This paper considers the design of an adaptive second order terminal observer for robust fault reconstruction of nonlinear Lipschitz systems with unknown upper bound of derivative fault.Firstly,a linear transforming matrix is introduced,which transforms the system into two subsystems,and thus to reduce the dimension of the system.One of the subsystem is affected by fault and disturbances,while the other is free,which simplifies the design of observer.Then,the design method of the observer gain matrix is transformed into a convex optimization problem under linear matrix inequalities (LMIs).A second order non-singular terminal sliding mode observer is designed for the transformed system to realize the accurate estimation of state and fault.Considering the unknown upper bound of derivative fault,an adaptive algorithm is designed in the equivalent output error injection signal to ensure the sliding mode motion reach the sliding surface within limited time.Finally,an example demonstrates the effectiveness of the proposed method in the paper.