{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T19:31:32Z","timestamp":1777491092542,"version":"3.51.4"},"reference-count":34,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,28]],"date-time":"2022-02-28T00:00:00Z","timestamp":1646006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61973169"],"award-info":[{"award-number":["61973169"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Jiangsu Natural Science Foundation","doi-asserted-by":"publisher","award":["BK20201392"],"award-info":[{"award-number":["BK20201392"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Qing Lan Project of Jiangsu Province","award":["R2021Q04"],"award-info":[{"award-number":["R2021Q04"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, a novel adaptive sliding-mode control algorithm is proposed for the attitude control of quadrotor unmanned aerial vehicles (UAVs) under the delta operator framework. First, the delta operator technique is used to discretize the attitude control systems of a quadrotor UAV. Then, based on the linear matrix inequality technique, a linear sliding surface is designed to ensure the asymptotical stability of the quadrotor UAV attitude control system during the sliding motion process. Second, by the estimated external disturbance using a radical basis function (RBF) neural network, an adaptive sliding-mode attitude controller is designed such that the states of the quadrotor UAV attitude systems can be driven towards the desired sliding surface, and thus the attitude control objective of the qudarotor UAV is achieved. Compared with the traditional adaptive sliding-mode control algorithm, the proposed adaptive sliding-mode control algorithm can effectively realize the attitude control of a quadrotor UAV subject to strong disturbances and couplings. Finally, comparisons of the simulation results verify the effectiveness and superiority of the control algorithm proposed in this paper.<\/jats:p>","DOI":"10.3390\/sym14030498","type":"journal-article","created":{"date-parts":[[2022,2,28]],"date-time":"2022-02-28T20:11:57Z","timestamp":1646079117000},"page":"498","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Adaptive Sliding Mode Attitude Control of Quadrotor UAVs Based on the Delta Operator Framework"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6142-6580","authenticated-orcid":false,"given":"Bochao","family":"Zheng","sequence":"first","affiliation":[{"name":"CICAEET, School of Automation, Nanjing University of Information Science and Technology, Nanjing 210044, China"}]},{"given":"Yuewen","family":"Wu","sequence":"additional","affiliation":[{"name":"CICAEET, School of Automation, Nanjing University of Information Science and Technology, Nanjing 210044, China"}]},{"given":"Hui","family":"Li","sequence":"additional","affiliation":[{"name":"CICAEET, School of Automation, Nanjing University of Information Science and Technology, Nanjing 210044, China"}]},{"given":"Zhipeng","family":"Chen","sequence":"additional","affiliation":[{"name":"CICAEET, School of Automation, Nanjing University of Information Science and Technology, Nanjing 210044, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1007\/s13369-020-04742-w","article-title":"Adaptive PID controller using sliding mode control approaches for quadrotor UAVs attitude and position stabilization","volume":"46","author":"Noordin","year":"2021","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wang, S.Y., Polyakov, A., and Zheng, G. (2022). Quadrotor stabilization under time and space constraints using implicit PID conttroler. J. Frankl. Institude.","DOI":"10.1016\/j.jfranklin.2022.01.002"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sun, C., Liu, M., Liu, C., Feng, X., and Wu, H. (2020). An industrial quadrotor UAVs control method based on fuzzy adaptive linear active disturbance rejection control. Aerosp. Sci. Technol., 10.","DOI":"10.3390\/electronics10040376"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1177\/01423312211031781","article-title":"Adaptive-based linear active disturbance rejection attitude control for quadrotor with external disturbances","volume":"44","author":"Wang","year":"2022","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Nie, Y., and Chen, L. (2021). Adaptive fuzzy fault-tolerant control against time-varying faults via a new sliding mode observer method. Symmetry, 13.","DOI":"10.20944\/preprints202107.0498.v1"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hong, Q., Shi, Y., and Chen, Z. (2020). Adaptive sliding mode control based on disturbance observer for placement pressure control system. Symmetry, 12.","DOI":"10.3390\/sym12061057"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3718","DOI":"10.1016\/j.ijleo.2016.01.010","article-title":"Discreti-time sliding mode control for a quadrotor UAV","volume":"127","author":"Xiong","year":"2016","journal-title":"Optik"},{"key":"ref_8","unstructured":"Wu, X.W., Xiao, B., and Qu, Y.H. (2019). Modeling and sliding mode-based attitude tracking control of a quadrotor UAVs with time-varying mass. ISA Trans."},{"key":"ref_9","first-page":"105716","article-title":"Nonsingular terminal sliding mode control for a quadrotor UAVs with a total rotor failure","volume":"98","author":"Hou","year":"2021","journal-title":"Electronics"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.isatra.2020.07.007","article-title":"Adaptive sliding mode attitude controller such that all attitudes of the quadrotor UAVs can be driven to the designed sliding surface","volume":"106","author":"Yang","year":"2021","journal-title":"ISA Trans."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1007\/s11071-019-05002-9","article-title":"Adaptive fuzzy global sliding mode control for trajectory tracking of quadrotor UAVs","volume":"97","author":"Zhang","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"104763","DOI":"10.1016\/j.conengprac.2021.104763","article-title":"Robust path tracking of a quadrotor using adaptive fuzzy terminal sliding mode control","volume":"110","author":"Nekoukar","year":"2021","journal-title":"Control. Eng. Pract."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s12555-019-0116-3","article-title":"Finite-time adaptive integral backstepping fast terminal sliding mode control applications on quadrotor UAV","volume":"18","author":"Eliker","year":"2020","journal-title":"Int. J. Control. Autom. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zhang, X., Wang, Y., Zhu, G., Chen, X., and Su, C.-Y. (2022). Discrete-time adaptive neural tracking control and its Enperiments for quadrotor unmanned aerial vehicle systems. IEEE\/ASME Trans. Mechatronics.","DOI":"10.1109\/TMECH.2021.3112470"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1109\/TAC.1986.1104162","article-title":"Improved finite word length characteristics in digital control using delta operators","volume":"31","author":"Middleton","year":"1986","journal-title":"IEEE Trans. Automat. Contr."},{"key":"ref_16","unstructured":"Middleton, R.H., and Goodwin, G.C. (1990). Digital Control and Estimation: A Unified Approach, Prentice Hall."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1002\/rnc.3882","article-title":"Quantized sliding mode control in delta operator framework","volume":"28","author":"Zheng","year":"2018","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Yang, H., Xia, Y., Shi, P., and Zhao, L. (2012). Analysis and Synthesis of Delta Operator Systems, Springer.","DOI":"10.1007\/978-3-642-28774-9"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1016\/j.nahs.2019.100827","article-title":"Stabilization of uncertain switched discrete-time systems against actuator faults and input saturation","volume":"35","author":"Sakthivel","year":"2020","journal-title":"Nonlinear-Anal.-Hybrid Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1002\/rnc.2915","article-title":"Insensitive output feedback H-infinity control of delta operator systems with insensitivity to sampling time jitter","volume":"24","author":"Guo","year":"2014","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1080\/00207179.2012.654512","article-title":"Low-sensitivity H\u221e filter design for linear delta operator systems with sampling time jitter","volume":"85","author":"Guo","year":"2012","journal-title":"Int. J. Control"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.neucom.2020.01.076","article-title":"Group consensus of multi-agent systems with cooperative-competitive interaction and communication delay in switching topologies networks based on the delta operator method","volume":"390","author":"Pu","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/j.neucom.2020.11.013","article-title":"Couple-group consensus for heterogeneous MASs under switched topologies in cooperative-competitive systems: A hybrid pinning and delta operator skills","volume":"441","author":"Pu","year":"2021","journal-title":"Neurocomputing"},{"key":"ref_24","unstructured":"Chen, G.S. (2020). Study on Anti-Jamming and Fault-Tolerant Control of Quadrotor UAV, Nanjing University of Information Science and Technology."},{"key":"ref_25","unstructured":"Zhang, Z.C. (2016). Sliding Mode Fault-Tolerant Control Algorithm for Delta Operator Time-Delay Systems, Nanjing University of Aeronautics and Astronautics."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2891","DOI":"10.1109\/TIE.2014.2364982","article-title":"Nonlinear robust adaptive tracking control of a quadrotor UAVs via immersion and invariance methodology","volume":"62","author":"Zhao","year":"2015","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.2514\/1.27882","article-title":"Real-time nonlinear embedded control for an autonomous quadrotor helicopter","volume":"30","author":"Lendoul","year":"2007","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_28","unstructured":"Zhao, D., Sun, C., Wang, Q., and Yang, W. Neural-network-based PID control for quadrotor aircraft. Proceedings of the International Conference on Intelligent Science and Big Data Engineering."},{"key":"ref_29","first-page":"587098","article-title":"Attitude adn altitude controller design for quad-rotor type MAVs","volume":"2013","author":"Wang","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_30","first-page":"324112","article-title":"Attitude active disturbance rejection control of UAVs attitude based on iterative learning control","volume":"41","author":"Wang","year":"2020","journal-title":"Acta Aeronaut. Astronaut. Sin."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/j.ast.2017.07.012","article-title":"Finite time formation control for a group of quadrotor aircraft","volume":"69","author":"Du","year":"2017","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_32","unstructured":"Li, H., Wu, B., Li, G., and Yang, C. (2005). Global Theory of Delta Operator Control and Its Robustness Control, National Defense Inddustry Press."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Society for Industrial and Applied Mathematics, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611970777"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Liu, J. (2018). Intelligent Control Design and MATLAB Simulation, Springer Nature Singapore Pte Ltd.","DOI":"10.1007\/978-981-10-5263-7"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/498\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:29:40Z","timestamp":1760135380000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/498"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,28]]},"references-count":34,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["sym14030498"],"URL":"https:\/\/doi.org\/10.3390\/sym14030498","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,28]]}}}