This paper investigates the problem of compensating for the slow-changing sensor drift failures occurring in digital PID systems with unknown dynamics. It is extremely difficult to ensure the tracking performance of the faulty systems, due to the system models unavailable, the measurements corrupted by the faults, and the impact of the sensor failures propagated under feedback control. Therefore, the existing data driven fault-tolerant control (FTC) methods are not capable of dealing with the above problem. Contrary to the current state-of-the-art, a novel residual generator structure is devised. Furthermore, its data-driven realization is accomplished, along with the convergence analysis of the proposed closed-loop recursive identification algorithm. On this basis, such failures can be estimated continuously from the residual signals, using an iterative estimation procedure, the convergence of whose mean-square estimation error is theoretically proven. As a result, the effect of the aforementioned failures on the tracking properties is eliminated with the aid of a fault detection mechanism and based on the tracking error signal corrected with the estimated faults. Finally, the effectiveness and merits of the resultant data-driven FTC algorithm are validated by the continuous stirred tank heater benchmark process. (C) 2017 Elsevier Ltd. All rights reserved.