Non-fragile filtering for fuzzy stochastic systems over fading channel

Neurocomputing 174 (2016) 553–559

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Non-fragile filtering for fuzzy stochastic systems over fading channel$ Renquan Lu a,b, Shuang Liu a, Hui Peng a, Yong Xu a,b,n, Kan Xie b a The Key Lab for IOT and Information Fusion Technology of Zhejiang, The Institute of Information and Control, Hangzhou Dianzi University, Hangzhou 310018, China b School of Automation, Guangdong University of Technology, and Guangdong Key Laboratory of IoT Information Processing, Guangzhou 510006, China

art ic l e i nf o

a b s t r a c t

Article history: Received 17 July 2015 Received in revised form 17 September 2015 Accepted 20 September 2015 Communicated by Hongyi Li Available online 1 October 2015

This paper investigates the problem of non-fragile filter design for nonlinear stochastic systems subject to channel fading. The nonlinear stochastic systems are represented by the Takagi–Sugeno (T–S) model and the channel fading is described by a random process. By using the Lyapunov method, a sufficient condition which guarantees that the filtering error system is stochastically stable and satisfies the passive performance is established. Then the parameters of the non-fragile filter are obtained by solving linear matrix inequalities (LMIs). Finally, numerical examples are employed to illustrate the effectiveness of the proposed approach. & 2015 Published by Elsevier B.V.

Keywords: Fuzzy stochastic systems Channel fading Non-fragile filter Passive performance

1. Introduction Recently, a lot of attentions have been paid to Networked Control Systems (NCSs) since they possess great advantages including flexibilities in the location, ease of maintenance, and high reliability [1–6]. However, as the result of the limited channel capacity, many new challenges appear, one of them is channel fading [7]. When the system measurements transmit through a fading channel, the filter and the controller design problems need to be reconsidered, otherwise, the stability and the performance of the systems will be influenced. Some pioneering works have been done on channel fading. In [8], the stabilization problem has been considered for the single-input systems over fading channel. Then the result has been extended to the multiple-input systems by using resources allocation methods [9]. In [10], the necessary and sufficient conditions have been derived for the LQ control problem over fading channels. However, due to the complexity of the nonlinear systems, the nonlinear stochastic systems with channel fading need to be investigated. Over the past three decades, T–S fuzzy model has been considered by many researchers, because it is a mighty tool to set up a bridge ☆ This work was supported in part by the China National Funds for Distinguished Young Scientists under Grant 61425009, in part by the National Natural Science Foundation of China under Grants 61320106009, 61320106010, 61503106. n Corresponding author at: The Key Lab for IOT and Information Fusion Technology of Zhejiang, the Institute of Information and Control, Hangzhou Dianzi University, Hangzhou 310018, China. E-mail addresses: [email protected] (R. Lu), [email protected] (Y. Xu).

http://dx.doi.org/10.1016/j.neucom.2015.09.078 0925-2312/& 2015 Published by Elsevier B.V.

between linear systems and nonlinear systems [11–13]. T–S fuzzy model adopts fuzzy rules to approximate any smooth nonlinear functions. As a result, the problem of analysis and synthesis for the complex nonlinear systems can be handled by using the linear system analysis tool, and many important results have been obtained, like estimator/controller design [14], fault detection [15], performance analysis [16], and model reduction [17]. On the other frontier, state estimation which is one of the most important problems in control fields has been widely studied by many researchers [18]. In the actual condition, the disturbances in the environment always influence the controller and the filter. As a result, non-fragile controller and filter design have become an important issue in practical applications and theoretical research [19,20], where the systems are still stable and achieve the expected performance under the condition that the controller and the filter are perturbed. So far this problem has been widely investigated by many researchers [21,22]. In [23], non-fragile H 1 filter has been proposed for nonlinear systems. In [24], non-fragile H 1 and H2 filter has been designed based on randomised algorithm. However, the non-fragile filter design problem for nonlinear stochastic systems with channel fading has not been studied yet. Motivated by the above discussions, this paper investigates the problem of non-fragile filter design for nonlinear stochastic systems with channel fading, where the considered nonlinear stochastic systems are represented by T–S fuzzy model. A sufficient condition which guarantees that the filtering error system is stochastically stable with passive performance is derived. Then, the parameters of the non-fragile filter are obtained by using the LMI approach. Finally, a numerical example is proposed to illustrate the

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R. Lu et al. / Neurocomputing 174 (2016) 553–559

effectiveness of the obtained results. The main contributions of this paper can be concluded as follows:

with ζ ij ðϕj ðkÞÞ representing the grade of membership of ϕj ðkÞ in ζij. P Obviously, we obtain ιi ¼ 1 g i ðϕðkÞÞ ¼ 1.

 The imperfect measurements induced by the channel fading is

2.2. Channel fading



considered for the T–S fuzzy systems, which is described by a random process. The nonfragile filter is designed to improve the robustness of the filter, which is always influenced by the environmental noises.

The rest of this paper is organized as follows. Section 2 describes fuzzy stochastic systems subject to fading channel. Section 3 analyzes the stochastic stability and the passivity performance of the filtering error system. Then, in Section 4, the nonfragile filter is designed. Illustrative examples are presented in Section 5, while Section 6 draws the conclusions. Notation: Throughout the paper, Rn denotes the n-dimensional real space; Rnm is the set of all n  m real matrices. X Z Y (X 4Y) means that the matrix X Y is positive semi-definite (positive definite). I stands for the identity matrix with appropriate dimension. MT represents the transposition of the matrix M, and diagfg denotes a block-diagonal matrix; EfxðkÞg and CfxðkÞg denote the expectation and the covariance of the random variable xðkÞ, respectively. l2 ½0; 1Þ represents the space of square summable infinite vector sequence. n represents a term which can be inferred by symmetry.

In the network environment, it is quite common that the measurement yðkÞ is transmitted over the fading channel, that is ~ yðkÞ ¼ δðkÞyðkÞ

ð3Þ

where δðkÞ is a random process and satisfies     E δðkÞ ¼ δ ; C δðkÞ ¼ σ 2 : 2.3. Non-fragile filter In this paper, we consider the following nonfragile filter: Plant rule i: IF ϕ1 ðkÞ is ζi1 and ϕ2 ðkÞ is ζi2 and ⋯ and ϕp ðkÞ is ζip, THEN 8   < xðk ^ þ ðK i þ ΔK i Þ yðkÞ ^ ~  δ C i xðkÞ ^ þ 1Þ ¼ Ai xðkÞ : zðkÞ ^ ^ ¼ L xðkÞ i

^ ^ A Rn and zðkÞ A Rq are the filter state and where i ¼ 1; 2; ⋯ι, xðkÞ the estimation of zðkÞ, respectively. Ki denotes the parameter to be designed, the uncertainty ΔK i in the filter gain is described as follows:

ΔK i ¼ Mi Δ~ i ðkÞN i 2. Preliminaries 2.1. System description Consider a nonlinear discrete-time stochastic system, which can be approximated by the following T–S fuzzy model: Plant rule i: IF ϕ1 ðkÞ is ζi1 and ϕ2 ðkÞ is ζi2 and ⋯ and ϕp ðkÞ is ζip, THEN 8 > < xðk þ1Þ ¼ Ai xðkÞ þ Bi νðkÞ þ ðEi xðkÞ þ F i νðkÞÞωðkÞ yðkÞ ¼ C i xðkÞ þDi νðkÞ > : zðkÞ ¼ L xðkÞ

ð1Þ

~ ðkÞ satisfies Δ ~ T ðkÞΔ ~ ðkÞ r I, M A Rn1 , N1m are known where Δ i i i i i matrices. For notational simplicity, in the rest part, we adopt gi and g iþ to denote g i ðϕðkÞÞ and g i ðϕðk þ 1ÞÞ, respectively. Then according to the fuzzy weighting function for each filter, we have 8 ι n  o X > > > xðk ^ þ 1Þ ¼ ^ þ ðK i þ ΔK i Þ yðkÞ ^ ~  δ C i xðkÞ g i ðϕðkÞÞ Ai xðkÞ > > < i¼1 ð4Þ ι X   > > > ^ ^ g i ðϕðkÞÞ Li xðkÞ zðkÞ ¼ > > : i¼1

i

where ϕ1 ðkÞ; ϕ2 ðkÞ; ⋯; ϕp ðkÞ denote the premise variables, ζ 1p ; ζ 2p ; ⋯; ζ ip represent the fuzzy sets, i A Ω 9 f1; 2; …; ιg, where ι stands for the number of fuzzy rules. xðkÞ A Rn , yðkÞ A Rm , and zðkÞ A Rq are the state, the measurement, and the signal to be estimated, respectively. νðkÞ A Rh which belongs to l2 ½0; 1Þ is a disturbance noise. Ai A Rnn , Bi A Rnh , Ei A Rnn , F i A Rnh , C i A Rmn ; Di A Rmh , and Li A Rqn are constant matrices, the Gaussian white process ω ðkÞ A R complies with     E ωðkÞ ¼ 0; C ωðkÞ ¼ 1: According to the fuzzy weighting function for each subsystem, the overall fuzzy system is 8 ι X   > > > g i ðϕðkÞÞ Ai xðkÞ þ Bi νðkÞ þ ðEi xðkÞ þ F i νðkÞÞωðkÞ > xðk þ1Þ ¼ > > i¼1 > > > > ι X <   yðkÞ ¼ g i ðϕðkÞÞ C i xðkÞ þ Di νðkÞ ð2Þ > i¼1 > > > > ι X >   > > zðkÞ ¼ > g i ðϕðkÞÞ Li xðkÞ > :

Remark 1. For the actual systems, parameter perturbation is unavoidable. This phenomenon influences the stability and the performance of the systems if they do not be handled appropriately. Therefore, the non-fragile filter is designed in this paper to improve the robustness of the filter. 2.4. Main objective From (2)–(4), the filtering error system is 8 ι X ι n X > > > η ðk þ1Þ ¼ g g Aij ηðkÞ þ δ~ ðkÞA~ ij ηðkÞ þ Bij νðkÞ > i j > > > i ¼ 1j ¼ 1 > > o <   þ δ~ ðkÞB~ ij νðkÞ þ E i ηðkÞ þ F i νðkÞ ωðkÞ > > > ι X ι > X   > > > g i g j Lij ηðkÞ eðkÞ ¼ > > : i ¼ 1j ¼ 1 where h

"

i¼1

Aij ¼

where

σ i ðϕðkÞÞ ; i ¼ 1 σ i ðϕðkÞÞ

g i ðϕðkÞÞ ¼ Pι

p

σ i ðϕðkÞÞ ¼ ∏ ζ ij ðϕj ðkÞÞ Z 0 j¼1

iT

^ T ; ηðkÞ ¼ xðkÞT xðkÞ

" A~ ij ¼

Ai

^ eðkÞ ¼ zðkÞ  zðkÞ # 0

δ ðK j þ ΔK j ÞC i Aj  δ ðK j þ ΔK j ÞC j 0 ðK j þ ΔK j ÞC i

# 0 ; 0

"

Bij ¼

Bi δ ðK j þ ΔK j ÞDi

#

ð5Þ

R. Lu et al. / Neurocomputing 174 (2016) 553–559

" B~ ij ¼

#

0

Ei

0

0

 Lij ¼ Li Lj :

0

ðK j þ ΔK j ÞDi

δ~ ðkÞ ¼ δðkÞ  δ ;

 Ei ¼

;



;

Fi ¼

Fi



Define

0

Φijl ¼ ATij P l Aij þ σ 2 A~ ij P l A~ ij þE i P l E i  P i : T

k¼0

holds for any initial conditions. To derive our main results, we introduce the following lemma: Definition 2. Given a scalar γ 4 0, the filtering error system in (5) is stochastically stable and passive in the sense of expectation, if it is stochastically stable and the inequality ( ) ( ) 1 1 X X 2E eðkÞT νðkÞ r γ 2 E νðkÞT νðkÞ k¼0

holds under zero initial condition. Lemma 1. Given real matrices

Υ 1 þ Υ 3 ΔðkÞΥ 2 þ Υ ΔðkÞ Υ T 2

T

Υ1, Υ2, Υ3, with Υ 1 ¼ Υ T1 ,

EfVðk þ 1Þg  VðkÞ r

holds for all ΔðkÞ satisfying ΔðkÞ ΔðkÞ r I, if and only if there is a scalar ε 4 0 such that

Υ 1 þ εΥ T2 Υ 2 þ ε  1 Υ 3 Υ T3 o0: 3. Main result In this section, a sufficient condition is derived which guarantees the stochastic stability and the passive performance of the filtering error system in (5), the result can be concluded as follows. Theorem 1. Given matrices Kj and a scalar γ 4 0, the filtering error system in (5) is stochastically stable and satisfies the passive performance, if there exist matrices P i 4 0 such that the following inequalities hold for all i; j; l A Ω: 2 3 T T  P i LTij ATij P l σ A~ ij P l E i P l 6 7 6 7 T 2 ~ T Pl F T Pl 7 6 n  γ I B P σ B l ij i ij 6 7 6 7 ð6Þ 6 n n  Pl 0 0 7 o 0: 6 7 6 n 7 n n  Pl 0 5 4 n n n n  Pl

l¼1i¼1j¼1

Proof. Firstly, in order to prove the stochastic stability of the filtering error system in (5) with νðkÞ ¼ 0, the following Lyapunov function is established: ! ι X g i P i ηðkÞ: VðkÞ ¼ ηðkÞT i¼1

Define the difference of the Lyapunov function as EfΔVðkÞg 9 EfVðk þ 1Þg V ðkÞ 8 ι X ι X ι X ι X ι
i   P l  Adt þ δ~ ðkÞA~ dt þ E d ωðkÞ  P i ηðkÞg

l¼1i¼1j¼1

  T T g lþ g i g j ηðkÞT ATij P l Aij þ σ 2 A~ ij P l A~ ij þ E i P l E i P i ηðkÞ: ð7Þ

g lþ g i g j ηðkÞT Φijl ηðkÞ o 0:

ð8Þ

By some iterations on both sides of (8), we obtain that EfVðk þ 1Þg  Vð0Þ 8 9

k X ι X ι X ι
  In view of E VðkÞ Z0, for any k Z 0, we naturally get ( ) k X   ηðsÞT ηðsÞ r ð min λmin ð  Φijl Þ Þ  1 Vð0Þ: E i;j;l A Ω

s¼0

As k approach to 1, we can obtain ( )

1 X   1 ηðkÞT ηðkÞ r min λmin ð  Φijl Þ ηð0ÞT E i;j;l A Ω

ι X

! g a ðϕð0ÞÞP a

a¼1

ηð0Þ rcηð0ÞT ηð0Þ:

    1    maxa A Ω λmax ðP a Þ . where c ¼ mini;j;l A Ω λmin ð  Φijl Þ According to Definition 2, the above inequality guarantees that the filtering error system in (5) is stochastically stable. Next, we will prove the passive performance of the filtering error system in (5). Define an index J as J¼

s n o X E 2eðkÞT νðkÞ  γ 2 νðkÞT νðkÞ :

ð9Þ

k¼0

Considering the fact that EfVðkÞ Z 0g and the zero initial condition, we have Jr

s n o X E  2eðkÞT νðkÞ  γ 2 νðkÞT νðkÞ þ ΔVðkÞ k¼0

r

ι X ι X ι X l¼1i¼1j¼1

g lþ g i g j

h

i

"

ηðkÞ ηðkÞ νðkÞ Ξ ðkÞ νðkÞ T

T

#

where 2

Ξ ðkÞ ¼ 4

ι X ι X ι X

ι X ι X ι X

k¼0

T 3 o0 T

r

T

Considering (6) and (7), we have

Definition 1. The filtering error system in (5) is stochastically stable with νðkÞ ¼ 0, if there is a scalar c 4 0, such that the inequality ( ) 1 X ηðkÞT ηðkÞ o cEfηð0ÞT ηð0Þg E

k¼0

555

ATij BTij

" þ

3

2



5P l Aij Bij

P i

 LTij

n

 γ2I



3 ~T h A ij 6 7 þ σ 2 4 T 5P l A~ ij ~ B ij

2 T3 i Ei

 ~ B ij þ 4 T 5 P l E i F i Fi

# :

By using Schur complement to (6), we can derive Ξ ðkÞ o 0, which implies that the system in (5) is passive in the sense of expectation. □

4. Non-fragile filter design In terms of Theorem 1, we begin to design the parameters of the non-fragile filter in this section. Theorem 2. Given a scalar γ 4 0, the filtering error system in (5) is stochastically h i stable with passive performance, if there exist matrices 4 0; W 1 ; W 2 ; W 3 ; K~ j ; scalars ε1 4 0; and ε2 4 0 such P i ¼ Pn11i PP 12i 22i

556

R. Lu et al. / Neurocomputing 174 (2016) 553–559

that the following inequalities hold for all i; j; l A Ω: 2 3 6 6 6 6 6 6 6 6 6 6 6 4

Γ 11

Γ 12 Γ 13 Γ 14 Γ 15  γ 2 I Γ 23 Γ 24 Γ 25 n Γ 33 0 0 n n Γ 44 0 n n n Γ 55

n n n n

Γ 16 Γ 17 Γ 26 0 7 7 7 Γ 36 Γ 37 7 7 Γ 46 0 7 7o0 0

n

n

n

n

n

Γ 66

n

n

n

n

n

n

7 0 7 7 0 7 5

Define the new variables T K~ j ¼ K Tj W T2 :

ð10Þ

Γ 77

The inequalities (12) are equal to 2 T ATij W T σ A~ ij W T P i  LTij 6 6 T 6 n  γ2I BTij W T σ B~ ij W T 6 6 6 n n Pl  W  W T 0 6 6 n n n P  W WT 4 l n

where "

Γ 11 ¼

 P 11i

 P 12i

n

 P 22i

;

Γ 12 ¼ 4

 LTi LTj

3

n

0 0

7 7 7 7 7 o0: 7 7 7 5

P l W W T

ðP l  WÞP l 1 ðP l  WÞT Z 0 3

which implies P l W W T Z  WP l 1 W T :

K j ¼ W 2 1 K~ j :

ð17Þ n o Define Λ ¼ diag I; I; P l W  1 ; P l W  1 ; P l W  1 , then, pre-multiplying (17) with Λ and post-multiplying (17) with ΛT, we obtain (6). So we can conclude that the system in (5) is stochastically stable with the passive performance γ, if conditions (10) hold.□ Remark 2. The numerical complexity of obtained results can be described by the total number ND of the scalar decision variables and the total row size NL of the LMIs. According to conditions (10), we have ND ¼ ð2ι þ 3Þn2 þ ðm þ 1Þιn þ2; NL ¼ 7n þh þ 4.

ð11Þ

Proof. Applying Lemma 1 and Schur complement to conditions (10), we can derive the following inequalities: 2 3

Γ 12

 γ2I n n n

Γ 13 Γ 14 Γ 15 Γ 23 Γ 24 Γ 25 7 7 7 Γ 33 0 0 7 7 n Γ 44 0 7 5 n n Γ 55

T ~ T ~ T ~ ðkÞT Σ þ Σ T Δ ~ þΣ Δ 2 j 2 j ðkÞΣ 1 þ Σ 3 Δ j ðkÞ Σ 4 þ Σ 4 Δ j ðkÞΣ 3 o 0 T 1

ð16Þ

Based on (15) and (16), we obtain the following inequalities: 2 3 T T P i  LTij ATij W T σ Ei WT A~ ij W T 6 7 6 7 T T 6 n 7  γ2I BTij W T σ B~ ij W T F i WT 6 7 6 7  1 T 6 n 7 o 0: n  WP W 0 0 l 6 7 6 7 1 T 6 n 7 n n  WP W 0 l 4 5 1 T n n n n  WP l W

Then, the parameters of the filter in (4) are given by

n

n

T

F i WT

3

Due to P l 40, we can construct the following inequality:

5

T T ATi W T1 þ δ C Ti K~ j ATi W T3 þ δ C Ti K~ j 7 Γ 13 ¼ 6 4 T T T T 5 Aj W 2  δ C Tj K~ j ATj W T2  δ C Tj T K~ j " # " # T ~T T ~T ET W T ET W T Γ 14 ¼ σ C i K j σ C i K j ; Γ 15 ¼ i 1 i 3 0 0 0 0 " # " # T T 0 0 εC N 0 Γ 16 ¼ 1 i j ; Γ 17 ¼ ε2 C Tj N Tj 0 0 0 h i h i T T Γ 23 ¼ BTi W T1 þ δ DTi K~ j BTi W T3 þ δ DTi K~ j ; Γ 26 ¼ ε1 DTi N Tj 0 h i h i T T Γ 24 ¼ σ DTi K~ j σ DTi K~ j ; Γ 25 ¼ F Ti W T1 F Ti W T3 2 3 2 3 0 δ W 2 Mj 0  δ W 2 Mj 4 5 4 5; ; Γ 37 ¼ Γ 36 ¼ 0 δ W 2 Mj 0  δ W 2 Mj " # 0 σ W 2 Mj Γ 46 ¼ 0 σ W M 2 j " # P 11l  W 1  W T1 P 12l  W 2  W T3 Γ 33 ¼ Γ 44 ¼ Γ 55 ¼ n P 22l  W 2  W T2 " # " #  ε1 I  ε2 I 0 0 Γ 66 ¼ ; Γ 77 ¼ : n  ε1 I n  ε2 I

Γ 11

n

T

Ei WT

ð15Þ 2

#

2

6 n 6 6 6 n 6 6 n 4

ð14Þ

ð12Þ

where

Σ 1 ¼ ½Nj C i 0 N j Di 0 0 0 0 0 0 Σ 2 ¼ ½0 0 0 δ M Tj W T2 δ MTj W T2 σ MTj W T2 σ MTj W T2 0 0 Σ 3 ¼ ½0 Nj C j 0 0 0 0 0 0 0 Σ 4 ¼ ½0 0 0  δ MTj W T2  δ MTj W T2 0 0 0 0: Define the partition of matrices Pi, W as follows, for all iA Ω: " # " # P 11i P 12i W1 W2 ; W¼ : ð13Þ Pi ¼ n P 22i W3 W2

Remark 3. Comparing with the existing works, the result obtained in Theorem 2 has two advantages. On the one hand, the channel fading problem is considered which covers the packet dropouts as a special case. On the other hand, the filter designed here is robust to the filter parameters perturbation.

5. Simulation example Example 1. We will give a numerical example to illustrate the feasibility of the proposed method. The discrete-time model is given as (ι ¼ 2) 2 3 2 3 2 3 0:7 0:1 0 0:01 0 0:03 0:3 6 7 6 7 6 7 0 5; B1 ¼ 4 0:1 5 A1 ¼ 4 0 0:8 0:1 5; E1 ¼ 4 0:02 0:02 0:1 0 0:5 0:02 0 0:03 0:2   0:3 1 0 0 ; D1 ¼ ; F 1 ¼ ½0:01 0 0:02T C1 ¼ 0:4 0 1 1 M 1 ¼ ½1 0:7 0:9T ; N 1 ¼ ½0:03 0:01; L1 ¼ ½1 1 1 2 3 2 3 2 3 0:8 0 0:2 0:02 0:02 0:03 0:2 6 7 6 6 7 0:03 0 7 A2 ¼ 4 0:1 0:6 0:1 5; E2 ¼ 4 0 5; B2 ¼ 4 0:1 5 0 0 0:7 0 0:02 0:02 0:3   0:4 1 1 0 ; D2 ¼ C2 ¼ ; F 2 ¼ ½0 0:02 0:01T 0:2 0 1 1 M 2 ¼ ½0:8 0:6 0:7T ;

N 2 ¼ ½0:01 0:03;

L 2 ¼ ½ 1 1 1

R. Lu et al. / Neurocomputing 174 (2016) 553–559 4

0.16

ω(k)

2

0.14

0

0.12

−2

0.1

−4

0

20

40

60

80

557

0.08

100

k

0.06

1.4

δ(k)

0.04

1.2

0.02

1

0

0.8 0

20

40

60

80

100

−0.02

0

20

40

60

80

100

80

100

k

k

Fig. 3. The states of the filter.

Fig. 1. The Gaussian white noise and the channel fading. 0.2

0.45 0.4

0.15

0.35 0.3

0.1

0.25 0.2

0.05

0.15 0.1

0

0.05 −0.05

0 0

20

40

60

80

100

k

−0.05

0

20

40

60 k

Fig. 2. The states of the system.

Fig. 4. The filtering error.

Δ~ 1 ðkÞ ¼ sin ðkÞ;

Δ~ 2 ðkÞ ¼ cos ðkÞ:

Assume the membership functions satisfy g 1 ðkÞ ¼

1  sin ðx1 ðkÞÞ ; 2

g 2 ðkÞ ¼ 1  g 1 ðkÞ:

Choosing the external disturbance as νðkÞ ¼ expð  0:2kÞ sin ð0:1kÞ, and considering δ ¼ 0:9; σ ¼ 0:1, then by solving the LMIs in Theorem 2, we get the parameters of the non-fragile filter as follows: 2 3 2 3 0:3405 0:2565 0:4117  0:1540 6 7 6 0:0266 7 K 1 ¼ 4 0:0703 0:2622 5; K 2 ¼ 4 0:1685 5; 0:2723 0:1845 0:2552  0:1778 and the minimal passive performance index γ 2min ¼ 2:7870. Assume the initial conditions of the system and the filter are all ^ zeros, i.e., xð0Þ ¼ xð0Þ ¼ ½0 0 0T , and the white noise ωðkÞ and the fading condition δðkÞ are shown in Fig. 1. Then the trajectories of the system states and the filter states are shown in Figs. 2 and 3, respectively. Fig. 4 shows the trajectory of the filtering error. Example 2. Consider a tunnel diode circuit shown in Fig. 5. The circuit is governed by the following nonlinear model: C x_ 1 ðtÞ ¼ 0:002x1 ðtÞ  0:01x31 ðtÞ þ x2 ðtÞ Lx_ 2 ðtÞ ¼ x1 ðtÞ  Rx2 ðtÞ þ νðtÞ

Fig. 5. Tunnel diode circuit.

yðtÞ ¼ ½1 0xðtÞ þ νðtÞ zðtÞ ¼ ½1 0xðtÞ where x1 ðtÞ ¼ V c ðtÞ, and x2 ðtÞ ¼ iL ðtÞ. The T–S fuzzy model is used to deal with this nonlinear system with the following parameters [32]:  A1 ¼  A2 ¼

0:9887

0:9024

 0:0180

0:8100

0:90337

0:8617

 0:0172

0:8103

C 1 ¼ C 2 ¼ ½1 0;



 ;

B1 ¼

B2 ¼

0:0091 0:0181

D1 ¼ D2 ¼ 1

E1 ¼ E2 ¼ F 1 ¼ F 2 ¼ 0;



0:0181 

;

0:0093

L1 ¼ L2 ¼ ½1 0:



558

R. Lu et al. / Neurocomputing 174 (2016) 553–559 1.4

−3

14

δ(k) 1.3

x 10

12

1.2

10

1.1

8

1

6 4

0.9

2

0.8

0 0.7

0

20

40

60

80

100

k

−2

0

20

40

60

80

100

80

100

k

Fig. 6. The channel fading.

Fig. 8. The states of the filter. 0.35

0.8

0.3 0.25

0.6

0.2

0.4

0.15

0.2

0.1

0 0.05

−0.2

0 −0.05

−0.4 0

20

40

60

80

100

k

−0.6

Fig. 7. The states of tunnel diode circuit.

20

40

60 k

Fig. 9. The filtering error.

The membership functions satisfy 8 1 > > x1 ðkÞ þ 1 if  3 r x1 ðkÞ o0 > > <3 1 h1 ðkÞ ¼  x1 ðkÞ þ 1 if 0 rx1 ðkÞ r 3 > > > > : 3 0 otherwise h2 ðkÞ ¼ 1  h1 ðkÞ: Assume   1 0:8 M1 ¼ ; M2 ¼ ; 0:7 0:6 Δ~ ðkÞ ¼ cos ðkÞ:

0

the original signal well. Fig. 9 shows the trajectory of the filtering error.

6. Conclusions N 1 ¼ N 2 ¼ 0:01;

Δ~ 1 ðkÞ ¼ sin ðkÞ;

2

Considering the external disturbance as νðkÞ ¼ expð  0:07kÞ sin ð0:1kÞ, and supposing δ ¼0.95, σ ¼0.1, by solving the LMIs in Theorem 2, we get the parameters of the non-fragile filter as follows:   1:4187 1:3327 K1 ¼ ; K2 ¼ ; 0:0279 0:0291 and the minimal passive performance index γ 2min ¼ 2:6488. Let the initial conditions of the tunnel circuit system and the ^ filter be zero, i.e., xð0Þ ¼ xð0Þ ¼ ½0 0T , and the channel fading be shown in Fig. 6. Then the states of the tunnel diode circuit are shown in Fig. 7, and from Fig. 8 we can see that the filter can track

In this paper, we have investigated the problem of passive filter design for nonlinear stochastic systems with channel fading. The nonlinear stochastic system has been represented by the T–S model and the channel fading has been described by a random process. Sufficient condition which guarantees that the filtering error system is stochastically stable with passive performance has been derived via Lyapunov method, then, the parameters of the non-fragile filter have been obtained by solving a set of LMIs. Examples have been given to illustrate the effectiveness of the results. How to analyze the blind signal obtained via the fading channel [25–27] and how to design the optimal controller with the imperfect signal [28–35] are interesting issues to be studied in the future.

R. Lu et al. / Neurocomputing 174 (2016) 553–559

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[30] J. Qiu, G. Feng, H. Gao, Static-output-feedback H1 control of continuous-time T–S fuzzy affine systems via piecewise Lyapunov functions, IEEE Trans. Fuzzy Syst. 21 (2) (2013) 245–261. [31] J. Qiu, H. Tian, Q. Lu, H. Gao, Nonsynchronized robust filtering design for continuous-time T–S fuzzy affine dynamic systems based on piecewise Lyapunov functions, IEEE Trans. Cybern. 43 (6) (2013) 1755–1766. [32] S. Nguang, W. Assawinchaichote, H 1 filtering for fuzzy dynamical systems with D stability constraints, IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50 (11) (2003) 1503–1508. [33] D. Liu, X. Yang, D. Wang, Q. Wei, Reinforcement-learningbased robust controller design for continuous-time uncertain nonlinear systems subject to input constraints, IEEE Trans. Cybern. 45 (7) (2015) 1372–1385. [34] D. Liu, H. Li, D. Wang, Error bounds of adaptive dynamic programming algorithms for solving undiscounted optimal control problems, IEEE Trans. Neural Netw. Learn. Syst. 26 (6) (2015) 1323–1334. [35] D. Liu, D. Wang, F. Wang, H. Li, X. Yang, Neural-networkbased online HJB solution for optimal robust guaranteed cost control of continuous-time uncertain nonlinear systems, IEEE Trans. Cybern. 44 (12) (2014) 2834–2847. Renquan Lu received the Ph.D. degree in Control Science and Engineering from Zhejiang University, Hangzhou, China, in 2004. Currently, he is a Full Professor in the Institute of Information and Control, Hangzhou Dianzi University, Hangzhou, China. His research interests include robust control, singular systems, and complex systems.

Shuang Liu was born in Shandong Province, China, in 1989. She received the B.S. degree in Electronic Engineering and Automation from Qufu Normal University, Rizhao, China, in 2013. She is currently working toward the M.S. degree in Hangzhou Dianzi University. Her main research interests include networked control systems and fuzzy systems.

Hui Peng was born in Hunan province, China, in 1990. She received the B.S. degree in Automation from Hangzhou Dianzi University, Hangzhou, China, in 2013. She is currently working toward the Ph.D. degree in Control Science and Engineering at the Institute of Information and Control, Hangzhou Dianzi University. Her research interests include Networked control systems, fuzzy systems and distributed sensor networks.

Yong Xu was born in Zhejiang Province, China, in 1983. He received the B.S. degree from Nanchang Hangkong University, Nanchang, China, in 2007, the M.S. degree from Hangzhou Dianzi University, Hangzhou, China, in 2010, and the Ph.D. degree from Zhejiang University, Hangzhou, China, in 2014. He was visiting internship student with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, from June 2013 to November 2013. Now he is a lecturer with School of Automation, at Guangdong University of Technology, Guangzhou, China. His research interests include networked controlsystems, state estimation, and Markov jump systems. Kan Xie was born in Hubei, China. He received the M.S. degree in Software Engineering from the South China University of Technology, Guangzhou, China, in 2009. Currently, he is pursuing the Ph.D. degree in Intelligent Signal and Information Processing at the Guangdong University of Technology, Guangzhou, China. His research interests include machine learning, nonnegative signal processing, blind signal processing, automation and biomedical signal processing.