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This paper investigates the use of canonical piecewise affine (PWA) functions for approximation and fast implementation of linear MPC controllers. The control law is approximated in an optimal way over a regular simplicial partition of a given set of states of interest. The stability properties of the resulting closed-loop system are analyzed by constructing a suitable PWA Lyapunov function. The main advantage of the proposed approach to the implementation of MPC controllers is that the resulting stabilizing approximate MPC controller can be implemented on chip with sampling times in the order of tens of nanoseconds.