A two-stage least squares based iterative parameter estimation
algorithm for feedback nonlinear systems
based on the model decomposition
Peipei Hu, Yongsong Xiao and Rui Ding
Abstract— A two-stage least squares based iterative pa-
rameter estimation algorithm is proposed for identifying a
feedback nonlinear system with the open-loop being a controlled
autoregressive moving average model from input-output data.
The identification model is bilinear on two unknown parameter
vectors. By decomposing a system into two subsystems, we
identify each subsystem, which is linear about a parameter
vector. The simulation example is provided.
I. INTRODUCTION
System identification is the theory and methods of estab-
lishing the mathematical models of systems [1]–[12]. The
mathematical models of dynamic systems can be obtained
from the given input-output data [13], [14]. The parameter
estimation is useful in system modeling [15]–[17]. For exam-
ple, Li et al used an iterative parameter identification methods
for nonlinear functions [18]; Ding studied a multi-innovation
methods for systems [19]–[22]; Wang et al presented the
maximum likelihood recursive least squares identification
methods for linear controlled autoregressive autoregressive
models [23] and for controlled autoregressive autoregressive
moving average systems [24].
Identifying the parameters of nonlinear systems has re-
ceived much attention in the processes of the dead-zone
nonlinearities and the valve saturation nonlinearities. Re-
cently, several estimation algorithms have been developed
for nonlinear systems, which can model block structure non-
linear dynamic processes [25]–[27], such as the Hammerstein
model that is a static nonlinear block followed by a linear
dynamic subsystem and the Wiener model that is a linear
subsystem followed by a static nonlinearity [28]–[30]. The
projection algorithm, the stochastic gradient identification
algorithm, the Newton recursive algorithm and the Newton
iterative algorithm for nonlinear systems have been studied
for these nonlinear systems [31]–[33].
The iterative methods can produce more accurate estimates
with the measured input-output data, have faster convergence
rates and can be used for solving matrix equations [34]–[36]
and for estimating the parameters of systems [37]–[39]. Liu
et al developed a least squares based iterative identification
approach for a class of multirate systems [40]; Zhang et
al presented a hierarchical least squares iterative estimation
This work was supported by the National Natural Science Foundation of
China (No. 61203111).
The authors are with the Key Laboratory of Advanced Process Control
for Light Industry (Ministry of Education), Jiangnan University, Wuxi,
214122, PR China. hppkuaile@yahoo.cn; xiaoys@jiangnan.edu.cn;
rding2003@yahoo.cn
algorithm for multivariable Box-Jenkins-like systems using
the auxiliary model [41]; Bao et al proposed a least squares
based iterative identification method for multivariable con-
trolled ARMA systems [42]; Ding et al presented the least
squares based iterative algorithms for Hammerstein nonlinear
ARMAX systems [43] and a two-stage least squares based
iterative parameter estimation for CARAMMA system mod-
elling [44]; Xiong et al discussed a least squares parameter
estimation algorithm and its convergence for a class of
input nonlinear systems [45]. On the basis of the multistage
identification methods in [46]–[48], this paper studies a two-
stage identification method for feedback nonlinear systems
using the hierarchical identification principle [52]–[55].
The paper is organized as follows. Section II describes
the identification problem formulation for feedback nonlinear
systems. Section III derives a two stage iterative parameter
estimation for feedback nonlinear systems. Section IV pro-
vides an illustrative example to show the effectiveness of
the proposed algorithm. Finally, we offer some concluding
remarks in Section V.
II. SYSTEM DESCRIPTION AND IDENTIFICATION MODEL
Let us introduce some notation first. The symbol I
n
stands
for an identity matrix of order n; the superscript T denotes
the matrix transpose; 1
n
represents an n-dimensional column
vector whose elements are 1; the norm of a matrix X is
defined as kXk
2
:= tr[XX
T
];
c
X(t) denotes the estimate of
X at time t.
Consider the feedback nonlinear system in Figure 1 [46],
where A(z), B(z) and D(z) are polynomials in the unit
backward shift operator z
−1
[i.e., z
−1
y(t) = y(t − 1)] with
the known orders n
a
, n
b
and n
d
, and
A(z) := 1 + a
1
z
−1
+ a
2
z
−2
+ · · · + a
n
a
z
−n
a
,
B(z) := b
1
z
−1
+ b
2
z
−2
+ · · · + b
n
b
z
−n
b
,
D(z) := 1 + d
1
z
−1
+ d
2
z
−2
+ · · · + d
n
d
z
−n
d
.
Assume that r(t) = 0, y(t) = 0, v(t) = 0 for t 6 0.
In Figure 1, the open-loop part is a controlled autore-
gressive moving average (CARMA) subsystem, {r(t)} and
{y(t)} are the reference input and output sequences of the
system, {v(t)} is a stochastic white noise sequence with zero
mean and variance σ
2
and it is supposed that the output of
the nonlinear block ¯y = f(y) is a linear combination of
a known basis f := (f
1
, f
2
, · · · , f
m
) in the system output
2013 American Control Conference (ACC)
Washington, DC, USA, June 17-19, 2013
978-1-4799-0176-0/$31.00 ©2013 AACC 5466
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