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This paper is concerned with the filtering problem for both discrete-time stochastic linear (DTSL) systems and discrete-time stochastic nonlinear (DTSN) systems. In DTSL systems, an linear optimal filter with multiple packet losses is designed based on the orthogonal principle analysis approach over unreliable wireless sensor networks (WSNs), and the experience result verifies feasibility and effectiveness of the proposed linear filter; in DTSN systems, an extended minimum variance filter with multiple packet losses is derived, and the filter is extended to the nonlinear case by the first order Taylor series approximation, which is successfully applied to unreliable WSNs. An application example is given and the corresponding simulation results show that, compared with extended Kalman filter (EKF), the proposed extended minimum variance filter is feasible and effective in WSNs.
In wireless networks, data losses, communication delay and constrained bandwidth are general problems across communication links because of collision and transmission errors. Especially, in WSNs, due to its limited resources, such as restricted computation, processing and communication ability, data/packet losses should be further studied.
Many researchers are interested in networked control systems with packet losses. The research on packet losses can be traced back to Nahi [1] and Hadidi [2]. Recently, the packet loss problem has been studied using jump linear systems (JLS), which is a hybrid system with model transitions modeled as Markov chains that switches among several discrete models [3–5]. JLS methods restrict their formulation to the steady-state case, where Kalman gain is constant. Furthermore, the transition probability and state error covariance matrices are required being computed exactly [6]. Sinopoli et al. [7] consider general case of time varying Kalman gain and discusses how packet dropouts can affect state estimation. They illustrate that there exists a certain threshold of the packet dropout rate. Under packet dropouts, their filter has a smaller linear minimum mean square error than static counterpart. Liu et al. [8] extend the idea to the case where there are multiple sensors and packets from different sensors dropping independently. In practice, it is assumed that packets are dropped independently, which is certainly not true in case where burst packets are dropped or in queuing networks where consecutive packets are not dropped [7,8]. They also use the expected value of the error covariance matrix as the measure of performance. This might ignore the fact that events with arbitrarily low probability can make the expected value diverge. So Epstein et al. [9] give a more complete characterization of the estimator performance instead of considering a probabilistic description of the error covariance. The optimal filtering problem is considered for systems where multiple packets are dropped in an unreliable network [10–12]. Different from [10–12], where only the multiple packet dropouts are considered, reference [13] investigates both the estimation problem for systems with bounded random measurement delays and packet dropouts, which are described by some binary random variables whose probabilities are only known. Schenato [14] proposes a probabilistic framework to design the minimum error covariance estimator in a generic digital communication network where sensor's observation packets are subject to random delays and packet losses. Speranzon et al. [15] analyze and design a distributed adaptive algorithm to estimate a time-varying signal, measured by WSNs, where measurement noise and packet losses are considered. But they do not consider multiple packet losses in WSNs.
On the other hand, mobile target tracking with multiple sensors measurement is an important application of WSNs in recent years. There are great deals of wireless sensor nodes deployed randomly in a monitored field. One node or several nodes are scheduled as tasking nodes in target tracking application at each time step. Some natural problems are that how to apply filters to WSNs and who are scheduled as current task sensor nodes. There are many sensor scheduling strategies, such as the nearest distance scheduling, where the nearest sensor node to the target is scheduled as task node, minimum trace scheduling [17], where minimum trace sensor node of the error covariance matrix is scheduled, adaptive sensor scheduling [18], which selects the next tasking sensor and determines sampling interval according to the predicted accuracy and tracking cost. We propose an improved dynamic-grouping scheduling strategy (DGSS) which considers not only energy consumption and predicted accuracy, but also the real-time property of tracking target.
In this paper, we discuss minimum variance filters (MVFs) with multiple packet losses for systems that are considered not only DTSL systems but also DTSN systems in WSNs. The MVFs with packet losses across an unreliable network are designed and packet losses are assumed to be random with a given i.i.d distribution. Unlike [14] and [16], where the estimator is computed depending on whether the current measurement is received, our MVFs can be computed only depending on the packet arrival rate pk at each time instant and do not need know if a measurement is received at a particular time instant. Furthermore, our filters do not require that the measurement is time-stamped.
Simulation results show that it is feasible and effective that DGSS is adopted to select next sensor node as task node, and MVFs with multiple packet losses are used to track mobile target.
The remainder of the paper is organized as follows. MVFs with multiple packet losses are formulated in Section 2. The linear MVF is designed and a numerical example shows that linear MVF is effective in Section 3. The nonlinear MVF is derived and a target tracking example is shown in WSNs in Section 4. Finally, some conclusions are drawn in Section 5.
Problem Formulation
In WSNs, mobile target tracking with multiple sensors measurement is an important application in recent years. In practice, sensor measurements are probably lost. How to deal with packet losses and how to make multiple sensors collaborate to complete common task? We are interested in these problems and discuss them in the following part.
In Figure 1, we assume that measurements from the plant are encapsulated into packets, but are not time-stamped, and then transmitted through WSNs, whose goal is to deliver packets from a plant to a filter.
In the same time instant, the scheduler selects only one sensor from N sensors to sample measurements according to sensor scheduling strategies, where measurements come probably from the same sensor, also come probably from different sensor at different time step. At the filter side, if the current measurement has one or more delays, or is lost, the filter utilizes the latest measurement in the stack to update the state estimation, where the stack retains the latest three measurements. The stack has property as follows: last input first output. For example, the order of measurements entering stack is
zk−2jk−2,
zk−1jk−1,
zkjk from a group in WSNs. While the order of coming out is
zkjk,
zk−1jk−1,
zk−2jk−2, respectively. Therefore, MVFs can use the latest measurement from the stack in Figure 1. If the current measurement is correctly received in time, the filter uses directly measurement received from a group in WSNs (About a group, please refer to Subsection 4.2.1 in p.12). In addition, we assume that packet losses are uncorrelated and there is not a retransmission in order to increase performance of real time once a packet is lost.
According to the above framework in Figure 1, we consider a time-varying DTSL system as follows:
xk+1=Akxk+Bkwkzkjk=Ckjkxk+vkjkyk=γkzkjk+(1−γk)γk−1zk−1jk−1+(1−γk)(1−γk−1)zk−2jk−2where xk ∈ Rn is the state vector at time step k; jk is sensor node number at time step k;
zkjk,
zk−1jk−1, and
zk−2jk−2 are measurement outputs of sensor node jk, jk−1, and jk–2 at time step k, k−1, and k−2 respectively; yk is measurement received by the filter at time step k; k=1,2,…,M, where M is total sensor number in a group; Ak, Bk and
Ckjk are time-varying linear matrices with appropriate dimensions respectively; wk is state noises with variance Qk and
vkjk is measurement noises of sensor node jk with variance
Rkjk at time step k; γk is a random variable taking value 0 or 1, where 0 stands for packet loss and 1 stands for received packet. The probability is as follows, respectively:
Pr[γk=1]=pkPr[γk=0]=1−pk
Taking expectation:
Eγk=pkEγkγi=pkpi,k≠iEγk2=pk
From (1c) we know that
zkjk is lost if γk = 0 at time step k, and yk depends on γk−1, where
yk=zk−1jk−1, if γk−1 =1. Otherwise
yk=zk−2jk−2.
Given the state vector xk+1 defined by (1a). It is desired to find the estimate of xk+1, denoted by x̂k+1|k+1, which is a linear function of observations y0,…,yk+1 minimizing:
ExEγ[xk+1−x^k+1|k+1]TL[xk+1−x^k+1|k+1]where L is a symmetric positive definite matrix; yk is given by (1c), Ex is the expectation with respect to xk, wk, and vk; and Eγ is the expectation with respect to γ.
Since state estimation x̂k+1|k+1 is a linear function of y0, y1,…, and yk+1, it can be written as:
x^k+1|k+1=∑i=0k+1aiyiwhere ai is an n × m matrix.
Our objective is to design MVFs x̂k+1|k+1 based on received measurement sequences y0, y1,…, yk+1. According to (1c), we know γ0, γ1,..., γk+1 are unknown stochastic variable sequences. That is, we desire to find E[xk+1 | yk+1, yk, yk−1,..., y0].
Definition: Error covariance matrix and state variance matrix at the time k respectively are defined as follows:
Error covariance matrix:
Pk|k≡ExEγ[xk−x^k|k]T[xk−x^k|k]and state variance matrix:
Sk≡E[xkTxk]
Assumption 1: For systems (1a) and (1b), wk and vk are uncorrelated white noises with zero means and variances Qk and Rk respectively. Furthermore, if k ≠ j, E[wkwj] = 0, otherwise E[wkwj] = Qk, and Qk > 0; if k ≠ j E[vkvj] = 0, otherwise E[vkvj] = Rk, and Rk > 0.
Assumption 2: The initial state x0 is independent of wk and vk with:
Ex0=m0,E[x0Tx0]=S0,E[x0−m0]T[x0−m0]=P0.
Linear Minimum Variance Filters with Multiple Packet Losses
In this section we will show our main results on linear minimum variance filters with multiple packet losses based on the orthogonal principle analysis approach. In time-varying DTSL systems, a linear filter with multiple packet losses is to be designed. Before giving our main results, firstly we will present the following Lemma.
Proof: Define estimation error: x̃k+1 ≡ xk+1 − x̂k+1|k+1. Because x̃k+1 and yi are orthogonal, i = 0,1,...,k+1, we attain easily (10) according to orthogonal principle [12].
Since x̂k+1|k+1 is a linear function of observations yi, i=0, 1,…, k, k+1. From (8) and (10) we may know:
ExEγ[xk+1−x^k+1|k+1]x^k+1|k+1T=0
From (9) it follows that:
Pk+1|k+1=ExEγ[xk+1−x^k+1|k+1][xk+1−x^k+1|k+1]T=ExEγ[(xk+1−x^k+1|k+1)xk+1T−(xk+1−x^k+1|k+1)x^k+1|k+1T]=ExEγ[xk+1−x^k+1|k+1]xk+1T
From (10) it follows that:
ExEγ[xk+1yiT]=ExEγ[x^k+1|k+1]yiT,i=0,1,⋯,k+1
Remark 1: Substituting (8) and (10) into (7), we may know that (7) attains minimum value. That is, (10) is the sufficient condition for (7) taking minimum value.
Design of Linear Filter with Multiple Packet Losses
Utilizing the above background knowledge, we will design a linear minimum variance filter with multiple packet losses.
Theorem 1: For systems (1) satisfying Assumption 1 and 2, the linear minimum variance filter (LMVF) is shown as follows:
x^k+1|k+1=Fk+1x^k|k+Kk+1yk+1where:
Fk+1=Ak−Kk+1JKk+1=[AkPk|kJT+pk+1BkQkBkT(Ck+1jk+1)T]×[JPk|kJT−JSkJT+pk+1Ck+1jk+1AkSkAkT(Ck+1jk+1)T+pk(1−pk+1)CkjkSk(Ckjk)T+(1−pk)(1−pk+1)Ck−1jk−1Ak−1−1SkAk−1−T(Ck−1jk−1)T+pk+1Rk+1jk+1+pk(1−pk+1)Rkjk+(1−pk)(1−pk+1)Rk−1jk−1+pk+1Ck+1jk+1BkQkBkT(Ck+1jk+1)T+(1−pk+1)(1−pk)Ck−1jk−1Ak−1−1Bk−1Qk−1Bk−1T(Ak−1−1)T(Ck−1jk−1)T]−1Pk+1|k+1=Fk+1Pk|kAkT+(In−pk+1Kk+1Ck+1jk+1)BkQkBkTJ=pk+1Ck+1jk+1Ak+pk(1−pk+1)Ckjk+(1−pk)(1−pk+1)Ck−1jk−1Ak−1−1Sk+1=AkSkAkT+BkQkBkT
Initial value x̂0|0 = m0, P0|0 = P0 and S0 = P0.
Proof: From the equation (1a) it follows that:
xk−1=Ak−1−1xk−Ak−1−1Bk−1wk−1where it assumes Ak−1 is invertible.
From (1) we can get:
yk+1=γk+1zk+1+(1−γk+1)γkzk+(1−γk−1)(1−γk)zk−1=γk+1[Ck+1jk+1Akxk+Ck+1jk+1Bkwk+vk+1jk+1]+(1−γk+1)γk[Ckjkxk+vkjk]+(1−γk+1)(1−γk)[Ck−1jk−1Ak−1−1xk−Ck−1jk−1Ak−1−1Bk−1wk−1+vk−1jk−1]
Let us substitute (1a), (13) and (20) into (10) respectively. Making use of (19), it follows that:
ExEγ[xk+1−x^k+1|k+1]yiT=ExEγ{Akxk+Bkwk−Fk+1x^k|k−Kk+1[γk+1Ck+1jk+1Akxk+(1−γk+1)γkCkjkxk+(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1xk+γk+1vk+1jk+1+(1−γk+1)γkvkjk+(1−γk+1)(1−γk)vk−1jk−1+γk+1Ck+1jk+1Bkwk−(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1Bk−1wk−1]}×[γiCijixi+(1−γi)γi−1Ci−1ji−1xi−1+(1−γi)(1−γi−1)Ci−2ji−2xi−2+γiviji+(1−γi)γi−1vi−1ji−1+(1−γi)(1−γi−1)vk−2ji−2]T=0,i=2,3,…,k+1;ji=1,2,…,N
Taking i = 2,3,…k−2, and making use of Assumption 1 and (12) in Lemma 1, the above equation follows that:
Eγ{Ak−Fk+1−Kk+1[γk+1Ck+1jk+1Ak+(1−γk+1)γkCkjk+(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1]}×Ex{xk[γk−2Ck−2jk−2xk−2+(1−γk−2)γk−3Ck−3jk−3xk−3+(1−γk−2)(1−γk−3)Ck−4jk−4xk−4]T}=0
Because:
Ex{xk[γk−1Ck−1jk−1xk−1+(1−γk−1)γk−2Ck−2jk−2xk−2+(1−γk−1)(1−γk−2)Ck−3jk−3xk−3]T}≠0we can obtain:
Eγ{Ak−Fk+1−Kk+1[γk+1Ck+1jk+1Ak+(1−γk+1)γkCkjk+(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1]}=0and know easily that:
Fk+1=Ak−Kk+1[pk+1Ck+1jk+1Ak+(1−pk+1)pkCkjk+(1−pk+1)(1−pk)Ck−1jk−1Ak−1−1]=Ak−Kk+1Jwhere (4)–(6) is used. Obviously, (14) and (17) can be attained by (22).
From Lemma 1 we take i=k+1 and then it follows that:
ExEγ[xk+1−x^k+1|k+1]yk+1T=ExEγ{Ak[xk−x^k|k]−Kk+1J[xk−x^k|k]+Kk+1JxkBkwk−Kk+1[[γk+1Ck+1jk+1Ak+(1−γk+1)γkCkjk+(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1]xk+[γk+1vk+1jk+1+(1−γk+1)γkvkjk+(1−γk+1)(1−γk)vk−1jk−1]+[γk+1Ck+1jk+1Bkwk−(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1Bk−1wk−1]}×{[γk+1Ck+1jk+1Ak+(1−γk+1)γkCkjk+(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1]xk+[γk+1vk+1jk+1+(1−γk+1)γkvkjk(1−γk+1)(1−γk)vk−1jk−1]+[γk+1Ck+1jk+1Bkwk−(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1Bk−1wk−1]}T=AkPk|kJT−Kk+1JPk|kJT+Kk+1JSkJT−Kk+1[pk+1Ck+1jk+1AkSkAkT(Ck+1jk+1)T+(1−pk+1)pkCkjkSk(Ckjk)T+(1−pk+1)(1−pk)Ck−1jk−1Ak−1−1SkAk−1−T(Ck−1jk−1)T]+pk+1BkQkBkT(Ck+1jk+1)T−Kk+1[pk+1Rk+1jk+1+(1−pk+1)pkRkjk+(1−pk+1)(1−pk)Rk−1jk−1+pk+1Ck+1jk+1BkQkBkT(Ck+1jk+1)T+(1−pk+1)(1−pk)Ck−1jk−1Ak−1−1Bk−1Qk−1Bk−1TAk−1−T(Ck−1jk−1)T]=0
From the above equation it may derive (15).
According to (11) in Lemma 1, it follows that:
Pk+1|k+1=ExEγ[xk+1−x^k+1|k+1]xk+1T=ExEγ{Ax(xk−x^k|k)−Kk+1J(xk−x^k|k)+Kk+1Jxk+Bkwk−Kk+1[[γk+1Ck+1jk+1Ak+(1−γk+1)γkCkjk+(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1]xk+γk+1vk+1jk+1+(1−γk+1)γkvkjk+(1−γk+1)(1−γk)vk−1jk−1+γk+1Ck+1jk+1Bkwk−(1−γk+1)(1−γk)Ck−1jk−1Ak−1−1Bk−1wk−1]}×{Akxk+Bkwk}T=(Ak−Kk+1J)Pk|kAkT+(In−pk+1Kk+1Ck+1jk+1)BkQkBkTwhere Assumption 1 is used and In is n × n identity matrix.
From Definition, the next equation is derived:
Sk+1=E[xk+1xk+1T]=AkSkAkT+BkQkBKT
Remark 2:
In special case 1, when packet arrival rate pk+1 = 0 and pk = 0:
Fk+1=Ak−Kk+1Ck−1jk−1Ak−1−1Kk+1=AkPk|kAk−1−T(Ck−1jk−1)T[Ck−1Ak−1−1Pk|kAk−1−T(Ck−1jk−1)T+Rk−1jk−1+Ck−1jk−1Ak−1−1Bk−1QkBk−1TAk−1−1(Ck−1jk−1)T]−1
From (23) we know that the information at the current step and the information at the latest previous step are consecutively lost. In this situation, the filter (13) uses the (k − 1)th step measurement to update the information of the (k + 1)th step.
In special case 2, when packet arrival rate pk+1 = 0 and pk = 1:
Fk+1=Ak−Kk+1CkjkKk+1=AkPk|k(Ckjk)T[CkjkPk|k(Ckjk)T+Rkjk]−1
It means that the information at the current step is lost. In this situation, the filter (13) uses the kth step measurement to update information of the (k + 1)th step.
In special case 3, when packet arrival rate pk+1 = 1:
Fk+1=Ak−Kk+1Ck+1jk+1AkKk+1=(AkPk|kAkT+BkQkBkT)(Ck+1jk+1)T×[Ck+1jk+1AkPk|kAkT(Ck+1jk+1)T+Rk+1jk+1+Ck+1jk+1BkQkBkT(Ck+1jk+1)T]−1
In this situation, the LMVF reduces the standard KF. An example for DTSL system is given in the following part.
An example for LMVF
We give a numerical example to verify the validity of Theorem 1. It is assumed
Ckjk is a constant in the following equation at each time step and:
xk+1=Akxk+Bkwkzkjk=Ckjkxk+vkjkyk=γkzkjk+(1−γk)γk−1zk−1jk−1+(1−γk)(1−γk−1)zk−2jk−2where
Ak=[0.600.80.5],
Bk=[0.60.5],
Ckjk=[11] and
xk=[xk1xk2]T.
Define estimation error
Erki at time step k and total estimation error Er respectively:
Erk1=|xk1−x^k|k1|,Erk2=|xk2−x^k|k2|Er1=∑k=1NErk1,Er2=∑k=1NErk2where (
(x^k|k1,x^k|k2)T is the estimate of xk; N denotes total time step.
In simulation, x̂0|0 = [0 0]T, P0|0 = diag([0.1 0.1]), S0 = P0|0, n = 1,000,
Rkjk=1 and
Qk=[0.360.300.300.25].
(x^k|k1,x^k|k2)T at each time step be calculated making use of Theorem 1. Estimation error Erk at each time step and total estimation error Er are computed by (27) and (28) respectively. 100 Monte Carlo simulations are tested and corresponding simulation results are shown in Figure 2, where the red line with symbol ‘*’ denotes total error Er1 and the blue line with symbol ‘o’ denotes total error Er2. From Figure 2 we see the estimation error decreases as packet arrival rate p increases from 0.1 to 1. When p = 0.1, Er1 = 0.5527 and Er2 = 0.7445; When p=1, Er1 = 0.3231 and Er2 = 0.3557.
The experience results show estimation error is very small. The simulation verifies feasibility and effectiveness of the proposed LMVF.
Extended Minimum Variance Filter with Multiple Packet Losses
In this section we will show our augmented results on extended minimum variance filters with multiple packet losses. In time-varying DTSN systems, an extended minimum variance filter is to be derived and it is extended to nonlinear case by Taylor series approximation in the following section.
Design of Extended Minimum Variance Filter with Multiple Packet Losses
In mobile target tracking of WSNs, the state models of the plant and measurement models are usually nonlinear, so we require making linearization for models.
Rewriting (1), let it become DTSN system:
xk+1=φ(xk)+Bkwkzkjk=hjk(xk)+vkjkyk=γkzkjk+(1−γk)γk−1zk−1jk−1+(1−γk)(1−γk−1)zk−2jk−2where φ(xk) and h(xk) are nonlinear functions with respective to the state xk and the time step k. The other conditions are similar to (1a) and (1b) in Section 2.
To obtain the estimated state x̂k+1|k+1, the nonlinear function in (29a) is expanded in Taylor series around the latest estimate x̂k|k with the first order to yield an extended minimum variance filter. Taylor series expansion of (29a) to the first order is approximately:
xk+1=φ(x^k|k)+Φk(xk−x^k|k)+Bkwkwhere
Φk=∂ϕ(xk)∂xk|xk=x^k|k is the Jacobian matrix of the function φ(xk).
Similarly, we make linearization of the nonlinear measurement function (29c) by Taylor series:
yk=γkhjk(x^k|k−1)+γkHkjk(xk−x^k|k−1)+γkvkjk+(1−γk)γk−1hjk(x^k−1|k−2)+(1−γk)γk−1Hk−1jk−1(xk−1−x^k−1|k−2)+(1−γk)γk−1vk−1jk−1+(1−γk)(1−γk−1)hjk−2(x^k−2|k−3)+(1−γk)(1−γk−1)Hk−2jk−2(xk−2−x^k−2|k−3)+(1−γk)(1−γk−1)vk−2jk−2where
Hkjk=∂hjk(xk)∂xk|xk=x^k|k−1 is the Jacobian matrix of the function hjk (•). Based on (30) and (31), Theorem 2 is shown as follows:
Theorem 2: For nonlinear systems (30) and (31) satisfying Assumption 1 and 2, extended minimum variance filter (EMVF) is shown as follows:
x^k+1|k+1=x^k+1|k+Kk+1[yk+1−pk+1hjk+1(x^k+1|k)−(1−pk+1)pkhjk(x^k|k−1)−(1−pk+1)(1−pk)hjk−1(x^k−1|k−2)]where:
x^k+1|k=φ(x^k|k,k)Kk+1=[ΦkPk|kJT+pk+1BkQkBkT(Hk+1jk+1)T]×{JPk|kJT−JSkJT+pk+1Hk+1jk+1ΦkSkΦkT(Hk+1jk+1)T+pk(1−pk+1)HkjkSk(Hkjk)T+(1−pk)(1−pk+1)Hk−1jk−1Φk−1−1SkΦk−1−T(Hk−1jk−1)T+pk+1Rk+1jk+1+pk(1−pk+1)Rkjk+(1−pk)(1−pk+1)Rk−1jk−1+(1−pk)(1−pk+1)Hk−1jk−1Φk−1−1Bk−1Qk−1Bk−1TΦk−1−T(Hk−1jk−1)T+pk+1Hk+1jk+1BkQkBkT(Hk+1jk+1)T}−1Pk+1|k+1=(Φk−Kk+1J)Pk|kΦkT+(In−pk+1Kk+1Hk+1jk+1)BkQkBkTJ=pk+1Hk+1jk+1Φk+pk(1−pk+1)Hkjk+(1−pk)(1−pk+1)Hk−1jk−1Φk−1−1Sk+1=ΦkSkΦkT+BkQkBkTwith:
x^0|−1=m0,P0|0=P0andS0=P0
The proof of Theorem 2 is similar to Theorem 1, so it is omitted here. Theorem 2 can be applied to nonlinear cases of the measurement model and state model. An application example for EMVF is given in the following subsection.
Application of EMVF in WSNs
In the above subsection a kind of EMVF is designed. In the following we apply the EMVF to track mobile target in WSNs. Usually multiple sensors are scheduled cooperatively to complete a common task. Firstly we give a practice state model and measurement model, and then provide a sensor scheduling strategy. At last, simulation results illustrate our EMVF is feasible and effective in WSNs.
Nonlinear state model
We use the nonlinear state model of reference [20]. The idea is to augment the state vector with a turning rate parameter ω, which is to be estimated along with the other system parameters, which in this example are the position
(xkt,ykt) and the velocity
(x˙kt,y˙kt) of the target. Thus, the state vector can be expressed as:
xk=[xktyktx˙kty˙ktωkt]Tand the dynamic model for the coordinated turns is:
xk+1=A⌢kxk+Bkwkwhere:
A⌢k=[10sin(Δtkωkt)ωktcos(Δtkωkt)−1ωkt001cos(Δtkωkt)−1ωktsin(Δtkωkt)ωkt000cos(Δtkωkt)−sin(Δtkωkt)000sin(Δtkωkt)cos(Δtkωkt)000001]Bk=[00001]Tand wk is white Gaussian process noise with zero mean and variance Qk. The Jacobian matrix of Ȃk is computed as:
Φk=[10sin(Δtkωkt)ωktcos(Δtkωkt)−1ωkt∂xk+1t∂ωkt01cos(Δtkωkt)−1ωktsin(Δtkωkt)ωkt∂yk+1t∂ωkt00cos(Δtkωkt)−sin(Δtkωkt)∂x˙k+1t∂ωkt00sin(Δtkωkt)cos(Δtkωkt)∂y˙k+1t∂ωkt00001]where:
∂xk+1t∂ωtk=Δtkωktcos(Δtkωkt)+sin(Δtkωkt)−1(ωkt)2x˙k−Δtkωktsin(Δtkωkt)+cos(Δtkωkt)(ωkt)2y˙k∂yk+1t∂ωkt=Δtkωktsin(kωkt)+cos(Δtkωkt)−1(ωkt)2x˙k−kωktcos(Δtkωkt)+sin(Δtkωkt)(ωkt)2y˙k∂x˙k+1t∂ωkt=Δtksin(Δtkωkt)x˙k−Δtkcos(Δtkωkt)y˙k∂y˙k+1t∂ωkt=Δtkcos(Δtkωkt)x˙k−Δtksin(Δtkωkt)y˙k
Nonlinear measurement model
Sonic sensors are used to measurement target and the measurement model for sensor jk at time step k is defined as follows:
zkjk=hjk(xk)+vkjk=(xkt−xjk)2+(ykt−yjk)2+vkjkwhere jk = 1,2,⋯, N, and N is the total number of sensor nodes;
(xkt,ykt) is the dynamic location of the mobile target and (xjk, yjk) is the known position of sensor jk ;
vkjk is the zero-mean Gaussian measurement noise with variance
Rkjk for sensor jk at the time step k; γk is a random variable with Eγk = pk.
Since measurement model is nonlinear function, it has to be linearized and Jacobian matrix
Hkjk of
hkjk is computed as follows:
Hkjk=[xkt−xjk(xkt−xjk)2+(ykt−yjk)2ykt−yjk(xkt−xjk)2+(ykt−yjk)200]
Distributed wireless sensor scheduling
In target tracking application of distributed WSNs, there are a number of wireless sensor nodes deployed randomly in a monitored field. One node or several nodes are scheduled as task nodes in target tracking application at each time step k.
We improve dynamic-group scheduling strategy (DGSS) [19]. The improved DGSS (IDGSS) is regarded as the scheduler to select next task sensor node during mobile target tracking, where predicted accuracy, predicted energy and real time are considered. We define cost function:
J(jk,Δtk)=w∅kjk+(1−w)E(jk,jk+1)Δtkwhere
∅kjk is the predicted accuracy, E(jk, jk+1) is the predicted energy, Δtk is sample interval and w ∈ (0,1) is weight parameter used to balance the tracking accuracy and energy consumption.
The task sensor node is scheduled in the following two situations:
After prediction, if none of the sensors can achieve the satisfactory tracking accuracy using any sampling interval between Tmax and Tmin, in this case, Δtk is set to the minimal sampling interval Tmin, and the sensor is selected by:
jk=argjk∈Gmin{J(jk,Tmin)}where G is the set of candidate sensors in the group which can be selected by sensor jk. The purpose of this mode is to drive the tracking accuracy to be satisfactory as soon as possible with consideration of the energy consumption.
After prediction, if at least one sensor can achieve the satisfactory tracking accuracy. In this case, the optimal (jk, Δtk) are selected by:
J(jk,Δtk)=argjk∈G*,ϕ(jk,jk+1)≤ϕ0min{E(jk,jk+1)Δtk}where G* is the set of sensors that can achieve the satisfactory tracking accuracy, and ∅0 is the threshold of tracking accuracy. Equation (44) above utilizes the objective function (42) with w=0. The basic idea of this mode is that when the predicted tracking accuracy is satisfactory, the sensors and the sampling interval are selected according to the energy efficiency.
The sample interval significantly affects tracking accuracy and energy efficiency of the whole network. For example, with a short sampling interval, the target can be tracked more accurately but lead to much energy consumption. While long sampling interval maybe load tracking accuracy to decrease or the target to be lost. In this paper we suppose the sampling interval is selected from a predefined N values
{Tt}1N, where T1=Tmin, T2=Tmax, and Tt1 < Tt2 if t1 < t2.
The operations of each task sensor node mainly include the following steps:
Measuring the distance
zkjk between the motion target and the current sensor node jk at the current time k, where (41) is used;
Performing extended minimum variance filtering algorithm with packet losses by Theorem 2 in WSNs, and calculating the prediction accuracy according to
∅kjk=[11000]Pk|k[11000]T;
Transmitting the predicted accuracy
∅kjk and sample interval Δtk according to (43) and (44) at time step k to the scheduler.
We suppose that all sensor nodes are usually in the sleeping mode and are awaked to perform sensing tasks by using an ultralow power channel when they are scheduled to perform the sensing tasks.
The scheduler of IDGSS is shown as follows:
First, the nearest node to the target (such as bold small circle in the group G1 in Figure 3) is awaked from sleep state and regarded as the task node at current time step k in the neighbor region;
The nearest sensor node is considered as the center node in a local neighbor area, total M nearest nodes to the target are awaked to form a group. When the target lies in the group, we consider both estimation accuracy and energy. If estimation accuracy is satisfactory, the sensor node, whose consumption energy is least, is selected to track the target by (44) at the next time step k+1, and changed sample interval Δtk is used in order to save energy; If estimation accuracy is not satisfactory, (43) is used to select next task node jk and Tmin is used as sample interval;
When the target moves out of the group, the sensor nodes in the group return sleep state, and a new task node is awaked from the sleep state in a new local neighbor region. Similarly, a new dynamic group is formed again, as seen from Figure 3, such as G2, G3, G4,…
Remark 3: we use dynamic grouping idea and firstly awake M sensor nodes in order to increase real-time performance of target tracking, and adopt the changed sampling intervals during target tracking in order to save energy consumption under the satisfactory estimation accuracy.
Simulation results
Nonlinear state model and measurement model in Section 4.2.1 and 4.2.2 are adopted to track the mobile target in WSNs.
Define two measurements received as follows respectively.
Model 1:
yk1=γkzkjk+(1−γk)γk−1zk−1jk−1
Model 2:
yk2=γkzkjk+(1−γk)γk−1zk−1jk−1+(1−γk)(1−γk−1)zk−2jk−2where
yk1 denotes that the filter uses the previous measurement
zk−1jk−1 to update if the current measurement
zkjk is lost, and the filter uses only zero to update if both the current measurement and the previous measurement are lost at time step k; Different from
yk1,
yk2 denotes that the filter can use
zk−2jk−2 to update when both the current measurement
zkjk and the previous measurement
zk−1jk−1 are lost at time step k.
The monitored field is 100 m×100 m and covered by 20 sensors randomly deployed in Figures 4, 5, 6 and 7, where the IDGSS is adopted in the target tracking, and the red circles represent random sensor nodes deployed in the monitored WSNs area, and the red circles with symbol ‘*’ represent sensor nodes scheduled in the process of the target tracking, and the associated tasking sensor of each estimated target position is indicated by a blue line between them.
In the monitored field the target moves along the circle trajectory, whose start position is (30, 70). For Theorem 2, initial state vector x̂0|−1 =[30 70 20 20 −1]T, initial covariance matrix P0|0 =2×diag([0.1 0.1 0.1 0.1 0.1]) and initial variance S0 = P0|0 in (38). Total time step N = 67, number of sensor in a group M = 5, q = 0.2 Rk = 0.001 and Qk = Bk × q × BkT, where Bk = [0 0 0 0 0.1]T. Minimum (maximum) time sampling interval Tmin = 0.1 (Tmax = 0.5). Weight parameter w = 0.7. Threshold of accuracy ∅0 = 8. The estimation errors of the EMVF under different packet arrival rate are shown in Figures 8, 9, 10 and 11, where estimation error is defined as the distance between the estimated target positions and the true position of the target, and γk (Gamma) is a random variable taking value 0 or 1.
For simplicity, in this simulation packet arrival rate pk = p. In Figure 4 when packet arrival rate p = 1, standard EKF, Model 1 and Model 2 have the approximately same tracking trajectory. Estimation error is approximate in Figure 8. Because packet arrival rate p = 1, we can verify that corresponding EMVF of Model 1 and Model 2 reduce standard EKF [20]. When packet loss rate p = 0.98, standard EKF starts to diverge once packet loss happens, while Model 1 and Model 2 still can track the mobile target in Figure 5. When the time step k = 8, packet loss happens, and then EKF has much higher estimation error than Model 1 and Model 2 in Figure 9. When packet arrival rate p is 0.95 in Figure 6, compared with Model 1, Model 2 has superior tracking trajectory. Moreover, Model 2 shows smaller estimation error than Model 1 in Figure 10. Model 2 has better performance than Model 1, because Model 1 can’t deal with consecutive two measurements dropped, while Model 2 still can work well. It is similar in Figure 7 and Figure 11. Moreover, From Figure 8 to Figure 11, we know that estimation error becomes more and more small when packet arrival rate p increases.
It is illustrated that sample interval becomes small and total time steps increase when packet arrival rate p decreases, such as the total 42, 44, 52 and 58 time steps in Figure 12, 13, 14, and 15 respectively, where it is also resulted in that consumed energy increases when packet arrival rate p decreases in WSNs.
The simulation results verify that derived EMVF is feasible and available in WSNs. Compared with EKF, EMVF has superior performance in tracking mobile target.
In order to illustrate further the effect of consumption energy and estimation accuracy to sampling intervals, we assume that packet arrival rate p=1 and estimation accuracy threshold ∅0 = 0.5.
Consumed energy (Ce) and estimation accuracy (Ec) are defined as follows respectively:
Ce=1mn∑i=1m∑k=0nE(jk,jk+1)Ec=1m∑t=1m(1n∑k=1n∅kjk)
Table 1 demonstrates that consumption energy is quite small and estimation accuracy is not also high when IDGSS is adopted. The experience shows that IDGSS make tradeoffs between consumed energy and estimation accuracy during target tracking in WSNs.
Conclusions
For DTSL systems and DTSN systems, a linear optimal filter and an extended minimum variance filter with packet losses are designed in this paper, respectively. Especially, the proposed EMVF is applied to WSNs for target tracking. A first application example is given and the corresponding simulation result verifies the effectiveness and advantages of the proposed LMVF. The second application example illustrates that the EMVF with multiple packet losses is feasible and available for target tracking in WSNs. In the future, we will further study filters with both multiple time delays and finite consecutive packet losses and in WSNs.
This work is supported by NSFC-Guangdong Joint Foundation Key Project U0735003, Oversea Cooperation Foundation under Grant 60828006, Fundamental Research Funds for the Central Universities under Grant No. 2009ZM0076 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
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MVFs with Multiple Packet Losses and scheduling in WSNs.
Estimation error of LMVFs under different packet arrival rate 0.1 ≤ <italic>p</italic> ≤ 1 based on 100 Monte Carlo simulations.
The framework of IDGSS.
Real trajectory and estimation with IDGSS under packet arrival rate <italic>p</italic> = 1.
Real trajectory and estimation with IDGSS under packet arrival rate <italic>p</italic> = 0.98.
Real trajectory and estimation with IDGSS under packet arrival rate <italic>p</italic> = 0.95.
Real trajectory and estimation with IDGSS under packet arrival rate <italic>p</italic> = 0.90.
Estimation error with IDGSS under packet arrival rate <italic>p</italic> = 1.
Different Filters with IDGSS under packet arrival rate <italic>p</italic> = 0.98.
Estimation error of different filters with IDGSS under packet arrival rate <italic>p</italic> = 0.95.
Estimation error of different filters with IDGSS under packet arrival rate <italic>p</italic> = 0.90.
Changed sampling interval under packet arrival rate <italic>p</italic> = 1.
Changed sampling interval under packet arrival rate <italic>p</italic> = 0.98.
Changed sampling interval under packet arrival rate <italic>p</italic> = 0.95.
Changed sampling interval under packet arrival rate <italic>p</italic> = 0.90.
Consumed energy and estimation accuracy under different sampling intervals with IDGSS based on 100 Monte Carlo simulations.