## 2 Theoretical Background and Preliminaries
### Multi-Loop Mrac
Assuming that the inner loop and reference model are designed stable, multi-loop MRAC-FOPID control systems are stable and convergent when model error \(e_{m}\) is bounded, the input is bounded, and output is stable and convergent. Therefore, it is sufficient to investigate the convergence condition of model error \(e_{m}\) to show the stability of multi-loop MRAC-FOPID control system. Lemma 1: _(Zero condition of the model error [11]): For multi-loop MRAC-FOPID control structure, the model error takes a zero value (\(e_{m}=0\)) when the adaptation gain \(\theta\) is equal to one and the transfer functions of reference model and inner loop are equal (\(T(s)=T_{m}(s)\))._  
Proof: By considering the model error given by Equation (12) for multi-loop MRAC-FOPID control structure, the model error \(e_{m}=y-y_{m}\) can be written for \(\theta=1\), and \(T(s)=T_{m}(s)\) as  
\[e_{m}=T(s)\theta r-T_{m}(s)r=T(s)r-T_{m}(s)r=0. \tag{17}\]  
Then, the model error becomes zero. This proves a zero condition of the model error as to be \(\theta=1\) and \(T(s)=T_{m}(s)\). Theorem 1: _(Stability of model error [11]): For a stable system models \(T(s)=T_{m}(s)\), multi-loop MRAC-FOPID control system is stable only if the model error function:_  
Figure 2:0  
_expresses a stable function._  
Proof: By using the adaptation rule of multi-loop MRAC-FOPID control system that was given by Equation (16) as \(\theta=-\gamma\frac{1}{s}y_{m}e_{m}\), the model error can be derived as  
Figure 2:1Output of reference model \(y_{m}\) can be written as \(y_{m}=T_{m}(s)r(s)\). For a set-point control stability, one considers step response of the system. Therefore, the reference input is assumed to be a step function by substituting \(r(s)=\frac{1}{s}\) in Equation (19). Figure 2:2  
According to Lemma 1, a zero value of model error (\(e_{m}=0\)) is possible for the case of \(\theta=1\), and \(T(s)=T_{m}(s)\). By applying this condition, the model error is obtained as  
Figure 2:3  
The function \(T(s)\) is transfer function of inner loop that is a closed-loop FOPID controller. Therefore, it can be expressed in the form of \(T(s)=\frac{1}{1+C(s)C(s)}\). When \(T(s)\) is used in Equation (21), one can write and reach to Equation (18). The multi-loop MRAC--FOPID control system is stable if the function \(e_{m}\) given Equation (18) is a stable function (check if the all characteristic roots of characteristic equation of \(e_{m}\) lies on the left half plane (LHP)). Then, one can infer that when \(e_{m}\) function is LHP stable function, the amplitude \(|e_{m}|=|y-y_{m}|<\epsilon\in R\) is bounded. Since the reference model was always configured to be stable function in design, the reference model output is bounded ( \(|y_{m}|<\epsilon\in R\)) and it leads output of plant is also bounded (\(|y|<\delta\in R\) ), and whole system is stable. Mathematical Model of the MI System: From a First Principles Model to a Narx Black Box Approximation  
The content of the following section conveys the mathematical modeling of the ML system using two different approaches. First, first-principles modeling is considered; then, the NARX model is developed. It is important to compare the accuracy of both models to ensure that the latter represents a coherent model of the ML system's dynamics. The Magnetic Levitation System (MLS) is a nonlinear, open-loop unstable, time-varying dynamical system with negligible air friction force. Figure 3 shows a picture of an experimental ML setup that was used for testing the proposed control structure. The objective in this experimental system is to control the terminal voltage of an electromagnet so that the ferromagnetic sphere can be levitated at the desired distance from the magnet. One of the promising applications of ML is the ML transportation systems. Specifically, physically contactless transportation on a guide-way can offer a number of advantages such as lower noise, less friction forces, less costly maintenance, higher efficiency and less emission of exhaust fumes.