An Overview of the Euler-Type Universal Numerical Integrator (E-TUNI): Applications in Non-Linear Dynamics and Predictive Control

A Universal Numerical Integrator (UNI) is a computational framework that combines a classical numerical integration method, such as Euler, Runge–Kutta, or Adams–Bashforth, with a universal approximator of functions, such as a feed-forward neural network (including MLP, SVM, RBF, among others) or a fuzzy inference system. The Euler-Type Universal Numerical Integrator (E–TUNI) is a particular case of UNI based on the first-order Euler integrator and is designed to model non-linear dynamic systems observed in real-world scenarios accurately. The UNI framework can be organized into three primary methodologies: the NARMAX model (Non-linear AutoRegressive Moving Average with eXogenous input), the mean derivatives approach (which characterizes E–TUNI), and the instantaneous derivatives approach. The E–TUNI methodology relies exclusively on mean derivative functions, distinguishing it from techniques that employ instantaneous derivatives. Although it is based on a first-order scheme, the E–TUNI achieves an accuracy level comparable to that of higher-order integrators. This performance is made possible by the incorporation of a neural network acting as a universal approximator, which significantly reduces the approximation error. This article provides a comprehensive overview of the E–TUNI methodology, focusing on its application to the modeling of non-linear autonomous dynamic systems and its use in predictive control. Several computational experiments are presented to illustrate and validate the effectiveness of the proposed method.