For a class of Takagi-Sugeno (T-S) fuzzy nonlinear systems, in this article, an improved fuzzy observer-based fault estimator is designed to estimate the fault signals on-line based on common Lyapunov function. Attention is focused on the linear matrix inequality (LMI)-based design method with specified performance constraints and less conservatism, which is suitable for complex system models with more rules. The multiobjective and H-infinity optimisation theory is applied to cope with the constraints on the disc-regional poles assignment index and the robustness to disturbances. Some developed relaxing techniques are utilised to reduce the conservatism and the number of LMI constraints which were generated by the conventional design method with common quadratic Lyapunov function. Thus, the resulting estimator is not only less conservative, but can also guarantee the desired fault estimation performance on the rapidity as well as robustness to neglected modelling dynamics, and the robustness against external disturbances with satisfied estimation accuracy to faults signals. The conventional design is also presented for comparative study. All the results are formulated in the form of LMIs. Simulative examples demonstrate that our proposed method is less conservative and effective for fault estimation.