kalman filter

kalman filter

The Kalman Filter and its extended version (EKF) are developments of the least-squares analysis method in the framework of a sequential data assimilation, in which each background is provided by a forecast that starts from the previous analysis. It is adapted to the real-time assimilation1 of observations distributed in time into a forecast model .The analysis equations of the linear Kalman Filter are exactly the ones already described in the least-squares analysis theorem. The notation is the same, except that the background (i.e. forecast) and analysis error covariance matrices are now respectively denoted and . The background state is a forecast denoted . Kalman filtering was developed in the 1960s, although it has its roots as far back as Karl Gauss in 1795. Kalman filtering has been applied in areas as diverse as aerospace, marine navigation, nuclear power plant instrumentation, demographic modeling, manufacturing, and many others.

Mathematics

Consider the problem of estimating the variables of some system. In dynamic systems (that is, systems which vary with time) the system variables are often denoted by the term state variables. Assume that the system variables, represented by the vector x, are governed by the equation xk+1 = Axk + wk where wk is random process noise, and the subscripts on the vectors represent the time step. For instance, if our dynamic system consists of a spacecraft which is accelerating with random bursts of gas from its reaction control system thrusters, the vector x might consist of position p and velocity v. Then the system equation would be given by Equation 1:

where ak is the random time-varying acceleration, and T is the time between step k and step k+1. Now suppose we can measure the position p. Then our measurement at time k can be denoted zk = pk + vk where vk is random measurement noise.

The question which is addressed by the Kalman filter is this: Given our knowledge of the behavior of the system, and given our measurements, what is the best estimate of position and velocity? We know how the system behaves according to the system equation, and we have measurements of the position, so how can we determine the best estimate of the system variables? Surely we can do better than just take each measurement at its face value, especially if we suspect that we have a lot of measurement noise.

Notation and hypothesesThey are the same as in the least-squares analysis theorem, except that:

• the background and analysis error covariance matrices and are respectively replaced by and to denote the fact that the background is now a forecast.

 

• The time index of each quantity is denoted by the suffix . The model forecast operator from dates to is denoted by

 

• forecast errors: the deviation of the forecast prediction from the true evolution, , is called the model error2 and we assume that it is not biased3 and that the model error covariance matrix is known.

 

• uncorrelated analysis and model errors: the analysis errors and ...