## 2 Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity
The ML-MR PID-I control structure behaves linearly and improves the disturbance rejection performance independent of the reference input amplitude. Results in Figure 10 demonstrate this effect. Although ML-MR PID-MIT-CGM is very effective around the unity set-point, its performance degrades at set-point 0, although it is slightly better than the optimal PID loop proposed by [PERSON] et al. However, ML-MR PID-I control presents the same performance as ML-MR PID-MIT control at the unity set-point and can maintain this performance improvement at set-point 0. Figure 11 validates this effect by showing the adaptive adjustment of the RDR spectrum at set-point  
\[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]2 (**a**) Evolution of the adaptation gain \(\theta\) of the ML-MR PID-MIT-CGM control structure. (**b**) Corresponding adaptive modification of the RDR spectrum of the \(C_{\theta}(s)\) controller in order to reject disturbances. levels 1 and 0. As a consequence, for linear control systems with almost zero time delay, the proposed ML-MR PID-I control structure is capable of improving the disturbance rejection performance without deteriorating the set-point performance. The simulation results verify the effectiveness of the ML-MR control loops by showing that the ML-MR control has a performance capacity beyond the typical disturbance rejection of classical single-loop control loops. This is a very useful property that improves the robust control performance in the case of disturbance interference to the control system. The ML-MR PID-MIT-CGM control structure is more preferable for set-point control of nonlinear systems, where linear ML-MR control structures are not an effective solution. \[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]3 Improvement of the disturbance rejection performance of the classical optimal PID loop by the proposed ML-MR PID-I control structure for the model. (**a**) Step response and (**b**) change of model error. Multi-Loop Model Reference PID Internal Model (ML-MR PID-IM) Control Structure: Internal Model Linear Adaptation Rule  
Internal model control (IMC) relies on a mathematical model of a plant or process, which is referred to as an internal model and performs the control law according to this internal model. It was reported that IMC can be an effective solution for disturbance rejection and model perturbation problems [32]. In this section, we consider IMC to design a ML-MR control structure. Previously, an IMC-PID controller was proposed for improved disturbance rejection of time-delayed processes by using the reference model of the plant [14]. IMC law can be designed by manipulating transfer function models of the control system. If the internal model is an accurate enough representation of the system, the derived control law in the s-domain becomes valid for the real system. In this section, we assume that the internal model is the PID control loop (the inner loop) and design an internal model PID controller, such that the outer loop can perform the IMC for the inner loop. Let us consider the multi-loop control structure that is depicted in \[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]4 To improve the disturbance rejection performance without influencing the control performance of the inner loop, the reference model, which is the internal model, is assumed to be the transfer function of the inner loop. \[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]5  
One can write the transfer function that involves the internal model controller loop \(C_{2}(s)\) and the theoretical reference model \(T_{1}(s)\) as:  
\[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]6  
Our design objective is to find an internal model controller \(C_{2}(s)\) so that the whole system behaves the same as the theoretical reference model \(T_{m}(s)\), which represents an optimal design of the PID control loop. In order to achieve this objective, the transfer function condition \(T_{2}(s)=T_{m}(s)\) should be satisfied. By using \(T_{m}(s)=T_{1}(s)\) (Equation (19)), the transfer function condition becomes \(T_{2}(s)=T_{m}(s)=T_{1}(s)\).