{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:36:00Z","timestamp":1760229360576,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,10]],"date-time":"2022-06-10T00:00:00Z","timestamp":1654819200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Technology, Republic of China","award":["MOST 110-2221-E-992-062"],"award-info":[{"award-number":["MOST 110-2221-E-992-062"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The slotless self-bearing motor (SSBM) is a motor with its self-bearing function. The mechanical structure of the motor is six symmetrical hexagonal shapes. The main control problem for this motor is disturbance and uncertainty rejection. Therefore, this paper proposes a new disturbance observer (DOB) based on an optimal fixed-time state observer (OFTSOB) and adaptive sliding mode control (SMC) for the motor. Firstly, the optimal state observer was used to construct to obtain the information of the states of the bearing-less motor system. Second, a new disturbance observer base on the fast speed reaching law is proposed for estimating the unknown dynamics and unpredicted uncertainty of the motor system. Third, the adaptive fast-reaching law-sliding mode control is designed to control the positions and rotational speed. Fourth, the proposed control system is proved via the Lyapunov theorem. Finally, the corrections of proposed method once again tested by using MATLAB simulation. The obtained results figured out that the proposed method is good at rejection disturbance and uncertainty and precision in control the movement and rotation. The novelties of the proposed method are that the gains of fixed-time observer were met by the support of optimal pole placement method, the disturbances were mostly rejected by a new reaching law of unknown input observer.<\/jats:p>","DOI":"10.3390\/sym14061206","type":"journal-article","created":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T06:31:59Z","timestamp":1655101919000},"page":"1206","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Inversed Model-Based Disturbance Observer Base on Adaptive Fast Convergent Sliding Mode Control and Fixed-Time State Observer for Slotless Self-Bearing Motor"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5301-7058","authenticated-orcid":false,"given":"Quang Dich","family":"Nguyen","sequence":"first","affiliation":[{"name":"Institute for Control Engineering and Automation, Hanoi University of Science and Technology, No. 1 Dai Co Viet, Hai Ba Trung, Ha Noi 100000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0670-874X","authenticated-orcid":false,"given":"Van Nam","family":"Giap","sequence":"additional","affiliation":[{"name":"School of Electrical & Electronic Engineering, Hanoi University of Science and Technology, No. 1 Dai Co Viet, Hai Ba Trung, Ha Noi 100000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shyh-Chour","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, National Kaohsiung University of Science and Technology, Kaohsiung 807618, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Maslen, E.H., and Schweitzer, G. (2009). Magnetic Bearings: Theory, Design, and Application to Rotating Machinery, Springer.","DOI":"10.1007\/978-3-642-00497-1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2644","DOI":"10.1109\/TIE.2010.2076309","article-title":"Analysis and control of non-salient permanent magnet axial-gap self-bearing motor","volume":"58","author":"Nguyen","year":"2011","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1109\/TMECH.2010.2045392","article-title":"Modeling and control of salient-pole permanent magnet axial gap self-bearing motor","volume":"16","author":"Nguyen","year":"2011","journal-title":"IEEE ASME Trans. Mechatron."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"El-Shourbagy, S.M., Saeed, N.A., Kamel, M., Raslan, K.R., Nasr, E.A., and Awrejcewicz, J. (2021). On the Performance of a Nonlinear Position-Velocity Controller to Stabilize Rotor-Active Magnetic-Bearings System. Symmetry, 13.","DOI":"10.3390\/sym13112069"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"012006","DOI":"10.1088\/1757-899X\/1113\/1\/012006","article-title":"Generalized proportional integral disturbance observer-based fuzzy sliding mode control for active magnetic bearing system","volume":"1113","author":"Giap","year":"2021","journal-title":"IOP Conf. Ser. Mat. Sci. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"934","DOI":"10.1177\/0020294020905044","article-title":"Effectiveness of fuzzy sliding mode control boundary layer based on uncertainty and disturbance compensator on suspension active magnetic bearing system","volume":"53","author":"Giap","year":"2020","journal-title":"Meas. Control"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Giap, V.N., Huang, S.-C., Nguyen, Q.D., and Trinh, X.T. (2020, January 12\u201315). Time varying disturbance observer based on sliding mode control for active magnetic bearing system. Proceedings of the 2nd Annual International Conference on Material, Machines and Methods for Sustainable Development (MMMS2020), Nha Trang, Vietnam.","DOI":"10.1007\/978-3-030-69610-8_123"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.1080\/00207721.2018.1443231","article-title":"Robust observer-based optimal linear quadratic tracker for five-degree-of-freedom sampled-data active magnetic bearing system","volume":"49","author":"Tsai","year":"2018","journal-title":"Int. J. Syst. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"9822","DOI":"10.1109\/TIE.2020.3020034","article-title":"State-Constraints Adaptive Backstepping Control for Active Magnetic Bearings With Parameters Nonstationarities and Uncertainties","volume":"68","author":"Yang","year":"2021","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4667","DOI":"10.1109\/TIE.2020.2989710","article-title":"Vibration Suppression for Magnetically Levitated High-Speed Motors Based on Polarity Switching Tracking Filter and Disturbance Observer","volume":"68","author":"Gong","year":"2021","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_11","unstructured":"Han, X., Liu, G., Chen, B., and Zheng, S. (2021). Surge Disturbance Suppression of AMB-Rotor Systems in Magnetically Suspension Centrifugal Compressors. IEEE Trans. Contr. Syst. Technol., 1\u201311."},{"key":"ref_12","first-page":"9245","article-title":"Disturbance Force Self-Sensing and Suppression Method for AMB-Rotor System Based on Disturbance Observer","volume":"20","author":"Sun","year":"2020","journal-title":"IEEE Sens. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1313","DOI":"10.1109\/TIE.2016.2612622","article-title":"Feedback Linearization and Extended State Observer-Based Control for Rotor-AMBs System with Mismatched Uncertainties","volume":"64","author":"Liu","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2753","DOI":"10.1109\/TPEL.2014.2352366","article-title":"Mismatched Disturbance Rejection Control for Voltage-Controlled Active Magnetic Bearing via State-Space Disturbance Observer","volume":"30","author":"Peng","year":"2015","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Nguyen, H.P., Nguyen, X.B., Bui, T.T., Ueno, S., and Nguyen, Q.D. (2019). Analysis and control of slotless self-bearing motor. Actuators, 8.","DOI":"10.3390\/act8030057"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5974","DOI":"10.1109\/TIE.2014.2311379","article-title":"Slotless bearingless disk drive for high-speed and high-purity applications","volume":"61","author":"Steinert","year":"2014","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1109\/TIA.2015.2466683","article-title":"Evaluation of One- and Two-Pole-Pair Slotless Bearingless Motors with Toroidal Windings","volume":"52","author":"Steinert","year":"2016","journal-title":"IEEE Trans. Ind. Appl."},{"key":"ref_18","first-page":"462","article-title":"Development of a Lorentz-Force-Type Slotless Self-Bearing Motor","volume":"3","author":"Ueno","year":"2009","journal-title":"J. Syst. Des. Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ueno, S., and Kato, T. (2009, January 2\u20135). A Novel Design of a Lorentz-Force-Type Small Self Bearing Motor. Proceedings of the 8th International Conference on Power Electronics and Drive Systems, Taipei, Taiwan.","DOI":"10.1109\/PEDS.2009.5385835"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"23980","DOI":"10.1109\/ACCESS.2022.3155121","article-title":"Robust Sliding Mode Control-Based a Novel Super-Twisting Disturbance Observer and Fixed-Time State Observer for Slotless-Self Bearing Motor System","volume":"10","author":"Nguyen","year":"2022","journal-title":"IEEE Access"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"932","DOI":"10.1109\/41.857974","article-title":"A nonlinear disturbance observer for robotic manipulators","volume":"47","author":"Chen","year":"2000","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2182","DOI":"10.1109\/TASE.2020.3015870","article-title":"Disturbance-compensation-based continuous sliding mode control for overhead cranes with disturbances","volume":"17","author":"Wu","year":"2020","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Li, X., Zhu, Z., Shen, G., and Tang, Y. (2021). Wire Tension Coordination Control of Electro-Hydraulic Servo Driven Double-Rope Winding Hoisting Systems Using a Hybrid Controller Combining the Flatness-Based Control and a Disturbance Observer. Symmetry, 13.","DOI":"10.3390\/sym13040716"},{"key":"ref_24","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_25","doi-asserted-by":"crossref","first-page":"225805","DOI":"10.1109\/ACCESS.2020.3045416","article-title":"Disturbance observer-based linear matrix inequality for the synchronization of Takagi\u2013Sugeno fuzzy chaotic systems","volume":"8","author":"Giap","year":"2020","journal-title":"IEEE Access"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"23907","DOI":"10.1109\/ACCESS.2021.3056413","article-title":"Synthetic adaptive fuzzy disturbance observer and sliding-mode control for chaos-based secure communication systems","volume":"9","author":"Giap","year":"2021","journal-title":"IEEE Access"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"133663","DOI":"10.1109\/ACCESS.2021.3114030","article-title":"Disturbance and uncertainty rejection-based on fixed-time sliding-mode control for the secure communication of chaotic systems","volume":"9","author":"Giap","year":"2021","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1509","DOI":"10.1177\/1461348420981181","article-title":"Robust controlbased disturbance observer and optimal states feedback for T\u2013S fuzzy systems","volume":"40","author":"Giap","year":"2020","journal-title":"J. Low Freq. Noise Vib. Act. Control"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"96390","DOI":"10.1109\/ACCESS.2021.3095465","article-title":"Robust observer based on fixed-time sliding mode control of position\/velocity for a T\u2013S fuzzy MEMS gyroscope","volume":"9","author":"Giap","year":"2021","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"116090","DOI":"10.1109\/ACCESS.2020.3004241","article-title":"Extended disturbance observer-based integral sliding mode control for nonlinear system via T\u2013S fuzzy model","volume":"8","author":"Hwang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1144","DOI":"10.1049\/iet-cta.2016.1256","article-title":"Finite- and fixed-time differentiators utilising HOSM techniques","volume":"11","author":"Basin","year":"2017","journal-title":"IET Control Theory Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2257","DOI":"10.1109\/9.811208","article-title":"Robust pole placement in LMI regions","volume":"44","author":"Chilali","year":"1999","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1109\/TAC.1977.1101446","article-title":"Variable structure systems with sliding modes","volume":"22","author":"Utkin","year":"1977","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"49793","DOI":"10.1109\/ACCESS.2018.2868785","article-title":"Global terminal sliding mode control with the quick reaching law and its application","volume":"6","author":"Xiu","year":"2018","journal-title":"IEEE Access"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Li, T., and Liu, X. (2021). Model-Free Non-Cascade Integral Sliding Mode Control of Permanent Magnet Synchronous Motor Drive with a Fast Reaching Law. Symmetry, 13.","DOI":"10.3390\/sym13091680"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1206\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:27:42Z","timestamp":1760138862000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1206"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,10]]},"references-count":35,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061206"],"URL":"https:\/\/doi.org\/10.3390\/sym14061206","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,6,10]]}}}