## 2 Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity
(2018) as:  
\[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]7  
We used this optimal PID controller for the AVR model in \[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]8 Figure 8a,b show responses of [PERSON] et al. 's optimal PID control loop and ML-MR PID-MIT-CGM control structure for square waveform input disturbances. The figure clearly shows disturbance rejection performance improvements for the ML-MR PID-MIT-CGM control at simulation times of 15 and 35 s, where the set-point is equal to 1. At these instances, the adaptation gain \(\theta\) of the outer loop actively responds to step-up and step-down disturbances, and it increases the RDR performance of the inner loop by increasing \(\theta\), as illustrated in \[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]9 This effect provides a better disturbance rejection control than the classical PID control loop. Due to the amplitude dependence of the MIT rule, contributions of the ML-MR PID-MIT-CGM control to the disturbance rejection performance begin to vanish in the vicinity of the set-point 0. Consequently, the system can respond slightly better for step-up and step-down disturbances at the simulation times of 65 and 85 s compared to the optimal PID loop proposed by [PERSON] et al. \[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]0 Improvement of the disturbance rejection performance of the classical optimal PID loop by the proposed ML-MR PID-MIT-CGM control structure for the AVR control system model. (**a**) Step response and (**b**) change of model error. Figure 9 describes how ML-MR PID-MIT-CGM control responds to disturbance incidents. For a disturbance incident, the gain \(\theta\) increases, as shown in Figure 9b. This results in an increase of PID coefficients, and accordingly an increase in the RDR rates in the spectrum, as shown in Figure 9b. Therefore, the RDR performance of the optimal PID control loop of [PERSON] et al. is adaptively modified by adaptation gain \(\theta\), while the disturbances affect the system response. This figure verifies an important asset of the proposed ML-MR PID-MIT-CGM control structure: It adaptively adjusts the disturbance rejection control performance and retains the set-point control performance of the optimal PID loops in absence of disturbance incidents. Thus, it overcomes the inherent shortcoming that is caused by the performance tradeoff between set-point control and disturbance rejection control. This property can significantly contribute to the practical robust control performance of classical control loops and can be viewed as an elegant solution for increasing the disturbance rejection performance without deteriorating the set-point performance of control systems. However, there will be a few drawbacks in the PID-MIT structures for linear control system applications. The nonlinearity of the MIT rule leads to dependence of disturbance rejection performance on the level of the set-point. Another drawback for linear control applications is that the performance of the PID-MIT-CGM control structure is very sensitive to the time delay of plants. A sufficiently large time delay may easily lead to instability in the systems. As mentioned in the previous section, the MIT rule does not behave linearly due to the terms of \(y_{m}e_{m}\). To eliminate this nonlinearity and to make structural contributions to the PID control loop independent of the level of signal amplitudes, one can update the term \(y_{m}e_{m}\) as \(c_{m}\), where the constant \(c\in R\) can be typically set to 1. This modification implies that the sensitivity derivative is constant and can be shown as \(\frac{d\epsilon_{m}}{d\theta}=c\). Hence, Equation (16) is simplified to an integral operator and the adaptation gain can be written as:  
\[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]1  
Using this modified adaptation gain--where Equation (18) replaces the MIT rule in Equation (16) --produces a new ML-MR control structure, which is called multi-loop model reference PID integral (ML-MR PID-I) control.