Applied Mathematics Letters 57 (2016) 13–19
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Applied Mathematics Letters
www.elsevier.com/locate/aml
Hierarchical parameter estimation for a class of MIMO
Hammerstein systems based on the reframed models
✩
Dongqing Wang
College of Automation Engineering, Qingdao University, Qingdao 266071, China
a r t i c l e i n f o
Article history:
Received 30 November 2015
Received in revised form 29
December 2015
Accepted 29 December 2015
Available online 6 January 2016
Keywords:
Parameter estimation
Least squares
Hierarchical identification principle
Hammerstein system
MIMO system
a b s t r a c t
Block-oriented Hammerstein systems consist of a nonlinear static block followed by
a linear dynamic block. For the identification of a complex class of multi-input
multi-output (MIMO) Hammerstein systems with different types of coefficients:
a matrix coefficient and scalar coefficients, it is difficult to express this class of
complex Hammerstein systems as a regression identification model in all parameters
of the nonlinear part and the linear part in which the standard least squares
method can be easily applied to implement parameter estimation. By the matrix
transformation, this paper reframes an MIMO Hammerstein system with different
types of coefficients into two models, each of which is expressed as a regression form
in the parameters of the nonlinear part or in the parameters of the linear part. Then
a hierarchical extended least squares algorithm is applied to these two models to
alternatively estimate the parameters of the nonlinear part and the linear part.
© 2016 Elsevier Ltd. All rights reserved.
1. Introduction
In the past decades, block-oriented structures as nonlinear models have been proven to be simple and
useful in theory [1–4] and applications [5–8], and a number of successful identification methods have been
developed. Based on the over-parametrization model, Ding et al. studied an auxiliary model based recursive
least squares algorithm for Hammerstein output-error systems [9]. By the iterative principle, Bai and Li pre-
sented an iterative identification algorithm for a general Hammerstein system and analyzed its convergence
property [10]. By the key term separation technique, V¨or¨os decomposed two or three block systems as special
formulas those are linear in parameters, and derived least squares algorithms to estimate parameters of these
systems [11,12]. By the hierarchical identification principle, Wang et al. presented hierarchical least squares
identification methods for a state space Hammerstein system and a dual-rate Hammerstein system [13,14].
By the maximum likelihood principle and the Newton optimization method, Li et al. discussed the identi-
fication problems of two types of Hammerstein systems [15,16], etc.
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This work was supported by the National Natural Science Foundation of China (Nos. 61573205, 61403217) and the Shandong
Provincial Natural Science Foundation of China under grant ZR2015FM017.
E-mail address: dqwang64@qdu.edu.cn.
http://dx.doi.org/10.1016/j.aml.2015.12.018
0893-9659/© 2016 Elsevier Ltd. All rights reserved.