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1998, with author Carravetta F., in the category IASI Research Reports
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IASI Research Report n. 489 Francesco Carravetta,

Alfredo Germani,

Shuakayev M.K.A new suboptimal approach to the filtering problem for cubic-sensor-like nonlinear systemsABSTRACT The filtering problem for a nice class of nonlinear stochastic multivariable systems is here considered. The state equation is an Ito bilinear stochastic differential equation, whereas the output process is given by a linear transformation of the state powers, corrupted by multiplicative noise. This class includes bilinear stochastic systems and, moreover, it is a generalization of the well known "cubic sensor", for which the nonexistence of finite-dimensional filters was proved. The aim of this paper is to present a new approach consisting in searching for suboptimal state-estimates instead of the conditional statistics. As a first result, a finite-dimensional optimal linear filter for the subclass of the bilinear stochastic systems is defined. Next, the more general problem of designing polynomial finite-dimensional filters is considered. The equations of a finite-dimensional filter are given, producing a state-estimate which is optimal in a class of polynomial transformations of the measurements with arbitrarily fixed degree. Key words: square integrable martingales, wide-sense Wiener processes, stochastic bilinear systems, cubic sensor, Kronecker algebra, Kalman-Bucy filtering, polynomial filtering, vector Ito formula.