978–1–5090–3511–316$31.00 c 2016 IEEE 2016 13th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), Mexico, City. Mexico. September 26-30, 2016. Improving Robustness of Distributed Filtering for Sensor Networks Using FIR Filtering Miguel Vazquez-Olguin, Yuriy S. Shmaliy, Oscar Ibarra-Manzano Department of Electronics Engineering Universidad de Guanajuato Salamanca, 36885, Mexico Email: {miguel.vazquez}{shmaliy}{ibarrao}@ugto.mx Abstract—Robustness is required from an estimator to provide better performance if a wireless sensor network (WSN) operates under harsh conditions with incomplete information about noise. This paper shows that robustness of the WSN can be improved by using the distributed unbiased finite impulse response (UFIR) filter rather than the traditional distributed Kalman filter (KF), both based on the average consensus. Unlike the KF, the UFIR filter completely ignores the noise statistics and initial values which are typically not well known. As an example, we consider a vehicle travelling along a circular trajectory under unpredictable impacts and errors in the noise statistics. A case of impulsive noise generated by manufacturing process is also considered. I. I NTRODUCTION Wireless sensor networks (WSNs) have found applications in last decades to serve for military needs, industrial processes, home security, etc. Distributed filtering [1], [2] was introduced to reduce computational complexity of centralized filtering and increase reliability of WSNs [3]–[5]. Here, each node interacts with a few neighbors, estimates a vector quantity  Q(t), and then cooperatively passes data through other nodes to a central station [6]. For distributed filtering to operate in real time, supporting consensus filtering was introduced using data fusion [7]. The consensus-based approach has gained currency in different forms and became an important part of the WSN design [5], [8], [9]. Developments of WSNs can be found in [7], [11] and references therein. Most of the practical algorithms for WSNs were designed using Kalman filtering [2], [7], [12]. However, the optimality of Kalman filtering does not always go along with robust- ness, scalability, and fault tolerance required by real-world applications. Robustness of the Kalman filter (KF) becomes particularly poor under the imprecisely defined noise statistics [15], [16]. Therefore, efforts were made to improve the KF performance [2]. But robustification of KF not always lead to essential progress [17] that has a lot to do with its infinite impulse response (IIR) [18]. Better robustness have filters with finite impulse response (FIR) [19], [20]. In recent years, several solutions on data processing over finite horizons were proposed for WSNs [21]–[25]. FIR filter- ing offers many other fast solutions which may efficiently be used in WSNs. A receding horizon Kalman FIR filter designed in [26] operates similarly to KF on finite horizons. For deter- ministic time-invariant control systems, a fast recursion-based algorithm was developed in [27]. An iterative p-shift unbiased FIR (UFIR) algorithm proposed in [16], [28] completely ignores the noise statistics and initial values while reducing the output noise variance as a reciprocal of the horizon length. This algorithm provides filtering with p =0, smoothing with p< 0, and prediction with p> 0. Note that prediction (p> 0) can efficiently be used to estimate data with misses measurements [29]. A fast Kalman-like algorithm was designed in [30] for optimal FIR (OFIR) filtering called optimal UFIR (OUFIR) filtering and in [31] for bias-constrained OFIR filtering. An important feature of a simple UFIR algorithm is that its estimate becomes practically optimal when the optimal horizon occurs to be large. Besides, the performance of the UFIR filter can be improved by adapting the generalized noise power gain to operation conditions [32]. Hence, using methods of fast FIR filtering may open new horizons in design of robust WSNs. II. SYSTEM MODEL AND DISTRIBUTED KALMAN FILTER Given a space environment represented with a quantity Q(t) (temperature, pressure, location, etc.) which dynamics is described with a linear K-state equation regarding the state vector x k ∈ R K . The environment is covered with a wireless sensor network (WSN) consisting of n nodes. Each ith, i ∈ [1,n], node provides linear measurements of Q(t) as y (i) k = H (i) k x k + v (i) k ∈ R p , where H (i) k ∈ R p×K , the number p of the measured states is p  K, and v (i) k is the measurement noise. In discrete time index k, the state-space model is x k = A k x k-1 + B k w k , (1) y k = H k x k + v k , (2) where y k = [ y (1) k T ...y (n) k T ] T ∈ R np is the measure- ment vector, A k ∈ R K×K , H k =[ H (1) k T ...H (n) k T ] T ∈ R np×K , and B k has proper dimensions. The mutually inde- pendent and uncorrelated white Gaussian noise vectors w k and v k =[ v (1) k T ...v (n) k T ] T ∈ R np have zero mean and covariances Q k = E{w k w T k } and R k = E{v k v T k } = diag[ R (1) k T ...R (n) k T ] T ∈ R np . We assign ˆ x k|r to be an estimate of x k at time index k via measurements from past up to and including at time-index r. We also employ the following variables: ˆ x - k  ˆ x k|k-1 , the a priori (prior or predicted) state estimate given measurements up to and including at time k −1; P - k  P k|k-1 = E{(x k − ˆ x - k )(x k − ˆ x - k ) T }, the a priori (prior or predicted) estimate covariance given measurements up to and including at time k − 1; ˆ x k  ˆ x k|k , the a posteriori or posterior state estimate given measurements up to and including at k; P k  P k|k = E{(x k − ˆ x k )(x k − ˆ x k ) T },