{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:14:46Z","timestamp":1760058886221,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,4]],"date-time":"2025-05-04T00:00:00Z","timestamp":1746316800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The Science and Technology Project of Henan Province","award":["232102241028"],"award-info":[{"award-number":["232102241028"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper investigates the problem of ensuring the stable operation of multiple high-speed train systems under the threat of False Data Injection (FDI) attacks. Due to the wireless communication characteristics of railway networks, high-speed train systems are particularly vulnerable to FDI attacks, which can compromise the accuracy of train data and disrupt cooperative control strategies. To mitigate this risk, we propose a Distributed Model-Free Adaptive Predictive Control (DMFAPC) scheme, which is data-driven and does not rely on an accurate system model. First, by using a dynamic linearization method, we transform the nonlinear high-speed train system model into a dynamically linearized model. Then, based on the above linearized model, we design a DMFAPC control strategy that ensures bounded train velocity tracking errors even in the presence of FDI attacks. Finally, the stability of the proposed scheme is rigorously analyzed using the contraction mapping method, and simulation results demonstrate that the scheme exhibits excellent robustness and stability under attack conditions.<\/jats:p>","DOI":"10.3390\/a18050267","type":"journal-article","created":{"date-parts":[[2025,5,4]],"date-time":"2025-05-04T20:10:27Z","timestamp":1746389427000},"page":"267","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Data-Driven Distributed Model-Free Adaptive Predictive Control for Multiple High-Speed Trains Under False Data Injection Attacks"],"prefix":"10.3390","volume":"18","author":[{"given":"Bin","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Intelligent Engineering, Huanghe Jiaotong University, Jiaozuo 454950, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Intelligent Engineering, Huanghe Jiaotong University, Jiaozuo 454950, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fuzhong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo 454000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3406","DOI":"10.1109\/TITS.2017.2776943","article-title":"Optimization for the following operation of a high-speed train under the moving block system","volume":"19","author":"Liu","year":"2017","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1109\/MCAS.2010.936782","article-title":"Automatic train control system development and simulation for high-speed railways","volume":"10","author":"Dong","year":"2010","journal-title":"IEEE Circuits Syst. Mag."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Zhao, Y., Sun, H., Yang, J., Hou, L., and Yang, D. (2025). Event-Triggered Fully Distributed Bipartite Containment Control for Multi-Agent Systems Under DoS Attacks and External Disturbances. IEEE Trans. Circuits Syst. I Regul. Pap., 1\u201314.","DOI":"10.1109\/TCSI.2025.3534241"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"10470","DOI":"10.1109\/TASE.2024.3524258","article-title":"Distributed Hybrid Dynamic Event-Triggered Consensus Control for Nonlinear Multi-Agent Systems","volume":"22","author":"Wang","year":"2025","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1109\/TITS.2019.2956162","article-title":"Robust distributed cruise control of multiple high-speed trains based on disturbance observer","volume":"22","author":"Wang","year":"2019","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3662","DOI":"10.1002\/rnc.5996","article-title":"Cooperative control for multiple high-speed trains with constraints and acceleration zone under moving block system","volume":"32","author":"Tian","year":"2022","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"8499","DOI":"10.1109\/TITS.2024.3386392","article-title":"Finite-time distributed adaptive coordinated control for multiple traction units of high-speed trains","volume":"25","author":"Wang","year":"2024","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3277","DOI":"10.1109\/TVT.2023.3328640","article-title":"Distributed adaptive fault-tolerant control for high-speed trains using multi-agent system model","volume":"73","author":"Guo","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"105768","DOI":"10.1016\/j.conengprac.2023.105768","article-title":"Adaptive cooperative control for multiple high-speed trains with uncertainties, input saturations and state constraints","volume":"142","author":"Guo","year":"2024","journal-title":"Control Eng. Pract."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1794","DOI":"10.1109\/TCYB.2023.3257876","article-title":"Distributed learning control for high-speed trains subject to operation safety constraints","volume":"54","author":"Gao","year":"2023","journal-title":"IEEE Trans. Cybern."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1241","DOI":"10.1109\/TIV.2022.3168550","article-title":"Distributed cooperative fault-tolerant control of high-speed trains with input saturation and actuator faults","volume":"8","author":"Zhu","year":"2022","journal-title":"IIEEE Trans. Intell. Veh."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4076","DOI":"10.1109\/TIE.2016.2636126","article-title":"An overview of dynamic-linearization-based data-driven control and applications","volume":"64","author":"Hou","year":"2016","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5119","DOI":"10.1109\/TSMC.2018.2866909","article-title":"Quantized Data Driven Iterative Learning Control for a Class of Nonlinear Systems with Sensor Saturation","volume":"50","author":"Bu","year":"2020","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"20006","DOI":"10.1109\/TITS.2024.3454428","article-title":"Resource-efficient model-free adaptive platooning control for vehicles with encrypted information","volume":"25","author":"Zhao","year":"2024","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4117","DOI":"10.1109\/TSMC.2021.3091422","article-title":"Resilient Model-Free Adaptive Iterative Learning Control for Nonlinear Systems Under Periodic DoS Attacks via a Fading Channel","volume":"52","author":"Yu","year":"2022","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1963","DOI":"10.1109\/TNNLS.2020.2995600","article-title":"Data-driven terminal iterative learning consensus for nonlinear multiagent systems with output saturation","volume":"32","author":"Bu","year":"2020","journal-title":"IEEE Trans. Neural Networks Learn. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6957","DOI":"10.1109\/TCYB.2024.3431290","article-title":"Disturbance observer dynamic linearization-based model-free adaptive control for discrete-time nonlinear systems","volume":"54","author":"Yang","year":"2024","journal-title":"IEEE Trans. Cybern."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"9597","DOI":"10.1109\/TCYB.2021.3058997","article-title":"Event-triggered model-free adaptive iterative learning control for a class of nonlinear systems over fading channels","volume":"52","author":"Bu","year":"2021","journal-title":"IEEE Trans. Cybern."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5690","DOI":"10.1109\/TVT.2022.3231712","article-title":"Resilient coordinated data-driven control of multiple high-speed trains under fading measurements and denial-of-service attacks","volume":"72","author":"Yu","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"14702","DOI":"10.1109\/TITS.2021.3131997","article-title":"Data-driven event-triggered cooperative control for multiple subway trains with switching topologies","volume":"23","author":"Wang","year":"2021","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2967","DOI":"10.1016\/j.automatica.2014.10.128","article-title":"Model predictive control: Recent developments and future promise","volume":"50","author":"Mayne","year":"2014","journal-title":"Automatica"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Li, H., Wang, C., Yuan, S., Zhu, H., Li, B., Liu, Y., and Sun, L. (2025). Energy Scheduling of Hydrogen Hybrid UAV Based on Model Predictive Control and Deep Deterministic Policy Gradient Algorithm. Algorithms, 18.","DOI":"10.3390\/a18020080"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4493","DOI":"10.1109\/TIV.2024.3358229","article-title":"Physical-informed neural network for MPC-based trajectory tracking of vehicles with noise considered","volume":"9","author":"Jin","year":"2024","journal-title":"IEEE Trans. Intell. Veh."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Pan, T., Hou, J., Jin, X., Yu, Z., Zhou, W., and Wang, Z. (2024). Distributed Control of Hydrogen-Based Microgrids for the Demand Side: A Multiagent Self-Triggered MPC-Based Strategy. Algorithms, 17.","DOI":"10.3390\/a17060251"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ates, C., Bicat, D., Yankov, R., Arweiler, J., Koch, R., and Bauer, H.-J. (2023). Model predictive evolutionary temperature control via neural-network-based digital twins. Algorithms, 16.","DOI":"10.3390\/a16080387"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3270","DOI":"10.1109\/TAC.2023.3321375","article-title":"Distributionally robust model predictive control with output feedback","volume":"69","author":"Li","year":"2023","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"7261726","DOI":"10.1155\/2019\/7261726","article-title":"Adaptive model predictive control for cruise control of high-speed trains with time-varying parameters","volume":"2019","author":"Xu","year":"2019","journal-title":"J. Adv. Transp."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2173","DOI":"10.1109\/TNN.2011.2176141","article-title":"Data-driven model-free adaptive control for a class of MIMO nonlinear discrete-time systems","volume":"22","author":"Hou","year":"2011","journal-title":"IEEE Trans. Neural Networks"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1325","DOI":"10.1109\/TSMC.2023.3326823","article-title":"Event-triggered model-free adaptive predictive control for networked control systems under deception attacks","volume":"54","author":"Li","year":"2023","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1812","DOI":"10.1016\/j.egyr.2023.04.140","article-title":"Event-triggered model-free adaptive predictive control for multi-area power systems under deception attacks","volume":"9","author":"Bu","year":"2023","journal-title":"Energy Rep."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Wang, D., and Wang, F. (2024). Model-Free Adaptive Predictive Tracking Control for High-Speed Trains Considering Quantization Effects and Denial-of-Service Attacks. Actuators, 13.","DOI":"10.3390\/act13080301"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1109\/TITS.2020.3017351","article-title":"Constrained model free adaptive predictive perimeter control and route guidance for multi-region urban traffic systems","volume":"23","author":"Hou","year":"2020","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1109\/JAS.2024.124929","article-title":"Broad-Learning-System-Based Model-Free Adaptive Predictive Control for Nonlinear MASs Under DoS Attacks","volume":"12","author":"Xiong","year":"2025","journal-title":"IEEE\/CAA J. Autom. Sin."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3027","DOI":"10.1080\/00207721.2022.2068693","article-title":"Distributed model-free adaptive predictive control for heterogeneous nonlinear multi-agent systems","volume":"53","author":"Pan","year":"2022","journal-title":"Int. J. Syst. Sci."},{"key":"ref_35","first-page":"32","article-title":"Distributed model-free adaptive predictive control for MIMO multi-agent systems with deception attack","volume":"10","author":"Pan","year":"2024","journal-title":"IEEE Trans. Signal Inf. Process. Netw."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4567","DOI":"10.1002\/asjc.3147","article-title":"Resilient distributed model-free adaptive predictive control for multiarea power systems under denial of service attack","volume":"25","author":"Bu","year":"2023","journal-title":"Asian J. Control"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"12427","DOI":"10.1109\/TVT.2021.3120695","article-title":"Resilient cooperative control for high-speed trains under denial-of-service attacks","volume":"70","author":"Zhao","year":"2021","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1066","DOI":"10.1109\/JAS.2023.124011","article-title":"Attack-resilient distributed cooperative control of virtually coupled high-speed trains via topology reconfiguration","volume":"11","author":"Xiao","year":"2024","journal-title":"IEEE\/CAA J. Autom. Sin."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1016\/j.ins.2020.08.012","article-title":"Event-triggered adaptive control for multiple high-speed trains with deception attacks in bottleneck sections","volume":"547","author":"Zhao","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2298","DOI":"10.1109\/TCST.2024.3420012","article-title":"Data-driven security consensus tracking of multiple high-speed trains under random topologies with data recovery mechanism","volume":"32","author":"Yu","year":"2024","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1554","DOI":"10.1109\/TVT.2023.3312439","article-title":"Sliding mode control for high-speed train via event-triggered strategy under energy-limited deception attacks","volume":"73","author":"Yu","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.sysconle.2016.05.017","article-title":"Iterative learning control with input sharing for multi-agent consensus tracking","volume":"94","author":"Yang","year":"2016","journal-title":"Syst. Control Lett."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/5\/267\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:27:00Z","timestamp":1760030820000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/5\/267"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,4]]},"references-count":42,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["a18050267"],"URL":"https:\/\/doi.org\/10.3390\/a18050267","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2025,5,4]]}}}