Extended Kalman Filter (EKF)
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An Extended Kalman Filter (EKF) is an kalman-like filter that linearizes about an estimate of the current mean and covariance.
References
2017a
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/extended_Kalman_filter Retrieved:2017-2-14.
- 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.
2017b
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/extended_Kalman_filter#Formulation Retrieved:2017-2-14.
- 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. : [math]\displaystyle{ \boldsymbol{x}_{k} = f(\boldsymbol{x}_{k-1}, \boldsymbol{u}_{k-1}) + \boldsymbol{w}_{k-1} }[/math] : [math]\displaystyle{ \boldsymbol{z}_{k} = h(\boldsymbol{x}_{k}) + \boldsymbol{v}_{k} }[/math] Where wk-1 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-1 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.
At each time step, the Jacobian is evaluated with current predicted states. These matrices can be used in the Kalman filter equations. This process essentially linearizes the non-linear function around the current estimate.
- 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. : [math]\displaystyle{ \boldsymbol{x}_{k} = f(\boldsymbol{x}_{k-1}, \boldsymbol{u}_{k-1}) + \boldsymbol{w}_{k-1} }[/math] : [math]\displaystyle{ \boldsymbol{z}_{k} = h(\boldsymbol{x}_{k}) + \boldsymbol{v}_{k} }[/math] Where wk-1 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-1 is the control vector.
2000
- (Wan & Merwe, 2000) ⇒ Eric A. Wan, and Rudolph Van Der Merwe. (2000). “The Unscented Kalman Filter for Nonlinear Estimation.” In: Adaptive Systems for Signal Processing, Communications, and Control Symposium
- ABSTRACT: This paper points out the flaws in using the extended Kalman filter (EKE) and introduces an improvement, the unscented Kalman filter (UKF), proposed by Julier and Uhlman (1997). A central and vital operation performed in the Kalman filter is the propagation of a Gaussian random variable (GRV) through the system dynamics. In the EKF the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF addresses this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, but is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any nonlinearity. The EKF in contrast, only achieves first-order accuracy. Remarkably, the computational complexity of the UKF is the same order as that of the EKF. Julier and Uhlman demonstrated the substantial performance gains of the UKF in the context of state-estimation for nonlinear control. Machine learning problems were not considered. We extend the use of the UKF to a broader class of nonlinear estimation problems, including nonlinear system identification, training of neural networks, and dual estimation problems. In this paper, the algorithms are further developed and illustrated with a number of additional examples.
1997
- (Boutayeb et al., 1997) ⇒ M. Boutayeb., H. Rafaralahy, and M. Darouach. (1997). “Convergence Analysis of the Extended Kalman Filter Used As An Observer for Nonlinear Deterministic Discrete-time Systems.” In: IEEE Transactions on automatic control 42, no. 4
- ABSTRACT: In this paper, convergence analysis of the extended Kalman filter (EKF), when used as an observer for nonlinear deterministic discrete-time systems, is presented. Based on a new formulation of the first-order linearization technique, sufficient conditions to ensure local asymptotic convergence are established. Furthermore, it is shown that the design of the arbitrary matrix plays an important role in enlarging the domain of attraction and then improving the convergence of the modified EKF significantly. The efficiency of this approach, compared to the classical version of the EKF, is shown through a nonlinear identification problem as well as a state and parameter estimation of nonlinear discrete-time systems.