Neural networks (NNs) have numerous applications to online processes, but the problem of stability is rarely discussed. This is an extremely important issue because, if the stability of a solution is not guaranteed, the equipment that is being used can be damaged, which can also cause serious accidents. It is true that in some research papers this problem has been considered, but this concerns continuous-time NN only. At the same time, there are many systems that are better described in the discrete time domain such as population of animals, the annual expenses in an industry, the interest earned by a bank, or the prediction of the distribution of loads stored every hour in a warehouse. Therefore, it is of paramount importance to consider the stability of the discrete-time NN. This paper makes several important contributions. 1) A theorem is stated and proven which guarantees uniform stability of a general discrete-time system. 2) It is proven that the backpropagation (BP) algorithm with a new time-varying rate is uniformly stable for online identification and the identification error converges to a small zone bounded by the uncertainty. 3) It is proven that the weights' error is bounded by the initial weights' error, i.e., overfitting is eliminated in the proposed algorithm. 4) The BP algorithm is applied to predict the distribution of loads that a transelevator receives from a trailer and places in the deposits in a warehouse every hour, so that the deposits in the warehouse are reserved in advance using the prediction results. 5) The BP algorithm is compared with the recursive least square (RLS) algorithm and with the Takagi-Sugeno type fuzzy inference system in the problem of predicting the distribution of loads in a warehouse, giving that the first and the second are stable and the third is unstable. 6) The BP algorithm is compared with the RLS algorithm and with the Kalman filter algorithm in a synthetic example.

In this article, an argument Kalman filter is exposed for the fast updating of a neural network. The argument Kalman filter is developed based on the extended Kalman filter, but the recommended scheme has the next two advantages: first, it has less computational complexity because it only employs the Jacobian argument instead of the full Jacobian, second, its gain is ensured to be uniformly stable based on the Lyapunov approach. The commented scheme is applied for the modeling of two Takagi-Sugeno fuzzy models.

In this study, a neuro‐controller with adaptive deadzone compensation for a class of unknown SISO non‐linear systems in a Brunovsky form with uncertain deadzone input is presented. Based on a proper smooth parameterisation of the deadzone, the unknown dynamics is identified by using a continuous time recurrent neural network whose weights are adjusted on‐line by stable differential learning laws. On the basis of this neural model so obtained, a feedback linearisation controller is developed in order to follow a bounded reference trajectory specified. By means of Lyapunov analysis, the boundedness of all the closed‐loop signals as well as the weights and deadzone parameter estimations is rigorously proven. Besides, the exponential convergence of the actual tracking error to a bounded zone is guaranteed. The effectiveness of this scheme is illustrated by a numerical simulation.