For descriptor systems with both input faults and measurement faults, proportional, multiple-integral, and derivative (PMID) observer technique is presented to simultaneously estimate the system states, the fault signals and the finite times derivatives of the faults. For descriptor systems with input faults, input disturbances (or modeling errors) and output faults, a robust PMID observer is designed to simultaneously estimate the system states, the fault signals, the finite times derivatives of the faults, and attenuate the input disturbances (or modeling errors) successfully. The derivative gain is selected to make the observer error dynamics internally proper, and the proportional gain and the multiple-integral gains are chosen such that the error dynamics are internally stable and the effect of the input disturbances (or modeling errors) are prevailed over. The considered faults are allowed to be unbounded, thus the fault-tolerant control scheme is another interest of this study. The separation property between the proposed observer gains and the state-feedback gain is proven. Using the linear matrix inequality (LMI) technique, an PMID observer-based fault-tolerant control scheme is addressed. In the framework of the proposed fault-tolerant design, the closed-loop plant is guaranteed to be internally proper stable, the action of the faults to the plant is removed completely, and the effect of the input disturbance (or modeling error) to the plant output is also attenuated as desired. Finally, a numerical example is given to illustrate the design procedures, and simulations show the satisfactory tracking and fault-tolerant control performance.