Starting from an examination of the existence of analytical redundancy, this paper provides a new approach to the fault detection and isolation observer design. The approach is based on an explicit decomposition of an extended matrix pencil containing all system matrices. The decomposition enables an easy check for the existence of analytical redundancy for fault detection and isolation and a straightforward design of the residual generator. Three important classes of faults, component, actuator and sensor faults can be treated in a unified manner. The proposed design is numerically reliable and has straightforward computer implementation. Several illustrative examples are included.