Neural networks to model dynamic systems with time delays

An algorithm is proposed to identify a neural network model that represents a nonlinear dynamic system with a multivariate time delay response. The al gorithm consists of two major parts. The first one identifies the time dela y vector for a given neural network structure. This task is accomplished by using an exhaustive integer enumeration algorithm that minimizes a statist ical parameter to assess the performance of the neural network model. The s econd part uses a cross-validation strategy to identify the best neural net work model. Since the structure that models a nonlinear system is usually u nknown, the identification strategy consists of selecting several neural ne twork structures and identifying the best time delay vector for each networ k. The modeling process starts with the simplest structure and progressivel y the complexity of the network is increased to end up with a complex struc ture. Finally, the network that offers the simplest structure with the best network performance is the one that exhibits the appropriate neural networ k structure with the corresponding optimal time delay vector. The Monte Car lo simulation technique was used to test the performance of the algorithm u nder the presence of linear and nonlinear relationships among several varia bles of dynamic systems and with a different time delay applied to each inp ut variable. The introduced algorithm is used to detect a chemical reaction delay among enriched amyl acetate, acetic acid, water, and the pH of eryth romycin salt. An appropriate neural network model was designed to model the pH of the erythromycin during a continuous extraction process. To the best of the authors knowledge the proposed algorithm is the only one currently available to identify time delay interactions in the multivariate input out put variables of a system. The major drawback of the introduced algorithm i s that it becomes very slow as the number of system inputs increases. This algorithm works efficiently in a system that involves five inputs or less.