## 2 MOMI Tuning Method for Integrating Processes
### Closed-Loop Configuration and Magnitude Optimum Multiple Integration
\begin{gathered} A_{0}=K_{PR}\\ A_{1}=K_{PR}\Big{(}a_{1}-b_{1}+T_{delay}\\ 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}b_{ 1}}{2!}+\frac{T_{delay}}{3!}\Big{)}+A_{2}a_{1}-A_{1}a_{2}.\\ A_{4}=K_{PR}\Big{(}-a_{4}+b_{4}-T_{delay}b_{3}+\frac{T_{delay}{}^{2}b_{ 2}}{2!}-\frac{T_{delay}{}^{3}b_{1}}{3!}+\frac{T_{delay}{}^{4}}{4!}\Big{)}+A_{3} a_{1}-A_{2}a_{2}+A_{1}a_{3}\\ \vdots\\ A_{k}=K_{PR}\Bigg{(}(-1)^{k+1}(a_{k}-b_{k})+\sum\limits_{i=1}^{k}(-1 )^{k+i}\frac{T_{delay}{}^{i}b_{k-i}}{i!}\Bigg{)}+\sum\limits_{i=1}^{k-1}(-1)^{ k+i-1}A_{i}a_{k-i}.\end{gathered}  \tag{8}\]  
This means that the characteristic areas can be obtained directly from the process transfer function. Additionally, the characteristic areas can also be calculated in the time domain, i.e., from the process time response. To this end, the steady state of the process should be changed first. Then, the multiple integrations of the process input (\(u(t)\)) and output (\(y(t)\)) signals can be calculated [119; 122]:  
\[ \begin{array}{ll}u_{0}=\frac{u(t)-u(0)}{\Delta U_{t}}\\ I_{U1}(t)=\int_{0}^{t}u_{0}(\tau)d\tau&I_{Y1}(t)=\frac{y(t)-y(0)-\dot{y}(0)\cdot t }{\Delta U}\,\\ I_{U2}(t)=\int_{0}^{t}I_{U1}(\tau)d\tau&I_{Y2}(t)=\int_{0}^{t}I_{Y1}(\tau)d \tau\\ &\vdots&\vdots\end{array}  \tag{9}\]  
where \(u_{0}\) represents the normalised process input signal, and  
\[\Delta U=u(\infty)-u(0). \tag{10}\]  
The characteristic areas are expressed as follows:  
\[ \begin{array}{ll}A_{0}=y_{0}(\infty);\ y_{1}=A_{0}I_{U1}(t)-I_{Y1}(t)\\ A_{1}=y_{1}(\infty);\ y_{2}=A_{1}I_{U1}(t)-A_{0}I_{U2}(t)+I_{Y2}(t)\\ A_{2}=y_{2}(\infty);\ y_{3}=A_{2}I_{U1}(t)-A_{1}I_{U2}(t)+A_{0}I_{U3}(t)-I_{Y3}( t)\.\\ \vdots\end{array}  \tag{11}\]  
Note that \(\ddot{y}(0)=\dddot{y}(0)=\cdots=0\) and that \(A_{0}\) equals the process steady-state gain, \(K_{\text{PR}}\). In practice, the integration in Expression (9) can end when the process signals settle. Thus, Expression (9) can be recursively and simply calculated, therefore the process Model (1) is not needed. Therefore, using the characteristic areas, the process can be easily parameterised from any change in the process steady state [121]. In practice, certain rules need to be followed to accurately calculate the areas from the process response corrupted by noise. A few practical guidelines for calculating the characteristic areas from the process time responses in practice, including the high-frequency process noise, are provided in Reference [123]. As will be shown in Section 7, the method was also tested on a real process with present noise. Moreover, the proposed method for the PI controller [1] has also been tested on several processes with measurement noise. In all the mentioned tests, it was shown that the calculated areas and, therefore, the controller parameters, are not extensively sensitive on high-frequency process noise. However, the method still requires deeper noise analysis to be carried out in the future. For ease of clarification, Figure 3 shows a graphical representation of the characteristic area \(A_{1}\). Signals \(u_{0}\) and \(y_{0}\) represent the process input and process output responses, respectively [121]. Figure 3: Characteristic area \(A_{1}\) (graphical representation). This area was measured after the change of the process steady state.