Mean-square filter design for nonlinear polynomial systems with Poisson noise

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

“…Applying the mean-square filter for incompletely measured polynomial states with a polynomial multiplicative noise over linear observations (see Basin and Maldonado (2011)) to the system (7),(1),(8) yields the desired filtering equations (4),(5). Finally, after representing the superior conditional moments of the system state as functions of the conditional expectation m(t) and error variance P (t) using the property of a Poisson random variable x(t) − m(t) of representing the superior conditional moments of the system state as functions of the variance P (t), (see Basin and Maldonado (2011) for details), a finite-dimensional system of the filtering equations, closed with respect to m(t) and P (t), can be obtained, if the initial condition [z 0 , x 0 ] for the extended state vector is conditionally Poisson.…”

“…Note that some particular cases of Theorem 1, like linear or bilinear systems with statedependent noises, were previously considered in Basin and Maldonado (2011), where the explicit mean-square finite-dimensional filtering equations were obtained. On the other hand, the general result of Theorem 1 allows one to design a suboptimal mean-square finitedimensional filter for any polynomial state confused with white Poisson noise disturbances over polynomial observations.…”

“…This filtering problem generalizes the cubic sensor problem stated in Hazewinkel et al (1983). The resulting filter yields a reliable and rapidly converging estimate, in spite of a significant difference in the initial conditions between the state and estimate, whereas the filter designed for systems with white Gaussian noises, constructed according to Basin et al (2009) and Basin and Maldonado (2011), behaves unsatisfactorily.…”

“…There also exists an extensive bibliography on robust, in particular, H ∞ filtering for linear (Xu and Chen (2003), Mahmoud and Shi (2003) and Xu et al (2005)) and nonlinear (Xie et al (1996), Nguang and Fu (1996), Fridman and Shaked (1997), Shi (1998), Fleming and McEneaney (2001), Yaz and Yaz (2001), Xu and van Dooren (2002), Wang et al (2003), Gao and Wang (2004), Zhang et al (2005), Gao et al (2005), Zhang et al (2007), Gao and Chen (2007), Wang et al (2008), Wang et al (2009), Wei et al (2009) and Shen et al (2009)) stochas-tic systems. Apart form the "general situation," the mean-square finite-dimensional filters have been designed for certain classes of polynomial system states with Gaussian noises over linear observations (Basin (2008), Basin et al (2008) and Basin et al (2009)) and a few results related to nonlinear Poisson systems can be found in Lu et al (2001), Kolmanovsky and Maizenberg (2002a), Hannequin and Mas (2002), Kolmanovsky and Maizenberg (2002b), Zhang et al (2008a), Dupé et al (2008), Zhang et al (2008b), and Basin and Maldonado (2011). Recently, the mean-square filtering problem for polynomial systems, where both, state and observation, equations include polynomial functions of the system state in the right-hand sides, was solved in Basin et al (2010); however, that paper did not consider systems corrupted with non-Gaussian noises.…”

“…On the other hand, it is well-known that there are a number of practical situations where dynamic system states are corrupted not with uniformly acting white Gaussian noises (like a static noise in a phone line) but with noises acting at random isolated time moments (like a series of electromagnetic impulses), which are referred to as white Poisson noises. This paper presents an approximate finitedimensional filter for polynomial system states confused with white Poisson noises over polynomial observations, continuing the research in the area of the mean-square filtering for polynomial systems with Gaussian (Basin (2008), Basin et al (2008) and Basin et al (2009)) and Poisson Basin and Maldonado (2011) noises. In contrast to the previously obtained results, the paper deals with the general case of nonlinear polynomial states and observations with white Poisson noises.…”