{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T18:40:28Z","timestamp":1780512028751,"version":"3.54.1"},"reference-count":44,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,2,10]],"date-time":"2018-02-10T00:00:00Z","timestamp":1518220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, a distributed Bayesian filter design was studied for nonlinear dynamics and measurement mapping based on Kullback\u2013Leibler divergence. In a distributed structure, the nonlinear filter becomes a challenging problem, since each sensor cannot access the global measurement likelihood function over the whole network, and some sensors have weak observability of the state. To solve the problem in a sensor network, the distributed Bayesian filter problem was converted into an optimization problem by maximizing a posterior method. The global cost function over the whole network was decomposed into the sum of the local cost function, where the local cost function can be solved by each sensor. With the help of the Kullback\u2013Leibler divergence, the global estimate was approximated in each sensor by communicating with its neighbors. Based on the proposed distributed Bayesian filter structure, a distributed cubature Kalman filter (DCKF) was proposed. Finally, a cooperative space object tracking problem was studied for illustration. The simulation results demonstrated that the proposed algorithm can solve the issues of varying communication topology and weak observability of some sensors.<\/jats:p>","DOI":"10.3390\/e20020116","type":"journal-article","created":{"date-parts":[[2018,2,12]],"date-time":"2018-02-12T10:50:38Z","timestamp":1518432638000},"page":"116","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Kullback\u2013Leibler Divergence Based Distributed Cubature Kalman Filter and Its Application in Cooperative Space Object Tracking"],"prefix":"10.3390","volume":"20","author":[{"given":"Chen","family":"Hu","sequence":"first","affiliation":[{"name":"Xi\u2019an Institute of High-Tech, Xi\u2019an 710025, Shaanxi, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haoshen","family":"Lin","sequence":"additional","affiliation":[{"name":"Xi\u2019an Institute of High-Tech, Xi\u2019an 710025, Shaanxi, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhenhua","family":"Li","sequence":"additional","affiliation":[{"name":"Xi\u2019an Institute of High-Tech, Xi\u2019an 710025, Shaanxi, China"},{"name":"Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bing","family":"He","sequence":"additional","affiliation":[{"name":"Xi\u2019an Institute of High-Tech, Xi\u2019an 710025, Shaanxi, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gang","family":"Liu","sequence":"additional","affiliation":[{"name":"Xi\u2019an Institute of High-Tech, Xi\u2019an 710025, Shaanxi, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,2,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1109\/MAES.2013.6477867","article-title":"Applying aerospace technologies to current issues using systems engineering: 3rd aess chapter summit","volume":"28","author":"Oliva","year":"2013","journal-title":"IEEE Aerosp. Electron. Syst. Mag."},{"key":"ref_2","unstructured":"Kennewell, J.A., and Vo, B.N. (2013, January 9\u201312). An overview of space situational awareness. Proceedings of the 2013 16th IEEE International Conference on Information Fusion (FUSION), Istanbul, Turkey."},{"key":"ref_3","unstructured":"Weeden, B., Cefola, P., and Sankaran, J. (2010, January 14\u201317). Global space situational awareness sensors. Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Conference, Maui, HI, USA."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Vladimirova, T., Bridges, C.P., Paul, J.R., Malik, S.A., and Sweeting, M.N. (2010, January 6\u201313). Space-based wireless sensor networks: Design issues. Proceedings of the 2010 IEEE Aerospace Conference, Big Sky, MT, USA.","DOI":"10.1109\/AERO.2010.5447031"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Teixeira, B.O., Santillo, M.A., Erwin, R.S., and Bernstein, D.S. (2008). Spacecraft tracking using sampled-data Kalman filters. IEEE Control Syst., 28.","DOI":"10.1109\/MCS.2008.923231"},{"key":"ref_6","unstructured":"Tian, X., Chen, G., Blasch, E., Pham, K., and Bar-Shalom, Y. (2013, January 9\u201312). Comparison of three approximate kinematic models for space object tracking. Proceedings of the 2013 16th International Conference on IEEE Information Fusion (FUSION), Istanbul, Turkey."},{"key":"ref_7","unstructured":"Anderson, B.D.O., and Moore, J.B. (1979). Optimal Filtering, Prentice-Hall."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1109\/9.754809","article-title":"Stochastic stability of the discrete-time extended Kalman filter","volume":"44","author":"Reif","year":"1999","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1741","DOI":"10.1109\/TIA.2003.818991","article-title":"Extended Kalman filter tuning in sensorless PMSM drives","volume":"39","author":"Bolognani","year":"2003","journal-title":"IEEE Trans. Ind. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1535","DOI":"10.1088\/0266-5611\/17\/5\/319","article-title":"Improving the singular evolutive extended Kalman filter for strongly nonlinear models for use in ocean data assimilation","volume":"17","author":"Carme","year":"2001","journal-title":"Inverse Probl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1109\/9.847726","article-title":"A new method for the nonlinear transformation of means and covariances in filters and estimators","volume":"45","author":"Julier","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Zhou, H., Huang, H., Zhao, H., Zhao, X., and Yin, X. (2017). Adaptive Unscented Kalman Filter for Target Tracking in the Presence of Nonlinear Systems Involving Model Mismatches. Remote Sens., 9.","DOI":"10.3390\/rs9070657"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1109\/TAC.2009.2019800","article-title":"Cubature kalman filters","volume":"54","author":"Arasaratnam","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4977","DOI":"10.1109\/TSP.2010.2056923","article-title":"Cubature Kalman filtering for continuous-discrete systems: theory and simulations","volume":"58","author":"Arasaratnam","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1016\/j.sigpro.2013.06.015","article-title":"Cubature information filters with correlated noises and their applications in decentralized fusion","volume":"94","author":"Ge","year":"2014","journal-title":"Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1177\/0142331214523032","article-title":"Multiple sensor estimation using a new fifth-degree cubature information filter","volume":"37","author":"Jia","year":"2015","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Olfati-Saber, R. (2009, January 15\u201318). Kalman-consensus filter: Optimality, stability, and performance. Proceedings of the Joint IEEE Conference on Decision and Control and Chinese Control Conference, Shanghai, China.","DOI":"10.1109\/CDC.2009.5399678"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Olfati-Saber, R. (2007, January 12\u201314). Distributed Kalman filtering for sensor networks. Proceedings of the IEEE Conference on Decision and Control, New Orleans, LA, USA.","DOI":"10.1109\/CDC.2007.4434303"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1016\/j.automatica.2013.11.042","article-title":"Kullback\u2013Leibler average, consensus on probability densities, and distributed state estimation with guaranteed stability","volume":"50","author":"Battistelli","year":"2014","journal-title":"Automatica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.automatica.2016.01.071","article-title":"Stability of consensus extended Kalman filter for distributed state estimation","volume":"68","author":"Battistelli","year":"2016","journal-title":"Automatica"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.1109\/TAC.2014.2357135","article-title":"Consensus-Based Linear and Nonlinear Filtering","volume":"60","author":"Battistelli","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4458","DOI":"10.1109\/TSP.2015.2424205","article-title":"Distributed Kalman Filtering With Dynamic Observations Consensus","volume":"63","author":"Das","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zhou, Z., Hong, Y., and Fang, H. (2012, January 17\u201320). Distributed estimation for moving target under switching interconnection network. Proceedings of the International Conference on Control Automation Robotics & Vision, Hanoi, Vietnam.","DOI":"10.1109\/ICARCV.2012.6485124"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2096","DOI":"10.1109\/TAC.2013.2246476","article-title":"Distributed estimation for moving target based on state-consensus strategy","volume":"58","author":"Zhou","year":"2013","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4334","DOI":"10.1109\/TSP.2012.2196697","article-title":"Likelihood consensus and its application to distributed particle filtering","volume":"60","author":"Hlinka","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1908","DOI":"10.1109\/TAES.2016.140506","article-title":"Cooperative space object tracking using space-based optical sensors via consensus-based filters","volume":"52","author":"Jia","year":"2016","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/j.automatica.2015.02.005","article-title":"Adaptive unscented Gaussian likelihood approximation filter","volume":"54","author":"Morelande","year":"2015","journal-title":"Automatica"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.sigpro.2016.07.007","article-title":"Kullback\u2013Leibler divergence approach to partitioned update Kalman filter","volume":"130","author":"Raitoharju","year":"2017","journal-title":"Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Li, Y., Cheng, Y., Li, X., Hua, X., and Qin, Y. (2017). Information Geometric Approach to Recursive Update in Nonlinear Filtering. Entropy, 19.","DOI":"10.3390\/e19020054"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Li, Y., Cheng, Y., Li, X., Wang, H., Hua, X., and Qin, Y. (2017). Bayesian Nonlinear Filtering via Information Geometric Optimization. Entropy, 19.","DOI":"10.3390\/e19120655"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3112","DOI":"10.1109\/TAC.2013.2277621","article-title":"Information weighted consensus filters and their application in distributed camera networks","volume":"58","author":"Kamal","year":"2013","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.sysconle.2004.02.022","article-title":"Fast linear iterations for distributed averaging","volume":"53","author":"Xiao","year":"2004","journal-title":"Syst. Control Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2882","DOI":"10.1109\/TIT.2009.2018176","article-title":"Sided and symmetrized Bregman centroids","volume":"55","author":"Nielsen","year":"2009","journal-title":"IEEE trans. Inf. Theory"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1109\/18.61115","article-title":"Divergence measures based on the Shannon entropy","volume":"37","author":"Lin","year":"1991","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_35","first-page":"1705","article-title":"Clustering with Bregman divergences","volume":"6","author":"Banerjee","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"ref_36","unstructured":"Battistelli, G., Chisci, L., and Selvi, D. (2016). Distributed averaging of exponential-class densities with discrete-time event-triggered consensus. IEEE Trans.Control Netw. Syst."},{"key":"ref_37","unstructured":"Nielsen, F., and Garcia, V. (arXiv, 2009). Statistical exponential families: A digest with flash cards, arXiv."},{"key":"ref_38","unstructured":"Casella, G., and Berger, R.L. (2002). Statistical Inference, Duxbury."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Curtis, H.D. (2013). Orbital Mechanics for Engineering Students, Butterworth-Heinemann.","DOI":"10.1016\/B978-0-08-097747-8.00006-2"},{"key":"ref_40","unstructured":"Battistelli, G., Chisci, L., and Selvi, D. (2016, January 5\u20138). Distributed Kalman filtering with data-driven communication. Proceedings of the International Conference on Information Fusion, Heidelberg, Germany."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Julier, S.J., and Uhlmann, J.K. (1997, January 6). A non-divergent estimation algorithm in the presence of unknown correlations. Proceedings of the 1997 IEEE American Control Conference, Albuquerque, NM, USA.","DOI":"10.1109\/ACC.1997.609105"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1879","DOI":"10.1109\/TAC.2002.804475","article-title":"Estimation under unknown correlation: covariance intersection revisited","volume":"47","author":"Chen","year":"2002","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1109\/TAC.2008.2009515","article-title":"Distributed subgradient methods for multi-agent optimization","volume":"54","author":"Nedic","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2069","DOI":"10.1109\/TAC.2010.2042987","article-title":"Diffusion strategies for distributed Kalman filtering and smoothing","volume":"55","author":"Cattivelli","year":"2010","journal-title":"IEEE Trans. Autom. Control"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/2\/116\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:54:33Z","timestamp":1760194473000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/2\/116"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,10]]},"references-count":44,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,2]]}},"alternative-id":["e20020116"],"URL":"https:\/\/doi.org\/10.3390\/e20020116","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,2,10]]}}}