A PARTICLE FILTER FOR NONLINEAR
FILTERING WITH LÉVY JUMPS
Erika Hausenblas1, Kistosil Fahim2, Pani W. Fernando3
1 Chair of Applied Mathematics
Montanuniversitat Leobe n
Leoben – 8700, AUSTRIA
2 Department of Mathematics
Institut Teknologi Sepuluh Nopember
Surabaya - 60111, INDONESIA
3 Department of Mathematics
University of Sri Jayewardenepura Gangodawila
Nugegoda – 10250, SRI LANKA

Abstract

Dynamical systems arise in engineering, physical sciences as well as in social sciences. If the state of a system is known, one also knows its properties, and may, e.g., stabilise the system and prevent it from blowing up, or predict its near future. However, the state of a system consists often on internal parameters which are not always accessible. Instead, often only an observation process $Y$, which is a transformation of the current state, is accessible. Furthermore, a system operates in real environments; hence, itself and its observation are affected by random noise and/or disturbances. So, in reality, the dynamics of the system and the observation are corrupted by noise. The problem of nonlinear filtering is estimating the state of the system $X(t)$ at a given time $t>0$ through the data of the observation $Y$ until time $t$ (i.e. $\{Y(s):0\le s\le t\}$).

Usually, one considers models where the state process and the observation process are perturbed by Gaussian noise. When these perturbations are known to exhibit extreme behaviour, as seen frequently in application from finance or environmental studies, a model relying on the Gaussian distribution is not appropriate. A suitable alternative could be a model based on a heavy-tailed distribution, as the stable distribution. In such a model, these perturbations are allowed to have extreme values with a probability which is significantly higher than in a Gaussian-based model.

In general, a stable process can not be simulated directly. In practice, we can approximate it by a Gaussian and a compound Poisson process. In particular, we replace the small jumps by a Gaussian process. Thus, we are interested in nonlinear filtering where the signal and observation processes are corrupted by a Gaussian and a compound Poisson process. To catch up the jumps, we use methods from control engineering and construct a so-called Luenberger observer. These methods are combined with particle filters to construct an estimator of the state process, respective, an estimator of the density process. We apply this method to a mathematical pendulum, a single-link flexible joint robot, and a Van der Pol oscillator.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 34
Issue: 5
Year: 2021

DOI: 10.12732/ijam.v34i5.1

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