Multi-InnovationStochasticGradientIdentificationAlgorithmforHammersteinControlledAutoregressiveAutoregressiveSystemsBasedontheKeyTermSeparationPrincipleandontheModelDecomposition资源-CSDN文库

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Hindawi Publishing Corporation

Journal of Applied Mathematics

Volume , Article ID ,  pages

http://dx.doi.org/.//

Research Article

Multi-Innovation Stochastic Gradient Identification Algorithm

for Hammerstein Controlled Autoregressive Autoregressive

Systems Based on the Key Term Separation Principle and on

the Model Decomposition

Huiyi Hu,

Xiao Yongsong,

and Rui Ding

Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, China

School of Internet of ings Engineering, Jiangnan University, Wuxi 214122, China

Correspondence should be addressed to Rui Ding; rding@.com

Received  June ; Revised  August ; Accepted  September 

Academic Editor: Reinaldo Martinez Palhares

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An input nonlinear system is decomposed into two subsystems, one including the parameters of the system model and the other

including the parameters of the noise model, and a multi-innovation stochastic gradient algorithm is presented for Hammerstein

controlled autoregressive autoregressive (H-CARAR) systems based on the key term separation principle and on the model

decomposition, in order to improve the convergence speed of the stochastic gradient algorithm. e key term separation principle

can simplify the identication model of the input nonlinear system, and the decomposition technique can enhance computational

eciencies of identication algorithms. e simulation results show that the proposed algorithm is eective for estimating the

parameters of IN-CARAR systems.

1. Introduction

ere exist many nonlinear systems in process control [–

].Anonlinearsystemcanbemodeledbyinputnonlinear

systems [] and output nonlinear systems [], input-output

nonlinear systems [], feedback nonlinear systems [], and so

on. Input nonlinear systems, which are called Hammerstein

systems [], include input nonlinear equation error type sys-

tems and input nonlinear output error type systems. Recently,

many identication algorithms have been developed for input

nonlinear systems, such as the iterative methods [–], the

separable least squares methods [, ], the blind methods

[], the subspace methods [], and the overparameteriza-

tion methods [, ]. Some methods require paying much

extra computation.

e stochastic gradient (SG) algorithm is widely applied

to parameter estimation. For example, Wang and Ding pre-

sented an extended SG identication algorithm for Hammer-

stein-Wiener ARMAX systems [],butitiswellknownthat

the SG algorithm has slower convergence rates. In order to

improvetheconvergencerateoftheSGalgorithm,Xiaoetal.

presented a multi-innovation stochastic gradient parameter

estimation algorithm for input nonlinear controlled autore-

gressive (IN-CAR) models using the over-parameterization

method []; Chen et al. proposed a modied stochastic

gradient algorithm by introducing a convergence index in

order to improve the convergence rate of the parameter

estimation []; Han and Ding developed a multi-innovation

stochastic gradient algorithm for multi-input single-output

systems [];Liuetal.studiedtheperformanceofthestochas-

tic gradient algorithm for multivariable systems [].

e decomposition identication techniques include

matrix decomposition and model decomposition. Hu and

Ding presented a least squares based iterative identication

algorithm for controlled moving average systems using the

matrix decomposition []; Ding derived an iterative least

squares algorithm to estimate the parameters of output

error systems, and the matrix decomposition can enhance

computational eciencies []. Ding also divided a Ham-

merstein nonlinear system into two subsystems based on the

 Journal of Applied Mathematics

model decomposition and presented a hierarchical multi-

innovation stochastic gradient algorithm for Hammerstein

nonlinear systems [].

is paper discusses identication problems of input

nonlinear controlled autoregressive autoregressive (IN-

CARAR) systems or Hammerstein controlled autoregressive

autoregressive (H-CARAR) systems, which is one kind of

input nonlinear equation error type systems. e basic idea

is using the key term separation principle []andthe

decomposition technique [] to derive a multi-innovation

stochastic gradient identication algorithm, which is dier-

ent from the work in [, , ].

e rest of this paper is organized as follows. Section 

gives the identication model for the IN-CARAR systems.

Section  introduces the SG algorithm for the IN-CARAR

system. Section  deduces a multi-innovation SG algorithm

for IN-CARAR system using the decomposition technique.

Section  provides a numerical example to show the eective-

nessoftheproposedalgorithm.Finally,Section  oers some

concluding remarks.

2. The System Identification Model

e paper focuses on the parameter estimation of a Ham-

merstein nonlinear controlled autoregressive autoregressive

(H-CARAR) system, that is, an input nonlinear controlled

autoregressive autoregressive (IN-CARAR) system, which

consists of a nonlinear block and a linear dynamic sub-

system. It is worth noting that Xiao and Yue discussed a

data ltering based recursive least squares algorithm for H-

CARAR systems [] and a multi-innovation stochastic gra-

dient parameter estimation algorithm for input nonlinear

controlled autoregressive (IN-CAR) systems.

An IN-CARAR system shown as Figure  is expressed as

[]



(



)



(



)

=

(



)



(



)



(



)

(



)

()



(



)



(



)

(



)

()



(



)

(



(



))

,

()

where ()and ()are the system input and output,

()is

the output of the nonlinear part, and V()is an uncorrelated

stochastic noise with zero mean. Here,

()is expressed as

[]



(



)

(



(



))

=

𝑛

𝛾



𝑖=1



𝑖



𝑖

(



(



))

=



(



(



))

+



(



(



))

+⋅⋅⋅+

𝑛

𝛾



𝑛

𝛾

(



(



))

()

where f (()):=[

(()),

(()),...,

𝑛

𝛾

(())]∈R

1×𝑛

𝛾

and

:=[

,

,...,

𝑛

𝛾

]

𝑇

∈R

𝑛

𝛾

is the parameters vector of the

nonlinear part.

u(t)

f(·)

u(t)

(t)

x(t)

B(z)

A(z)

w(t)

y(t)

A(z)C(z)

F : e IN-CARAR system.

In (), (), (),and()are the polynomials, of known

orders 

𝑎

, 

𝑏

,and

𝑐

,intheunitbackwardshioperator

−1

[

−1

()=(−1)], dened by



(



)

:=1+



−1

+



−2

+⋅⋅⋅+

𝑛

𝑎



−𝑛

𝑎



(



)

:=1+



−1

+



−2

+⋅⋅⋅+

𝑛

𝑏



−𝑛

𝑏



(



)

:=1+



−1

+



−2

+⋅⋅⋅+

𝑛

𝑐



−𝑛

𝑐

()

Assume ()= 0, () =0,andV() =0for 0. 

𝑖

, 

𝑖

,and



𝑖

are the parameters to be estimated from measured input–

output data {(),()}.

Dene the parameter vectors:

𝜃 :=𝜃

𝑇

𝑠

,𝜃

𝑇

𝑛



𝑇

∈R

𝑛

𝑎

+𝑛

𝑏

+𝑛

𝛾

+𝑛

𝑐

𝜃

𝑠

:=a

𝑇

,

𝑇



𝑇

∈R

𝑛

𝑎

+𝑛

𝑏

+𝑛

𝛾

a :=

,

,...,

𝑛

𝑎



𝑇

∈R

𝑛

𝑎

b :=

,

,...,

𝑛

𝑏



𝑇

∈R

𝑛

𝑏

𝜃

𝑛

:=

,

,...,

𝑛

𝑐



𝑇

∈R

𝑛

𝑐

()

and the information vectors:

𝜑

(



)

:=

𝜑

𝑠

(



)

𝜑

𝑛

(



)

∈R

𝑛

𝑎

+𝑛

𝑏

+𝑛

𝛾

+𝑛

𝑐

𝜑

𝑠

(



)

:= −

(

−1

)

,−

(

−2

)

,...,−−

𝑎

,

(−1),...,(−

𝑏

),f(())

𝑇

∈R

𝑛

𝑎

+𝑛

𝑏

+𝑛

𝛾

𝜑

𝑛

(



)

:=−

(

−1

)

,−

(

−2

)

,...,−−

𝑐



𝑇

∈R

𝑛

𝑐

()

Equation ()canbewrittenas



(



)

[

1−

(



)

]



(



)

(



)

=−

𝑛

𝑐



𝑖=1



𝑖



(

−

)

(



)

=𝜑

𝑇

𝑛

(



)

𝜃

𝑛

(



)

()

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