## 2 Theoretical Background and Preliminaries
### Multi-Loop Mrac  
Direct use of the MIT rule for control action brings out some instability concerns because the control action of the MIT rule is very sensitive to output amplitude of reference models [5]. This shortcoming is the main motivation for the development of two-loop MRAC-FOPID control structures, where an outer loop, which is an MRAC loop with MIT rule, only performs for adaptation to reference model. The two-loop MRAC-FOPID control has been proposed to improve the robust control performance of the conventional FOPID control loop [8; 9]. For this purpose, an additional outer loop to improve adaptation characteristics of the control system is appended to the FOPID control loop. In other words, the outer loop with a classical integer-order MIT rule encloses the inner loop and it is called the adaptation loop. This simplifies the design task of the outer loop. Thus, stability and set-point control performance are managed by the inner loop that is a FOPID control loop, and the disturbance rejection control to improve robust control performance is provided by the outer loop that is an MRAC loop with MIT rule. The design task of this control structure includes the following basic steps [8; 9]:  
Step 1: The inner loop (the FOPID control loop) is designed optimally by using any optimal FOPID tuning method. Robust stability and a satisfactory set-point control performance should be achieved by using a suitable optimal controller tuning method. Step 2: A model of the inner loop is obtained and used as a reference model of MRAC structure. Step 3: The outer loop is connected to the inner loop according to MIT rule. The final structure of the two-loop MRAC-FOPID control is shown in Figure 2. In the figure, the adaptation gain \(\theta\) is multiplied to the reference input \(r\) in order to shape the reference input when the response of the reference model \(T_{m}(s)\) diverges from the response of FOPID loop due to disturbance incidents. Therefore the outer loop, which is also known as the adaptation loop, enforces the inner loop, which is called the control loop, to behave similarly to the reference model that describes disturbance-free and optimal responses of the inner loop. Specifically, Step 2 ensures the correct establishment of this mechanism. It should be noted that in this work, Step 2 is modified. This will be clear from the following discussion. Specifically, the reference model is obtained independently from the inner loop. The closed-loop retuning FOPID control system was implemented and the FOPID controller was optimally tuned by using FOMCON toolbox [47]. The theoretical background of MIT rule to implement the outer loop was summarized as follows [8; 9]:  
Figure 2: Block diagram of multi-loop Model Reference Adaptive Control (MRAC)-FOPID system [9]. The MIT rule works for the minimization of model error that is defined according to difference between outputs of inner loop (FOPID control loop) and the reference model as \(e_{m}=y-y_{m}\). Therefore, the error function to be minimized is written in form of square of instant model error  
\[J=\frac{1}{2}e_{m}^{2}. \tag{10}\]  
According to Figure 2, to perform adaptation by applying input-shaping technique, the input of closed-loop system is modified as \(\bar{u}=\theta\,r\), where the adaptation gain \(\theta\) is determined according to the MIT rule. The MIT rule performs the continuous time gradient descent method to minimize the error function, it was expressed for MRAC [2; 3; 4; 5; 6] as  
\[\frac{\mathrm{d}\theta}{\mathrm{d}t}=-\gamma\frac{\mathrm{d}J}{\mathrm{d} \theta}=-\gamma e_{m}\frac{\mathrm{d}e_{m}}{\mathrm{d}\theta}. \tag{11}\]  
By considering multi-loop MRAC-FOPID structure in Figure 2, the model error can be expressed as [8; 9]  
\[e_{m}=y-y_{m}=T(s)\theta r-T_{m}(s)r. \tag{12}\]  
Then, the sensitivity derivative \(de_{m}/d\theta\) is written by  
\[\frac{\mathrm{d}e_{m}}{\mathrm{d}\theta}=T(s)r, \tag{13}\]  
where the transfer function \(T(s)\) represents the current model of the inner loop.