**Streszczenie**

This paper has two main objectives. First, it gives an overview on the identification of MIMO nonlinear systems using NARX models. It covers the classical approach of the FROLS method, as well as the SEMP method. The second is to present some new useful results in model structure selection for NARX polynomial models applied to MIMO systems. It shows how to make a representation of MIMO systems from NARX polynomial models and the application of classical methods to identify these models. The study case used is a real didactic quadruple tank system manufactured by Quanser.

**Słowa kluczowe:**

*Nonlinear system identification, MIMO systems, Model Structure Selection, Coupled Tanks, quadruple tanks.*

**Abstract**

Artykuł ma dwa cele. Po pierwsze przedstawia przegląd metod identyfikacji nieliniowych systemów MIMO przy użyciu modelu NARX. Przedstawiono klasyczną metodę FROLS a także metodę SEMP. Po drugie przedstawiono użyteczne wyniki selekcji struktury wielomianowego modelu NARX zastosowanego do systemów MIMO. Nieliniowy system identyfikacji systemu

**Keywords:**

*nieliniowy system identyfikacji, system MIMO, model NATRX*

System identification is a knowledge area with the main objective of developing methods and techniques to find accurate and reliable mathematical representations for dynamic systems from observed data and available knowledge [1, 2]. A way to build these models is using information from collected system data to build ratios between inputs and outputs from the process, thus describing their dynamics. This method is called black-box identification since no information about internal details is used. Some real applications, especially industrial applications, have multiple-inputs and multiple-outputs (MIMO) systems. Such systems present some difficulties in obtaining mathematical models by physical laws (white-box modelling) or even by black-box methods to represent their dynamic behaviors. Therefore, linear models are used to represent the dynamics of such MIMO systems but they present lower quality than non-linear models. In black-box system identification, the representation of the polynomial nonlinear autoregressive moving average with exogenous input (NARMAX), proposed by [3], has a great performance in its ability to represent a nonlinear input-output relationship. In cases where the deterministic input-output relationship is the focus, a nonlinear autoregressive with exogenous input (NARX) model can be employed, using a simplification of the disturbance model [4]. Other researchers demonstrate these capabilities improved with applied techniques of structure selection and parameter estimation, enabling to find models with better representation and most robust performance, as explored in [5, 6]. Some papers propose the use of Artificial Neural Networks models [7, 8] or more elaborate methods [9, 10] for the task of identifying MIMO systems. However, NARX model-based identification methods, although generating less accurate models than the previously mentioned methods, have the advantage of generating simpler and easier [...]

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