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In this paper, a decentralized sliding-mode control approach is applied to the control tasks of vehicles in platoons. Using the well-known bicycle model, a robust nonlinear observer is introduced to facilitate the controller design, which needs full-state measurements. The vehicles in platoons can be treated as an interconnected system with a special form. Observer gain and controller gain are properly designed. In addition, appropriate linear matrix inequality (LMI) stability conditions by the Lyapunov method are derived to ensure the stability of the system. The main advantages can be summarized as follows: (1) The linear approximation of the nonlinear vehicle model enables various advanced robust control possibilities. (2) The proposed robust control approach with the nonlinear observer ensures the convergence of the whole interconnected system, given that the system is operated within the stable region of linearization. (3) Stability conditions in the form of LMIs for both observer and controller are rigorously derived. Finally, simulation results for three identical vehicles based on the relative bicycle model are demonstrated to show the performance of the approach.