## 4 Modeling the Target Tracking Process Using Kalman Filters
Using [PERSON]'s expressions [28]:  
\[\begin{array}{c}\varphi(x)=\sum\limits_{k=1}^{8}\text{Res}[\text{e}^{xx} \Phi(z)]=e^{(a+bi)x}\frac{\tau+i\beta}{\gamma+i\beta}+e^{(a+bi)x}\frac{\tau-i \delta}{\gamma-i\delta}=\\ =\  
Figure 7 illustrates the \"rate\" at which the probability values decrease for the radar system to complete interaction with the object within a specified time frame. The graph helps assess how much time the radar system will need to finish tracking the object under different interference scenarios.  
At a high intensity of external influences (\(\lambda=9.0\)), the probability of completing the interaction reaches its maximum within shorter time intervals. As the intensity of external influences decreases, the probability of completing the interaction also decreases. These results confirm the practical applicability of the model and its utility for studies at different interference intensity levels.  
Similar results can be observed in Figure 8. This figure shows that at \(\lambda=9.0\), the probability density peak occurs within a shorter time span, indicating rapid completion of the interaction. At \(\lambda=5.0\), the peak occurs later, reflecting an increase in interaction time. At \(\lambda=2.0\), the probability is maximized over longer time intervals.  
For the example presented above, the coefficients of the polynomial at \(\lambda_{1}=9.0\):  
\[[1.28835972\text{e}+06,\,-1.84190646\text{e}+06,\,1.13882976\text{e}+06,\]  
\[-3.97856880\text{e}+05,\,8.59725600\text{e}+04,\,-1.17760000\text{e}+04,\]  
\[9.99000000\text{e}+02,\,-4.80000000\text{e}+01,\,1.00000000\text{e}+00]\]  
The coefficients of the polynomial at \(\lambda_{1}=5.0\):  
\[[7.15755398\text{e}+05,\,-1.08779337\text{e}+06,\,7.17163200\text{e}+05,\]  
\[-2.68071600\text{e}+05,\,6.22052000\text{e}+04,\,-9.18400000\text{e}+03,\]  
\[8.43000000\text{e}+02,\,-4.40000000\text{e}+01,\,1.00000000\text{e}+00]\]  
The coefficients of the polynomial at \(\lambda_{1}=2.0\):  
\[[2.86302159\text{e}+05,\,-5.22208547\text{e}+05,\,4.00913280\text{e}+05,\]  
\[\text{f}(z)=\frac{d_{m}z^{m}+d_{m-1}z^{m-1}+\ldots+d_{1}z+d_{0}}{l_{m}z^{m}+l_{m -1}z^{m-1}+\ldots+l_{1}z+l_{0}},\ d_{m}\
eq 0,\ l_{m}\
eq 0 \tag{12}\]0  
\[\text{f}(z)=\frac{d_{m}z^{m}+d_{m-1}z^{m-1}+\ldots+d_{1}z+d_{0}}{l_{m}z^{m}+l_{m -1}z^{m-1}+\ldots+l_{1}z+l_{0}},\ d_{m}\
eq 0,\ l_{m}\
eq 0 \tag{12}\]1  
Let us evaluate the reliability of the results obtained as a result of mathematical modeling. To check this hypothesis about the compliance of the results will use reliability criteria according to the Pearson \(\chi^{2}\) criterion [29; 30; 31]:  
\[\text{f}(z)=\frac{d_{m}z^{m}+d_{m-1}z^{m-1}+\ldots+d_{1}z+d_{0}}{l_{m}z^{m}+l_{m -1}z^{m-1}+\ldots+l_{1}z+l_{0}},\ d_{m}\
eq 0,\ l_{m}\
eq 0 \tag{12}\]2  
where \(k\) is the number of digits (intervals) of the statistical series;  
\(P_{i}^{*}\) and \(P_{i}\) are the \"statistical\" and theoretical probabilities of the event. As a result of the experiment, theoretical values, \(\chi^{2}\), and a tabular value, \(\overline{\chi^{2}}\), were obtained as the inverse of the right-hand probability distribution, \(\chi^{2}\). The conducted verification showed that the proposed hypothesis can be considered plausible or, at least, does not contradict the results obtained in mathematical modeling. This is confirmed by the fact that with a sufficiently large value of the confidence probability \(Q=0.95\) for all considered \(\lambda_{1}(\lambda_{1}=9;\,\lambda_{1}=5;\,\lambda_{1}=2)\), the corresponding values \(\chi^{2}\) (\(\chi^{2}_{1}=19.1,\,\chi^{2}_{2}=14.5,\,\chi^{2}_{3}=12.2\)) are much less than \(\overline{\chi^{2}}=99.1\), which allows us to recognize the discrepancies between the \"statistical\" (\(P_{i}^{*}\)) and theoretical (\(P_{i}\)) probabilities of the event occurrence as insignificant. The accuracy of the developed model can be evaluated through a comparative study with mathematical models using similar formalization approaches. One such approach is mathematical modeling with Markov networks.