## 2 Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity
Multi-Loop Model Reference MIT-PID with Plant Function Adaptation (ML-MR MIT-PID-PFA): Nonlinear Adaptation Rule for Time-Delayed Systems  
To improve the disturbance rejection capacity of time delay systems, model reference control is adopted for plant functions in a similar manner to classical IMC [14]. Any mismatch in responses from the real system and mathematical models of plants is assumed to be disturbance and is rejected by contributions of the MIT rule. In this configuration, the MIT rule forms the inner loop. The outer loop deals with the stability and set-point control. Therefore, this type of ML-MR control structure can be more effective for time delay systems because the adaptation loop only handles model mismatches between real plants and their mathematical models. Given a first order plus dead time plant function in the form of \(G(s)=\frac{K}{\tau s+1}e^{-Ls}\), an analytical design scheme for the ML-MR MIT-PID-PFA control structure can be suggested as follows. Step 1: Identify a model of the plant function and take the reference model of the MRC loop as the plant function (\(T_{m}(s)=G(s)\)). Step 2: Enclose the MRC loop with an optimal closed loop PID controller loop, as shown in \[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]7 Use the Tavakoli-Tavakoli PID tuning rule, which is expressed by Equation (10), for optimal operation of the PID controller function \(C(s)\). An example design for the first order plus time delay plant model is given in the following section. Multi-Loop Model Reference PID-PID with Plant Function Adaptation (ML-MR PID-PID-PFA): Linear Adaptation Rule for Time-Delayed Systems  
To obtain a linear adaptation rule, the MIT rule, which leads to nonlinearity, is changed to classical PID controller. ML-MR PID-PID-PFA control is implemented, as illustrated in \[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]8  
The design steps of the ML-MR PID-PID-PFA control structure for a first order time delay plant in the form of \(G(s)=\frac{K}{\tau s+1}e^{-Ls}\) can be carried out as follows. Step 1: Identify the model of the plant function and take the reference model of the MRC loop as the plant function (\(T_{m}(s)=G(s)\)). Step 2: Enclose the MRC loop with the optimal closed PID controller loop, as shown in \[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]9 Tavakoli-Tavakoli PID tuning rule, which is expressed by Equation (10), is used for the both PID controller functions. \[\theta=-\gamma\frac{1}{s}\frac{dJ}{d\theta}=-\gamma\frac{1}{s}{e_{m}}\frac{{ de_{m}}}{d\theta} \tag{5}\]0 Block diagram of the ML-MR MIT-PID-PFA control structure. To demonstrate the effectiveness of the ML-MR MIT-PID-PFA and ML-MR PID-PID-PFA control structures, we used them to enhance the first order plus deadtime plant model, which was given by [PERSON] et al. [34]. \[\theta=-\gamma\frac{1}{s}\frac{dJ}{d\theta}=-\gamma\frac{1}{s}{e_{m}}\frac{{ de_{m}}}{d\theta} \tag{5}\]1  
This plant function is a first order dynamical model of the Basic Process Rig 38-100 Feedback Unit (produced by Feedback Instruments Ltd) experimental platform [34]. This system presents a 50 s time delay with a time constant of 433.33 s and is considered a large time delay process. [PERSON] et al. designed an optimal fractional order PID (FOPID) controller for this system as follows:  
\[\theta=-\gamma\frac{1}{s}\frac{dJ}{d\theta}=-\gamma\frac{1}{s}{e_{m}}\frac{{ de_{m}}}{d\theta} \tag{5}\]2  
We implemented the ML-MR MIT-PID-PFA and ML-MR PID-PID-PFA control structures for this system by following the design steps given above. For control performance comparison purposes, we implemented a classical optimal PID control system with the Tavakoli-Tavakoli PID tuning rule (\(k_{p}=1.1862\), \(k_{i}=0.0026\), \(k_{d}=22.8885\)), a classical optimal PID control loop with MATLAB optimal tuning (\(k_{p}=0.8029\), \(k_{i}=0.002\), \(k_{d}=-35.3930\)), and an optimal FOPID control system with [PERSON] et al. 's tuning rule.