## 2 Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity
Then, the sensitivity derivative of the system is obtained:  
\[\frac{{de_{m}}}{d\theta}=T(s)r \tag{6}\]  
Here, one can substitute the reference input with \(r=y_{m}/T_{m}(s)\) in Equation (6) and rearrange the sensitivity derivative as:  
\[\frac{{de_{m}}}{d\theta}=\frac{T(s)}{T_{m}(s)}y_{m} \tag{7}\]  
By using it in Equation (5), the MIT rule for the update of the adaptation coefficient \(\theta\) to minimize cost function \(J\) is obtained as [25; 26; 28]:  
\[\theta=-\gamma\frac{1}{s}\bigg{(}\frac{T(s)}{T_{m}(s)}y_{m}e_{m}\bigg{)} \tag{8}\]  
When one configures \(T_{m}(s)=T(s)\), the adaptation gain \(\theta\) is simplified to:  
\[\theta=-\gamma\frac{1}{s}y_{m}e_{m}, \tag{9}\]  
which is implemented in the outer loop in Figure 1[25; 26; 28]. Application of ML-MR PID-MIT control for the existing control loop was defined as follows [25; 26; 28]:Step 1: Identify the transfer function model of the existing closed loop PID control system by means of closed loop model identification methods and use this transfer function as a reference model. Step 2: Enclose the existing closed loop control system via the outer loop. The outer loop employs Equation (9) and the reference model control. To create a design from scratch for a first order system model in the form of \(G(s)=\frac{K}{\tau s+1}e^{-Ls}\), an analytical tuning scheme for the ML-MR PID-MIT control structure can be proposed as follows:  
Step 1: Design a closed loop PID control system according to the Tavakoli-Tavakoli PID tuning rule that can be rearranged for a standard PID controller function in parallel form as follows [29]:  
\[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]0  
This optimal tuning rule implements ITAE and its control performance have been shown previously [29]. The optimal PID controller function of the inner loop can be written for the Tavakoli-Tavakoli optimal PID tuning rule as:  
\[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]1  
Step 2: The reference model is a theoretical model that represents the optimally tuned closed loop control system. Therefore, the transfer function of reference model can be expressed as:  
\[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]2  
In MATLAB and Simulink control system simulations, we implemented this reference model as a numerical model of the PID control loop, as illustrated in the diagram in \[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]3  
Step 3: Implement the ML-MR PID-MIT control structure as shown in \[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]4 The learning rate \(\gamma=|k_{i}|\) can be selected to comply with integral dynamics of the PID control loop. It is important to note two characteristic features of this structure. Firstly, the performance of the ML-MR PID-MIT control structure is sensitive to the time delay component of the system. As the time delay of the system increases, the adaptation performance of the MIT rule decreases because large delay in the system response easily misleads gradient descent optimization of the MIT rule. Figure 3 shows this effect for the ML-MR PID-MIT control structure design for control of the plant functions with zero time delay (\(G_{1}(s)=\frac{2}{s+1}\)) and 0.3 s time delay (\(G_{2}(s)=\frac{2}{s+1}e^{-0.3s}\)) [29]. The system responses in the figure demonstrate that the ML-MR PID-MIT control structure can significantly improve the  
\[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]5 Block diagram of the ML-MR PID-MIT control structure and a solution for numerical realization of reference models. disturbance rejection performance of classical PID control for zero delay systems. However, when the time delay of the plant increases, improvement in disturbance rejection control decreases. Secondly, due to multiplication terms \(u_{r}=\theta r\) and \(y_{m}e_{m}\), the ML-MR PID-MIT control structure presents a nonlinear control characteristic. The origin of the nonlinearity in the adaptation gain \(\theta\) is the nonlinear cost function, which involves the square of model error in Equation (3).