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We consider linear control systems with saturation in the input for which we know two stabilizing output feedback controllers. One is a nonlinear globally asymptotically stabilizing controller, while the other one is a linear only locally asymptotically stabilizing controller. We look for a composite output feedback control law that is equal to the linear controller for initial conditions in a neighborhood of the origin and that is globally asymptotically stabilizing. We suggest a constructive approach to solve this uniting problem and to modify the linear local controller into a nonlinear global controller. Moreover since we want some robustness with respect to measurement noise, actuator errors and external disturbances, we consider hybrid output feedback controllers, following developments in the literature on robustness in hybrid systems. We illustrate our main result by means of numerical examples.