## 1 Introduction  
Real-life control systems are subject to unpredictable disturbances that may severely decrease control performance. Therefore, control systems, which operate in real-world conditions, should be designed according to certain robustness criteria to deal with impacts of disturbance incidents on the control system performance. A major weakness of a typical control system is the dependence of the controller on a mathematical formulation of the plant--the so called model-based control design process. For consideration of disturbance dynamics, classical controller tuning requires modeling of disturbance incidences in order to ensure that the impact of disturbances on the system's performance is minimal. However, since disturbances are of unpredictable and uncertain character in many real-world applications, a classical controller may not respond optimally at the onset of a disturbance signal in a control loop. Another factor, which limits inherent disturbance rejection control performance of classical control loops, is the tradeoff between disturbance rejection control and set-point tracking performance. The controller tuning for higher disturbance rejection performance results in deterioration of the tracking performance [1]. Intelligent control techniques should be used to solve these problems. These techniques can detect disturbance incidences and adapt the control system response to reduce the negative impacts of disturbances on the control performance. The motivation for the current study comes from this requirement. In this manner, a two-loop control structure that involves specialized loops is considered: a fractional-order proportional integral derivative (FOPID) control loop for improved stability and set-point control performance, and a Model Reference Adaptive Control (MRAC) loop for improvement of disturbance rejection performance. These specialized control loops work in harness to increase disturbance rejection capability of the control system while preserving set-point tracking performance. MRAC is one of the widely used approaches for adaptive control of systems [2; 3; 4]. The MRAC structure uses a reference model that reinforces the control system according to a predefined reference model. Here, the reference model describes a desired response of the control system, and the control system is designated to resemble responses of the reference model [2]. In the current study, the outer loop is implemented by using Massachusetts Institute of Technology (MIT) rule. The MIT rule is a straightforward technique for performing the MRAC [4; 5; 6; 7] process. The MIT rule is essentially based on the decrease of difference between responses of the reference model and the control system by implementing a gradient descent method in adaptation loop (the outer loop). Direct use of the MIT rule for control action brings out some stability concerns because the control action of the MIT rule is very sensitive to output amplitude of the reference model [5]. This shortcoming is a major reason for the development of multi-loop MRAC-FOPID control structures, where the MRAC with MIT (outer loop) rule only activates in case of disturbance incidents. The set-point control and system stability are achieved by the closed-loop control system (the inner loop) [8; 9; 10; 11]. On the other hand, fractional control has attracted significant interest in the past few decades [12; 13; 14]. Several works have reported robust control performance improvements by using fractional-order controllers [15; 16; 17; 18]. In the current study, the FOPID controller is implemented by using a retuning FOPID controller method that was suggested as an effective approach to implementing FOPID controllers while keeping original PID control loops intact [17; 18]. The retuning FOPID control loop is used in case of this work for set-point control of the experimental magnetic levitation (ML) system and this loop is nestled into the inner loop of the proposed multi-loop MRAC-FOPID control structure [9].