## 4 Modeling the Target Tracking Process Using Kalman Filters
The predicted state is being refined in the light of new data:  
\[\hat{x}_{k|k}=\hat{x}_{k|k-1}+K_{k}\Big{(}z_{k}-H_{k}\hat{x}_{k|k-1}\Big{)}\,,\]  
where  
\(z_{k}\) is the vector of measurements at step \(k\);  
\(z_{k}-H_{k}\hat{x}_{k|k-1}\) is the difference between measurements and the predicted state (observation error);  
\(K_{k}\) is the Kalman gain. The error covariance matrix is updated to reflect the reduced non-certainty:  
\[P_{k|k}\ =(I-K_{k}\ H_{k}\ )P_{k|k-1}\,\]  
where  
\(P_{k|k}\) is the updated state covariance matrix;  
\(I\) is the unit matrix. These stages are implemented based on the state and measurement equations, where the transition matrices, process noise covariance, and measurement noise covariance play a key role. A more detailed description of the algorithm can be found, for example, in works [24; 25; 26; 27], where the corresponding formulas and application examples are presented. In the context of our study, the Kalman filter is used to refine the trajectories obtained after the initial data processing. During the prediction stage, the target motion model, including its maneuverability, is taken into account. The correction stage reduces the impact of noise arising during the measurement of target coordinates and velocities. The use of the Kalman filter in combination with GERT models allows the integration of the advantages of both approaches, providing both macro-level process analysis and micro-level data detailing. This integration opens up opportunities for developing a flexible and accurate target tracking model in radar systems, especially under conditions of high uncertainty and noise. The connection between the two approaches can be represented as a sequence of stages and operations. The GERT model enables the following operations:  
* Models the high-level process of target detection and tracking;
* Describes probabilistic state transitions and time delays. The Kalman filter takes data generated by the GERT model and uses it as initial conditions for its operation. After the first step, the Kalman filter refines the local target trajectories, minimizing the influence of noise. This integration accounts for both macro-level and micro-level aspects of radar systems. At the macro level, the overall process structure is analyzed, including the impact of different states and probabilities. At the micro level, tracking accuracy is improved through detailed data processing. The proposed approach offers several advantages:  
* Improved accuracy. The Kalman filter reduces uncertainty caused by noise and interference, improving the quality of data provided by the GERT model;
* Robustness to interference. The GERT model accounts for complex probabilistic scenarios, while the Kalman filter adapts to changing real-time conditions;
* Efficiency. The combination of both approaches ensures a balanced workload: the GERT model manages the overall process structure, while the Kalman filter focuses on local tasks. Figure 3 presents a structural diagram illustrating the relationship between the GERT model and the Kalman filter. As an example of implementing the mathematical model complex, the following algorithm is presented and implemented:  
Step 1. Target Detection (GERT). The GERT model simulates the detection process and determines the probabilities and time characteristics of target detection. Step 2. Initial State Estimation (GERT). The GERT model provides a probabilistic estimate of the target's position and its initial trajectory. Step 3. Trajectory Refinement (Kalman Filter). The Kalman filter refines the trajectory based on the data provided by the GERT model, minimizing errors and ensuring continuous tracking. The results of the implementation and evaluation of the interaction between the two modeling approaches are presented in Figure 4.  
The simulation represented in Figure 4 is based on specific parameters and operational conditions that reflect real-world scenarios.