{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T16:45:06Z","timestamp":1769013906650,"version":"3.49.0"},"reference-count":53,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T00:00:00Z","timestamp":1619654400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["51977153, 51977161 , 51577046"],"award-info":[{"award-number":["51977153, 51977161 , 51577046"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this study, an intelligent computing paradigm built on a nonlinear autoregressive exogenous (NARX) feedback neural network model with the strength of deep learning is presented for accurate state estimation of an underwater passive target. In underwater scenarios, real-time motion parameters of passive objects are usually extracted with nonlinear filtering techniques. In filtering algorithms, nonlinear passive measurements are associated with linear kinetics of the target, governing by state space methodology. To improve tracking accuracy, effective feature estimation and minimizing position error of dynamic passive objects, the strength of NARX based supervised learning is exploited. Dynamic artificial neural networks, which contain tapped delay lines, are suitable for predicting the future state of the underwater passive object. Neural networks-based intelligence computing is effectively applied for estimating the real-time actual state of a passive moving object, which follows a semi-curved path. Performance analysis of NARX based neural networks is evaluated for six different scenarios of standard deviation of white Gaussian measurement noise by following bearings only tracking phenomena. Root mean square error between estimated and real position of the passive target in rectangular coordinates is computed for evaluating the worth of the proposed NARX feedback neural network scheme. The Monte Carlo simulations are conducted and the results certify the capability of the intelligence computing over conventional nonlinear filtering algorithms such as spherical radial cubature Kalman filter and unscented Kalman filter for given state estimation model.<\/jats:p>","DOI":"10.3390\/e23050550","type":"journal-article","created":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T10:30:41Z","timestamp":1619692241000},"page":"550","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Design of Nonlinear Autoregressive Exogenous Model Based Intelligence Computing for Efficient State Estimation of Underwater Passive Target"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0868-334X","authenticated-orcid":false,"given":"Wasiq","family":"Ali","sequence":"first","affiliation":[{"name":"School of Marine Science and Technology, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"Department of Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, Attock 43600, Pakistan"}]},{"given":"Wasim Ullah","family":"Khan","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Wuhan University, Wuhan 430072, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9953-822X","authenticated-orcid":false,"given":"Muhammad Asif Zahoor","family":"Raja","sequence":"additional","affiliation":[{"name":"Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan"}]},{"given":"Yigang","family":"He","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Wuhan University, Wuhan 430072, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5973-9780","authenticated-orcid":false,"given":"Yaan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Marine Science and Technology, Northwestern Polytechnical University, Xi\u2019an 710072, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ge, B., Zhang, H., Jiang, L., Li, Z., and Butt, M.M. (2019). Adaptive unscented Kalman filter for target tracking with unknown time-varying noise covariance. Sensors, 19.","DOI":"10.3390\/s19061371"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Li, Y., Chen, X., Yu, J., and Yang, X. (2019). NA fusion frequency feature extraction method for underwater acoustic signal based on variational mode decomposition, duffing chaotic oscillator and a kind of permutation entropy. Electronics, 8.","DOI":"10.3390\/electronics8010061"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107345","DOI":"10.1016\/j.apacoust.2020.107345","article-title":"Generalized pseudo Bayesian algorithms for tracking of multiple model underwater maneuvering target","volume":"166","author":"Ali","year":"2020","journal-title":"Appl. Acoust."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1049\/iet-smt.2017.0529","article-title":"Adaptive state estimation for tracking of civilian aircraft","volume":"12","author":"Patra","year":"2018","journal-title":"IET Sci. Meas. Technol."},{"key":"ref_5","first-page":"6671","article-title":"Weak target detection technology of passive radar on the navigation satellite signal","volume":"2019","author":"Fan","year":"2019","journal-title":"J. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hong, J., Laflamme, S., Dodson, J., and Joyce, B. (2018). Introduction to state estimation of high-rate system dynamics. Sensors, 18.","DOI":"10.3390\/s18010217"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Wirnshofer, F., Schmitt, P.S., Meister, P., Wichert, G.V., and Burgard, W. (2019, January 20\u201324). State estimation in contact-rich manipulation. Proceedings of the 2019 International Conference on Robotics and Automation (ICRA), Montreal, QC, Canada.","DOI":"10.1109\/ICRA.2019.8793572"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Luo, J., Han, Y., and Fan, L. (2018). Underwater Acoustic Target Tracking: A Review. Sensors, 18.","DOI":"10.3390\/s18010112"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ahmed, H., Ullah, I., Khan, U., Qureshi, M.B., Manzoor, S., Muhammad, N., Shahid Khan, M.U., and Nawaz, R. (2019). Adaptive Filtering on GPS-Aided MEMS-IMU for Optimal Estimation of Ground Vehicle Trajectory. Sensors, 19.","DOI":"10.3390\/s19245357"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Li, X., Zhao, C., Yu, J., and Wei, W. (2019). Underwater Bearing-Only and Bearing-Doppler Target Tracking Based on Square Root Unscented Kalman Filter. Entropy, 21.","DOI":"10.3390\/e21080740"},{"key":"ref_11","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_12","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.sigpro.2016.06.004","article-title":"Multistatic pseudolinear target motion analysis using hybrid measurements","volume":"130","author":"Nguyen","year":"2017","journal-title":"Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1145\/3363294","article-title":"An elementary introduction to kalman filtering","volume":"62","author":"Pei","year":"2019","journal-title":"Commun. ACM"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1516028","DOI":"10.1155\/2018\/1516028","article-title":"Kalman Filtering Algorithm for Systems with Stochastic Nonlinearity Functions, Finite-Step Correlated Noises, and Missing Measurements","volume":"2018","author":"He","year":"2018","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhang, M., Li, K., Hu, B., and Meng, C. (2019). Comparison of Kalman Filters for Inertial Integrated Navigation. Sensors, 19.","DOI":"10.3390\/s19061426"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1117\/12.280797","article-title":"New extension of the Kalman filter to nonlinear systems","volume":"3068","author":"Julier","year":"1997","journal-title":"Signal Process. Sens. Fusion Target Recognit. VI"},{"key":"ref_17","first-page":"106837","article-title":"An aperiodic phenomenon of the extended Kalman filter in filtering noisy chaotic signals","volume":"143","author":"Leung","year":"2020","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"14609","DOI":"10.1073\/pnas.1617398113","article-title":"State estimation and prediction using clustered particle filters","volume":"113","author":"Lee","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ali, W., Li, Y., Tanoli, S.A.K., Raja, M.A.Z., Javaid, K., and Ahmed, N. (2019, January 20\u201322). SConvergence Analysis of Unscented Transform for Underwater Passive Target Tracking in Noisy Environment. Proceedings of the 2019 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC), Dalian, China.","DOI":"10.1109\/ICSPCC46631.2019.8960777"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7905690","DOI":"10.1155\/2017\/7905690","article-title":"An improved unscented Kalman filter for discrete nonlinear systems with random parameters","volume":"2017","author":"Wang","year":"2017","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ali, W., Li, Y., Chen, Z., Raja, M.A.Z., Ahmed, N., and Chen, X. (2019). Application of Spherical-Radial Cubature Bayesian Filtering and Smoothing in Bearings Only Passive Target Tracking. Entropy, 21.","DOI":"10.3390\/e21111088"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"107581","DOI":"10.1016\/j.sigpro.2020.107581","article-title":"Underwater angle-only tracking with propagation delay and time-offset between observers","volume":"176","author":"Su","year":"2020","journal-title":"Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1388","DOI":"10.1049\/iet-rsn.2016.0594","article-title":"Maximum correntropy sparse Gauss\u2013Hermite quadrature filter and its application in tracking ballistic missile","volume":"11","author":"Qin","year":"2017","journal-title":"IET Radar Sonar Navig."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.automatica.2011.08.057","article-title":"Sparse-grid quadrature nonlinear filtering","volume":"48","author":"Jia","year":"2012","journal-title":"Automatica"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Hou, J., Yang, Y., and Gao, T. (2019). Variational Bayesian based adaptive shifted Rayleigh filter for bearings-only tracking in clutters. Sensors, 19.","DOI":"10.3390\/s19071512"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Zhang, Z., Fu, K., Sun, X., and Ren, W. (2019). Multiple target tracking based on multiple hypotheses tracking and modified ensemble Kalman filter in multi-sensor fusion. Sensors, 19.","DOI":"10.3390\/s19143118"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"016019","DOI":"10.1117\/1.JRS.12.016019","article-title":"Developing the fuzzy c-means clustering algorithm based on maximum entropy for multitarget tracking in a cluttered environment","volume":"12","author":"Chen","year":"2018","journal-title":"J. Appl. Remote Sens."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1016\/j.cjph.2019.04.015","article-title":"Design of computational intelligent procedure for thermal analysis of porous fin model","volume":"59","author":"Ahmad","year":"2019","journal-title":"Chin. J. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1080\/09540091.2015.1092499","article-title":"Nature-inspired computing approach for solving non-linear singular Emden\u2013Fowler problem arising in electromagnetic theory","volume":"27","author":"Khan","year":"2015","journal-title":"Connect. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1140\/epjp\/i2018-12013-3","article-title":"Neuro-evolutionary computing paradigm for Painlev\u00e9 equation-II in nonlinear optics","volume":"133","author":"Ahmad","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.jtice.2018.05.046","article-title":"Design of neuro-computing paradigms for nonlinear nanofluidic systems of MHD Jeffery\u2013Hamel flow","volume":"91","author":"Mehmood","year":"2018","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1140\/epjp\/i2018-12153-4","article-title":"A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head","volume":"133","author":"Raja","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.matcom.2020.01.005","article-title":"Novel design of Morlet wavelet neural network for solving second order Lane\u2013Emden equation","volume":"172","author":"Sabir","year":"2020","journal-title":"Math. Comput. Simul."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.aej.2019.12.011","article-title":"Neuro-fuzzy modeling and prediction of summer precipitation with application to different meteorological stations","volume":"59","author":"Bukhari","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1016\/j.asoc.2018.07.023","article-title":"Bio-inspired computational heuristics for Sisko fluid flow and heat transfer models","volume":"71","author":"Raja","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1007\/s00521-016-2400-y","article-title":"Neural network methods to solve the Lane\u2013Emden type equations arising in thermodynamic studies of the spherical gas cloud model","volume":"28","author":"Ahmad","year":"2017","journal-title":"Neural Comput. Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/j.neucom.2016.09.032","article-title":"An intelligent computing technique to analyze the vibrational dynamics of rotating electrical machine","volume":"219","author":"Raja","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/s00521-016-2547-6","article-title":"Intelligent computing to solve fifth-order boundary value problem arising in induction motor models","volume":"29","author":"Ahmad","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/j.asoc.2018.01.009","article-title":"Neuro-heuristics for nonlinear singular Thomas-Fermi systems","volume":"65","author":"Sabir","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s00521-016-2530-2","article-title":"Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch\u2019s problem arising in plasma physics","volume":"29","author":"Raja","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1866","DOI":"10.1186\/s40064-016-3517-2","article-title":"Bio-inspired computational heuristics to study Lane\u2013Emden systems arising in astrophysics model","volume":"5","author":"Ahmad","year":"2016","journal-title":"SpringerPlus"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2319","DOI":"10.1007\/s00382-018-4252-x","article-title":"Rainfall prediction methodology with binary multilayer perceptron neural networks","volume":"52","author":"Esteves","year":"2019","journal-title":"Clim. Dyn."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1509","DOI":"10.1016\/j.aej.2017.03.050","article-title":"Prediction of the true harmonic current contribution of nonlinear loads using NARX neural network","volume":"57","author":"Hatata","year":"2018","journal-title":"Alex. Eng. J."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Solanki, V., and Joshi, M. (2018, January 18\u201319). Energy Efficient NARX Model for Target Tracking in Wireless Sensor Network. Proceedings of the 2018 3rd IEEE International Conference on Recent Trends in Electronics, Information & Communication Technology (RTEICT), Bangalore, India.","DOI":"10.1109\/RTEICT42901.2018.9012333"},{"key":"ref_45","unstructured":"Peng, J., and Tang, Q. (2019). Application of NARX Dynamic Neural Network in Quantitative Investment Forecasting System. International Symposium on Intelligence Computation and Applications, Proceedings of the 11th International Symposium, ISICA 2019, Guangzhou, China, 16\u201317 November 2019, Springer."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1016\/j.engappai.2009.04.002","article-title":"PForecasting peak air pollution levels using NARX models","volume":"22","author":"Pisoni","year":"2009","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1080\/17445302.2019.1661619","article-title":"Application of NARX neural network for predicting marine engine performance parameters","volume":"15","author":"Raptodimos","year":"2020","journal-title":"Ships Offshore Struct."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"130922","DOI":"10.1109\/ACCESS.2020.3007848","article-title":"NARX prediction-based parameters online tuning method of intelligent PID system","volume":"8","author":"Liu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Rangel, E., Cadenas, E., Campos-Amezcua, R., and Tena, J.L. (2020). Enhanced Prediction of Solar Radiation Using NARX Models with Corrected Input Vectors. Energies, 13.","DOI":"10.3390\/en13102576"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"106982","DOI":"10.1016\/j.anucene.2019.106982","article-title":"Fractional-order particle swarm based multi-objective PWR core loading pattern optimization","volume":"135","author":"Zameer","year":"2020","journal-title":"Ann. Nucl. Energy"},{"key":"ref_51","first-page":"1","article-title":"Design of fractional swarming strategy for solution of optimal reactive power dispatch","volume":"11","author":"Muhammad","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"3681","DOI":"10.1007\/s00521-017-3318-8","article-title":"Novel application of FO-DPSO for 2-D parameter estimation of electromagnetic plane waves","volume":"31","author":"Akbar","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s00521-017-2991-y","article-title":"Fractional neural network models for nonlinear Riccati systems","volume":"31","author":"Lodhi","year":"2019","journal-title":"Neural Comput. Appl."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/5\/550\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:55:39Z","timestamp":1760162139000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/5\/550"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,29]]},"references-count":53,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["e23050550"],"URL":"https:\/\/doi.org\/10.3390\/e23050550","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,29]]}}}