sensors Article
Multi-Sensor Consensus Estimation of State, Sensor Biases and Unknown Input Jie Zhou, Yan Liang ∗ , Feng Yang, Linfeng Xu and Quan Pan Key Laboratory of Information Fusion Technology, Ministry of Education, School of Automation, Northwestern Polytechnical University, Xi’an 710129, China;
[email protected] (J.Z.);
[email protected] (F.Y.);
[email protected] (L.X.);
[email protected] (Q.P.) * Correspondence:
[email protected]; Tel.: +86-29-8843-1305 Academic Editors: Xue-Bo Jin, Feng-Bao Yang, Shuli Sun and Hong Wei Received: 12 March 2016; Accepted: 23 August 2016; Published: 1 September 2016
Abstract: This paper addresses the problem of the joint estimation of system state and generalized sensor bias (GSB) under a common unknown input (UI) in the case of bias evolution in a heterogeneous sensor network. First, the equivalent UI-free GSB dynamic model is derived and the local optimal estimates of system state and sensor bias are obtained in each sensor node; Second, based on the state and bias estimates obtained by each node from its neighbors, the UI is estimated via the least-squares method, and then the state estimates are fused via consensus processing; Finally, the multi-sensor bias estimates are further refined based on the consensus estimate of the UI. A numerical example of distributed multi-sensor target tracking is presented to illustrate the proposed filter. Keywords: bias estimation; state estimation; sensor registration; network consensus
1. Introduction The well-known Kalman filter is optimal given a linear Gaussian model, but its performance will deteriorate in the presence of unknown biases, such as the unknown and time-varying delays in chemical processes [1–4], faults or failures in fault-tolerant diagnosis and control systems [5,6], registration errors in multi-sensor fusion [7–10], or inertial drift in navigation [11–14]. In general, such a bias is represented as an unknown input (UI) to the nominal model. To the best of our knowledge, approaches to UI modeling and the design of corresponding filters typically fall into one of the following categories. The first type of UI is zero-mean random noise with unknown covariance. In the case of the stationary noise processes in linear dynamic systems, the covariance can be identified via Bayesian or maximum likelihood estimation. A corresponding filter has been applied for orbit determination for near-earth satellites [15,16]. Recently, this type of filter design has been extended to time-varying covariance [17] and jump Markov stochastic systems [18]. Furthermore, an M-robust estimator [19] has been derived for the simultaneous adaptive estimation of unknown states and observation noise statistics. An auto-covariance least-squares method [20] has been presented to achieve lower-variance covariance estimates along with the necessary and sufficient conditions for the uniqueness of the estimated covariances. By treating unknown covariances as missing data, the adaptive estimation problem has been transformed into a problem of the joint optimization of state estimation and parameter identification, allowing it to be solved in an expectation-maximization (EM) iterative processing framework [21]. The second type of UI is unknown but deterministic. Using least-squares estimation and moving-window hypothesis testing, the corresponding filters can cope with cases in which the UI is piecewise constant [22] or a sum of basis functions with piecewise-constant weights [23]. The problem Sensors 2016, 16, 1407; doi:10.3390/s16091407
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of state estimation with UIs in both dynamic and measurement models has been addressed using a joint EM optimization scheme for state estimation, parameter identification and iteration termination decision-making [24], and this approach has been extended to a distributed EM algorithm for sensor networks [25], and further extended to an adaptive divided difference filter for nonlinear system with multiplicative parameters [26]. The third type of UI is completely arbitrary, without the availability of any prior knowledge about its evolution. An asymptotically stable and UI-decoupled observer has been derived [27] based on the condition that the rank of its distribution matrix must be less than that of the UI. For the case in which the UI appears only in the process equation, a combination of least-squares estimation and the Kalman filter has been proposed for joint state and UI estimation, and the necessary and sufficient condition for the existence of such a joint estimator has also been presented [28]. In addition, this result has been extended to the case in which the UI appears in both the process and measurement models [29]. The fourth type of UI is norm-bounded, and robust offline filters have been designed to minimize the gain of the transfer function from the UI to the estimation error [30]. As an alternative approach, a method of linear minimax regret estimation has been developed to minimize the worst-case regret over all bounded data [31]. The fifth type of UI is characterized by a randomly switching parameter obeying a known Markov chain. The corresponding solutions fall within the scope of multiple model estimators [32–37], such as the interacting multiple model (IMM), which is well known in maneuvering-target tracking. Motivated by the concept of establishing a general framework for joint state estimation and data association in clutters for the tracking of a maneuvering target, the linear minimum-mean-square-error (MMSE) estimator has been derived for discrete-time Markovian jump linear systems with stochastic coefficient matrices [38]. In general, the filters discussed above are UI-specific because of the significant differences between both the different types of UIs and the solutions designed to address them. However, in many practical applications, much more complicated UIs may be encountered. Recently, the minimum upper bound filter (MUBF) was proposed for a linear stochastic system corrupted by a generalized UI representing an arbitrary linear combination of dynamic UIs, random UIs, and deterministic UIs [39]. The result was further extended to a discrete-time non-linear stochastic system with a UI in its measurements, and an iterative optimization method for joint estimation and parameter identification was derived [40]. In multi-sensor bias estimation [41], the concept of a generalized sensor bias (GSB) has been proposed, which is represented by a dynamic model driven by a structured UI. By deriving an equivalent state-free measurement model and the corresponding UI-free dynamic model for the GSB, the LMMSE can be obtained via the orthogonal principle. However, the application of the resultant filter is limited to the case in which all sensors have an identical measurement matrix, and only the GSB can be estimated. This problem motivates us to present and solve the problem of the joint estimation of the system state and the GSB under a UI in the case of GSB evolution. This paper presents the joint estimation of the system state and GSB under a common UI in the case of bias evolution in a heterogeneous sensor network. To the best of the author’s knowledge, this is the first attempt to achieve this kind of the joint estimation. In detail, a two-stages cooperative strategy for pursuing consistent estimates in the distributed network is proposed. First, a joint UI-free evolution model is derived, along with its necessary and sufficient conditions for existence. In the derived model, the process noise and measurement noise are correlated. And the optimal local MMSE estimator with correlated noise is obtained. Second, based on the local estimates, the global estimation of the target state and UI are further achieved by network consensus. Through utilizing the global estimation of UI, the GSB is refined. The optimal estimates of the target state, GSB and UI are finally obtained. Throughout the paper, the superscripts “−1” , “T” and “+” represent the inverse, transpose, and Moore-Penrose operations, respectively; “I” and “0” represent the identity matrix and the zero matrix, respectively, of the proper dimensions; “0m×n ” represents the zero matrix with dimensions of m × n;
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diag{·} indicates a diagonal matrix; col{·} denotes column augmentation; E{·} denotes the operator of mathematical expectation; the superscript i indicates a matrix or variable related to the ith sensor; rank{·} denotes the rank of the specified matrix; the subscripts “i”and “j” indicate the ith and jth subblocks of a matrix, respectively; the superscripts “∧” and “˜” indicate the estimate and residual, respectively, of a random vector; the subscripts “k|k + 1”represents the estimate of a vector at the time N k using measurements up to time k + 1 and denotes the Kronecker product. 2. Problem Formulation Sensor bias (SB) is widespread in multi-sensor systems and originates from two types of sources: SBs of the first type are local SBs (LSBs), which are independent among sensors, such as calibration error, navigation bias and sensor faults/drift, whereas sensor biases of the second type are global SB (GLSB, also called UI in this work),which are common to all sensors, such as electronic countermeasures (ECMs) in target tracking. A GSB model has been proposed based on dynamic evolution driven by both independent LSBs and a common UI [41]. However, the corresponding filter has two shortcomings: first, all sensors must have an identical measurement matrix, and second, the system state and the UI cannot be estimated simultaneously because the SB estimation is obtained by deriving the equivalent GSB model neglecting system state and UI. Here, we present the two assumptions adopted in our work. Assumption 1. All measurements made by sensors in the network are continuously interrupted by the GSB, which includes LSBs and GLSB. Assumption 2. The network topology is fixed, and the target can be monitored by all sensors throughout the entire process. There is no fusion centre or lead sensor in this sensor network. In other words, all sensors are equal in status, and they reach a network consensus by exchanging information with their neighbours. Consider a scenario of collaborative target tracking within a sensor network, as shown in Figure 1. A target is continuously moving within a surveillance zone. Each sensor node obtains raw measurements of the target corrupted by the GSB. At each step, each sensor obtains local estimates of the target state and GSB, exchanges its local system-state estimate with its neighbors, and obtains fused estimates of the target state and UI via network consensus. In other words, this sensor fusion process can be regarded as the joint distributed estimation of the target state, GSB and UI without the limitation that all sensor measurement matrices must be identical.
Target Unknown Input
Neighbor Sensor j Neighbor Sensor 1
Noise 1
Noise i
e mat esti UI tate f s o t e e Target stat mat Targ e estimate i t s le Local estim Loca ate of UI Target state es timate Local e Sensor i stimate of UI
Noise j
Noise 2 Neighbor Sensor 2
Figure 1. Collaborative target tracking in sensor network.
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Consider a network of s sensor nodes. The network topology is represented by an undirected graph G = (V , E , A), where V = {1, 2, ..., s} denotes the set of nodes; E ⊂ V × V is the set of permissible communication links; and A = aij is the graph topology matrix, the elements of which are defined as ( aij =
1, 0,
(i, j) ∈ E or i = j /E (i, j) ∈
Let Ni = i ∈ V : aij > 0, i 6= j denote the set of neighbors of the ith sensor, and let Ji = Ni ∪ {i } denote the inclusive neighbor set of sensor i. Here, the presented network is assumed to be a connected graph. Otherwise, at least two separate sub-networks would exist, and hence, information could only be fused within each sub-network instead of throughout the entire network because information could not be exchanged between the two. Here, we formulate the following problem of distributed sensor fusion in the presence of time-varying sensors subject to a common UI: x k +1 yik+1 bki +1
= Φk xk + ζ k , = =
(1)
Hki +1 xk+1 + Nki +1 bki +1 Fki bki + Gki dk + wik
+ vik+1
(2) (3)
where xk ∈
