Chapter 1: Extended Kalman filter
The extended Kalman filter (EKF), which linearizes about an estimate of the current mean and covariance, is the nonlinear variant of the Kalman filter used in estimation theory. The EKF has been taken into account for well-defined transition models.
Between 1959 and 1961, the publications laying the mathematical groundwork for Kalman type filters were released. For linear system models with additive independent white noise in both the transition and measurement systems, the Kalman filter is the best linear estimator. Unfortunately, nonlinear systems make up the majority of engineering systems, hence efforts were made to apply this filtering method to them; the majority of this work was done at NASA Ames. The EKF linearized a model around a working point by using multivariate Taylor series expansions, an adaptation of calculus techniques. If the system model (as stated below) is unknown or unreliable, estimate is done using Monte Carlo methods, particularly particle filters. Although Monte Carlo approaches were used before the EKF, they are more computationally expensive for any state-space of modest dimension.
The state transition and observation models in the extended Kalman filter don't have to be linear functions of the state alternatively they could 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 functions f and h can be used to compute the anticipated state from the prior estimate and the predicted measurement from the expected state, respectively. F and H, however, cannot be simply applied to the covariance. Instead, the Jacobian, a matrix of partial derivatives, is calculated.
The Jacobian is assessed using the most recent projected states at each time step. The Kalman filter equations can use these matrices. Essentially, the non-linear function around the present estimate is linearized by this method.
For notational notes, see the article on the Kalman Filter.
Notation
represents the estimate of
at time n given observations up to and including at time m = n.
When the following definitions are made for the state transition and observation matrices: Jacobians
The extended Kalman filter, unlike its linear version, is generally not the best estimator (it is optimal if the measurement and the state transition model are both linear, as in that case the extended Kalman filter is identical to the regular one). Additionally, because of its linearization, the filter may rapidly diverge if the initial estimate of the state is erroneous or if the process is improperly represented. The calculated covariance matrix tends to underestimate the true covariance matrix, which raises the possibility of statistical inconsistency without the addition of "stabilizing noise," which is another issue with the extended Kalman filter.
After stating this, it can be said that the extended Kalman filter can provide acceptable performance and is perhaps the de facto norm in GPS systems.
Model
Initialize
Predict-Update
The prediction and update processes are coupled in the continuous-time extended Kalman filter, in contrast to the discrete-time extended Kalman filter.
The majority of physical systems are modeled as continuous-time systems, while state estimation by a digital processor typically requires discrete-time measurements. The system model and measurement model are so provided by
where
.
Initialize
Predict
where
Update
where
The update equations are the same as those for the extended discrete-time Kalman filter.
A first-order extended Kalman filter is used in the recursion above (EKF). More terms from the Taylor series expansions may be kept in order to produce higher order EKFs. For instance, descriptions of second and third order EKFs have been made. But higher order EKFs often only improve performance when measurement noise is minimal.
The assumption of additive process and measurement noise is a common component in the formulation of the EKF. But the execution of EKF does not require this presumption. Instead, think about a more extensive framework of the form:
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.
then the equations for covariance prediction and innovation become
where the matrices
and
are Jacobian matrices:
The mean of the process and measurement noise terms, which is assumed to be zero, is used to evaluate the anticipated state estimate and measurement residual. Otherwise, the additive noise EKF is used to implement the non-additive noise formulation.
In some instances, the observation model of a nonlinear system cannot be solved for
, nonetheless, it can be articulated using the implicit function:
where
are the noisy observations.
The following changes can be made to the traditional extended Kalman filter::
where:
Here the original observation covariance matrix
is transformed, and the innovation
is defined differently.
The Jacobian matrix
is defined as before, but determined from the implicit observation model
.
By repeatedly altering the Taylor expansion's central point, the iterated extended Kalman filter enhances the linearization of the extended Kalman filter. This decreases the linearization error but necessitates more computation.
The signal model for the current state estimate is linearized, and the linear Kalman filter is then used to forecast the following estimate, resulting in the extended Kalman filter. The underlying Riccati equation's solutions are not necessarily positive definite, therefore while this aims to create a locally optimal filter, it is not necessarily stable. The fictitious algebraic Riccati strategy, which trades off optimality for stability, is one method of enhancing performance. The extended Kalman filter's recognizable structure is kept, but stability is attained by choosing a positive definite solution to a fictitious algebraic Riccati equation for the gain design.
Using the H-infinity findings from robust control is another method for enhancing the performance of extended Kalman filters. By including a positive definite term in the design Riccati equation, robust filters are produced. A scalar that the designer can adjust to accomplish a trade-off between mean-square-error and peak error performance criterion parametrizes the new term.
For nonlinear systems with symmetries, the invariant extended Kalman filter (IEKF) is a modified version of the EKF (or invariances). It combines the benefits of both the recently developed symmetry-preserving filters and the EKF. The gain matrix is not updated from a linear state error, but from an invariant state error; similarly, the IEKF utilizes a geometrically adapted correction term based on an invariant output error instead of a linear correction term based on a linear output error. The key advantage is that, unlike equilibrium points for the EKF, the gain and covariance equations converge to constant values on a significantly larger set of trajectories, leading to better convergence of the estimation.
The unscented Kalman filter is a nonlinear Kalman filter that appears to be an improvement over the EKF (UKF). In the UKF, the underlying Gaussian distribution is represented by a deterministic sampling of points that approximates the probability density. The posterior distribution's moments can then be deduced from the transformed samples by using the nonlinear transformation of these points as an estimation of the posterior distribution. The process is referred to as the unscented transform. When estimating error in both directions, the UKF is typically more reliable and accurate than the EKF.
"The extended Kalman filter (EKF), a nonlinear system estimation technique, is perhaps the most popular one. But more than 35 years of experience in the estimation field have demonstrated that it is challenging to implement, challenging to tune, and only trustworthy for systems that are nearly linear on the time scale of the updates. Its use of linearization is largely to blame for many of these issues."
The Second Order Extended Kalman Filter (SOEKF), also known as the enhanced Kalman filter, is believed to be more accurate than some UKF variations that have been published, according to simulation results from a 2012 research. However, because the UKF also employs linearization-specifically, linear regression-it is not exempt from this challenge. The square root of the covariance matrix numerical approximation typically causes stability problems for the UKF, whereas potential problems with the Taylor Series approximation along the trajectory cause stability problems for both the EKF and the SOEKF.
The Ensemble Kalman filter, developed by Evensen in 1994, really predated the UKF. The advantage over the UKF is that applications in very high-dimensional systems, like weather prediction, with state-space sizes of a billion or more, are possible since the number of ensemble members utilized can be much smaller than the state dimension.
In order to create a true possibilistic filter, probability distributions have recently been proposed to be replaced by possibility distributions in a fuzzy Kalman filter. This allows for the use of non-symmetric process and observation noises as well...
