By Michael Basin
0. 1 creation even though the final optimum resolution of the ?ltering challenge for nonlinear country and remark equations burdened with white Gaussian noises is given by means of the Kushner equation for the conditional density of an unobserved nation with recognize to obser- tions (see  or , Theorem 6. five, formulation (6. seventy nine) or , Subsection five. 10. five, formulation (5. 10. 23)), there are a really few identified examples of nonlinear structures the place the Ku- ner equation will be lowered to a ?nite-dimensional closed approach of ?ltering eq- tions for a definite variety of decrease conditional moments. the main recognized outcome, the Kalman-Bucy ?lter , is expounded to the case of linear nation and statement equations, the place simply moments, the estimate itself and its variance, shape a closed process of ?ltering equations. notwithstanding, the optimum nonlinear ?nite-dimensional ?lter should be - tained in another situations, if, for instance, the nation vector can take just a ?nite variety of admissible states  or if the commentary equation is linear and the glide time period within the 2 2 kingdom equation satis?es the Riccati equation df /dx + f = x (see ). the full classi?cation of the “general scenario” situations (this signifies that there are not any unique - sumptions at the constitution of kingdom and remark equations and the preliminary conditions), the place the optimum nonlinear ?nite-dimensional ?lter exists, is given in .
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Extra info for New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems
55). 2, a closed system of the filtering equations for a system state with polynomial drift and state-independent diffusion over linear observations can be obtained if the observation matrix A(t) is invertible for any t ≥ t0 . 56), implies that the random variable x(t) − m(t) is conditionally Gaussian with respect to the observation process y(t) for any t ≥ t0 . 58). 59) E(g(x,t)gT (x,t) | FtY ) − P(t)AT (t)(B(t)BT (t))−1 A(t)P(t))dt, with the initial condition P(t0 ) = E[(x(t0 ) − m(t0 )(x(t0 ) − m(t0 )T | FtY0 ].
On the contrary, the conventionally designed extended Kalman-Bucy estimates diverge from the real values. 51), as it should be in the optimal filter. Although this conclusion follows from the developed theory, the numerical simulation serves as a convincing illustration. 1 Problem Statement Let (Ω , F, P) be a complete probability space with an increasing right-continuous family of σ -algebras Ft ,t ≥ t0 , and let (W1 (t), Ft ,t ≥ t0 ) and (W2 (t), Ft ,t ≥ t0 ) be independent Wiener processes. 3 Filtering Problem for Polynomial State with Multiplicative Noise 25 and a linear differential equation for the observation process dy(t) = (A0 (t) + A(t)x(t))dt + B(t)dW2(t).
In the next subsection, performance of the designed optimal filter for a bilinear state with a bilinear multiplicative noise over linear observations is verified against the optimal filter for a bilinear state with a state-independent noise and a conventional extended Kalman-Bucy filter. 3 Example This subsection presents an example of designing the optimal filter for a quadratic state with a quadratic multiplicative noise over linear observations and comparing it to the optimal filter for a quadratic state with a state-independent noise and a conventional extended Kalman-Bucy filter.
New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems by Michael Basin