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State Estimation of Discrete-Time Takagi–Sugeno Fuzzy Systems in a Network Environment Hui Zhang and Junmin Wang, Senior Member, IEEE

Abstract—In this paper, we investigate the H ∞ filtering problem of discrete-time Takagi–Sugeno (T-S) fuzzy systems in a network environment. Different from the well used assumption that the normalized fuzzy weighting function for each subsystem is available at the filter node, we consider a practical case in which not only the measurement but also the premise variables are transmitted via the network medium to the filter node. For the network characteristics, we consider the multiple packet dropouts which are described by using a Markov chain. It is assumed that the filter uses the most recent packet. If there are packet dropouts occurring, the filter adopts the information for the last received packet. Suppose that the mode of the Markov chain is ordered according to the number of consecutive packet dropouts from zero to a preknown maximal value. For each mode of the Markov chain, it only has at most two jumping actions: 1) jump to the first mode and the current packet is transmitted successfully and 2) jump to the next mode and the number of consecutive packet dropouts increases by one. We aim to design mode-dependent and fuzzy-basis-dependent T-S fuzzy filter by using the transmitted packet subject to the described network issue. With the augmentation technique, we obtain a stochastic filtering error system in which the filter parameters and the Markovian jumping variable are all involved. A sufficient condition which guarantees the stochastic stability and the H ∞ performance is derived with the Lyapunov method. Based on the sufficient condition, we propose the filter design method and the filter parameters can be determined by solving a set of linear matrix inequalities (LMIs). A tunnel-diode circuit in a network environment is presented to show the effectiveness and the advantage of the proposed design approach. filter design, linear matrix Index Terms—H ∞ inequalities (LMIs), multiple packet dropouts, networked control systems (NCSs), Takagi–Sugeno (T-S) fuzzy system.

I. I NTRODUCTION ETWORKED control systems (NCSs) have attracted considerable attention during the past decade since NCSs have significant advantages, such as the flexibilities in the location and diagnosis, over the traditional point-to-point control

N

Manuscript received December 3, 2013; revised May 19, 2014 and August 18, 2014; accepted August 31, 2014. The work of H. Zhang was supported by the National Science Foundation of China under Grant 61403252. This paper was recommended by Associate Editor H. Zhang. H. Zhang is with the Merchant Marine College, Shanghai Maritime University, Shanghai 201306, China, and also with the Department of Mechanical and Aerospace Engineering, Ohio State University, Columbus, OH 43210 USA (e-mail: [email protected]). J. Wang is with the Department of Mechanical and Aerospace Engineering, Ohio State University, Columbus, OH 43210 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2354431

schemes (see [1]–[7]). However, due to the limited bandwidth of the communication medium and the reliability of the communication, the network would also lead to some issues, among which the network-induced delay and the packet dropouts are the prominent ones [8]. To deal with the cons, a lot of efforts have been paid to the NCSs. Generally, the efforts can be classified into two categories: active one and passive one. In the active way, the bandwidth of the communication medium can be increased or the network protocol can be redesigned and optimized to reduce the networkinduced issues [9]–[11]. Whereas, for the passive direction, network-issue-tolerant controllers and filters can be designed to compensate for, or reduce the effects of the network-induced issues [12], [13]. Since the passive methods usually do not increase the system’s cost, the second direction has received more attention. To apply the second direction, the network characteristics should be studied firstly and the effects of the network-induced issues should be modeled. It can be seen from [12] and [14] that the Markov chain was utilized to model the network-induced delays such that the jumping information between two different modes can be used to facilitate the controller design. In [15] and [16], the network-induced delays were assumed to be within a range and the range can be obtained before the design. For the packet dropouts, a simple method is to model the packet transmission as a switch [5]. If the packet is transmitted successfully, the switch is on and if the packet is missing, the switch is off. Then, a Bernoulli process can be used to represent the on/off packet transmission. To extract more information for the Bernoulli process, the Markov chain can be also applied to model the packet dropouts [13] in which the stabilization controller design method was proposed. The filtering problem is an important branch of the modern control theory since a soft sensor is applied to estimate the unknown state or a filter is designed to recover the signal from the noise-contaminated measurements. The Kalman filter [17] became well known after the successful application to aerospace systems. Since the network-induced issues would degrade the performance of the designed filter if the issues are not considered during the design, in a network environment, the Kalman filter has also been modified and improved (see [18], [19]). Beside the Kalman filter, another kind of well-known filters are named the H∞ filters which are powerful to reduce the effect of unknown disturbances to the desired signals (see [20], [21]). Considering the networkinduced delays and multiple sensor faults, Zhang et al. [4] studied the H∞ filtering problem for linear discrete-time systems. The multiple sensor faults were modeled via multiple

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independent Bernoulli processes. Multiple packet dropouts were considered for the optimal H∞ filtering in NCSs in [22]. Though a novel skill was proposed to handle the multiple packet dropouts [22], [23], the upper bound of the maximal consecutive packet dropouts was not considered in the analysis. Since many unreasonable cases such as the infinite consecutive packet dropouts were also included in the modeling, the result may be conservative. In the above introduction, we have focused on linear systems. However, nonlinearities are common in complex practical systems. It infers from [24]–[27] that smooth nonlinearities can be approximated via the Takagi–Sugeno (T-S) fuzzy model in which each subsystem is linear [28]. Since some theories from linear control systems can be adapted to T-S fuzzy systems, the study of T-S fuzzy systems has been a popular topic since it was first developed. The stability and design issues for T-S fuzzy system was discussed in [29]. As the stability conditions are only sufficient, relaxed and improved conditions can be seen in [30] and [31]. Jiang et al. [32] exploited the integrated fault estimation and accommodation design for discrete-time T-S fuzzy systems subject to actuator faults. Moreover, a bunch of results on delayed T-S fuzzy systems can be seen in [2] and [33]. With respect to the filter and observer design, Li and Yang [34] investigated the H∞ filter design for T-S fuzzy systems with unknown or partially unknown membership functions. The robust input observer design for unknown inputs T-S system was developed in [35]. New results on H∞ filter design problem for T-S fuzzy delayed models were claimed in [36]. The decentralized filtering for discretetime interconnected T-S fuzzy systems subject to time-varying delay was reported in [37]. In addition, the network control and filtering for T-S fuzzy systems can be seen in [38] and [39]. Here, we take the missing measurements also into the network framework. For most of the work on the networked T-S fuzzy systems, it is generally assumed that the measurements are subject to the network-induced issues but the normalized fuzzy weighting functions are ideal for the controllers or the filters. This assumption is unreasonable since the normalized fuzzy weighting functions or the premise variables are also transmitted via the network medium if the controllers or the filters need this kind of information. Therefore, the normalized fuzzy weighting functions or the premise variables are subject to the network-induced issues simultaneously with the measurements. Due to the existing problems of the filter design for the networked T-S fuzzy system, we investigate the H∞ filtering of discrete-time T-S fuzzy systems by considering multiple packet dropouts. The multiple packet dropouts are modeled via a homogenous Markov chain. Since the mode of the Markov chain is finite, the upper bound of the maximal consecutive packet dropouts is accommodated. A reasonable and practical data transmission strategy is that the measurements and the premise variables are packed together and sent to the filter. The filter always uses the most recent packet to estimate the signal. Since the values of the normalized fuzzy weighting functions may be different at the filter node and at the sensor node due to the multiple packet dropouts, the values at different nodes are treated differently. In order to design the

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fuzzy-basis-dependent filter, another difference from the published work is that the input for each sub-filter is not the overall measurement but the measurement for the corresponding original subsystem. With the modeling and the communication strategy, we derive a stochastic filter error system. Based on the analysis of the system, the mode-dependent and fuzzybasis-dependent filter parameters can be computed by solving a sequence of linear matrix inequalities (LMIs). Simulation results verify the correctness of the theory analysis and the effectiveness of the proposed design method. II. P ROBLEM F ORMULATION In this paper, the plant considered is a discrete-time complex smooth nonlinear system which is approximated by the following discrete-time T-S fuzzy model. Plant Rule i: IF θ1,k is Mi1 and θ2,k is Mi2 . . . and θs,k is Mis , THEN the T-S subsystem is represented by xk+1 = Ai xk + Bi ωk yk = Ci xk + Di ωk (1) zk = Ei xk + Fi ωk i = 1, . . . , κ where k is the sampling instant, θk = [θ1k , θ2k , . . . , θsk ] is the premise parameter value vector which is assumed to be a function of the measurements, Mij for j = 1, . . . , s denotes the fuzzy set, xk ∈ Rn stands for the system state vector at the time instant k, ωk ∈ Rm represents the exogenous input which is assumed to be energy bounded, yk ∈ Rq is the measurement of the system, and zk ∈ Rp indicates the signal to be estimated. Ai , Bi , Ci , Di , Ei and Fi are known matrices with appropriate dimensions. It can be seen that there are s premise parameters and κ T-S fuzzy rules. In order to study the overall T-S fuzzy system, the normalized fuzzy weighting function for each subsystem should be calculated. Here, the normalized fuzzy weighting functions are defined as s j=1 μij (θj,k ) (2) hi (θk ) = κ  s μ (θ ) ij j,k j=1 i=1

where μij (θj,k ) is the grade of membership of the premise parameter θj,k belonging to the Mij in the ith fuzzy rule. Since the grade of membership is nonnegative, the normalized fuzzy weighting functions satisfy 0 ≤ hi (θk ) ≤ 1 κ  (3) hi (θk ) = 1. i=1

Then, the overall T-S fuzzy system is represented by κ  (hi (θk )(Ai xk + Bi ωk )) xk+1 = i=1

= A(h¯ k )xk + B(h¯ k )ωk κ  yk = (hi (θk )(Ci xk + Di ωk )) i=1

= C(h¯ k )xk + D(h¯ k )ωk

(4)

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Fig. 1. Schematic of the filtering problem of the T-S fuzzy system in a network environment (zk is not transmitted via the network medium).

zk =

κ  (hi (θk )(Ei xk + Fi ωk )) i=1

= E(h¯ k )xk + F(h¯ k )ωk   h¯ k = h1 (θk ) h2 (θk ) · · · hκ (θk ) ∈ ρ. Here, ρ denotes the set of h¯ k . As the network transmission brings considerable advantages, we study the filtering problem in a network environment, that is, the filter to be designed and the plant are located at different places. The signals between them are transmitted via the network medium. Different from the T-S fuzzy networked system in [40] and [41], as shown in Fig. 1, not only the measurement but also the premise parameter vector is transmitted via the network medium to the filter. We assume that measurement yk and the vector θk are in the same packet. Suppose that the network transmission capacity is limited and the packet dropouts occur during the transmission. To better design the filter, the network characteristics can be studied before the design. In order to facilitate the filter design, we further make the following assumptions. Assumption 1: The multiple packet dropouts occur randomly but the probabilities of successful transmission and data missing at each sampling instant satisfy a Markov chain. Assumption 2: If the packet is missing at the time instant k, the filter uses the previously received data. Under these assumptions, we have the following facts. 1) If the packet is transmitted successfully from the plant to the filter, the information to be used for the filter at the time instant k is     yˆ k y = k (5) θk θˆk where yˆ k denotes the measurement to be used at the time instant k for the filter and θˆk is the premise parameters vector to be used at the time instant k for the filter. 2) If the packet is not transmitted successfully from the plant to the filter, the information to be used for the filter at the time instant k is     yˆ k−1 yˆ k = ˆ . (6) θˆk θk−1 It is necessary to mention that if the packet is missing, the information to be used for the filter is the previously utilized information. However, the previously utilized information may be the successfully transmitted information (updated packet)

Fig. 2.

3

Illustration of the packet dropouts and the equivalent effects.

at k − 1 or may be the packet at several instants before. The challenge here is that how to connect the yˆ k−1 with the measurement expression. To deal with the challenge, we analyze the packet dropouts and the equivalent effects firstly. As shown in Fig. 2, if there is only a single packet dropout, the information to be used at the time instant k is     yˆ k yk−1 = . (7) θk−1 θˆk If there are two consecutive packet dropouts, the information   y to be used for the two consecutive sampling instants is k−1 θk−1   yk−2 , respectively. Thus, it infers from the observations and θk−2 that the effect of the multiple packet dropouts is similar with the packet delays. Consequently, a generalized form of the information to be used for the filter is expressed as     yˆ k yk−τk = (8) θk−τk θˆk where τk is a variable taking values in the set D = {0, 1, . . . d} in which d is the maximal consecutive packet dropouts. Obviously, the value of d is bounded and can be derived by studying the network characteristics. The next challenge is how to employ the delay information to help the filter design. For a random packet, there are only two choices: successful transmission and failure transmission. If the packet is successfully transmitted, the value of τk equals to zero. If the packet is missing, the value of τk equals to τk−1 + 1. With Assumption 1, Fig. 3 illustrates the induced delay issue of the multiple Markovian packet dropouts. pij denotes the probability of successful transmission at the current instant if there are i consecutive packet dropouts just occurred. 1 − pij is the probability of unsuccessful transmission if there are i consecutive packet dropouts just occurred, that is, the probability of i + 1 consecutive packet dropouts if there are i consecutive packet dropouts just occurred. Therefore, τk is a Markovian jump parameter taking value in the set D with the transition probability matrix  = [πij ]i,j∈C ={1,2,...,d+1} ⎡ p00 1 − p00 0 ⎢ p10 0 1 − p10 ⎢ ⎢ .. .. = ⎢ ... . . ⎢ ⎣ p(d−1)0 0 0 1 0 0

··· ··· .. .

0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥. ⎥ · · · 1 − p(d−1)0 ⎦ ··· 0

(9)

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zˆk =

κ 

hj (θk−τk )Cfj (τk )ˆxk

j=1

+ Fig. 3.

Illustration of the effect of Markovian packet dropouts.

In this paper, we aim to design a full-order mode-dependent and fuzzy-basis-dependent filter. With the latest updated information, the filter is described by Filter Rule i: IF θ1,k−τk is Mi1 , and θ2,k−τk is Mi2 , . . . and θs,k−τk is Mis , THEN the subsystem of the filter is expressed as yk−τk = Ci xk−τk + Di ωk−τk xˆ k+1 = Afi (τk )ˆxk + Bfi (τk )yk−τk zˆk = Cfi (τk )ˆxk + Dfi (τk )yk−τk

(10)

i = 1, . . . , κ, τk ∈ D where xˆ k ∈ denotes the state vector of the full-order modedependent filter and zˆk ∈ Rp is the estimation zk . In addition, Afi (τk ), Bf (τk ), Cfi (τk ), and Df (τk ) are filter parameters to be determined. Note that there are 4κ(d + 1) parameters to be designed. Note that the input of the filter is subject to the delay τk . To deal with the delay, we augment the overall T-S fuzzy system in (1) and obtain the augmented system as ¯ h¯ k )¯xk + B( ¯ h¯ k )ωk (11) x¯ k+1 = A( ⎡

⎤

⎡

⎤ B(h¯ k ) ⎢ xk−1 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ .. ⎥ ¯ ¯ ⎢ ⎥ x¯ k = ⎢ . ⎥, B(hk ) = ⎢ ... ⎥ ⎥ ⎢ ⎢ ⎥ ⎣ xk−d+1 ⎦ ⎣ 0 ⎦ xk−d 0 ⎡ ⎤ A(h¯ k ) 0 · · · 0 0 ⎢ I 0 ··· 0 0⎥ ⎢ ⎥ . . . .⎥ ¯ h¯ k ) = ⎢ A( ⎢ .. .. . . . .. .. ⎥. ⎢ ⎥ ⎣ 0 0 ··· 0 0⎦ 0 0 ··· I 0 xk

With the augmented system, the utilized measurement yk−τk for each subsystem of the filter is expressed as yk−τk = Ci xk−τk + Di ωk−τk ¯ k )¯xk + Di ωk−τk = Ci C(τ (12) where

 ¯ k) = 0 · · · 0 C(τ (τ

I

k +1)th block

0 ··· 0



i =1, . . . , κ. Then, the overall T-S fuzzy system of the filter is written as κ  xˆ k+1 = hj (θk−τk )Afj (τk )ˆxk j=1

+ +

κ 

hj (θk−τk )Bfj (τk )

κ 

j=1

j=1

κ 

κ 

j=1

hj (θk−τk )Bfj (τk )

j=1

hj (θk−τk )Dfj (τk )

κ 

j=1

j=1

κ 

κ 

hj (θk−τk )Dfj (τk )

j=1

¯ k )¯xk hj (θk−τk )Cj C(τ hj (θk−τk )Dj ωk−τk . (13)

j=1

The main function of the filter is to estimate the signal zk . To evaluate the performance, we define the filtering error as ek := zk −ˆzk and obtain the filtering error system represented by     ξk+1 = Aˆ h¯ k , h¯ k−τk , τk ξk + Bˆ h¯ k , h¯ k−τk , τk ωˆ k (14)     ek = Eˆ h¯ k , h¯ k−τk , τk ξk + Fˆ h¯ k , h¯ k−τk , τk ωˆ k where

   x¯ k ωk , ωˆ k = xˆ k ωk−τk     ¯ h¯ k   0 A ˆA h¯ k , h¯ k−τk , τk =     ¯ k ) Af h¯ k−τk , τk B1f h¯ k−τ , τk C(τ     k   B¯ h¯ k  0  Bˆ h¯ k , h¯ k−τk , τk = 0 B2f h¯ k−τk , τk        ¯ k) Eˆ h¯ k , h¯ k−τk , τk = E¯ h¯ k − D1f h¯ k−τk , τk C(τ   − Cf h¯ k−τk , τk      Fˆ h¯ k , h¯ k−τk , τk = F(h¯ k ) −D2f h¯ k−τk , τk       E¯ h¯ k = E h¯ k 0 · · · 0 0 κ    Af h¯ k−τk , τk = hi (θk−τk )Afi (τk ) 

ξk =

Rn

where

+

κ 

¯ k )¯xk hj (θk−τk )Cj C(τ hj (θk−τk )Dj ωk−τk

i=1

  B1f h¯ k−τk , τk =

κ 

hi (θk−τk )Bfi (τk )

κ 

i=1

hi (θk−τk )Ci

i=1

κ κ     B2f h¯ k−τk , τk = hi (θk−τk )Bfi (τk ) hi (θk−τk )Di i=1





Cf h¯ k−τk , τk =   D1f h¯ k−τk , τk =

κ  i=1 κ 

i=1

hi (θk−τk )Cfi (τk ) hi (θk−τk )Dfi (τk )

i=1





D2f h¯ k−τk , τk =

κ  i=1

κ 

hi (θk−τk )Ci

i=1

hi (θk−τk )Dfi (τk )

κ 

hi (θk−τk )Di .

i=1

In order to reduce the filtering error, the filtering error should be bounded, that is, the filtering error system in (14) should be stable firstly. However, since a stochastic variable τk is involved, it is a stochastic system and the traditional asymptotical stability is not applicable. Therefore, we introduce the definition of the stochastic stability. Definition 1 [12], [42]: For the unforced system in (14), it is stochastically stable if for every finite initial values ξ0 and τ0 , there exists a finite W = W T > 0 such that the following condition holds:  ∞  2 ξk  |ξ0 , τ0 < ξ0T Wξ0 . (15) E k=0

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We can see from (14) that the value of the filtering error is not only affected by the stability of the filtering error system, but also by the exogenous input vector ωˆ k . Since the value of the exogenous input vector is unknown but l2 -bounded, in order to attenuate the effect of the exogenous input vector and minimize the filtering error, we adopt the H∞ filtering strategy and define the stochastic stability with an H∞ performance. Definition 2: Given a scalar γ > 0, the filtering error system in (14) is stochastically stable with an H∞ performance index γ if the filtering error system is stochastically stable and, under zero initial conditions, the following condition holds:   eE2 < γ ωˆ 2 (16) for ωˆ k ∈ l2 [0, ∞), where eE2 : =  all nonzero  T  ∞ E e e is the expected energy value of the filtering k=0 k k    ∞  T  ˆ k ωˆ k represents the energy error ek and ωˆ 2 := k=0 ω of the external input ωˆ k . III. M AIN R ESULTS In this section, first we study the stochastic stability and H∞ performance for the filtering error system in (14) by assuming that the parameters of filters are given. Based on the derived conditions, the filter design method will be proposed and the calculation of the filter parameters will be given. A. Stochastic Stability and H∞ Performance Analysis In order to facilitate the stability and H∞ performance analysis, we assume that the parameters of the fuzzy filter in the form of (10) are given. By using a mode-dependent and fuzzy-weighting-function-dependent Lyapunov function and Definition 1, we derive sufficient condition which have the capacity to guarantee the stochastic stability and the H∞ performance of the filtering error system in (14). Theorem 1: Given a constant γ , the filtering error system in (14) is stochastically stable with an H∞ performance level γ , if there exist matrices P(h¯ k , i) = PT (h¯ k , i) > 0, G(h¯ k−τk , i), any h¯ k ∈ ρ, any h¯ k+1 ∈ ρ, and any h¯ k−τk ∈ ρ such that the following condition is achievable: ⎡ ˆ h¯ k , h¯ k−τk , i − 1) 1 0 G(h¯ k−τk , i)A( ⎢ ∗ −I ˆ h¯ k , h¯ k−τk , i − 1) E( ⎢ ⎣ ∗ ∗ −P(h¯ k , i) ∗ ∗ ∗ ⎤ ˆ h¯ k , h¯ k−τk , i − 1) G(h¯ k−τk , i)B( ⎥ ˆ h¯ k , h¯ k−τk , i − 1) F( ⎥ < 0 (17) ⎦ 0 2 −γ I where ¯ k+1 , i) − G(h¯ k−τk , i) − GT (h¯ k−τk , i) 1 = P(h ¯ k+1 , j) = P(h

d+1 

πij P(hk+1 , j)

j=1

i = 1, . . . , d + 1.

5

Moreover, h¯ k in P(h¯ k , i) means that the matrix is a function of h¯ k and i in P(h¯ k , i) means that the matrix is modedependent. Similarly, G(h¯ k−τk , i) is also mode-dependent and fuzzy-basis-dependent. The explicit expressions will be determined later. ¯ k+1 , i), the Proof: If G(h¯ k , i) in (17) is chosen as P(h condition in (17) becomes ⎡ ˆ h¯ k , h¯ k−τk , i − 1) ¯ k+1 , i) 0 P(h ¯ k+1 , i)A( −P(h ⎢ ¯ ¯ k−τk , i − 1) ˆ ∗ −I E( h , k h ⎢ ⎣ ∗ ∗ −P(h¯ k , i) ∗ ∗ ∗ ⎤ ¯ k+1 , i)B( ˆ h¯ k , h¯ k−τk , i − 1) P(h ⎥ ˆ h¯ k , h¯ k−τk , i − 1) F( ⎥ < 0. ⎦ 0 2 −γ I

(18)

On the other hand, if the condition in (18) holds, performing a congruence transformation to the condition in (18) by diag{P¯ −1 (hk+1 , i), I, I, I}, we have ⎡ ˆ h¯ k , h¯ k−τk , i − 1) −P¯ −1 (hk+1 , i) 0 A( ⎢ ˆ h¯ k , h¯ k−τk , i − 1) ∗ −I E( ⎢ ⎣ ∗ ∗ −P(h¯ k , i) ∗ ∗ ∗ ⎤ ˆ h¯ k , h¯ k−τk , i − 1) B( ˆ h¯ k , h¯ k−τk , i − 1) ⎥ F( ⎥ < 0. ⎦ 0 2 −γ I

(19)

It infers from 1 < 0 that the matrix G(h¯ k , i) is nonsingular. Then, performing a congruence transformation to the condition in (19) by diag{GT (h¯ k , i), I, I, I}, we get the following equivalence: ⎡ −G(h¯ k , i)P¯ −1 (hk+1 , i)GT (h¯ k , i) ⎢ ∗ ⎢ ⎣ ∗ ∗ ˆ h¯ k , h¯ k−τk , i − 1) 0 G(h¯ k , i)A( ˆ h¯ k , h¯ k−τk , i − 1) −I E( ∗ −P(h¯ k , i) ∗ ∗ ⎤ ˆ h¯ k , h¯ k−τk , i − 1) G(h¯ k , i)B( ⎥ ˆ h¯ k , h¯ k−τk , i − 1) F( ⎥ < 0. (20) ⎦ 0 2 −γ I ¯ k+1 , i) − With the fact that (P(h −1 ¯ ¯ ¯ ¯ ≥ 0, it G(hk , i))P (hk+1 , i)(P(hk+1 , i) − G(hk , i)) ¯ k+1 , i) − G(h¯ k , i) − GT (h¯ k , i) ≥ is easy to get P(h −1 −G(h¯ k , i)P¯ (hk+1 , i)GT (h¯ k , i). Then, if the condition in (18) holds, the condition in (17) is also satisfied. Therefore, the condition in (18) is equivalent with the condition (17). If we can prove that the condition in (18) can guarantee

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the stochastic stability with an H∞ attenuation level γ , Theorem 1 is also proved. Consider the following mode-dependent and fuzzy-basisdependent Lyapunov function for the filtering error system: V(ξk , hk , τk = i − 1) = ξkT P(hk , i)ξk , ∀ i = 1, . . . , d + 1 (21) where P(hk , i) = PT (hk , i) > 0 is a Lyapunov weighting matrix. The difference of the Lyapunov function is defined as V(ξk+1 , τk+1 = j − 1, hk+1 , ξk , τk = i − 1, hk ) = E {V(ξk+1 , hk+1 , τk+1 = j − 1)|ξk , τk = i − 1, hk } − V(ξk , hk , τk = i − 1)

(22)

i, j = 1 . . . , d + 1. For the unforced filtering error system, the difference of the Lyapunov function is evaluated as V(ξk+1 , τk+1 = j − 1, hk+1 , ξk , τk = i − 1, hk )     ¯ k+1 , i)Aˆ h¯ k , h¯ k−τk , i − 1 ξk = ξkT Aˆ T h¯ k , h¯ k−τk , i − 1 P(h − ξkT P(hk , i)ξk      ¯ k+1 , i)Aˆ h¯ k , h¯ k−τk , i − 1 = ξkT Aˆ T h¯ k , h¯ k−τk , i − 1 P(h − P(hk , i)) ξk .

(23)

By using Schur complement, it follows from (18) that:    −P¯ (hk+1 , i) P¯ (hk+1 , i) Aˆ h¯ k , h¯ k−τk , i − 1 < 0 (24) ∗ −P(hk , i) which implies that L = Aˆ T (h¯ k , h¯ k−τk , i − 1) ˆ h¯ k , h¯ k−τk , i − 1) − P(hk , i) ¯P(hk+1 , i)A( < 0 and V(ξk+1 , τk+1 = j − 1, hk+1 , ξk , τk = i − 1, hk ) is negative. Then, we have V (ξk+1 , τk+1 = j − 1, hk+1 , ξk , τk = i − 1, hk )   ≤ − min −L(h+ , h, i) ξkT ξk < −αξkT ξk = −αξk 2 (25) which min (·) denotes the minimum eigenvalue of (·) and α = inf{ρmin (−L, i = 1, · · · , d + 1, k ≥ 0}. For any K > 0, summing up the left side and the right side from 0 to K, respectively, we have   K    2 ξk  |ξ0 , τ0 < V ξ0 , τ0 , h¯ 0 /α. (26) E k=0

If K approaches to infinity, we obtain the following inequality: ∞     2 E ξk  |ξ0 , τ0 < V ξ0 , τ0 , h¯ 0 /α ≤ ξ0T Wξ0 (27) k=0

where W is defined as max(P(h¯ 0 , i)/α), ∀h¯ 0 ∈ ρ, i = 1, . . . , d + 1. According to Definition 1, the filtering error system is stochastically stable. Subsequently, to establish the H∞ performance for the filtering error system, it is assumed that the state initial values of the plant and the estimator are all zeros. For nonzero external input, consider the following function for the filtering error system: J := V (ξk+1 , τk+1 = j − 1, hk+1 , ξk , τk = i − 1, hk )   (28) +E eTk ek |τk = i − 1 − γ 2 ωˆ kT ωˆ k .

Considering the trajectory of the filtering error system in (14), the function in (28) is evaluated as    T  1 2 ξk ξk (29) J := ωk ∗ 3 ωk where     1 = Aˆ T h¯ k , h¯ k−τk , i − 1 P¯ (hk+1 , i) Aˆ h¯ k , h¯ k−τk , i − 1     − P¯ (hk , i) + Eˆ T h¯ k , h¯ k−τk , i − 1 Eˆ h¯ k , h¯ k−τk , i − 1     2 = Aˆ T h¯ k , h¯ k−τk , i − 1 P¯ (hk+1 , i) Bˆ h¯ k , h¯ k−τk , i − 1     + Eˆ T h¯ k , h¯ k−τk , i − 1 Fˆ h¯ k , h¯ k−τk , i − 1     3 = Bˆ T h¯ k , h¯ k−τk , i − 1 P¯ (hk+1 , i) Bˆ h¯ k , h¯ k−τk , i − 1     + Fˆ T h¯ k , h¯ k−τk , i − 1 Fˆ h¯ k , h¯ k−τk , i − 1 − γ 2 I. (30) By using the Schur complement again, it follows from (18) that the function J is negative if the condition in (18) holds. With similar steps as in [40], if the value of J is negative, the H∞ performance in (16) is satisfied. According to Definition 2, the filtering error system in (14) is stochastically stable with an H∞ performance index γ . The proof is completed. In the problem formulation, we have summarized the objectives of the design in this paper. To achieve these objectives, we should design the filter which would not only make the filtering error system stochastically stable, but also make the H∞ performance guaranteed. However, since the filter parameters are unknown, the stability and the performance of the filtering error system are not easy to be analyzed. Therefore, we assumed the parameters of the filter were given and tried to derive the condition under which the filtering error system with specific filter parameters would be stochastically stable with an H∞ performance index γ . The sufficient condition was given in Theorem 1. In the proof of Theorem 1, we used the mode-dependent and fuzzy-basis-dependent Lyapunov function. After defining the difference of the Lyapunov function, we obtained the condition which can guarantee the negative difference of the Lyapunov function. For the H∞ performance analysis, we defined a cost function and got the condition which ensured the cost function negative. The proof of the H∞ performance is similar with the one in [7]. However, the system to be analyzed in this paper is a stochastic system which is more complicated than the one in [7]. In addition, we introduced a slack matrix G(h¯ k−τk , i) in Theorem 1. The slack matrix facilitates the filter design and leads to less conservative result. The slack matrix was firstly introduced in [43] for discrete-time linear time invariant systems. Due to the significant advantage on reducing the conservativeness, this technique has been applied to a lot of different plants and different studies (see [44]–[47]). B. Filter Design By assuming all the filter parameters are given, Theorem 1 provides the sufficient condition which guarantees the stochastic stability and the H∞ attenuation performance of the filtering error system. Based on the condition in Theorem 1, we propose the filter design method in this subsection.

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7

⎡

A¯ f B¯ 1f B¯ 2f

⎤ 1 0 2 3 ⎢ ∗ −I E( ˆ h¯ k , h¯ k−τk , i − 1) F( ˆ h¯ k , h¯ k−τk , i − 1) ⎥ ⎥ 0, G(h¯ k−τk , i) = ∗ P3 (h¯ k , i)   G1 (h¯ k−τk , i) HG2 (i) , A¯ f (h¯ k−τk , i − 1), B¯ 1f (h¯ k−τk , i − 1), G3 (h¯ k−τk , i) G2 (i) ¯ 1f (h¯ k−τk , i − 1), D ¯ 2f (h¯ k−τk , i − 1) for any C¯ f (h¯ k−τk , i − 1), D ¯hk ∈ ρ, any h¯ k+1 ∈ ρ, and any h¯ k−τk ∈ ρ such that the condition given in (31) and (32), as shown at the top of the page, is achievable. Moreover, i denotes the Markovian mode and there are d+1 modes, and i − 1 represents the corresponding value of τk . Proof: Note that G(h¯ k , i) is an unknown matrix and it is ˆ h¯ k , h¯ k−τk , i − 1) and B( ˆ h¯ k , h¯ k−τk , i − 1) which coupled with A( contain the filter parameters to be designed in Theorem 1. i) is assumed to In order to decouple these terms, G(h¯ k ,  G1 (h¯ k−τk , i) HG2 (i) . With this spehave a special form G3 (h¯ k−τk , i) G2 (i) ˆ h¯ k , h¯ k−τk , i − 1) and cial structure, 2 equals to G(h¯ k , i)A( ˆ h¯ k , h¯ k−τk , i − 1) if A¯ f (h¯ k−τk , i − 1), 3 equals to G(h¯ k , i)B( B¯ 1f (h¯ k−τk , i−1), and B¯ 2f (h¯ k−τk , i−1) are defined in (32). Thus, the condition in (31) is a sufficient representation of the one in (17). Therefore, if the condition in (31) holds, the filtering error system is stochastically stable with an H∞ performance level γ . Theorem 2 also offers the sufficient condition of the stochastic stability and the H∞ performance of the filtering error system. Note that the filter parameters are not assumed to be given. However, Theorem 2 cannot be applied to the filter parameter design due to the following challenges. 1) The filter parameters are involved in the unknown variables A¯ f (h¯ k−τk , i−1), B¯ 1f (h¯ k−τk , i−1), B¯ 1f (h¯ k−τk , i−1), ¯ 1f (h¯ k−τk , i − 1), and D ¯ 2f (h¯ k−τk , i − 1). C¯ f (h¯ k−τk , i − 1), D Since G(h¯ k−τk , i) is also an unknown variable to be determined, the condition in Theorem 2 is a bilinear matrix inequality which is not easy to be solved [7]. 2) Since h¯ k ∈ ρ, h¯ k+1 ∈ ρ, and h¯ k−τk ∈ ρ are unknown and time-varying, there are infinite choices for h¯ k , h¯ k+1 ,

(31)

(32)

and h¯ k−τk . We cannot enumerate all the choices to verify the feasibility of the condition. Due to the above challenges, we develop the following theorem and project the condition in Theorem 2 to each subsystem. Theorem 3: Given a constant γ and a matrix H, the filtering error system in (14) is stochastically stable  per with an H∞ P1,o1 ,i P2,o1 ,i > 0, formance level γ , if there exist matrices ∗ P3,o1 ,i   G1,o2 ,i HG2,i , Aˆ f ,o2 ,i , Bˆ f ,o2 ,i , Cf ,o2 ,i , and Df ,o2 ,i such that G3,o2 ,i G2,i the following condition is achievable: o1 ,o2 ,o3 ,o4 ,i + o1 ,o4 ,o3 ,o2 ,i < 0 where

⎡

o1 ,o2 ,o3 ,o4 ,i

⎢ =⎢ ⎣

¯ 1 (o2 , o3 , i)  ∗ ∗ ∗

(33)

0 −I ∗ ∗

¯ 3 (o1 , o2 , o4 , i) ⎤ ¯ 2 (o1 , o2 , o4 , i)   ¯ 4 (o1 , o2 , o4 , i)  ¯ ⎥    5 (o1 , o2 , o4 , i) ⎥ ⎥ 3 h2k = 1 − h1k .

Fig. 6. Monte Carlo tests of the filtering result for the tunnel-diode circuit without any disturbance (solid line: original signal; dashed line: estimated signal).

Since h1k + h2k = 1, the grades of the membership functions do not need to be normalized. The self-normalized membership function is depicted in Fig. 5. For the network characteristics, it is assumed that there are maximally two consecutive packet dropouts and the transition probability matrix for the Markov chain is ⎡ ⎤ 0.5 0.5 0  = ⎣ 0.6 0 0.4 ⎦. 1 0 0 By applying the proposed filter design method to this circuit example, the obtained minimum H∞ performance index is 0.0422. The value for matrix H is ⎤ ⎡ 0.7386 0.6609 ⎢ 0.5860 0.7298 ⎥ ⎥ ⎢ ⎢ 0.2467 0.8908 ⎥ ⎥ H=⎢ ⎢ 0.6664 0.9823 ⎥. ⎥ ⎢ ⎣ 0.0835 0.7690 ⎦ 0.6260 0.5814 And the corresponding parameters of the mode-dependent estimator are     1.3682 −1.1049 −2.0062 Af ,1,1 = , Bf ,1,1 = 1.1208 −0.9157 −2.9985 Cf ,1,1 = [ − 0.5478 0.4251], Df ,1,1 = 0.3850     5.0857 −4.7237 −6.0353 Af ,1,2 = , Bf ,1,2 = 6.4815 −6.0363 −8.5978 Cf ,1,2 = [ − 0.4024 0.0729], Df ,1,2 = −0.2339     3.7706 −1.9539 −1.3686 Af ,1,3 = , Bf ,1,3 = 4.7188 −2.1849 −2.0430 Cf ,1,3 = [ − 1.5795 1.0691], Df ,1,3 = 0.7276     1.3730 −1.0768 −2.3710 Af ,2,1 = , Bf ,2,1 = 1.1279 −0.8737 −3.5437 Cf ,2,1 = [ − 0.5487 0.4197], Df ,2,1 = 0.4550     4.9663 −4.5754 −7.1326 Af ,2,2 = , Bf ,2,2 = 6.3115 −5.8251 −10.1611

Fig. 7. Bounded external disturbance in the simulation of the tunnel-diode circuit.

Cf ,2,2 = [ − 0.4070 0.0787], Df ,2,2 = −0.2765     3.7064 −1.9028 −1.6174 Af ,2,3 = , Bf ,2,3 = 4.6229 −2.1087 −2.4145 Cf ,2,3 = [ − 1.5451 1.0419], Df ,2,3 = 0.8599. In the simulation, first we aim to show the stability of the designed mode-dependent filter. It is assumed that the initial state value of the tunnel-diode circuit system is [0.5 0.5]T and the external disturbance is zero. With the designed filter and zero external disturbance, Monte Carlo tests are shown in Fig. 6 in which the solid-blue curve is the original signal to be estimated and the dashed-red curves depict the estimated signals in dozens of random simulations. We can see that the estimated signals catch the original signal in the first several steps. After that, the estimated signals track the original signal consistently well though there are multiple random packet dropouts. Particularly, the tracking errors converge to zeros after 40 s. It infers from the Monte Carlo tests that the filtering error system is stochastically stable. To verify the second design objective, we assume that the initial states of the tunnel-diode circuit are zeros. The bounded external disturbance ωk shown in Fig. 7 is added to drive the system. For the circuit in the network environment and the designed filter, we have conducted 100 random

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Fig. 8. Calculated H∞ performance indices in 100 random simulation runs.

Fig. 9. Estimated signal and the original signal in a random simulation run.

simulation runs in which all the estimated signals can follow the signal to be estimated. With the simulation results, we calculate the actual H∞ performance index which is computed ˆ 2 . Fig. 8 shows the calculated H∞ performance as e2 /ω indices in 100 random simulation runs. Note that all the calculated indices are around 0.0163 and smaller than the derived optimal value 0.0422. Therefore, we can claim that the second objective is satisfied for the disturbance in Fig. 7. With nonzero external disturbance, we randomly choose one run to show the filtering performance. Fig. 9 depicts the estimated signal and the original signal in a random simulation run. Though there are some fluctuations, the estimation can track the original signal well. We use 1 to denote a successful signal transmission and 0 to denote a packet dropout. Fig. 10 is the corresponding packet transmissions in the random simulation run. The packet dropouts are obtained based on the transition probability matrix. We can see from the zoomed-in figure that there are two consecutive packet dropouts from the second sampling instant to the third sampling instant. It is necessary to mention that the calculated H∞ performance index is related to the frequency of the disturbance. Figs. 11 and 12 depict the calculated H∞ performance indices in 100 random simulation runs with low-frequency and high-frequency disturbances, respectively. Obviously, the calculated H∞ performance indices in Fig. 11

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Fig. 10. Illustration of the packet dropouts in a random simulation run. 1 denotes that the signal is transmitted successfully and 0 means that the packet is missing. If two consecutive values are zeros, two consecutive packets are missing.

Fig. 11. Calculated H∞ performance indices in 100 random simulation runs with low-frequency disturbance.

Fig. 12. Calculated H∞ performance indices in 100 random simulation runs with high-frequency disturbance.

(low-frequency case) are much larger than the ones in Fig. 12 (high-frequency case), which means that the designed filter is sensitive to the low-frequency disturbances. V. C ONCLUSION In this paper, we exploited the filtering problem of networked T-S fuzzy systems. The multiple packet dropouts modeled using a Markov chain were considered for the

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signal communication from the sensor node to the filter node. Different from the published work in the literature, the measurements and the normalized fuzzy weighting functions were sent to the filter together, that is, the normalized fuzzy weighting functions were also subject to the packet dropouts. Defining a filtering error variable, we obtained a stochastic filtering error system in which the Markov variable was involved. By studying the stochastic stability and the H∞ performance of the filtering error system, we developed the filter design method and the filter parameters were calculated through solving LMIs. Finally, an example was utilized to illustrate the design procedure and to show the effectiveness of the designed filter. R EFERENCES [1] M. Posthumus-Cloosterman, “Control over communication networks: Modeling, analysis, and synthesis,” Ph.D. dissertation, Dept. Mech. Eng., Eindhoven Univ. Technol., Eindhoven, The Netherlands, 2008. [2] H. Zhang, Y. Shi, and M. X. Liu, “H∞ step tracking control for networked discrete-time nonlinear systems with integral and predictive actions,” IEEE Trans. Ind. Informat., vol. 9, no. 1, pp. 337–345, Feb. 2013. [3] H. Gao, T. Chen, and T. Chai, “Passivity and passification for networked control systems,” SIAM J. Control Optim., vol. 46, no. 4, pp. 1299–1322, 2007. [4] H. Zhang, Y. Shi, and A. Saadat Mehr, “Robust weighted H∞ filtering for networked systems with intermitted measurements of multiple sensors,” Int. J. Adapt. Control Signal Process., vol. 25, no. 4, pp. 313–330, 2011. [5] M. Yu, L. Wang, and T. Chu, “Sampled-data stabilisation of networked control systems with nonlinearity,” IEE Proc. Control Theory Appl., vol. 152, no. 6, pp. 609–614, Nov. 2005. [6] H. Zhang, Y. Shi, and J. Wang, “On energy-to-peak filtering for nonuniformly sampled nonlinear systems: A Markovian jump system approach,” IEEE Trans. Fuzzy Syst., vol. 22, no. 1, pp. 212–222, Feb. 2014. [7] H. Zhang, Y. Shi, and A. Saadat Mehr, “Robust H∞ PID control for multivariable networked control systems with disturbance/noise attenuation,” Int. J. Robust Nonlinear Control, vol. 22, no. 2, pp. 183–204, 2012. [8] Y. Tipsuwan and M.-Y. Chow, “Control methodologies in networked control systems,” Control Eng. Pract., vol. 11, no. 10, pp. 1099–1111, 2003. [9] W. Lawrenz, CAN System Engineering: From Theory to Practical Applications. New York, NY, USA: Springer-Verlag, 1997. [10] D. Necic and A. R. Teel, “Input-to-state stability of networked control systems,” Automatica, vol. 40, no. 12, pp. 2121–2128, 2004. [11] D. B. Daˇci´c and D. Neši´c, “Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design,” Automatica, vol. 43, no. 7, pp. 1145–1155, 2007. [12] L. Zhang, Y. Shi, T. Chen, and B. Huang, “A new method for stabilization of networked control systems with random delays,” IEEE Trans. Autom. Control, vol. 50, no. 8, pp. 1177–1181, Aug. 2005. [13] J. Wu and T. Chen, “Design of networked control systems with packet dropouts,” IEEE Trans. Autom. Control, vol. 52, no. 7, pp. 1314–1319, Jul. 2007. [14] Y. Shi and B. Yu, “Output feedback stabilization of networked control systems with random delays modeled by Markov chains,” IEEE Trans. Autom. Control, vol. 54, no. 7, pp. 1668–1674, Jul. 2009. [15] H. Gao, X. Meng, and T. Chen, “Stabilization of networked control systems with a new delay characterization,” IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 2142–2148, Oct. 2008. [16] H. Gao and T. Chen, “H∞ estimation for uncertain systems with limited communication capacity,” IEEE Trans. Autom. Control, vol. 52, no. 11, pp. 2070–2084, Nov. 2007. [17] R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME J. Basic Eng., vol. 82, no. series D, pp. 35–45, 1960. [18] L. Shi, M. Epstein, and R. M. Murray, “Kalman filtering over a packetdropping network: A probabilistic perspective,” IEEE Trans. Autom. Control, vol. 55, no. 3, pp. 594–604, Mar. 2010.

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Hui Zhang received the B.Sc. degree in mechanical design manufacturing and automation from Harbin Institute of Technology at Weihai, Weihai, China, in 2006, the M.Sc. degree in automotive engineering from Jilin University, Changchun, China, in 2008, and the Ph.D. degree in mechanical engineering from the University of Victoria, Victoria, BC, Canada, in 2012. He is currently a Research Associate with the Department of Mechanical and Aerospace Engineering, Ohio State University, Columbus, OH, USA. His current research interests include diesel engine aftertreatment systems, vehicle dynamics and control, mechatronics, robust control and filtering, networked control systems, and signal processing. He has authored/co-authored over 50 peer-reviewed papers in journals and conference proceedings. Dr. Zhang has served on the IFAC Technical Committee on Automotive Control, ASME Automotive and Transportation Systems Technical Committee, SAE Commercial Vehicle Committee, and the International Program Committee for IASTED International Conference on Control and Applications.

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Junmin Wang (M’06–SM’14) received the B.E. degree in automotive engineering and the M.S. degree in power machinery and engineering from Tsinghua University, Beijing, China, in 1997 and 2000, respectively, the M.S. degrees in electrical engineering and mechanical engineering from the University of Minnesota, Minneapolis, MN, USA, in 2003, and the Ph.D. degree in mechanical engineering from the University of Texas at Austin, Austin, TX, USA, in 2007. He was a full-time Industrial Researcher with Southwest Research Institute, San Antonio, TX, USA, from 2003 to 2008. In 2008, he joined Ohio State University, Columbus, OH, USA, where he founded the Vehicle Systems and Control Laboratory. His current research interests include control, modeling, estimation, and diagnosis of dynamical systems, specifically for engine, powertrain, aftertreatment, hybrid, flexible fuel, alternative/renewable energy, (electric) ground vehicle, transportation, sustainable mobility, energy storage, and mechatronic systems. He has authored/co-authored over 180 peer-reviewed journal and conference papers and holds 11 U.S. patents. Dr. Wang was the recipient of the SAE Ralph R. Teetor Educational Award in 2012, the National Science Foundation CAREER Award in 2012, the 2009 SAE International Vincent Bendix Automotive Electronics Engineering Award in 2011, the Office of Naval Research Young Investigator Program Award in 2009, and the ORAU Ralph E. Powe Junior Faculty Enhancement Award in 2009. He serves as an Associate Editor for the IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY, IFAC Control Engineering Practice, ASME Transactions Journal of Dynamic Systems, Measurement and Control, and SAE International Journal of Engines.