A lot of effort has been devoted to the unknown input observer (UIO) research over the past years. However, the strong disturbance decoupling assumption (manifested as some rank constraint) is often implicitly embedded in much of the existing UIO work. With the purpose of state and fault estimation, this fact motivates us to investigate the viability of the UIO research when the strong disturbance decoupling is not possible, i.e., a “degenerate” problem of UIO decoupling exists. Inspired by the scheme of reducing the effect of external disturbance on estimation error, this paper incorporates the relaxed UIO (RxUIO) concept by means of the H∞H2, and mixed H2¡H∞ techniques. Necessary and sufficient conditions for the existence of different RxUIOs are presented in the tractable linear matrix inequality (LMI) form. Numerical experiments are presented to illustrate the effectiveness of the suggested method.