Fault detection (FD) is the process of monitoring a system to check if some malfunctions or abnormalities occur in the system. One of the widely used fault detection mechanism is the so-called model based fault detection, where a model of the target system is involved to estimate the expected output of the system under healthy condition. Then a fault can be detected by comparing the actual measured output the estimated healthy output. By making use of the state estimation capability of observers, various observer-based fault detection schemes have been proposed to estimate the system output for the purpose of fault detection. However, it is worth noting that, different from state-feedback control, observer-based fault detection only concerns the output and it is not necessary to have a full state estimation. In this paper, a new type of observers, referred to as output observer, is proposed for fault detection, thus the unnecessary state estimation in observer-based fault detection can be avoided. First, an input/output system representation is developed to enable us build the output observer. Then a new approach of output observer design is presented, in which only the output variables are estimated instead of estimating any other system state variables. The application of output observer to fault detection in systems under disturbance is presented and systematic approach to design the FD scheme based on an optimisation method is given. This FD scheme is validated by simulating a model of a three wheels robot.