###### Abstract
Combined with the above analysis, we integrate the prediction mechanism into the MFAC framework, which can enable the controller to predict future system behavior and enhance robustness to disturbances and uncertainties. Model-Free Adaptive Predictive Control (MFAPC), an advanced control methodology that adjusts controller parameters online using real-time input-output data without relying on complete system models, has drawn significant scholarly attention [28; 29]. Recent studies have applied MFAPC across various domains to achieve resilient control under challenging conditions. For instance, ref. [30] proposed an event-triggered MFAPC method for the secure operation of multi-area power systems under spoofing attacks. In [31], a robust predictive tracking control strategy was developed for high-speed trains to address quantization errors and DoS attacks. Additionally, ref. [32] introduced a constraint-aware MFAPC strategy for perimeter control and path guidance in multi-area urban traffic systems, achieving traffic flow optimization without accurate models. Based on generalized learning systems, ref. [33] developed an MFAPC method for nonlinear MAS to improve re silience and responsiveness under DoS attacks. For heterogeneous nonlinear MAS, ref. [34] implemented a distributed control framework enabling cooperative behavior without model information. Further extending this approach to MIMO MAS, ref. [35] proposed a distributed robust predictive control scheme to counter deception attacks. In [36], a robust distributed control method with anti-DoS capabilities was designed for multi-area power systems, ensuring coordination and stability under adversarial disruptions. Meanwhile, the increasing digitalization and networking of high-speed trains have made their open communication systems more vulnerable to cybersecurity threats [37]. To address this, ref. [38] introduced an anti-attack distributed cooperative control strategy based on topological reconfiguration, enhancing resilience through a virtual coupling mechanism. In [39], an event-triggered adaptive control method was developed for bottleneck sections, capable of resisting deception attacks. Furthermore, in ref. [40], aiming at the problem of random topology change and data loss that may occur in high-speed train operation, a security consistency tracking and control method with a data recovery mechanism was constructed. Combining sliding mode control with event-triggered mechanisms, ref. [41] achieved secure and energy-efficient control of high-speed trains under energy constraints and malicious attacks. In summary, this study presents a systematic exploration of distributed control, model-free adaptive predictive control, and cyber-attack defense strategies. Inspired by the above discussion, we investigate the cooperative control problem of MHSTs under FDI attack. The main contributions of this paper are as follows:  
1. A Distributed Model-Free Adaptive Predictive Control (DMFAPC) method is developed for the cooperative control of multiple high-speed trains (MHSTs) under false data injection (FDI) attack conditions. A dynamic linearization approach is introduced to convert the nonlinear train dynamics into a data-driven model, eliminating the need for an accurate system model. 2. Compared to the existing literature [19; 20; 33; 34], this paper investigates the impact of FDI attacks on the performance of MHSTs by combining model-free adaptive algorithms with predictive algorithms to achieve enhanced stability and control performance. 3. Unlike existing model-free adaptive predictive control schemes [23; 24; 29], this paper proposes a distributed consensus tracking control scheme for MHSTs in the FDI attack environment. In this paper, some relevant mathematical symbols are defined as shown in Table 1.