A fuzzy relational sliding mode observer (FRSMO) and proportional integral observer (FRPIO) are proposed to estimate the magnitude of slowly evolving faults in information-poor, i.e., difficult to model and non-linear systems. Where there is no fault in the system, the mismatch between the actual output and the model output can be shown to be zero. When the fault occurs in the system, the error is not zero, which is called a residual and can be used to diagnose the fault. In the fuzzy PI observer, the size of the fault can be obtained from the error passing the PI feedback compensation. In the fuzzy sliding mode observer, the equivalent injection signal is used to compensate for the fault thus obtaining the magnitude of the fault. To reduce modelling errors, an on-line learning fault identification scheme (OLFIS) is used to update the model and identify the fault in a periodical mode with different time intervals during the whole procedure. The selection of the intervals between model update and fault identification, convergence and speed of the scheme are investigated. The performance of the proposed methods is evaluated using a cooling-coil subsystem of an air-conditioning plant in a simulation environment. An actuator fault in the valve and a flow reduction fault are two typical incipient faults in the cooling coil system. The results of the actuator fault estimation and flow reduction fault estimation confirm the effectiveness of the methods.