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The design of an adaptive cruise controller for a Smart car, a type of small car, is proposed as a benchmark setup for several model predictive control methods for nonlinear and piecewise affine systems. Each of these methods has been already applied to specific case studies, different from method to method. This paper has therefore the purpose of implementing and comparing them over a common benchmark, allowing us to assess the main properties, characteristics, and strong/weak points of each method. In the simulations, a realistic model of the Smart car, including gear box and engine nonlinearities, is considered. A description of the methods to be compared is presented, and the comparison results are collected in a table. In particular, the tradeoffs between complexity and accuracy of the solution, as well as computational aspects are highlighted.