IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 6, JUNE 2010 1287
Recursive Identification for Nonlinear ARX Systems
Based on Stochastic Approximation Algorithm
Wen-Xiao Zhao, Han-Fu Chen, Fellow, IEEE, and Wei Xing Zheng, Senior Member, IEEE
Abstract—The nonparametric identification for nonlinear au-
toregressive systems with exogenous inputs (NARX) described
by
+1
= ( ...
+1
...
+1
)+
+1
is
considered. First, a condition on
( )
is introduced to guarantee
ergodicity and stationarity of
. Then the kernel function based
stochastic approximation algorithm with expanding truncations
(SAAWET) is proposed to recursively estimate the value of
( )
at any given
[
(1)
...
( ) (1)
...
( )
] R
2
.
It is shown that the estimate converges to the true value with
probability one. In establishing the strong consistency of the
estimate, the properties of the Markov chain associated with the
NARX system play an important role. Numerical examples are
given, which show that the simulation results are consistent with
the theoretical analysis. The intention of the paper is not only to
present a concrete solution to the problem under consideration
but also to profile a new analysis method for nonlinear systems.
The proposed method consisting in combining the Markov chain
properties with stochastic approximation algorithms may be of
future potential, although a restrictive condition has to be imposed
on
( )
, that is, the growth rate of
( )
should not be faster than
linear with coefficient less than 1 as
tends to infinity.
Index Terms—Kernel function, Markov chain, nonlinear ARX
system, recursive identification, stochastic approximation.
I. INTRODUCTION
I
DENTIFICATION for linear stochastic systems (see, e.g.,
[7], [16]) has been extensively studied for many years, and
it is relatively mature in comparison with that for nonlinear sys-
tems. The system consisting of a linear subsystem cascaded
with a static nonlinearity, called the Hammerstein or Wiener
system depending on the order of their connection, probably
is the simplest nonlinear system. Identification of Hammerstein
and Wiener systems has been attracting a great attention from
researchers in recent years (see, e.g., [5], [6], [14], [15], [17],
[29], [30] and references therein). For this kind of systems, the
Manuscript received July 24, 2008; revised March 05, 2009. First published
February 05, 2010; current version published June 09, 2010. This work was
supported in part by NSFC 60821091 and partly by 60625305, 60721003, 973
Program 2009CB320602, by NSFC 60821091, 60874001, by a grant from the
National Laboratory of Space Intelligent Control, and by the Australian Re-
search Council. Recommended by Associate Editor B. Ninness.
W.-X. Zhao is with the Department of Automation, Tsinghua University, Bei-
jing 100084, China (e-mail: wxzhao@mail.tsinghua.edu.cn; wxzhao@amss.ac.
cn).
H.-F. Chen is with the Key Laboratory of Systems and Control of CAS, Insti-
tute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190,
China (e-mail: hfchen@iss.ac.cn).
W. X. Zheng is with the School of Computing and Mathematics, University of
Western Sydney, Sydney, NSW 1797, Australia (e-mail: w.zheng@uws.edu.au).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2010.2042236
structure information greatly simplifies the work when identi-
fying the systems.
Let us consider the problem of identifying the following
single-input single-output (SISO) nonlinear autoregressive
systems with exogenous inputs (NARX)
(1)
where
and are the system input and output, respectively,
is the system noise, is the known system order and is
the unknown function to be identified. The NARX system (1)
is a straightforward generalization of the linear ARX system
and belongs to the class of nonlinear systems without a priori
structure information. The problem under study in this paper is
to recursively identify the nonlinear function
on the basis
of observations
and to prove the strong consistency of
estimates.
Identification of the NARX system (1) is a topic of many
papers, e.g., [2], [3], [13], [26], [27] among others. According
to the description of
, the methods for identifying (1) can
roughly be divided into two categories: the parametric approach
[27], and the nonparametric approach [2], [3], [13], [26]. In the
parametric approach, the unknown
is expanded to a sum of
known functions (for example, polynomials, neural networks,
wavelets and so on) with unknown coefficients [27]. Then, iden-
tification of NARX systems becomes a parameter estimation
problem.
It is noticed that the system (1) is representable as
with
In the nonparametric approach, the value of is estimated
for any fixed
. It is of practical significance to con-
sider the nonparametric identification, since it may be difficult in
advance to choose an appropriate basis of functions to approxi-
mate the unknown
, and the assumptions made on nonlinear
systems for such kind of methods are, hopefully, weaker than
those for parametric methods. In this paper the nonparametric
method is adopted.
We now briefly review the existing works with nonparametric
methods [2], [3], [13], [26]. In [26] a so-called direct weight
optimization (DWO) method is proposed, which later is also
considered in [2]. Let
be the collected data set. The
DWO method proposes to estimate
by
(2)
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