Optimal Two-Stage Algorithm for Estimation of Nonlinear Functionals of State in Linear Stochastic System

Abstract

Aim: This paper focuses on the optimal estimation of a nonlinear functional of state (NFS) in continuous-time stochastic systems. The NFS represents a multivariate functional of state variables which carries useful information of a target system for control. The aim of this paper is to develop an optimal estimator for an arbitrary NFS in a linear dynamical systems and study its accuracy. Materials and Methods: This optimal solution is based on a minimum mean square error (MMSE) approach. The proposed estimation algorithm includes two stages: the optimal Kalman estimate of a state vector computed at the first stage is nonlinearly transformed at the second stage based on the NFS and the MMSE criterion. Results: We derive and study the optimal MMSE estimator for a general NFS. Some challenging theoretical aspects of analytic calculation of the optimal MMSE estimate are solved by usage of the multivariate Gaussian integrals for special functionals such as the Euclidean norm, maximum and absolute value. The polynomial functionals are studied in detail. The polynomial MMSE estimator has a simple closed form and it is easy to implement in practice. Effective matrix formulas for a true mean square error of the optimal and suboptimal quadratic estimators are derived. Conclusion: In wide engineering applications an NFS plays a fundamental role in control. The entire development of the optimal nonlinear estimator is based on the two-stage MMSE estimation algorithm involving the Kalman estimator. Performance of the MMSE estimator on examples with different types of NFS demonstrates its effectiveness.