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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 [

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 [

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 [_{k}

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.

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

In the same time instant, the scheduler selects only one sensor from

According to the above framework in _{k}^{n}_{k}_{k}_{k−1}, and _{k–2} at time step _{k}_{k}_{k}_{k}_{k}_{k}_{k}

Taking expectation:

From _{k}_{k}_{k−1}, where
_{k−1} =1. Otherwise

Given the state vector _{k+1} defined by _{k+1}, denoted by _{k+1|k+1}, which is a linear function of observations _{0},…,_{k+1} minimizing:
_{k}_{x}_{k}_{k}_{k}_{γ}

Since state estimation _{k+1|k+1} is a linear function of _{0}, _{1},…, and _{k+1}, it can be written as:
_{i}

Our objective is to design MVFs _{k+1|k+1} based on received measurement sequences _{0}, _{1},…, _{k+1}. According to _{0}, _{1},..., _{k+1} are unknown stochastic variable sequences. That is, we desire to find _{k+1} | _{k+1}, _{k}_{k−1},..., _{0}].

Error covariance matrix:

_{k}_{k}_{k}_{k}_{k}w_{j}_{k}w_{j}_{k}_{k}_{k}v_{j}_{k}v_{j}_{k}_{k}

_{0} is independent of _{k}_{k}

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.

_{k+1} ≡ _{k+1} − _{k+1|k+1}. Because _{k+1} and _{i}

Since _{k+1|k+1} is a linear function of observations _{i}

From

From

Utilizing the above background knowledge, we will design a linear minimum variance filter with multiple packet losses.

Initial value _{0|0} = _{0}, _{0|0} = _{0} and _{0} = _{0}.

_{k−1} is invertible.

From

Let us substitute

Taking

Because:

From Lemma 1 we take

From the above equation it may derive

According to _{n}

From Definition, the next equation is derived:

In special case 1, when packet arrival rate _{k+1} = 0 and _{k}

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 (^{th}^{th}

In special case 2, when packet arrival rate _{k+1} = 0 and _{k}

It means that the information at the current step is lost. In this situation, the filter ^{th}^{th}

In special case 3, when packet arrival rate _{k+1} = 1:

In this situation, the LMVF reduces the standard KF. An example for DTSL system is given in the following part.

We give a numerical example to verify the validity of Theorem 1. It is assumed

Define estimation error
_{k}

In simulation, _{0|0} = [0 0]^{T}, _{0|0} = _{0} = _{0|0}, _{k}^{1} and the blue line with symbol ‘o’ denotes total error ^{2}. From ^{1} = 0.5527 and ^{2} = 0.7445; When ^{1} = 0.3231 and ^{2} = 0.3557.

The experience results show estimation error is very small. The simulation verifies feasibility and effectiveness of the proposed LMVF.

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.

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:
_{k}_{k}_{k}

To obtain the estimated state _{k+1|k+1}, the nonlinear function in _{k|k} with the first order to yield an extended minimum variance filter. Taylor series expansion of _{k}

Similarly, we make linearization of the nonlinear measurement function ^{jk} (•). Based on

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.

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.

We use the nonlinear state model of reference [_{k}_{k}_{k}

Sonic sensors are used to measurement target and the measurement model for sensor _{k}_{k}^{jk}, y^{jk}_{k}_{k}_{k}_{k}_{k}

Since measurement model is nonlinear function, it has to be linearized and Jacobian matrix

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

We improve dynamic-group scheduling strategy (DGSS) [_{k}_{k}_{+1}) is the predicted energy, Δ_{k}

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 _{max} and _{min}, in this case, Δ_{k}_{min}, and the sensor is selected by:
_{k}

After prediction, if at least one sensor can achieve the satisfactory tracking accuracy. In this case, the optimal (_{k}_{k}^{*} is the set of sensors that can achieve the satisfactory tracking accuracy, and ∅_{0} is the threshold of tracking accuracy.

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 _{1}_{min}_{2}_{max}_{t1} < _{t2} if _{1} < _{2}.

The operations of each task sensor node mainly include the following steps:

Measuring the distance
_{k}

Performing extended minimum variance filtering algorithm with packet losses by Theorem 2 in WSNs, and calculating the prediction accuracy according to

Transmitting the predicted accuracy
_{k}

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

The nearest sensor node is considered as the center node in a local neighbor area, total _{k}_{k}_{min} 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

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.

The monitored field is 100 m×100 m and covered by 20 sensors randomly deployed in

In the monitored field the target moves along the circle trajectory, whose start position is (30, 70). For Theorem 2, initial state vector _{0|−1} =[30 70 20 20 −1]^{T}, initial covariance matrix _{0|0} =2×_{0} = _{0|0} in _{k}_{k}_{k}_{k}^{T}_{k}^{T}_{min}_{max}_{0} = 8. The estimation errors of the EMVF under different packet arrival rate are shown in _{k}

For simplicity, in this simulation packet arrival rate _{k}

It is illustrated that sample interval becomes small and total time steps increase when packet arrival rate

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 _{0} = 0.5.

Consumed energy (

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.

Consumed energy and estimation accuracy under different sampling intervals with IDGSS based on 100 Monte Carlo simulations.

Sampling interval (s) | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | IDGSS |

Consumed energy (mJ) | 14.8564 | 9.1176 | 7.0780 | 6.1009 | 5.6857 | 6.0649 |

Estimation accuracy(m) | 0.3161 | 0.4407 | 0.4973 | 0.6450 | 0.7114 | 0.6729 |