## 2 MOMI Tuning Method for Integrating Processes  
The process in a closed-loop configuration with a 2-DOF controller is presented in Figure 1. Signals \(d\), \(y\), \(r\), and \(u\) represent input disturbance, process output, controller reference, and controller output, respectively. The stable process was modelled with the rational transfer function:  
\[G_{P}(s)=\frac{K_{PR}}{s}\frac{1\ +\ b_{1}s\ +\ b_{2}s^{2}\ +\cdots+\ b_{m}s^{m}}{1 \ +\ a_{1}s\ +\ a_{2}s^{2}\ +\cdots+\ a_{m}s^{m}}e^{-sT_{delay}}, \tag{1}\]  
where \(T_{delay}\) represents the time-delay. According to [58], \"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). This technique is called magnitude optimum (MO), modulus optimum, or Betragsoptimum, and results in a fast and non-oscillatory closed-loop time response for a large class of process models [60]. If we have closed-loop transfer function  
\[G_{CL}(s)=\frac{Y(s)}{R(s)}=\frac{G_{P}(s)G_{CR}(s)}{1\ +\ G_{P}(s)G_{CY}(s)}, \tag{2}\]  
the controller is determined in such a way that  
\[G_{CL}(0)=1 \tag{3}\]  
Figure 1: General closed-loop scheme with a two-degrees-of-freedom (2-DOF) controller. Figure 2: “Closed-loop amplitude (magnitude) response over frequency”. Adapted from [58]. \[\lim_{\omega\to 0}\left[\frac{d^{2k}\big{|}G_{CL}(j\omega)\big{|}^{2}}{d \omega^{2k}}\right]=0;k=1,2,\cdots,k\_{\max} \tag{4}\]  
for as many \(k\) as possible [58; 61]. The fulfilment of Equation (3) is simple. When using the controller structure containing an integral term, the steady-state control error becomes zero (under the condition that the closed-loop response is stable). Controller order (number of controller parameters) impacts the number of satisfied conditions in Equation (4). If the closed-loop transfer function is described by 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 Expressions (4) can be met by satisfying the following conditions [60]:  
\[\sum_{i=0}^{2n}(-1)^{i+n}\big{(}f_{i}f_{2n-i}e_{0}^{\ 2}-e_{i}e_{2n-i}f_{0}^{\ 2} \big{)}=0;n=1,2,\ldots \tag{6}\]  
The next step is the calculation of the 2-DOF PI controller parameters. A controller may be defined by the next transfer functions:  
\[ \begin{array}{l}G_{CR}(s)=bK_{P}+\frac{K_{i}}{s}\\ G_{CY}(s)=K_{P}+\frac{K_{i}}{s},\end{array}  \tag{7}\]  
where \(b\), \(K_{P}\), and \(K_{i}\) are the reference weighting factor, proportional gain, and integral gain, respectively. In order to simplify subsequent derivations, let us develop Process (1) into an infinite Taylor series [61]:  
\[G_{P}(s)=\frac{A_{0}-A_{1}s\ +\ A_{2}s^{2}-A_{3}s^{3}+\cdots}{s}, \tag{8}\]  
where \(A_{i}\) represent the so-called \"characteristic areas\" of the process [60]. The areas can be expressed by process Parameters (1) in the following way [58]:  
\[ \begin{array}{l}A_{0}=K_{PR}\\ A_{1}=K_{PR}\Big{(}a_{1}-b_{1}+T_{delay}\Big{)}\\ A_{2}=K_{PR}\Big{(}b_{2}-a_{2}-T_{delay}b_{1}+\frac{T_{delay}}{2!}\Big{)}+A_{1} a_{1}\\ A_{3}=K_{PR}\Big{(}a_{3}-b_{3}+T_{delay}b_{2}-\frac{T_{delay}}{2!}+\frac{T_{delay}}{3!} \Big{)}+A_{2}a_{1}-A_{1}a_{2}.\\ \vdots\end{array}  \tag{9}\]  
By applying Expressions (7) and (1) to Expression (2), the following closed-loop transfer function parameters (5) are obtained:  
\[ \begin{array}{l}e_{0}=A_{0}K_{i}\\ e_{1}=-A_{1}K_{i}+A_{0}K_{P}\\ e_{2}=A_{2}K_{i}-A_{1}K_{P}+1\\ e_{3}=-A_{3}K_{i}+A_{2}K_{P}\\ e_{4}=A_{4}K_{i}-A_{3}K_{P}\\ \vdots\end{array}  \tag{10}\]  
\[ \begin{array}{l}f_{0}=A_{0}K_{i}\\ f_{1}=-A_{1}K_{i}+A_{0}bK_{P}\\ f_{2}=A_{2}K_{i}-A_{1}bK_{P}\\ f_{3}=-A_{3}K_{i}+A_{2}bK_{P},\\ \vdots\end{array}  \tag{11}\]In order to calculate two PI controller parameters, the first two equations in Conditions (6) should be satisfied.