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Research ArticleCombined Parameter and State Estimation Algorithms forMultivariable Nonlinear Systems Using MIMO Wiener Models

Houda Salhi and Samira Kamoun

University of Sfax, National Engineering School of Sfax (ENIS), Laboratory of Sciences and Technique ofAutomatic Control and Computer Engineering (Lab-SAT), BP 1173, 3038 Sfax, Tunisia

Correspondence should be addressed to Houda Salhi; [email protected]

Received 29 November 2015; Revised 20 April 2016; Accepted 12 May 2016

Academic Editor: James Lam

Copyright © 2016 H. Salhi and S. Kamoun. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper deals with the parameter estimation problem for multivariable nonlinear systems described by MIMO state-spaceWienermodels. Recursive parameters and state estimation algorithms are presented using the least squares technique, the adjustablemodel, and the Kalman filter theory. The basic idea is to estimate jointly the parameters, the state vector, and the internal variablesof MIMOWiener models based on a specific decomposition technique to extract the internal vector and avoid problems related toinvertibility assumption. The effectiveness of the proposed algorithms is shown by an illustrative simulation example.

1. Introduction

Over the last years, modeling, identification, and parameterestimation theories have received much attention by variousresearch teams [1–4]. Blocks-oriented nonlinear models, inparticular, which consist of interconnected linear dynamicsubsystems and memory less nonlinear elements, have beenwidely used for modeling a large variety of nonlinear sys-tems in such different fields as mechanical dynamics [5],chemical process [6], biotechnologies [7], signal filtering[8], and so on. In fact, this class of nonlinear models isable to describe the dynamic of complex systems, witha relatively simple structure. They can even simplify theidentification, control, or diagnostic problems [9–12]. Addedto that, several approaches developed in the linear casecan be applied with an appropriate practical implementa-tions [13–19]. In recent years, many identification methodshave been studied for blocks-oriented systems and a largeamount of works have been published in the literature.For example, Voros [20] proposed a least squares basediterative algorithm for Hammerstein-Wiener systems witha backlash output; Hu et al. [21] developed an extendedleast squares parameter estimation algorithm for Wienersystems based on the overparameterization method; Dinget al. [22] used the hierarchical identification principal to

identify Hammerstein systems; Mao and Ding [8] proposeda multi-innovation stochastic gradient algorithm for Ham-merstein systems using the key term separation principal; Li[23] studied the maximum likelihood estimation algorithmfor Hammerstein CARARMA systems; Guo and Bretthauer[24] proposed a recursive identification method for Wienermodels, based on the prediction error method. Furthermore,Chaudhary et al. [25] explored an adaptive algorithm basedon fractional signal processing for parameter estimation ofHammerstein autoregressivemodels;Wu et al. [26] presenteda robust Hammerstein adaptive filtering algorithm based onthe Maximum Correntropy Criterion, which aims at maxi-mizing the similarity between the model output and the reelresponse; Falck et al. [27] proposed an identification methodof NARX Wiener-Hammerstein models using kernel-basedestimation technique; Kibangou and Favier [28] developeda new approach for estimating a Parallel-Cascade WienerSystemusing a joint diagonalization of the𝑝th-order Volterrakernel slices to identify linear zsubsystems and using theleast square algorithm to identify nonlinear subsystems. Thisapproach has been extended to other blocks-oriented modelswith polynomials nonlinearities [29].

Recently, much attention has been paid to blocks-oriented state-space systems which have been successfully

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016, Article ID 9614167, 12 pageshttp://dx.doi.org/10.1155/2016/9614167

2 Journal of Control Science and Engineering

used for control algorithms, identification schemes, and sig-nal filtering [30, 31]. However, the parameter estimation hasbecome more difficult because the blocks-oriented modelsnot only include the unknown parameter of linear andnonlinear subsystems but also include the unmeasurablestate variables [32–35]. In this framework, Wang and Dingproposed a recursive parameter and state estimation forHammerstein state-space systems [36] and forHammerstein-Wiener state-space systems [37], using the hierarchical prin-cipal; Wang et al. [38] discussed an iterative identificationalgorithm Hammetstein state-space system, by combiningthe iterative least square and the hierarchical identificationmethod. However, for Wiener state-space models, there is alittle contribution in the literature that addresses the param-eter estimation problems or the state estimation problems.In fact, Westwick and Verhaegen [39] proposed a subspaceidentification method for MIMO Wiener subsystems withodd and even nonlinearities and a Gaussian input system;Bruls et al. [40] derived separable least squares algorithmsfor a state-space Wiener model with Chebyshev polynomialsnonlinearity; Lovera et al. [41] developed a recursive subspaceidentification method for Wiener state-space models usingthe singular-value decomposition technique; Glaria Lopezand Sbarbaro [42] proposed an observer design for a Wienermodel with known parameters.

Themain difficulty in the identification ofWienermodelsis that the internal variables, acting between linear andnonlinear blocks, are almost unavailable and the input-outputavailable data are not enough to provide all information onthese unknown variables. To overcome this difficulty, mostpublished works, addressing the identification of Wienersystems, assume one of these assumptions: the invertibility ofthe unknown nonlinear element [43], an a priori knowledgeof the nonlinearity [44], an approximation of the nonlinearityas a piecewise linear function [44], and a specific inputsignal [39]. However, these assumptions, and especially theinvertibility assumption, severely limit the applicability ofWiener models because the output nonlinearity, in severalreal cases, is noninvertible or is quite complicated to find theinverse nonlinearity especially for multivariable systems.

This paper introduces a recursive identification methodfor MIMO Wiener model. This model is characterised bya linear dynamic block as an observer state-space modeland a nonlinear block as combined and arbitrary (reversibleor irreversible) nonlinearities. A recursive algorithm whichcombines the least squares technique, the adjustable model,and the Kalman filter principle is developed to resolve theparameters and state estimation problem with less compu-tational effort and a fast convergence rate. Indeed, in theproposed method, the parameters of the linear part andnonlinear part of the MIMO Wiener model are estimatedseparately in order to decrease the dimension of the unknownparameters matrices and reduce the parameter redundancy.Moreover, a modified Kalman filter and a specific decom-position technique are developed to extract and estimate theunknown internal vector without any research of the inversenonlinear functions.

The remainder of this paper is organized as follows. Sec-tion 2 describes the problem formulation for MIMOWiener

state-space models. The least squares based and adjustablemodel based recursive parameter estimation algorithm anda new recursive state estimation algorithm based on Kalmanfilter theorem are presented in Section 3. Section 4 providesan illustrative example to show the efficiency of the proposedalgorithms. Finally, some concluding remarks are given inSection 5.

2. Problem Formulation

Consider the MIMO discrete-time Wiener model Figure 1where the linear dynamic part is given by the following state-space equation:

𝑥 (𝑘 + 1) = 𝐴 (𝑘) 𝑥 (𝑘) + 𝐵 (𝑘)𝑈 (𝑘) + 𝑊 (𝑘)

𝑍 (𝑘) = 𝐶 (𝑘) 𝑥 (𝑘) + 𝐷 (𝑘)𝑈 (𝑘) + 𝑉 (𝑘) ,(1)

where 𝑥𝑇(𝑘) = [𝑥1(𝑘) 𝑥2(𝑘) ⋅ ⋅ ⋅ 𝑥𝑛𝑋

(𝑘)], 𝑈𝑇(𝑘) =

[𝑢1(𝑘) 𝑢2(𝑘) ⋅ ⋅ ⋅ 𝑢𝑛𝑈(𝑘)], and 𝑍

𝑇(𝑘) =

[𝑧1(𝑘) 𝑧2(𝑘) ⋅ ⋅ ⋅ 𝑧𝑛𝑍(𝑘)] are, respectively, the state

vector, the input vector, and the internal vector at thediscrete-time 𝑘,𝑊𝑇(𝑘) = [𝑤1(𝑘) ⋅ ⋅ ⋅ 𝑤𝑛𝑋

(𝑘)] and 𝑉𝑇(𝑘) =[V1(𝑘) ⋅ ⋅ ⋅ V

𝑛𝑍(𝑘)] are two noise vectors, 𝐴(𝑘) ∈ 𝑅

𝑛𝑋×𝑛𝑋 ,𝐵(𝑘) ∈ 𝑅

𝑛𝑋×𝑛𝑈 , and 𝐶(𝑘) ∈ 𝑅𝑛𝑍×𝑛𝑋 and 𝐷(𝑘) ∈ 𝑅

𝑛𝑍×𝑛𝑈 aredefined, respectively, by

𝐴 (𝑘) =

[[[[[[[

[

𝑎11 (𝑘) 𝑎

12 (𝑘) ⋅ ⋅ ⋅ 𝑎1𝑛𝑋

(𝑘)

𝑎21 (𝑘) 𝑎

22 (𝑘) ⋅ ⋅ ⋅ 𝑎2𝑛𝑋

(𝑘)

.

.

.... d

.

.

.

𝑎𝑛𝑋1

(𝑘) 𝑎𝑛𝑋2(𝑘) ⋅ ⋅ ⋅ 𝑎𝑛𝑋𝑛𝑋

(𝑘)

]]]]]]]

]

,

𝐵 (𝑘) =

[[[[[[[

[

𝑏11 (𝑘) 𝑏

12 (𝑘) ⋅ ⋅ ⋅ 𝑏1𝑛𝑈

(𝑘)

𝑏21 (𝑘) 𝑏

22 (𝑘) ⋅ ⋅ ⋅ 𝑏2𝑛𝑈

(𝑘)

.

.

.... d

.

.

.

𝑏𝑛𝑋1

(𝑘) 𝑏𝑛𝑋2(𝑘) ⋅ ⋅ ⋅ 𝑏𝑛𝑋𝑛𝑈

(𝑘)

]]]]]]]

]

,

𝐶 (𝑘) =

[[[[[[[

[

𝑐11 (𝑘) 𝑐

12 (𝑘) ⋅ ⋅ ⋅ 𝑐1𝑛𝑋

(𝑘)

𝑐21 (𝑘) 𝑐

22 (𝑘) ⋅ ⋅ ⋅ 𝑐2𝑛𝑋

(𝑘)

.

.

.... d

.

.

.

𝑐𝑛𝑍1

(𝑘) 𝑐𝑛𝑍2(𝑘) ⋅ ⋅ ⋅ 𝑐𝑛𝑍𝑛𝑋

(𝑘)

]]]]]]]

]

,

𝐷 (𝑘) =

[[[[[[[

[

𝑑11 (𝑘) 𝑑

12 (𝑘) ⋅ ⋅ ⋅ 𝑑1𝑛𝑈

(𝑘)

𝑑21 (𝑘) 𝑑

22 (𝑘) ⋅ ⋅ ⋅ 𝑑2𝑛𝑈

(𝑘)

.

.

.... d

.

.

.

𝑑𝑛𝑍1

(𝑘) 𝑑𝑛𝑍2(𝑘) ⋅ ⋅ ⋅ 𝑑𝑛𝑍𝑛𝑈

(𝑘)

]]]]]]]

]

.

(2)

Assume that the degrees 𝑛𝑋, 𝑛𝑈, and 𝑛

𝑍are known

and the internal vector 𝑍(𝑘) and the state vector 𝑥(𝑘) are

Journal of Control Science and Engineering 3

Dynamic linearelement

Static nonlinearelement

u1(k)

up(k)

z1(k)

zm(k)

y1(k)

yl(k)

......

...

Figure 1: The MIMOWiener model.

unmeasurable. The static nonlinear function of the MIMOWiener model is defined as

𝑌 (𝑘) = 𝑓 [𝑍 (𝑘)] + 𝐸 (𝑘)

=

[[[[[[[

[

𝑓1[𝑧1 (𝑘) , 𝑧2 (𝑘) , . . . , 𝑧𝑛𝑍

(𝑘)]

𝑓2[𝑧1 (𝑘) , 𝑧2 (𝑘) , . . . , 𝑧𝑛𝑍

(𝑘)]

.

.

.

𝑓𝑛𝑌[𝑧1 (𝑘) , 𝑧2 (𝑘) , . . . , 𝑧𝑛𝑍

(𝑘)]

]]]]]]]

]

+

[[[[[[

[

𝑒1 (𝑘)

𝑒2 (𝑘)

.

.

.

𝑒𝑛𝑌(𝑘)

]]]]]]

]

,

(3)

where 𝑌𝑇(𝑘) = [𝑦1(𝑘) 𝑦2(𝑘) ⋅ ⋅ ⋅ 𝑦𝑛𝑌(𝑘)] is the output

system vector, 𝑓[𝑍(𝑘)] is the nonlinear function vector thatdepends on the unknown internal vector𝑍(𝑘), and 𝐸(𝑘) is anerror vector.

In the rest of this paper, we propose to rewrite the systemoutput vector 𝑌(𝑘) into two submodels.

The first submodel is given by the following equation:

𝑌 (𝑘) = ΘΨ (𝑘, 𝑍 (𝑘)) + 𝐸 (𝑘) , (4)

where Θ and Ψ(𝑘, 𝑍(𝑘)) are defined by

Θ =

[[[[[[

[

𝜃11

⋅ ⋅ ⋅ 𝜃1𝑛Ψ

𝜃21

⋅ ⋅ ⋅ 𝜃2𝑛Ψ

.

.

. d...

𝜃𝑛𝑌1

⋅ ⋅ ⋅ 𝜃𝑛𝑌𝑛Ψ

]]]]]]

]

,

Ψ (𝑘, 𝑍 (𝑘)) =(

(

𝜓1(𝑘, 𝑧1 (𝑘) ; . . . ; 𝑧𝑛𝑍

(𝑘))

𝜓2(𝑘, 𝑧1 (𝑘) ; . . . ; 𝑧𝑛𝑍

(𝑘))

.

.

.

𝜓𝑝(𝑘, 𝑧1 (𝑘) ; . . . ; 𝑧𝑛𝑍

(𝑘))

)

)

.

(5)

However, the second submodel is based on a decomposi-tion technique of nonlinear functions 𝑓

𝑖[⋅], given in (3):

𝑓𝑖[𝑧1 (𝑘) , 𝑧2 (𝑘) , . . . , 𝑧𝑛𝑍 (

𝑘)]

= 𝑧𝑗 (𝑘) + 𝑓

∗

𝑖[𝑧1 (𝑘) , 𝑧2 (𝑘) , . . . , 𝑧𝑛𝑍

(𝑘)] ,

(6)

with 𝑖 = 1, . . . , 𝑛𝑌and 𝑗 = 1, . . . , 𝑛

𝑍.

Using (6), the system output vector𝑌(𝑘) can be written as

𝑌 (𝑘) = Γ𝑍 (𝑘) + Θ⋆Ψ⋆(𝑘) + 𝐸 (𝑘) , (7)

where Γ ∈ 𝑅𝑛𝑌×𝑛𝑍 (if 𝑛

𝑌= 𝑛𝑍, Γ = 𝐼

𝑛𝑌×𝑛𝑌; 𝐼 is an identity

matrix) and Θ⋆Ψ⋆(𝑘) is defined by

Θ⋆Ψ⋆(𝑘)

=

[[[[[[[

[

𝜃11

∗⋅ ⋅ ⋅ 𝜃

1𝑛Ψ

∗

𝜃21

∗⋅ ⋅ ⋅ 𝜃

2𝑛Ψ

∗

.

.

. d...

𝜃𝑛𝑌1

∗⋅ ⋅ ⋅ 𝜃𝑛𝑌𝑛Ψ

∗

]]]]]]]

]

(

(

𝜓1

∗(𝑘, 𝑧1 (𝑘) ; . . . ; 𝑧𝑛𝑍

(𝑘))

𝜓2

∗(𝑘, 𝑧1 (𝑘) ; . . . ; 𝑧𝑛𝑍 (

𝑘))

.

.

.

𝜓𝑝

∗(𝑘, 𝑧1 (𝑘) ; . . . ; 𝑧𝑛𝑍

(𝑘))

)

)

.

(8)

It is worth noting that the application of this decompositiontechnique to Wiener model has led to useful modelisationbecause it allows the extraction of the inaccessible internalvector 𝑍(𝑘) and then the identification and control schemesbecome easier and more efficient.

From (1) and (4), it is clear that the number of unknownparameters is quite large, causing difficulties in the identifi-cation algorithm implementation. In order to reduce thesedifficulties and have a precise estimation quality, we proposeto estimate the parameters of the linear dynamic part andthose of the static nonlinear part, in two separate andrecursive steps.The objective of this paper is to present a newrecursive method to estimate jointly the parameters (𝑎

𝑖𝑗, 𝑏𝑖𝑗,

𝑐𝑖𝑗,𝑑𝑖𝑗), the state vector𝑥(𝑘), and the intermediate vector𝑍(𝑘)

using the measured input-output data {𝑈(𝑘), 𝑌(𝑘)}.

3. The Recursive Parametric andState Estimation Algorithm

In order to simplify the formulation of the parametric andstate estimation problem, this section is divided into twosubsections. The basic idea is to estimate recursively thedynamic linear part parameters and the static nonlinear partparameters of the consideredMIMOWiener model and thento estimate the state vector 𝑥(𝑘) and the internal vector 𝑍(𝑘).

3.1. The Parameter Estimation Algorithm. If the state vector𝑥(𝑘) and the internal vector 𝑍(𝑘) are known, then we canapply the following adjustable model to generate the lineardynamic subsystem estimate (1):

𝑥𝑝 (𝑘 + 1) = �� (𝑘 + 1) 𝑥 (𝑘) + �� (𝑘 + 1)𝑈 (𝑘)

𝑍𝑝 (𝑘) = �� (𝑘) 𝑥 (𝑘) + �� (𝑘)𝑈 (𝑘) .

(9)

Using the least squares principle and minimizing the predic-tion errors functions

𝛿𝑥 (𝑘) = 𝑥 (𝑘) − 𝑥𝑝 (𝑘) ,

𝛿𝑍 (𝑘) = 𝑍 (𝑘) − 𝑍𝑝 (𝑘) ,

(10)


we can obtain the following recursive algorithm:

�� (𝑘) = �� (𝑘 − 1) + 𝜉𝑥 (𝑘 − 1) 𝐺𝑥𝛿𝑥 (𝑘 − 1) 𝑥𝑇(𝑘 − 2)

�� (𝑘) = �� (𝑘 − 1) + 𝜉𝑥 (𝑘 − 1) 𝐺𝑥𝛿𝑥 (𝑘 − 1)𝑈𝑇(𝑘 − 2)

�� (𝑘) = �� (𝑘 − 1) + 𝜉𝑍 (𝑘 − 1) 𝐺𝑍𝛿𝑍 (𝑘 − 1) 𝑥𝑇(𝑘 − 1)

�� (𝑘)

= �� (𝑘 − 1) + 𝜉𝑍 (𝑘 − 1) 𝐺𝑍𝛿𝑍 (𝑘 − 1)𝑈𝑇(𝑘 − 1)

𝛿𝑥 (𝑘 − 1)

= 𝑥 (𝑘 − 1) − �� (𝑘 − 1) 𝑥 (𝑘 − 2)

− �� (𝑘 − 1)𝑈 (𝑘 − 2)

𝛿𝑍 (𝑘 − 1)

= 𝑍 (𝑘 − 1) − �� (𝑘 − 1) 𝑥 (𝑘 − 1)

− �� (𝑘 − 1)𝑈 (𝑘 − 1)

𝜉𝑥 (𝑘 − 1)

=𝑙𝑥

𝜆𝐺𝑥[𝑥 (𝑘 − 2) 𝑥𝑇 (𝑘 − 2) + 𝑈 (𝑘 − 2)𝑈𝑇 (𝑘 − 2)]

𝜉𝑍 (𝑘 − 1)

=𝑙𝑍

𝜆𝐺𝑍[𝑥 (𝑘 − 1) 𝑥𝑇 (𝑘 − 1) + 𝑈 (𝑘 − 1)𝑈𝑇 (𝑘 − 1)]

,

(11)

where 𝐺𝑥and 𝐺

𝑍are definite positive symmetrical matrices

and 𝜆𝐺𝑥

and 𝜆𝐺𝑍

are, respectively, the maximum eigenvaluesof 𝐺𝑥and 𝐺

𝑍. In order to guarantee the convergence of the

parameter estimate matrices ��(𝑘), ��(𝑘), ��(𝑘), and ��(𝑘), thegains parameter 𝑙

𝑥and 𝑙𝑍must be chosen such that 0 < 𝑙

𝑥< 2

and 0 < 𝑙𝑍< 2. It should be noted that these gains can be

chosen as time-varying parameters in order to improve theparameter estimation quality.

To avoid the parametric redundancy problem and con-struct the estimate parameters of the static nonlinear part (4),we suggest using the following recursive least squares (RLS)algorithm:

Θ (𝑘) = Θ (𝑘 − 1) + 𝑃 (𝑘)Ψ (𝑘, 𝑍 (𝑘)) Ξ (𝑘)

𝑃 (𝑘)

= 𝑃 (𝑘 − 1)

−𝑃 (𝑘 − 1)Ψ (𝑘, 𝑍 (𝑘)) Ψ

𝑇(𝑘, 𝑍 (𝑘)) 𝑃 (𝑘 − 1)

1 + Ψ𝑇 (𝑘, 𝑍 (𝑘)) 𝑃 (𝑘 − 1)Ψ (𝑘, 𝑍 (𝑘))

Ξ (𝑘) = 𝑌 (𝑘) − Θ (𝑘 − 1)Ψ (𝑘, 𝑍 (𝑘)) ,

(12)

where Θ(𝑘) represent the estimate of Θ in the regressionequation (4) and 𝑃(𝑘) = 𝑝

0𝐼𝑝×𝑝

is a definite positivesymmetrical matrix where 𝐼

𝑝×𝑝is an identity matrix and 𝑝

0

is generally taken as a large positive number, for example,𝑝0= 105.

In reality, the regression matrix Ψ(𝑘, 𝑍(𝑘)) contains theunmeasurable internal vector𝑍(𝑘), so the parametric estima-tion algorithms (11) and (12) cannot be implemented directly.Added to that, the state vector 𝑥(𝑘) is also supposed to beunknown. The proposed solution is to replace 𝑥(𝑘) and 𝑍(𝑘)in (11) and (12) with their estimated vectors ��(𝑘) and ��(𝑘) andthen to define

�� (𝑘) =

[[[[[[[

[

��1 (𝑘)

��2 (𝑘)

.

.

.

��𝑛𝑋(𝑘)

]]]]]]]

]

,

�� (𝑘) =

[[[[[[[

[

��1 (𝑘)

��2 (𝑘)

.

.

.

��𝑛𝑍(𝑘)

]]]]]]]

]

.

(13)

3.2. Modified Kalman Filter. In the area of state estimationalgorithms, the extended Kalman filter (EKF) is the mostwidely used nonlinear estimation method [45]. The EKF isbased on the linearization of nonlinear models and calcu-lation of the Jacobian matrices with each iteration, whichmay cause significant implementation difficulties and highestimation errors.Therefore, we propose to use a newfilteringtechnique, developed in our previous work [15], based onthe linear Kalman filter (KF). This filter yields the sameperformance as KF and can be applied to blocks-orientedmodels without any linearization steps. Let ��(𝑘) and ��0(𝑘)represent, respectively, the a priori and a posteriori estimateof 𝑥(𝑘) at a discrete-time 𝑘 and ��(𝑘) represents the a prioriestimate of 𝑍(𝑘).

This part is based essentially on the Kalman filter princi-ple, which is defined by the following.

Theorem 1. Define 𝛼 and 𝛽 as two random vectors, withΛ = [

𝛼

𝛽 ] a multinomial distribution vector with a mean vector[𝛼

𝛽] and a variance-covariance matrix [

Σ𝛼 Σ𝛼𝛽

Σ𝑇

𝛼𝛽Σ𝛽]. Thus, the

conditional probability density 𝑝(𝛼, 𝛽) is a Gaussian function,where its mean and variance are given, respectively, by

�� = 𝛼 + Σ𝛼𝛽Σ−1

𝛽(𝛽 − 𝛽)

Σ𝛼\𝛽

= Σ𝛼− Σ𝛼𝛽Σ−1

𝛽Σ𝑇

𝛼𝛽.

(14)

In a first step, applying this theorem to the linear state-space equation (1) yields

��0(𝑘)

= �� (𝑘)

+ 𝐾𝑥 (𝑘) [𝑍 (𝑘) − �� (𝑘) �� (𝑘) − �� (𝑘)𝑈 (𝑘)]


𝐾𝑥 (𝑘) =

𝑃𝑥 (𝑘) �� (𝑘)

𝑄𝑍+ �� (𝑘) 𝑃𝑥 (𝑘) ��

𝑇

(𝑘)

𝑃0

𝑥(𝑘) = 𝑃𝑥 (𝑘) − 𝐾𝑥 (𝑘) �� (𝑘) 𝑃𝑥 (𝑘)

�� (𝑘 + 1) = �� (𝑘) ��0(𝑘) + �� (𝑘)𝑈 (𝑘)

𝑃𝑥 (𝑘 + 1) = �� (𝑘) 𝑃

0

𝑥(𝑘) ��𝑇

(𝑘) + 𝑄𝑥,

(15)

where 𝑄𝑥and 𝑄

𝑍are variance-covariance matrices of noise

vectors𝑊(𝑘) and 𝑉(𝑘), respectively, and 𝑃𝑥(𝑘) = 𝐸[(𝑥(𝑘) −

��(𝑘))(𝑥(𝑘) − ��(𝑘))𝑇] is the variance-covariance matrices.

However, the a posteriori estimate ��0(𝑘) contains theunknown internal vector 𝑍(𝑘), so the algorithm (15) isimpossible to implement. The solution is to replace theunknown internal vector𝑍(𝑘)with its estimate ��(𝑘). For thisreason, we propose to use the second form (7) of the systemoutput vector 𝑌(𝑘) and apply the KF to the following state-space model:

𝑍 (𝑘 + 1) = 𝐶 (𝑘)𝑋 (𝑘 + 1) + 𝐷 (𝑘)𝑈 (𝑘 + 1)

+ 𝑊 (𝑘 + 1)

𝑌 (𝑘) = Γ𝑍 (𝑘) + Θ⋆Ψ⋆(𝑘) + 𝐸 (𝑘) .

(16)

UsingTheorem 1 gives the following equations to generate therecursive estimate vector 𝑍(𝑘):

�� (𝑘 + 1) = �� (𝑘) �� (𝑘) ��0(𝑘) + �� (𝑘) �� (𝑘)𝑈 (𝑘)

+ �� (𝑘)𝑈 (𝑘 + 1) + 𝐾𝑍 (𝑘) [𝑌 (𝑘) − Γ�� (𝑘) �� (𝑘)

− Γ�� (𝑘)𝑈 (𝑘) − Θ⋆

(𝑘) Ψ⋆

(𝑘)]

𝐾𝑍 (𝑘)

=�� (𝑘) �� (𝑘) 𝑃𝑥 (𝑘) Γ��

𝑇

(𝑘)

Γ�� (𝑘) 𝑃𝑥 (𝑘) Γ��𝑇

(𝑘) + 𝑄𝑌 + Θ⋆

(𝑘) 𝐺 (𝑘) Θ⋆𝑇

(𝑘)

,

(17)

where 𝑄𝑌is the variance-covariance matrix of the noise

vector 𝐸(𝑘) and 𝐺(𝑘) = 𝐸[(𝑌(𝑘) − ��(𝑘))(𝑌(𝑘) − ��(𝑘))𝑇] is

the variance-covariance matrix.Combining (15) and (17), we can summarize the recursive

algorithm to generate the estimates ��(𝑘) and ��(𝑘) of the statevector 𝑥(𝑘) and the internal vector 𝑍(𝑘):

��0(𝑘) = �� (𝑘) + 𝐾𝑥 (𝑘) [�� (𝑘) − �� (𝑘) �� (𝑘)

− �� (𝑘)𝑈 (𝑘)]

𝐾𝑥 (𝑘) =

𝑃𝑥 (𝑘) �� (𝑘)

𝑄𝑍+ �� (𝑘) 𝑃𝑥 (𝑘) ��

𝑇

(𝑘)

𝑃0

𝑥(𝑘) = 𝑃𝑥 (𝑘) − 𝐾𝑥 (𝑘) �� (𝑘) 𝑃𝑥 (𝑘)

�� (𝑘 + 1) = �� (𝑘) ��0(𝑘) + �� (𝑘)𝑈 (𝑘)

𝑃𝑥 (𝑘 + 1) = �� (𝑘) 𝑃

0

𝑥(𝑘) ��𝑇

(𝑘) + 𝑄𝑥

�� (𝑘 + 1) = �� (𝑘) �� (𝑘) ��0(𝑘) + �� (𝑘) �� (𝑘) 𝑈 (𝑘)

+ �� (𝑘)𝑈 (𝑘 + 1) + 𝐾𝑍 (𝑘) [𝑌 (𝑘) − Γ�� (𝑘) �� (𝑘)

− Γ�� (𝑘)𝑈 (𝑘) − Θ⋆

(𝑘) Ψ⋆

(𝑘)]

𝐾𝑍 (𝑘)

=�� (𝑘) �� (𝑘) 𝑃𝑥 (𝑘) Γ��

𝑇

(𝑘)

Γ�� (𝑘) 𝑃𝑥 (𝑘) Γ��𝑇

(𝑘) + 𝑄𝑌 + Θ⋆

(𝑘) 𝐺 (𝑘) Θ⋆𝑇

(𝑘)

.

(18)

The details and the convergence analysis of (18) is treated in[15].

Thus, we can replace 𝑥(𝑘) and 𝑍(𝑘) in (11) and (12) withtheir estimates ��(𝑘) and ��(𝑘).

Combining (11), (12), and (18), we can form a recursivestate and parameter estimation algorithm for state-spaceMIMO Wiener systems (which is abbreviated as the RPSEalgorithm). To initialize the RPSE algorithm, we take ��(0),��(0), ��(0), ��(0), ��(0), ��(0), and Θ(0) as small real matricesand vectors; for example, ��(0) = 10

−3𝚤𝑛𝑥×𝑛𝑥

with 𝚤𝑛𝑥×𝑛𝑥

is amatrix whose all elements are equal to 1.We also take𝐺

𝑥(𝑘) =

𝑔𝑥𝐼, 𝐺𝑍(𝑘) = 𝑔

𝑍𝐼, 𝑃(𝑘) = 𝑝

0𝐼, 𝑃𝑥(𝑘) = 𝑝

𝑥𝐼 with 𝑔

𝑥, 𝑔𝑍, 𝑝0,

and 𝑝𝑥as large positive numbers and 𝐼 is an identity matrix

with appropriate sizes. It should be noted that the choice ofthe initial conditions and the different gains must be chosenadequately in order to improve the estimation quality and theconvergence rapidity of the various parameters.

The procedure for computing the parameter, the state,and the internal vector estimates using the RPSE algorithmis listed as follows:

(1) To initialize, let 𝑘 = 1, ��(1), ��(2), ��(1), ��(1), ��(1),��(1), ��(1), Θ(1), 𝐺

𝑥, 𝐺𝑍, 𝑃(1), 𝑙

𝑥, 𝑙𝑍, 𝑄𝑥, 𝑄𝑍, 𝑃𝑥(1),

and 𝐺.

(2) Collect the input-output vectors 𝑈(𝑘) and 𝑌(𝑘).

(3) Form 𝜉𝑥(𝑘 − 1) and 𝜉

𝑍(𝑘 − 1), compute 𝛿

𝑥(𝑘 − 1) and

𝛿𝑍(𝑘−1), and update ��(𝑘), ��(𝑘), ��(𝑘), ��(𝑘) using (11).

(4) Compute the covariance matrix 𝑃(𝑘) and constructΘ(𝑘) using (12).

(5) Compute the state gains 𝐾𝑥(𝑘) and 𝐾

𝑍(𝑘) and the

covariance matrix 𝑃𝑥(𝑘 + 1) and update the state

estimate ��(𝑘 + 1) and the internal vector estimate��(𝑘 + 1) using (18).

(6) Increase 𝑘 by 1 and go to step (2).

The flowchart of the recursive learning algorithms whichare used for the parameters estimation of nonlinear MIMOWiener models is shown in Figure 2.


Start

End

Initialize x(k), Z(k), A(k), B(k), C(k),D(k), Θ(k), Gx(k), GZ(k), P(k), Px(k)

lx, lz for k = 1

Collect input-output data {U(k), Y(k); k = 1, . . . ,M}

Form 𝜉x(k − 1), 𝜉Z(k − 1)

Compute 𝛿x(k − 1), 𝛿Z(k − 1)

Update A(k), B(k), C(k)

Compute P(k) and Ξ(k)

Update Θ(k)

Update x(k + 1) , and Z(k + 1)

Obtain the estimate A(k), B(k), C(k),D(k) , and Θ(k)

Compute Kx(k), KZ(k) , and Px(k)

, and D(k)

Figure 2: Flowchart for recursive estimation of MIMOWiener models.

4. Example

Consider the following state-space model:

𝑥 (𝑘 + 1) = [0 0.5

−0.4 0] 𝑥 (𝑘)

+ [0 √0.85

−√0.6 0]𝑈 (𝑘) + 𝑊 (𝑘)

𝑍 (𝑘) = [1 0

0 1] 𝑥 (𝑘) + 𝑉 (𝑘)

(19)

and the system outputs are defined as

𝑦1 (𝑘) = 𝑧1 (𝑘) + 0.3𝑧2 (𝑘) + 𝑒1 (𝑘) .

𝑦2 (𝑘) = 0.3𝑧1

2(𝑘) + 𝑧2 (𝑘) + 1 + 𝑒2 (𝑘)

(20)

These two outputs can be grouped into the following matrixform:

𝑌 (𝑘) = ΘΨ (𝑘) + 𝐸 (𝑘) , (21)


−1

−0.5

0

0.5

1

840 860 880 900 920 940 960 980 1000820k

z1(k)

z1(k)

−1

−0.5

0

0.5

1

840 860 880 900 920 940 960 980 1000820k

z2(k)

z2(k)

−1

−0.5

0

0.5

1

840 860 880 900 920 940 960 980 1000820k

y1(k)

y1(k)

0.8

1

1.2

1.4

1.6

840 860 880 900 920 940 960 980 1000820k

y2(k)

y2(k)

Figure 3:The internal variables 𝑧1(𝑘) and 𝑧

2(𝑘) and their estimates ��

1(𝑘) and ��

2(𝑘) and the outputs 𝑦

1(𝑘) and 𝑦

2(𝑘)with the predicted outputs

��1(𝑘) and ��

2(𝑘).

where Θ and Ψ(𝑘) are given by

Θ = [1 0.3 0 0 0

0 0 0.3 1 1] ,

Ψ𝑇(𝑘) = [𝑧

1 (𝑘) 𝑧2 (𝑘) 𝑧2

1(𝑘) 𝑧2 (𝑘) 1]

(22)

Using the decomposition technique, (21) can be written in asecond matrix form:

𝑌 (𝑘) = 𝑍 (𝑘) + Θ∗Ψ∗(𝑘) + 𝐸 (𝑘) , (23)

where Θ∗ and Ψ∗(𝑘) are defined as

Θ∗= [

0 0.3 0

0.3 0 1]

Ψ∗𝑇(𝑘) = [𝑧

2

1(𝑘) 𝑧2 (𝑘) 1]

(24)

In simulation, the inputs 𝑢1(𝑘) and 𝑢

2(𝑘) are taken as

two square sequences of levels [−1.2 1.2] and [−1.5 1.5],

respectively; the variance-covariance matrices are 𝑄𝑥

=

[ 0.005 00 0.004

], 𝑄𝑍= [ 0.015 00 0.015

], and 𝑄𝑌= 𝜎𝑒

2[ 1 00 1]. Applying

the RPSE algorithm to estimate the parameters, the state,and internal variables of this system, the gain parametersand the initial conditions are chosen properly. The internalvariables 𝑧

1(𝑘) and 𝑧

2(𝑘) and their estimates ��

1(𝑘) and ��

2(𝑘)

and the outputs 𝑦1(𝑘) and 𝑦

2(𝑘) with the predicted outputs

��1(𝑘) and ��

2(𝑘) are shown in Figure 3. The estimation errors

𝛿𝑧1(𝑘), 𝛿𝑧2(𝑘), 𝛿𝑦1(𝑘), and 𝛿

𝑦2(𝑘) are shown in Figure 4. The

evolution curve of the variance 𝜎𝑦1

2(𝑘) and 𝜎

𝑦2

2(𝑘) of the

system outputs 𝑦1(𝑘) and 𝑦

2(𝑘) is given in Figure 5, where

𝜎𝑦𝑖

2(𝑘) =

Σ [𝛿𝑦𝑖(𝑘) − 𝑚𝛿𝑦𝑖

]2

𝑘𝑓− 𝑘𝑖+ 1

, (25)

with 𝑚𝛿𝑦𝑖

the statistical mean of the output prediction errorand 𝑘𝑖and 𝑘𝑓the initial and final discrete time.Theparameter

estimates and their estimation errors with different datalength and different noise variances are shown in Tables 1 and2, where the parameter estimation error is defined by


−1

−0.5

0

0.5

1×10−3

820 840 860 880 900 920 940 960 980 1000800k

820 840 860 880 900 920 940 960 980 1000800k

−2

−1

0

1

2×10−3

−0.4

−0.2

0

0.2

0.4

−1

−0.5

0

0.5

1

𝛿z1 (k) 𝛿z2 (k)

𝛿y1 (k) 𝛿y2 (k)

820 840 860 880 900 920 940 960 980 1000800k

820 840 860 880 900 920 940 960 980 1000800k

Figure 4: The estimation errors 𝛿𝑧1(𝑘), 𝛿

𝑧2(𝑘), 𝛿

𝑦1(𝑘), and 𝛿

𝑦2(𝑘).

𝛿 (𝑘)

=[[

[

[∑2

𝑖=1[∑2

𝑗=1(𝑎𝑖𝑗− ��𝑖𝑗 (𝑘))

2

] + ∑2

𝑖=1[∑2

𝑗=1(𝑏𝑖𝑗− ��𝑖𝑗 (𝑘))

2

] + ∑2

𝑖=1[∑2

𝑗=1(𝑐𝑖𝑗− ��𝑖𝑗 (𝑘))

2

] + ∑5

𝑟=1[∑2

𝑠=1(𝜃𝑟𝑠− ��𝑟𝑠 (𝑘))

2

]]

[∑2

𝑖=1∑2

𝑗=1(𝑎𝑖𝑗)2

+ ∑2

𝑖=1∑2

𝑗=1(𝑏𝑖𝑗)2

+ ∑2

𝑖=1∑2

𝑗=1(𝑐𝑖𝑗)2

+ ∑2

𝑟=1∑2

𝑠=1(𝜃𝑟𝑠)2]

]]

]

0.5

× 100.

(26)

From the simulation results in Tables 1, 2, and 3 andFigures 3, 4, 5, and 6, we can draw the following conclusions:

(i) The estimated parameters converge to real ones andthe parameter estimation errors given by the RPSEalgorithm become smaller when 𝑘 increases and theoutput variances decrease; see Tables 1 and 2.

(ii) The estimated internal outputs ��1(𝑘) and ��

2(𝑘) and

the predicted system outputs ��1(𝑘) and ��

2(𝑘) can

track the actual outputs without the computationstep of the inverse nonlinear function and with smallestimation errors; see Figures 3 and 4.

(iii) The output variances 𝜎𝑦1

2(𝑘) and 𝜎

𝑦2

2(𝑘) rapidly drop

to law values when the noise variances decrease; seeFigure 5.

(iv) The estimation quality is better when the parametricgains 𝑙

𝑥and 𝑙𝑍are chosen as a time-varying parame-

ters; see Figure 6.(v) The variances of the parameters estimates, using the

Monte Carlo simulation, are small which improvesthe effectiveness of the RPSE algorithm; see Table 3.

(vi) The proposed algorithm can achieve a satisfactoryestimation quality through appropriately choosingthe parametric gains and the innovation length.

5. Conclusions

This paper presents a recursive parameter and state esti-mation algorithm by combining the least square technique,the adjustable model, and the Kalman filter principal for


𝜎y12(k), (𝜎e

2(k) = 0.01)

𝜎y12(k), (𝜎e

2(k) = 0.05)

𝜎y22(k), (𝜎e

2(k) = 0.01)

𝜎y22(k), (𝜎e

2(k) = 0.05)

00.10.20.30.40.50.60.70.80.9

1

200 300 400 500 600 700 800 900 1000100

Figure 5: The output variances 𝜎𝑦1

2(𝑘) and 𝜎

𝑦2

2(𝑘) with 𝜎

𝑒

2= 0.01

and 𝜎𝑒

2= 0.05.

−2

0

2

4

955 960 965 970 975 980 985 990 995 1000950k

x1(k)

−2

0

2

4

955 960 965 970 975 980 985 990 995 1000950k

x2(k)

x1(k); (lx, lZ) x1(k); (lx(k), lZ(k))

x2(k); (lx, lZ) x2(k); (lx(k), lZ(k))

Figure 6: The state estimates ��1(𝑘) and ��

2(𝑘) with constant gains

(𝑙𝑥, 𝑙𝑍) and variable parametric gains (𝑙

𝑥(𝑘), 𝑙𝑍(𝑘)).

estimating jointly the parameters, the state vector, and theinternal variables of MIMO Wiener state-space models. Byestimating the parameters of the linear and nonlinear partsseparately and using a specific decomposition technique, wecan remove the redundant parameters and avoid problemsrelated to computing the inverse nonlinear functions. Theproposed algorithm can be combined with adaptive controlschemes and extended to other blocks-oriented models.

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper.

Acknowledgments

This work was supported by the ministry of higher educationand scientific research of Tunisia.

Table 1: The recursive parameter estimates and errors with 𝜎𝑒

2=

0.01.𝑀 100 200 500 1000𝑎11(𝑘) = 0.0000 −0.0166 −0.0008 −0.0002 −0.0001

𝑎12(𝑘) = 0.5000 0.4769 0.4989 0.5000 0.5001

𝑎21(𝑘) = −0.4000 −0.3965 −0.3998 −0.4000 −0.4000

𝑎22(𝑘) = 0.0000 0.0048 0.0002 0.0001 0.0000

𝑏11(𝑘) = 0.0000 0.001 0.0022 0.0001 0.0000

𝑏12(𝑘) = 0.9220 0.8976 0.8991 0.9108 0.9219

𝑏21(𝑘) = −0.7746 −0.2654 −0.5653 −0.7731 −0.7744

𝑏22(𝑘) = 0.0000 −0.1022 0.0101 0.0105 −0.0002

𝑐11(𝑘) = 1.0000 1.0012 1.0000 1.0000 1.0000

𝑐12(𝑘) = 0.0000 0.0019 0.0000 0.0000 0.0000

𝑐21(𝑘) = 0.0000 −0.0007 −0.0000 −0.0000 −0.0000

𝑐22(𝑘) = 1.0000 0.9989 1.0000 1.0000 1.0000

𝜃11(𝑘) = 1.0000 0.9573 0.9902 1.0027 1.000

𝜃12(𝑘) = 0.3000 0.1502 0.1853 0.2889 0.3001

𝜃13(𝑘) = 0.0000 0.0167 −0.0027 0.0016 0.0010

𝜃14(𝑘) = 0.0000 0.1202 0.1153 0.1108 0.0979

𝜃15(𝑘) = 0.0000 0.0027 0.00620 0.0015 0.0001

𝜃21(𝑘) = 0.0000 −0.0793 −0.0046 0.0003 0.0000

𝜃22(𝑘) = 0.0000 0.0036 0.0043 0.0014 0.0002

𝜃23(𝑘) = 0.3000 0.2306 0.2490 0.2913 0.3001

𝜃24(𝑘) = 1.0000 0.3797 0.5472 0.9894 0.9989

𝜃25(𝑘) = 1.0000 0.9941 0.9526 1.0008 1.0001

𝛿(𝑘) (%) 31.6697 19.9767 4.2708 3.6902

Table 2: The recursive parameter estimates and errors with 𝜎𝑒

2=

0.05.𝑀 100 200 500 1000𝑎11(𝑘) = 0.0000 −0.0166 −0.0008 0.0000 0.0000

𝑎12(𝑘) = 0.5000 0.4769 0.4989 0.5002 0.5000

𝑎21(𝑘) = −0.4000 −0.3965 −0.3998 −0.4000 −0.4000

𝑎22(𝑘) = 0.0000 0.0048 0.0002 0.0000 0.0001

𝑏11(𝑘) = 0.0000 0.0012 0.0011 0.0010 0.0010

𝑏12(𝑘) = 0.9220 0.8974 0.8996 0.8999 0.9105

𝑏21(𝑘) = −0.7746 −0.2654 −0.5651 −0.7659 −0.7699

𝑏22(𝑘) = 0.0000 −0.1022 −0.0101 −0.0101 −0.0091

𝑐11(𝑘) = 1.0000 1.0012 1.0000 1.0000 1.0000

𝑐12(𝑘) = 0.0000 0.0019 0.0000 0.0000 0.0000

𝑐21(𝑘) = 0.0000 −0.0007 0.0000 0.0000 0.0000

𝑐22(𝑘) = 1.0000 0.9989 0.9995 1.0000 1.0000

𝜃11(𝑘) = 1.0000 0.9071 0.9567 1.0082 0.9976

𝜃12(𝑘) = 0.3000 0.1710 0.1773 0.2705 0.2995

𝜃13(𝑘) = 0.0000 0.0123 −0.0052 0.0013 0.001

𝜃14(𝑘) = 0.0000 0.1410 0.0473 0.1205 0.0992

𝜃15(𝑘) = 0.0000 −0.0597 0.00731 −0.00384 −0.0025

𝜃21(𝑘) = 0.0000 −0.0016 −0.0075 −0.0010 0.0013

𝜃22(𝑘) = 0.0000 0.0044 0.0044 0.0014 0.0001

𝜃23(𝑘) = 0.3000 0.4817 0.3529 0.2921 0.2997

𝜃24(𝑘) = 1.0000 0.4528 0.5773 0.9541 0.9903

𝜃25(𝑘) = 1.0000 0.9679 0.9486 1.0023 1.0010

𝛿(𝑘) (%) 31.4375 18.7664 5.1010 3.8043


Table3:Th

eMon

teCa

rloestim

ates

andvaria

nces

with

𝜎𝑒

2=0.01.

𝑀𝑎11

𝑎12

𝑎21

𝑎22

��11

��12

��21

500

−0.00

02±0.00

010.5000±0.00

45−0.40

00±0.0135

0.00

01±0.00

060.00

01±0.00

040.9108±0.0089

−0.7731±0.0077

1000

−0.00

01±0.00

005

0.5001±0.0026

−0.40

00±0.00256

0.00

00±0.00

020.00

00±0.00

001

0.9219±0.00

6897

−0.7744±0.00755

True

values

0.00

000.5000

−0.40

000.00

000.00

000.9220

−0.7746

𝑀��22

��11

��12

��21

��22

𝜃11

𝜃12

500

0.0105±0.00

061.0

000±0.00

900.00

00±0.00

018

−0.00

00±0.00

041.0

000±0.00

9814

1.0027±0.02173

0.2889±0.03178

1000

−0.00

02±0.00

011.0

000±0.00

900.00

00±0.00

01−0.00

00±0.00

008

1.000

0±0.00

968

1.000

0±0.0211

0.3001±0.0233

True

values

0.00

001.0

000

0.00

000.00

001.0

000

1.000

00.3000

𝑀𝜃13

𝜃14

𝜃15

𝜃21

𝜃22

𝜃23

𝜃24

𝜃25

500

0.0016±0.00

080.1108±0.0086

0.0015±0.0160

0.00

03±0.0039

0.0014±0.00

050.2913±0.0253

0.9894±0.099

1.0008±0.0195

1000

0.0010±0.00

020.0979±0.0081

0.00

01±0.0100

0.00

00±0.0033

0.00

02±0.00

020.3001±0.0230

0.9989±0.017

1.0001±

0.0130

True

values

0.00

000.00

000.00

000.00

000.00

000.3000

1.000

01.0

000


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