## 2 MOMI Tuning Method for Integrating Processes
The first condition in (6) gives the following result:  
\[K_{i}=0.5A_{0}\Big{(}1-b^{2}\Big{)}K_{P}{}^{2}, \tag{12}\]  
and the second condition in (6) results in  
\[K_{i}=\frac{\Big{(}A_{0}A_{2}-0.5A_{1}{}^{2}\Big{)}\Big{(}1-b^{2}\Big{)}K_{P}{} ^{2}\ +\ A_{1}K_{P}-0.5}{A_{2}}. \tag{13}\]  
The controller gain (\(K_{P}\)) can be calculated by equating Expressions (12) and (13):  
\[K_{P}=\frac{-A_{1}+\sqrt{A_{1}{}^{2}+\xi}}{\xi}, \tag{14}\]  
where  
\[\xi=\Big{(}1-b^{2}\Big{)}\Big{(}A_{0}A_{2}-A_{1}{}^{2}\Big{)}. \tag{15}\]  
**Remark 1**. : _If either reference weighting factor \(b\) is close or equal to 1, or the value of \(\xi\) in Equation (15) approaches zero, proportional gain \(K_{P}\) can be calculated by developing the expression under the square root in Equation (14) into a Taylor series. In this case, the proportional gain becomes_  
\[K_{P}=\frac{0.5}{A_{1}}. \tag{16}\]  
_However, if \(b\) = 1 (1-DOF PI controller), integral Gain (12) becomes \(K_{i}\) = 0, and we obtain a proportional (P) controller. Therefore, disturbance-rejection performance is degraded when increasing the value of factor \(b\). The influence of factor \(b\) on tracking and disturbance-rejection performance is shown through an illustrative example in Section 2.1._  
**Remark 2**. : _While Expressions (12) and (14) result in stable and fast closed-loop responses for a large majority of IP models, closed-loop stability is still not guaranteed for an arbitrary process model. Closed-loop stability and robustness are discussed in detail in Section 3._  
Therefore, PI controller parameters can be expressed in terms of characteristic areas or process parameters by using Expression (9). However, characteristic areas can also be calculated from the process time response while changing the process steady state. To this end, the multiple integrations of the process input (\(u(t)\)) and output (\(y(t)\)) signals should be calculated [58; 61]:  
\[ \begin{array}{l}u_{0}=\frac{u(t)-u(0)}{\Delta U}\\ I_{U1}(t)=\int_{0}^{t}u_{0}(\tau)d\tau\ I_{Y1}(t)=\frac{y(t)-y(0)-\dot{y}(0) \cdot t}{\Delta U},\\ I_{U2}(t)=\int_{0}^{t}I_{U11}(\tau)d\tau\ I_{Y2}(t)=\int_{0}^{t}I_{Y1}(\tau)d \tau\\ \vdots\end{array}  \tag{17}\]  
where \(u_{0}\) is normalised process input signal, and  
\[\Delta U=u(\infty)-u(0). \tag{18}\]The areas are expressed as follows:  
\[ \begin{array}{l}A_{0}=y_{0}(\infty);\ y_{1}=A_{0}l_{\rm UI}(t)-I_{\gamma 1}(t)\\ A_{1}=y_{1}(\infty);\ y_{2}=A_{1}l_{\rm UI}(t)-A_{0}l_{\rm UI2}(t)+I_{\gamma 2}(t)\\ A_{2}=y_{2}(\infty);\ y_{3}=A_{2}l_{\rm UI}(t)-A_{1}l_{\rm UI2}(t)+A_{0}l_{\rm IJ }(t)-I_{\gamma 3}(t),\\ \vdots\end{array}  \tag{19}\]  
where \(\ddot{y}(0)=\dddot{y}(0)=\cdots=0\). In practice, integration time can be limited. The integration can be terminated when the process input and output signals in Equation (17) settle. Therefore, Equation (17) can be easily and recursively calculated in practice, and process Model (1) is not required [58; 62; 63]. The controller tuning steps are as follows:  
* Determine characteristic areas from Expression (9) if the process model is known. Otherwise, modify the process steady state. During the process transient response, sample the controller and process output signals. Areas \(A_{0}\) to \(A_{2}\) are calculated from Expressions (17)-(19). The initial values of the process input and output signals can be estimated by averaging them before the change of the controller output signal. * Choose appropriate reference weighting factor \(0\leq b<1\). * Calculate controller parameters according to Equations (12) and (14). All MATLAB and Simulink files required for controller parameters calculation are electronically accessible [63]. **Remark 3**.