Kalman filtering with empirical noise models

Posterior linearisation filter for non-linear state transformation noises

“…With PLF the linearization changes between each iteration and the PLF can be used with transforming functions that effectively allow to use non-Gaussian noises with a Kalman type filter. The formulation in [10] uses an augmented state, where the measurement noise variables ε i are augmented with the state dimensions in the update to obtain posterior estimate and the posterior linearization for those too. In the augmented state formulation, the posterior covariance may be singular, which causes problems in computation of the statistical linearization of the PLF.…”

“…where Φ is the Cumulative Distribution Function (CDF) of a standard normal distribution and F −1 is the inverse CDF of the desired distribution. In [10] was also presented a method to determine function g i (•) from samples. Function g i (•) was built using piecewise cubic Hermite interpolating polynomials.…”

“…Filtering problems with (18) could be solved with the PLF presented in [10], but (17) would not get any help from using PLF as it does the prediction as GGF. PLS has different linearization points for the state transition model between iterations, but it is only for additive Gaussian noises.…”

“…In [10] the measurement noise ε i was augmented with the state so that both were estimated in iterations. In this paper we will also augment the state transition noise in the augmented state…”

“…This augmented state can be updated with the version of PLF that was introduced in [10]. The iterations involving the augmented state (19) and measurement model (20) produce the xi |y 1:i , but we have to note that the first elements of the augmented state are x i−1 |y 1:i , which is not the posterior of the time instance i, but a single step fixed-lag smoothed state.…”