With the development of automatic driving technology and the internet of vehicles, platooning based on control of connected autonomous vehicles has become one of the most promising approaches to improve traffic efficiency. This paper studies the control problem of mixed vehicle platoons consisting of human-driven vehicles and connected autonomous vehicles. Firstly, we propose a data-driven method to model mixed vehicle platoons based on Koopman operator theory. This method gives a way to represent the mixed vehicle platoon by a linear model in a high-dimensional space, the approximation of which is obtained by a neural network framework. Secondly, we employ model predictive control (MPC) to address the platoon control problem of mixed vehicle platoons, where both centralized MPC and distributed MPC algorithms are designed. Finally, the effectiveness of the data-driven modeling method and the centralized/distributed MPC algorithms is verified by numerical simulations. It is revealed that the proposed data-driven DMPC algorithm exhibits comparable control performance with less computation cost compared with the centralized MPC algorithm, and it shows faster convergence speed than the nonlinear model based DMPC algorithm.