Residual generation for linear differential-algebraic systems is considered. A new systematic method for observer-based residual generation is presented. The proposed design method places no restrictions on the system to be diagnosed. If the fault of interest can be detected in the system, the output from the design method is a residual generator in statespace form that is sensitive to the fault of interest. The method is iterative and relies only on constant matrix operations such as multiplications, null-space calculations and equivalence transformations, and thereby straightforward to implement. An illustrative numerical example is included, where the design method is applied to a non-observable model of a robot manipulator.