Friction stir welding (FSW) is an innovative solid-state welding technology that produces high quality joints and is widely used in the aerospace industry. Previous studies have revealed welding temperature to be a decisive factor for joint quality. Consequently, several temperature control systems for FSW have been developed. These output feedback control systems usually require delicate and expensive temperature measuring equipment, which reduces their suitability for industrial practice. This paper presents a novel state feedback system of the welding temperature to remedy this shortcoming. The system uses a physical model of the FSW process (digital twin) for the determination of the welding temperature signal from the process torque signal. The digital twin is based on a multi-input nonlinear time invariant model, which is fed with the torque signal from the spindle motor. A model-based L1 adaptive controller was employed for its robustness with respect to model inaccuracies and fast adaption to fluctuations in the controlled system. The experimental validation of the feedback control system showed improved weld quality compared to welded joints produced without temperature control. The achieved control accuracies depended on the results of the temperature calculation. Control deviations of less than 10 K could be achieved for certain welding parameters, and even for a work piece geometry, which deliberately caused heat accumulation.