Kalman filter information

For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.

This digital filter is sometimes termed the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, nonlinear filter developed somewhat earlier by the Soviet mathematician Ruslan Stratonovich.^[1][2][3][4] In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before the summer of 1961, when Kalman met with Stratonovich during a conference in Moscow.^[5]

Kalman filtering^[6] has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically.^[7] Furthermore, Kalman filtering is a concept much applied in time series analysis used for topics such as signal processing and econometrics. Kalman filtering is also one of the main topics of robotic motion planning and control^[8][9] and can be used for trajectory optimization.^[10] Kalman filtering also works for modeling the central nervous system's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback, the use of Kalman filters^[11] provides a realistic model for making estimates of the current state of a motor system and issuing updated commands.^[12]

The algorithm works by a two-phase process having a prediction phase and an update phase. For the prediction phase, the Kalman filter produces estimates of the current state variables, along with their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise) is observed, these estimates are updated using a weighted average, with more weight being given to estimates with greater certainty. The algorithm is recursive. It can operate in real time, using only the present input measurements and the state calculated previously and its uncertainty matrix; no additional past information is required.

Optimality of Kalman filtering assumes that errors have a normal (Gaussian) distribution. In the words of Rudolf E. Kálmán: "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear."^[13] Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible linear estimator in the minimum mean-square-error sense,^[14] although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature^[where?]) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian.^[15]

Extensions and generalizations of the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The basis is a hidden Markov model such that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in multi-sensor fusion,^[16] and distributed sensor networks to develop distributed or consensus Kalman filtering.^[17]