We present a proportional, multiple-integral and derivative (PMID) observer technique that can simultaneously estimate system states, fault signals and the finite times derivatives of the faults for a descriptor system with input and measurement faults. Furthermore for a descriptor system with input and measurement faults and unknown disturbances (including modelling errors), a robust PMID observer is designed to simultaneously estimate system states, fault signals, the derivatives of the faults, and attenuate disturbances successfully. Fault-tolerant design is another important issue in this study. By using the obtained estimates of states and faults, and linear matrix inequality technique, a fault-tolerant control scheme is addressed, which ensures the closed-loop plant to be internally proper stable with prescribed H1 performance index even as unbounded faults occur. Finally, a numerical example is given to illustrate the design procedures, and simulations show satisfactory tracking and fault-tolerant control performance.