In this paper a method is proposed to design an anti-windup scheme for discrete time linear systems with input saturations and actuator additive failures. This method provides a fault tolerant system reconfiguration mechanism with a control law which compensates for the estimated actuator additive faults and maintains the overall system stability in spite of actuator saturations. The design approach is derived from the solution of linear matrix inequalities (LMI) to guarantee the stability regions. For that purpose the fault tolerant control method is based on a linear quadratic regulator (LQR) and a fault estimator for compensation purposes. This method was tested in realistic simulations with the software Carins (CNES) on a pressure and mass flow rate model of a cryogenic test bench cooling circuit.