## 2 MOMI Tuning Method for Integrating Processes
### Extension to Integrating Processes  
Figure 4 presents the closed-loop system with a 2-DOF PID controller. Signals \(y\), \(d\), \(u\), and \(r\) represent the process output, disturbance, controller output, and controller reference, respectively. Parameters \(b\) and \(c\) are the proportional and derivative weighting factors, while \(K_{D}\), \(K_{P}\), \(K_{I}\), and \(T_{F}\) are the derivative gain, proportional gain, integral gain, and filter time constant, respectively. The controller presented in Figure 4 is defined by the following transfer functions:  
\[ \begin{split} G_{\text{CR}}(s)&=\frac{hK_{D}s^{2} +hK_{D}s+K_{I}}{s(1+sT_{F})}\\ G_{\text{CY}}(s)&=\frac{K_{D}s^{2}+K_{D}s+K_{I}}{s(1 +sT_{F})}.\end{split}  \tag{12}\]  
To simplify the derivations, \(c=b\) is chosen. Thereafter, this controller will be denoted PID\({}_{\text{b}}\). While the controller filter  
\[G_{F}=\frac{1}{1+sT_{F}} \tag{13}\]  
is actually part of the PID\({}_{\text{b}}\) controller transfer function. To simplify the derivation of the PID\({}_{\text{b}}\) controller parameters, the controller filter is considered part of the Process (1). Therefore, the aforementioned filter must be added to Expression (1) every time the PID\({}_{\text{b}}\) controller parameters are calculated with the final formula. The controller filter is used for reduction of the controller's output high-frequency noise. The controller's high-frequency gain equals to \(K_{HF}=\frac{K_{D}}{T_{F}}\)[18]. Besides defining the controller filter time constant in advance, it can also be calculated based on the desired high-frequency gain as follows:  
\[T_{F}=\frac{K_{D}}{K_{HF}} \tag{14}\]  
Initially, the controller parameters (including derivative gain \(K_{D}\)) should be calculated with \(T_{F}\) = 0. Next, the controller parameters can be calculated again by applying Equation (14). The procedure can be repeated until the \(T_{F}\) value settles. In practice, up to three iterations are usually sufficient. In the following derivation, the PID\({}_{\text{b}}\) controller parameters will be calculated. The IP has been modelled by the transfer Function (1). Next, Process (7) and the Controller (12) transfer functions are inserted into a closed-loop transfer Function (2). In this way, the following closed-loop transfer Function (5) parameters are obtained:  
Figure 4: The closed-loop system with a 2-DOF proportional–integral–derivative (PID) controller. \[ \begin{array}{c}e_{0}=A_{0}K_{I}\\ e_{1}=-A_{1}K_{I}+A_{0}K_{P}\\ e_{2}=A_{2}K_{I}-A_{1}K_{P}+A_{0}K_{D}+1\\ e_{3}=-A_{3}K_{I}+A_{2}K_{P}-A_{1}K_{D}\\ e_{4}=A_{4}K_{I}-A_{3}K_{P}+A_{2}K_{D}\\ \vdots\\ \end{array}  \tag{15}\]  
\[ \begin{array}{c}f_{0}=A_{0}K_{I}\\ f_{1}=-A_{1}K_{I}+A_{0}bK_{P}\\ f_{2}=A_{2}K_{I}-A_{1}bK_{P}+A_{0}bK_{D}\\ f_{3}=-A_{3}K_{I}+A_{2}bK_{P}-A_{1}bK_{D}\\ f_{4}=A_{4}K_{I}-A_{3}bK_{P}+A_{2}bK_{D},\\ \vdots\\ \end{array}  \tag{16}\]  
To simplify the derivation of the PID\({}_{\rm b}\) controller parameters, in the next step, all closed-loop transfer function parameters, represented by Expressions (15) and (16), are divided by \(A_{0}K_{I}\). Consequently, the following closed-loop transfer function Parameters (5) are obtained:  
\[ \begin{array}{c}e_{0}=1\\ e_{1}=G_{10}+T_{I}\\ e_{2}=G_{20}+G_{10}T_{I}+K_{DI}+G_{00}K_{I1}\\ e_{3}=G_{30}+G_{20}T_{I}+G_{10}K_{DI}\\ e_{4}=G_{40}+G_{30}T_{I}+G_{20}K_{DI}\\ \vdots\\ \end{array}  \tag{17}\]  
\[ \begin{array}{c}f_{0}=1\\ f_{1}=G_{10}+bT_{I}\\ f_{2}=G_{20}+G_{10}bT_{I}+bK_{DI}\\ f_{3}=G_{30}+G_{20}bT_{I}+G_{10}bK_{DI}\\ f_{4}=G_{40}+G_{30}bT_{I}+G_{20}bK_{DI},\\ \vdots\\ \end{array}  \tag{18}\]  
where  
\[T_{I}=\frac{K_{P}}{K_{I}},\ K_{i1}=\frac{1}{K_{I}},\ K_{Di}=\frac{K_{D}}{K_{I} },\ G_{00}=\frac{1}{A_{0}},\ G_{10}=-\frac{A_{1}}{A_{0}},\ G_{20}=\frac{A_{2}} {A_{0}},\ G_{30}=-\frac{A_{3}}{A_{0}},\ G_{40}=\frac{A_{4}}{A_{0}}. \tag{19}\]  
For calculation of the three PID\({}_{\rm b}\) controller parameters, the first three equations in Condition (6) should be solved.