This research is to solve a problem of sustainability of a balancing robot controlled by an artificial neural network. The mentioned network acts as a regulator and calculates at its output layer a control action for the plant. Online training of such a network is necessary to improve the quality of the robot control since it changes its parameters or a mode of functioning in the course of operation. Implementing such training, the question of the learning rate limitation arises sharply. It is directly related to the assessment of sustainability of the control system under consideration. That is why a method based on the second Lyapunov approach is proposed to calculate the upper allowable limit of the online learning rate for the neural network controller under various conditions at each moment of its functioning. This method does not require the plant mathematical model. The efficiency of the approach is proved by experiments with a real balancing robot based on the EV3 platform.