## 2 MOMI Tuning Method for Integrating Processes
### Extension to Integrating Processes
The first condition of Expression (6) gives the following result:  
\[K_{Di}=\frac{0.5\big{(}1-b^{2}\big{)}T_{I}{}^{2}-\frac{G_{00}}{K_{I}}}{(1-b)}, \tag{20}\]  
and the second condition in Expression (6) provides the following result:  
\[K_{I}=-\frac{G_{00}(b-1)^{2}\big{(}-2G_{10}{}^{2}-2G_{10}T_{I}+2G_{20}+bT_{I}{ }^{2}(1+b)\big{)}}{0.25(b^{2}-1)^{3}T_{I}{}^{4}}. \tag{21}\]  
The third condition of Expression (6) gives the following fourth-order polynomial:  
\[\alpha T_{I}{}^{4}+\beta T_{I}{}^{3}+(\gamma_{1}+\gamma_{2})T_{I}{}^{2}+\delta T _{I}+\varepsilon=0, \tag{22}\]where  
\[ \begin{array}{l}\alpha=(1-b)(1+b)^{2}\big{(}{G_{10}}^{2}(1-b)+2b{G_{20}}\big{)},\\ \beta=8\Big{(}\big{(}1-b^{2}\big{)}{G_{10}}^{3}+\big{(}2b^{2}+b-1\big{)}{G_{10}} {G_{20}}-b(1+b){G_{30}}\big{)},\\ \gamma_{1}=8\Big{(}-\big{(}b^{2}-3\big{)}{G_{10}}^{4}+\big{(}3b^{2}+b-6\big{)}{G _{10}}^{2}{G_{20}}-2\big{(}b^{2}+b-1\big{)}{G_{10}}{G_{30}}\big{)},\\ \gamma_{2}=8\Big{(}\big{(}1-b^{2}\big{)}{G_{20}}^{2}+b(1+b){G_{40}}\big{)},\\ \delta=16\big{(}2{G_{10}}^{5}-6{G_{10}}^{3}{G_{20}}+3\big{(}{G_{10}}^{2}{G_{30 }}+{G_{10}}{G_{20}}^{2}\big{)}-{G_{10}}{G_{40}}-{G_{20}}{G_{30}}\big{)},\\ \varepsilon=16\big{(}{G_{10}}^{6}-4{G_{10}}^{4}{G_{20}}+2{G_{10}}^{3}{G_{30}}+ 4{G_{10}}^{2}{G_{20}}^{2}-{G_{10}}^{2}{G_{40}}-2{G_{10}}{G_{20}}{G_{30}}-{G_{2 0}}^{3}+{G_{20}}{G_{40}}\big{)}.\end{array}  \tag{23}\]  
To calculate the PID\({}_{\text{b}}\) parameters, the integral time constant \(T_{I}\), represented by Expressions (22) and (23), should be solved first. The correct result is the highest positive real number. In the next step, integral gain, \(K_{I}\), should be calculated using Equation (21). In the last step, the proportional, \(K_{P}\), and derivative, \(K_{D}\), gains can be calculated using the following expression:  
\[ \begin{array}{l}K_{P}=K_{I}T_{I},\\ K_{D}=\frac{0.5\big{(}1-b^{2}\big{)}T_{I}^{2}K_{I}-{G_{00}}}{(1-b)}.\end{array}  \tag{24}\]  
Since calculating the PID\({}_{\text{b}}\) controller parameters requires solving the fourth-order polynomial, the final derivation of the tuning formula is not presented. Regardless, the tuning formula remains analytical. To facilitate the implementation of the proposed tuning method and save the reader's time and effort, all MATLAB files used for the calculation of the PID\({}_{\text{b}}\) parameters are available online in Reference [124]. **Remark 1**. : _Weighting factor \(b\) influences the disturbance-rejection and tracking performance. When \(b\) approaches 1, the integral gain \(K_{I}\) (21) approaches 0 [1]. For \(b=1\), we obtained the following PID controller:_  
\[ \begin{array}{l}K_{P}=\frac{K_{D}A_{0}+0.5}{A_{1}},\\ K_{I}=0,\\ K_{D}=\frac{A_{3}}{2(A_{1}A_{2}-A_{0}A_{3})}.\end{array}  \tag{25}\]  
_Thus, we obtained a PD controller that cannot reject the process input disturbances. For this reason, increasing the value of weighting factor \(b\) degrades the disturbance-rejection performance. In practice, the value of \(b\) should be limited within the interval \(0<b<0.9\)[1]. An illustrative example showing the tracking and disturbance rejection performance and its relationship to factor \(b\) is presented in Section 2.3._  
**Remark 2**. : _The PID\({}_{b}\) controller parameters calculated by Expressions (21) and (24) result in stable and fast closed-loop responses for most IP models. However, the closed-loop stability of an arbitrary process model is still not guaranteed. Section 5 provides additional information on closed-loop stability and robustness._  
As explained in Section 2.1, the characteristic areas \(A_{0}\) to \(A_{4}\) can be obtained directly from the process transfer function or calculated from the process time response when changing the process steady state. The following steps are required for PID\({}_{\text{b}}\) controller tuning:  
1. Choose the appropriate filter time constant, \(T_{F}\), of the PID\({}_{\text{b}}\) controller. 2. Determine the characteristic areas \(A_{0}\)-\(A_{4}\). If the process is given in a Laplace form, use Expression (8) (frequency domain).