Constrained State Estimation for Nonlinear Systems with Unknown Input
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Constrained State Estimation for Nonlinear Systems with Unknown Input Zhen Luo · Huajing Fang · Yuanhao Luo
Received: 16 May 2012 / Revised: 24 January 2013 © Springer Science+Business Media New York 2013
Abstract This paper extends the problem of state estimation for linear discrete-time systems with unknown input to the nonlinear systems. Based on physical consideration, the constraints of state are also considered. And the constraints which can improve the quality of estimation are imposed on individual updated sigma points as well as the updated state. The advantage of algorithm is that it is able to deal with arbitrary constraints on the states during the estimation procedure, Least-squares unbiased estimation algorithm can be used to obtain unknown input, and the unknown input which can be any signal affects both the system and the outputs. The state estimation problem is transformed into a standard Unscented Kalman filter problem which can easily be solved. Simulations are provided to demonstrate the effectiveness of the theoretical results. Keywords Unscented Kalman filter · Constrained · Unknown input · Direct feedthrough · Least-squares 1 Introduction State estimation is a broad field which has been widely studied during the past decades. The main task of state estimation is to design the filter. H∞ filter technique Z. Luo · H. Fang () Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China e-mail: [email protected] Z. Luo e-mail: [email protected] Y. Luo No 1 Middle School of Huang Long Town, Xiangyang, 441109, China e-mail: [email protected]
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is an effective way to deal with the state estimation problems and much research effort has been paid to the H∞ filtering problems, especially for the discrete-time systems with a state delay [4, 17] and time delay [30]. There are several approaches to solve the H∞ filtering problems, such as frequency-domain approaches [22, 23], LMI-based approaches [24, 31]. It should be noted that most the H∞ filter design methods mentioned above are focus on linear systems. However, in most applications of interest the system dynamics and observation equations are nonlinear. Nonlinear state estimation includes many algorithms such as Moving Horizon Estimation, the Particle filter, the Ensemble Kalman filter, the Unscented Kalman filter (UKF) and the Extended Kalman filter (EKF). Among various methods, UKF are the most widely used approach to analyze the stochastic nonlinear system and show good performance in many cases [8]. UKF as an improvement to EKF was proposed by Julier et al. This method is based on the unscented transform (UT) technique, a mechanism for propagating mean and covariance through a nonlinear transformation [9, 10]. The state vector is represented by a minimal set of carefully chosen sample points, called sigma points, which approximate the posterior mean and covariance of the Gaussian random variable with a second order accuracy [11, 12]. In contrast, the
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