J(19) = 1.0258
Jth = 1.0983 1.0981 = J(22) < Jth < J(23) = 1.1308
Jth = 0.6945 0.6269 = J(13) < Jth < J(14) = 0.7338
research topics would include the extension of the main results developed in this paper to more general complex systems such as networked systems with quantization, general fractional uncertainties, polynomial nonlinear systems and functional differential equations of the neutral type. Acknowledgments The authors would like to thank the editor and the reviewers for their helpful suggestions to improve the quality of this correspondence. This paper is partially supported by the National Natural Science Foundation of China (Grant Nos.
Y. Zhang et al. / Information Sciences 221 (2013) 534–543
543
61104027, 61174107, 61034006), China Postdoctoral Science Foundation (Grant No. 2011M500899), National Natural Science Foundation of Mathematical Tianyuan Fund (Grant No. 11126278). 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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