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Explicit model predictive controllers computed exactly by multi-parametric optimization techniques often lead to piecewise affine (PWA) state feedback controllers with highly complex and irregular partitionings of the feasible set. In many cases complexity prohibits the implementation of the resulting MPC control law for fast or large-scale system. This paper presents a new approach to synthesize low-complexity PWA controllers on regular partitionings that enhance fast on-line implementation with low memory requirements. Based on a PWA control-Lyapunov function, which can be obtained as the optimal cost for a constrained linear system corresponding to a stabilizing MPC setup, the synthesis procedure for the low-complexity control law boils down to local linear programming (LP) feasibility problems, which guarantee stability, constraint satisfaction, and certain performance requirements. Initially, the PWA controllers are computed on a fixed regular partitioning. However, we also present an automatic refinement procedure to refine the partitioning where necessary in order to satisfy the design specifications. A numerical example show the effectiveness of the novel approach.