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Mathematical Problems in Engineering
Volume 2015, Article ID 217253, 10 pages
Research Article

Estimation of Nonlinear Functions of State Vector for Linear Systems with Time-Delays and Uncertainties

1Department of Sensor Systems, Hanwha Corporation R&D Center, 52-1 Oesam-dong, Yuseong-gu, Daejeon 305-106, Republic of Korea
2Department of Applied Mathematics, Saint Petersburg State Polytechnic University, 29 Polytechnicheskaya Street, Saint Petersburg 195251, Russia
3Department of Information and Statistics, Research Institute of Natural Science, Gyeongsang National University, 501 Jinjudaero, Jinju, Gyeongsangnam-do 660-701, Republic of Korea

Received 4 June 2014; Revised 1 August 2014; Accepted 2 August 2014

Academic Editor: Yuxin Zhao

Copyright © 2015 Il Young Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper focuses on estimation of a nonlinear function of state vector (NFS) in discrete-time linear systems with time-delays and model uncertainties. The NFS represents a multivariate nonlinear function of state variables, which can indicate useful information of a target system for control. The optimal nonlinear estimator of an NFS (in mean square sense) represents a function of the receding horizon estimate and its error covariance. The proposed receding horizon filter represents the standard Kalman filter with time-delays and special initial horizon conditions described by the Lyapunov-like equations. In general case to calculate an optimal estimator of an NFS we propose using the unscented transformation. Important class of polynomial NFS is considered in detail. In the case of polynomial NFS an optimal estimator has a closed-form computational procedure. The subsequent application of the proposed receding horizon filter and nonlinear estimator to a linear stochastic system with time-delays and uncertainties demonstrates their effectiveness.