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This paper focuses on estimation of a nonlinear functional of state vector (NFS) in discrete-time linear stochastic systems. The NFS represents a nonlinear multivariate functional of state variables, which can indicate useful information of a target system for control. The optimal mean-square estimator of a general NFS represents a function of the Kalman estimate and its error covariance. The polynomial functional of state vector is studied in detail. In this case an optimal estimation algorithm has a closed-form computational procedure. The novel mean-square quadratic estimator is derived. For a general NFS we propose to use the unscented transformation to calculate an optimal estimate. The obtained results are demonstrated on theoretical and practical examples with different types of NFS. Comparative analysis with suboptimal estimators for NFS is presented. The subsequent application of the proposed estimators to linear discrete-time systems demonstrates their practical effectiveness.

Mean-Square Estimation of Nonlinear Functionals via Kalman Filtering