ean-Square Filtering for Polynomial System States Confused with Poisson Noises over Polynomial Observations

Abstract

In this paper, the mean-square filtering problem for polynomial system states confused with white Poisson noises over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, the paper deals with the general case of nonlinear polynomial states and observations with white Poisson noises. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third order sensor filtering problem for a quadratic state, assuming a conditionally Poisson initial condition for the extended third order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.

Mean-square filter design for stochastic polynomial systems with Gaussian and Poisson noises

“…It is known that the mean-square filter for linear systems with Poisson white noises coincides with the Kalman-Bucy filter (Liptser and Shiryayev 1989;Pugachev and Sinitsyn 2001). Other results related to non-linear Poisson systems can be found in Lu, Liang, and Chen (2001), Kolmanovsky and Maizenberg (2002a), Hannequin and Mas (2002), Kolmanovsky and Maizenberg (2002b), Zhang, Fadili, Starck, and Dige (2008b), Dupé, Fadili, and Starck (2008), Zhang, Fadili, and Starck (2008a), , Basin, Maldonado, and Karimi (2011), Basin and Maldonado (2012). However, to the best of authors' knowledge, no filtering algorithms solving the mean-square filters * Corresponding author.…”

Mean-square filtering problem for stochastic polynomial systems with Gaussian and Poisson noises

“…As shown in [13], [14], [24], a closed system of the filtering equations for a polynomial system state over linear observations can be obtained in case of Gaussian or Poisson white noises in the state and observation equations. In case of Gaussian and Poisson white noises, a special transformation of the observation equation [14] is first applied to cast the observation equation in a new form with an invertible observation matrix.…”

“…Performance of the designed optimal filter is verified for a third degree polynomial 2D state over scalar linear observations against the mean-square filters for stochastic polynomial systems with Gaussian [14] and Poisson [24] white noises. It can be observed that all components of the estimate produced by the designed filter rapidly converge to the state, whereas this is not so for the other estimates.…”

Robust Adaptive Filter for Small Satellite Attitude Estimation Based on Magnetometer and Gyro

“…It can estimate online the statistical characteristic of the noise in real time, so as to adjust the filter parameter [8]. Many results on estimation and adaptive filtering design for different kinds of dynamic systems have been obtained [9][10][11][12][13][14][15][16][17][18]. Adaptive filtering can be divided into three basic types: multimode self-adaptive filtering, self-adaptive filtering based on the innovation, and adaptive filtering based on the residual [19].…”