A Two-step Method For Nonlinear Polynomial Model Identification Based on
Yu CHENG, Lan WANG and Jinglu HU
Graduate School of Information, Production and Systems, Waseda University
Hibikino 2-7, Wakamatsu-ku, Kitakyushu-shi, Fukuoka, JAPAN
Email: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
A two-step identification method for nonlinear polynomial
model using Evolutionary Algorithm (EA) is proposed in
this paper, and the method has the ability to select a
parsimonious structure from a very large pool of model
terms. In a nonlinear polynomial model, the number of
candidate monomial terms increases drastically as the order
of polynomial model increases, and it is impossible to obtain
the accurate model structure directly even with state-of-
art algorithms. The proposed method firstly carries out a
pre-screening process to select a reasonable number of
important monomial terms based on the importance index.
In the next step, EA is applied to determine a set of
significant terms to be included in the polynomial model. In
this way, the whole identification algorithm is implemented
very efficiently. Numerical simulations are carried out to
demonstrate the effectiveness of the proposed identification
System identification is concerned with building a model
from input-output observations of a system, creating a
mathematical description of the system . In these years,
various NARX (nonlinear AutoRegressive with eXogenous
inputs) models have been reported in the literature . As
a polynomial expansion of NARX models, nonlinear poly-
nomial models have shown great potential in their ability in
approximating complex nonlinear input-output relationships
and have attracted much attention because it is linear-in-
parameter , , .
However, nonlinear polynomial model identification re-
mains a difficult task, because a very large pool of model
terms has to be considered initially , , from which a
useful model is then generated based on the parsimonious
principle, of selecting the smallest possible model .
What’s more, the number of candidate terms grows dras-
tically with increasing the order of the model and the
maximum delays of the input and output signals.
Some famous methods for identifying nonlinear poly-
nomial model have been proposed in the last few years.
The Orthogonal Least-Squares method (OLS) is usually
considered as an effective approach , . However, it has
been pointed out that the algorithm cannot guarantee that
the resultant model is globally optimized . In addition,
it will become prohibitively expensive as its cost increases
super linearly with the number of candidates.
In recent years, Genetic Algorithm (GA) based methods
have been proposed extensively , , , and it can
effectively search many local optima and thereby increases
the possibility of finding the global optimum. However, GA
based approaches are so far containing all possible terms.
When the model order and the maximum delays of input
and output signals are large, it becomes time-consuming and
easily trap into a local optimum.
Since a GA based approach is efficient when the searching
space is small, it is highly motivated to combine a GA
based approach with a pre-screening process, in which a low
accuracy identification method is used to reduce searching
space . Based on this consideration, a two-step method
for nonlinear polynomial model identification is proposed.
Firstly candidate pool is reduced in a low accuracy according
to the importance index of all the terms. If the candidate pool
is still too large, a further selection would be processed with
the help of AP clustering method  on the correlation
coefficient between each pairs of monomial terms, and
important terms in each cluster are selected. In the second
step, single and multi-objective GA are applied to determine
the nonlinear polynomial model. In order to differ from
method using GA directly, Evolutionary Algorithm (EA) is
used to represent both the single and multi-objective GA
applied in the second step.
The paper is organized as follows: Section 2 briefly
describes the background of problem to be solved. Section
3 discusses the two-step identification approach in detail.
Section 4 provides numerical simulations and section 5
presents the discussions and conclusions.
Consider the following SISO NARX system whose input-
output relation described by:
푦(푡) = 푔(휑(푡)) + 푒(푡)
978-1-4244-5612-3/09/$26.00 c ?2009 IEEE
Test Result of GA method
Test Result of OLS2 and TSO
Test Result of TMO
Figure 2. Simulation of obtained polynomial models on
can find that the obtained model terms are all ranked top
according to the importance index, and a quick and accurate
identification is proceeded by single objective optimization.
When heavy noise is added, or the nonlinear system is too
complex, multi-objective optimization could also determine
the model structure well. Therefore, it is very efficient
to carry out the identification in two steps for nonlinear
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