This paper deals with the design of an unknown input observer (UIO) with the assumption that the well-known observer matching condition is not satisfied. The proposed method can be used for fault detection problems with the use of residual vector. The basis of method is to compensate the unmatched uncertainties with the use of a set of auxiliary outputs. The introduced auxiliary outputs are obtained from successive integration of the system measurements and known inputs. Then, an unknown input observer is proposed which estimates exponentially the outputs. Therefore, the residual vector, generated from the estimated outputs and the actual outputs, will be obtained which insensitive to the unmatched disturbances. At the same time, the sensitivity of the proposed residual vector to the fault in sensors is investigated. The generated residual vector will be more robust against the presence of noise in the measurements. It is shown through numerical simulations that the proposed residual vector is sensitive to the presence of fault in sensors while it is insensitive to the presence of the unknown input. In addition, a comparison with a derivative based method is presented.