The Role of Kalman Gain and Noise Covariance Selection on the Convergence in State Estimation

In state estimation problems, the Kalman filter (KF) algorithm considers the noise in the measurements and the systems facilitating convergence to the true state. This paper presents the Bayesian derivation of the discrete-time KF algorithm for a simple example known as the random walk model. However, if the KF coefficients are not well-tuned, it can significantly impact the estimation accuracy and may lead to algorithmic inconsistency. The Kalman gain is a quantitative measure which plays a crucial role in achieving the optimum convergence and stability. In this study, we evaluate the importance of the Kalman gain in the KF algorithm across several choices of the error covariance within the context of the random walk model. Furthermore, we demonstrate that the optimal Kalman gain is determined by minimizing the mean squared error (MSE), producing an unbiased and efficient estimate. This adaptive adjustment enables the KF to tune parameters easily. The theoretical and numerical investigations were carried out using the random walk plus noise model.