## 2 Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity
The set-point of this experimental system was originally 0.47 [34] and a step disturbance with an amplitude of 0.47 at 1000 s was applied to the input of the plant model. The results of the control system simulations are shown in \[\theta=-\gamma\frac{1}{s}\frac{dJ}{d\theta}=-\gamma\frac{1}{s}{e_{m}}\frac{{ de_{m}}}{d\theta} \tag{5}\]3 The simulation results reveal that both the set-point and disturbance rejection performances of the ML-MR PID-PID-PFA structure re better than other controllers. The lowest overshoot and fastest settling results were achieved by the ML-MR PID-PID-PFA structure. It is noteworthy to observe that the control performance of the ML-MR PID-PID-PFA structure is superior to the optimal FOPID control system. This finding is a clear indication of the potential for multi-loop PID controllers to surpass the control performance of the FOPID controllers. The FOPID controllers have been considered as substitutes for classical PID control and have been extensively studied in the last decade, showing performance improvements of FOPID controllers over classical PID controllers. A major complication in practice for FOPID controllers is their realization complexity. Fractional order controllers can be implemented by using integer order models because near-ideal realization of fractional order derivatives and integral elements in digital systems is highly computationally expensive [2]. Recently, analog circuit realizations of fractional order controllers have been shown, with results promising low-cost and low-complexity solutions for practical realization of FOPID controllers [35; 36]. Recent works have shown the robust control performance improvement of MR-ML control structures with FOPID controllers [25; 26], indicating an avenue for robust control system research. \[\theta=-\gamma\frac{1}{s}\frac{dJ}{d\theta}=-\gamma\frac{1}{s}{e_{m}}\frac{{ de_{m}}}{d\theta} \tag{5}\]4 Block diagram of ML-MR PID-PID-PFA control structure. It is useful to investigate the frequency dependence of disturbance rejection performance. For this purpose, the performances of the classical closed loop PID control structure and ML-MR PID-PID-PFA control structure were compared in the control problem of the Basic Process Rig 38-100 Feedback Unit experimental platform [34] using sinusoidal waveform disturbance signals at several frequencies. The PID controller was tuned according to the ITAE design rule proposed by [PERSON] et al. Table 1 shows the mean absolute error (MAE) of the control error signal (\(\frac{1}{T}\int\limits_{0}^{T}\big{|}e_{c}(t)\big{|}dt\)) from simulations. For step disturbance, the ML-MR control structure reduces 50% the MAE of classical PID control system. In general, environmental disturbances are altered slowly and involve low frequency components. For very low frequency disturbance signals, the MAE of the ML-MR PID-PID-PFA control structure decreases down to 25% of the MAE of the classical PID control system. As the disturbance frequency increases, the MAE performance of the ML-MR PID-PID-PFA control structure equals the classical PID control structure. Figure 18 shows some of the simulation results and confirms these findings. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline Angular Frequencies & Step & 0.001 & 0.005 & 0.01 & 0.05 & 0.1 & 0.5 \\ \(\omega\) (rad/s) & Disturbance & & & & & & \\ \hline Classical PID Control & & & & & & & \\ ([PERSON] et al.) & 0.0470 & 0.0868 & 0.1748 & 0.1605 & 0.0560 & 0.0267 & 0.0145 \\ \hline ML-MR PID-PID-PFA & 0.0234 & 0.0216 & 0.1037 & 0.1465 & 0.0690 & 0.0250 & 0.0140 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mean absolute error (MAE) of control error signals obtained from control simulations for step and sinusoidal waveform disturbances. Figure 17: Control performance of various control systems for robust performance optimal control of a large time delay plant model [34].