## 2 Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity
In order to adjust the RDR performance of the PID control loop adaptively, a variable gain PID controller is proposed by multiplying the PID controller function with the adaptation coefficient \(\theta\). This modified PID controller function allows adjustment of the DC gain of the classical PID controller by means of the coefficient \(\theta\). The transfer function of this controller can be expressed as:  
\[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]9  
The RDR spectrum of this RDR-adjustable PID controller can be obtained as:  
\[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]0  
This expression of the RDR spectrum apparently shows that the proposed variable gain PID controller \(C_{\theta}(s)\) allows modification of the RDR spectrum via adaptation gain \(\theta\). This enables adaptive adjustment of the disturbance rejection performance of the inner loop by the outer loop. Since the gain \(\theta\) amplifies the control error inside the inner loop, the value of the adaptation gain \(\theta\) should be configured to remain positive, and hence is written by:  
\[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]1  
Figure 5 shows the RDR spectrum of the controller \(C_{\theta}(s)\) for several values of the gain \(\theta\). This figure reveals that the proposed variable gain PID controller \(C_{\theta}(s)\) enables modification of the RDR spectrum via the gain \(\theta\), and this asset allows adaptive adjustment of disturbance rejection performance in the control loop. For \(\theta=1\), the \(C_{\theta}(s)\) controller exhibits disturbance rejection performance equal to the performance of the classical PID controllers. When \(\theta>1\), the disturbance rejection performance of \(C_{\theta}(s)\) increases, as illustrated in the RDR spectrum in \[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]2 It should be noted that the variable gain PID control may cause instability, particularly for time delay systems, because higher values of the gain \(\theta\) result in greater amplification of the integral gain, which may cause instability in the control systems as a consequence of accumulation of integral element control errors during the delay period of the system response. This limits the application domain of the proposed method to stable and almost-zero time delay plant functions. The application steps for ML-MR PID-MIT with controller gain modification (ML-MR PID-MIT-CGM) for existing control loops can be given as follows:  
Step 1: Obtain a transfer function model of an optimal control loop and use this transfer function as a reference model of the MRC loop (\(T_{m}(s)=T_{1}(s)\)). Step 2: Enclose the existing closed loop control system by the outer loop, as shown in \[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]3 Apply the MIT rule for controller gain modification of the closed loop PID control system to adjust the RDR. To show an application of the ML-MR PID-MIT-CGM control structure, an Automatic Voltage Regulator (AVR) control example is presented in this section. Figure 7 shows a block diagram of the proposed ML-MR PID-MIT-CGM control structure for the AVR control example. The learning rate \(\gamma\) is set to 10. \[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]4 RDR spectra of \(C_{\theta}(s)\) controllers for several \(\theta\) values. This figure reveals the adjustment of RDR performance according to several \(\theta\) configurations. \[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]5 Block diagram of the ML-MR PID-MIT with controller gain modification (ML-MR PID-MIT-CGM) control structure and numerical realization of the reference model. \[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]6 ML-MR PID-MIT-CGM control structure with control gain shaping that is implemented for improvement of the disturbance rejection performance of an optimal closed loop PID control of an AVR model. [PERSON] et al. implemented an improved harmony search algorithm for optimal tuning of closed loop PID control of an AVR model. The optimal PID controller was obtained by [PERSON] et al. in [PERSON] et al.