ICIC Express Letters ICIC International
c
2016 ISSN 1881-803X
Volume 10, Number 10, October 2016 pp. 2305–2312
COMBINED HIERARCHICAL BASED RECURSIVE LEAST SQUARES
ALGORITHM AND KALMAN FILTER FOR HAMMERSTEIN
STATE SPACE SYSTEMS
Hong Wang, Wei Li, Shuqing Zhang, Xusheng Wang and Chao Ge
Institute of Information Engineering
North China University of Science and Technology
No. 46, Xinhua Road, Tangshan 063009, P. R. China
gechao365@126.com
Received March 2016; accepted June 2016
Abstract. This paper is concerned with the parameter estimation of nonlinear dynamic
Hammerstein systems in state space form. More specifically, a hierarchical based recur-
sive least squares algorithm is employed to estimate the parameters of the proposed Ham-
merstein systems and a Kalman filter is derived to estimate the states of the canonical
state space subsystem. Simulation examples demonstrate the effectiveness of the proposed
method.
Keywords: Hammerstein system, Parameter estimation, Hierarchical based recursive
least squares algorithm, Kalman filter
1. Introduction.
System modeling and parameter estimation have received much at-
tention in recent years [1]. All physical systems are nonlinear and can be modelled by
block-oriented nonlinear models. System identification has been widely used in many ar-
eas [2]. This paper focuses on the identification of Hammerstein model which consists of a
nonlinear static block followed by a linear dynamic subsystem. The linear dynamic block
has been assumed to be ARMAX [3], state space model [4], and so on.
Many approaches have been proposed to estimate parameters of Hammerstein systems.
The iterative method was first proposed to estimate Hammerstein system in Narendra
and Gallman [5], which is a very simple and efficient algorithm. Over-parameterization
method is one direct method for the identification of Hammerstein models, in which the
output of Hammerstein system is linear on the parameter space. However, the resulting
parameter vector contains cross-products between the parameters of the non-linear part
and the linear part, which increases the dimension of the parameter vector and leads to
many redundant parameter estimates [6]. V¨or¨os used the key term separation technique
to deal with identification problems of Hammerstein systems with discontinuous nonlin-
earities [7]. For the key term separation technique, the output of system can be expressed
as the linear regressive of all parameters which can be directly estimated. The blind
identification method is presented by Bai to identify the Hammerstein models. By using
the blind approach, Bai discussed the the Hammerstein and Hammerstein-Wiener model
identification problems [8]. Hierarchical identification is inspired by the decomposition-
coordination principle based hierarchical control for large-scale systems, which uses sub-
system decomposition in identification [9].
In the area of state space system identification, Wills et al. addressed the problem of
estimating parameters in state space model from observed frequency domain data [10].
Merc`ere and Bako presented a new subspace-based identification method to estimate the
multi-inputs multi-outputs (MIMO) canonical state space model directly from data, which
does not require the so-called observability/controllability indices [11]. Ding discussed a
recursive least squares algorithm for a canonical state space dynamic system, in which the
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