2500 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 16, NO. 4, APRIL 2020
A Novel EM Identification Method for
Hammerstein Systems With Missing Output Data
Dongqing Wang , Shuo Zhang , Min Gan , and Jianlong Qiu
Abstract—This article concerns a novel auxiliary-model-
based expectation maximization (EM) estimation method for
Hammerstein systems with data loss by extending the EM
method to estimate models with multiple parameter vectors.
The novel EM method relaxes the requirements on an au-
toregression model with one parameter vector, interactively
maximizes the expectation over multiple parameter vectors
in a more general model, and uses the output of an auxiliary
model to substitute the missing outputs in the information
vector in iteration processes. A numerical simulation is
employed to demonstrate the effectiveness of the proposed
novel EM method.
Index Terms—Data loss, expectation maximization (EM)
algorithm, Hammerstein systems, parameter identification.
I. INTRODUCTION
D
UE TO the widespread nonlinear characteristics of actual
systems, a nonlinear model is a reasonable modeling
method for a natural complex system that is absolutely nec-
essary [1]–[5]. For decades, block-oriented nonlinear systems
are popular with their ability of elastic arrangement of functional
modules and elastic representation of each module. The thought
of block-oriented nonlinear systems is originated from the
innovative idea of dividing comprehensive problems into simple
small ones. A whole nonlinear dynamic structure can be divided
into static nonlinearities and linear dynamics, i.e., one com-
prehensive structure is parted into two or more simple blocks.
The investigated two-block-oriented Hammerstein system is the
simplest among the family of the block-oriented structure.
The identification methods for the block-oriented system can
be classified into different types according to different aspects.
From the viewpoint of the basic identification principle, the
identification strategies can be divided into the least squares
methods [6], [7], the stochastic gradient methods [8], [9], the
probability-based maximum likelihood methods [10], [11], and
Manuscript received June 24, 2019; revised July 15, 2019; accepted
July 22, 2019. Date of publication July 29, 2019; date of current version
January 17, 2020. This work was supported in part by the National
Natural Science Foundation of China under Grant 61873138, Grant
61573205, Grant 61673155, and Grant 61877033, and in part by the
Taishan Scholar Project Fund of Shandong Province of China. Paper no.
TII-19-2677. (Corresponding author: D. Q. Wang.)
D. Q. Wang, S. Zhang, and M. Gan are with the College of Electri-
cal Engineering, Qingdao University, Qingdao 266071, China (e-mail:,
dqwang64@qdu.edu.cn; a3b412@163.com; aganmin@aliyun.com).
J. L. Qiu is with the School of Automation and Electrical Engineering,
Linyi University, Linyi 276000, China (e-mail:,qiujianlong@lyu.edu.cn).
Color versions of one or more of the figures in this article are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TII.2019.2931792
the orthogonal matching pursuit identification methods [12],
[13]. From the viewpoint of computational style, the identifica-
tion strategies can be divided into the recursive estimation meth-
ods [14]–[16] and the iterative estimation methods [17]–[20].
From the viewpoint of the parameterization mode, there exist
the overparameterization estimation methods [21]–[24], the hi-
erarchical estimation methods [25]–[27], and the key-term sepa-
ration estimation methods [28]–[30]. From the viewpoint of the
assistant technique, there exist the filtering technique [31], the
multi-innovation theory [33], and the auxiliary model idea [34].
In discrete-time systems, usually, the update of t he system
input and output at the same period, i.e., single-rate systems,
when the measurement technology is limited, or the system de-
lay or congestion appears, different input–output sampling rates
or missing output data are occurred [35]–[37], i.e., dual-rate,
multirate, or miss data systems.
Several identification methods for missing data systems are
published. Through the polynomial transformation technique
[38], a dual-rate sampling system can be transformed into a
dual-rate regression model that does not involve the missing
output data; through the lifting t echnique, the multirate s ystems
can be reformulated into a lifted state space model with a
slow-rate sampling [39], [ 40]; and through the auxiliary model
idea, parameter estimation of the multirate systems can be
implemented by substituting the missing output with the output
of an auxiliary model [41].
Recently, the likelihood-function-based expectation maxi-
mization (EM) estimation methods have captured much atten-
tion in the estimation area of missing data [42]–[44]. Xiong
et al. investigated a likelihood-function-based EM algorithm for
a nonlinear block-oriented Wiener structure with missing output
data [42], Chen et al. explored a modified Bayesian-based EM
algorithm for a linear missing data system [43], and Zhang et al.
applied a likelihood-function-based EM algorithm to handle
missing data of a discrete distribution system [44].
Most EM algorithms like in papers [42]–[44] aim to transform
linear/nonlinear systems into a regression identification model
with only one parameter vector, e.g.,
y(k)=Φ
T
Θ
where y(k) ∈ R is the system output, Φ ∈ R
n
is the measured
vector formed by observed data, and Θ ∈ R
n
is the parameter
vector [42]–[44]. The typical EM method is to maximize the
expectation of a logarithmic likelihood function of the complete
data over the parameter vector Θ to compute the parameter
estimates.
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