## 4 Multi-Loop Mrac-Fopid Control with Nax Reference Model for MI System Control Application  
This section presents the experimental results of multi-loop MRAC-FOPID control of ML system by using a NARX reference model. The design steps of Multi-loop MRAC-FOPID control was carried out for experimental ML system:  
1. The closed-loop retuning FOPID control system was implemented and FOPID controller was optimally tuned by using FOMCON toolbox [47]. 2. The control signal and sphere position data from the designed closed-loop control system described above were collected and these data were used to train the NARX model in the virtual closed-loop control system. Thus, the virtual closed-loop PID control loop with the NARX model is used as reference model to represent closed-loop retuning FOPID control system. 3. The outer loop is connected to inner loop according to MIT rule as shown in Figure 10. \begin{table}
\begin{tabular}{l l} \hline \hline
**Compare Type** & **MSE** \\ \hline ANN Model of-line (YEXP-YANN) & \(2.32\times 10^{-7}\) \\ Real-Time Simulation (YEXP-UEXP) & \(2.12\times 10^{-7}\) \\ Mathematical Model (YMATH-UEXP) & \(8.02\times 10^{-8}\) \\ ANN Model (YANN-UEXP) & \(2.83\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MSE ball position comparison among models. Figure 9: A comparison of ANN model performance and mathematical model performance. Regarding the configuration of the experimental system, in Figure 11, the complete control diagram used to implement both simulation and real time experiments is presented. For the implementation of the experimental configuration, the MATLAB/Simulink environment is used with FOMCON toolbox for implementing fractional-order control. The tuning of the controllers used in the retuning loop is detailed in [9]. However, compared to that work, the value of \(\gamma\) in the MRAC loop was tuned down from \(-40,000\) to \(-20,000\) in order to minimize high frequency components entering the control system and causing excessive vibrations of the sphere. Figure 12 shows a simulation result to demonstrate disturbance rejection performance of multi-loop MRAC-FOPID with NARX model. After settling of the sphere to a height of \(0.0109\) m, a step-down disturbance and then step-up input disturbances were applied at the simulation times \(10\) s and \(40\) s, respectively. The responses of the proposed multi-loop MRAC-FOPID and the conventional FOPID control are shown in the figure. The conventional FOPID control is the inner loop that was performed by disabling contribution of outer loop (MRAC with MIT rule). Thus we can test control improvements of the multi-loop MRAC-FOPID with the NARX model compared to the FOPID control loop alone. At disturbance incidents, multi-loop MRAC-FOPID with NARX model settles back to the set-point faster with lower ripples. These results indicate considerable improvements indisturbance rejection control performance of the FOPID control loop when the outer loop (MRAC) is enabled. Armed with promising simulation results, real-time verification of the multi-loop control system was performed. The Simulink model was built according to Figure 11 with additive disturbance injected into the system using the control input \(u(t)\) in the original PID control loop. The model runs on a Desktop PC (3.40 GHz Processor and 8.00 GB RAM) and the computational complexity of the control scheme allows real-time operation with 0.001 data sampling rate. Figures 13 and 14 show experimental results to demonstrate disturbance rejection performance of the multi-loop MRAC-FOPID control with NARX reference in comparison with the conventional FOPID control (MRAC is disabled). In Figure 13, after settling to the set-point of 0.0109 m high, a step-down input disturbance signal was applied at \(10\) s. Responses of control systems to this disturbance signal are compared in the figure. The multi-loop MRAC-FOPID control improves disturbance responses of ML system with faster resettling and lower ripples around the set-point.