## 2 Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity
This yields a nonlinear term \(y_{m}e_{m}=y_{m}(y-y_{m})=y_{m}y-y_{m}^{2}\). Therefore, ML-MR PID-MIT control can exhibit some characteristic properties that do not appear in linear control systems. A noteworthy property is that the time response of the PID-MIT control shows amplitude dependence. In the case of reference input \(r(t)=0\), this results in \(y_{m}\cong 0\); in this case, the outer loop is not functional, so that \(y_{m}e_{m}=0\) in Equation (9). As the amplitude of the reference input signal \(r(t)\) grows, the amplitude of \(y_{m}\) increases, and this effect indeed amplifies the model error due to the term of \(y_{m}e_{m}\). Accordingly, this makes the MIT rule of the outer loop more sensitive to model errors and the system responds more effectively in cases where the model mismatches between reference model and control loops. To demonstrate this effect, an illustrative example for control of the plant function \(G(s)=\frac{2}{s+1}\) is carried out. Figure 4 shows time response ML-MR PID-MIT control structure that is designed for plant function \(G(s)=\frac{2}{s+1}\)[29]. The learning rate \(\gamma\) is set to \(-4\), and the PID controller is configured to \(k_{p}=4\), \(k_{i}=4\), and \(k_{d}=0\) according to the Tavakoli-Tavakoli PID tuning rule in this simulation. When the amplitude of \(r(t)\) moves to one (\(r(t)=1\)), the ML-MR PID-MIT control structure becomes more effective and contributes to the disturbance rejection control performance of the inner loop (classical PID control loop). However, when the amplitude of \(r(t)\) moves to zero (\(r(t)=0\)), the ML-MR PID-MIT control structure is not functional, and its disturbance rejection control performance becomes the same as the disturbance rejection performance of the classical PID controller. The overlapping of time responses at set-point zero in the figure apparently confirms this effect. \[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]6 Comparison of disturbance rejection performance of ML-MR PID-MIT control and classical PID control for \(G_{1}(s)\) and \(G_{2}(s)\) plants. Multi-Loop Model Reference PID-MIT with Controller Gain Modification (ML-MR PID-MIT-CGM): Adaptive RDR Adjustment for Dynamic Disturbance Rejection and the Linear Adaptation Rule  
The disturbance rejection capacity of a closed loop control system can be expressed with the reference to disturbance ratio (RDR) spectrum, which presents a measure for the additive input disturbance rejection performance of a control system for each frequency component [4,30]. The analysis demonstrated that the RDR performance of a closed loop control system depends on the spectral power density of the controller function [4]. The RDR spectrum of the PID controller is expressed in the form of [4]:  
\[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]7  
The increase of low frequency RDR values improves the disturbance rejection control for low frequency and steady disturbance signals. An increase in high frequency RDR values improves the transient disturbance rejection performance. However, as mentioned previously, the tradeoff between  
\[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]8 Comparison of disturbance rejection performances of ML-MR PID-MIT control and classical PID control for the cases of \(r(t)=1\) and \(r(t)=0\). (**a**) Step response and (**b**) change of model error. disturbance rejection control and set-point control performance bounds the RDR performance of the closed loop PID control systems [4]. A large increase in RDR values deteriorates the step response performance; this results in very high overshoots and consecutive ripples, which delay the convergence of the controlled system output to the set-point. Further increases in RDR values finally lead to instability of the control systems [4]. Therefore, the unstable state of the closed loop control systems limits the improvement of their RDR performances. A solution for this problem is adaptive adjustment of RDR performance; RDR performance should be increased for disturbance incidents in the inner loop.