The detectability by conventional step-hypothesized generalized-likelihood-ratio (SHGLR) method for detection of a parameter change (fault detection) in a linear discrete dynamic system is analysed and it is shown that a weakly-diagnosable-space (WDS) exists for dynamics and sensor faults. Based on the fault detectability, a reduced order SHGLR method is then developed which highly improves the detection rate and speed. In the same framework of the GLR method, another reduced order detection scheme is given, which makes the most use of the information about the input and the state of the system to raise the detectability for faults for the case where the step hypothesis cannot be applied effectively.