Extended Kalman filter

In estimation theory, the extended Kalman filter (EKF) is the nonlinear version of the Kalman filter which linearizes about an estimate of the current mean and covariance. In the case of well defined transition models, the EKF has been considered the de facto standard in the theory of nonlinear state estimation, navigation systems and GPS.'The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization.' In estimation theory, the extended Kalman filter (EKF) is the nonlinear version of the Kalman filter which linearizes about an estimate of the current mean and covariance. In the case of well defined transition models, the EKF has been considered the de facto standard in the theory of nonlinear state estimation, navigation systems and GPS. The papers establishing the mathematical foundations of Kalman type filters were published between 1959 and 1961. The Kalman Filter is the optimal linear estimate for linearsystem models with additive independent white noise in both the transition and the measurement systems.Unfortunately, in engineering, most systems are nonlinear, so some attempt was immediately made to applythis filtering method to nonlinear systems. Most of this work was done at NASA Ames. The EKF adapted techniques from calculus, namely multivariate Taylor Series expansions, to linearize a model about a working point. If the system model (as described below) is not well known or is inaccurate, then Monte Carlo methods, especially particle filters, are employed for estimation. Monte Carlo techniques predate the existence of the EKF but are more computationally expensive for any moderately dimensioned state-space. In the extended Kalman filter, the state transition and observation models don't need to be linear functions of the state but may instead be differentiable functions. Here wk and vk are the process and observation noises which are both assumed to be zero mean multivariate Gaussian noises with covariance Qk and Rk respectively. uk is the control vector. The function f can be used to compute the predicted state from the previous estimate and similarly the function h can be used to compute the predicted measurement from the predicted state. However, f and h cannot be applied to the covariance directly. Instead a matrix of partial derivatives (the Jacobian) is computed.