By Michael Basin
0. 1 advent even supposing the overall optimum answer of the ?ltering challenge for nonlinear nation and remark equations careworn with white Gaussian noises is given via the Kushner equation for the conditional density of an unobserved country with recognize to obser- tions (see [48] or [41], Theorem 6. five, formulation (6. seventy nine) or [70], Subsection five. 10. five, formulation (5. 10. 23)), there are a really few identified examples of nonlinear platforms the place the Ku- ner equation should be diminished to a ?nite-dimensional closed approach of ?ltering eq- tions for a definite variety of decrease conditional moments. the main well-known consequence, the Kalman-Bucy ?lter [42], is expounded to the case of linear kingdom and remark equations, the place basically moments, the estimate itself and its variance, shape a closed process of ?ltering equations. although, the optimum nonlinear ?nite-dimensional ?lter will be - tained in another circumstances, if, for instance, the kingdom vector can take just a ?nite variety of admissible states [91] or if the statement equation is linear and the float time period within the 2 2 country equation satis?es the Riccati equation df /dx + f = x (see [15]). the full classi?cation of the “general state of affairs” instances (this implies that there aren't any distinct - sumptions at the constitution of kingdom and commentary equations and the preliminary conditions), the place the optimum nonlinear ?nite-dimensional ?lter exists, is given in [95].
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Additional info for New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems
Example text
48) confused with independent and identically distributed disturbances modeled as white Gaussian noises. 5. 2P22(t) − P12 (t), with the initial condition P(0) = E((x(0) − m(0))(x(0) − m(0))T | y(0)) = P0 . 2PK22(t) − PK12 (t), with the initial condition PK (0) = E((x(0) − m(0))(x(0) − m(0))T | y(0)) = P0 . 54). 47). 1, P011 = P012 = P022 = 1. 48) is realized using the built-in MatLab white noise function. 5 6 4 2 0 Fig. 4. 54). 5 (Fig. 4). 51). 26. 54). Discussion The simulation results show that the values of the estimate calculated by using the obtained optimal filter for a quadratic-linear state with unmeasured linear part over linear observations are noticeably closer to the real values of the reference variable than the values of the estimate given by the conventional extended Kalman-Bucy filter.
Mn (t) Pn1 (t) Pn2 (t) · · · Pnn (t) ⎤ ⎡ m1 (t)P11 (t) m1 (t)P12 (t) · · · m1 (t)P1n (t) ⎥ ⎢ ⎢ m2 (t)P21 (t) m2 (t)P22 (t) · · · m2 (t)P2n (t) ⎥ ⎥, ⎢ .. .. ⎥ ⎢ . ⎦ ⎣ . . mn (t)Pn1 (t) mn (t)Pn2 (t) · · · mn (t)Pnn (t) and the transposed matrix P(t) ∗ mT (t) = (m(t) ∗ P(t))T is defined as the matrix whose columns are equal to columns of P(t) multiplied by the same corresponding element of m(t). 19) for the error variance matrix P(t) form a closed system of filtering equations in the case of a third-order polynomial state equation and linear observations.
26. 54). Discussion The simulation results show that the values of the estimate calculated by using the obtained optimal filter for a quadratic-linear state with unmeasured linear part over linear observations are noticeably closer to the real values of the reference variable than the values of the estimate given by the conventional extended Kalman-Bucy filter. 47) itself is unstable and the nonlinear component x1 (t) goes to infinity for a finite time. On the contrary, the conventionally designed extended Kalman-Bucy estimates diverge from the real values.