{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T14:46:28Z","timestamp":1767192388745,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,11]],"date-time":"2022-08-11T00:00:00Z","timestamp":1660176000000},"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 111-2221-E-992-068"],"award-info":[{"award-number":["MOST 111-2221-E-992-068"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper mainly proposes a new disturbance observer (DO) for a secure communication system (SCS) of the chaos-based system (CBS). First, the fractional-order (FO) Chen chaotic system is remodeled by a Takagi\u2013Sugeno (T\u2013S) fuzzy system with the aim of softening in calculation. Second, the robust fixed-time was designed to synchronize two nonidentical chaotic systems. Third, a new disturbance observer was proposed to compensate for the disturbance and uncertainty of the secure communication system. Fourth, the proof of the proposed method based on Lyapunov condition together with simulation are given to illustrate the correctness and effectiveness of the proposed theory. The tested disturbance on the public channel was mostly compensated by the appropriately estimated disturbance value. The states of master and slave systems (MSSs) were closed to each other in fixed-time. These factors are used to confirm that the symmetry of two chaotic systems were obtained by the proposed control methods.<\/jats:p>","DOI":"10.3390\/sym14081668","type":"journal-article","created":{"date-parts":[[2022,8,11]],"date-time":"2022-08-11T23:05:49Z","timestamp":1660259149000},"page":"1668","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A Novel Disturbance Rejection Method Based on Robust Sliding Mode Control for the Secure Communication of Chaos-Based System"],"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":"Van Huy","family":"Tran","sequence":"additional","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"}]},{"given":"Duc-Hung","family":"Pham","sequence":"additional","affiliation":[{"name":"Faculty of Electrical and Electronic Engineering, Hung Yen University of Technology and Education, Hai Duong 160000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7158-3720","authenticated-orcid":false,"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":[[2022,8,11]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"86345","DOI":"10.1109\/ACCESS.2020.2992635","article-title":"Disturbance attenuation for surface-mounted PMSM drives using nonlinear disturbance observer-based sliding mode control","volume":"8","author":"Nguyen","year":"2020","journal-title":"IEEE Access"},{"key":"ref_3","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_4","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-Sugeno fuzzy chaotic systems","volume":"8","author":"Giap","year":"2020","journal-title":"IEEE Access"},{"key":"ref_5","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_6","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_7","doi-asserted-by":"crossref","unstructured":"Giap, V.N., Vu, H.S., and Huang, S.-C. (2022). Time-varying disturbance observer based on regulating boundary layer thickness sliding mode control for microelectromechanical systems gyroscope. Meas. Control.","DOI":"10.1177\/00202940221083547"},{"key":"ref_8","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_9","doi-asserted-by":"crossref","unstructured":"Giap, V.N., Nguyen, Q.D., Trung, N.K., Huang, S.C., and Trinh, X.T. (2022). Disturbance Observer Based on Terminal Sliding-Mode Control for a Secure Communication of Fractional-Order Takagi-Sugeno Fuzzy Chaotic Systems. Proceedings of the International Conference on Advanced Mechanical Engineering, Automation and Sustainable Development, Springer.","DOI":"10.1007\/978-3-030-99666-6_137"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"103251","DOI":"10.1016\/j.engappai.2019.103251","article-title":"A polynomial-fuzzy-model-based synchronization methodology for the multi-scroll Chen chaotic secure communication system","volume":"87","author":"Chen","year":"2020","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.aeue.2018.03.008","article-title":"Secure communication with a chaotic system owning logic element","volume":"88","author":"Kocamaz","year":"2018","journal-title":"AEU-Int. J. Electron. Commun."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2001","DOI":"10.1109\/TAES.2017.2680661","article-title":"An electronic implementation of Lorenz chaotic oscillator synchronization for bistatic radar applications","volume":"53","author":"Pappu","year":"2017","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Mahmoud, E.E., Higazy, M., and Althagafi, O.A. (2020). A novel strategy for complete and phase robust synchronizations of chaotic nonlinear systems. Symmetry, 12.","DOI":"10.3390\/sym12111765"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"21332","DOI":"10.1109\/ACCESS.2021.3055580","article-title":"Finite time chaos synchronization in time-delay channel and its application to satellite image encryption in OFDM communication systems","volume":"9","author":"Vaseghi","year":"2021","journal-title":"IEEE Access"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107484","DOI":"10.1016\/j.sigpro.2020.107484","article-title":"A novel image encryption scheme based on conservative hyperchaotic system and closed-loop diffusion between blocks","volume":"171","author":"Zhou","year":"2020","journal-title":"Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.ins.2019.08.041","article-title":"Image encryption algorithm for synchronously updating Boolean networks based on matrix semi-tensor product theory","volume":"507","author":"Wang","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1140\/epjp\/i2018-11872-8","article-title":"A new 4-D chaotic hyperjerk system, its synchronization, circuit design and applications in RNG, image encryption and chaos-based steganography","volume":"133","author":"Vaidyanathan","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"54175","DOI":"10.1109\/ACCESS.2020.2979827","article-title":"An efficient image encryption scheme based on S-boxes and fractional-order differential logistic map","volume":"8","author":"Zhang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1007\/s11071-019-04828-7","article-title":"A chaotic secure communication scheme based on synchronization of double-layered and multiple complex networks","volume":"96","author":"Zhou","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2369","DOI":"10.1007\/s11071-019-05408-5","article-title":"Hidden extreme multistability with hyperchaos and transient chaos in a Hopfield neural network affected by electromagnetic radiation","volume":"99","author":"Lin","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2558","DOI":"10.1109\/TNNLS.2017.2700321","article-title":"Exponential synchronization of networked chaotic delayed neural network by a hybrid event trigger scheme","volume":"29","author":"Fei","year":"2018","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2358","DOI":"10.1109\/TNNLS.2018.2884620","article-title":"Global synchronization of coupled fractional-order recurrent neural networks","volume":"30","author":"Liu","year":"2019","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2434","DOI":"10.1109\/TNNLS.2018.2884954","article-title":"Passivity and synchronization of coupled uncertain reaction-diffusion neural networks with multiple time delays","volume":"30","author":"Wang","year":"2019","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_24","first-page":"442","article-title":"An adaptive event-triggered synchronization approach for chaotic Lur\u2019e systems subject to aperiodic sampled data","volume":"66","author":"Wang","year":"2019","journal-title":"IEEE Trans. Circuits Syst. II Exp. Briefs"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.aeue.2018.03.007","article-title":"A novel digital programmable multi-scroll chaotic system and its application in FPGA-based audio secure communication","volume":"88","author":"Chang","year":"2018","journal-title":"AEU-Int. J. Electron. Commun."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Giap, V.-N., Huang, S.-C., and Nguyen, Q.D. (2020, January 23\u201325). Synchronization of 3D chaotic system based on sliding mode control: Electronic circuit implementation. Proceedings of the 2020 IEEE Eurasia Conference on IOT, Communication and Engineering (ECICE), Yunlin, Taiwan.","DOI":"10.1109\/ECICE50847.2020.9301998"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"37989","DOI":"10.1109\/ACCESS.2019.2906770","article-title":"Generalized chaos synchronization circuit simulation and asymmetric image encryption","volume":"7","author":"Wu","year":"2019","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.chaos.2018.07.011","article-title":"Dynamic analysis, circuit realization, control design and image encryption application of an extended L\u00fc system with coexisting attractors","volume":"114","author":"Lai","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Lendek, Z., Guerra, T.M., Babuska, R., and De Schutter, B. (2011). Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models, Springer.","DOI":"10.1007\/978-3-642-16776-8"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1109\/TSMC.1985.6313399","article-title":"Fuzzy identification of systems and its applications to modeling and control","volume":"1","author":"Takagi","year":"1985","journal-title":"IEEE Trans. Syst. Man Cybern. B Cybern."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1423","DOI":"10.1142\/S0218127402005224","article-title":"Circuitry implementation and synchronization of Chen\u2019s attractor","volume":"12","author":"Zhong","year":"2002","journal-title":"Inter. J. Bifurcation Chaos"},{"key":"ref_32","unstructured":"Tepljakov, A., Petlenkov, E., and Belikov, J. (2011). FOMCON: Fractional-order modeling and control toolbox for MATLAB. Fractional-Order Modeling and Control of Dynamic Systems, Springer."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2106","DOI":"10.1109\/TAC.2011.2179869","article-title":"Nonlinear feedback design for fixed-time stabilization of linear control systems","volume":"57","author":"Polyakov","year":"2012","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Giap, V.N., Nguyen, Q.D., Trung, N.K., and Huang, S.C. (2022). Time-varying disturbance observer based on sliding-mode observer and double phases fixed-time sliding mode control for a TS fuzzy micro-electro-mechanical system gyroscope. J. Vib. Control.","DOI":"10.1177\/10775463211073199"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Truc, L.N., Vu, L.A., Thoan, T.V., Thanh, B.T., and Nguyen, T.L. (2022). Adaptive Sliding Mode Control Anticipating Proportional Degradation of Actuator Torque in Uncertain Serial Industrial Robots. Symmetry, 14.","DOI":"10.3390\/sym14050957"},{"key":"ref_36","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_37","doi-asserted-by":"crossref","unstructured":"Petr\u00e1\u0161, I. (2011). Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Science & Business Media.","DOI":"10.1007\/978-3-642-18101-6"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1668\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:07:29Z","timestamp":1760141249000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1668"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,11]]},"references-count":37,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081668"],"URL":"https:\/\/doi.org\/10.3390\/sym14081668","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,11]]}}}