Central suboptimal H_∞ filter design for nonlinear polynomial systems

Abstract

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...