Optimal filtering for polynomial system states with polynomial multiplicative noise

Circuits Syst Signal Process

2008

Self Cite

In this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations with an arbitrary, not necessarily invertible, observation matrix is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. Thus, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. A transformation of the observation equation is introduced to reduce the original problem to the previously solved one with an invertible observation matrix. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of linear and bilinear state equations. In an example, the performance of the designed optimal filter is verified against those of the optimal filter for a quadratic state with a stateindependent noise and a conventional extended Kalman-Bucy filter.

Section: Filtering Problem For Polynomial State Over Linear Observationscontrasting

confidence: 49%

Section: Filtering Problem For Polynomial State Over Linear Observationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal Filtering for Incompletely Measured Polynomial Systems with Multiplicative Noise

Circuits Syst Signal Process

2008

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Optimal filtering for incompletely measured polynomial states over linear observations

Basin

Adaptive Control & Signal

Skliar

2007

Self Cite

In this paper, the optimal filtering problem for polynomial system states over linear observations with an arbitrary, not necessarily invertible, observation matrix is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. As a result, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. A transformation of the observation equation is introduced to reduce the original problem to the previously solved one with an invertible observation matrix. The procedure for obtaining a closed system of the filtering equations for any polynomial state over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular case of a third-order state equation. In the example, performance of the designed optimal filter is verified against a conventional extended Kalman-Bucy filter. 483closed system of filtering equations for a certain number of lower conditional moments. The most famous result, the Kalman-Bucy filter [2], is related to the case of linear state and observation equations, where only two moments, the estimate itself and its variance, form a closed system of filtering equations. However, the optimal nonlinear finite-dimensional filter can be obtained in some other cases, if, for example, the state vector can take only a finite number of admissible states [3] or if the observation equation is linear and the drift term in the state equation satisfies the Riccati equation d f /dx + f 2 = x 2 (see [4]). The complete classification of the 'general situation' cases (this means that there are no special assumptions on the structure of state and observation equations and the initial conditions), where the optimal nonlinear finite-dimensional filter exists, is given in [5]. The last two papers actually deal with specific types of polynomial filtering systems. There also exists a considerable bibliography on robust filtering for the 'general situation' systems (see, for example, [6-11]). Apart from the 'general situation', the optimal finite-dimensional filters have recently been designed [12][13][14] for certain classes of polynomial system states with Gaussian initial conditions over linear observations with an invertible observation matrix.This paper presents the optimal finite-dimensional filter for polynomial system states over linear observations with an arbitrary, not necessarily invertible, observation matrix, thus generalizing the results of [12][13][14]. Designing the optimal filter for polynomial systems with a non-invertible observation matrix presents a significant advantage in the filtering theory and practice, since it enables one to address the joint state and parameter optimal identification problems for polynomial systems. The optimal filtering problem is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance [15]. As the first ...

Central suboptimal H_∞ filter design for nonlinear polynomial systems

Basin

Shi

Adaptive Control & Signal

2008

Self Cite

This paper presents the central finite-dimensional H ∞ filter for nonlinear polynomial systems, which is suboptimal for a given threshold with respect to a modified Bolza-Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, the paper reduces the original H ∞ filtering problem to the corresponding optimal H 2 filtering problem, using the technique proposed in (IEEE Trans. Automat. Control 1989; 34:831-847). The paper presents the central suboptimal H ∞ filter for the general case of nonlinear polynomial systems based on the optimal H 2 filter given in (Int. J. Robust Nonlinear Control 2006; 16:287-298). The central suboptimal H ∞ filter is also derived in a closed finite-dimensional form for third (and less) degree polynomial system states. Numerical simulations are conducted to verify performance of the designed central suboptimal filter for nonlinear polynomial systems against the central suboptimal H ∞ filter available for the corresponding linearized system. CENTRAL SUBOPTIMAL H ∞ FILTER DESIGN 927 established a background for consistent treatment of filtering/controller problems in the H ∞framework. The H ∞ filter design implies that the resulting closed-loop filtering system is robustly stable and achieves a prescribed level of attenuation from the disturbance input to the output estimation error in L 2 /l 2 -norm. A large number of results on this subject have been reported for systems in the general situation, linear, or nonlinear (see [5][6][7][8][9][10][11][12][13][14][15][16][17]). The sufficient conditions for existence of an H ∞ filter, where the filter gain matrices satisfy Riccati equations, were obtained for linear systems in [4] and linear systems with state delay in [18] or with measurement delay in [19]. However, the criteria of existence and suboptimality of solution for the central H ∞ filtering problems based on the reduction of the original H ∞ problem to the induced H 2 one, similar to those obtained in [1, 4] for linear systems, remain yet unknown for nonlinear polynomial systems.Although the general optimal solution of the filtering problem for nonlinear state and observation equations confused with white Gaussian noises is given by the equation for the conditional density of an unobserved state with respect to observations [20], there are a very few known examples of nonlinear systems where that equation can be reduced to a finite-dimensional closed system of filtering equations for a certain number of lower conditional moments (see [21][22][23] for more details). The complete classification of the 'general situation' cases (this means that there are no special assumptions on the structure of state and observation equations and the initial conditions), where the optimal nonlinear finite-dimensional filter exists, is given in [24]. Apart from the 'general situation', the optimal finite-dimensional filters have been designed for certain classes of polynomial system states over linear observations [25][26][2...