Extended Kalman filter for nonlinear state estimation, basic theory outline

This video outlines some theoretical considerations that give rise to the "extended Kalman filter." Such a filter is the generalization of the Kalman filter for linear time-invariant systems to nonlinear systems, based on the time-varying linearization around the estimated state trajectory. A quick recap on the non-stationary Kalman filter equations for linear time-invariant systems is carried out in the first four minutes of the video. Next, the nonlinear extension is described. Basically, if the "true" state $x_k$ is close to the estimated state $\hat x_k$, then the linearisation of the state and output equations will allow us to estimate the next state or the measurement in both mean and variance. With that, we have the information to set up a standard Kalman filter. The result is a non-stationary filter with model process matrices that change as the state and input change. The basic drawback is that it is an approximation: with strongly nonlinear systems or inaccurate initial conditions for the state estimate, the filter ceases to be "close to optimal," and, in fact, it can even be unstable. However, it is widely used in applications due to its simplicity, efficient computational implementation if Jacobians are available, and reasonably good behavior in "almost linear" systems. ________________ PDF/code/notes at: http://personales.upv.es/asala/YT/V/ekfteoEN.html #kalman #nonlinear #robotics #controlengineering _________________ Antonio Sala Full collection of videos at: http://personales.upv.es/asala/YT/indexEN.html