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Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 2367042, 7 pages
https://doi.org/10.1155/2017/2367042
Research Article

Convergent Properties of Riccati Equation with Application to Stability Analysis of State Estimation

1Engineering Research Center of Digitized Textile & Apparel Technology, Ministry of Education, College of Information Science and Technology, Donghua University, Shanghai 201620, China
2Department of Automation, Shanghai Jiao Tong University and Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai 200240, China

Correspondence should be addressed to Y. S. Ding

Received 7 April 2017; Accepted 18 May 2017; Published 12 June 2017

Academic Editor: Weihai Zhang

Copyright © 2017 X. Cai 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.

Abstract

Since the recursive nature of Kalman filtering always results in a growing size of the optimization problem, state estimation is usually realized by use of finite-memory, receding horizon, sliding window, or “frozen” techniques, which causes difficulties on stability analysis. This paper proposes a novel method on selection of an initial covariance matrix and a horizon for the Kalman filter to make sure that a sequence of the closed-loop Kalman filters are stable as time-invariant filters at subsequent time instant. Convergent properties of Riccati Difference Equation (RDE) are first exploited. Based on these properties, sufficient conditions for stability of a sequence of Kalman filters are obtained. Compared with the existent literature, the convergent properties and the stability conditions are less conservative since they provide analytic results and are applicable to more common cases where the RDEs are not monotonic.