Distributed Fusion Filter for Networked Multi-sensor Systems with Unknown Measurement Interferences and Packet Dropouts

This paper is concerned with the design of distributed fusion filter for networked systems with unknown measurement interferences and packet dropouts. A Bernoulli distributed random variable is used to depict the phenomenon of packet dropouts. Without any prior information about the interference, a recursive Kalman-type state filter independent of the unknown interferences is designed for each sensor subsystem by applying the linear unbiased minimum variance estimation criterion. Based on the state filters of individual subsystems, the estimation error cross-covariance matrices between any two subsystems are derived. Then, the distributed fusion filter is designed by using the matrix-weighted fusion estimation algorithm in the linear minimum variance sense. Simulation results show the effectiveness of the proposed algorithms. Introduction With the rapid development of electronics, communication and computer technologies, networked control systems have been gradually applied to all aspects of productions and lives due to the convenient connection mode and high-speed transmission. The state estimation problems for networked control systems have attracted the interests of many scholars [1-4]. However, the networks bring convenience meanwhile introduce a lot of uncertainties. Due to the limitation of network bandwidths, there are the transmission delays and losses of data. There are many reports about the state estimation problems for systems with time delays and packet dropouts [5-8]. A linear minimum variance filter dependent on the probability of missing measurements is presented in [5]. The optimal linear estimation problem about multiple packet dropouts is studied in [6]. The missing measurements and time-varying delays existing in uncertain stochastic networks are considered in [7]. Ref. [8] proposes a state filter for networked systems with multiple random delays and packet losses. In addition, the external interferences and syntheses of device failures have effect on the sensor outputs, which results in the uncertainties of measurement outputs. In recent years, the state estimation for systems with unknown inputs has also become a hot research topic [9-11], especially in the application of fault diagnosis. A fault diagnosis method based on the optimal unknown input observer is presented in [12]. Ref. [13] proposes a fault detection filter for linear discrete time-varying systems with multiple packet dropouts. Ref. [14] gives a fusion predictor for multi-sensor systems with missing measurements and unknown measurement interferences, where the computation of the state second-order moment is required. Based on the results of references above, for the multi-sensor networked systems with unknown measurement interferences and packet dropouts, we present a recursive Kalman-type local state filter independent of unknown interferences for each sensor subsystem based on the linear unbiased minimum variance estimation criterion [15]. Further, the estimation error cross-covariance matrices between any two subsystems are derived. At last, we give the distributed fusion filter weighted by matrices in the linear minimum variance sense. Differently from [14], the computation of the state second-order moment is avoided. 6th International Conference on Machinery, Materials, Environment, Biotechnology and Computer (MMEBC 2016) © 2016. The authors Published by Atlantis Press 2113 Problem formulation Consider a multi-sensor linear discrete stochastic system with unknown measurement interferences and packet dropouts: ( 1) ( ) ( ) x t x t w t + = Φ +Γ (1) ( ) ( ) ( ) ( ), 1, 2, , i i i i i y t H x t v t D t i L θ = + + =  (2) ( ) ( ) ( ), 1, 2, , i i i z t u t y t i L = =  (3) In the type: ( ) n x t R ∈ is the state, ( ) i m i y t R ∈ is the measured output of the ith sensor, which will be transmitted to the filter through networks, ( ) i m i z t R ∈ is the measurement received by the filter, and ( ) i p i t R θ ∈ signifies the unknown sensor measurement interference. L is the number of sensors. ( ) r w t R ∈ and ( ) i m i v t R ∈ are the process and measurement noises. { ( )} i u t is a known white Bernoulli distributed stochastic sequence taking values 1 and 0 with the probability { } Prob ( ) 1 i i u t α = = , { } Prob ( ) 0 1 i i u t α = = − , 0 1 i α ≤ ≤ , and independent of other stochastic variables. If ( ) 1 i u t = , the measurement of the th i sensor is received during transmissions, otherwise, the filter receives nothing, which means packet dropout. Moreover, Φ , Γ , i H and i D are constant matrices with suitable dimensions. We will present our main results based on the following assumptions. Assumption 1: ( ) w t and ( ) i v t are uncorrelated white noises with mean 0 and the variances 0 w Q ≥ and 0 i v Q > . Assumption 2: The initial state (0) x is uncorrelated with ( ) w t , ( ) i v t and ( ) i u t , and satisfies T 0 0 0 0 E{ (0)} ,E{[ (0) ][ (0) ] } x x x P μ μ μ = − − = (4) where symbol E denotes the mathematical expectation, T is the transpose operator. Assumption 3: rank[ ] i i i D p m = < , rank[ ] ∗ denotes the rank of matrix ∗ . The objectives of this paper are to design the recursive Kalman-type local state filter ˆ ( ) i x t by applying the linear unbiased minimum variance estimation criterion based on the received measurements ( ( ), ( 1), , (1)) i i i z t z t z −  of the ith sensor and the distributed fusion filter ˆ ( ) o x t . Local state filter design In this section, since there is not any prior information about the interferences, a recursive Kalman-type local state filter independent of the unknown interference will be designed. Theorem 1: For system (1)-(3) under Assumptions 1-3, the recursive local state filter is calculated as follows: ˆ ˆ ˆ ( 1) ( ) ( 1) ( 1)[ ( 1) ( )] i i i i i i i x t x t u t K t z t H x t + = Φ + + + + − Φ (5) where T T T 1 ( 1) [ ( ) ( 1) ] ( 1) i i i i i i K t P t H t D C t − + = −Λ + + (6) T T 1 T 1 1 ( 1) ( ) ( 1) [ ( 1) ] i i i i i i i i t P t H C t D D C t D − − − Λ + = + + (7) T ( 1) ( ) i i i i i v C t H P t H Q + = + (8) The filtering error variance matrix is given as T ( 1) ( ) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( ) i i i i i i i i i i P t P t u t K t C t K t u t K t H P t + = + + + + + − + + T T T ( 1) ( ) ( 1) i i i i u t P t H K t − + + (9) where T T ( ) ( ) i i w P t P t Q = Φ Φ +Γ Γ (10) with the initial value 0 ˆ (0) i x μ = and 0 (0) i P P = .