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A model predictive controller based on linear matrix inequalities (LMIs) is presented. As in standard model predictive control (MPC) algorithms, at each (sampling) time, a convex optimisation problem is solved to compute the control law. The optimisation involves constraints written as LMIs, including those normally associated with MPC problems, such as input and output limits. Even though a state-space representation is used, only the measurable output and the extreme values of the unmeasurable states are used to determine the controller, hence, it is an output feedback control design method. Stability of the closed-loop system is demonstrated. Based on this MPC, a Lyapunov matrix is built and the controller computation is set in a more standard MPC framework. The design techniques are illustrated with numerical examples.