Robust Kalman Filter with Application to State Estimation of a Nuclear Reactor

Abstract

State estimation algorithm deals with recovering some desired state variables of a dynamic system from available noisy measurements, and estimation of the state variables is one of the fundamental and significant problems in control and signal processing areas, and many significant progresses have been made in this area. In 1940s, Wiener, the founder of the modern statistical estimation theory, established the Wiener filtering theory which solves the minimum variance estimation problem for stationary stochastic processes. It was not until late 1950s and early 1960s that Kalman filtering theory, a novel recursive filtering algorithm, was developed by Kalman and Bucy which did not require the stationarity assumption (Kalman, 1960; Kalman & Bucy, 1961). Since Kalman filter theory is only applicable for linear systems and almost all practical dynamic systems are nonlinear, Bucy and some other researchers were engaged in extending Kalman filtering theory to nonlinear systems in the following 10 years (Bucy & Senne, 1971). The most celebrated and widely used nonlinear filtering algorithm is the extended Kalman filter (EKF), which is essentially a suboptimal nonlinear filter. The key idea of the EKF is using the linearized dynamic model to calculate the covariance and gain matrices of the filter. The Kalman filter (KF) and the EKF are all widely used in many engineering areas, such as aerospace, chemical and mechanical engineering. However, it is well known that both the KF and EKF are not robust against modelling uncertainties and disturbances. This has motivated many studies on extending the Kalman filtering theory to its robust version, which may yield a suboptimal state estimator with respect to the nominal system model, but will guarantee an upper bound to the estimation error variance in spite of large disturbances and modelling uncertainties. In recent years, several results have been made on the design of such robust state estimators. Based on the fuzzy dynamic programming technique, a bounding estimator (Jian, 1975) for uncertain nonlinear systems was developed, which gave an upper bound to the error for any allowed system parameter variation. Petersen & McFarlane (1991, 1994) converted a robust Kalman filter design problem to a H∞ controller design problem of a certain linear continuous timeinvariant system based on the concept of quadratic stability of linear continuous timeinvariant systems. Xie & Soh (1994) converted the existence of robust Kalman filters to the existence of two differential Riccati equations, and then established a design approach of robust estimators for linear continuous time systems with uncertainties in the state and the Source: Kalman Filter, Book edited by: Vedran Kordic, ISBN 978-953-307-094-0, pp. 390, May 2010, INTECH, Croatia, downloaded from SCIYO.COM