{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T08:48:53Z","timestamp":1767084533291,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2018,5,11]],"date-time":"2018-05-11T00:00:00Z","timestamp":1525996800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Basic Research and Development (973) Program of China","award":["2014CB049404"],"award-info":[{"award-number":["2014CB049404"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51575511"],"award-info":[{"award-number":["51575511"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018621","name":"Program for Changjiang Scholars and Innovative Research Team in University","doi-asserted-by":"publisher","award":["IRT_16R68"],"award-info":[{"award-number":["IRT_16R68"]}],"id":[{"id":"10.13039\/501100018621","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Fundamental Research Funds for the Central Universities in China","award":["2015XKMS025"],"award-info":[{"award-number":["2015XKMS025"]}]},{"DOI":"10.13039\/501100012246","name":"Priority Academic Program Development of Jiangsu Higher Education Institutions","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100012246","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Qing Lan Project in Jiangsu province"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, a nonlinear adaptive fuzzy backstepping controller combined with an adaptive backstepping controller and an adaptive fuzzy controller is proposed for real-time tracking control of an electro-hydraulic force loading system. Firstly, a nonlinear dynamic model for the electro-hydraulic force loading system is built with consideration of parameter uncertainties and external disturbances. Then, the adaptive backstepping controller is employed to obtain desired control output for the force loading control system considering parameter uncertainties and external disturbances. Furthermore, an adaptive fuzzy control scheme is designed to adjust uncertain control parameters based on adaptive fuzzy system to cope with the chattering condition that results from the overwhelming external disturbances. The stability of the overall system with the proposed control algorithm can be proved by Lyapunov stability theory. Finally, an electro-hydraulic force loading experimental system with xPC rapid prototyping technology is carried out to verify the effectiveness of the proposed nonlinear adaptive fuzzy backstepping controller. Experimental results verify that the proposed control method exhibit excellent performances on force loading tracking control of the electro-hydraulic force loading experimental system compared with a conventional proportional-integral-derivative (PID) controller with velocity feedforward and adaptive backstepping control schemes.<\/jats:p>","DOI":"10.3390\/sym10050155","type":"journal-article","created":{"date-parts":[[2018,5,14]],"date-time":"2018-05-14T02:57:20Z","timestamp":1526266640000},"page":"155","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Force Loading Tracking Control of an Electro-Hydraulic Actuator Based on a Nonlinear Adaptive Fuzzy Backstepping Control Scheme"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6657-6411","authenticated-orcid":false,"given":"Xiang","family":"Li","sequence":"first","affiliation":[{"name":"School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhen-Cai","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guang-Chao","family":"Rui","sequence":"additional","affiliation":[{"name":"Zhengzhou Institute of Mechanical and Electrical Engineering, Zhengzhou 450015, China"},{"name":"Henan Key Laboratory of Underwater Intelligent Equipment, Zhengzhou 450000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dong","family":"Cheng","sequence":"additional","affiliation":[{"name":"Zhengzhou Institute of Mechanical and Electrical Engineering, Zhengzhou 450015, China"},{"name":"Henan Key Laboratory of Underwater Intelligent Equipment, Zhengzhou 450000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gang","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,5,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1109\/TCST.2007.908195","article-title":"Nonlinear adaptive robust control of single-rod electro-hydraulic actuator with unknown nonlinear parameters","volume":"16","author":"Guan","year":"2008","journal-title":"IEEE Trans. Control. Syst. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1111\/j.1747-1567.2012.00837.x","article-title":"Development of a versatile hybrid testing system for seismic experimentation","volume":"38","author":"Shao","year":"2014","journal-title":"Exp. Tech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1128","DOI":"10.1016\/j.conengprac.2013.03.007","article-title":"Adaptive feed-forward compensation for hybrid control with acceleration time waveform replication on electro-hydraulic shaking table","volume":"21","author":"Shen","year":"2013","journal-title":"Control Eng. Pract."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1002\/eqe.2207","article-title":"Effective force testing using a robust loop shaping controller","volume":"42","author":"Nakata","year":"2013","journal-title":"Earthq. Eng. Struct. D"},{"key":"ref_5","first-page":"717","article-title":"Robust electrohydraulic force control","volume":"221","author":"Plummer","year":"2007","journal-title":"Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"539","DOI":"10.12989\/eas.2014.6.5.539","article-title":"Investigation on the performance of the six DOF CGS, Algeria, shaking table","volume":"6","author":"Aknouche","year":"2014","journal-title":"Earthq. Struct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.engfailanal.2013.01.009","article-title":"Fatigue strength evaluation of a bogie frame for urban maglev train with fatigue test on full-scale test rig","volume":"31","author":"Han","year":"2013","journal-title":"Eng. Fail. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1951","DOI":"10.1177\/0954406215584810","article-title":"Design and experimental evaluation of feedforward controller integrating filtered-x LMS algorithm with applications to electro-hydraulic force control systems","volume":"230","author":"Tang","year":"2015","journal-title":"Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Alleyne, A., Rui, L., and Wright, H. (1998, January 21\u201326). On the limitations of force tracking control for hydraulic active suspensions. Proceedings of the American Control Conference, Philadelphia, PA, USA.","DOI":"10.1109\/ACC.1998.694625"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/j.mechatronics.2008.07.007","article-title":"Force control for hydraulic load simulator using self-tuning grey predictor\u2014Fuzzy PID","volume":"19","author":"Truong","year":"2009","journal-title":"Mechatronics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s12239-008-0005-x","article-title":"Design of a forced control system for a dynamic road simulator using QFT","volume":"9","author":"Kim","year":"2008","journal-title":"Int. J. Automot. Technol."},{"key":"ref_12","first-page":"161","article-title":"Feel force control incorporating velocity feedforward and inverse model observer for control loading system of flight simulator","volume":"227","author":"Zhao","year":"2013","journal-title":"Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1177\/0142331216651326","article-title":"Force tracking control of an electro-hydraulic control loading system on a flight simulator using inverse model control and a damping compensator","volume":"40","author":"Zhao","year":"2016","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.mechatronics.2015.10.009","article-title":"Real-time electro-hydraulic hybrid system for structural testing subjected to vibration and force loading","volume":"33","author":"Shen","year":"2015","journal-title":"Mechatronics"},{"key":"ref_15","first-page":"1","article-title":"Real-time tracking control of electro-hydraulic force servo systems using offline feedback control and adaptive control","volume":"11","author":"Shen","year":"2016","journal-title":"ISA Tran."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3904","DOI":"10.1016\/j.cnsns.2011.02.031","article-title":"Robust sliding-mode backstepping design for synchronization control of cross-strict feedback hyperchaotic systems with unmatched uncertainties","volume":"16","author":"Li","year":"2011","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"895","DOI":"10.1016\/j.conengprac.2009.02.011","article-title":"Observer-based backstepping force control of an electro hydraulic actuator","volume":"17","author":"Prut","year":"2009","journal-title":"Control Eng. Pract."},{"key":"ref_18","first-page":"728","article-title":"High dynamic feedback linearization control of hydraulic actuators with backstepping","volume":"229","author":"Yao","year":"2015","journal-title":"Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1016\/j.chaos.2005.05.001","article-title":"Synchronization of Genesio chaotic system via backstepping approach","volume":"27","author":"Park","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"6285","DOI":"10.1109\/TIE.2014.2304912","article-title":"Extended-State-Observer-Based Output Feedback Nonlinear Robust Control of Hydraulic Systems with Backstepping","volume":"61","author":"Yao","year":"2014","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1482","DOI":"10.1109\/TMECH.2014.2369454","article-title":"\u03bc-synthesis based adaptive robust control of linear motor driven stages with high-frequency dynamics: A case study with comparative experiments","volume":"20","author":"Chen","year":"2015","journal-title":"IEEE-ASME Trans. Mechatron."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1109\/TMECH.2014.2319355","article-title":"Vibration Isolation for Active Suspensions with Performance Constraints and Actuator Saturation","volume":"20","author":"Sun","year":"2015","journal-title":"IEEE-ASME Trans. Mechatron."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.mechatronics.2013.11.005","article-title":"Nonlinear adaptive torque control of electro-hydraulic load system with external active motion disturbance","volume":"24","author":"Wang","year":"2014","journal-title":"Mechatronics"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1499","DOI":"10.1007\/s12206-014-0137-z","article-title":"Nonlinear adaptive robust backstepping force control of hydraulic load simulator: Theory and experiments","volume":"28","author":"Yao","year":"2014","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.ifacol.2015.12.024","article-title":"Adaptive Robust Backstepping Force Control of 1-DOF Joint Exoskeleton for Human Performance Augmentation","volume":"48","author":"Chen","year":"2016","journal-title":"IFAC-PapersOnLine"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1016\/j.actaastro.2016.04.015","article-title":"Robust adaptive backstepping control for reentry reusable launch vehicles","volume":"126","author":"Wang","year":"2016","journal-title":"Acta Astronaut."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1109\/TMECH.2013.2252360","article-title":"High-accuracy tracking control of hydraulic rotary actuators with modelling uncertainties","volume":"19","author":"Yao","year":"2014","journal-title":"IEEE-ASME Trans. Mechatron."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4415","DOI":"10.1016\/j.jfranklin.2014.06.002","article-title":"High dynamic adaptive robust control of load emulator with output feedback signal","volume":"351","author":"Yao","year":"2014","journal-title":"J. Frankl. Inst."},{"key":"ref_29","first-page":"751","article-title":"Fuzzy integral sliding mode based on backstepping control synthesis for an autonomous helicopter","volume":"227","author":"Kadda","year":"2012","journal-title":"Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng."},{"key":"ref_30","first-page":"1044","article-title":"High-performance motion control of the hydraulic press based on an extended fuzzy disturbance observer","volume":"230","author":"Wei","year":"2016","journal-title":"Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng."},{"key":"ref_31","first-page":"1","article-title":"Adaptive complementary fuzzy self-recurrent wavelet neural network controller for the electric load simulator system","volume":"8","author":"Wang","year":"2016","journal-title":"Adv. Mech. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.neucom.2017.01.011","article-title":"Adaptive fuzzy tracking control for nonlinear strict-feedback systems with unmodeled dynamics via backstepping technique","volume":"235","author":"Zhang","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/j.neucom.2016.12.030","article-title":"A novel meta-cognitive fuzzy-neural model with backstepping strategy for adaptive control of uncertain nonlinear systems","volume":"230","author":"Rong","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.ifacol.2016.07.1005","article-title":"An Adaptive Resonance Regulator for an Actuator using Periodic Signals in Camless Engine Systems","volume":"49","author":"Mercorelli","year":"2016","journal-title":"IFAC-PapersOnLine"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3413","DOI":"10.1109\/TIE.2016.2606091","article-title":"An Adaptive Resonance Regulator Design for Motion Control of Intake Valves in Camless Engine Systems","volume":"64","author":"Mercorelli","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_36","unstructured":"Magdi, S. (2018). Mahmoud, Fuzzy Control, Estimation and Diagnosis, Springer International Publishing."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/5\/155\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:03:57Z","timestamp":1760195037000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/5\/155"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,11]]},"references-count":36,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2018,5]]}},"alternative-id":["sym10050155"],"URL":"https:\/\/doi.org\/10.3390\/sym10050155","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,5,11]]}}}