In this paper, we present a new sensor fault tolerant control scheme for linear, discrete-time systems. The scheme consists of a bank of estimators, each associated with a sensor or group of sensors, a mechanism for the detection and identification of sensor transitions from healthy-to-faulty and faulty-to-healthy operation, an estimate reconfiguration module, and an estimate-based feedback controller with reference tracking. The detection and identification approach is based on the separation of "healthy" and "faulty" sets, where appropriately selected residual variables remain under healthy or faulty operation, from "after-fault" and "after-recovery" sets, towards which the residual variables "jump" when abrupt sensor faults or recoveries occur in one or more groups of sensors. This "set-separation" approach provides pre-checkable fault tolerance guarantees whenever certain conditions, given in terms of system's known data such as plant and estimator dynamics and bounds on reference and disturbance signals, are satisfied.