Complete Example of Kalman Filter Implementation Program

Here we present a complete example of a Kalman filter program that can be utilized for both one-dimensional and two-dimensional calculations. Kalman filtering is a widely adopted technique in control and navigation systems for estimating unknown states, particularly those contaminated by noise. As a recursive filter, it calculates the current state through optimal weighted averaging of past states and observation results. The filter operates based on a state-space model where states are affected by random noise disturbances.

The implementation typically involves two main phases: prediction and update. In the prediction step, the algorithm projects the current state forward using the state transition matrix and estimates the error covariance. The update phase then incorporates new measurements by calculating the Kalman gain, which determines how much weight to give to the new measurement versus the prediction. Key functions in the implementation include state prediction using matrix multiplication, covariance propagation, and measurement residual calculation.

For one-dimensional cases, the implementation simplifies to scalar operations, while two-dimensional scenarios require matrix operations for state transitions and measurement models. The program demonstrates proper handling of process noise covariance (Q) and measurement noise covariance (R) matrices, which are crucial for filter performance. This filter not only smoothes measurement data but also enables future state prediction, making it particularly valuable for real-time applications where accurate state estimation is critical.