## 5 Results
Figure 3:2 Approximation of the time constant by mode 3 (the blue line indicates pressure dynamics; the black line denotes the approximation of the time constant \(T_{3}\)). where \(E_{1},E_{2},E_{4}\)--gain coefficients of distributed proportional, integrating and differentiating blocks; \(n_{1},n_{2},n_{4}\)--weight coefficient, \(\
abla^{2}\)--laplacian. Application of DHC in this system is necessary to regulate the pressure distribution in the reservoir during field operation to the required value, which is set at the beginning of modelling. This will result in a closed-loop system for controlling the pressure field during gas production in structurally complex fields. Assume that the phase shift introduced into the system by the controller is zero. Then, we define the cut-off points of the open-loop module. Figure 3:3  
where \(W(G,j\omega)\)--complex transfer coefficient of the control object. Using Equation (21) for the selected spatial modes (\(G_{1}\) H \(G_{3}\)), we determine the cut-off frequencies \(\omega_{1}=1.155,\ \omega_{3}=1.18673\). At this frequency, according to the relationship:  
Figure 3:4  
Define \(k_{ob,1}=0.1,k_{ob,3}=0.15\). By the ratio, we find the regulator coefficient we obtain \(k_{reg,1}=10,k_{reg,3}=6.67\). To determine all parameters of the DHC, we calculate the values of generalised coordinates \(G_{1}\) and \(G_{3}\) based on the numbers of spatial modes:  
\[ \begin{split} G_{1}&=\Big{(}\tfrac{\pi\cdot i}{L \cdot x}\Big{)}^{2}+\Big{(}\tfrac{\pi\cdot j}{L\cdot y}\Big{)}^{2}=0.00045.\\ G_{3}&=\Big{(}\tfrac{\pi\cdot i}{L\cdot x}\Big{)}^{2 }+\Big{(}\tfrac{\pi\cdot j}{L\cdot y}\Big{)}^{2}=0.00403.\end{split} \]  
Define \(E_{1}\) and \(n_{1}\) by the system of equations  
\[ \begin{split}\tfrac{1}{K_{ob,1}}&=E_{1}\Big{(} \tfrac{n_{1}-1}{n_{1}}-\tfrac{G_{1}}{n_{1}}\Big{)},\\ \tfrac{1}{K_{ob,3}}&=E_{1}\Big{(}\tfrac{n_{1}-1}{n_ {1}}-\tfrac{G_{1}}{n_{1}}\Big{)}.\end{split}  \tag{22}\]  
From the system (22) express \(n_{1}\), for this purpose, we divide the equations into each other:  
\[n_{1}=\ \frac{-1+\Delta K-\Delta K\cdot G_{1}+G_{3}}{\Delta K-1}. \tag{23}\]  
Considering that \(\Delta K=\tfrac{K_{reg,3}}{K_{reg,1}}=\tfrac{6.67}{10}=0.67\), and substituting the values of \(G_{1}\) and \(G_{3}\), we obtain the value of  
\[n_{1}=\ \frac{-1+0.67-0.67\cdot 0.00045+0.00403}{0.67-1}=0.98. \tag{24}\]  
Considering that \(n_{1}\geq 1,n_{2}\geq 1,n_{4}\geq 1\), we assume \(n_{1}=1\). Substituting \(n_{1}\) into the first Equation (22), we obtain  
\[E_{1}=\frac{K_{reg,1}}{\tfrac{n_{1}-1}{n_{1}}+\tfrac{G_{1}}{n_{1}}}=22,341. \tag{25}\]  
Based on the equation of the inflection line, we obtain the equation at our cut-off frequencies:  
\[\log\Delta\omega^{2}=\log\biggl{(}\frac{n_{4}-1+\ G_{3}}{n_{4}-1+\ G_{1}} \biggr{)}-\log\biggl{(}\frac{n_{2}-1+\ G_{3}}{n_{2}-1+\ G_{1}}\biggr{)}. \tag{26}\]Define \(\Delta\omega^{2}=\frac{\omega_{1}^{2}}{\omega_{1}^{2}}=\frac{1.18673^{2}}{1.155^{ 2}}=1.06\). It is known from the theory of DCS that \(\Delta\omega^{2}>1\); the value of the coefficient is assumed to be \(n_{2}=\infty\)[57]. From here, \(n_{4}\) is determined by the formula  
\[n_{4}=\frac{\Delta\omega^{2}-1+\ G_{3}-\Delta\omega^{2}}{\Delta\omega^{2}-1}= \frac{1.06-1+0.00403-1.06\cdot 0.00405}{1.06-1}=1.037. \tag{27}\]  
The relationship between the parameters of the considered links and the parameter \(\Delta\) is defined as follows. \[\text{lg}\omega_{1}=\text{lg}\bigg{(}\frac{1}{K_{2}(G_{1})}\bigg{)},\text{lg} \omega_{2}=\text{lg}\bigg{(}\frac{1}{K_{4}(G_{1})}\bigg{)},\Delta(G_{1})= \text{lg}\omega_{1}-\text{lg}\omega_{2}, \tag{28}\]  
where \(K(G)=E_{i}\Big{(}\frac{n_{i}-1}{n_{i}}-\frac{\alpha}{n_{i}}\Big{)},(i=2,4)\). By performing simple transformations, we obtain the following result:  
\[E_{4}=\bigg{(}\frac{\omega_{1}^{2}}{((n_{4}-1)/n_{4}+G_{1}/n_{4})^{2}\cdot 10 ^{\Delta}}\bigg{)}^{0.5}. \tag{29}\]  
The coefficient \(E_{2}\) is determined from the formula  
\[E_{2}=1/(10^{\Delta}\cdot K_{4}), \tag{30}\]  
where \(10^{\Delta}=50\).