Fault diagnosis over wireless sensor networks using distributed Kalman and distributed Particle Filtering
The paper considers the problem of distributed fault detection and isolation for continuous time dynamical systems. Such a fault diagnosis procedure involves the transmission of measurements to local processing units over a wireless sensor network and the fusion of local state estimates with the use of distributed filtering algorithms. Most of the existing fault diagnosis methods are centralized, which means that all sensing data are collected and processed at one unit. This not only causes excessive communication and computational burdens, but also creates a single point of failure. To overcome these problems, distributed detection and data fusion methods are being developed. Distributed fault diagnosis can be performed by: (a) running local fault diagnosers (hypotheses tests) at distributed processing units, obtaining local diagnosis results and fusing the local diagnoses so as to reach the final decision on the existence of a fault ,, (b) fusing the individual state estimates provided by distributed filters and using the aggregate state vector in a fault diagnosis algorithm to generate residuals ,. In this paper, the problem of distributed fault diagnosis will be studied according to second approach.
In many applications it is possible to obtain the state vector of a dynamical system simultaneously from distributed sources. For example: (i) in the aerospace and maritime systems when transmitting measurements about the system’s condition to distributed processing units, (ii) in chemical processes when measuring the system’s state variables with multiple sensors and dispatching these measurements to local processing units. Based on these measurements, filtering algorithms running on different processing units produce estimates of the system’s state vector while to improve the estimation accuracy and the reliability of data processing, fusion of the distributed state estimates is performed by an aggregation filter. The overall state estimation procedure is known as distributed _ltering .
The paper proposes first the Extended Information Filter (EIF) and the Unscented Information Filter (UIF) as possible distributed filtering approaches which enable to obtain an estimation of the state vector of the monitored system, under the assumption of Gaussian noises. The Extended Information Filter is a generalization of the Information Filter in which the local filters do not exchange raw measurements but send to an aggregation filter their local information matrices (local inverse covariance matrices) and their associated local information state vectors (products of the local information matrices with the local state vectors) ,,. In the case of the Unscented Information Filter there is no linearization of the system’s observation equation. However the application of the Information Filter algorithm is possible through an implicit linearization which is performed by approximating the Jacobian matrix of the system’s output equation by the product of the inverse of the state vector’s covariance matrix (information matrix) with the cross-correlation covariance matrix between the system’s state vector and the system’s output ,. Again, the local information matrices and the local information state vectors are transferred to an aggregation filter which produces the global estimation of the system’s state vector. The EIF and UIF estimated state vector can in turn be used by a FDI algorithm for residuals generation. After residuals have been obtained a remaining problem in the FDI procedure is the definition of an effective fault threshold based on the Likelihood Ratio or the Generalized Likelihood Ratio ,.
Next, the Distributed Particle Filter (DPF) is proposed as a distributed filtering method which is well-suited for providing estimates of the monitored system’s state vector in the case of non-Gaussian measurements . Difficulties in implementing distributed particle filtering stem from the fact that particles from one particle set (which correspond to a local particle filter) do not have the same support (do not cover the same area and points on the samples space) as particles from another particle set (which are associated with another particle filter) ,,. This can be resolved by transforming the particles sets into Gaussian mixtures, and defining the global probability distribution on the common support set of the probability density functions associated with the local filters . The state vector which is estimated with the use of the DPF can be used again by the FDI algorithm for residuals generation. After residuals have been produced a remaining problem in particle filtering-based FDI algorithms is the definition of an effective fault threshold based on a suitable reformulation of the Generalized Likelihood Ratio ,.
The structure of the paper is as follows: in Section 2 the Extended Information Filter (Distributed Extended Kalman Filter) is studied. In Section 3, the Unscented Information Filter (Distributed Unscented Kalman Filter) is analyzed and its use for fusing distributed state estimates is explained. In Section 4 Distributed Particle Filtering is proposed for fusing the state estimates produced by the distributed (local) processing units. In Section 5 fault diagnosis based on the Generalized Likelihood Ratio (GLR) is analyzed in case that the Extended Information Filter or the Unscented Information Filter is used for residuals generation. In Section 6 FDI with the use of the GLR is formulated in case that the Distributed Particle Filter is employed for residuals generation. In Section 7 simulation experiments are carried out to evaluate the performance of the distributed Kalman Filters and of the distributed Particle Filter in estimating the state of the monitored system (e.g. UAV sensors and actuators) over a wireless sensor network and subsequently in performing fault diagnosis. Finally, in Section 8 concluding remarks are stated.
Keywords: Extended Information Filter, Unscented Information Filter, Distributed Particle Filter, distributed fault diagnosis, Generalized Likelihood Ratio.
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