One-channel networked data fusion with communication constraint

Available online at www.sciencedirect.com

Journal of the Franklin Institute 351 (2014) 156–173 www.elsevier.com/locate/jfranklin

One-channel networked data fusion with communication constraint Xiaolei Bian, Yuanqing Xian, Zhihong Deng, Mengyin Fu School of Automation, Key Laboratory of Intelligent Control and Decision of Complex Systems, Beijing Institute of Technology, Beijing 100081, China Received 4 November 2012; received in revised form 6 June 2013; accepted 14 August 2013 Available online 29 August 2013

Abstract Multi-sensor data fusion over one channel is studied in this paper. The communication constraint considered here is medium access constraint. When the synchronous time division multiplexing (STDM) mechanism is used to address this problem, collective delay emerges. Collective delay time depends upon the channel capacity and traffic flow assigned to the communication channel, causing contradiction between traffic flow and delay time (the number of transmitted sensors and delay steps). A new model is developed that can truly reflect this contradiction by introducing a stochastic process θ. Based on the obtained system model, the optimal data fusion filter is designed. It also gives the upper bounds of the expected estimation error covariance and estimation error covariance with one-step delay. Two illustrative examples are given in the last section to show the influence of θ on estimation performance. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction The interest in networked data fusion has increased tremendously in recent years, see eg., [1–4]. The networked data fusion makes it possible for remote transmission, estimation, monitoring and control, so it is applied in both military and nonmilitary fields broadly. However, the insertion of network also introduces some disadvantages such as data transmission delay, fading and loss [5], Out-Of-Sequence Problem (OOSP) [6–8], and energy constraint [9–11]. These problems directly constrain the development of data fusion and draw more and more attention of researchers. n

Corresponding author. Tel./fax: þ86 68914506. E-mail addresses: [email protected] (X. Bian), [email protected], [email protected] (Y. Xia), [email protected] (Z. Deng), [email protected] (M. Fu). 0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.08.013

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In networked data fusion applications, usually a group of (identical or nonidentical) sensors take noisy observations of the system state and send them to a fusion center over a shared networked channel. The fusion center processes all the information from the sensors and outputs a state estimate. It is shown schematically in Fig. 1. For the multi-sensor data fusion over one channel, the communication constraint problems that the network brings fall into two categories: (i) common packet dropout, OOSP, delay, etc.; (ii) the medium access constraint [12], i.e., nodes are not allowed to access the shared medium simultaneously. About (i), there exist lots of results, see eg., [13–26]. Paper [13] considers the problem of performing Kalman filtering with intermittent observations. It shows that, depending on the eigenvalues of the state matrix and the structure of the observation matrix, there exists a critical value λc which is related to the stability of the filter. Meanwhile, paper [13] gives explicit upper and lower bounds on λc , and shows that they are tight in some special cases. Paper [14] presents a set of new centralized algorithms for estimating the state of linear dynamic Multiple-Input Multiple-Output (MIMO) control systems with asynchronous, nonsystematically delayed and corrupted measurements provided by a set of sensors. Paper [15] studies optimal estimation design for sampled linear systems where the sensor measurements are subject to random delay or might even be completely lost, the architecture of this estimator is independent of the communication protocol and can be implemented by using a finite memory buffer if the delivered packets have a finite maximum delay. Paper [16] proposes a flight path update algorithm for a sequence with arbitrary delayed OOSMs. It is a globally optimal recursive algorithm for arbitrary delayed OOSMs including the case of interlaced OOSMs with less storage, compared with the optimal state update algorithm in [17]. It can also update the current whole flight path in addition to the current single state with less computation. About (ii), we can handle it via the well-known STDM mechanism, i.e., all the sensors use the shared networked channel to transmit their packets in turn during a sampling interval. Some results revealing the relationship between estimation performance and channel accessing probabilities or priorities in time-multiplexing mechanism can be found, e.g., [27– 29]. Paper [27] designs a specified media access control protocol to address the channel accessing processes of various transmission nodes in time-multiplexing mechanism. It considers the relationship between the estimation performance and the channel accessing probabilities, and designs an optimal linear filter that critically depends on them. Paper [29] supposes that, at each time step, only a subset of all sensors are selected to send their observations to the fusion center due to channel capacity constraint or limited energy budget.

Sensor 2 system

. . .

channel

Sensor 1 . . .

The fusion center

Sensor N

Fig. 1. The one-channel networked data fusion system.

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It proposes a multi-step sensor selection strategy to schedule sensors to transmit for the next T steps of time with the goal of minimizing an objective function related to the Kalman filter error covariance matrix. The aforementioned results have made some progresses in the networked estimation with communication constraint. However, many critical issues remain to be addressed. For example, to the system using the STDM mechanism, there exists the collective delay. Collective delay time depends upon the channel capacity and traffic flow assigned to the communication channel (the number of transmitted sensors) [30,31]. Obviously, the more the transmitted sensors are selected in a sampling interval, the better the estimation performance is. However, the increase of the number of transmitted sensors necessarily brings the increase of delay time, which in turn degrades estimation performance. Indeed, delay time especially depends, in a nonlinear fashion, upon the number of transmitted sensors. To the best of the authors' knowledge, these behaviors are not modeled in the current fusion algorithms. This paper gives a description of the relationship between traffic flow assigned to the communication channel (the number of transmission sensors) and delay time (delay steps) in the STDM mechanism. Through introducing a stochastic process to describe the selection strategy related to the number of transmitted sensors, a new data fusion model is given. This model truly reflects the relationship between the delay steps and the number of transmitted sensors in the STDM mechanism. It also truly reflects their influences on estimation performance. Based on the obtained system model and inspired by IF given in [14], the optimal data fusion filter is designed. It also gives the upper bounds of the expected estimation error covariance and estimation error covariance of the modified data fusion filter with one-step delay. The paper is organized as follows. Section 2 formalizes the problem of one-channel networked data fusion with communication constraint. Section 3 gives details about how data delay depends upon the channel capacity and the number of transmitted sensors. In Section 4, a strategy to select the number of transmitted sensors is modeled. Based on the novel model, an optimal data fusion filter is designed. The upper bounds of the expected estimation error covariance and estimation error covariance with one-step delay are then obtained in Section 5. In Section 6, two examples are included to discuss the influence of the strategy modeled in this paper on estimation performance. Finally, Section 7 draws conclusions and directions for future work. Notations: R, Z, and N define the set of real numbers, non-negative integer numbers, and natural numbers respectively. Given that A; B A Rn are the positive definitive (positive semidefinite) matrix, denoted as A; Bmac; sc; ð≽Þ 0, then Amac; sc; B denotes ABmac; sc; 0. A random variable χ, which is Gaussian with mean value μ and covariance matrix Γmac; sc; 0 is denoted as χ  N ðμ; ΓÞ. A\B denotes that set B is subtracted from set A.

2. Problem formulation Consider a discrete time linear stochastic system: xtþ1 ¼ Axt þ wt ;

t A Z;

ð1Þ

where xt A Rn is the system state and A A Rnn , nA N, is the system matrix. The initial system state x0  N ð0; Π 0 Þ. The driving noise wt  N ð0; QÞ.

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To remotely estimate the system state, N A N sensors are used to measure xt in Eq. (1). Each sensor i AN, where N ¼ f1; 2; …; Ng, provides a noisy measurement signal: yit ¼ ci xt þ vit ;

ð2Þ

where yit A Rmi is the measurement and ci A Rmi n is the output matrix. The measurement noise vit  N ð0; r i Þ. Assumption 1. x0, wt and vit are mutually uncorrelated for any time t. Assumption 2. ðA; ci Þ is observable, ðA; Q1=2 Þ is controllable. Let γ it , iA N be the indicator function whose value (1 or 0) indicates whether sensor i is selected to use the shared single channel at time t. Let yt ¼ fyit jγ it ¼ 1; iA Ng. Obviously, yt denotes all the measurements selected to be transmitted at time t. Let mðtÞ ¼ ∑Ni¼ 1 γ it be the number of transmitted sensors at time t. Fig. 2 depicts the overall configuration of the system to be studied. The distinguishing aspect of the problem at hand lies in that the use of the communication channel introduces data delay. Data delay depends upon the channel capacity, but is also affected by the traffic flow assigned to the communication channel, i.e., the amount of transmitted data over the communication channel. In fact, for the given channel capacity, shorter data delay can be achieved by less transmitted data. However, less transmitted data, unfortunately, also lead to degrading of state estimation performance. Obviously, the amount of transmitted data is intimately related to the selection strategy of transmitted sensors γ it . Hence, the design of the state estimation scheme in Fig. 2 involves the selection strategy of transmitted sensors. Remark 1. One of our motivating applications is a distributed sensor network that collects observations and sends them to one central unit that is responsible for estimation and control. For example, in a pursuit evasion game in which mobile pursuers perform their control actions based on the current estimate of the positions of both pursuers and evaders. The usage of multiple sensors including position and velocity sensors is essential for providing a larger overall view of the terrain and benefits noise suppression and estimation accuracy. However, the increase of the number of sensors necessarily brings about the increase of delay time. The results that we present here can aid to design the sensor network in the choice of the number of sensors to obtain the optimal estimation performance.

Fig. 2. The one-channel networked data fusion system with selected sensors.

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3. Transmission effects In this section, the details about how data delay depends upon the channel capacity and the selection strategy related to the number of transmitted sensors m(t) are given. We will model transmission effects by introducing the arrival processes dt ¼ fdt gt A Z : 8 0 if yt arrives without delay at time t; i:e:; DðtÞ r T 0 > > > > < 1 if yt arrives with one  step delay at time t; i:e:; T 0 oDðtÞr 2T 0 ð3Þ dt ¼ ⋮ > > > > : d if y arrives with dstep delay at time t; i:e:; ðd1ÞT 0 oDðtÞr dT 0 t where d is the maximally allowed delay steps by the system, T0 denotes the sampling interval, D(t) denotes the delay time of transmitted data at time t. Remark 2. For the multi-sensor data fusion over one channel, there exists the medium access constraint [12], i.e., nodes are not allowed to access the shared medium (or channel) simultaneously. For the sensors which are spatially distributed in a large physical area, this problem can be dealt with via the STDM mechanism. With STDM, each sampling instant T0 over the shared channel is divided into many slots, and the measurement yti of each transmitted sensor at time t is coded into multiple packets and then uses these slots in turn. Since the data can only be decoded and used to estimate the system state until all packets of a sensor at time t are received by the remote fusion center, collective delay may occur. The classic metric for delay over the communication channel, neglecting processing and propagation delay, is D¼

f ; Cf

ð4Þ

where f is the traffic flow assigned to the communication channel and C is its capacity, where f, C units are bits/second (bps), see [30,31]. This formula provides a good metric in practice, since the closer the flow is to the maximum data rate over a given communication channel, the more likely the channel will get congested and incur delay. Although the above model is only valid in the time-invariant case, we shall adopt it also for time-varying delay time and traffic flow. For that purpose, we introduce the instantaneous traffic flow over the channel via: N

f ðtÞ ¼ ∑ γ it bit  mðtÞb;

ð5Þ

i¼1

where bit denotes the packet length of data of sensor i at time t. For simplicity, we assume that the packet length of each sensor bit is equal to constant b. Remark 3. The channel metrics (3) assume a fixed channel capacity C. However, the capacity C is a function of the signal-to-noise power ratio (SNR) over the channel as well as the bandwidth allocated to that channel. Consider a discrete-time additive white Gaussian noise (AWGN) channel, the capacity of this channel is given by the Shannon well-known formula [30]: C ¼ B log 2 ð1 þ rÞ, where B denotes the channel bandwidth and r denotes SNR.

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From Eq. (5), clearly f(t) depends on the number of transmission sensors m(t). Combining Eqs. (4) and (5), it is easy to obtain DðtÞ ¼

mðtÞb : CmðtÞb

ð6Þ

Expression (3) then gives 8 T 0C > > 0 if mðtÞr > > ð1 þ T 0 Þb > > > > > T C 2T 0 C 0 < 1 if omðtÞ r ð1 þ T 0 Þb ð1 þ 2T 0 Þb dt ¼ > > ⋮ > > > > ðd1ÞT 0 C dT 0 C > > > : d if ð1 þ ðd1ÞT Þb omðtÞr ð1 þ dT Þb 0

ð7Þ

0

In fact, m(t) influences the estimation performance in two ways. On the one hand, more transmitted sensors must result in the improvement of estimation performance. On the other hand, the more the transmitted sensors in a sampling instant, the more the steps delay will be caused and then result in further larger estimation errors. Expression (7) gives the specific correlation between dt and m(t). However, the sensor selection algorithm is related to the estimation performance. Consequently, the sensor selection algorithm should be considered carefully. This forms one of the main themes of the present work. 4. State estimation over one channel Inspired by the relationship between delay steps and the number of transmitted sensors, we assume that the number of transmitted sensors has dþ1 choices denoted as mi ðiA f0; 1; 2; …; dgÞ. When mðtÞ ¼ mi , yt arrives at the fusion center with i step delay. To ensure estimation errors as small as possible, it is obvious that: mi ¼ ⌊

iT 0 C ⌉; ð1 þ iT 0 Þb

ð8Þ

where ⌊  ⌉ denotes rounding to the nearest integer. Obviously, m0 r m1 r ⋯ r md . We will model the selection strategy related to the number of transmitted sensors by introducing a d þ 1-valued random variable θðtÞ A f0; 1; 2; …; dg, where θðtÞ ¼ i denotes mðtÞ ¼ mi , i.e., d t ¼ i. θðtÞ has a probability mass function: PrðiÞ ¼ Prob½θðtÞ ¼ i: ∑di ¼ 1 PrðiÞ ¼ 1.

ð9Þ

For simplicity, we assume that θðt 1 Þ and θðt 2 Þ are independent for Obviously, any time t 1 a t 2 . We made a further assumption that θðtÞ carries no information about the state, i. e., θðtÞ is independent of wt, vit and the initial state x0. Let θ denotes a stochastic scheduling scheme that defines the value of θðtÞ at each time t. Clearly the set of all scheduling schemes, which is denoted as Θ, consists of ðd þ 1Þt different schemes up to time t. Let Y t;k ¼ fyt received at time k; k Z tg. Obviously, Y t;k depends on yt and dt, i.e., θðtÞ. ( 0 if kot þ θðtÞ Y t;k ¼ ð10Þ yt if k Z t þ θðtÞ

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For a given θ, the data fusion problem is as follows: x^ tjt1;k ¼ Efxt jY 1;k ; …; Y t1;k g; x^ t;k ¼ Efxt jY 1;k ; …; Y t;k g; Ptjt1;k ¼ Efðxtjt1;k ^x tjt1;k Þðxtjt1;k ^x tjt1;k Þ′jY 1;k ; …; Y t1;k g; Pt;k ¼ Efðxt;k ^x t;k Þðxt;k ^x t;k Þ′ Y 1;k ; …; Y t;k g:

ð11Þ

Finite horizon optimal data fusion filters for system (1)–(11) will be developed in this section by using the method in paper [14]. Theorem 1. For system (1)–(11) satisfying Assumption1 and the given θ, the optimal estimate x^ t;t and the estimation error covariance Pt;t can be computed as x^ tjt1;k ¼ A^x t1;k ;

ð12Þ

Ptjt1;k ¼ APt1;k A′ þ Q;

ð13Þ

1 P1 t;k ¼ Ptjt1;k þ I t;k ;

ð14Þ

^ tjt1;k þ it;k Þ; x^ t;k ¼ Pt;k ðP1 tjt1;k x

ð15Þ

I t;k ¼

it;k ¼

8 > <0

ðkot þ θðtÞÞ c′i r 1 i ci

ðk Z t þ θðtÞÞ

ð16Þ

i c′i r 1 i yt

ðkot þ θðtÞÞ ðk Z t þ θðtÞÞ

ð17Þ

∑ > : i A ImθðtÞ 8 > <0

∑ > : i A ImθðtÞ

In ¼ f~i 1 ; ~i 2 ; …; ~i n ; n A Ng;

ð18Þ

~i 1 ¼ arg maxc′i r 1 i ci ;

ð19Þ

~i 2 ¼ arg max c′i r 1 i ci ;

ð20Þ

⋮ ~i N ¼ arg

ð21Þ

iAN

i A N\~i 1

max c′i r 1 i ci ; i A N\~i 1 \~i 1 …\~i N1

where x^ 0;k ¼ x0 , P0;k ¼ P0 . Proof. Based on the KF, Eqs. (12) and (13) can be easily obtained. According to [14], the Kalman filter (KF) in the state space (^x tjt1;k , x^ t;k , Ptjt1;k , Pt;k ) and the Information filter (IF) in the information space (^z tjt1;k , z^ t;k , Z tjt1;k , Z t;k ) are related by the state projection operation 1 ^ tjt1;k , z^ t;k ¼ P1 ^ t;k , Z tjt1;k ¼ P1 {^z tjt1;k ¼ P1 tjt1;k x t;k x tjt1;k , Z t;k ¼ Pt;k }. Based on IF, we have Z t;k ¼ Z tjt1;k þ I t;k ;

ð22Þ

X. Bian et al. / Journal of the Franklin Institute 351 (2014) 156–173

z^ t;k ¼ z^ tjt1;k þ it;k :

ð23Þ

Clearly, if kot þ θðtÞ, Y t;k ¼ 0; if k Z t þ θðtÞ, Y t;k ¼ yt . Then 8 ðkot þ θðtÞÞ > <0 N I t;k ¼ γ it c′i r 1 ðk Z t þ θðtÞÞ > i ci : i∑ ¼1

it;k ¼

8 > <0

ðkot þ θðtÞÞ

γ it c′i r 1 > i yi : i∑ ¼1

ðk Z t þ θðtÞÞ

N

163

ð24Þ

ð25Þ

where ∑Ni¼ 1 γ t ¼ mθðtÞ . From definition of In , it can be obtained that N

∑ γ it c′i r i ci r ∑ c′i r 1 i ci :

i¼1

i A ImθðtÞ

Then Eqs. (16) and (17) are obtained.

□

Combining Eqs. (22) and (23) and the relationship between KF and IF, we have Eqs. (14) and (15). Remark 4. Since Theorem 1 is based on KF and IF, the estimate x^ t;k given by Theorem 1 is unbiased, and it is optimal in the means of minimum mean squared error (MMSE). If without Assumption 1, Theorem 1 would not work, i.e., x^ t;k would not be the optimal solution to the data fusion problem. And the method of performance analysis given in the following section does not work. Remark 5. In Theorem 1, the estimation error covariance is given in the form of converse. In the case of d ¼ 1, θðtÞA f0; 1g, and yt1 must arrive at time t whether θðt1Þ ¼ 0 or 1. Since k Z t, Ptjt1;k can be directly expressed with Ptjt1 . And the estimation error covariance can also be rewritten as follows: if k ¼ t; Pt;k ¼ Ptjt1 ð1θðtÞÞPtjt1  C′0 ðC 0 Ptjt1 C ′0 þ R0 Þ1 C 0 Ptjt1 ;

ð26Þ

if k4t; Pt;k ¼ Ptjt1 ð1θðtÞÞPtjt1  C′0 ðC 0 Ptjt1 C ′0 þ R0 Þ1 C 0 Ptjt1 θðtÞPtjt1 C′1 ðC 1 Ptjt1;k C ′1 þ R0 Þ1  C 1 Ptjt1 ; where

0

ci~1

1

B c~ C B i2 C C C0 ¼ B B ⋮ C; @ A ci~m 0

2

r ~i 1

6 0 6 R0 ¼ 6 6 ⋮ 4 0

0

⋯

r ~i 2

⋯

⋮ 0

⋱ ⋯

3

0

0 7 7 7; ⋮ 7 5

r i~m

0

ð27Þ

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X. Bian et al. / Journal of the Franklin Institute 351 (2014) 156–173

0

1

c~i 1

2

B c~ C B i2 C C C1 ¼ B B ⋮ C; @ A c~i m 1

C0 A R

∑i A Im0 mi

, R0 A R

r i~1

6 0 6 R1 ¼ 6 6 ⋮ 4 0

3

0

⋯

0

r i~2 ⋮

⋯ ⋱

0 ⋮

0

⋯

r ~i m

ð∑i A Im0 mi Þð∑i A Im0 mi Þ

7 7 7; 7 5

1

∑ i A Im 1 m i

, C1 A R

, R1 A Rð∑i A Im1 mi Þð∑i A Im1 mi Þ .

Remark 6. The performance of the optimal filter given in Theorem 1 is explicitly dependent on θðtÞ. Generally, there are two scenarios one needs to consider when the filter is designed. The first scenario is that θðtÞ is priori for the remote filter. In this case, by using the equations in Theorem 1, the optimal estimate of the system state can be easily obtained. The second scenario is that θðtÞ is unknown for the remote filter. In this case, one has to allocate the θðtÞ for the transmitted sensors to obtain the optimal estimation performance, which is indeed the sensor scheduling problem [32,33] and our research direction in the future. Remark 7. Comparing to paper [14], our main differences and contributions are as follows: the paper [14] presents a set of simple memories and computational efficient algorithms designed to estimate the state of system with random delayed and corrupted measurements provided by a set of sensors. The contradiction between the collective delay time and traffic flow assigned to the communication channel (the number of transmitted sensors) has not yet been considered in paper [14] and other relevant papers. In this paper a new model has been developed which can truly reflect this contradiction. The data fusion algorithm given by Theorem 1 considers this relationship between the two. 5. Estimation performance In this section, we give an upper bound of the expected estimation error covariance and estimation error covariance with d¼ 1 for a random schedule. 1 Theorem 2. For any t Z 1, Pt;t r Ptjt1 r M , where M ¼ AðC ′0 R1 0 C 0 Þ A′ þ Q.

Proof. From Eqs. (14) and (16), when k4t, i~m0

i~m1

i ¼ i~1

i ¼ i~1

1 ′ 1 ′ 1 P1 t;k ¼ Ptjt1 þ ð1θðtÞÞ ∑ ci r i ci þ θðtÞ ∑ ci r i ci ~i m 0

~i m 1

i ¼ ~i 1

i ¼ ~i m0 þ1

′ 1 ¼ P1 tjt1 þ ∑ ci r i ci þ θðtÞ

∑

c′i r 1 i ci

i~m0

′ 1 Z P1 tjt1 þ ∑ ci r i ci ; i ¼ i~1

~i m 0

′ 1 Pt;k r P1 tjt1 þ ∑ ci r i ci i ¼ ~i 1

!1 :

ð28Þ

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Applying the matrix inversion lemma, we observe that 0 !1 11 i~m0 A Ptjt1 Pt;k r Ptjt1 Ptjt1 @Ptjt1 þ ∑ c′ r 1 ci i ¼ ~i 1

i i

1 1 ¼ Ptjt1;k Ptjt1 ðPtjt1 þ ðC′0 R1 0 C 0 Þ Þ Ptjt1 ¼ Ptjt1 Ptjt1 C ′0 ðC0 Ptjt1 C′0 þ R0 Þ1 C0 Ptjt1 : 1 1′ ′ 1 According to Lemma A.2 in [34], Pt;k rC 1 for any k4t. Combining 0 R0 C 0 ¼ ðC 0 R0 C 0 Þ Eq. (13) and Lemma A.1 in [34], 1 Ptjt1 ¼ APt1;k A′ þ Q r AðC ′0 R1 0 C 0 Þ A′ þ Q; 1 ′ 1 Pt;t r Ptjt1 ¼ AðC 0 R0 C0 Þ A′ þ Q:

The proof is complete.

□

Let us define gPr ðXÞ ¼ AXA′ð1Prð1ÞÞAXC ′0 ðC 0 XC ′0 þ R0 Þ1  C 0 XA′ Prð1ÞAXC ′1 ðC1 XC ′1 þ R1 Þ1 C1 XA′ þ Q:

ð29Þ

Then Ptjt1;k satisfy E½Ptjt1  ¼ E½gPr ðPt1jt2 Þ: Lemma 1. gPr(X) defined in Eq. (29) is a concave function in X for X Z 0. Thus E½gPr ðXÞ r gPr ðE½XÞ. Proof. The concavity of the function gPr(X) is a fundamental property. The inequality in the lemma is derived by directly applying Jensen's inequality. It gives an upper bound of E½Ptjt1  as a function of E½Pt1jt2 . □ Theorem 3. For any t Z 1, E½Pt;t r E½Ptjt1 r N Pr , where N Pr ¼ AðC′1 B1 C1 Þ1 A′ þ Q, 8 iA Im0 < ri ; ri bi ¼ : Prð1Þ; iA f~i m0 þ 1; …; ~i m1 g 3 2 ~ bi 1 0 ⋯ 0 7 6 6 0 bi~2 ⋯ 0 7 7; 6 B¼6 ⋮ ⋱ ⋮ 7 5 4 ⋮ i~m1 0 0 ⋯ b where B A Rð∑i A Im1 mi Þð∑i A Im1 mi Þ . Proof. From Eq. (29), we derive 0 i~m0

i~m1

i ¼ i~1

i ¼ i~m0 þ1

gPr ðXÞ ¼ A@X 1 þ ∑ c′i r 1 i ci þ Prð1Þ i~m1

¼ A X 1 þ ∑ c′i b1 i ci i ¼ i~1

!1

∑

A′ þ Q

11 A A′ þ Q c′i r 1 i ci

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X. Bian et al. / Journal of the Franklin Institute 351 (2014) 156–173

0

0

B ¼ A@XX @X þ

1 !1 11 A XC ∑ c′i b1 AA′ þ Q i ci

~i m 1

i¼1

¼ AðXXðX þ ðC′1 B1 C1 Þ1 Þ1 XÞA′ þ Q:

ð30Þ

According to Lemmas A.1 and A.2 in [34], we have, gPr ðXÞr AðC′1 B1 C 1 Þ1 A′ þ Q; Therefore, E½Ptjt1 r gPr ðE½Pt1jt2 Þ r AðC ′1 B1 C 1 Þ1 A′ þ Q; ptE½Pt;t r E½Ptjt1  rAðC′1 B1 C1 Þ1 A′ þ Q: The proof is complete.

□

Remark 8. The observability of ðA; C0 Þ, ðA; C1 Þ can prevent the estimation error from becoming unbounded. Based on that, the upper bound of the expected error covariance and error covariance is given. However, the components of C0, C1 are undetermined. So we give the observability of ðA; ci Þ to guarantee the observability of ðA; C 0 Þ, ðA; C 1 Þ. That is a sufficient condition but not a necessary condition. 6. Illustrative example In this section, we illustrate two examples to support the results in this paper and discuss the influence of θ on estimation performance. 6.1. Example 1 Consider a multi-sensors target tracking system. The system model is given by Eqs. (1) and (2). The state vector xt ¼ ½x1t ; x2t ; x3t ′, where x1t , x2t , x3t are the position, velocity, acceleration of moving target (e.g., truck) at time tT0 respectively. yit, i A f1; 2; 3g, is the measurement of the target from sensor i. The system parameters are given as follows: 2 3 2 3 0:1 0 0 1 T 0 0:5T 0 6 7 6 7 T 0 5; Q ¼ 4 0 0:2 0 5; A¼40 1 0 0 2 1 0 6 C1 ¼ 4 0 1 0

0

1 3

0 7 0 5; 1

0 

1 C2 ¼ 0

0 1

0  0 ; 0

0; 

1 C3 ¼ 0

 0 0 ; 1 0

where T 0 ¼ 0:04 is the sampling period. r 1 ¼ 0:35I 3 , r 2 ¼ 0:4I 2 , r 3 ¼ 0:4I 2 , P0 ¼ 0:2I 3 , x0 ¼ ½1; 2; 1′, d ¼ 1, m0 ¼ 1 and m1 ¼ 3. According to the definition of ~i and In , ~i 1 ¼ 1, ~i 2 ¼ 2, ~i 3 ¼ 3, I1 ¼ f1g, I2 ¼ f1; 2; 3g. First we consider the data fusion with Prð1Þ equalling to 0.1, 0.5 and 0.9. By applying Theorem 1, the data fusion filter is obtained and shown in Fig. 3, where the true values and the optimal estimate of the position of moving target x1t are depicted by red solid and blue dashed lines respectively. It can be seen from Fig. 3 that the filter provides satisfactory performance.

X. Bian et al. / Journal of the Franklin Institute 351 (2014) 156–173

167

100 50 0

0

50

100

150

200

0

50

100

150

200

0

50

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Fig. 3. The estimate with Prð1Þ ¼ 0:1; 0:5; 0:9. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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Fig. 4. Relationship between Prð1Þ and trace of Pt;t .

In what follows, the influence of Prð1Þ on estimation performance is discussed. We show that the relationship between Prð1Þ and the trace of estimation error covariance matrix is shown in Fig. 4. Let us consider scenarios where Prð1Þ ¼ 0:1, 0.5 and 0.9. Then, by applying stochastic generated θðtÞ based on the different probability Prð1Þ and Theorem 1, the trace of the estimation error covariance Pt;t is obtained and shown in Fig. 4. It can be seen from Fig. 4 that if Prð1Þ ¼ 0:1 or 0.9, the trace of estimation error covariance matrix has little fluctuation. If Prð1Þ ¼ 0:5, the minimum value of the trace of estimation error covariance matrix is smaller compared with the scenario when Prð1Þ ¼ 0:1 or 0.9, but has violent fluctuation. Now, we consider the influence of Prð1Þ on the expected estimation error covariance. Fig. 5 shows the trace of expected estimation error covariance E½Pt;t  by blue dashed line. N Pr is shown

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Fig. 6. Relationship between θ and trace of Pt;t . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

by the red solid line. It can be seen from Fig. 5 that the trace of E½Pt;t  increases with Prð1Þ and they are all smaller than N Pr , i.e., Theorem 3 is illustrated. For the given probability Prð0Þ ¼ 0:7 ðPrð1Þ ¼ 0:3Þ, we define different period scheduling schemes θ1 with period {0 0 1 0 0 1 0 0 0 1}, θ2 with period {0 0 0 0 1 1 0 0 0 1} and θ3 with period {0 0 0 0 0 0 0 1 1 1}. Fig. 6 shows the traces of the corresponding estimation error covariance matrix of θ1 , θ2 and θ3 blue dotted dashed, red solid and green dashed lines respectively. It can be seen from Fig. 6 that the trace of the estimation error covariance matrix of different scheduling schemes is different even with the same probability. In comparing to the case θ2 and θ3 , the performance under the scheduling scheme θ1 is best.

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6.2. Example 2 Consider the system (1) and (2). The state vector xt ¼ ½x1t ; x2t ; x3t ′ . yit, iA f1; 2; 3g, is the measurement of the target from sensor i. The system parameters are given as follows: A ¼ 0:8I 3 , Q ¼ 0:1I 3 , c1 ¼ ½1; 0; 1, c2 ¼ ½0; 1; 0, c3 ¼ ½1; 0; 0, r 1 ¼ 0:01, r 2 ¼ 0:01, r 3 ¼ 0:02, P0 ¼ 0:2I 3 , x0 ¼ ½0:1; 0; 0′, d ¼ 1, m0 ¼ 1 and m1 ¼ 3. According to the definition of ~i and In , ~i 1 ¼ 3, ~i 2 ¼ 2, ~i 3 ¼ 1, I1 ¼ f3g, I2 ¼ f3; 2; 1g. First we consider the data fusion with Prð1Þ equalling to 0.1, 0.5 and 0.9. By applying Theorem 1, the data fusion filter is obtained and shown in Fig. 7, where the true values and the optimal estimate of the system state x1t are depicted by red solid and blue dotted lines respectively. It can be seen from Fig. 7 that the filter provides satisfied performance. Next, the influence of Prð1Þ on estimation performance is discussed. The relationship between Prð1Þ and the trace of estimation error covariance matrix is shown in Fig. 8. Let us consider scenarios where Prð1Þ ¼ 0:1, 0.5 and 0.9. Then, by applying stochastic generated θðtÞ based on the different probability Prð1Þ and Theorem 1, the trace of the estimation error covariance Pt;t is obtained and shown in Fig. 8. It can be seen from Fig. 8 that if Prð1Þ ¼ 0:1 or 0.9, the trace of estimation error covariance matrix has little fluctuation. If Prð1Þ ¼ 0:5, the trace of estimation error covariance matrix is smaller compared with the scenario when Prð1Þ ¼ 0:1 or 0.9, but has violent fluctuation. Now, we consider the influence of Prð1Þ on the expected estimation error covariance. Fig. 9 shows the trace of expected estimation error covariance E½Pt;t  by the blue dashed line. N Pr is shown by the red solid line. It can be seen from Fig. 9 that the trace of E½Pt;t  decreases with Prð1Þ and they are all smaller than N Pr , i.e., Theorem 3 is illustrated. For the given probability Prð0Þ ¼ 0:7 ðPrð1Þ ¼ 0:3Þ, define different period scheduling schemes θ1 with period {0 0 1 0 0 1 0 0 0 1}, θ2 with period {0 0 0 0 1 1 0 0 0 1} and θ3 with period {0 0 0 0 0 0 0 1 1 1}. Fig. 10 shows the traces of the corresponding error covariance matrix of θ1 , θ2 and θ3 with blue dotted dashed, red solid and green dashed lines respectively. It can be seen from Fig. 10 that the traces of the estimation error covariance matrix of different scheduling schemes 2 0 −2

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Fig. 7. The estimate with Prð1Þ ¼ 0:1; 0:5; 0:9. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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Fig. 9. Relationship between Prð1Þ and trace of E½Pt;t . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

are different even with the same probability. In comparing to the case θ2 and θ3 , the performance under the scheduling scheme θ1 is best. 7. Conclusion The one-channel networked data fusion with communication constraint, which is medium access constraint, is considered in this paper. When the STDM mechanism is used to address this communication constraint, there exists the collective delay. Collective delay time depends upon the number of transmitted sensors. Hence, the contradiction between the number of transmitted sensors and delay steps will be caused, then the number of transmitted sensors has to be carefully

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selected to attain the optimal estimate. A new model is developed that can truly reflect this relationship by introducing a stochastic process θ. Based on this model, the optimal data fusion filter is designed for a given θ. It also gives the upper bounds of estimation error covariance and the expected estimation error covariance dependence on Pr (i) (the probability of θ). Through simulation, the influence of Pr (i) on estimation performance is discussed. There are many interesting directions along the line of this work including analyzing estimation performance with multiple-step delay, the data fusion problem over multiple communication channels and scheduling sensors in Pr (i) fixed scenario. Acknowledgment The authors would like to thank the referees for their valuable and helpful comments which have improved the presentation. The work of Yuanqing Xia was supported by the National Basic Research Program of China (973 Program) (2012CB720000), the National Natural Science Foundation of China (61225015, 61127004, 60974011), Program for New Century Excellent Talents in University of China (NCET-11-0784), the Ph.D. Programs Foundation of Ministry of Education of China (20091101110023, 20111101110012), Beijing Municipal Natural Science Foundation (4102053, 4101001), and CAST Foundation (CAST201210).

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