## 2 Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity  
The ML-MR PID-MIT control structure has been proposed to enhance the robust control performance of existing closed loop control systems. Its application has been discussed in several works [25; 26; 27; 28]. This structure is essentially composed of two hierarchical control loops: the inner loop is a classical PID control loop that is a stable and well-tuned control loop, while the outer loop is a adaptation loop that performs the MIT rule. The outer loop is appended to the inner loop to enhance the robust control performance of the inner loop according to regulation of a reference model. Figure 1 shows a block diagram of the ML-MR PID-MIT control structure. It has two error signals per loop:  
1. Control error: When the controller of the inner loop is tuned properly, the value of the control error, which is written as \(e_{c}=u_{r}-y\), moves to zero, and thus ensures the control system output settles to the reference input (the signal \(u_{r}\) is the modified reference input controlled by the MIT rule and \(y\) is the system output). 2. Model error: Discrepancy between the reference model output and controlled system output is defined as the model error, which is written as \(e_{m}=y-y_{m}\) (the signal \(y_{m}\) is the output of reference model). The objective of the outer loop is to shape the reference input signal (\(r\)) such that the model error converges to zero. The convergence of the model error to zero implies that the inner loop resembles the response of the reference model \(T_{m}(s)\). The reference model describes the desired response of the control system that forms the inner loop. Let us consider a PID control system as the inner control loop. The PID controllers are three-coefficient standard industrial controllers and optimal tuning of the PID coefficients (\(k_{p}\), \(k_{i}\)  
Figure 1: Block diagram of multi-loop model reference PID-MIT (ML-MR PID-MIT) control structure with input shaping. \(k_{d}\)) enhances step performance by reducing overshoots and settling time. The transfer function of PID controllers is widely expressed in the parallel form as:  
\[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{1}\]  
For a given plant (process) function \(G(s)\), the transfer function of the inner loop can be written by:  
\[T(s)=\frac{C(s)G(s)}{1+C(s)G(s)} \tag{2}\]  
Previous studies on the ML-MR PID-MIT control structure aim to maintain the initial well-tuned control performance of an existing closed loop control system. For this reason, the reference model is taken as the transfer function of the inner loop; that is, \(T_{m}(s)=T(s)\). Accordingly, any performance deterioration in the inner loop leads to an increase in model errors. The recovery of temporal performance deterioration is carried out by shaping the reference input of the inner loop via \(u_{r}=\theta r\). The adaptation gain \(\theta\) is determined according to the well-known MIT rule [21], which implements continuous gradient descent optimization of the cost function:  
\[J=\frac{1}{2}{e_{m}}^{2} \tag{3}\]  
The feed-forward MIT rule is expressed in the continuous domain as [18; 19; 20; 21]:  
\[\frac{d\theta}{dt}=-\gamma\frac{dJ}{d\theta}, \tag{4}\]  
where the parameter \(\gamma\) is the learning rate, which is an important parameter for the convergence speed and stability of the gradient descent optimization. Taking Laplace transform of Equation (4), one obtains:  
\[\theta=-\gamma\frac{1}{s}\frac{dJ}{d\theta}=-\gamma\frac{1}{s}{e_{m}}\frac{{ de_{m}}}{d\theta} \tag{5}\]  
To find the sensitivity derivative \(\frac{d{e_{m}}}{d\theta}\), the model error is written in the form of \(e_{m}=y-y_{m}=T(s)\theta r-T_{m}(s)r\).