Sequential covariance intersection‐based Kalman consensus filter with intermittent observations

IET Signal ProcessingVolume 14, Issue 9 p. 624-633 Research Article Free Access Sequential covariance intersection-based Kalman consensus filter with intermittent observations Ning Wang, School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorYinya Li, Corresponding Author liyinya@njust.edu.cn School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorJinliang Cong, School of Electric and Automatic Engineering, Changshu Institute of Technology, Changshu, 215500 People's Republic of ChinaSearch for more papers by this authorAndong Sheng, School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this author Ning Wang, School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorYinya Li, Corresponding Author liyinya@njust.edu.cn School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorJinliang Cong, School of Electric and Automatic Engineering, Changshu Institute of Technology, Changshu, 215500 People's Republic of ChinaSearch for more papers by this authorAndong Sheng, School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this author First published: 29 October 2020 https://doi.org/10.1049/iet-spr.2019.0547Citations: 1 AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract This paper investigates the distributed state estimation for a class of linear time-varying systems with intermittent observations in sensor networks. Unlike the existing studies in distributed state estimation, this work considers the scenario where the cross-covariances between different sensors are unavailable and the measurements for state estimation encounter intermittent observations and/or random losses. For this practical scenario, a new sequential covariance intersection-based Kalman consensus filer (SCIKCF) is then developed. We show that, with the proposed SCIKCF, each sensor can achieve consensus estimates regardless of the order of fusion. Furthermore, the stability of the SCIKCF as well as the boundedness of the estimation error and the corresponding error covariances are analysed. Finally, three examples are performed to verify the effectiveness of the proposed SCIKCF. 1 Introduction As one of the most important issues in the estimation field, the distributed state estimation has been a focus of research with the extensive application of sensor networks in recent decades [1 ]. It is widely applied in various military and civilian areas, such as factory automation, integrated navigation, target tracking and so on [2, 3 ]. The objective of distributed state estimation is to obtain an improved fusion estimation accuracy and reliable fusion results by combining the individual information from different sensor nodes in a systematic manner. The distributed estimation generally has a less communication burden and higher survivability compared with the centralised estimation. Due to its distributed topological structure, this type of estimation technique is more reliable and flexible [4 ]. In some practical applications, an agreement on estimation is required among sensor nodes in distributed style, and the consensus filter was thus firstly studied in [5 ]. Based on this consensus filter, each sensor can reach a relatively high consistent estimation accuracy. In the past decades, many distributed filters have been proposed by adopting the consensus technique [6, 7 ]. Within the Kalman filtering framework, the reported consensus filtering approaches can be classified into four forms: consensus on estimates, consensus on measurements, consensus on information and consensus [8 ]. The approach of consensus on information was first established in [9 ], which can guarantee the filter stability even for a single consensus step. Then, the Kullback–Leibler average was introduced into the consensus filter in [10 ], where the authors exploited the consensus on local posterior probability density functions. Considering non-linear discrete time systems, a weighted average consensus approach was proposed in [11 ] under the framework of the unscented Kalman filter (UKF), and a hybrid consensus-based cubature Kalman filter (CKF) was proposed in [12 ]. By using the variance-constrained approach, Li et al. [13 ] proposed a variance-constrained distributed extended Kalman filter (EKF) which can optimise an upper bound matrix despite the linearisation errors and the edge-covariance matrices for the distributed state estimation of non-linear systems over sensor networks. Considering the non-linear systems with non-Gaussian noise, the linear fusion of unbiased and independent partial estimators was addressed [14 ], which focuses on the scenario where no communication is allowed among them. In order to reduce the communication cost, the consensus Monte Carlo algorithm was proposed by running a separate Monte Carlo algorithm on each machine, and then averaging individual Monte Carlo draws across machines [15 ]. More recently, Neiswanger et al. [16 ] presented a parallel Markov chain Monte Carlo algorithm, where the subsets of data are processed independently with very little communication. However, the earliest randomly intermittent observations may occur at the sensor node in practical applications because of the shelter of obstacles, a high noise environment, accidental failures of sensors and so on [17 ]. In this case, an individual sensor not only may fail to receive the estimation information from other sensors due to unreliable communications, but also may not sense the current state of the system by itself, which makes it more challenging for state estimation. To the best of our knowledge, few literatures have been focused on the distributed filters consensus on information with intermittent observations. However, it is an inevitable common phenomenon for practical distributed estimation in sensor networks. During the process of distributed information fusion, it is well-known that the information from different sensor nodes is always correlated [18, 19 ]. However, due to the lacking knowledge of real systems or unbearable computation burden [20 ], the cross-correlations between different sensor nodes are usually unavailable in practice. In view of this challenge, Julier and Uhlmann [20 ] firstly presented a covariance intersection (CI) algorithm to handle the problem. By utilising this algorithm, it is able to yield a consistent estimate irrespective of whether the cross-correlations are known or not. Through the development in the past two decades, a number of important results based on the CI algorithm have been reported [1, 19, 4, 21 ]. Recently, Deng et al. [19 ] proposed a sequential CI Kalman filter algorithm by solving an optimisation problem consisting of several one-dimensional non-linear functions. However, this algorithm may generate different estimation accuracy with different information fusion orders (i.e. the orders of the sequential processing). To handle this problem, a modified sequential fast CI (SFCI) scheme was then developed in [21 ]. With this modified SFCI, the computational complexity was reduced, and more importantly, the estimation accuracy is independent on the order of fusion at the cost of a slight decrease in accuracy. Combining CI and consensus on information, Wang and Ren [22 ] proposed a distributed hybrid information fusion (DHIF) algorithm, where the consensus weights are calculated via CI. The DHIF algorithm was then extended to solve the problem of dynamic state estimation where the target model, the sensor model and the communication topology might subject to time-varying changes [23 ]. For more general case, a CI-based Kalman consensus filtering (CIKCF) was presented in [24 ]. Furthermore, a systematic way to prove the boundedness of the estimation error and the estimation error covariance for distributed Kalman filters in general time-varying systems was presented. More recently, Chang and Mehta [25 ] investigated the relationship between effective observability in each agent and system topology, and discovered that some CI links may not provide an improvement on estimation performance but generate additional uncertainty. However, these consensus weights calculated by CI are based on the batch processing technique, which inevitably increases the computational burden and degenerate the performance for some practical applications. For the estimation of intermittent observations, the stochastic stability analysis is a significant issue in the sector of Kalman filtering research, and some important results can be found in the literature. For example, in the estimation of non-linear systems, the stability analyses of EKF, UKF and CKF have been investigated in [26-28 ], respectively. Due to the constrained communication energy and limited bandwidth, Li and Xia [27 ] established an event-triggered UKF for non-linear dynamic systems, and derived a critical bound of package delivery rate to guarantee the convergence. In [29 ], an extended filter (EF) for non-linear systems with measurement packet losses was proposed, and the conditions for guaranteeing stochastic stability of estimation error of the proposed EF were also given. In [17 ], the Bernoulli distribution was firstly introduced to describe the intermittent observations, and the stability and convergence properties of the estimation process were studied comprehensively. Sequentially, Huang and Dey [30 ] used a Markovian chain to formulate the loss of measurements, and derived some sufficient conditions for the stability of the peak covariance process in vector form. However, most of the results in the literature only focus on the stability analysis for centralised estimation, and few studies have been considered the stability analysis in distributed estimation with intermittent observations. Motivated by the aforementioned discussions, this paper is devoted to handling the issue of distributed state estimation for a class of linear time-varying systems with intermittent observations. By employing the CI method to the Kalman consensus filter (KCI), a new sequential CI-based Kalman consensus filer (SCIKCF) is developed. With this SCIKCF algorithm, the state estimates calculated by each sensor can reach consensus with intermittent observations. More importantly, the proposed SCIKCF can generate the fusion results sequentially and the estimation accuracy is insensitive to the order of fusion. This feature can effectively reduce the computation burden, and is preferred by users and has great competition potential for further applications in practice. Moreover, the estimation error covariances of the SCIKCF are rigorously proved to be bounded with some reasonable conditions. Combining the idea from the stochastic stability problem in [26 ], the exponential boundedness of the corresponding estimation errors in the sense of mean square is also analysed. The outline of this paper is organised as follows. In Section 2, the problem of the distributed estimation with intermittent observations is formulated. Then, the SCIKCF algorithm is expounded in Section 3. Section 4 gives the stability analysis of the proposed algorithm, including the boundedness of the estimation error covariances and estimation errors. Furthermore, three examples are given in Section 5 to validate the effectiveness of the established results. Finally, we state our conclusions and give directions for future works in Section 6. Notation: stands for the n -dimensional Euclidean space. is the expectation operator. For a matrix , and separately represent its inverse and transpose. represents that the matrix is positive definite (positive semi-definite), and indicates that is positive definite (positive semi-definite). denotes a block-diagonal matrix with matrices on the diagonal. stands for the trace of the matrix in the bracket. means the probability of a random event. is the n -dimensional identity matrix. is equivalent to and . 2 Problem formulation Consider a class of linear time-varying systems with intermittent observations as below: (1) (2) where is the current system state, and denotes the measurement of sensor i at time instant k. and are known matrices with appropriate dimensions. The process noise and measurement noise are assumed to be independent zero-mean white Gaussian, whose covariance matrices are and , respectively. is an observable binary variable and used to describe whether the observation is obtained or not in sensor i at time instant k. For sensor i, indicates that the system observation is available, otherwise it is not. Actually, here can be viewed as a Bernoulli random variable, and it is widely used to model the phenomenon of intermittent observations or incomplete measurements in practical applications [17, 31, 32 ]. The probability of can then be described as (3) where denotes the observation arrival possibility, . In this work, we consider a more practical application scenario, where the measurements are obtained by a distributed sensor network, and each sensor acts not only as an observer but also an estimator with both sensing and computation capabilities. In this paper, we use an directed graph to describe the topological structure of the estimation network. Here, denotes the set of sensors, and represents the set of edges. The set of sensors, which can communicate with sensor i, , is defined as the neighbours of sensor i and designated as , where sensor i is viewed as a special neighbour of itself and thus also included in . Here, the number of is denoted as . For any sensor network topology, one can observe that there is at least one sensor in , i.e. . Before the communication occurs between sensors, the modified Kalman filter algorithm [17 ] is applied in each sensor node to calculate the local state estimate. First, the one-step predicted state and covariance are calculated by (4) (5) Next, the processing of measurement update and Kalman gain matrix are computed from the following equations: (6) (7) (8) where and denote the updated estimate and covariance before consensus filtering separately. Remark 1.Under the framework of Bayesian estimation, the partial filter (PF) based on Monte Carlo method is adequate for both Gaussian and non-Gaussian probability densities [33-35 ]. But the computational complexity of PF increases exponentially with the dimension of the state vector, which is much larger than that of Kalman filter and reduces the filtering real-time performance inevitably [36 ]. Besides, the particle degradation problem usually occurs in the iteration of PF, and the corresponding estimation performance is generally degraded. Moreover, the Kalman filter is optimal in the linear and Gaussian systems [37 ]. Thus, the modified Kalman filter is utilised for the local estimation. In order to make use of this individual estimation information, the fusion algorithm is required. Moreover, during this process, one significant issue is that the local estimates are often correlated. Noting that the cross-covariance is usually unavailable in practice, the CI algorithm [20 ] is used in the fusion process for the consistency of the fused estimates. It can be described as follows: (9) (10) where is the consensus weight, and . Based on the CI algorithm, the standard Kalman-consensus filtering algorithm is outlined in Algorithm 1 [38 ]. Here the notations and denote the error covariance and estimate after the consensus update in Algorithm 1, respectively. Algorithm 1.Standard Kalman-consensus filtering algorithm [38 ] Input: the set of measurement Output: the covariance and the estimate 1: For each sensor , calculate the local estimates and the covariances using the modified Kalman filter (4 )–(8 ).2: Initialise the parameters and , respectively.3: for , do (11)end for4: Update the covariance and the estimate by (12)5: return and . Based on the standard Kalman-consensus filtering algorithm shown in Algorithm 1, one can achieve the consistent fusion result. However, this algorithm is a batch method for distributed estimation, which is hard to handle the asynchronous information collection problem and may result in a large computational burden. Besides, when the intermittent observations occur during the measurement processing, the fused estimates may be divergent. Naturally, three interesting problems arise for the distributed estimation in sensor networks with intermittent observations: How to calculate the suitable fusion coefficients with unknown cross-covariance? How to design a sequential consensus filtering algorithm which is insensitive to the fusion orders? Can the developed consensus filtering algorithm guarantee the fused estimation errors bounded in face of the intermittent observations? This paper is devoted to tackling these problems in the following sections. 3 Sequential covariance intersection-based Kalman consensus filter For obtaining optimal fusion results, the consensus weights are generally determined by minimising the trace of the error covariance of . Unfortunately, the minimisation of is a complex non-linear optimisation problem and suffers from a large computational burden [4, 19 ]. In order to handle this problem, a new fast batch CI fusion method [39 ] is introduced as follows: (13) Using (13 ) to Step 3 in Algorithm 1, we can obtain a new batch CI-based Kalman consensus filter (BCIKCF) algorithm. Compared with other existing optimised CI algorithms [4, 19 ], the presented CI algorithm gives a straightforward method to calculate the consensus weights, and needs less computation burden. However, this new fusion algorithm is also a batch method, which can only calculate the fused results after collecting a series of sensors' information. Based on the BCIKCF and motivated by the idea of cumulative moving average, the proposed SCIKCF algorithm is established. Embedding the proposed SCIKCF into sensor nodes, one can generate the fusion result sequentially. The formulas of computing the consensus weights are given by (14) where and are the sequential consensus weights at instant k. Therein, represents the consensus weight of the fused estimate, where l is the number of local estimates fused by sensor i, and is the weight of the local estimate generated by sensor i 's neighbour . It is noticed that , which can ensure that the proposed algorithm satisfies the requirement of unbiased fusion estimation [1 ]. Hence, the SCIKCF algorithm is presented in Algorithm 2. Remark 2.According to the pseudo-codes of BCIKCF and SCIKCF presented in Algorithms 1 and 2, respectively, one can observe that the consensus weights of SCIKCF and BCIKCF are both based on the individual information . Utilising the theory of cumulative moving average, in the next section, we show that SCIKCF can obtain the same fused estimation as BCIKCF if the same information (including the local estimate and the estimate error covariance) is collected. On the other hand, compared with the existing KCF algorithms based on the batch processing [10, 24 ], SCIKCF can achieve the consensus fused estimates sequentially, and especially, it is insensitive to the fusion orders of local estimates. Due to these characteristics, SCIKCF is more suitable for the distributed estimation in practical applications. Remark 3.In practice, L is always a preset finite constant to guarantee the termination of Algorithm 2. Generally, with a larger value of L, the accuracy of the fused estimates is higher. However, in this case, the computational burden will increase with more information collected. Therefore, the compromise between the estimation accuracy and the computational burden can be adjusted by choosing a suitable value of L. As the local estimate of sensor can just be used one time at each estimation period k, the maximum of L is . Therefore, one can obtain the highest fusion estimation accuracy when L is set to be . Algorithm 2.Sequential CI-based Kalman-consensus filtering algorithm Input: the set of measurement Output: the covariance and the estimate 1: For each sensor , calculate the local estimates and the covariances by the modified Kalman filter (4 )–(8 ).2: Initialise the parameters and , respectively.3: for , do (15)end for4: Update the covariance and the estimate by (16)5: return and . Remark 4.Based on our previous work in [21 ], the calculation of consensus weights in the SCIKCF algorithm is inspired by the SFCI scheme. When the SFCI algorithm is employed in the sensor network, each sensor generates the local estimate through an unaided filter algorithm. However, sensors in the SCIKCF update their local estimates based on the consensus fused estimates, instead of the local estimates. For this feature, one can expect that the proposed SCIKCF has a faster convergent speed than SFCI with the same computational burden and communication cost. 4 Stability analysis Stability analysis is an important topic in the research of the distributed Kalman filter, and has been studied consistently in the past decades. In this section, we will give the stability analysis of the proposed SCIKCF with intermittent observations. The boundedness of error covariance will be first discussed. Then, we will prove that the estimation errors of SCIKCF remain bounded. Different from the existing results (see, e.g. [24, 26-29, 40 ]), the analyses in this paper are involved in a distributed estimation manner with intermittent observations. Before demonstrating these analyses, the following assumptions are necessary. Assumption 1.There exist positive real constants , , such that the following bounds on the system parameter matrices are satisfied for each : (17) Assumption 2.The initial estimation error covariances ( ) are positive definite. Assumption 3.There exist positive scalars , such that the local estimation error covariances are bounded (18) Remark 5.In mathematics, since the noises in the considered system are assumed to be Gaussian, the limited eigenvalues of the noise covariances make Assumption 1 inadequate. However, in practical applications, the covariances of noise cannot be infinitely large because of the physical and energy constraints. Therefore, these assumptions are reasonable. Similar conditions have also been used in literature (see, e.g. [24, 29, 40 ]) to deal with the stability analysis issues for Kalman-type filters. Remark 6.In [17, 29 ], a sufficient condition has been provided to guarantee the boundedness of the matrix in Assumption 3, i.e. the observation arrival possibilities . 4.1 Boundedness of error covariance Theorem 1.Considering the linear time-varying systems (1 ) and (2 ) with Assumptions 1–3, there exist positive scalars and , such that the error covariances ( ) given by Algorithm 2 are uniformly bounded for all (19) Proof.We will give the proof in two steps: an upper bound and a lower bound for the error covariances. An upper boundFirstly, define to describe the difference between the inverse of l steps fused estimation error covariance and sensor j 's local estimation error covariance. In this proof, let us denote the superscript new as the l th sensor collected by sensor i. Then, the mathematical induction is used to prove for all .Base case : For , sensor i can just fuse its own estimates, so the following equation holds: (20) Inductive step : Assume that when , . Consider , the following equation can be obtained: (21) According to the mathematical induction, it follows immediately that for all . By Assumption 2, the following equation can be obtained: (22) A lower bound As explained in Remark 2, we will use the consensus weights calculated by BCIKCF to prove the existence of the lower bound. According to (7 ) and Algorithm 2, one can obtain (23) Since , then we have (24) Combining the results of (22 ) and (24 ), one can conclude that the error covariances obtained in Algorithm 2 are bounded. The proof is thus completed. □ 4.2 Boundedness of estimation errors In Theorem 1, the error covariance has been proved to be bounded. Now, we will turn to prove the boundedness of the estimation errors. This problem is another challenge in this paper because the correlations between sensors are completely unknown. Before analysing the boundedness of the estimation errors, some lemmas are required and listed as below. Lemma 1.Assume there is a stochastic process as well as real numbers and such that (25) (26) Then the stochastic process is exponentially bounded in mean square, i.e. (27) and the stochastic process is bounded with probability 1. Proof.Lemma 1 has been proved in [26 ]. Lemma 2.Given an integer , N positive definite matrices , and N vectors , the following inequality holds: (28) With the above-mentioned auxiliary lemmas set up, it is able to present another main result of this paper in the following. Proof.Lemma 2 has been proved in [10 ]. Theorem 2.Consider the linear stochastic systems described by (1 ) and (2 ) and the algorithm given by Algorithms 2. Under Assumptions 1–3, the estimation error is exponentially bounded in mean square with probability 1 for all provided that the initial estimation error is bounded. Proof.The estimation error of Algorithm 2 with intermittent observations is given by (29) According to the local measurement update (6 ), one can obtain (30) where , and . Define a stochastic process as follows: (31) According to the result of Theorem 1, one has (32) Since (33) It follows immediately that (34) where .Taking the expectation operation on , it yields (35) For the first term on the right side of (35 ), we have (36) Using Lemma 2, one can derive that (37) Meanwhile, for the covariance , from (5 ) and , it follows that (38) Let us define . Substituting (38 ) into (37 ) yields (39) For the second term in (35 ), the following inequality is derived: (40) According to Lemma 2 and , the following inequality holds: (41) Considering the third term in (35 ) (42) Overall, combining (39 )–(42 ), it can be obtained that (43) In (43 ), it is readily observed that and always hold, which satisfies the conditions of Lemma 1. Hence, one can conclude that the estimation error of BCIKCF is exponentially bounded in mean square.Refer to (11 ) and (13 ), the trace of can be decomposed as (44) where is the number of . Thus, can be expanded in detail as below: (45) Assume sensor is an arbitrary neighbour of sensor i, then (45 ) is transformed as follows: (46) From (46 ), since sensor g is an arbitrary neighbour of sensor i, it can be observed that the estimation error covariance obtained by BCIKCF is equal to that of SCIKCF. In a similar manner, the state estimate is also equal to the one of BCIKCF. As previously noted in Remark 2, if the same information is collected, the estimation result of BCIKCF is the same as that of SCIKCF. In addition, the estimation error of BCIKCF has been proved to be exponentially bounded in mean square, thus one can readily conclude that the estimation error of SCIKCF is also exponentially bounded in mean square. The proof is thus completed □. Remark 7.Generally, the observation arrival possibility of sensor nodes is designed and tested to satisfy the desired level described in Remark 6 for engineering applications in real-world systems. But in some complex application scenarios, some observation arrival possibilities may be lower than . In this case, these sensor nodes' information can be removed from the state estimation process to ensure the stability of the system, i.e. does not contain these sensors whose observation arrival possibilities are less than the desired level. Thus, Theorems 1 and 2 still hold with this treatment. Remark 8.Resort to the results presented in [17 ], the local estimation covariance can converge with any initial condition when . Ther