Distributed Filter with Consensus Strategies for Sensor Networks

Abstract

Consensus algorithm for networked dynamic systems is an important research problem for data fusion in sensor networks. In this paper, the distributed filter with consensus strategies known as Kalman consensus filter and information consensus filter is investigated for state estimation of distributed sensor networks. Firstly, an in-depth comparison analysis between Kalman consensus filter and information consensus filter is given, and the result shows that the information consensus filter performs better than the Kalman consensus filter. Secondly, a novel optimization process to update the consensus weights is proposed based on the information consensus filter. Finally, some numerical simulations are given, and the experiment results show that the proposed method achieves better performance than the existing consensus filter strategies.

1. Introduction

In recent years, there has been a surge of interests in the area of distributed sensor networks. The advantages of distributed sensor networks lie in their low processing power, cheap memory, scalable sensing features, and fault tolerance capabilities.

One of the most basic problems for distributed sensor networks is to develop distributed algorithms [1] for the state estimation of a process of interest. When a process is observed by a group of sensors organized in a network, the goal of each sensor node is to obtain the accurate state estimation for the process. Kalman filtering has been proved to be an effective algorithm for state estimation of dynamic processes [2, 3]. Because of this, most papers focusing on distributed estimation propose different mechanisms by combining the Kalman filter with a consensus filter in order to ensure that the estimates asymptotically converge to the same value, schemes which will be henceforth called consensus based distributed filtering algorithms. Based on the idea mentioned above, a scheme for distributed Kalman filtering (DKF) was proposed in [4] based on reaching an average-consensus [5, 6], and in [7] Olfati-Saber proposed a scalable and distributed Kalman filtering algorithm based on reaching a dynamic average consensus [8]. Olfati-Saber’s algorithm [7] has been further developed by other researchers [9] with similar algorithms. However, methods based on such kind of algorithms produce relatively weak performances. According to [10], the performance is compared with the collective estimation error of 𝑛 noncooperative local Kalman filters, which is a trivial base performance level for distributed estimation in sensor networks. To solve this, Olfati-Saber developed the Kalman consensus filter (KCF) in [11], where a consensus filter runs directly on the estimator state space variables. In addition, a formal derivation followed by optimality and stability analysis of KCF in discrete-time has been elaborated in [10]. However, in distributed implementations, there is a correlation between local estimates [12] in KCF. In general distributed networks, it is not possible to exactly determine this correlation [13] and it results in nonoptimal local estimates. Other techniques to accomplish distributed estimation for dynamic systems that rely on the inverse covariance filter or information filter have been around for many years [14, 15]. An information consensus filter (ICF) is presented in [16] that applies consensus filters to an information filter. This method does not exactly solve the problem of correlation between local estimates but it gives insight into the statistical effects of the correlation and is working much well in distributed sensor networks. Based on the ICF, we focus on designing the consensus weights to improve its performance.

In this paper, we firstly describe the existing distributed filter with consensus strategies. Then we make an in-depth comparison between the KCF and the ICF. Based on the ICF, we propose the consensus weights optimization for better performance of the system and refer this method as weights optimized information consensus filter (WO-ICF). We show experimentally that the proposed method achieves the best performance and it is closest to the optimal centralized performance.

The structure of this paper is organized as follows. In Section 2, the background knowledge on Kalman filter and the centralized information filter are provided. In Section 3, the consensus strategies are discussed. In Section 4, Kalman consensus filter is presented. In Section 5 we discuss the information consensus filter, and an ICF based weights optimization method is proposed. Simulation results and performance comparisons between the KCF and ICF algorithm are provided in Section 6. In Section 7, we make a brief summary.

2. Kalman Filter: Information Form

2.1. Kalman Filter

Consider a dynamic process with the linear time-varying model as follows: 𝑥()(𝑘)𝑥(𝑘)(𝑘)𝑤(𝑘);𝑥(0)𝑘+1=𝐴+𝐵∈𝑁𝑥(0),𝑃₀, (1) where 𝑥(𝑘)∈𝑅^𝑛 and 𝑤(𝑘)∈𝑅^𝑚 are the state and input noise of the process at time 𝑘∈{0,1,2…}, respectively; 𝑥(0) is the initial state with a Gaussian distribution; 𝐴(𝑘) is the model matrix, 𝐵(𝑘) is the state noise matrix. We are interested in tracking the state of this target by the use of a sensor network with 𝑛 sensors and the communication topology 𝐺=(𝑉;𝐸).

The observations at sensor 𝑖 and time 𝑘 are 𝑧_𝑖(𝑘)=𝐻_𝑖(𝑘)𝑥(𝑘)+𝑣_𝑖(𝑘), (2) where 𝑧_𝑖(𝑘)∈𝑅^𝑝𝑖 with ∑^𝑛_𝑖=1𝑝_𝑖=𝑝, 𝐻_𝑖(𝑘)∈𝑅^{𝑝𝑖×𝑛} is the local observation matrix for sensor 𝑖, and 𝑣_𝑖(𝑘) is the local observation noise. We refer to 𝑧_𝑖(𝑘) as sensor data. Assume that 𝑤(𝑘) and 𝑣_𝑖(𝑘) are zero mean white Gaussian noise with the following statistics: 𝐸𝑤(𝑘)𝑤(𝑙)^𝑇=𝑄(𝑘)𝛿_𝑘𝑙,𝐸𝑣_𝑖(𝑘)𝑣_𝑖(𝑙)^𝑇=𝑅_𝑖(𝑘)𝛿_𝑘𝑙, (3) where 𝛿_𝑟𝑠=1 if 𝑟=𝑠 and 𝛿_𝑟𝑠=0, otherwise. We stack the observations at all 𝑛 sensors in the sensor network to get the global observation model as follows.

Let the global observation vector 𝐳(𝑘)∈𝑅^𝑝, the global observation matrix 𝐻(𝑘)∈𝑅^𝑝×𝑛, and the global observation noise vector 𝑣(𝑘)∈𝑅^𝑛 be 𝐳(𝑘)=𝑧₁(𝑘),𝑧₂(𝑘),…,𝑧_𝑛(𝑘)^𝑇,𝐻(𝑘)=𝐻₁(𝑘),𝐻₂(𝑘),…,𝐻_𝑛(𝑘)^𝑇,𝑣(𝑘)=𝑣₁(𝑘),𝑣₂(𝑘),…,𝑣_𝑛(𝑘)^𝑇. (4)

Then the global observation model is given by 𝐳(𝑘)=𝐻(𝑘)𝑥(𝑘)+𝑣(𝑘). (5)

Since observation noises of different sensors are mutually independent, we can combine 𝑅_𝑖(𝑘) into one global observation noise covariance matrix 𝑅(𝑘) as 𝑅(𝑘)=blockdiag𝑅_𝑖(𝑘),…,𝑅_𝑛(𝑘). (6)

Given the collective information 𝑍(𝑘)={𝐳(0),𝐳(1),…,𝐳(𝑘)}, the estimation of the state of the process can be expressed as 𝑥(𝑘)∶=𝑥(𝑘∣𝑍(𝑘))=𝐸[𝑥(𝑘)∣𝑍(𝑘)],𝑥(𝑘)∶=𝑥(𝑘∣𝑍(𝑘−1))=𝐸[𝑥(𝑘)∣𝑍(𝑘−1)]. (7)

We refer to 𝑥(𝑘) and 𝑥(𝑘) as estimate and prior estimate (or prediction) of the state 𝑥(𝑘), respectively. Then, the error covariance matrices associated with the estimates 𝑥(𝑘) and 𝑥(𝑘) are given by 𝑀(𝑘)∶=𝐸(𝑥(𝑘)−𝑥(𝑘))(𝑥(𝑘)−𝑥(𝑘))^𝑇=𝐸𝜂(𝑘)𝜂(𝑘)^𝑇,𝑃(𝑘)∶=𝐸(𝑥(𝑘)−𝑥(𝑘))(𝑥(𝑘)−𝑥(𝑘))^𝑇=𝐸𝜂(𝑘)𝜂(𝑘)^𝑇, (8) where 𝜂(𝑘)=𝑥(𝑘)−𝑥(𝑘) and 𝜂(𝑘)=𝑥(𝑘)−𝑥(𝑘) denote the estimate errors and 𝑃(0)=𝑃₀. Then, the Kalman filter is a linear estimator in the form 𝑥(𝑘)=𝑥(𝑘)+𝐾(𝑘)(𝐳(𝑘)−𝐻(𝑘)𝑥(𝑘)) (9) with the Kalman gain 𝐾(𝑘).

Remark 1. Throughout this paper, due to the importance of the node indices, we adopt a notation that is free of the time-index 𝑘 and call it an index-free notation to represent all estimators [10]. The index-free form of the above estimator can be written as 𝑥=𝑥+𝐾(𝐳−𝐻𝑥). (10)

We also use the update operation {⋅⁺} defined in [10] to rewrite the sensing model of node 𝑖 of the sensor network as 𝑥⁺=𝐴𝑥+𝐵𝑤,𝑧_𝑖=𝐻_𝑖𝑥+𝑣_𝑖. (11)

Then, we get the index-free recursive equations of a centralized Kalman filter (CKF) for system (11): 𝑥=𝑥+𝐾(𝐳−𝐻𝑥),𝐾=𝑃𝐻^𝑇𝑅+𝐻𝑃𝐻^𝑇−1,𝑀=𝑃−𝑃𝐻^𝑇𝑅+𝐻𝑃𝐻^𝑇−1𝑃𝐻𝑃,₊=𝐴𝑀𝐴^𝑇+𝐵𝑄𝐵^𝑇,𝑥⁺=𝐴𝑥. (12)

2.2. Information Form: Centralized Information Filter

Using the matrix inversion lemma 𝐼+𝐴𝐵𝐶⁻¹𝐵^𝑇−1𝐴=𝐴⁻¹+𝐵𝐶⁻¹𝐵^𝑇−1=𝐴−𝐴𝐵𝐵^𝑇𝐴𝐵+𝐶⁻¹𝐵^𝑇𝐴. (13) By use of the identity 𝐵^𝑇𝐴𝐵𝐶⁻¹=[(𝐵^𝑇𝐴𝐵+𝐶)𝐶⁻¹−𝐼], we have 𝐼+𝐴𝐵𝐶⁻¹𝐵^𝑇−1𝐴𝐵𝐶⁻¹=𝐴⁻¹+𝐵𝐶⁻¹𝐵^𝑇−1𝐵𝐶⁻¹=𝐴𝐵𝐵^𝑇𝐴𝐵+𝐶⁻¹. (14)

From (14), we have 𝐾=𝑃𝐻^𝑇𝑅+𝐻𝑃𝐻^𝑇−1=𝑃⁻¹+𝐻^𝑇𝑅⁻¹𝐻⁻¹𝐻^𝑇𝑅⁻¹. (15)

Using the matrix inversion lemma (𝐴+𝐵𝐶𝐷)⁻¹=𝐴⁻¹−𝐴⁻¹𝐵𝐶⁻¹+𝐷𝐴⁻¹𝐵⁻¹𝐷𝐴⁻¹, (16) we have 𝑃⁻¹+𝐻^𝑇𝑅⁻¹𝐻⁻¹=𝑃−𝑃𝐻^𝑇𝑅+𝐻𝑃𝐻^𝑇−1𝐻𝑃=𝑀. (17) Then we can rewrite (15) as 𝐾=𝑀𝐻^𝑇𝑅⁻¹. (18)

Based on the above derivation, the recursive equations of the Kalman filter can be rewritten as 𝑥=𝑥+𝐾(𝐳−𝐻𝑥),𝐾=𝑀𝐻^𝑇𝑅⁻¹,𝑀=𝑃⁻¹+𝐻^𝑇𝑅⁻¹𝐻⁻¹,𝑃⁺=𝐴𝑀𝐴^𝑇+𝐵𝑄𝐵^𝑇,𝑥⁺=𝐴𝑥. (19) From (19), the estimate 𝑥 can be expressed as 𝑥=𝑥+𝐾(𝐳−𝐻𝑥)=𝑥+𝑀𝐻^𝑇𝑅⁻¹(𝐳−𝐻𝑥)=𝑥+𝑀𝐻^𝑇𝑅⁻¹𝐳−𝐻^𝑇𝑅⁻¹𝐻𝑥. (20)

Now we define the 𝑛-dimensional global observation variables as 𝑦=𝐻^𝑇𝑅⁻¹𝐳,𝑆=𝐻^𝑇𝑅⁻¹𝐻 (21) and the 𝑛-dimensional local observation variables at sensor 𝑖 as 𝑦_𝑖=𝐻^𝑇_𝑖𝑅_𝑖⁻¹𝑧_𝑖,𝑆_𝑖=𝐻^𝑇_𝑖𝑅_𝑖⁻¹𝐻_𝑖. (22)

When the observations are distributed among the sensors, the KF can be implemented by collecting all the sensor observations at a central location, or with observation fusion by realizing that the global observation variables in (21), it can be written as 𝐲=𝐻^𝑇𝑅⁻¹𝐳=𝐻^𝑇₁𝑅₁⁻¹𝐳₁+⋯+𝐻^𝑇_𝑁𝑅_𝑁⁻¹𝐳_𝑁=^𝑁_𝑖=1𝑦_𝑖. (23) Similarly, 𝑆=𝐻^𝑇𝑅⁻¹𝐻=𝐻^𝑇₁𝑅₁⁻¹𝐻₁+⋯+𝐻^𝑇_𝑁𝑅_𝑁⁻¹𝐻_𝑁=^𝑁_𝑖=1𝑆_𝑖. (24)

Recall (20) where the estimate 𝑥 could be written as 𝑥=𝑥+𝑀(𝐲−𝑆𝑥). (25) Multiplication on the left by 𝑀⁻¹ yields a variation of (25) as 𝑀⁻¹𝑥=𝑀⁻¹𝑥+𝐲−𝑆𝑥=𝑃⁻¹+𝑆𝑥+𝐲−𝑆𝑥=𝑃⁻¹𝑥+𝐲. (26)

Let the inverse of 𝑀 and 𝑃 be the information matrices, 𝐼̂ and 𝐼. Let 𝑖̂ and 𝑖 be the information vectors. We have the following relations: 𝐼̂=𝑀⁻¹,𝐼=𝑃⁻¹,𝑖̂=𝑀⁻¹𝑥,𝑖=𝑃⁻¹𝑥. (27) Then (26) can be rewritten as 𝑖̂=𝑖+𝐲, (28) and the update of predicted estimate 𝑥 can be expressed as 𝑥⁺=𝐴𝑥=𝐴𝑀𝐼̂𝑥=𝐴𝑀𝐼𝑥̂=𝐴𝐼̂⁻¹𝑖,̂ (29) and we have 𝑖⁺=𝑃⁻¹𝑥⁺=𝐼⁺𝐴𝐼̂⁻¹𝑖.̂ (30)

Now we get the following simpler form of the filter in (19) and call it centralized information filter (CIF): 𝐼̂=𝐼+𝑆,𝑖̂=𝑖+𝐲,𝐼⁺=𝐴𝐼̂⁻¹𝐴^𝑇+𝐵𝑄𝐵^𝑇−1,𝑖⁺=𝐼⁺𝐴𝐼̂⁻¹𝑖,̂ (31) where (31) is the filter step and prediction step (or update step) of the CIF, respectively.

3. Consensus Strategy

Consensus strategy defines a set of rules for a team of agents to agree on specific consensus states. With these rules each agent exchanges information with its neighboring agents and finally reaches an agreement (or consensus) concerning the consensus state over time [17, 18]. Furthermore, average consensus occurs when the final consensus state is the average of the initial values.

Consider a team of 𝑛 agents to agree on specific consensus states, and at any discrete-time instant 𝜏, the communication topology between 𝑛 agents can be described by the graph 𝐺[𝜏]=(𝑉,𝐸[𝜏]), the graph 𝐺 is undirected, 𝑉={1,2,…,𝑛} is the vertex set, and 𝐸[𝜏]⊆𝑉×𝑉 is the edge set. In the consensus algorithm, each agent in the network maintains a local copy of the consensus state 𝜁_𝑖∈𝑅^𝑛 and updates this value using its neighbors’ consensus states according to the rule: 𝜁_𝑖[𝜏+1]=𝜁_𝑖[𝜏]+^𝑛_𝑗=1𝛽_𝑖𝑗[𝜏]𝜁_𝑗[𝜏]−𝜁_𝑖[𝜏], (32) where 𝜏 indicates the consensus filter iteration step. To choose the weights 𝛽_𝑖𝑗[𝜏], we can use the Maximum-degree weights or the Metropolis weights [19]. Here we use the latter which preserves the averaging in consensus filters and can be computed by 𝛽_𝑖𝑗[𝜏]=1+max𝑑_𝑖[𝜏],𝑑_𝑗[𝜏]⁻¹if𝑖,𝑗∈𝐸[𝜏],1−_{(𝑖,𝑙)∈𝐸[𝜏]}𝛽_𝑖𝑙[𝜏]if𝑖=𝑗,0otherwise, (33) where 𝑑_𝑖[𝜏] is the degree of agent 𝑖 in the graph 𝐺[𝜏]. Arrange the local consensus states into the vector 𝜻[𝜏]=[𝜁^𝑇₁[𝜏],…,𝜁^𝑇_𝑛[𝜏]]^𝑇, and define the matrix (𝐵[𝜏])_𝑖𝑗=𝛽_𝑖𝑗[𝜏] for 𝑖≠𝑗; otherwise (𝐵[𝜏])_𝑖𝑖=1−∑_{(𝑖,𝑙)∈𝐸[𝜏]}𝛽_𝑖𝑙[𝜏], and we can rewrite the update in (32) as 𝜻[𝜏+1]=(𝐵[𝜏]⊗𝐼)𝜻[𝜏], (34) where 𝐼 is the appropriate size identity matrix and ⊗ denotes the matrix Kronecker product.

The 𝑖𝑗th element of 𝐵[𝜏] in (34) satisfies the following four conditions: (1) (𝐵[𝜏])_𝑖𝑗≥0, (2) ∑_𝑖(𝐵[𝜏])_𝑖𝑗=1, (3) ∑_𝑗(𝐵[𝜏])_𝑖𝑗=1, (4) and each nonzero entry is both uniformly upper and lower bounded. Based on these conditions, we have the following results [20] for average consensus.

Lemma 2. Under switching interaction topologies, if there exists a finite 𝑇≥0 that for every interval [𝜏,𝜏+𝑇] the union of the interaction graph across interval is strongly connected, then consensus protocol (34) achieves average consensus asymptotically; that is, 𝜁_𝑖[𝜏]→(1/𝑛)∑^𝑛_𝑖=1𝜁_𝑖[0] as 𝜏→∞.

Remark 3. In order to calculate the metropolis weights in (33), we assume undirected communication throughout this paper. Therefore, if the graph 𝐺[𝜏] is connected, the matrix 𝐵[𝜏] is a doubly stochastic matrix, and the four conditions on 𝐵[𝜏] are satisfied. This implies that average consensus is achieved asymptotically as long as every graph is connected [21].

4. Distributed Kalman Filter: Consensus on Estimate

In this section, we discuss an alternative approach to distribute the Kalman filtering that relies on communicating state estimates between neighboring nodes and refer to it as Kalman consensus filter (KCF). Before presenting the KCF algorithm, we first need to discuss a more primitive DKF algorithm called local Kalman filter (LKF) which forms the basis of the KCF.

4.1. Local Kalman Filter

In local Kalman filtering, let 𝑁_𝑖={𝑗∶(𝑖,𝑗)∈𝐸} be the set of neighbors of node 𝑖 on graph 𝐺. Each node 𝑖 of the sensor network communicates its measurement 𝑧_𝑖, covariance information 𝑅_𝑖, and observation matrix 𝐻_𝑖 with its neighbors 𝑁_𝑖. For node 𝑖, we assume that the information flow from nonneighboring nodes to node 𝑖 is prohibited if there is no nodes except for its neighbors 𝑁_𝑖 exist. Therefore, node 𝑖 can use a central Kalman filter that only utilizes the observation vectors and observation matrices of the nodes in 𝐽_𝑖=𝑁_𝑖∪{𝑖} [11]. This leads to the following primitive DKF algorithm with no consensus on state estimation.

LKF Iterations. Assume that node 𝑖 only receives information from its neighbors. Then, we have the iterations of node 𝑖 in local Kalman filtering as 𝑦^𝑖=_{𝑗∈𝐽𝑖}𝐻^𝑇_𝑗𝑅_𝑗⁻¹𝑧_𝑗=_{𝑗∈𝐽𝑖}𝑦_𝑗,𝑆^𝑖=_{𝑗∈𝐽𝑖}𝐻^𝑇_𝑗𝑅_𝑗⁻¹𝐻_𝑗=_{𝑗∈𝐽𝑖}𝑆_𝑗,𝑥_𝑖=𝑥_𝑖+𝑀_𝑖𝑦^𝑖−𝑆^𝑖𝑥,𝑀_𝑖=𝑃_𝑖⁻¹+𝑆^𝑖−1,𝑃_𝑖⁺=𝐴𝑀_𝑖𝐴^𝑇+𝐵𝑄𝐵^𝑇,𝑥_𝑖⁺=𝐴𝑥_𝑖, (35) where 𝑦^𝑖 and 𝑆^𝑖 are local aggregate information vector and matrix, respectively and node 𝑖 locally computes both 𝑦^𝑖 and 𝑆^𝑖.

4.2. Kalman Consensus Filter

We now present the Kalman consensus filter (KCF). The KCF uses consensus strategy (32) on the state estimate in a distributed Kalman filter, where each node maintains a local Kalman filter. Corresponding to (32), let 𝜁_𝑖[𝜏]=𝑥_𝑖 be the prior estimate at time 𝜏 and 𝜁_𝑖[𝜏+1]=𝑥_𝑖^𝑐 be fused prior estimate; each node fuses the prior estimates from its neighbors according to the rule: 𝑥_𝑖^𝑐=𝑥_𝑖+_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]𝑥_𝑗−𝑥_𝑖. (36) Using the fused prior estimate 𝑥_𝑖^𝑐, the filter estimate at node 𝑖 could be implemented by 𝑥_𝑖=𝑥_𝑖^𝑐+𝑀_𝑖𝑦^𝑖−𝑆^𝑖𝑥_𝑖^𝑐=𝑥_𝑖+𝑀_𝑖𝑦^𝑖−𝑆^𝑖𝑥_𝑖+𝐼−𝑀_𝑖𝑆^𝑖_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]𝑥_𝑗−𝑥_𝑖. (37)

The local KCF is summarized in Algorithm 1, where 𝜏 is the time index for the consensus strategy and 𝑇_𝑝∈𝑍⁺ is the time interval between prediction updates. One-time step 𝑘−1→𝑘 is equivalent to 𝑇_𝑝 time steps of the consensus time index 𝜏→𝜏+1; that is, for each node, the information exchanges between neighboring nodes occurred faster than the prediction update step. The three steps in KCF prediction, local filter estimate, and consensus update are not necessarily sequential.

Initialization (for node 𝑖):         𝑥𝑖=𝑥(0)  𝑃𝑖=𝑃0         𝜏=1     𝜏𝑝=𝜏+𝑇𝑝 Loop  {Local iteration on node 𝑖} (1) Consensus update         𝑥𝑖𝑐=𝑥𝑖+∑𝑗∈𝑁𝑖𝛽𝑖𝑗[𝜏](𝑥𝑗−𝑥𝑖)         𝑦𝑖=∑𝑗∈𝐽𝑖𝐻𝑗𝑇𝑅𝑗−1𝑧𝑗 𝑆𝑖=∑𝑗∈𝐽𝑖𝐻𝑗𝑇𝑅𝑗−1𝐻𝑗 𝜏←𝜏+1 (2) If new observations are taken then the Kalman consensus state estimate are computed    𝑥𝑖=𝑥𝑖𝑐+𝑀𝑖𝑦𝑖−𝑆𝑖𝑥𝑖𝑐=𝑥𝑖+𝑀𝑖𝑦𝑖−𝑆𝑖𝑥𝑖+𝐼−𝑀𝑖𝑆𝑖∑𝑗∈𝑁𝑖𝛽𝑖𝑗[𝜏](𝑥𝑗−𝑥𝑖)         𝑀𝑖=(𝑃𝑖−1+𝑆𝑖)−1 (3) If time for a predication step (i.e., 𝜏=𝜏𝑝) then prediction step         𝑃𝑖+=𝐴𝑀𝑖𝐴𝑇+𝐵𝑄𝐵𝑇         𝑥𝑖+=𝐴𝑥𝑖         𝜏𝑝=𝜏+𝑇𝑝 End loop

Algorithm 1 

Kalman consensus filter.

The last term in (37) is the correction of filter estimate 𝑥_𝑖 compared to the standard Kalman estimator. Intuitively, adding the consensus term in (37) will force local estimators to reach a consensus regarding state estimates. The structure of node 𝑖 in the KCF algorithm is shown in Figure 1.

Figure 1 

The algorithm structure of node 𝑖 in the KCF.

In [11], the author proposed the following Kalman consensus estimator which is in the form of 𝑥_𝑖=𝑥_𝑖+𝑀_𝑖𝑦^𝑖−𝑆^𝑖𝑥_𝑖+𝐶_{𝑖𝑗∈𝑁𝑖}𝑥_𝑗−𝑥_𝑖, (38) where 𝐶_𝑖 is named consensus gain of node 𝑖. The choice of the consensus gain 𝐶_𝑖 is free. A poor choice of 𝐶_𝑖 leads to either the lack of consensus on estimates (e.g., setting 𝐶_𝑖=0, for all 𝑖) or the lack of stability of the error dynamics of the filter. One possible choice is to let 𝐶_𝑖=𝛾𝑃_𝑖𝑃=𝜀^𝑖1+𝑃_𝑖𝐹, (39) where 𝜀>0 is a relative small constant and ‖⋅‖_𝐹 denotes the Frobenius norm of a matrix. The derivation of the optimal Kalman consensus filter can be found in [10], where we can also find that the computational complexity of updating the error covariance 𝑃_𝑖⁺=𝐴𝑀_𝑖𝐴^𝑇+𝐵𝑄𝐵^𝑇 of the optimal Kalman consensus filter is not scalable in 𝑛. To obtain a suboptimal approximation of the KCF which is distributed and scalable in 𝑛, we make an assumption that the consensus gains 𝐶_𝑖=𝑂(𝜀) are of the order of 𝜀. Then we get the stable suboptimal KCF summarized in Algorithm 2.

Initialization:         𝑥𝑖=𝑥(0), 𝑃𝑖=𝑃0, and message 𝑚𝑗={𝑦𝑗,𝑆𝑗,𝑥𝑗} While new data exists do (1) Compute local observation vector and matrix of node 𝑖:         𝑦𝑖=𝐻𝑖𝑇𝑅𝑖−1𝑧𝑖  𝑆𝑖=𝐻𝑖𝑇𝑅𝑖−1𝐻𝑖 (2) Broadcast message 𝑚𝑖={𝑦𝑖,𝑆𝑖,𝑥𝑖} to neighbors. (3) Receive messages from all neighbors. (4) Compute the local aggregate information vector and matrix:         𝑦𝑖=∑𝑗∈𝐽𝑖𝑦𝑗  𝑆𝑖==∑𝑗∈𝐽𝑖𝑆𝑗 (5) Compute the Kalman consensus state estimate         𝑥𝑖=𝑥𝑖+𝑀𝑖(𝑦𝑖−𝑆𝑖𝑥𝑖)+𝛾𝑃𝑖∑𝑗∈𝑁𝑖(𝑥𝑗−𝑥𝑖)         𝑀𝑖=(𝑃𝑖−1+𝑆𝑖)−1         𝜀𝛾=1+𝑃𝑖,  ‖𝑋‖=tr(𝑋𝑇𝑋)1/2 (6) Update the state of the Kalman consensus filter:         𝑃𝑖+=𝐴𝑀𝑖𝐴𝑇+𝐵𝑄𝐵𝑇         𝑥𝑖+=𝐴𝑥𝑖 End While

Algorithm 2 

Suboptimal Kalman consensus filter: DKF Algorithm with an estimator that has a rigorously derived consensus term (message passing during one time cycle for node).

5. Distributed Filter: Consensus on Information Matrix

The KCF discussed previously applies consensus strategy on the prior estimate to the Kalman filter and improves the state estimate of KF. However, the error covariance matrix 𝑀_𝑖 is not improved because each node in KCF only fuses the prior estimates from its neighbors but neglects the helpful information about the error covariance matrix. In the next section, we adopt an information matrix weighted consensus strategy to improve the consensus based distributed Kalman filter algorithm in the estimation fusion of sensor networks. We refer to this method as information consensus filter (ICF) [16]. Before presenting the ICF algorithm, we need to discuss a more primitive DKF algorithm called local information filter (LIF) which forms the basis of the ICF.

5.1. Local Information Filter

To distribute the estimation of the global state vector, 𝑥 in CIF, we implement local information filter (LIF) at each sensor 𝑖, which is based on the sensor model (11) and can be derived from local Kalman filter (LKF) in (35) after we use its information form. Each LIF computes local objects (matrices and vectors) which are then fused (if required) by exchanging information among the neighbors. In LIF, there is no centralized knowledge of the estimation of the global state that exists in CIF; however, it can be obtain by fusing the local state vector.

Let the inverses of 𝑀_𝑖 and 𝑃_𝑖 be the local information matrices, 𝐼_𝑖̂ and 𝐼_𝑖. Let 𝑖_𝑖̂ and 𝑖_𝑖 be the local information vector. We have the following relations: 𝐼_𝑖̂=𝑀_𝑖⁻¹,𝐼_𝑖=𝑃_𝑖⁻¹,𝑖_𝑖̂=𝑀_𝑖⁻¹𝑥_𝑖,𝑖_𝑖=𝑃_𝑖⁻¹𝑥_𝑖. (40)

Then we get the following simpler form of the filter in (35).

LIF Iterations. The local information filtering iterations for node 𝑖 are in the form 𝐼_𝑖̂=𝐼_𝑖+𝑆^𝑖,𝑖_𝑖̂=𝑖_𝑖+𝑦^𝑖,𝐼_𝑖⁺=𝐴𝐼̂_𝑖⁻¹𝐴^𝑇+𝐵𝑄𝐵^𝑇−1,𝑖_𝑖⁺=𝐼_𝑖⁺𝐴𝐼̂_𝑖⁻¹𝑖_𝑖̂. (41)

5.2. Information Consensus Filter

We now present the information consensus filter (ICF). The ICF uses consensus strategy (32) on both the information state and the information matrix in a distributed Kalman filter, where each node maintains a local information filter. Recalling (32), let 𝑖_𝑖^𝑐 be the fused local information vector and 𝐼_𝑖^𝑐 the fused local information matrix; each node fuses the local information from its neighbors according to the rule: 𝑖_𝑖^𝑐=𝑖_𝑖+_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]𝑖_𝑗−𝑖_𝑖,𝐼_𝑖^𝑐=𝐼_𝑖+_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]𝐼_𝑗−𝐼_𝑖. (42)

Using the fused local information vector and matrix, 𝑖_𝑖^𝑐 and 𝐼_𝑖^𝑐, the local ICF is summarized in Algorithm 3.

Initialization (for node 𝑖):         𝑖𝑖=𝑖(0)   𝐼𝑖=𝐼(0)         𝜏=1    𝜏𝑝=𝜏+𝑇𝑝 Loop  {Local iteration on node 𝑖} (1) Consensus update         𝑖𝑖𝑐=𝑖𝑖+∑𝑗∈𝑁𝑖𝛽𝑖𝑗[𝜏](𝑖𝑗−𝑖𝑖)         𝐼𝑖𝑐=𝐼𝑖+∑𝑗∈𝑁𝑖𝛽𝑖𝑗[𝜏](𝐼𝑗−𝐼𝑖)         𝑦𝑖=∑𝑗∈𝐽𝑖𝐻𝑗𝑇𝑅𝑗−1𝑧𝑗 𝑆𝑖=∑𝑗∈𝐽𝑖𝐻𝑗𝑇𝑅𝑗−1𝐻𝑗 𝜏←𝜏+1 (2) If new observations are taken then the information consensus estimate are computed         𝑖𝑖̂=𝑖𝑖𝑐+𝑦𝑖=𝑖𝑖+𝑦𝑖+∑𝑗∈𝑁𝑖𝛽𝑖𝑗[𝜏](𝑖𝑗−𝑖𝑖)         𝐼𝑖̂=𝐼𝑖𝑐+𝑆𝑖=𝐼𝑖+𝑆𝑖+∑𝑗∈𝑁𝑖𝛽𝑖𝑗[𝜏](𝐼𝑗−𝐼𝑖) (3) If time for a predication step (i.e., 𝜏=𝜏𝑝) then prediction step         𝐼𝑖+=(𝐴𝐼̂𝑖−1𝐴𝑇+𝐵𝑄𝐵𝑇)−1         𝑖𝑖+=𝐼𝑖+𝐴𝐼̂𝑖−1𝑖𝑖̂         𝜏𝑝=𝜏+𝑇𝑝 End Loop

Algorithm 3 

Information consensus filter.

Now we make a comparison between ICF and KCF based on the state estimate 𝑥_𝑖 and the error covariance matrix 𝑀_𝑖. Let 𝛽_𝑖𝑖[𝜏]=1−∑_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]; we can rewrite (42) as 𝑖_𝑖^𝑐=𝛽_𝑖𝑖[𝜏]𝑖_𝑖+_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]𝑖_𝑗,𝐼_𝑖^𝑐=𝛽_𝑖𝑖[𝜏]𝐼_𝑖+_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]𝐼_𝑗. (43) Then we have 𝑥_𝑖^𝑐=𝐼_𝑖^𝑐−1𝑖_𝑖^𝑐=𝐼_𝑖^𝑐−1𝛽_𝑖𝑖[𝜏]𝑖_𝑖+_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]𝑖_𝑗=𝑊_𝑖𝑖𝑥_𝑖+_{𝑗∈𝑁𝑖}𝑊_𝑖𝑗𝑥_𝑗. (44) Here we use 𝑖_𝑖=𝐼_𝑖𝑥_𝑖, 𝑖∈𝐽_𝑖 and 𝑊_𝑖𝑙=𝛽_𝑖𝑙[𝜏](𝐼_𝑖^𝑐)⁻¹𝐼_𝑙, 𝑙∈𝐽_𝑖. Then we finally get 𝑥_𝑖=𝑥_𝑖^𝑐+𝑀_𝑖𝑦^𝑖−𝑆^𝑖𝑥_𝑖^𝑐=𝑊_𝑖𝑖𝑥_𝑖+𝑀_𝑖𝑦^𝑖−𝑆^𝑖𝑊_𝑖𝑖𝑥_𝑖+𝐼−𝑀_𝑖𝑆^𝑖_{𝑗∈𝑁𝑖}𝑊_𝑖𝑗𝑥_𝑗=𝐼_𝑖^𝑐−1𝐼_𝑖𝑥_𝑖+𝑀_𝑖𝑦^𝑖−𝑆^𝑖𝐼_𝑖^𝑐−1𝐼_𝑖𝑥_𝑖+𝐼−𝑀_𝑖𝑆^𝑖_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]𝐼_𝑖^𝑐−1𝐼_𝑗𝑥_𝑗−𝐼_𝑖^𝑐−1𝐼_𝑖𝑥_𝑖,𝑀_𝑖=𝐼̂_𝑖⁻¹=𝐼_𝑖+𝑆^𝑖+_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏]𝐼_𝑗−𝐼_𝑖⁻¹, (45) where the error covariance matrix 𝑀_𝑖 has been improved by a consensus term compared with 𝑀_𝑖=(𝐼_𝑖̂)⁻¹=[𝐼_𝑖+𝑆^𝑖]⁻¹ in KCF, and the state estimate 𝑥_𝑖 is also corrected by a factor ((𝐼_𝑖^𝑐)⁻¹𝐼_𝑖) compared with the 𝑥_𝑖 in KCF.

5.3. The Optimization of Consensus Weights in ICF

The weights 𝛽_𝑖𝑗[𝜏] are important parameters of ICF and we can use the metropolis weights or the maximum-degree weights to determine it. In fact, the more reasonable approach is to choose different weights according to the fused local information 𝐼_𝑖 and 𝑖_𝑖. Here, a nice scheme to optimize the consensus weights is proposed base on the following objective function: 𝐹_𝑖=𝛼¹_𝑖tr𝐼_𝑖^𝑐−1tr𝐼_𝑖⁻¹+𝛼²_𝑖𝑖_𝑖^𝑐−𝑖^𝑐_{𝑖,𝑎𝑣}²+∑_{𝑗∈𝑁𝑖}𝑖_𝑗−𝑖^𝑐_{𝑖,𝑎𝑣}²∑_{𝑗∈𝐽𝑖}𝑖_𝑗−𝑖_{𝑖,𝑎𝑣}², (46) where 𝑖^𝑐_{𝑖,𝑎𝑣}=11+𝑑_𝑖𝑖^𝑐_𝑖+_{𝑗∈𝑁𝑖}𝑖_𝑗,𝑖_{𝑖,𝑎𝑣}=11+𝑑_𝑖𝑖_𝑖+_{𝑗∈𝑁𝑖}𝑖_𝑗, (47)𝛼¹_𝑖, 𝛼²_𝑖 are the weight coefficients, and 𝛼¹_𝑖+𝛼²_𝑖=1(0<𝛼¹_𝑖<1,0<𝛼²_𝑖<1).

The first term in the objective function 𝐹_𝑖 is used to assess the prior estimate error covariance of node 𝑖 after fusing the local information of its neighbors, and the second term is used to assess the consensus of the fused local information vector in node 𝑖 and the prior estimates of its neighbors. Base on the objective function 𝐹_𝑖, the consensus weights optimization problem can be described as 𝜷^∗_𝑖=argmin_𝛽𝑖𝐹_𝑖s.t.𝛽_𝑖𝑗[𝜏]𝛽≥0,𝑖,𝑗∈𝐸_𝑖𝑗[𝜏]𝜷=0,𝑖,𝑗∉𝐸_𝑖1=1, (48) where 𝜷_𝑖=[𝛽_𝑖1,𝛽_𝑖2,…,𝛽_𝑖𝑛]. To solve the optimization problem (48),we only need the local information of node 𝑖 and that of its neighbors. We refer to this method as weights optimized information consensus filter (WO-ICF).

6. Numerical Simulations

Let the linear system under consideration be represented by a second-order discrete time-varying model: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑤, (49) where 𝐴=𝐼₂+𝛿𝐴₀+(𝛿²/2)𝐴²₀+(𝛿³/6)𝐴³₀ with 𝐴₀=2⁰¹₁₀, 𝛿=0.015 and 𝐵=𝛿𝐵₀ with 𝐵₀=𝐼₂, 𝑄=25𝐼₂. The initial conditions are 𝑥₀=[15,−10], 𝑃₀=20𝐼₂. A sensor network with 20 randomly located nodes is used in this experiment (see Figure 2). The local observation matrix for sensor 𝑖 is 𝐻_𝑖=⁰¹₁₀, and the local observation noise covariance is 𝑅_𝑖=100𝐼₂ for 𝑖≤10 otherwise 𝑅_𝑖=3000𝐼₂.

Figure 2 

A sensor network with 20 nodes and 51 links.

Define the averaged estimation error 𝐸(𝑘) and the averaged consistency estimation error 𝐷(𝑘) as the algorithm performance metrics, which can be computed as follows: 𝐸(𝑘)=1𝑛^𝑛_𝑖=1𝑥_𝑖(𝑘)−𝑥(𝑘)^𝑇𝑥_𝑖(𝑘)(𝑘),𝐷(𝑘)=−𝑥1𝑛^𝑛_𝑖=1𝑥_𝑖(𝑘)−𝑥_𝑎𝑣(𝑘)^𝑇𝑥_𝑖(𝑘)−𝑥_𝑎𝑣(𝑘), (50) where 𝑥_𝑎𝑣(𝑘)=(1/𝑛)∑^𝑛_𝑖=1𝑥_𝑖(𝑘) is the averaged estimation of state.

Figure 3 demonstrates the averaged estimation error using different algorithms. We can see that the ICF and the WO-ICF behave in a similar manner (with comparable performances), and the averaged estimation accuracy in ICF and WO-ICF is improved compared to KCF. After 50 iterations, their performances are very close to CKF; this is because the average consensus is achieved after constantly information exchanging, fusing, and filtering.

Figure 3 

The averaged estimation errors of different algorithms.

Figure 4 shows that our WO-ICF performs significantly better than both KCF and ICF, it was the fastest converged, and the consistency of estimates between different nodes in WO-ICF was improved by optimizing the consensus weights.

Figure 4 

The averaged consistency estimation errors of different algorithms.

Figure 5 demonstrates the comparisons of different algorithms on the traces of the averaged estimation covariance matrices tr((1/𝑛)∑^𝑛_𝑖=1𝑀_𝑖(𝑘)). A quick look at Figure 5 reveals that both ICF and our WO-ICF perform significantly better than KCF, of which the reason is that the information matrix weighted consensus strategy is adopted. Furthermore, by optimizing the consensus weights, the error covariance matrix 𝑀_𝑖 is improved significantly compared to ICF.

Figure 5 

The traces of the averaged estimation covariance matrices of different algorithms.

7. Conclusions

In this paper, a description about the existing distributed Filter with consensus strategies is presented, including Kalman consensus filter (KCF) and information consensus filter (ICF). In addition, an in-depth comparison between the KCF and the ICF is made. Based on the ICF, the weights optimized information consensus filter (WO-ICF) is proposed to optimize the consensus weights. Simulation shows that both ICF and WO-ICF perform better than KCF; they improve not only the state estimate but also the error covariance matrix, and the proposed WO-ICF achieves better consistency estimation performance than ICF. Compared with the existing consensus filter, our WO-ICF achieves the best performance and is closest to the optimal centralized performance.

Acknowledgments

This work is supported in part by NSFC (Natural Science Foundation of China) Projects 61202400, Natural Science Foundation of Zhejiang Q12F020078, Shaoxing Project of Science and Technology 2011A22013, and Wenzhou Project of Science and Technology H20090054, S20100029, and H20100095.