## 4 Modeling the Target Tracking Process Using Kalman Filters  
Radar systems, especially under interference conditions, face challenges in accurately tracking targets. During the modeling of these systems, GERT models provide a powerful tool for analyzing complex probabilistic processes, such as multipath delays, interactions between states, and temporal dependencies. However, for precise target tracking, where local parameter estimation--such as position and velocity--is critical, GERT models have the following limitations:  
- Macro-level nature of the GERT model: GERT networks excel at describing overall temporal and probabilistic characteristics of processes. However, they are not designed to estimate micro-level parameters, such as precise real-time target coordinates;  
- Limitations in dynamic data processing: GERT models do not support the iterative refinement process necessary for real-time data processing. Under noisy and highly uncertain conditions, a method capable of adapting to changes is required;* Error minimization challenge: For target tracking tasks, minimizing the root mean square error (RMSE) and ensuring robustness to false alarms are critical. GERT is not capable of effectively accounting for measurement noise and dynamic changes. While GERT networks offer macro-level insights, they lack the precision required for real-time trajectory refinement, necessitating the use of complementary techniques like Kalman filtering. In this regard, the use of Kalman filters becomes essential. They offer powerful capabilities for solving recursive filtering problems, accounting for noise, predicting target states, and refining estimates. The GERT model provides a probabilistic and temporal foundation for analyzing general processes, while the Kalman filter complements it. The Kalman filter is a powerful recursive algorithm used to estimate the state of a dynamic system under noisy and uncertain conditions. The operation of the filter involves two key stages:  
* Prediction. This stage predicts the next state of the object based on its motion model. * Correction. At this stage, the prediction is adjusted using the incoming measurements, minimizing errors and accounting for system noise. The prediction stage is used to compute the future state of the object based on the current estimate and the dynamics model. At this stage, the state of the object (e.g., its position and velocity) is updated according to the equation:  
\[\hat{x}_{k|k-1}=A_{k}\hat{x}_{k-1|k-1}+B_{k}u_{k}\]  
where  
\(\hat{x}_{k|k-1}\) is the predicted state of the system at step \(k\);  
\(A_{k}\) is the state transition matrix, which describes the dynamics of the object;  
\(\hat{x}_{k-1|k-1}\) is the previous adjusted status;  
\(B_{k}u_{k}\) is the influence of the control action, where \(B_{k}\) is the control matrix and \(u_{k}\) is the vector of control parameters (if any). Prediction of the error covariance: The state error covariance is updated to account for the process noise:  
\[P_{k|k-1}\ =A_{k}P_{k-1|k-1}\ A_{k}^{T}\ +Q_{k}\,\]  
where  
\(P_{k|k-1}\) is the error covariance matrix for the predicted state;  
\(Q_{k}\) is the process noise covariance matrix (characterizes dynamic uncertainty of the system). This stage provides a prediction of the future state of the object, which will subsequently be refined during the correction stage. The correction stage is used to adjust the predicted state based on new measurement data. This allows the filter to account for observation errors and minimize deviations. Calculation of the Kalman gain: The Kalman gain determines the extent to which the measurement data should influence the state update:  
\[K_{k}\ =P_{k|k-1}H_{k}^{T}\ (H_{k}\ P_{k|k-1}H_{k}^{T}\ +R_{k}\ )^{-1},\]  
where  
\(H_{k}\) is the observation matrix (relates the state of the system to the measured quantities);  
\(R_{k}\) is the measurement noise covariance matrix (describes the uncertainty of measurement data).