In this paper a stable neural network-based identification and control method for a class of first-order nonlinear plants is suggested. The method is based on neural networks built using a rational activation function. The paper discusses a method for stable adjustment of the network parameters and robustness issues in the resulting closed-loop system. The overall control scheme also includes a robust global controller that guarantees that the state of the system is confined to a region in the state space. The combination of robust global and neural network-based controllers appears to be well suited for performance improvement in complex nonlinear systems.