MPC has become the accepted standard for complex constrained multivariable control problems in the process industries. One of those control problems is traction control, which is used to improve a driver’s ability to control a vehicle under adverse external conditions such as wet or icy roads. By maximizing the tractive force between the vehicle’s tire and the road, a traction controller prevents the wheel from slipping and at the same time improves vehicle stability and steering ability. In most control schemes the wheel slip, i.e.,the difference between the normalized vehicle speed and the speed of the wheel is chosen as the controlled variable. The objective of the controller is to maximize the tractive torque while preserving the stability of the system. The relation between the tractive force and the wheel slip is nonlinear and is a function of the road condition. The presence of nonlinearities and constraints on one hand, and the simplicity needed for real-time implementation on the other, have discouraged the use of optimal control theory for designing practical traction control systems. In this thesis, a multiparametric quadratic programming(mp-QP) method is reviewed that moves all the computations necessary for the implementation of MPC off-line, while it preserves all other characteristics of the MPC controller. The mp-QP is successfully implemented both for reference tracking and state regulation problems. Due to the nonlinear nature of the wheel slip dynamics, a linear model is inadequate and standard MPC methodologies can not be applied to the traction control problem. We show how a Mixed Logical Dynamical(MLD) model for the traction system can be obtained using the HYSDEL compiler. Two efficient method are presented for computing the explicit MPC for MLD and Piecewise Affine (PWA) systems. We apply our methodology to a piecewise affine model of the traction system and compute an explicit nonlinear traction control law with very good performance in simulations. Key words Multiparametric Linear Programming; Multiparametric Quadratic Programming; Mixed Logical Dynamical system(MLD); Hybrid System; Piecewise Affine system(PWA); HYbrid System DEscription Language(HYSDEL)