## 1 Introduction
In another work using ANNs [30], authors suggested a radial basis function (RBF) network to perform control and another RBF network for model identification of the system. As a third component, a PID controller was used for the stabilization of this three component control system. In addition to neural control approaches, some recent works also demonstrated the use of other control approaches such as observer-based adaptive control [31] and the Takagi-Sugeno fuzzy controller [32] a for magnetic levitation control problem. However, these works have not purposely focused on disturbance rejection control performance of magnetic levitation control system.. The current study specifically addresses disturbance rejection control of a magnetic levitation system and experimentally validates performance improvements. To improve the performance of classical integer-order MRAC with the MIT rule, fractional-order MRAC approaches have been suggested in several works [3]. [PERSON] et al. have introduced fractional-order adjustment rules and use of fractional order reference models. They indicated the improvement in adaptation performance by using fractional order MIT rule [3]. Then, [PERSON] et al. suggested the use of fractional-order derivative at the controlled system output as a feedback to fractional-order MIT rule [33]. This modification was shown to improve noise performance of fractional-order MIT rule in simulation examples [33]. Experimental validation of fractional order MITrule was demonstrated for an experimental rotor control system [7]. These methods are named direct MRAC because the MIT rule is directly used to control a class of plant functions. The main difference of the multi-loop MRAC from the direct MRAC is that the MIT rule in multi-loop MRAC does not account for stability and control actions of the plant. Stability and optimal control are ensured by an inner control loop, which is commonly an optimal closed-loop control system. Several fractional-order adaptive control approaches have been proposed to improve control performance of fractional order MRAC systems such as a composite model reference adaptive control approach using tracing and prediction errors [34], fractional adaptive controller method with controller parameter adjustment for automatic voltage regulator [35], MRAC with Bode ideal loop reference model [36]. The motivation behind the present research effort has thus been established. The contribution of this research work can be stated as follows: a multiloop control method is proposed based on the FOPID retuning approach and computational intelligence in the form of an artificial neural network-based reference model used in the MRAC scheme for endowing the complete control system with greater robustness without deteriorating set-point tracking performance. The complete control system is initially simulated using a model of a magnetic levitation system and then real-time experiments are carried out confirming the improvement in the performance of the control system. Organization of the study is as follows: Section 2 provides a theoretical background for all control methods considered in this study. In Section 3, the approaches to modeling the magnetic levitation system are discussed including the first principles approach and the artificial neural network based black box approach. Section 4 introduces multi-loop MRAC-FOPID control with the obtained NARX reference model and presents results stemming from simulation and experimental studies. Finally, conclusions are drawn in Section 5.