Tillståndsrumkalmanfilter

Tillståndsrumkalmanfilter, often referred to as a state-space Kalman filter, is a mathematical algorithm used for estimating the state of a linear dynamic system from a series of noisy measurements. It is a recursive estimator, meaning it only requires the current measurement and the previous estimate to compute the new estimate. This makes it computationally efficient and suitable for real-time applications.

The Kalman filter operates by modeling the system's behavior and the measurement process using linear equations. The system's state is represented by a vector, and its evolution over time is described by a state transition matrix. The measurements are related to the state through a measurement matrix. Both the system and measurement processes are assumed to be subject to random noise, which is characterized by covariance matrices.

The algorithm proceeds in two main steps: prediction and update. In the prediction step, the filter uses the system model to predict the next state and its associated uncertainty. In the update step, it incorporates the latest measurement to refine the predicted state, taking into account the uncertainty of both the prediction and the measurement. The output of the Kalman filter is an optimal estimate of the system's state and its covariance, representing the uncertainty in the estimate. This optimal estimate minimizes the mean-squared error of the estimation. The tillståndsrumkalmanfilter is widely used in navigation, control systems, signal processing, and econometrics.