In this study, a fixed-time adaptive cooperative controller by a self-structuring neural network is proposed, and actuator faults are considered for multi-robot systems. First, a novel fixed-time leader state observer is developed to estimate the state information of the leader and pass on to other followers without measuring the leader's velocity. Second, a fixed-time cooperative controller is designed to achieve fast response and high precision. Third, a fixed-time convergent self-structuring neural network is designed to improve the approximation accuracy affected by system uncertainties and actuator faults. A new neuronal splitting strategy is designed to avoid excessive computational burden caused by too many neurons. Next, the Lyapunov stability theorem is employed to demonstrate that the whole error closed-loop system can globally converge to a small region around zero in a fixed time. Finally, a simulation example on multi-robot systems shows that the proposed fixed-time adaptive cooperative controller is able to obtain satisfactory performances in the presence of uncertainties from external disturbances, actuator faults and other causes.