Unscented Kalman filtering for target tracking systems with packet dropout compensation

IET Control Theory & ApplicationsVolume 13, Issue 12 p. 1901-1908 Research ArticleFree Access Unscented Kalman filtering for target tracking systems with packet dropout compensation Kemao Ma, Kemao Ma Department of Control Science, Engineering, Control, Simulation Center, Harbin Institute of Technology, 2 Yikuang Street, Harbin, People's Republic of ChinaSearch for more papers by this authorLong Xu, Corresponding Author Long Xu lx_8803@126.com Department of Control Science, Engineering, Control, Simulation Center, Harbin Institute of Technology, 2 Yikuang Street, Harbin, People's Republic of ChinaSearch for more papers by this authorHongxia Fan, Hongxia Fan Department of Control Science, Engineering, Control, Simulation Center, Harbin Institute of Technology, 2 Yikuang Street, Harbin, People's Republic of China School of Basic Science, Harbin University of Commerce, 1 Xuehai Street, Harbin, People's Republic of ChinaSearch for more papers by this author Kemao Ma, Kemao Ma Department of Control Science, Engineering, Control, Simulation Center, Harbin Institute of Technology, 2 Yikuang Street, Harbin, People's Republic of ChinaSearch for more papers by this authorLong Xu, Corresponding Author Long Xu lx_8803@126.com Department of Control Science, Engineering, Control, Simulation Center, Harbin Institute of Technology, 2 Yikuang Street, Harbin, People's Republic of ChinaSearch for more papers by this authorHongxia Fan, Hongxia Fan Department of Control Science, Engineering, Control, Simulation Center, Harbin Institute of Technology, 2 Yikuang Street, Harbin, People's Republic of China School of Basic Science, Harbin University of Commerce, 1 Xuehai Street, Harbin, People's Republic of ChinaSearch for more papers by this author First published: 27 June 2019 https://doi.org/10.1049/iet-cta.2018.5537Citations: 4AboutSectionsPDF 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 onFacebookTwitterLinkedInRedditWechat Abstract The unscented Kalman filtering problem is investigated for a class of non-linear discrete-time stochastic systems subject to random packet dropout. Here, a random variable, which obeys Bernoulli distribution with known conditional probability, is introduced to depict the phenomenon of packet dropout occurring in a stochastic way. Based on the compensation technology of packet dropout, the one-step prediction value of observer instead of zero-input is used as a compensator when packet dropout occurs. Owing to taking the phenomenon of random packet dropout into account, the authors need to compute parameters to reduce the effects of the compensator. Then, based on the minimum mean square error principle, a new unscented Kalman filtering algorithm is proposed such that, for the random packet dropout, the filtering error is minimised. By solving the recursive matrix equations, the filter gain matrices and error covariance matrices can be obtained and the proposed results can be easily verified by using the standard numerical software. They finally provide two examples, one is a numerical example to show the performance of the proposed approach, and the other is to solve the problem of target estimation for a tracking system with random packet dropout. 1 Introduction It is well known that Kalman first provided a new approach (Kalman filtering) to investigate the estimation problem in [1] for the linear discrete-time stochastic systems by employing the principle of minimum mean square error and the projection theory. Owing to the disturbance of random noises and a variety of other uncertain factors, the systems cannot receive accurate data. The Kalman filter is to estimate the systems state as accurately as possible based on the observation sequence. Also, the Kalman filtering algorithm is convenient for computer programming and real-time data update processing. Hence, it is the most efficient and most widely filtering method, which is applied in many practical systems, such as communications, power, aviation, and many other industrial and military fields [2–8]. Based on the traditional Kalman filtering algorithm, unscented Kalman filter (UKF), as a new method for non-linear filtering, has been constructed for non-linear systems in [9]. Also, an unscented transformation approach was utilised to approximate the mean and covariance in order to satisfy the minimum mean square error principle. Thus, the unscented Kalman filtering algorithm has also attracted wide attention [10–14]. In recent years, the filtering problem is extensively investigated due to its wide application in practical systems, such as the target tracking systems [15–22]. In the target tracking systems, the filtering method is used to estimate the position of moving target at the every time instant. Hence, a great deal of effort was devoted to addressing the estimation problems of moving target for tracking systems. More concretely, in [23], a new adaptive unscented Kalman filtering algorithm has been proposed to improve filtering precision and restrain divergence, which caused sensor faults or model mismatches for target tracking and the necessary and sufficient condition with third-order accuracy of the standard unscented Kalman filtering algorithm has been given. For bearings-only manoeuvring target tracking systems, a novel auxiliary truncated UKF has been constructed in [24], where compared with other non-linear filtering approaches, the experimental results of the proposed algorithm showed a significant improvement for both the univariate non-stationary growth model case and bearings-only target tracking case. Based on the information fusion technology, a sequential fusion estimation method has been presented in [25] to estimate the state of manoeuvring target tracking systems over asynchronous wireless sensor networks. In [26], for underwater bearings-only passive target tracking systems, an UKF has been constructed. For the multiple targets tracking problem, in [27], the Kalman filter has been designed to track the multiple objects simultaneously and an improved optical flow algorithm has been presented, which not only gave better accuracy but also handled occlusion in a video. With the rapid development of computer technology, the network phenomena are taken into account in various systems [28–33]. Owing to the long distance transmission and unreliability of the communication network, the sensor measurement of the system may experience the unexpected missing measurements (packet dropout) in the transmission process [34–37]. Hence, the problem of packet dropout has become one of the mainstream research subjects in networked control systems. It is generally known that the phenomenon of packet dropout has been described by the Bernoulli distributed random variable taking on values of 0 and 1. When the sensor exhibits the packet dropout, the value of zero-input, as a compensator, is usually used to estimate the state of the system. To mention a few, in [38], a time-varying filter has been constructed for a class of non-linear stochastic systems in the presence of event-triggered transmissions and multiple missing measurements with uncertain missing probabilities, where an upper bound of the filtering error covariance has been obtained and then minimised by properly designing the filter gain. In [39], a new non-linear Kalman filter has been designed for aircraft engine systems with time delay and packet dropout and the proposed methodology not only reduces the computational time but also obtains a satisfactory accuracy for state estimation in the cases of engine gas path measurement uncertainty. By using the unscented transformation approach, in [40], a new unscented Kalman filtering scheme has been proposed for unreliable communication networks with Markovian packet dropouts. The event-triggered non-linear filtering problem has been investigated in [41] for non-linear dynamic systems over a wireless sensor network with packet dropout. In [42], the non-linear filtering problem was addressed for a class of non-linear discrete time stochastic systems with missing measurements by applying the extended and unscented Kalman filtering approach, respectively, and it showed that the UKF is more effective than the extended Kalman filter. However, to the best of authors' knowledge, these researches do not pay much attention to the problem of the unscented Kalman filtering algorithm for non-linear discrete-time stochastic systems subject to packet dropout compensation in the measurement. Based on the above discussions, in our paper, the purpose is to solve the non-linear estimation problem for discrete-time stochastic systems subject to random packet dropout. Generally speaking, we know that zero-input is usually used as a compensation result when the system exhibits packet dropout. However, the state of systems is estimated by using the observation sequence. Here, we take the one-step prediction value of the observer as a compensator instead of zero-input when the packet dropout is used to estimate the state. Based on the minimum mean square error principle and the unscented transformation approach, the UKF is designed, which can address the effects of packet dropout in a unified framework. Here, we make the first attempt to propose the non-linear filter for systems with packet dropout compensation and establish a new recursive algorithm to obtain the optimal non-linear filter by employing the compensation technology of packet dropout. Hence, we need to compute some parameters to reduce the impact of the compensator on the performance of the filter and two sets of sigma points are selected to improve the accuracy of the unscented Kalman filtering algorithm. Then, we can recursively compute the filter gain matrices and error covariance matrices by using the new algorithm and Matlab software. Finally, we give a numerical example to verify the performance of the proposed filtering algorithm and a tracking system with random packet dropout to show the feasibility and effectiveness of the developed main results. The contribution of this paper: (i) the one-step prediction value of the observer is employed as a compensator to estimate the state of the system. (ii) We select two sets of sigma points to compute parameters to reduce the impact of the compensator on the performance of the filter. (iii) A new recursive algorithm is established to obtain the optimal non-linear filter, which is suitable for online applications. Notation: The symbols used in the paper are standard. denotes the n -dimensional Euclidean space. represents the transpose of a matrix A. is the expectation of the random variable x. The identity matrix and zero matrix are expressed by I and 0 with appropriate dimensions, respectively. If the dimensions of the matrices are not definitely stated, they are considered to be well-matched for algebraic operations. 2 Problem formulation We consider the following class of non-linear systems: (1) (2)where is the state vector, is the observation output. and are known non-linear functions. and are uncorrelated zero-mean Gaussian white noises with covariances and , respectively. It is generally known that there are two ways to describe the phenomenon of packet dropout. One is that the sensor only receives the communication noise when packet dropout occurs; the other is that the sensor receives nothing (zero) when packet dropout occurs. In our paper, the second scheme is adopted. Hence, a random variable , which obeys the Bernoulli distribution is employed to describe the phenomenon of the packet dropout as follows: If , , it represents that the sensor receives the data successfully at time instant k. If , , it stands for that the sensor occurs the packet dropout. Therefore, when packet dropout occurs, the value of one-step prediction is added in the following model as a compensator [43]: (3)where is the measured output of the sensor. The random variable satisfies the following statistical properties: where is a known scalar. In our paper, we assume that and other noise signals are mutually independent. Substituting (2) into (3), systems (1)–(3) can be equivalently rewritten as follows: (4) (5) The purpose of this study is to design the UKF for non-linear discrete-time stochastic systems (4) and (5) based on the minimum mean square error principle and observation sequence . 3 Main results In order to facilitate the subsequent developments, the following definition is introduced. Definition 1.Define the one-step prediction error , where is the one-step prediction and the error covariance matrix of one-step prediction . Similarly, define the state estimation error , where is the state estimation and error covariance matrix of state estimation . Now, based on the observation sequence , we are ready to design the optimal filter for non-linear discrete-time stochastic systems (4) and (5) by employing the projection theory and unscented transformation approach. Theorem 1.The recursive optimal filter for systems (4) and (5) is given as follows: (6) (7) (8) (9) (10)where (11) (12) (13) (14) is the innovation sequence and is the filter gain matrix to be determined. Proof.By using the method in [1], the optimal estimation can be computed as follows: (15)where is calculated by Definition 1 and is derived as follows: (16)Note that is uncorrelated with other terms. Then, we have (17)where (18)Therefore, (7), (9) and (14) can be obtained by (16)–(18).Based on the method in [1], can be calculated as follows: (19)Then, we have (20)From Definition 1, one has (21)Then, (12) is true.By using (16), (20) and , , it can be easily deduced that (22)where (23)We define gain matrix as follows: (24)From (17) and (22), we know (25)where . Substituting (24) into (15) yields (6).Subsequently, the following derivations are given to obtain . By Definition 1, it has (26)Then, we have (27)Hence, (10) is true. The proof of this theorem is now complete. □ According to Theorem 1, a new recursive algorithm can be established to obtain the UKF for the addressed discrete stochastic-time non-linear systems subject to random packet dropout. The following algorithm shows how to design the UKF in Theorem 1. Algorithm. The steps of the design of the UKF are shown as follows: Step 1 : Choose the sigma points. We choose points as a sigma points set, i.e. where is the scaling factor and is either the s th row or the s th column of the matrix square root of . Now, compute the transformed values of the sigma points by using the non-linear function (28) Then, one-step prediction and error covariance matrix can be calculated by recombining the weighted sigma points as follows: (29) (30)according to weights (31) On the other hand, we choose another points as another sigma points set. Then By using non-linear function , we calculate the transformed values of the sigma points (32) We define . Then, one has (33)where is computed by (31). Step 2 : Compute the value of parameters. Equations (11) and (12) are calculated by (29) and (30). Parameters and can be obtained as follows: (34) (35) Step 3 : Compute the value of estimation. (1) Substituting (35) into (9), we can obtain . Also, can be calculated from (11). (2) Computing by substituting (33) into (7). (3) Substituting (16) and (33) into (15), we obtain the optimal estimation . (4) The error covariance matrix of state estimation can be calculated by substituting (8), (12) and (34) into (10). Remark 1.It is worth noting that in Theorem 1, the recursive optimal estimator is designed for the addressed non-linear discrete-time stochastic systems with packet dropout compensation. Owing to the long distance transmission and unreliability of the communication network, the phenomenon of packet dropout exists commonly. Therefore, in this study, we consider the non-linear discrete-time stochastic systems subject to random packet dropout. Here, we need to compute parameters and to reduce the impact of the packet dropout compensator. In the following, two illustrative examples will be provided to show the feasibility of the proposed filtering scheme. Remark 2.In our study, for simplicity, the choice of the weighted sigma points is based on [9]. Also, the method to calculate the matrix square root is the Cholesky decomposition adopted in part of the selection of sigma points set, such as, and . In general, if the square root A of matrix P is the form of , the sigma points are formed from the row of A. Otherwise, the columns of A are employed if [9]. Therefore, we utilise the s th column of the matrix square root of in our algorithm. Remark 3.Compared with the previous researches, we take the one-step prediction value of the observer instead of zero-input as a compensator to estimate the state of the system. When the system exhibits packet dropout, we take the value of the one-step prediction as data compensation instead of zero to estimate the state and we need to compute parameters to reduce the impact of the compensator on the performance of the filter. Hence, we select two sets of sigma points in our algorithm to compute the parameters of the filter. Compared with the case of only one set of sigma pints, in our algorithm, we choose the first set sigma points based on the value of estimation . By using non-linear function , we compute the transformed values of sigma points . Then, one-step prediction and error covariance matrix can be calculated by recombining weighted sigma points . Next, we choose the second set sigma points based on the value of one-step prediction . By using non-linear function , we calculate the transformed values of the sigma points . Based on the value of and , we can obtain the value of estimation by computing parameters , and . In the above discussion, is computed based on second set sigma points instead of the value of and second set sigma points is chosen by using the value of one-step prediction . In this way, the performance of the filter can be improved and the accuracy of the algorithm will be improved. At the same time, it also increases the complexity of the filter and increases the computational burden. 4 Illustrative example In this section, we provide two examples, one is a numerical example to show the performance of the proposed approach; the other is to solve the problem of target estimation for a tracking system with random packet dropout. Example 1.Consider the following non-linear discrete-time stochastic systems: where and are uncorrelated Gaussian white noises with covariances , .Let and denote the error for the estimation of , i.e. , where .According to Theorem 1, the UKF can be constructed for non-linear systems with random packet dropout by applying the unscented transformation approach and minimum mean-square error (MMSE) estimation principle. Based on the given unscented Kalman filtering algorithm and Matlab software, filter gain matrices and error covariance matrices at every time step can be recursively computed. The results are shown in Figs. 1-5–6. Figs. 1, 2–3 plot the filtering errors . From the simulations, we can see that the range of error fluctuation in our study is relatively small compared with UKF with the zero-input compensation strategy and UKF with one set of sigma points. The actual system states and their estimates are plotted in Figs. 4, 5–6. It is easily seen that due to making a lot of efforts to reduce the effects from packet dropout compensation, the proposed filter can estimate the system state effectively and the recursive algorithm is feasible.In our study, we compare the filter estimation performance with the different rates of packet dropout (i.e. ). The corresponding simulation results are given in Figs. 7, 8–9. From the simulation results, we can conclude that the different rates of packet dropout have a great impact on the accuracy of the filtering algorithm. Compared with the zero-input compensation strategy, the filter, which is designed by us, can estimate the system states more accurately and effectively. The reason is that we utilise the one-step prediction value of the observer instead of zero-input as a compensator and two sets of sigma points are selected to compute parameters to reduce the impact of the compensator on the performance of the filter, which can improve the accuracy of the unscented Kalman filtering algorithm. Example 2.Consider the following target tracking systems with random packet dropout: where T is the sampling time. stands for the position, velocity and acceleration of target. The position of the observer is . is the distance between the target and the observer and is the angle of the target relative to the observer. and are uncorrelated Gaussian white noises with covariances , .Let initial state and sampling time . Then, the following simulation results are obtained and we can see that the proposed filter, which can effectively estimate the position of the target, is suitable for the target tracking systems with random packet dropout Figs. 10 and 11. Fig. 1Open in figure viewerPowerPoint Error Fig. 2Open in figure viewerPowerPoint Error Fig. 3Open in figure viewerPowerPoint Error Fig. 4Open in figure viewerPowerPoint trajectories of and Fig. 5Open in figure viewerPowerPoint Trajectories of and Fig. 6Open in figure viewerPowerPoint trajectories of and Fig. 7Open in figure viewerPowerPoint Trajectories of Fig. 8Open in figure viewerPowerPoint Trajectories of Fig. 9Open in figure viewerPowerPoint Trajectories of Fig. 10Open in figure viewerPowerPoint Position trajectory of horizontal direction Fig. 11Open in figure viewerPowerPoint Position trajectory of vertical direction 5 Conclusion The problem of non-linear estimation has been investigated for a class of discrete-time stochastic systems subject to random packet dropout. Firstly, by employing the compensation technology of packet dropout, the one-step prediction value of the observer is used as a compensator when packet dropout occurs. Owing to taking the phenomenon of random packet dropout into account, we need to compute parameters to reduce the effects of the packet dropout compensator. Then, based on the unscented transformation approach and MMSE principle, the UKF has been constructed, which can estimate the system state effectively. Finally, a recursive algorithm has been given to design the non-linear filter and a simulation example have been given to show the feasibility and usefulness of the proposed approach. What is more, we also have given a practical system example, a tracking system with random packet dropout, to show the feasibility and effectiveness of the developed main results. 6 Acknowledgments This work was supported in part by the National Natural Science Foundation of China under grant nos. 61174001 and 61321062. 7 References 1Kalman R.E.: 'A new approach to linear filtering and prediction problems', Trans. ASME J. Basic Eng., 1960, 82, pp. 35– 45 2Deshmukh R. 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