Energy-efficient H∞ filtering for networked systems with stochastic signal transmissions

Signal Processing 101 (2014) 134–141

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Energy-efficient H 1 filtering for networked systems with stochastic signal transmissions Dan Zhang a,b, Wenjian Cai b,n, Qing-Guo Wang c a

Department of Automation, Zhejiang University of Technology, Hangzhou 310032, PR China EXQUISITUS, Centre for E-City, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore c School of Electrical and Electronic Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b

a r t i c l e in f o

abstract

Article history: Received 8 November 2013 Received in revised form 20 January 2014 Accepted 30 January 2014 Available online 7 February 2014

This paper is concerned with the energy-efficient filtering for a class of wireless sensor networks (WSNs). Due to the power limitation of WSNs, the measurement signal is transmitted to the remote filter infrequently and stochastically. A stochastic framework is proposed to formulate the filtering problem for such systems. A sufficient condition is established such that the filtering error system is mean-square stable and achieves a prescribed disturbance attenuation level in the H 1 sense. The optimal filter design is presented to determine the filter gains. Relations between the transmission parameters, e.g., transmission probability, transmission intervals and the filtering performance are obtained. Finally, a continuous stirred tank reactor (CSTR) system is employed to evaluate the effectiveness of the proposed design. & 2014 Elsevier B.V. All rights reserved.

Keywords: Energy-efficient H 1 filtering Stochastic transmission Networked systems Sensor networks

1. Introduction Wireless Sensor Networks (WSNs) have received much attention in the last decade due to the extensive applications in target surveillance, information collection, environment monitoring and so on [1]. A typical WSN consists of a large number of wireless sensor nodes, and each sensor is equipped with limited computing and communication resources. These constraints have brought new challenges in developing control and estimation algorithms over networks. Specifically, the state estimation of dynamical systems based on the incomplete measurements is known to be challenging [2–6]. For example, the authors in [2] studied the H 1 filtering problem for a class of networked systems with repeated scalar nonlinearities under unreliable communication links, and they showed that the optimal filter gain

n

Corresponding author. Tel.: þ65 67906862; fax: þ 65 67933318. E-mail addresses: [email protected] (D. Zhang), [email protected] (W. Cai), [email protected] (Q.-G. Wang). http://dx.doi.org/10.1016/j.sigpro.2014.01.032 0165-1684 & 2014 Elsevier B.V. All rights reserved.

design can be determined by solving a linear matrix inequality. The authors in [5] investigated the H 1 filtering problem for a class of discrete-Time Systems with stochastic incomplete measurement and mixed delays, and a unified modeling was presented to capture the random discrete delay and distributed delay. Based on the Lyapunov stability theory and some stochastic system analysis, the filter gains were finally determined by solving an optimization problem. Sensor nodes in WSN are usually equipped with battery. Hence, power constraint in WSNs becomes a major problem that limits the application of WSNs. Experiments show that the major power consumption in sensor nodes comes from the communication process [7]. In the last decades, many efforts have been devoted to designing the energy efficient communication protocol for WSNs, e.g., multi-rate transmission protocol. Under such a protocol, each sensor transmits its measurement at the pre-given time instant. Though the transmission instants for different sensors are different, each sensor must transmit the measurement with a sequence (which is pre-given) and is not allowed to vary. For example, sensor 1 transmits its measurement with the 1st instant,

D. Zhang et al. / Signal Processing 101 (2014) 134–141

sensor 2 transmits its measurement with the 2nd instant, and so on. Related state estimation algorithms for such transmission protocol can be found in [8–15]. It is noted that the above multi-rate transmission protocol is a deterministic one as the transmission instant is pre-set. This may lead to poor estimation performance. In order to reduce the conservatism of the deterministic transmissions proposed in [8–15]. Recently, the authors in [16–18] proposed a stochastic transmission one and the optimal estimator designs were also given. However, it should be pointed out that there still exist some limitations in the results [16–18], e.g., they only considered the Kalman filtering problem under the stochastic transmission protocol. It is well known that the standard Kalman filter is sensitive to modeling errors, and the distribution of the noise should be Gaussian white noise. In practice, modeling error is inevitable and one may not always know the distribution of the noise. In these scenarios, estimator design methods in [16–18] fail to work and one should resort to other estimator design, e.g., the H 1 filter design [19,20], which is much robust than the Kalman filter. Further, in [16–18], they only used the transmission rate to design the filter, while other statistic information, e.g., the probability of successive non-transmissions has not been used, which leads to much conservatism. Hence, the energy-efficient filtering for networked systems with stochastic transmission protocol deserves research attention, which motivates the present study. In this paper, the H 1 filtering problem is considered for a class of networked systems, where the measurement signal is transmitted to the remote filter stochastically. Unlike the design results in [16–18], more statistic information is utilized to reduce the design conservatism. A new formulation is proposed to address such a filtering problem. A sufficient condition is obtained such that the filtering error system is asymptotically stable in the meansquare sense and achieves a prescribed H1 performance level. A case study on CSTR system is given to show the effectiveness of the proposed design. The main contributions of this paper are summarized as follows:

 Most authors have considered the Kalman filtering



problem with stochastic signal transmissions, see, e.g., [16–18], while it is well known that the standard Kalman filter is sensitive to modeling errors, and the distribution of the noise must be Gaussian white noise. In practice, modeling error is inevitable and one may not always know the distribution of the noise. In these scenarios, estimator design methods in these works fail to work and one should resort to other estimator design, e.g., the H 1 filter design, which is much robust than the Kalman filter. So, this motivates us to investigate the H 1 filtering problem, and there has been no related paper reported on the H 1 filtering with the stochastic signal transmissions. In the existing results, they only used the transmission rate to design the filter, while other statistic information, e.g., the probability of successive non-transmissions has not been used, which leads to much conservatism.





135

In this paper, more information is used to design the filter, and one can see the effectiveness of our design in the Example Section. Though some papers have been reported on the eventbased H 1 filtering for networked systems, e.g., reference [21]. In [21], the time delay system approach was used to study the event-based filtering problem, it should be pointed out that the main results are only be applicable onto the continuous-time systems and the single sensor case. Unlike [21], our attention is focused on the energy-efficient filtering for discretetime networked systems with multiple distributed sensors. Moreover, our analysis and design method is much simpler than that of [21]. In the filter design with the stochastic signal transmission, one of the main difficulties is how to quantify the effect of transmission parameters into the filtering performance. In our work, interesting relations are established, specifically, the more frequently the measurement is transmitted, the better the filtering performance one obtains. Moreover, the filtering performance level is a monotonic function of the largest transmission span of sensors. Such relations have not been established in the literature, e.g., [21].

Notation: The notation is fairly standard. We use W T to denote the transpose of any square matrix W. We use W 40 to denote a positive-definite matrix W and I n to denote the n  n identity matrix. Let Rn denote the n dimensional Euclidean space. Rmn is the set of all m  n real matrices. The notation l2 ½0; 1Þ refers to the space of square summable infinite vector sequences with the norm ‖‖2 . The symbol “n” will be used in some matrix expressions to represent the symmetric terms. Moreover, Efg stands for the mathematical expectation. 2. Problem formulation Consider the following discrete-time system: xðk þ 1Þ ¼ AxðkÞ þ BwðkÞ nx

ð1Þ nw

where xðkÞ A R is the state, wðkÞ A R is the unknown disturbance signal belonging to l2 ½0; þ 1Þ. A and B are the constant matrices with appropriate dimensions. Suppose there are m distributed sensors for the system and the pth sensor produces its measurement as yp ðkÞ ¼ C p xðkÞ þ Dp wðkÞ 1

ð2Þ

where yp A R is a scale signal, Cp and Dp are the constant matrices with appropriate dimensions. To save energy, the measurement may not be transmitted to the remote filter at each time instant. On the other hand, to estimate the state reasonably well, the filter needs recent measurements. Thus, a reasonable communication protocol is that the measurement signal is transmitted at least once over Np time steps, where N p 40 is an integer. No other restrictions are made on the protocol, implying that the transmission can happen at any time in these Np steps and is random. One possible transmission scenario is depicted in Fig. 1, where yp ðkÞ is the measurement signal sequence and y~ p ðkÞ is the transmitted one with Np ¼ 3.

136

D. Zhang et al. / Signal Processing 101 (2014) 134–141



αp;s ðkÞ ¼ 0 for s¼0,2. It follows from (3) that y p ð3Þ ¼ yp ð3  1Þ ¼ yp ð2Þ, which is true from Fig. 1. When k ¼4, the measurement is not transmitted again, so that αp:2 ðkÞ ¼ 1 and αp;s ðkÞ ¼ 0 for s¼0,1. It follows from (3) that y p ð4Þ ¼ yp ð4 2Þ ¼ yp ð2Þ, which is true from Fig. 1.

Remark 1. In the traditional estimator designs for single sensor systems, it may not be necessary to update the estimator state when no new measurement arrives at the estimator side. However, in the large scale wireless sensor networks, one cannot know how many sensor measurements may arrive at the filter side at certain time instant, hence, in this paper we require the estimator to update the state at each time instant.

Fig. 1. Transmission process.

It follows from the transmission mechanism that there could be no transmission at some time instants. In the case of no transmissions, the filter has to use the last transmitted measurement signal as its input. Then, at each time instant k, the filter input, y p ðkÞ, will be the most recent member of the transmitted subset of fyp ðkÞ; yp ðk  1Þ; …; yp ðk  Np þ 1Þg. To reflect this random selection of one member, a set of stochastic variables, αp;s ðkÞ A f0; 1g, s ¼ 0; 1; …; Np  1, is introduced such that αp;s ðkÞ ¼ 1 if yp ðk  sÞ is selected at k as the filter input, which happens when there is transmission at k  s but none at k  s þ1; k  s þ 2; …; k; αp;s ðkÞ ¼ 0, otherwise. For the scenario considered before, the selected y p ðkÞ is shown in Fig. 1 in the 3rd line. The above filter input selection rule implies that at each k, there is one and only one measurement point (most recent one) which is selected as the filter input, and equivalently, there is one and only one αp;s ðkÞ ¼ 1 with all Np  1 other variables being zero, so that ∑s ¼ 0 αp;s ðkÞ ¼ 1. In this paper, the probabilities Efαp;s ðkÞ ¼ 1g ¼ α p;s are assumed to be known, which is reasonable as the data on actual transmissions could be collected on any given system and the estimates of these probabilities can be obtained with Np  1 their frequencies. Obviously, there holds ∑s ¼ 0 α p;s ¼ 1. Moreover, different sensors are assumed to operate independent of each other and their transmission sequences are then also independent. It follows from the above transmission protocol and selection rule that the filter input can be expressed as

Remark 2. The signal may not be transmitted to the remote estimator, e.g., the measurement signal varies only a little compared with the ones transmitted in the last time instant. Hence, the measurement is transmitted intermittently. We formulate such a transmission as a stochastic one as in [16–18]. However, compared with the filtering results in [16–18], our follow-up filtering results would be much robust and more applicable as we do not assume the Gaussian white noise here. Remark 3. It is seen that (3) reduces to measurement models in [16–18] when αp;s ðkÞ ¼ 0 for all s ¼ 1; …; N p 1. Incorporating these statistics information to our new filter design can achieve a better estimation performance level than that of [16–18], as will be shown in simulation later. Remark 4. In this paper, we pay our attention on the stochastic transmission issue, however the packet may be lost under the networked environment. It should be pointed out that the proposed modeling (3) has also captured the successive packet dropout phenomenon. When one further considers the packet dropout problem, one may replace Np in (3) by M p þN p , where Mp is the upper bound of successive packet dropout and N p is the largest transmission span. Let N ¼ maxp ¼ 1;2;…;m fNp g, and define XðkÞ ¼ ½xT ðkÞ xT ðk 1Þ ⋯ xT ðk  N þ 1ÞT ; WðkÞ ¼ ½wT ðkÞ wT ðk  1Þ ⋯ wT ðk  N þ 1ÞT : Substituting (2) in (3) yields Np  1

y p ðkÞ ¼ ∑ αp;s ðkÞfC p Eps XðkÞ þDp H ps WðkÞg

ð4Þ

s¼0

y p ðkÞ ¼ αp;0 ðkÞyp ðkÞþ αp;1 ðkÞyp ðk 1Þ þ ⋯þ αp;Np  1 ðkÞyp ðk N p þ 1Þ

ð3Þ We now verify (3) for the scenario in Fig. 1. With N p ¼ 3, one readily sees that

 When k¼2, the measurement signal is transmitted imme-



diately so that αp;0 ðkÞ ¼ 1 and αp;s ðkÞ ¼ 0 for s¼1,2. It follows from (3) that y p ð2Þ ¼ yp ð2Þ, which is true from Fig. 1. When k ¼3, the measurement is not transmitted but was done at the last step, so that αp;1 ðkÞ ¼ 1 and

where Eps ¼ ½0 ⋯ 0 I nx 0 ⋯ 0, and H ps ¼ ½0 ⋯ 0 I nw 0 ⋯ 0 in which all elements are zeros except for the ðs þ 1Þ th block is I nx and I nw , respectively. Let αp ðkÞ ¼ ½αp;0 ðkÞ; αp;1 ðkÞ; …; αp;Np  1 ðkÞ, then the stochastic vector αp ðkÞ has Np possible realizations as ð5Þ αp ðkÞ A f½1; 0; 0; …; 0; …; ½0; 0; 0; ⋯; 1g Define αðkÞ ¼ ½α1 ðkÞ; α2 ðkÞ; …; αm ðkÞ. The total number of possible realizations of αðkÞ is N ¼ N1  N2  ⋯  N m . Let E αðkÞ ¼ ½ET1;α1 ðkÞ ET2;α2 ðkÞ ⋯ ETm;αm ðkÞ T ; H αðkÞ ¼ ½H T1;α1 ðkÞ H T2;α2 ðkÞ ⋯ H Tm;αm ðkÞ T :

D. Zhang et al. / Signal Processing 101 (2014) 134–141

One particular realization of αðkÞ means one sequence of αðkÞ, which specifies one particular case of ðE αðkÞ ; H αðkÞ Þ. For ease of exposition, define an integer set Γ ¼ f1; 2; …; Ng, and introduce a new set of stochastic variables, βi ðkÞ A f1; 0g; iA Γ, one variable for one possible realization of αðkÞ , such that β1 ðkÞ ¼ 1 if and only if αp ðkÞ ¼ ½1; 0; …; 0; p ¼ 1; 2; …; m; β2 ðkÞ ¼ 1 if and only if αp ðkÞ ¼ ½1; 0; …; 0; p ¼ 1; 2; …; m 1; αm ðkÞ ¼ ½0; 1; …; 0; and so on. By our construction, there is one and only one realization of αðkÞ at any time instant, so that ∑N i ¼ 1 βi ðkÞ ¼ 1. The probability, Efβi ðkÞ ¼ 1g ¼ β i , can be computed by the probabilities of sensor transmissions, α p;s . For example, we have two sensors and each sensor transmits the measurement within two time steps stochastically with their probabilities being α 1;0 , α 1;1 and α 2;0 , α 2;1 , respectively. Then, it follows from simple probability rules that β 1 ¼ α 1;0 α 2;0 , β 2 ¼ α 1;0 α 2;1 , β 3 ¼ α 1;1 α 2;0 and β 4 ¼ α 1;1 α 2;1 . In view of the above development, we can express the filter input vector as N

yðkÞ ¼ ∑ βi ðkÞfC E i XðkÞ þ D H i WðkÞg

ð6Þ

i¼1

where C ¼ diagfC 1 ; C 2 ; …; C m g, D ¼ diagfD1 ; D2 ; …; Dm g. In this paper, we aim to estimate the following signal: zðkÞ ¼ LxðkÞ

ð7Þ

where zðkÞ A Rnz and L is a constant matrix with appropriate dimension. In order to estimate the signal in (7), we propose the following filter: ( xf ðk þ 1Þ ¼ Af xf ðkÞ þ Bf yðkÞ ð8Þ zf ðkÞ ¼ C f xf ðkÞ where xf ðkÞ A Rnx is the state of the filter and zf ðkÞ A Rnz is the estimate of zðkÞ. Af ; Bf , and C f are the filter parameters to be designed. In order to derive the filtering error system, we rewrite state equation in (1) and the estimation equation in (7) as ( Xðk þ 1Þ ¼ AXðkÞ þ BWðkÞ ð9Þ zðkÞ ¼ LXðkÞ where " A¼

A

0

I nN

0

#

 B¼

;

 0 ; 0

B 0

L ¼ ½L 0:

Define the augmented state as ηðkÞ ¼ ½X T ðkÞ xTf ðkÞT and the estimation error as eðkÞ ¼ zðkÞ  zf ðkÞ. Then the filtering error system is given by 8 N > ~ ~ < ηðk þ 1Þ ¼ AηðkÞ þ BWðkÞ þ ∑ ðβi ðkÞ β i Þ½A^ i ηðkÞ þ B^ i WðkÞ > :

i¼1

~ eðkÞ ¼ LηðkÞ

where " A A~ ¼ Bf C E " A^ i ¼

#

0 ; Af

0

0

Bf C E i

0

" B~ ¼

#

B

" ;

B^ i ¼

0 Bf D H i

i¼1

N

H ¼ ∑ β iH i: i¼1

Our later development makes use of the following definitions. Definition 1. The system (10) with wðkÞ ¼ 0 is said to be asymptotically stable in the mean-square sense, if the solution ηðkÞ of system (10) satisfies limk-1 Ef‖ηðkÞ‖g ¼ ηð0Þ for any initial condition. Definition 2. For given scalars γ 4 0, system (10) is said to be asymptotically stable in the mean-square sense and achieves a prescribed H 1 performance γ 4 0, if it is asymptotically stable and under zero initial condition, ∑sþ¼10 EfeT ðsÞeðsÞg r∑sþ¼10 γ 2 wT ðsÞwðsÞ holds for all nonzero wðkÞ A l2 ½0; 1Þ. The filtering problem is summarized as follows: Filtering problem: Design a filter in form of (8) such that the filtering error system (10) is asymptotically stable in the mean-square sense and achieves a prescribed H 1 performance level in the presence of stochastic transmissions. 3. The proposed solution Based on the stochastic analysis method, a sufficient condition is presented for the solvability of considered filtering problem in the following theorem. Theorem 1. For given scalars τ and β i ; i ¼ 1; 2; …; N, if there exists a positive-definite matrix P such that the following inequality 2 3 qffiffiffiffiffiffi qffiffiffiffiffiffi Φ Φ2 β 1Ξ1 ⋯ β N Ξ N Φ3 6 1 7 6 7 6 n P 1 7 0 0 0 0 6 7 6 7 1 n P 0 0 0 7o0 6 n ð11Þ 6 7 6 n 7 n n ⋱ 0 0 6 7 6 7 n n n P 1 0 5 4 n n n n n n I holds, then the filtering error system (10) is asymptotically stable in the mean-square p sense ffiffiffiffi and achieves a prescribed H 1 performance level γ ¼ τ N. where   P 0 ~ T ; Ξ i ¼ ½A^ i B^ i T ; Φ1 ¼ ; Φ2 ¼ ½A~ B 2 0 τ I Φ3 ¼ ½L~ 0T : Proof. We first prove the stability of the filtering error system (10) with wðkÞ ¼ 0. Suppose the following Lyapunov function for system (10): VðkÞ ¼ ηT ðkÞPηðkÞ

#

Bf D H

ð10Þ

N

E ¼ ∑ β iE i;

137

L~ ¼ ½L C f ;

; # ;

ð12Þ

Then, one sees that T ~ ~ P½AηðkÞ ηT ðkÞPηðkÞ EfVðk þ 1Þ VðkÞg ¼ ½AηðkÞ 8" #T " #9 < N = N ^ ^ þE ∑ ϕ ðkÞA i ηðkÞ P ∑ ϕi ðkÞA i ηðkÞ : i¼1 i ; i¼1

ð13Þ

138

D. Zhang et al. / Signal Processing 101 (2014) 134–141

where ϕi ðkÞ ¼ βi ðkÞ  β i . On the other hand, we have 8" #T " #9 < N = N E ∑ ϕi ðkÞA^ i ηðkÞ P ∑ ϕi ðkÞA^ i ηðkÞ : i¼1 ; i¼1 9 8 N > > > > > > ∑ ðϕi ðkÞÞ2 ½A^ i ηðkÞT P½A^ i ηðkÞ > > = > > > > þ ∑ ∑ ϕi ðkÞϕj ðkÞ½A^ i ηðkÞT P½A^ j ηðkÞ > > > ; : i ¼ 1j ¼ 1;j a i

Hence, system (10) achieves a prescribed H 1 performance level as well. This completes the proof. □ Remark 5. In the filtering of stochastic transmitted systems, one of the main difficulties is how to quantify the transmission parameters into the filtering performance. In Theorem 1, it is interesting to see that the filtering performance level is a monotonic function of the largest transmission span N. It should be pointed out that theorem 1 cannot be used to determine the filter gain directly due to the co-existence of P and P  1 . In the following theorem, we present the filter gain design algorithm.

N

¼ ∑ β i ½A^ i ηðkÞT P½A^ i ηðkÞ i¼1

" 

#T "

N

∑ β i A^ i ηðkÞ

#

N

P ∑ β i A^ i ηðkÞ

i¼1

i¼1

N

r ∑ β i ½A^ i ηðkÞT P½A^ i ηðkÞ

ð14Þ

i¼1

Hence, T ~ ~ P½AηðkÞ EfVðk þ 1Þ  VðkÞg r½AηðkÞ N

ηT ðkÞPηðkÞ þ ∑ β i ½A^ i ηðkÞT P½A^ i ηðkÞ

ð15Þ

i¼1

It is seen that the right-hand side of (15) is negative under (11), and then system (10) is asymptotically stable in the mean-square sense. Now we consider the H 1 performance of the filtering error system (10). It follows the above analysis method, and we have EfVðk þ 1Þ  VðkÞ þ eT ðkÞeðkÞ  τ2 W T ðkÞWðkÞg T ~ ~ ~ ~ P½AηðkÞ þ BWðkÞ  ηT ðkÞPηðkÞ ¼ ½AηðkÞ þ BWðkÞ 9 8" #  T N  > >  ^ > > > > ^ > > ∑ βi ðkÞ  β i A i ηðkÞ þ B i WðkÞ > > = < i¼1 " # þE > > N > > >  P ∑ ðβ ðkÞ  β ÞðA^ ηðkÞ þ B^ WðkÞÞ > > > > > i i i i ; :

T

rη ðkÞ

Φ1 þ Φ2 PΦT2 þ

N

∑

i¼1

β i Ξ i PΞ Ti

2 1 Φ2

# þ Φ3 ΦT3

ηðkÞ

holds, then the considered filtering performance is solvable. Moreover, the filter gains are determined by Af ¼ G3 T AF , Bf ¼ G3 T BF , C f ¼ C F , where 2 3 2 3 " # 1 1 1 Φ2 Ξi Φ3 Φ 2 ¼ 4 2 5; Ξ i ¼ 4 2 5; Φ 3 ¼ ; P ¼ P  G GT ; Φ2 Ξi 0 with

i¼1

T ~ ~ þ ½LηðkÞ ½LηðkÞ  τ2 W T ðkÞWðkÞ "

Theorem 2. For given scalars τ and β i ; i ¼ 1; 2; …; N, if there exist a positive-definite matrix P and a matrix G with appropriate dimensions, such that the following inequality 2 3 qffiffiffiffiffiffi qffiffiffiffiffiffi Φ Φ β Ξ ⋯ β Ξ Φ N 1 2 1 3 1 N 6 7 6 7 6 n P 0 0 0 0 7 6 7 6 n n P 0 0 0 7 6 7o0 ð21Þ 6 7 6 n n n ⋱ 0 0 7 6 7 6 7 n n n P 0 5 4 n n n n n n I

ð16Þ

¼4

where ηðkÞ ¼ ½η ðkÞ W ðkÞT . By using the Schur Complement [22], it is easy to see EfVðk þ 1Þ  VðkÞ þ eT ðkÞeðkÞ  τ2 W T ðkÞWðkÞg o 0

T

A G2 þET BTF

ATF M

ATF

h T T 2 Φ 2 ¼ B G1 þ H BTF M "

T

T

T

A G1 þ ET BTF M

1 Ξi

¼

T

E i C BTF

0

0

T

5;

i T T B G2 þ H BTF ;

E i C BTF M

T

3

T

# ;

ð17Þ

Summing both side of (17) from k¼ 0 to k ¼T, then ( ) o T n E Vðk þ1Þ  Vð0Þ þ ∑ eT ðkÞeðkÞ  τ2 W T ðkÞWðkÞ o0 k¼0

ð18Þ which implies, by the zero initial condition and positiveness of Vðk þ1Þ, that ( ) o T n E ∑ eT ðkÞeðkÞ  τ2 W T ðkÞWðkÞ o0 ð19Þ

T

2

T

T

Ξ i ¼ ½H i D BTF M " 1 Φ3

¼ "

G¼

L

T

C TF

#

" ;

G1

G2

G3 M

G3

T

H i D BTF ;

P¼

P1

P2

n

P3

# ;

# ;

M ¼ ½I ⋯ I:

k¼0

Let T- þ1, then þ1

T

Proof. By pre- and post-multiplying (11) with diag 2

T

2

T

∑ Efe ðkÞeðkÞg r τ W ðkÞWðkÞ ¼ γ w ðkÞwðkÞ

k¼0

ð20Þ

fI; GT ; GT ; …GT ; Ig and its transpose respectively, (11) is |fflfflfflfflffl{zfflfflfflfflffl} N

D. Zhang et al. / Signal Processing 101 (2014) 134–141

equivalent to 2 qffiffiffiffiffiffi β 1Ξ1G 6 Φ1 Φ2 G 6 ~ 6 n P 0 6 6 n P~ 6 n 6 6 n n n 6 6 n n 4 n n

n

n

where P~ ¼ G P T

1

n

n

n

series–parallel reactions: ⋯

qffiffiffiffiffiffi β N ΞN G

0

0

0 ⋱ n

0 0 P~

n

n

3

Φ3 7 7 0 7 7 7 0 7 o0 7 0 7 7 7 0 5 I

A-B-C

ð22Þ

always hold for any matrix G. Then,

⋯

qffiffiffiffiffiffi β N ΞN G

0

0

0

0

⋱

0

n

P

n

n

which is the same as (21) with " # " # G2 G1 P1 P2 ; G¼ ; P¼ n P3 G3 M G3

ð23Þ

V_ dϑ kw A R ¼ ðϑ0  ϑÞ þ ðϑK  ϑÞ VR dt ζC p V R

AF ¼ GT3 Af ;

BF ¼ GT3 Bf ;

□

Remark 6. In order to obtain the minimum H 1 performance γ n , one can solve the following optimization problem: ν

subject to

Eq: ð21Þ with ν ¼ τ2

and find the minimum H 1

dcB V_ ¼ cB þ k1 cA k2 cB dt VR



and C F ¼ C f . This completes the proof.

min

where A ¼ cyclopentadiene, B ¼ cyclopentenol, C ¼ cyclope ntanediol and D ¼ dicyclopentadiene. The reactor inflow contains only the educt A in low concentration cA0 . The desired product is the component B, the intermediate component in the series reaction. Assuming constant density and an ideal residence time distribution within the reactor, the balance equations are given as follows [23]: dcA V_ ¼ ðcA0  cA Þ k1 cA  k3 c2A dt VR

3

Φ3 7 7 0 7 7 0 7 7 o0 7 0 7 7 7 0 5 I

ð25Þ

2A-D

G. On other hand, it is easy to see that

GT P  1 GT r P  G  GT (22) holds if 2 qffiffiffiffiffiffi Φ G β 1Ξ1G Φ 1 2 6 6 6 n P 0 6 6 n n P 6 6 6 n n n 6 6 n n 4 n

139

ð24Þ p ffiffiffiffiffiffiffiffi performance γ n by γ n ¼ νn N.

4. Illustrative examples In this section, a modified CSTR system [23] is given to show the usefulness of the proposed filter design. The CSTR is depicted in Fig. 2, and there is the following

BC AD 2 k1 cA ΔH AB R þk2 cB ΔH R þk3 cA ΔH R ζC p

ð26Þ

where cA and cB are the concentrations of educt A and the desired product B within the reactor, ϑ denotes the reactor temperature. The rate factors k1  k3 depend exponentially on the reactor temperature ϑ via Arrhenius law

  EAi ð27Þ ki ðϑÞ ¼ k0i exp Rϑ For the reaction system at hand, we let k1 ¼ k2 . The values of model parameters and the steady-state values of the main operating point of the reactor are listed in Table 1. Detailed background of CSTR can be found in [23]. Linearizing (26) at the operating point, we have the following state space model: _ ¼ Ap xðtÞ þ Bp uðtÞ xðtÞ

ð28Þ

where 2 3 2 3 x1 cA cAs 6 x 7 6 c c 7 x¼4 25¼4 B Bs 5; x3 ϑ  ϑs

" u¼

u1 u2

#

" ¼

V_  V_ s cA0 cA0s

# ;

Table 1 Model parameters and main operating point. Model parameters k01;2 ¼ 1:287  1012 h

Main operating point

k03 ¼ 9:043  10 l=mol h EA1;2 =R ¼ 9758:3 K EA3 =R ¼ 8560:0 K ΔHAB R ¼ 4:2 kJ=mol ΔHBC R ¼  11 kJ=mol

ΔHAD ¼  41:85 kJ=mol R ρ ¼ 0:9342 kg=l C P ¼ 3:01 kJ=kg K AR ¼ 0:215 m2 V ¼10.01 ϑ0 ¼ 403:15 K kw ¼ 4032 kJ=h m2 K Fig. 2. Continuous stirred tank reactor.

C As ¼ 1:235 mol=l

1

9

1

C Bs ¼ 0:9 mol=l ϑs ¼ 407:29 K 1 V_ =V R ¼ 18:83 h C A0s ¼ 5:1 mol=l

140

D. Zhang et al. / Signal Processing 101 (2014) 134–141

Ap and Bp are Jacobian matrices with 2 3  86:0962 0 4:2077 6 50:6146  69:4446 0:9974 7 Ap ¼ 4 5; 172:2263 197:9985  65:5149 2

0:3861

6 Bp ¼ 4 0:0899 0:4136

18:83 0

3 7 5:

0

In the state estimation problem, one may also treat the control input as the unknown input signal. Since CSTR is open loop stable based on our parameters, we choose u ¼ 0 for simplicity. By further considering the unknown disturbance in CSTR, we have the following discrete-time state space model by sampling period T 0 ¼ 1 min: xðk þ1Þ ¼ AxðkÞ þ BwðkÞ

0:0345

0:0206

3

0:3152

0:0033 7 5;

1:1042

0:3671

Fig. 3. Trajectories of z(k) and zf(k). 1.15

2 3 1 6 7 B ¼ 4 1 5: 1

1.1 1.05

In practice, it may be necessary for one to know the product concentration cB for other use, but the direct measure of concentration cB by using traditional chemical approaches is usually expensive. An alternative yet non-expensive approach is to use signal processing approaches to estimate the concentration. In this example, we deploy two sensors to measure the educt concentration cA and the reactor temperature ϑ, and the purpose is to estimate the product concentration cB . Hence, C 1 ¼ ½1 0 0; C 2 ¼ ½0 0 1, D1 ¼ 0:3, D2 ¼ 0:2 and L ¼ ½0 1 0. Suppose that the noise varies in wðkÞ A ½  1; 1. Note that this assumption is only for simulation purpose. One may find the up and lower bounds of the noise in real industrial fields. The estimation task is carried out for 100 min. Suppose that the transmission occurs if one of the following events happens: ‖yp ðkÞ  ylast;p ‖ Z δy;p

ð30Þ

k  klast;p 4 θk;p

ð31Þ

where ylast;p is the last transmitted signal of sensor p, klast;p is the last transmitted time instant of sensor p. δy;p and θk;p are the measurement threshold and the time threshold, respectively. In this example, we assume that no packet dropout occurs only for simulation simplicity, and we set δy;1 ¼ 0:1; δy;2 ¼ 0:2; θk;1 ¼ θk;2 ¼ 1. It implies that N 1 ¼ N 2 ¼ 2. By doing 200 random samples for the transmissions, we have α 1;0 ¼ 0:69; α 1;1 ¼ 0:31; α 2;0 ¼ 0:44; α 2;1 ¼ 0:56. With all information available, we solve the optimization (24) to get γ n ¼ 1:9069. The corresponding filter gains are 2 3 0:2171 0:0420 0:0167 6 7 Af ¼ 4 0:4788  0:0179 0:0159 5; 6:0850  4:3532 0:6279

true performance

where 2 0:2747 6 A ¼ 4 0:2323 1:2566

ð29Þ

1 0.95 0.9 0.85 0.8

0

100

200

300

400

500

samples Fig. 4. True performance level of 500 samples.

Table 2 Relation between α 1;0 and γ n . δy;1 α 1;0 γn

0.1 0.69 1.9069

2

1:2479 6 1:4373 Bf ¼ 4 4:2860

0.15 0.58 2.0926

0:1254

0.2 0.49 2.1378

0.25 0.43 2.1670

0.3 0.39 2.1859

3

0:1444 7 5; 0:4305

C f ¼ ½ 1:5297 0:7602 0:0040: One sample trajectories of z(k) and zf ðkÞ are shown in Fig. 3. By simple calculation, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 100 100 T T ∑k ¼ 0 e ðkÞeðkÞ=∑k ¼ 0 w ðkÞwðkÞ ¼ 0:9342 oγ ¼ 1:9069. In order to see how robust our method is, we run 500 simulations. In each simulation, the transmission process is randomly generated and the transmission probabilities of each simulation are different (also different from the statistical one). It is shown in Fig. 4 that the true performance levels are smaller than the optimal one γ n . Hence,

D. Zhang et al. / Signal Processing 101 (2014) 134–141

the estimator design is robust to the probability uncertainties from the simulation point of view. Note that if the transmission protocol in [16–18] is used, the optimal H 1 performance is γ n ¼ 2:0362, which is much larger than that of our ones. We are now on the position to see how the transmission parameters affect the filtering performance. Here, we increase the measurement threshold δy;1 . The relation between transmission probability and the filtering performance is listed in Table 2. It is seen that the more frequently the measurement is transmitted, the better the filtering performance one obtains. In this scenario, more energy would be consumed by the sensors. The tradeoff between the energy consumption and the filtering performance should be considered in design of the WSNs, for example, if one requires that the filtering performance level is below 2.1, then, one may set δy;1 ¼ 0:15 but not necessary to choose δy;1 ¼ 0:1. 5. Conclusions In this paper, the energy-efficient filtering for a class of networked systems has been investigated. A new stochastic transmission protocol is proposed and formulated. A sufficient condition is given such that the filtering error system is asymptotically stable in the mean-square sense and achieves a prescribed H 1 performance level. A filter design is presented and illustrated with a case study of CSTR system. In our work, relations between the transmission parameters, e.g., the transmission interval, the transmission probability and the filtering performance have been established. Acknowledgement The authors would like to thank the Editor-in-Chief, the Associate Editor, and anonymous reviewers for their constructive comments that have helped to improve this paper significantly. The work was jointly funded by AnSTAR-Ministry of National Development Green Building Joint Grant (1121760027) and Energy Research Institute @ NTU. References [1] V. Savic, H. Wymeersch, S. Zazo, Belief consensus algorithms for fast distributed target tracking in wireless sensor networks, Signal Process. 95 (February 2014) 149–160.

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