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The distributed proportional-integral-derivative (PID) controller has the potential to enhance the robustness of the platoon control and improve the dynamic tracking ability of the following vehicles. Considering that time delays degrade the performance and robustness of the vehicle platoon, in this paper, the distributed PID controller is designed for the vehicle platoon with communication delay and input delay, and the internal stability and the string stability of the vehicle platoon are investigated. First, in the internal stability analysis, the strong stability conditions are derived to guarantee the existence of the stable region. Then, the exact time delay bounds (ETDBs) of the internal stability are obtained by the cluster treatment of characteristic roots (CTCR) paradigm. Second, in the string stability analysis, the string stability conditions are given by analyzing the error transfer function to avoid disturbances amplifying along the string of the platoon. The input delay bound of string stability is derived in terms of the PID controller gains and the model parameters, which reveals the advantages of the PID controller for vehicle platoon control. Finally, simulations illustrate that the proposed platoon stability analysis methods are correct and efficient.