## 2 Theoretical Background and Preliminaries
### Multi-Loop Mrac
When the reference input is substituted with \(r=y_{m}/T_{m}(s)\) in Equation (13), the sensitivity derivative can be obtained  
\[\frac{\mathrm{d}e_{m}}{\mathrm{d}\theta}=\frac{T(s)}{T_{m}(s)}y_{m}. \tag{14}\]  
By using the sensitivity derivative in Equation (11), MIT rule for the update of adaptation gain is obtained as  
\[\theta=-\gamma\frac{1}{s}\left(\frac{T(s)}{T_{m}(s)}y_{m}e_{m}\right). \tag{15}\]  
To investigate the contribution of the adaptation gain \(\theta\) to system response, let us consider the response of the system in disturbance-free state and in case of a disturbance incident. (i) Disturbance-free state: In this state, one can assume \(T(s)=T_{m}(s)\) because reference model is configured as mathematical model of optimally tuned inner model. Accordingly, the adaptation gain can be written as,  
\[\theta=-\gamma\frac{1}{s}y_{m}e_{m}. \tag{16}\]  
In the disturbance-free state, outputs of reference model and the inner loop match each other, hence \(e_{m}=0\). When \(e_{m}=0\) and \(T(s)=T_{m}(s)\) conditions are used in Equation (12), one obtains \(e_{m}=T_{m}(s)(\theta-1)r=0\). This arithmetically infers the state of \(\theta=1\) because the reference model \(T_{m}(s)\) is non-zero function and the reference signal \(r\) is the independent input variable. Therefore, adaptation gain settles the value of \(\theta=1\) in the disturbance-free state. It does not shape the reference input. This property validates that outer loop does not affect control performance of the inner loop in disturbance free case. (ii) Disturbance state: In case of a disturbance incident, one can assume a perturbation of the system model due to additional dynamics of the additive interference of disturbance. This effect results in differentiation of inner loop model from the reference model, and let us assume \(T(s)\
eq T_{m}(s)\). The MIT rule is based on gradient descent perform optimization of the adaptation gain \(\theta\) to minimize error function and it leads to \(\lim\limits_{t\rightarrow\infty}e_{m}(t)\to 0\). Therefore, one can consider the case \(e_{m}=T(s)\theta\,r-T_{m}(s)r\to 0\), where the adaptation should change \(\theta\rightarrow\frac{T_{m}(s)}{T(s)}\) to reach the state \(e_{m}=0\) in time. When the MIT rule achieves the state \(e_{m}=0\), response of the reference model and response of the inner loop become the same, and impacts of disturbance incident at the system output are rejected by the change of \(\theta\rightarrow\frac{T_{m}(s)}{T(s)}\). This analysis reveals the fact that the performance of the multi-loop MRAC-FOPID control system depends on the capability of the reference model to represent the response of the closed-loop FOPID control system. Therefore, identifying an accurate model of the closed-loop FOPID control system is of vital importance. This requirement constitutes the central motivation of this study. Therefore, practical use of NARX modelling for multi-loop MRAC-FOPID control is tested in a relatively difficult control problem that is the disturbance rejection control of a nonlinear, unstable dynamical system, namely the ML system. To address stability concerns related to this configuration, the relevant discussion is provided next. In particular, it is important to address to components of stability analysis: the theoretical one and the empirical one. In the former case, stability conditions are derived mathematically, in the latter one, experiments are conducted and the results analyzed to ensure that the control system is indeed stable. Stability and convergence conditions of the multi-loop MRAC-FOPID control system was summarized as follows in [11]:  
The reference model and inner loop are designed as a stable system. Therefore, a sufficient condition for stability of the multi-loop MRAC-FOPID control system can be written based on stability and convergence of model error \(|e_{m}|=|y-y_{m}|<\epsilon\in R\).