This paper proposes a model predictive control strategy for solving the control problem affecting vehicle platoons:the limited reading range of on-board sensors and driving constraints. Considering that the relative velocity and acceleration of the longitudinal platoon model are not measurable, a state observer method is adopted to estimate the state of motion of the vehicle platoon. Moreover, the classical PD control strategy is applied to parameterize a set of platoon switching controllers, whose parameter matrices are calculated by solving the finite-horizon optimal control problem that affects the tracking and ride comfort performance. Based on this parameterized control strategy, the terminal constraints and the common Lyapunov functions are employed to establish the recursive feasibility of the parameterized platoon predictive controller, the cruise stability, and the string stability of the overall vehicle platoon system. Finally, the effectiveness of the proposed method is assessed by simulating the two typical scenarios of acceleration and deceleration.