Extended Kalman Filters for Continuous-time Nonlinear Fractional-order Systems Involving Correlated and Uncorrelated Pro
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Extended Kalman Filters for Continuous-time Nonlinear Fractionalorder Systems Involving Correlated and Uncorrelated Process and Measurement Noises Fanghui Liu, Zhe Gao*, Chao Yang, and Ruicheng Ma Abstract: In order to improve the estimation accuracy of the state information and save the computing time for fractional-order systems containing correlated and uncorrelated process and measurement noises, this paper investigates fractional-order extended Kalman filters for continuous-time nonlinear fractional-order systems using the method of fractional-order average derivative. Compared with Grünwald-Letnikov difference, the estimation accuracy is improved via the fractional-order average derivative method. Meanwhile, the computing time in the state estimation is saved. To deal with the correlated and uncorrelated process and measurement noises, two kinds of extended Kalman filters for nonlinear fractional-order systems are given. Finally, the effectiveness of the proposed fractional-order extended Kalman filters based on fractional-order average derivative is validated by two examples. Keywords: Fractional-order average derivative, Kalman filter, nonlinear fractional-order system, state estimation, Tustin generating function.
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INTRODUCTION
Compared with traditional integral-order calculus, fractional-order calculus reveals the development of novel applications in science and engineering fields [1, 2]. In [3] and [4], the state estimators for fractional-order memristive neural networks (FMNNs) with time delays and the adaptive synchronization problem of fractional-order memristor based neural networks with time delay were proposed, respectively. Due to the use of fractional-order operators, the designs of fractional-order controllers become more flexible, such as fractional-order PID controllers [5], fractional-order sliding mode controllers [6], fractional-order iterative learning controllers [7] and fractional-order H∞ controllers [8]. Because of the memory property of fractional-order differential operators, a great number of historical data of input and measurement signals need to be provided to estimate the state information of continuous-time linear and nonlinear fractionalorder systems. Hence, the treatment on the historical information is the key problem needed to be solved, which is different from integer-order systems. The measurement signals must be filtered to get effective state estimation
because of the noises existing in the measurement signals. Although the state information can be measured directly, it is necessary to design some kinds of filters to estimate the system state since the process noise and measurement noise generate disturbances in measurement signals. Kalman filter is an effective robust state observer [9, 10]. Many scholars have studied a various of Kalman filters in the past decades, such as the extended Kalman filter (EKF) [11], the sigma point Kalman filter (SPKF) [12], the particle filter (PF) [13], the cubature Kalm
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