## 2 MOMI Tuning Method for Integrating Processes  
### Closed-Loop Configuration and Magnitude Optimum Multiple Integration  
In this section, we present a method for the non-parametric identification of the process using so-called characteristic areas (process moments) [18]. These areas can be identified from a closed- or open-loop time response (in the time domain) or directly from an arbitrary rational transfer function with a pure time-delay (in the frequency domain). It is worth mentioning that both approaches (time and frequency domain) are equivalent. Later, the characteristic areas will be used for the calculation of the controller and the reference filters' parameters. Figure 1 shows a 2-DOF controller in a closed-loop configuration with a process. Signals \(u,d,y,\) and \(r\) represent the process output, input disturbance, controller output, and controller reference, respectively. The process is defined by the stable transfer function:  
\[G_{P}(s)=\frac{K_{PR}}{s}\frac{1+b_{1}s+b_{2}s^{2}+\cdots+b_{m}s^{m}}{1+a_{1}s+ a_{2}s^{2}+\cdots+a_{n}s^{n}}e^{-sT_{delay}}, \tag{1}\]  
where the process time-delay is represented by \(T_{delay}\). As stated in Reference [119], \"one possible design aim is to maintain the closed-loop magnitude response curve as flat and as close to unity for as large a bandwidth as possible\" (see Figure 2). Thus, the goal is to find a controller that has a low-frequency closed-loop response as close to unity as possible [121]. Figure 1: Universal closed-loop control scheme using the two-degrees-of-freedom (2-DOF) controller. Figure 2: The closed-loop magnitude frequency response according to the magnitude optimum (MO) method [1]. This approach is called modulus optimum, Betragsoptimum, or magnitude optimum (MO), and results in a relatively optimal closed-loop time response for a majority of process models, when considering the speed of its response and robustness [121]. With a 2-DOF controller (in Figure 1), the closed-loop transfer function becomes  
\[G_{CL}(s)=\frac{Y(s)}{R(s)}=\frac{G_{P}(s)G_{CR}(s)}{1+G_{P}(s)G_{CY}(s)}. \tag{2}\]  
The controller is configured so that  
\[G_{CL}(0)=1 \tag{3}\]  
\[\lim_{\omega\to 0}\left[\frac{d^{2k}\big{|}G_{CL}(j\omega)\big{|}^{2}}{d \omega^{2k}}\right]=0;k=1,2,\cdots,k_{max} \tag{4}\]  
is fulfilled for as many \(k\) as possible [119; 122]. Equation (3) is thus easily fulfilled. If the controller structure contains integral term and the closed-loop response is stable, the steady-state control error is zero; hence, Equation (3) holds. The number of satisfied conditions in Equation (4) is correlated to the number of controller parameters (controller order). For the PID controller, \(k_{\text{max}}=3\). Let us define the closed-loop transfer function by the following equation:  
\[G_{CL}(s)=\frac{f_{0}+f_{1}s+f_{2}s^{2}+\cdots}{e_{0}+e_{1}s+e_{2}s^{2}+\cdots}. \tag{5}\]  
Then, the following conditions must be satisfied to fulfil Expression (4) [121]:  
\[\sum\
olimits_{i=0}^{2n}(-1)^{i+n}\Big{(}f_{i}f_{2n-i}{e_{0}}^{2}-e_{i}e_{2n-i }{f_{0}}^{2}\Big{)}=0;n=1,2,\ldots \tag{6}\]  
For the calculation of the PI controller parameters, the first two equations in Condition (6) should be fulfilled, and for the calculation of the PID controller parameters, the first three equations in Condition (6) should be fulfilled. To simplify further derivations, Process (1) is developed into an infinite Taylor series [122]:  
\[G_{P}(s)=\frac{G_{P1}}{s}=\frac{A_{0}-A_{1}s+A_{2}s^{2}-A_{3}s^{3}+\cdots}{s}\, \tag{7}\]  
where \(A_{i}\) represents the \"characteristic areas\" of the process [121], which can be expressed by Process Parameters (1) [119]:  
\[