TY - JOUR

T1 - A PARTICLE FILTER FOR NONLINEAR FILTERING WITH LÉVY JUMPS

AU - Hausenblas, Erika

AU - Fahim, Kistosil

AU - Fernando, Pani W.

N1 - Publisher Copyright: © 2021. Academic Publications. All Rights Reserved.

PY - 2021

Y1 - 2021

N2 - 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 < s < 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.

AB - 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 < s < 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.

KW - feedback algorithm

KW - Levy copula

KW - Levy processes

KW - nonlinear filtering

KW - particle systems

UR - http://www.scopus.com/inward/record.url?scp=85122565617&partnerID=8YFLogxK

U2 - 10.12732/ijam.v34i5.1

DO - 10.12732/ijam.v34i5.1

M3 - Article

AN - SCOPUS:85122565617

VL - 34.2021

SP - 817

EP - 872

JO - International journal of applied mathematics

JF - International journal of applied mathematics

SN - 1311-1728

IS - 5

ER -