###### Abstract  
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  
adaptive coordination control method for multi-traction units, ensuring coordination within a limited time. Building on this, ref. [8] designed a distributed adaptive fault-tolerant control strategy with failure-handling capabilities for MAS, and [9] addressed system uncertainties, input saturation, and state constraints by constructing an adaptive cooperative control mechanism. In [10], a distributed learning control method integrating safety constraints was developed to improve both safety and intelligence in train systems, while [11] focused on distributed cooperative fault-tolerant control under actuator faults and input saturation conditions. Traditional control system designs often rely on accurate mathematical models grounded in physical principles. However, due to the complexity of high-speed rail systems--including strong nonlinearity, interconnection of subsystems, and operational fluctuations--model-based approaches can become cumbersome or infeasible. As complexity increases, model-free techniques have become a more attractive alternative, particularly when operating conditions deviate from quantifiable norms or system parameters vary over time. Model-Free Adaptive Control (MFAC) has emerged as a promising data-driven approach, deriving control laws directly from real-time input-output (I/O) data without requiring explicit system models [12; 13; 14; 15; 16]. For example, ref. [17] introduced a model-free adaptive control approach combining disturbance observers and dynamic linearization to enhance robustness against external disturbances. In [18], event-triggered mechanisms and iterative learning control were integrated to address communication constraints in fading channels, achieving resource-efficient adaptive control. It is worth emphasizing that MFAC, as a representative data-driven method for nonlinear systems, has attracted considerable attention in train control for its effectiveness and practicality. In [19], a resilient data-driven coordination control method for multiple high-speed trains was developed to improve robustness and stability under harsh network conditions such as measurement fading and denial-of-service (DoS) attacks. Similarly, ref. [20] addressed the issue of dynamic topological switching in subway train networks by designing an event-triggered data-driven coordination strategy that reduces communication costs while maintaining coordination performance. Predictive control has received much attention for its ability to improve system performance by optimizing future control inputs and effectively handling physical constraints on inputs and outputs by using system models to predict future outputs based on current and past inputs [21; 22; 23; 24; 25]. Ref. [26] proposed a distributionally robust MPC framework with output feedback to enhance control system performance under uncertainty. Ref. [27] developed an adaptive MPC method for cruise control in high-speed trains, incorporating time-varying parameters to improve real-time adaptability and robustness in dynamic conditions such as varying speeds and external disturbances.