## 2 Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity
It is used in Equation (20) to satisfy the transfer function condition, which gives:  
\[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]7  
Then, the controller function \(C_{2}(s)\) is written as:  
\[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]8  
\[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]9 (**a**) Evolution of adaptation gain \(\theta\) of the ML-MR PID-I control structure. (**b**) Adaptive modification of the RDR spectrum of the \(C_{\theta}(s)\) controller for disturbance rejection. By using Equation (19) in Equation (22), the \(C_{2}(s)\) function is found:  
\[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]0  
where the \(L(s)\) function is the open loop transfer function of the inner loop, which is expressed as \(L(s)=C_{1}(s)G(s)\). In this form, the \(C_{2}(s)\) function can be directly implementable. Application steps for existing control loops are summarized as follows:  
Step 1: The transfer function model of an optimal control loop is obtained, and this transfer function is used as the reference model (\(T_{m}(s)=T_{1}(s)\)). Step 2: The optimal closed loop control system is enclosed by the outer loop, as shown in \[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]1  
To demonstrate a possible application for the ML-MR PID-IM control, an example design for level control problem is presented. The plant model is an approximate model that represents the liquid level in the reboiler of the steam-heated distillation equipment [14; 33]. The liquid level is controlled by adjusting the control valve on the steam line and the process model is given as \(G(s)=\frac{-1.6(-0.5s+1)}{3s(s+1)}\)[14; 33]. The optimal PID controller for improved disturbance rejection was given as [33]:  
\[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]2  
The system model is unstable, and the controller function was obtained with negative coefficients. By considering Equation (23), the \(C_{2}(s)\) function of the ML-MR PID-IM control structure is obtained as:  
\[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]3  
We implemented the ML-MR PID-IM control structure for this system, as shown in Figure 13, and performed control simulations for square waveform input disturbance. Figure 14 shows responses of the ML-MR PID-IM control structure and [PERSON] et al. 's optimal PID control loop. Results in Figure 14 clearly demonstrate that the ML-MR PID-IM control structure improves the disturbance rejection performance of [PERSON] et al. 's optimal PID control loop without influencing its set-point performance. In the absence of disturbance, responses of the ML-MR PID-IM control structure overlap with responses of the optimal PID control loop. The set-point performance of the ML-MR PID-IM control is almost the same as the performance of the optimal PID control loop proposed by [PERSON] et al. This is an important advantage for robust control performance so that the proposed ML-MR control structures can improve the disturbance rejection while maintaining the  
\[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]4 Block diagram of the ML-MR PID-IM control structure. set-point performance of the existing PID control loops. A limitation of the ML-MR PID-IM control is that it is not an effective solution for large time delay plant dynamics and inaccurate plant models. High time delays can severely affect the performance of the ML-MR PID-IM control scheme. \[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]5 Improvement of the disturbance rejection performance of the classical optimal PID loop by the proposed ML-MR PID-IM control structure for liquid level control in the reboiler model. (**a**) Step response and (**b**) change of model error. \[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]6 Implementation of the ML-MR PID-IM control structure for disturbance rejection control of the liquid level of the reboiler model.