{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T13:37:16Z","timestamp":1776260236659,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T00:00:00Z","timestamp":1750204800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T00:00:00Z","timestamp":1750204800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100018777","name":"Nile University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100018777","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Complex Intell. Syst."],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>This paper presents a novel Adaptive Relaxed Sliding Mode Control (ARSMC) framework for nonlinear systems, enhanced by an actor-critic reinforcement learning algorithm. ARSMC addresses the limitations of traditional sliding mode control by providing a chattering-free control signal, minimizing steady-state error, and effectively managing unbounded uncertainties and disturbances. The algorithm\u2019s stability is rigorously proven via the Lyapunov theorem. Simulations on an electronic throttle system demonstrate ARSMC\u2019s superior performance over traditional and relaxed sliding mode controllers under various disturbances and uncertainties. Performance metrics, including integral absolute error (IAE) and integral squared error (ISE), highlight ARSMC\u2019s improved control precision and robustness. The results also showcase enhancements in the rise time and maximum overshoot, further validating its effectiveness. The proposed ARSMC framework marks a significant advancement in nonlinear control, providing a robust solution for systems with complex dynamics and parameter uncertainties. It improves upon traditional sliding mode controllers by minimizing steady-state error and chattering while maintaining the ability to handle unbounded external disturbances.<\/jats:p>","DOI":"10.1007\/s40747-025-01978-7","type":"journal-article","created":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:38:14Z","timestamp":1750221494000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Reinforcement learning-enhanced adaptive sliding mode control for nonlinear systems"],"prefix":"10.1007","volume":"11","author":[{"given":"Omnia","family":"Youssef","sequence":"first","affiliation":[]},{"given":"Mohamed","family":"Wafa","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3639-3665","authenticated-orcid":false,"given":"Raafat","family":"Shalaby","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,6,18]]},"reference":[{"key":"1978_CR1","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1515\/nleng-2016-0077","volume":"6","author":"J Iqbal","year":"2017","unstructured":"Iqbal J, Ullah M, Khan S, Khelifa B, \u0106ukovi\u0107 S (2017) Nonlinear control systems-a brief overview of historical and recent advances. Nonlinear Eng 6:301\u2013312. https:\/\/doi.org\/10.1515\/nleng-2016-0077","journal-title":"Nonlinear Eng"},{"key":"1978_CR2","doi-asserted-by":"crossref","unstructured":"Hoang Q, Tuan L, Thong T, Gam N, Nghiem L, Cuong N (2023) Kinematic-based nonlinear control for an omni-directional robot. 2023 International Conference on Control, Robotics and Informatics (ICCRI). pp. 30\u201333","DOI":"10.1109\/ICCRI58865.2023.00013"},{"key":"1978_CR3","doi-asserted-by":"publisher","first-page":"514","DOI":"10.3390\/e25030514","volume":"25","author":"I Hameed","year":"2023","unstructured":"Hameed I, Abbud L, Abdulsaheb J, Azar A, Mezher M, Jawad A, Abdul-Adheem W, Ibraheem I, Kamal N (2023) A new nonlinear dynamic speed controller for a differential drive mobile robot. Entropy 25:514","journal-title":"Entropy"},{"key":"1978_CR4","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-020-02829-0","author":"M Aghababa","year":"2020","unstructured":"Aghababa M, Saif M (2020) Adaptive control realization for canonic Caputo fractional-order systems with actuator nonlinearity: application to mechatronic devices. Adv Differ Equ. https:\/\/doi.org\/10.1186\/s13662-020-02829-0","journal-title":"Adv Differ Equ"},{"key":"1978_CR5","doi-asserted-by":"publisher","DOI":"10.1155\/2022\/2504907","author":"Y Gurefe","year":"2022","unstructured":"Gurefe Y, Pandir Y, Akturk T (2022) On the nonlinear mathematical model representing the coriolis effect. Math Problems Eng. https:\/\/doi.org\/10.1155\/2022\/2504907","journal-title":"Math Problems Eng"},{"key":"1978_CR6","first-page":"16950","volume":"13","author":"S Gomathy","year":"2018","unstructured":"Gomathy S, Maheswari M (2018) A survey on control techniques to stabilize and control the non-linear system. Int J Appl Eng Res 13:16950\u201316953","journal-title":"Int J Appl Eng Res"},{"key":"1978_CR7","unstructured":"Koelewijn P (2023) Analysis and control of nonlinear systems with stability and performance guarantees: a linear parameter-varying approach. (Eindhoven University of Technology), Proefschrift"},{"key":"1978_CR8","unstructured":"Index H. (2019) International Journal of Scientific and Technology Research. Development. pp. Q4"},{"key":"1978_CR9","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1016\/j.ast.2016.01.017","volume":"51","author":"A Nair","year":"2016","unstructured":"Nair A, Selvaganesan N, Lalithambika V (2016) Lyapunov based PD\/PID in model reference adaptive control for satellite launch vehicle systems. Aerosp Sci Technol 51:70\u201377","journal-title":"Aerosp Sci Technol"},{"key":"1978_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.oceaneng.2021.109010","volume":"232","author":"P Gong","year":"2021","unstructured":"Gong P, Yan Z, Zhang W, Tang J (2021) Lyapunov-based model predictive control trajectory tracking for an autonomous underwater vehicle with external disturbances. Ocean Eng 232:109010","journal-title":"Ocean Eng"},{"key":"1978_CR11","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2021.105956","volume":"103","author":"A Tocino","year":"2021","unstructured":"Tocino A, Mart\u00edn del Rey A (2021) Local stochastic stability of SIRS models without Lyapunov functions. Commun Nonlinear Sci Numer Simul 103:105956","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"1978_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.isatra.2019.07.016","volume":"97","author":"A Haruna","year":"2020","unstructured":"Haruna A, Mohamed Z, Efe M, Basri M (2020) Improved integral backstepping control of variable speed motion systems with application to a laboratory helicopter. ISA Trans 97:1\u201313","journal-title":"ISA Trans"},{"key":"1978_CR13","doi-asserted-by":"publisher","first-page":"2304","DOI":"10.1109\/TMECH.2019.2930211","volume":"24","author":"G Yu","year":"2019","unstructured":"Yu G, Cabecinhas D, Cunha R, Silvestre C (2019) Nonlinear backstepping control of a quadrotor-slung load system. IEEE\/ASME Trans Mechatron 24:2304\u20132315","journal-title":"IEEE\/ASME Trans Mechatron"},{"key":"1978_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2021.107769","volume":"158","author":"W Deng","year":"2021","unstructured":"Deng W, Yao J, Wang Y, Yang X, Chen J (2021) Output feedback backstepping control of hydraulic actuators with valve dynamics compensation. Mech Syst Signal Process 158:107769","journal-title":"Mech Syst Signal Process"},{"key":"1978_CR15","first-page":"1","volume":"65","author":"D Bhattacharyya","year":"2018","unstructured":"Bhattacharyya D, Padhee S, Pati K (2018) Modeling of DC-DC converter using exact feedback linearization method: a discussion. IETE J Res 65:1\u201312","journal-title":"IETE J Res"},{"key":"1978_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10846-020-01265-2","volume":"101","author":"L Martins","year":"2021","unstructured":"Martins L, Cardeira C, Oliveira P (2021) Feedback linearization with zero dynamics stabilization for quadrotor control. J Intell Robot Syst 101:1\u201317","journal-title":"J Intell Robot Syst"},{"key":"1978_CR17","doi-asserted-by":"publisher","first-page":"6286","DOI":"10.1016\/j.jfranklin.2018.06.014","volume":"355","author":"M Hamdy","year":"2018","unstructured":"Hamdy M, Shalaby R, Sallam M (2018) A hybrid partial feedback linearization and deadbeat control scheme for a nonlinear gantry crane. J Frankl Instit 355:6286\u20136299","journal-title":"J Frankl Instit"},{"key":"1978_CR18","unstructured":"Alyoussef F, Kaya I (2019) A review on nonlinear control approaches: sliding mode control, back-stepping control, and feedback linearization control"},{"key":"1978_CR19","unstructured":"Zhao P, Snyder S, Hovakimyana N, Cao C (2020) Robust adaptive control of linear parameter-varying systems with unmatched uncertainties. ArXiv Preprint ArXiv:2010. 04600"},{"key":"1978_CR20","doi-asserted-by":"publisher","first-page":"222077","DOI":"10.1109\/ACCESS.2020.3043453","volume":"8","author":"X Dang","year":"2020","unstructured":"Dang X, Do V, Nguyen X (2020) Robust adaptive fuzzy control using genetic algorithm for dynamic positioning system. IEEE Access 8:222077\u2013222092","journal-title":"IEEE Access"},{"key":"1978_CR21","first-page":"1","volume":"1","author":"N Abo Mosali","year":"2021","unstructured":"Abo Mosali N, Shamsudin S (2021) Issues of designing a model adaptive controller without a state observer. Progress Aerosp Aviat Technol 1:1\u20136","journal-title":"Progress Aerosp Aviat Technol"},{"key":"1978_CR22","doi-asserted-by":"publisher","first-page":"2965","DOI":"10.1109\/TAC.2016.2605043","volume":"62","author":"G Incremona","year":"2017","unstructured":"Incremona G, Rubagotti M, Ferrara A (2017) Sliding mode control of constrained nonlinear systems. IEEE Trans Autom Control 62:2965\u20132972","journal-title":"IEEE Trans Autom Control"},{"key":"1978_CR23","doi-asserted-by":"publisher","first-page":"3275","DOI":"10.1109\/TIE.2009.2027531","volume":"56","author":"X Yu","year":"2009","unstructured":"Yu X, Kaynak O (2009) Sliding-mode control with soft computing: a survey. IEEE Trans Industr Electron 56:3275\u20133285","journal-title":"IEEE Trans Industr Electron"},{"key":"1978_CR24","doi-asserted-by":"publisher","first-page":"10244","DOI":"10.1016\/j.jfranklin.2020.06.018","volume":"357","author":"V Utkin","year":"2020","unstructured":"Utkin V, Poznyak A, Orlov Y, Polyakov A (2020) Conventional and high order sliding mode control. J Franklin Inst 357:10244\u201310261","journal-title":"J Franklin Inst"},{"key":"1978_CR25","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.mechatronics.2018.04.005","volume":"52","author":"Y Pan","year":"2018","unstructured":"Pan Y, Li X, Wang H, Yu H (2018) Continuous sliding mode control of compliant robot arms: a singularly perturbed approach. Mechatronics 52:127\u2013134","journal-title":"Mechatronics"},{"key":"1978_CR26","doi-asserted-by":"publisher","first-page":"172988142098708","DOI":"10.1177\/1729881420987082","volume":"18","author":"A Azzabi","year":"2021","unstructured":"Azzabi A, Nouri K (2021) Design of a robust tracking controller for a nonholonomic mobile robot based on sliding mode with adaptive gain. Int J Adv Rob Syst 18:1729881420987082","journal-title":"Int J Adv Rob Syst"},{"key":"1978_CR27","doi-asserted-by":"publisher","first-page":"101765","DOI":"10.1109\/ACCESS.2019.2931324","volume":"7","author":"F Zaihidee","year":"2019","unstructured":"Zaihidee F, Mekhilef S, Mubin M (2019) Application of fractional order sliding mode control for speed control of permanent magnet synchronous motor. IEEE Access 7:101765\u2013101774","journal-title":"IEEE Access"},{"key":"1978_CR28","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1016\/j.jprocont.2019.11.008","volume":"85","author":"M Herrera","year":"2020","unstructured":"Herrera M, Camacho O, Leiva H, Smith C (2020) An approach of dynamic sliding mode control for chemical processes. J Process Control 85:112\u2013120","journal-title":"J Process Control"},{"key":"1978_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3182\/20060607-3-IT-3902.00003","volume":"39","author":"V Utkin","year":"2006","unstructured":"Utkin V (2006) Chattering problem in sliding mode control systems. IFAC Proceed Vol 39:1\u20131","journal-title":"IFAC Proceed Vol"},{"key":"1978_CR30","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.cnsns.2014.12.015","volume":"24","author":"F Zhong","year":"2015","unstructured":"Zhong F, Li H, Zhong S, Zhong Q, Yin C (2015) An SOC estimation approach-based on adaptive sliding mode observer and fractional order equivalent circuit model for lithium-ion batteries. Commun Nonlinear Sci Numer Simul 24:127\u2013144","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"1978_CR31","doi-asserted-by":"publisher","first-page":"6831","DOI":"10.1016\/j.aej.2021.12.030","volume":"61","author":"N Mazhar","year":"2022","unstructured":"Mazhar N, Malik F, Raza A, Khan R (2022) Predefined-time control of nonlinear systems: a sigmoid function based sliding manifold design approach. Alex Eng J 61:6831\u20136841","journal-title":"Alex Eng J"},{"key":"1978_CR32","doi-asserted-by":"publisher","first-page":"934","DOI":"10.1177\/0020294020905044","volume":"53","author":"V Giap","year":"2020","unstructured":"Giap V, Huang S (2020) Effectiveness of fuzzy sliding mode control boundary layer based on uncertainty and disturbance compensator on suspension active magnetic bearing system. Measure And Control 53:934\u2013942","journal-title":"Measure And Control"},{"key":"1978_CR33","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1109\/OJIES.2020.3040412","volume":"2","author":"X Yu","year":"2021","unstructured":"Yu X, Feng Y, Man Z (2021) Terminal sliding mode control - an overview. IEEE Open J Ind Electron Soc 2:36\u201352","journal-title":"IEEE Open J Ind Electron Soc"},{"key":"1978_CR34","doi-asserted-by":"publisher","DOI":"10.1609\/aaai.v32i1.11694","author":"P Henderson","year":"2018","unstructured":"Henderson P, Islam R, Bachman P, Pineau J, Precup D, Meger D (2018) Deep reinforcement learning that matters. Proceed AAAI Conf Artif Intell. https:\/\/doi.org\/10.1609\/aaai.v32i1.11694","journal-title":"Proceed AAAI Conf Artif Intell"},{"key":"1978_CR35","doi-asserted-by":"publisher","first-page":"209320","DOI":"10.1109\/ACCESS.2020.3038605","volume":"8","author":"M Naeem","year":"2020","unstructured":"Naeem M, Rizvi S, Coronato A (2020) A gentle introduction to reinforcement learning and its application in different fields. IEEE Access 8:209320\u2013209344","journal-title":"IEEE Access"},{"key":"1978_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3301273","volume":"3","author":"W Koch","year":"2019","unstructured":"Koch W, Mancuso R, West R, Bestavros A (2019) Reinforcement learning for UAV attitude control. ACM Trans Cyber-Phys Syst 3:1\u201321","journal-title":"ACM Trans Cyber-Phys Syst"},{"key":"1978_CR37","doi-asserted-by":"publisher","unstructured":"Z. Guo, Y. Shen, C. Chakraborty, F. Alblehai and K. Yu, \"Industrial 6G-IoT and Machine-Learning-Supported Intelligent Sensing Framework for Indicator Control Strategy in Sewage Treatment Process,\" in IEEE Internet of Things Journal, vol. 11, no. 18, pp. 29308\u201329320, 15 Sept. 15, 2024, https:\/\/doi.org\/10.1109\/JIOT.2024.3429341.","DOI":"10.1109\/JIOT.2024.3429341"},{"key":"1978_CR38","doi-asserted-by":"crossref","unstructured":"Tan, Z. & Karak\u00f6se, M. Comparative evaluation for effectiveness analysis of policy-based deep reinforcement learning approaches. International Journal of Computer and Information Technology. 10 (2021).","DOI":"10.24203\/ijcit.v10i3.104"},{"key":"1978_CR39","doi-asserted-by":"publisher","first-page":"1180","DOI":"10.1016\/j.neucom.2007.11.026","volume":"71","author":"J Peters","year":"2008","unstructured":"Peters J, Schaal S (2008) Natural actor-critic. Neurocomputing 71:1180\u20131190","journal-title":"Neurocomputing"},{"key":"1978_CR40","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1016\/j.compchemeng.2019.05.029","volume":"127","author":"J Shin","year":"2019","unstructured":"Shin J, Badgwell T, Liu K, Lee J (2019) Reinforcement learning-overview of recent progress and implications for process control. Computers Chemical Engineering 127:282\u2013294","journal-title":"Computers Chemical Engineering"},{"key":"1978_CR41","doi-asserted-by":"crossref","unstructured":"Hu, Y., Wang, H., Cao, Z., Zheng, J., Ping, Z., Chen, L. & Jin, X. Extreme learning-machine-based FNTSM control strategy for electronic throttle. Neural Computing and Applications. 32(2020,9).","DOI":"10.1007\/s00521-019-04446-9"},{"key":"1978_CR42","doi-asserted-by":"crossref","unstructured":"YADAV, A. & GAUR, P. AI-based adaptive control and design of autopilot system for nonlinear UAV. Sadhana. 39, 765\u2013783 (2014,7).","DOI":"10.1007\/s12046-014-0275-0"},{"key":"1978_CR43","doi-asserted-by":"crossref","unstructured":"Youssef, O. & Shalaby, R. Optimizing Automotive Electronic Throttle Control with a Modified Grey Wolf Algorithm. 2023 5th Novel Intelligent and Leading Emerging Sciences Conference (NILES). pp. 222\u2013227 (2023).","DOI":"10.1109\/NILES59815.2023.10296740"},{"key":"1978_CR44","doi-asserted-by":"crossref","unstructured":"Ye, M. & Wang, H. A Robust Adaptive Chattering-Free Sliding Mode Control Strategy for Automotive Electronic Throttle System via Genetic Algorithm. IEEE Access. PP. 1\u20131 (2019,8).","DOI":"10.1109\/ACCESS.2019.2934232"},{"key":"1978_CR45","doi-asserted-by":"publisher","unstructured":"\u00c1lvarez Rodr\u00edguez, S. Robustness vs Chattering-effect study for the Sliding Mode Control. Nova Scientia. 11, 65\u201386 (2019,11), https:\/\/doi.org\/10.21640\/ns.v11i23. 1972","DOI":"10.21640\/ns.v11i23"},{"key":"1978_CR46","doi-asserted-by":"publisher","first-page":"1291","DOI":"10.1109\/TSMCC.2012.2218595","volume":"42","author":"I Grondman","year":"2012","unstructured":"Grondman I, Busoniu L, Lopes G, Babuska R (2012) A survey of actor-critic reinforcement learning: standard and natural policy gradients. IEEE Trans Syst, Man, Cybern Part C (Applications and Reviews) 42:1291\u20131307","journal-title":"IEEE Trans Syst, Man, Cybern Part C (Applications and Reviews)"},{"issue":"3","key":"1978_CR47","doi-asserted-by":"publisher","first-page":"95","DOI":"10.3390\/drones8030095","volume":"8","author":"W Hu","year":"2022","unstructured":"Hu W, Yang Y, Liu Z (2022) Deep deterministic policy gradient (DDPG) agent-based sliding mode control for quadrotor attitudes. Drones 8(3):95","journal-title":"Drones"},{"key":"1978_CR48","doi-asserted-by":"publisher","first-page":"2347","DOI":"10.1007\/s00521-022-07710-7","volume":"35","author":"R Shalaby","year":"2023","unstructured":"Shalaby R, El-Hossainy M, Abo-Zalam B, Mahmoud T (2023) Optimal fractional order PID controller based on fractional-order actor-critic algorithm. Neural Comput Appl 35:2347\u20132380","journal-title":"Neural Comput Appl"},{"issue":"3","key":"1978_CR49","doi-asserted-by":"publisher","first-page":"2462","DOI":"10.1109\/TSG.2020.3042498","volume":"12","author":"B Liu","year":"2020","unstructured":"Liu B, Murat A, Mcdermott T (2020) Automated control of transactive HVACs in energy distribution systems. IEEE Trans Smart Grid 12(3):2462\u20132471","journal-title":"IEEE Trans Smart Grid"},{"key":"1978_CR50","doi-asserted-by":"publisher","DOI":"10.1016\/j.fraope.2024.100077","volume":"6","author":"S Djeffal","year":"2024","unstructured":"Djeffal S, Morakchi M, Ghoul A, Kargin T (2024) DDPG-based reinforcement learning for controlling a spatial three section continuum robot. Franklin Open 6:100077","journal-title":"Franklin Open"}],"container-title":["Complex &amp; Intelligent Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-025-01978-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40747-025-01978-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-025-01978-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T20:31:03Z","timestamp":1757190663000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40747-025-01978-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,18]]},"references-count":50,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["1978"],"URL":"https:\/\/doi.org\/10.1007\/s40747-025-01978-7","relation":{},"ISSN":["2199-4536","2198-6053"],"issn-type":[{"value":"2199-4536","type":"print"},{"value":"2198-6053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,18]]},"assertion":[{"value":"19 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 May 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"351"}}