Kalmanfiltritele

Kalmanfiltritele, often referred to as Kalman filters, are a set of recursive equations that provide an efficient computational method for estimating the state of a linear dynamic system from a series of incomplete and noisy measurements. Developed by Rudolf E. Kálmán in 1960, these filters are widely used in various fields, including navigation, control systems, econometrics, and signal processing.

The Kalman filter operates in two main steps: prediction and update. In the prediction step, it forecasts the next state of the system based on the current state estimate and the system's dynamic model. This prediction is then refined in the update step, where it incorporates new incoming measurements. The filter weighs the prediction and the measurement based on their respective uncertainties, producing an optimal estimate of the system's state.

The core assumption of the standard Kalman filter is that the system dynamics and the measurement process are linear, and that the noise associated with them is Gaussian. However, extensions such as the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) exist to handle non-linear systems. The EKF linearizes the non-linear functions around the current estimate, while the UKF uses a deterministic sampling approach to approximate the probability distribution.

The primary advantage of the Kalman filter is its optimality under certain conditions, meaning it provides the minimum mean-square error estimate of the system's state. Its recursive nature also makes it computationally efficient, as it only requires the previous state estimate and the current measurement to compute the new estimate, rather than needing to store the entire history of observations.