## 1 Introduction
The outer loop with the MIT rule encloses the inner loop and it is called the adaptation loop. The adaptation loop (the outer loop) is designed to detect the disturbance incidents and perform an adaptation of the inner loop in order to decrease negative effects of disturbance on the control performance. An experimental study of a multi-loop MRAC-FOPID control structure was carried out for magnetic levitation (ML) control problem by [PERSON] et al. [9]. Authors observed that the performance of multi-loop MRAC-FOPID control with MIT rule largely depends on capability of the reference model to represent leading closed-loop dynamics of the experimental ML system. In [9], a linear model of an experimental ML system was used to obtain the transfer function ofthe closed-loop retuning FOPID control system. This function was used as the reference model. This modeling approach produces two drawbacks for the multi-loop MRAC-FOPID control structure. 1. Linearization of a nonlinear ML system makes the reference model valid at the modeling conditions such as operating temperature, sphere motion ranges etc. This limitation decreases the accuracy of the reference model. 2. Identification of the inner loop is an initial process and this static reference model is not adaptive for changes in condition and behavior of the real systems. In the current study, to deal with these drawbacks, nonlinear autoregressive neural network with external inputs (NARX) modeling is employed to obtain a more accurate reference model. The use of neural networks in system identification allows the utilization of machine learning techniques in control, and this makes the multi-loop MRAC-FOPID control structure with the NARX model more adaptive compared to previous configurations. Recurrent neural networks have been effectively used for intelligent control and online system modeling [19; 20; 21; 22; 23; 24]. Autoregressive neural network models can online learn the dynamical responses of linear and nonlinear systems from sampled input and output data [21; 22; 23; 24]. Therefore, it gains significant flexibility for multi-loop MRAC-FOPID control structures to employ in real applications. NARX modeling can automatically learn responses of a wide range of system models as a black-box model and this reduces the need for guessing the model structure prior to model identification. The reference model is automatically identified from data that are captured from the input and output of controlled systems online. This point is an important contribution of this study to facilitate the use of multi-loop MRAC-FOPID control structures in practical control applications. Preliminary studies on neural network control have discussed the use of ANNs in control practice and neural control concepts in [25; 26]. Nonlinearity in the behavior of real systems is one of the major factors that complicates the control design task and reduces practical control performance [27]. The static linear control based on linearized models may not always yield satisfactory results for control of nonlinear systems because of unpredictable alterations in operation conditions in the real world systems. Several nonlinear control methods have been proposed to address control problems of nonlinear systems in [27]. However today's intelligent system paradigm requires highly adaptive, model-free control systems. ANNs have emerged as a possible candidate for evolution of control systems to meet these requirements [28]. For magnetic levitation control problems, several control approaches have been proposed in the literature. In some research, neural networks were successfully used. For instance, a recurrent neural network-based adaptive optimal backstepping strategy was proposed--in the work, authors used a dynamic model of the recurrent neural network (RNN) to solve the constrained optimization problem to achieve improved control performance [29].