{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T18:55:47Z","timestamp":1780599347554,"version":"3.54.1"},"reference-count":31,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T00:00:00Z","timestamp":1754092800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Laboratory on Ship Vibration and Noise","award":["JCKY2023207CI03"],"award-info":[{"award-number":["JCKY2023207CI03"]}]},{"name":"National Key Laboratory on Ship Vibration and Noise","award":["202350F030"],"award-info":[{"award-number":["202350F030"]}]},{"name":"university independent research and development project","award":["JCKY2023207CI03"],"award-info":[{"award-number":["JCKY2023207CI03"]}]},{"name":"university independent research and development project","award":["202350F030"],"award-info":[{"award-number":["202350F030"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Based on the principle of symmetry, we propose a variable step-size FxLMS algorithm with double nonlinear functions cooperative coupling (DNVSS-FxLMS), aiming to optimize the contradiction between convergence rate and steady-state error in the active pressure pulsation control system of hydraulic systems. The algorithm innovatively couples two types of nonlinear mechanisms (rational-fractional and exponential-function-based), constructing a refined error-step mapping relationship to achieve a balance between rapid convergence and low steady-state error. Simulation experiments were conducted considering the complex time-varying operating environment of a simulation-based hydraulic system. The results demonstrate that, when the system undergoes unstable random changes, the DNVSS-FxLMS algorithm converges at least twice as fast as traditional and existing variable step size algorithms, while reducing steady-state error by 2\u20135 dB. The proposed DNVSS-FxLMS algorithm exhibits significant advantages in convergence rate, steady-state error reduction, and tracking capability, providing a highly efficient and robust solution for real-time active control of hydraulic system pressure pulsation under complex operating conditions.<\/jats:p>","DOI":"10.3390\/sym17081222","type":"journal-article","created":{"date-parts":[[2025,8,4]],"date-time":"2025-08-04T09:41:17Z","timestamp":1754300477000},"page":"1222","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Variable Step-Size FxLMS Algorithm Based on Cooperative Coupling of Double Nonlinear Functions"],"prefix":"10.3390","volume":"17","author":[{"given":"Jialong","family":"Wang","sequence":"first","affiliation":[{"name":"Institute of Noise & Vibration, Naval University of Engineering, Wuhan 430033, China"},{"name":"Nation Key Laboratory on Ship Vibration & Noise, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jian","family":"Liao","sequence":"additional","affiliation":[{"name":"Institute of Noise & Vibration, Naval University of Engineering, Wuhan 430033, China"},{"name":"Nation Key Laboratory on Ship Vibration & Noise, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lin","family":"He","sequence":"additional","affiliation":[{"name":"Institute of Noise & Vibration, Naval University of Engineering, Wuhan 430033, China"},{"name":"Nation Key Laboratory on Ship Vibration & Noise, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaopeng","family":"Tan","sequence":"additional","affiliation":[{"name":"Institute of Noise & Vibration, Naval University of Engineering, Wuhan 430033, China"},{"name":"Nation Key Laboratory on Ship Vibration & Noise, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zongbin","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Noise & Vibration, Naval University of Engineering, Wuhan 430033, China"},{"name":"Nation Key Laboratory on Ship Vibration & Noise, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1186\/s10033-021-00532-z","article-title":"Research Status, Critical Technologies, and Development Trends of Hydraulic Pressure Pulsation Attenuator","volume":"34","author":"Wang","year":"2021","journal-title":"Chin. J. Mech. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"111375","DOI":"10.1016\/j.ymssp.2024.111375","article-title":"Active Control Method for Fluid-Borne Noise in Aerospace Fluid Systems of Variable Operation Statuses","volume":"214","author":"Wu","year":"2024","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Pan, M., Hillis, A., and Johnston, N. (2014, January 8\u201311). Active control of fluid-bome noise in hydraulic systems using in-series and by-pass structures. Proceedings of the 2014 UKACC International Conference on Control (CONTROL), Loughborough, UK.","DOI":"10.1109\/CONTROL.2014.6915166"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1282","DOI":"10.1109\/TASLP.2020.2982030","article-title":"Robust Generalized Maximum Correntropy Criterion Algorithms for Active Noise Control","volume":"28","author":"Zhu","year":"2020","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"091006","DOI":"10.1115\/1.4049734","article-title":"Novel Integrated Active and Passive Control of Fluid-Borne Noise in Hydraulic Systems","volume":"143","author":"Pan","year":"2021","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1109\/78.558478","article-title":"A Robust Variable Step Size LMS-Type Algorithm: Analysis and Simulations","volume":"45","author":"Aboulnasr","year":"2002","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3121","DOI":"10.1007\/s00034-013-9598-z","article-title":"Variable Step size LMS Algorithm Based on Function Control","volume":"32","author":"Li","year":"2013","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_8","first-page":"24","article-title":"Review and Comparison of Variable Step Size LMS Algorithms","volume":"21","author":"Bismor","year":"2016","journal-title":"Int. J. Acoust. Vib."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1109\/TCS.1986.1085982","article-title":"Adaptive Filters with Individual Adaptation of Parameters","volume":"33","author":"Mikhail","year":"1986","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1109\/78.143435","article-title":"A Variable Step size LMS Algorithm","volume":"40","author":"Kwong","year":"1992","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_11","first-page":"145","article-title":"A New Variable Step Size LMS Method and Its Application in DOA Estimation of OFDMA Signals","volume":"36","author":"Zhao","year":"2020","journal-title":"J. Southeast Univ."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"He, D., Wang, M., Han, Y., and Hui, S. (, January 28\u201330). Variable Step size LMS Adaptive Algorithm Based on Exponential Function. Proceedings of the 2019 IEEE International Conference on Information Communication and Signal Processing (ICICSP), Weihai, China.","DOI":"10.1109\/ICICSP48821.2019.8958492"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1186\/s13634-020-00672-9","article-title":"A Variable Step Size Diffusion LMS Algorithm with a Quotient Form","volume":"2020","author":"Zerguine","year":"2020","journal-title":"EURASIP J. Adv. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4885","DOI":"10.1109\/ACCESS.2019.2961162","article-title":"A Class of Diffusion Zero Attracting Stochastic Gradient Algorithms with Exponentiated Error Cost Functions","volume":"8","author":"Luo","year":"2019","journal-title":"IEEE Access"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Yu, Y., and Zhao, H. (2013, January 5\u20138). An Improved Variable Step Size NLMS Algorithm Based on a Versiera Function. Proceedings of the 2013 IEEE International Conference on Signal Processing, Communication and Computing (ICSPCC), Kunming, China.","DOI":"10.1109\/ICSPCC.2013.6663959"},{"key":"ref_16","first-page":"335","article-title":"A Variable Step Size Normalized Adaptive Filtering Algorithm Based on a Versiera-Like Function","volume":"43","author":"Huo","year":"2021","journal-title":"J. Electron. Inf. Technol."},{"key":"ref_17","first-page":"1967","article-title":"A Robust Family of Algorithms for Adaptive Filtering Based on the Arctangent Framework","volume":"69","author":"Kumar","year":"2022","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3040","DOI":"10.1007\/s00034-015-0175-5","article-title":"A Normalized Least Mean Square Algorithm Based on the Arctangent Cost Function Robust Against Impulsive Interference","volume":"35","author":"Zeng","year":"2016","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s11760-023-02729-2","article-title":"A Filter Proportionate LMS Algorithm Based on the Arctangent Framework for Sparse System Identification","volume":"18","author":"Rosalin","year":"2024","journal-title":"Signal Image Video Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.sigpro.2019.06.007","article-title":"Spline Adaptive Filter with Arctangent-Momentum Strategy for Nonlinear System Identification","volume":"164","author":"Yang","year":"2019","journal-title":"Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/j.ymssp.2014.10.002","article-title":"Active Control of Impulsive Noise with Symmetric \u03b1-Stable Distribution Based on an Improved Step Size Normalized Adaptive Algorithm","volume":"56","author":"Zhou","year":"2015","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_22","first-page":"116","article-title":"An Improved Variable Step Size LMS Algorithm Based on Hyperbolic Tangent Function","volume":"41","author":"Zhang","year":"2020","journal-title":"J. Commun."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1206","DOI":"10.1109\/LAWP.2020.2995244","article-title":"Fast and Robust Variable Step Size LMS Algorithm for Adaptive Beamforming","volume":"19","author":"Jalal","year":"2020","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_24","first-page":"7","article-title":"An Improved Variable Step Size LMS Adaptive Filtering Algorithm","volume":"26","author":"Zhang","year":"2021","journal-title":"J. Xi\u2019an Univ. Posts Telecommun."},{"key":"ref_25","first-page":"1306","article-title":"An Improved Variable Step Size LMS Algorithm for Vehicle Interior Noise Control","volume":"44","author":"Qian","year":"2021","journal-title":"J. Hefei Univ. Technol. (Nat. Sci.)"},{"key":"ref_26","first-page":"3294674","article-title":"Improved Variable Step size Least Mean Square Algorithm for Pipeline Noise","volume":"2022","author":"Zhang","year":"2022","journal-title":"Sci. Program."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4415","DOI":"10.1007\/s00034-023-02303-8","article-title":"Variable Step Size LMS Algorithm Based on Hyperbolic Tangent Function","volume":"42","author":"Li","year":"2023","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2464","DOI":"10.1177\/10775463241260110","article-title":"Research on Active Vibration Control of Blade Using Variable Step Size Filtered-x Least Mean Square Algorithm","volume":"31","author":"Jiang","year":"2024","journal-title":"J. Vib. Control"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1109\/79.248551","article-title":"Active Noise Control","volume":"10","author":"Elliott","year":"1993","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Wang, C., He, L., Li, Y., Zhang, X., and Zuo, L. (2016, January 9\u201311). An Anti-Interference Variable-Step Adaptive Algorithm and Its Application in Active Vibration Control. Proceedings of the 2016 IEEE\/OES China Ocean Acoustics (COA), Harbin, China.","DOI":"10.1109\/COA.2016.7535766"},{"key":"ref_31","unstructured":"Sayed, A.H. (2011). Adaptive Filters, John Wiley & Sons."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1222\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:21:49Z","timestamp":1760034109000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1222"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,2]]},"references-count":31,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081222"],"URL":"https:\/\/doi.org\/10.3390\/sym17081222","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,2]]}}}