A distributed nonlinear fault observer is devised for a directed topological structure nonlinear multi-agent system with disturbances. First, the method in this paper simplifies the derivation process by transforming the original system into an augmented system. Second, the method in this paper relaxes the condition of the observer gain matrix, which makes the observer gain matrix have solutions in a wider range, and the LMI (Linear Matrix Inequality) for solving the observer gain matrix does not have equality constraints. Third, in this paper, there is no restriction on the derivatives of faults, and faults with derivatives approaching infinity can be tracked quickly. Finally, we verify the effectiveness of the adjustable parameter distributed nonlinear fault observer proposed in this paper through simulation.