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Mathematical Problems in Engineering
Volume 2016, Article ID 9548428, 10 pages
Research Article

Extended Stochastic Gradient Identification Method for Hammerstein Model Based on Approximate Least Absolute Deviation

1Department of Automation, China University of Petroleum, Beijing 102200, China
2Beijing Urban Construction Design & Development Group Co., Limited, Beijing 100037, China

Received 24 February 2016; Revised 19 May 2016; Accepted 19 May 2016

Academic Editor: Mitsuhiro Okayasu

Copyright © 2016 Bao-chang Xu 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.


In order to identify the parameters of nonlinear Hammerstein model which are contaminated by colored noise and peak noise, the least absolute deviation (LAD) is selected as the objective function to solve the problem of large residual square when the identification data is disturbed by the impulse noise which obeys symmetrical alpha stable () distribution. However, LAD cannot meet the need of differentiability required by most algorithms. To improve robustness and to solve the nondifferentiable problem, an approximate least absolute deviation (ALAD) objective function is established by introducing a deterministic function to replace absolute value under certain situations. The proposed method is derived from ALAD criterion and extended stochastic gradient method. Due to the differentiability of the objective function, we can get a recursive identification algorithm which is simple and easy to calculate compared with LAD. The convergence of the proposed identification method is also proved by Lyapunov stability theory, and the simulation experiments show that the proposed method has higher accuracy and stronger robustness than the least square (LS) method in the identification of Hammerstein model with colored noise and impulse noise. The impact of impulse noise can be restrained effectively.