Decomposition of an ARX model on Laguerre orthonormal bases

ISA Transactions 51 (2012) 848–860

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ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans

Decomposition of an ARX model on Laguerre orthonormal bases
´
Kais Bouzrara a,n, Tarek Garna a, Jose Ragot b, Hassani Messaoud a
a
b

´
´
´
Unite de Recherche Automatique, Traitement de Signal et Image, Ecole Nationale d’Ingenieurs de Monastir, Universite de Monastir, Rue Ibn Eljazzar, 5019 Monastir, Tunisie
Centre de Recherche en Automatique de Nancy, CNRS UPRES-A 7039, 2, Avenue de la forˆt de Haye, 54516 Vandoeuvre Cedex, France
e

a r t i c l e i n f o

a b s t r a c t

Article history:
Received 21 February 2012
Received in revised form
22 May 2012
Accepted 8 June 2012
Available online 10 July 2012

In this paper, we propose a new reduced complexity model by expanding a discrete-time ARX model on
Laguerre orthonormal bases. To ensure an efficient complexity reduction, the coefficients associated to
the input and the output of the ARX model are expanded on independent Laguerre bases, to develop a
new black-box linear ARX-Laguerre model with filters on model input and output. The parametric
complexity reduction with respect to the classical ARX model is proved theoretically. The structure and
parameter identification of the ARX-Laguerre model is achieved by a new proposed approach which
consists in solving an optimization problem built from the ARX model without using system input/
output observations. The performances of the resulting ARX-Laguerre model and the proposed
identification approach are illustrated by numerical simulations and validated on benchmark manufactured by Feedback known as Process Trainer PT326. A possible extension of the proposed model to a
multivariable process is formulated.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords:
ARX model
Reduced parametric complexity
Laguerre bases
Optimization

1. Introduction
Over the last few years, the modelling of the stable linear time
invariant (LTI) systems by Laguerre filters has been particularly
attractive as the resulting model prides on its parameter complexity reduction and on its simple structure given by a linear
combination of the Laguerre filters. This approach was suggested
and initiated by Arnold [1] and was widely used in the context of
system identification [10,6,9] and control [8,12,13]. The concept is
based on the expansion of the impulse response of the LTI system
on an orthonormal basis of Laguerre. This decomposition leads to
a network of Laguerre filters which consists of a first-order lowpass filter connected to several all-pass filters. These all-pass
filters are favourable in terms of numerical sensitivity, and they
are often recommended to use in filter design. In this case, we
obtain the filtering of the input by Laguerre orthonormal functions. Thereafter, the resulting model is entitled the Laguerre
model characterized by a linear representation with respect to the
input filters. These filters are characterized by the Laguerre pole,
the optimal identification of which is achieved by exploiting a
priori information about the dominating pole of the system. This
identification guarantees a parametric reduction which can be
significant when the considered system is linear with a dominant
first-order dynamic. However, since the Laguerre model handles
only a real pole, it suffers from some drawbacks in the case of
scattered poles and an oscillating system. In fact, in the latter

n

Corresponding author. Tel.: þ216 73 500 511; fax: þ256 73 500 514.
E-mail address: kais.bouzrara@enim.rnu.tn (K. Bouzrara).

case, the Laguerre model requires a huge number of Laguerre
functions and then a large number of parameters to represent
systems with various representative modes.
To circumvent this drawback, other orthonormal function
bases (OFB) suitable to the representation of the complex linear
systems have been proposed in the literature such as Kautz
orthonormal basis [11] and generalized orthonormal basis
(GOB) [7]. We note that these bases are characterized by a set
of poles the choice of which strongly influences the parsimony of
the expansion. The Laguerre basis is a special case of the Kautz
basis, which in turn, is a particular realization of GOB, and the
Laguerre functions have the desirable property to be completely
determined by a single parameter (Laguerre pole). For this reason,
the optimization of a Laguerre basis is easier than the Kautz basis
or the GOB and the development on the Laguerre basis is
generally less parsimonious. In this context, we propose and
develop, in this paper, an alternative solution to represent any
complex dynamics linear systems with a reduced parameter
complexity model using the Laguerre bases. This contribution is
based on the development of the ARX model (Auto Regressive
model structure with an eXternal input) on two independent
Laguerre orthonormal bases. This new solution exploits the idea
of collecting the maximum of information about the system and
then extending the principle of the input filtering by Laguerre
functions to filter both the input and the output of the system
represented by the ARX model. It consists in decomposing the
parameters of the ARX model associated with the input and the
output on two independent Laguerre orthonormal bases. This
decomposition can be realized since the coefficients of the ARX
model are absolutely summable on ½0, þ 1½ in the sense of the

0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.06.005

K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860

BIBO (Bounded-Input Bounded-Output) stability criterion of the
system. The resulting model, called ARX - Laguerre model, is a
new linear representation with respect to the filtered input and
the filtered output. We prove theoretically that the proposed
model that enables to represent accurately any complex linear
system is characterized by a reduced parametric complexity
compared to the ARX model. We deduce that the ARX-Laguerre
parameter number never exceeds that of the ARX model. This
result depends essentially on the value of the Laguerre pole of
each basis. We also prove that the ARX-Laguerre model resulting
from the expansion of the ARX model on two independent
Laguerre bases is equivalent to the development of both polynomials of the ARX model on these bases. Furthermore, the
theoretical study and the proof of the parametric complexity
reduction are used to propose a new technique to estimate both
Laguerre poles and the model coefficients. This approach is based
on the resolution of an optimization problem without needing a
set of input/ouput measurements.
The paper is organized as follows: in Section 2 the principle of
the OFB expansion of the ARX coefficients is exposed. In Section 3
the proposed ARX-Laguerre model will be presented. We also
present the input and the output filters network of the ARXLaguerre model as well as its recursive representation. In Section
4, the theoretical analysis and the proof of the parametric
reduction of ARX-Laguerre with respect to the ARX model are
given. Besides, we present the proposed identification approach of
the ARX-Laguerre parameters (Fourier coefficients and Laguerre
poles). In Section 5 we briefly review the Laguerre model with the
input filters which will be used in the next simulations section. In
Section 6, to confirm the supremacy of the ARX-Laguerre model
with respect to classical Laguerre one we compare both models
on the three numerical examples with gradual complexity. It
resorts that the Laguerre model is no longer living once the
dynamics are not real and near each other. We validate the
parametric reduction of the ARX-Laguerre with respect to the
ARX model on benchmark system manufactured by Feedback
known as Process Trainer PT326. Section 7 presents a possible
extension of the proposed model to a multivariable process.

849

is the vector containing the poles defining the orthonormal basis
Ii . Taking into account the stability condition (2), the ARX model
defined by the relation (1) can be written as
1
X

yðkÞ ¼

ha ðjÞyðkÀjÞ þ

j¼1

1
X

hb ðjÞuðkÀjÞ

ð4Þ

j¼1

with
(
ha ðjÞ ¼ 0
hb ðjÞ ¼ 0

if j 4 na
if j 4 nb

ð5Þ

and by substituting the relation (3) in the ARX model defined by
the relations (4) and (5), the model resulting from the description
of the linear ARX model by discrete orthonormal filters is given by
0
1
0
1
1
1
1
1
X
X
X a
X b
yðkÞ ¼
g n,a @
B ðj, x ÞyðkÀjÞA þ
g n,b @
B ðj, x ÞuðkÀjÞA
n

n¼0
1
X

¼

n¼0

n

a

n¼0

j¼1

g n,a xn,y ðk, x a Þ þ

1
X
n¼0

b

j¼1

g n,b xn,u ðk, x b Þ

ð6Þ

where xn,y and xn,u are respectively the output and the input
filtered by the nth orthonormal function, we have
1
X

xn,y ðk, x a Þ ¼

Ba ðj, x a ÞyðkÀjÞ ¼ Ba ðk, x a ÞnyðkÞ
n
n

ð7Þ

Bb ðj, x b ÞuðkÀjÞ ¼ Bb ðk, x b ÞnuðkÞ
n
n

ð8Þ

j¼1

xn,u ðk, x b Þ ¼

1
X
j¼1

where n denotes the convolution product. According to the
condition (2) characterizing the stability condition BIBO of the
ARX model, the infinite series (3) can be truncated to a finite order
Na and Nb respectively
ha ðjÞ ¼

NaÀ1
X
n¼0

g n,a Ba ðj, x a Þ,
n

hb ðjÞ ¼

NbÀ1
X
n¼0

g n,b Bb ðj, x b Þ
n

ð9Þ

and the relation (6) can be rewritten as
2. OFB expansion of the ARX coefficients

yðkÞ ¼

NaÀ1
X
n¼0

g n,a xn,y ðk, x a Þ þ

NbÀ1
X
n¼0

g n,b xn,u ðk, x b Þ

ð10Þ

A strictly causal discrete time system can be represented by
the following ARX model
yðkÞ ¼

na
X

ha ðjÞyðkÀjÞ þ

j¼1

nb
X

hb ðjÞuðkÀjÞ

ð1Þ

j¼1

where uðkÞ A R is the system input, yðkÞ A R is its output and ni
and hi(j) are respectively the orders and the model parameters for
i¼a, b such that na Z nb. Considering the stability condition BIBO
of the ARX model, a necessary and sufficient condition is that the
model parameters are absolutely summable such that
1
X

9hi ðjÞ9 o1,

i ¼ a,b

ð2Þ

j¼0

Therefore, these coefficients belong to the Lebesgue space ‘2 ½0,1½,
so that they can be represented by means of OFB which form an
orthonormal basis in such space. We propose to decompose ha ðjÞ
and hb ðjÞ on two independent OFB Ia ¼ fBa g1¼ 0 and Ib ¼ fBb g1¼ 0
n n
n n
with respect to the output and the input respectively as follows:
ha ðjÞ ¼

1
X
n¼0

g n,a Ba ðj, x a Þ,
n

hb ðjÞ ¼

1
X
n¼0

g n,b Bb ðj, x b Þ
n

ð3Þ

where Bi ðj, x i Þ, i¼a,b, is the nth orthonormal function and g n,a and
n
g n,b are the Fourier coefficients of ha ðjÞ and hb ðjÞ expansion and x i

3. ARX-Laguerre model
In the literature [1,10], the discrete Laguerre functions can be
obtained by the application of the procedure of Gram–Schmidt
orthonormalization to the sequence of following independent
functions:
nþ1 k

f n þ 1 ðk, xÞ ¼ k

x , for k ¼ 1; 2, . . . and n ¼ 0; 1, . . .

ð11Þ

where 9x9 o 1 is called the (real-valued) Laguerre pole. The
orthonormal Laguerre function ‘n ðk, xÞ of order n is defined by:
qffiffiffiffiffiffiffiffiffiffiffiffi X
n
2l þ kÀðn þ 1Þ
‘n ðk, xÞ ¼ 1Àx2
C ln C nþ kÀ1 ðÀ1Þl x
ð12Þ
l
l¼0

Cj
i

¼ i!=j!ðiÀjÞ! if i Zj and C j ¼ 0 else. The Z-transform of
with
i
‘n ðk, xÞ is:
qffiffiffiffiffiffiffiffiffiffiffiffi

2
1Àx 1Àxz n
, n ¼ 0; 1,2, . . .
ð13Þ
Ln ðzÞ ¼
zÀx
zÀx
Then, the linear model represented by discrete orthonormal filters
(10) can be developed on two independent Laguerre bases

850

a

b

Ia ¼ f‘n gNaÀ1 and Ib ¼ f‘n gNbÀ1 , where
n¼0
n¼0
n
2l þ kÀðn þ 1Þ
‘in ðk, xi Þ ¼ 1Àx2
C ln C nþ kÀ1 ðÀ1Þl xi
i
l

The identification between (19) and (21) provides:
ð14Þ

l¼0

Li ðzÞ ¼
n

ð15Þ

zÀxi

and xi , i¼a,b, is the pole of the independent Laguerre basis Ii
associated to the output if i¼a and to the input if i¼ b. The
resulting model obtained from the new representation (10) is
entitled ARX-Laguerre model:
yðkÞ ¼

NaÀ1
X

g n,a xn,y ðk, xa Þ þ

n¼0

NbÀ1
X

g n,b xn,u ðk, xb Þ

ð16Þ

n¼0

and from the relations (7) and (8) xn,y ðk, xa Þ and xn,u ðk, xb Þ are given
by
xn,y ðk, xa Þ ¼

1
X

‘a ðj, xa ÞyðkÀjÞ ¼ ‘a ðk, xa ÞnyðkÞ
n
n

ð17Þ

‘b ðj, xb ÞuðkÀjÞ ¼ ‘b ðk, xb ÞnuðkÞ
n
n

ð18Þ

j¼1

xn,u ðk, xb Þ ¼

1
X
j¼1

By assuming that the initial conditions are equal to zero and using
Z-transform, the relation (16) leads to
YðzÞ ¼

NaÀ1
X

g n,a X n,y ðzÞ þ

n¼0

¼

NaÀ1
X

NbÀ1
X

g n,b X n,u ðzÞ

n¼0

g n,a La ðzÞ Á YðzÞ þ
n

n¼0

NbÀ1
X

g n,b Lb ðzÞ Á UðzÞ
n

X n,y ðzÞ ¼ La ðzÞ Á YðzÞ,
n

X n,u ðzÞ ¼ Lb ðzÞ Á UðzÞ
n

ð20Þ

where X n,y ðzÞ, X n,u ðzÞ, Y(z) and U(z) are the Z-transforms of
xn,y ðk, xa Þ, xn,u ðk, xb Þ, y(k) and u(k) respectively. We note that
X n,y ðzÞ and X n,u ðzÞ can be obtained from relation (17) and (18).
In the same way, the ARX model (1) yields:
na
X

ha ðjÞzÀj YðzÞ þ

j¼1

nb
X

À1

À1

where Aðz Þ and Bðz
and nb respectively:
na
X

ha ðjÞzÀj ,

hb ðjÞzÀj UðzÞ

NbÀ1
X

g n,b Lb ðzÞ
n

ð26Þ

n¼0

Therefore, the development of the ARX model on Laguerre bases
leads to the decomposition of both polynomials AðzÀ1 Þ and BðzÀ1 Þ
on two independent orthonormal Laguerre bases.
3.1. Discrete-time ARX-Laguerre filters network
From the relation (15) the orthonormal functions defining
i
each of the independent Laguerre bases Ii ¼ f‘n ðj, xi ÞgNiÀ1 , i ¼ a,b,
n¼0
can be described in a recurrent way of the Z-transform as
8
2

1Àxi
i
L ðzÞ ¼
0
zÀxi
ð27Þ

i
L ðzÞ ¼ 1Àxi zL ðzÞ, n ¼ 1; 2, . . .
n
:
nÀ1
zÀxi
By substituting the previous expression of Li ðzÞ, i¼ a,b, in relation
n
(20), we can formulate the following recurrent relations for the
filters X n,y ðzÞ and X n,u ðzÞ, n ¼1,2,y:
8
2

1Àxa

X 0,y ðzÞ ¼

YðzÞ
zÀxa
ð28Þ

1Àxa z

X n,y ðzÞ ¼
X nÀ1,y ðzÞ
:
zÀxa
8
2

1Àxb

X 0,u ðzÞ ¼
UðzÞ
zÀxb

1Àxb z

X n,u ðzÞ ¼
X nÀ1,u ðzÞ
:
zÀxb

ð29Þ

Then, according to the relations (19), (28) and (29), we give in
Fig. 1 the discrete-time ARX-Laguerre filter network.

3.2. Recursive representation
ð21Þ

Þ are the two polynomials with degrees na

BðzÀ1 Þ ¼

j¼1

nb
X

hb ðjÞzÀj

ð22Þ

j¼1

From relations (9) and (14), we have:
#!
na
n
X NaÀ1
X
2
l n
l 2l þ jÀðn þ 1Þ Àj
À1
g n,a 1Àxa
C n C l þ jÀ1 ðÀ1Þ xa
z
Aðz Þ ¼
j¼1

BðzÀ1 Þ ¼

j¼1

¼ AðzÀ1 ÞYðzÞ þBðzÀ1 ÞUðzÞ

AðzÀ1 Þ ¼

g n,a La ðzÞ,
n

ð19Þ

n¼0

with

YðzÞ ¼

NaÀ1
X
n¼0


2
1Àxi 1Àxi z n
zÀxi

AðzÀ1 Þ ¼

n¼0

l¼0

According to the filter network of Fig. 1 we can establish the
following recurrent equations:
8
x ðk þ1Þ ¼ x x ðkÞ þ 1Àx2 yðkÞ
0,y
a 0,y

a


x1,y ðk þ1Þ ¼ ð1Àx2 Þx0,y ðkÞ þ xa x1,y ðkÞÀxa 1Àx2 yðkÞ

a
a

^
ð30Þ
nÀ2
X

xn,y ðk þ1Þ ¼ xa xn,y ðkÞ þ ð1Àx2 Þ
ðÀxa ÞnÀjÀ1 xj,y ðkÞ

a

j¼0


2
2
:
þ ð1Àxa ÞxnÀ1,y ðkÞ þ ðÀxa Þn 1Àxa yðkÞ

ð23Þ

nb
X
j¼1

BðzÀ1 Þ ¼

#!
NbÀ1
n
X
2
2l þ jÀðn þ 1Þ Àj
g n,b 1Àxb
C ln C nþ jÀ1 ðÀ1Þl xb
z
l
n¼0

l¼0

ð24Þ
and from (21) the transfer function G(z) is:
GðzÞ ¼

YðzÞ
BðzÀ1 Þ
¼
UðzÞ
1ÀAðzÀ1 Þ

ð25Þ

8
x ðk þ 1Þ ¼ x x ðkÞ þ 1Àx2 uðkÞ
0,u
b 0,u

b


2
x1,u ðk þ 1Þ ¼ ð1Àx Þx0,u ðkÞ þ xb x1,u ðkÞÀxb 1Àx2 uðkÞ

b
b

^
nÀ2
X

xn,u ðk þ 1Þ ¼ xb xn,u ðkÞ þ ð1Àx2 Þ
ðÀxb ÞnÀjÀ1 xj,u ðkÞ

b

j¼0


2
2
:
þ ð1Àx ÞxnÀ1,u ðkÞ þ ðÀxb Þn 1Àx uðkÞ
b

b

ð31Þ


851

Fig. 1. Discrete-time ARX-Laguerre filters network.

and the filters outputs xn,j , j ¼u, y, can be computed recursively as
follows:
XðkÞ ¼ AXðkÀ1Þ þ bu uðkÀ1Þ þ by yðkÀ1Þ

ð32Þ

with
X(k) the (Na þNb)-dimensional discrete vector contains the
filters outputs defined as:
XðkÞ ¼ ½x0,y ðkÞ, . . . ,xNaÀ1,y ðkÞ,x0,u ðkÞ, . . . ,xNbÀ1,u ðkÞŠT

ð33Þ
where for i¼a, b, T i ¼
dimension Ni.

A the (NaþNb)- dimensional matrix defined as:
!
Ay
0Na,Nb
A ¼ bloc diag½Ay Au Š ¼
0Nb,Na
Au

ð34Þ

where 0Nb,Na and 0Na,Nb are the null matrices of dimensions
(Nb Â Na) and (Na Â Nb) respectively and Ay and Au are two
square matrices of dimension Na and Nb respectively:
2
3
xa
0
ÁÁÁ 0
6
7
2
1Àxa
xa
ÁÁÁ 0 7
6
6
7
2
2
6
1Àxa
. . . 0 7 ð35Þ
Ay ¼ 6 Àxa ð1Àxa Þ
7
6
7
6
^
^
^ 7
4
5
2
2
ðÀxa ÞNaÀ1 ð1Àxa Þ ðÀxa ÞNaÀ2 ð1Àxa Þ Á Á Á xa
2
6
6
6
6
Au ¼ 6
6
6
4

0

xb
2
1Àxb

...

xb
2

...
2

Àxb ð1Àxb Þ

1Àxb

...

^

^

NbÀ1

ðÀxb Þ

2
ð1Àxb Þ

NbÀ2

ðÀxb Þ

2
ð1Àxb Þ

...

by and bu the (NaþNb)-dimensional vectors defined as:
2
3
2
3
1
0Na,1
6 Àx
7
6
7
1
7
7
6
6
a
6
7
6
7
2 7
6 ðÀxa Þ
6 Àxb
7
6
7
6
7
ð37Þ
by ¼ T a 6
7, bu ¼ T b 6
7
^
7
6
6 ðÀxb Þ2 7
7
7
6
6
6 ðÀxa ÞNaÀ1 7
6
7
^
4
5
4
5
0Nb,1
ðÀxb ÞNbÀ1

0

3

7
07
7
07
7
7
^ 7
5

xb

2
1Àxi and 0Ni,1 is the null vector of

Then, by defining the (Naþ Nb)-dimensional parameters vector c
containing all the Fourier coefficients g n,a for n ¼ 0, . . . ,NaÀ1 and
g n,b for n ¼ 0, . . . ,NbÀ1,
c ¼ ½g 0,a , . . . ,g NaÀ1,a ,g 0,b , . . . ,g NbÀ1,b ŠT

ð38Þ

the ARX-Laguerre model can be represented by the following
recursive representation:
(
XðkÞ ¼ AXðkÀ1Þ þ bu uðkÀ1Þ þ by yðkÀ1Þ
ð39Þ
yðkÞ ¼ cT XðkÞ
According to the recursive vector representation (39) the parameter vector c can be computed by minimizing the criterion Jab:
Jab ¼

H
X

ðym ðkÞÀcT XðkÞÞ2

ð40Þ

k¼1

ð36Þ
where ym(k) is the measured output data over a measurement
horizon H. As the ARX-Laguerre model is linear and the criterion
Jab is quadratic with respect to c respectively and its minimization

852


provides a global minimum. Therefore, the standard parameter
estimation methods like LMS or RLS methods can be used to
b
compute the estimated parameter vector c and the estimated
b
transfer function GðzÞ of the system as:
T
T
b
c
GðzÞ ¼ b ðIÀðA þ by b ÞzÀ1 ÞÀ1 bu zÀ1
c

ð41Þ

with I the identity matrix of dimension Na þNb.
4. Performance of the ARX-Laguerre model with respect to
the ARX model
The parameter number of the new black-box linear ARXLaguerre model is:
C ¼ Na þNb

ð42Þ

Then, comparing the relations (1) and (16), a significant reduction
of the parameter number is satisfied if Na þ Nbo na þnb. This
parametric reduction is due to the property of the completeness
of the orthonormal bases in the Lebesgue space ‘2 ½0,1½ where the
orthonormal function’s number depends essentially on the system’s dynamical behavior. This latter depends mainly on the
poles xa and xb characterizing each basis so that the provided
model better describes the linear system dynamics with small
truncating orders. An important issue concerns how to select the
Laguerre poles in order to minimize the truncating orders Na and
Nb i.e. the number of functions which provide a given approximation accuracy. This problem will be handled in this section
where we propose a theoretical study and a proof to highlight the
parametric reduction. Furthermore, we propose an alternative
approach to estimate both the Fourier coefficients and the two
Laguerre poles based on the resolution of an optimization
problem.

4.2. Proof of parameter number reduction
As stated above, the parameter number reduction is ensured
once Na o na and Nbo nb. In the following, we prove that if
Na Zna or Nb Znb, the parameter number is not increased. From
the relation (46), we note that for Ni Zni, i¼ a,b, the unknown
number Niþ1 characterizing the Fourier coefficients and the
Laguerre pole are superior to the equation number. Therefore,
we must add (Niþ1) À ni null equations to the equation system
(46) so that the equation number is equal to the unknown
number Niþ1. To conclude, we have the following condition from
the relation (5) of ARX model:
hi ðjÞ ¼ 0

From the relation (9), the coefficients ha and hb are expressed
in function of the Laguerre functions as follows:
ha ðjÞ ¼

NaÀ1
X

a

g n,a ‘n ðj, xa Þ,

n¼0

hb ðjÞ ¼

NbÀ1
X

b

g n,b ‘n ðj, xb Þ

ð43Þ

n¼0

By substituting the orthonormal functions of Laguerre by their
definition (14) in the relation (43), we can write:
!
NaÀ1
n
X
X l n
2l þ jÀðn þ 1Þ
g n,a
C n C l þ jÀ1 ðÀ1Þl xa
ha ðjÞ ¼ T a
ð44Þ
n¼0

hb ðjÞ ¼ T b

NbÀ1
X

l¼0

g n,b

n¼0

n
X

!
2l þ jÀðn þ 1Þ

C ln C nþ jÀ1 ðÀ1Þl xb
l

ð45Þ

l¼0

In this case, when the transfer function G(z) is known, i.e. the
orders na and nb as well as the coefficients ha ðjÞ, j¼1,y,na, and
hb(j), j¼ 1,y,nb, are known, we can deduce from the relations (44)
and (45) the equation system for each truncating order Na and Nb
associated with the orders na and nb respectively:
!
8
n
X
NiÀ1
h ð1Þ
2lÀn
Xg

ðÀ1Þl C ln C n xb
¼ i

l

n ¼ 0 n,i l ¼ 0
Ti

!
NiÀ1
X
n

X

h ð2Þ
2lÀn þ 1

g n,i
ðÀ1Þl C ln C nþ 1 xb
¼ i

l

Ti
n¼0

l¼0

^

^

!

NiÀ1
n
X
X
h ðniÞ

2l þ niÀðn þ 1Þ
l l n

g n,i
ðÀ1Þ C n C l þ niÀ1 xb
¼ i

:
Ti
n¼0

l¼0

i ¼ a,b

ð46Þ

if NiZ ni

ð47Þ

In this case, we obtain from (46) the following equation system
8Ni Zni:
!
8 NiÀ1
n
X
X
h ð1Þ

l l n 2lÀn

¼ i
g
ðÀ1Þ C n C l xb

Ti
n ¼ 0 n,i l ¼ 0

!
NiÀ1
X
n

X

h ð2Þ
2lÀn þ 1

g n,i
ðÀ1Þl C ln C nþ 1 xb
¼ i

l

Ti
n¼0

l¼0

^
i ¼ a,b

^

!
NiÀ1
n
X
X
h ðniÞ
ð48Þ
2l þ niÀðn þ 1Þ

g n,i
ðÀ1Þl C ln C nþ niÀ1 xb
¼ i
l

Ti
n¼0
l¼0

!

NiÀ1
n
X
X

2l þ niÀn

g n,i
ðÀ1Þl C ln C nþ ni xb
¼0

l

n¼0

l¼0

^

!
NiÀ1
X
n
X

2l þ NiÀn
l l n

g n,i
ðÀ1Þ C n C l þ Ni þ 1 xb
¼0

:
n¼0

4.1. Theoretical study

8j ¼ ni þ1, . . . ,Niþ 1

l¼0

Example, for Ni¼5 and ni¼ 3 we have six unknowns, the five
Fourier coefficients g 0, i , g 1, i , g 2, i , g 3, i , g 4, i and the Laguerre
pole xi :
8

g 0,i Àg 1,i xi þ g 2,i x2 Àg 3,i x3 þg 4,i x4 ¼ hi ð1Þ

i
i
i

Ti

2
3
g 0,i xi Àg 1,i ð2xi À1Þ þg 2,i ð3xi À2xi ÞÀg 3,i ð4x4 À3x2 Þ

i
i

h ð2Þ

þ g 4,i ð5x5 À4x3 Þ ¼ i

i
i

Ti

g x2 Àg ð3x3 À2x Þ þ g ð6x4 À6x2 þ 1Þ
0,i i
i

1,i
2,i
i
i
i

5
3
Àg 3,i ð10x À12x þ 3xi Þ þ g 4,i ð15x6 À20x4 þ6x2 Þ ¼ hi ð3Þ

i
i
i
i
i

Ti

g x3 Àg ð4x4 À3x2 Þ þ g ð10x5 À12x3 þ3x Þ

0,i i
i
1,i
2,i
i
i
i
i
ð49Þ
6
4
2
Àg 3,i ð20xi À30xi þ12xi À1Þ

þ g ð35x7 À60x5 þ 30x3 À4x Þ ¼ 0

i
4,i
i
i
i

g x4 Àg ð5x5 À4x3 Þ þ g ð15x6 À20x4 þ6x2 Þ
0,i i

1,i
2,i
i
i
i
i
i

7
5
3

Àg 3,i ð35xi À60xi þ 30xi À4xi Þ

þ g 4,i ð70x8 À140x6 þ 90x4 À20x2 þ 1Þ ¼ 0

i
i
i
i

0,i i
1,i
2,i
i
i
i
i
i

Àg ð56x8 À105x6 þ 60x4 À10x2 Þ

3,i
i
i
i
i

: þ g ð126x9 À280x7 þ210x5 À60x3 þ 5x Þ ¼ 0
i
4,i
i
i
i
i
Therefore, in the system (48) the (Niþ1)th equation is always null
if NiZ ni:
!
NiÀ1
n
X
X
2l þ NiÀn
g n,i
ðÀ1Þl C ln C nþ Ni þ 1 xi
¼0
ð50Þ
l
n¼0

l¼0

From (50) we note that the smallest value of the exponent of xi is
equal to 1 and is obtained when n ¼NiÀ 1 and l ¼0. Therefore, the


polynomial of the left hand side of (50) does not contain a
constant term and the solution of this equation is xi ¼ 0. We
obtain:

xi ¼ 0 8NiZ ni

ð51Þ

Consequently, the Laguerre base is reduced to the FIR base, the
ARX-Laguerre model becomes the ARX model and the Fourier
coefficients are given by 8NiZ ni:
(
hi ðjÞ ¼ g jÀ1,i j ¼ 1, . . . ,ni
i ¼ a,b
ð52Þ
g jÀ1,i ¼ 0 j 4ni

853

to the classic case of the Laguerre model. However, it is easy to
identify the dynamics of the system from the model ARXLaguerre by determining the estimated transfer function from
the relation (41) and computing its poles.

5. Laguerre model with filters on model input
In the literature, the impulse response h(k) and the transfer
function G(z) of a stable LTI system are linear combinations of the
Laguerre orthonormal functions:

To conclude, we confirm that the ARX-Laguerre coefficient number is always smaller or equal but never bigger than the number
of coefficients engaged in the classical ARX model.

hðkÞ ¼

4.3. Optimization problem

where gn are the Fourier coefficients. In practice, the infinite series
in relation (56) are truncated at an order N and the Laguerre
model is given by:

As the ARX model is well-known, the systems (46) or (48) can
be solved to provide the values of Fourier coefficients and
Laguerre poles without collecting system observations. Since the
equations of both systems are non-linear with respect to the pole
xi , the resolution can be transformed into an optimization
problem for i¼ a,b as follows:

qffiffiffiffiffiffiffiffiffiffiffiffi NiÀ1
P
n
X
X l
X
2
min
hi ðjÞÀ 1Àxi
g n,i
Cn
g 0,i ,...,g NiÀ1,i , xi

n¼0

1
X

g n ‘n ðk, xÞ,

GðzÞ ¼

n¼0

NÀ1
X

yðkÞ ¼

1
X

g n Ln ðzÞ

ð56Þ

n¼0

g n xn ðkÞ

ð57Þ

n¼0

with xn (k) the input filtered by the nth orthonormal function:
xn ðkÞ ¼

1
X

‘n ðj, xÞuðkÀjÞ

ð58Þ

j¼0

with respect to

From Malti [5] the application of the bilinear transformation
computes directly the Laguerre coefficients when the transfer
function G(z) is known. Indeed, by applying the bilinear transformation to the relation (56) defined by this variable change:

9xi 9 o 1

wÀ1 ¼

j¼1

2l þ jÀðn þ 1Þ
C nþ jÀ1 ðÀ1Þl xi
l

l¼0

i2

ð53Þ

where P¼ni if Nio ni or P¼Niþ1 if NiZ ni with hi(j) ¼0 for
j ¼ni þ1,y,Niþ 1.
This problem can be solved by numerical optimization methods such as Pattern Search (PS) method [2]. Furthermore, according to the relations (23), (24) and (26), describing the
decomposition of the polynomials AðzÀ1 Þ and BðzÀ1 Þ respectively
on the independent Laguerre bases, the problem (53) can be
written:
For i ¼ a:

2

#2
NaÀ1
P
NaÀ1

X
X
X

À1

a
Aðz ÞÀ
g n,a La ðzÞ
¼
ha ðjÞÀ
g n,a ‘n ðj, xa Þ

n

n¼0
n¼0
j¼1

#
qffiffiffiffiffiffiffiffiffiffiffiffi NaÀ1
P
n
2
X
X l n
X
2
2l þ jÀðn þ 1Þ
¼
ha ðjÞÀ 1Àxa
g n,a
C n C l þ jÀ1 ðÀ1Þl xa
n¼0

j¼1

l¼0

ð54Þ
Pna

PP

such as Aðz Þ ¼ j ¼ 1 ha ðjÞz if Na o na or Aðz Þ ¼ j ¼ 1 ha ðjÞzÀj
if Na Z na with ha ðjÞ ¼ 0 for j¼ naþ1,y,P.
For i ¼ b:

2

#2

NbÀ1
P
NbÀ1
X
X
X

À1

b
g n,b Lb ðzÞ
¼
hb ðjÞÀ
Bðz ÞÀ

n

n¼0
n¼0
j¼1

#
qffiffiffiffiffiffiffiffiffiffiffiffi NbÀ1
P
n
2
X
X l n
X
2
2l þ jÀðn þ 1Þ
¼
hb ðjÞÀ 1Àxb
g n,b
C n C l þ jÀ1 ðÀ1Þl xb
À1

Àj

n¼0

j¼1

À1

l¼0

ð55Þ
Pnb

PP

such as Bðz Þ ¼ j ¼ 1 hb ðjÞz if Nb onb or Bðz Þ ¼ j ¼ 1 hb ðjÞzÀj
if Nb Znb with hb ðjÞ ¼ 0 for j ¼nbþ 1,y, P.
From the relations (54) and (55), the minimization of the
criterion of the optimization problem (53) leads to decompose the
polynomials AðzÀ1 Þ and BðzÀ1 Þ on the Laguerre orthonormal bases.
So, the computed values of the Laguerre poles xa and xb do not
give any information about the dynamics of the system contrary
À1

Àj

À1

1Àxz
1 þ xwÀ1
3z ¼
zÀx
wÀ1 þ x

we obtain:

1
X
1 þ xwÀ1
2
g n wÀn
¼ ðwÀ1 þ xÞ
1Àx G
wÀ1 þ x
n¼0

#
1
X
1
À1
Àk
) ðw Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi xg 0 þ
ðg kÀ1 þ xg k Þw
2
k¼1
1Àx

ð59Þ

ð60Þ

with ðwÀ1 Þ ¼ Gðð1 þ xwÀ1 Þ=ðwÀ1 þ xÞÞ. Therefore, according to (60)
a linear relation exists between the Laguerre coefficients and
those of the developed function ðwÀ1 Þ in series of wÀ1 . So, a term
to term identification permits the direct calculation of the Fourier
coefficients. In this case, we determine the expression of ðwÀ1 Þ
by applying the bilinear transformation (59) to G(z), which is
followed by a polynomial division according to the powers of wÀ1
between the numerator and the denominator of ðwÀ1 Þ. This
identification approach of Fourier coefficients is used in the
following simulation section.

6. Simulation results
6.1. Numerical simulations
In this section, we present a comparative study between the
performances of the ARX-Laguerre model and the Laguerre model.
This comparative study is illustrated through three different and
strictly causal systems (Damped system, oscillating system and
highly oscillating system) where each ARX model is represented
by its associated transfer function. The performances of the two
models are evaluated by the following criteria:
The criterion Ja for the ARX-Laguerre model characterizing the
estimation of the decomposition of the polynomial AðzÀ1 Þ on

854


the Laguerre basis
Ja ¼

na
X

NaÀ1
X

!2

1
a
ha ðjÞÀ
g n,a ‘n ðj, xa Þ
na j ¼ 1
n¼0

ð61Þ

The Laguerre pole computed by the Fu and Dumont algorithm
is x ¼ 0:63. For the bilinear transformation, the variable change
(59) is written
wÀ1 ¼

The criterion Jb for the ARX-Laguerre model characterizing the
estimation of the decomposition of the polynomial BðzÀ1 Þ on
the Laguerre basis
!2
nb
NbÀ1
X
1 X
b
Jb ¼
hb ðjÞÀ
ð62Þ
nb j ¼ 1
n¼0

1À0:63z
1 þ 0:63wÀ1
3z ¼
zÀ0:63
wÀ1 þ0:63

Then, the bilinear transformation of G1(z) is given by
ðwÀ1 Þ ¼

À36:7647wÀ2 þ 101:4951wÀ1 þ 78:5338
wÀ2 À7:3135wÀ1 þ 12:8255

and it can be rewritten in series of wÀ1 as follows:
ðwÀ1 Þ ¼ 6:1233þ 11:4052wÀ1 þ 3:1596wÀ2
þ0:9125wÀ3 þ 0:274wÀ4 þ0:0851wÀ5

The PRR (Parametric Reduction Rate) of the model (Laguerre
or ARX-Laguerre) given by

N or ðNa þ NbÞ
ð63Þ
PRR ¼ 1À
na þ nb
The NMSE (Normalized Mean Squared Error) given by
PH
2
¼ 1 ½y ðkÞÀyðkÞŠ
NMSE ¼ kPH m
2
k ¼ 1 ½ym ðkÞŠ

ð64Þ

We represent, in Fig. 2, the sequence of gaussian input used to
generate the output signal ym to compute the NMSE over
H¼1000 measurements.
We specify that for the identification phase of the models, the
Laguerre poles and the Fourier coefficients are computed by
exploiting only the transfer function G(z) as:
For the ARX-Laguerre model: we apply the resolution of the
optimization problem (53).
For the Laguerre model:
J we apply the Fu and Dumont algorithm [3] to compute the
Laguerre pole.
J we apply the bilinear transformation of G(z) and the
identification term to term in the relation (60) to compute
the Fourier coefficients.
Example 1 (Damped system). We consider the ARX model with
4 parameters given by the following transfer function in Malti [5]:
G1 ðzÞ ¼

3
2
5zÀ3:9
þ
¼ 2
zÀ0:75 zÀ0:8
z À1:55z þ0:6

i.e. na¼nb ¼2
hb ð2Þ ¼ À3:9.

with ha ð1Þ ¼ 1:55,ha ð2Þ ¼ À0:6 and hb ð1Þ ¼ 5,

2
1.5

Amplitude

1
0.5

þ0:0272wÀ6 þ0:0088wÀ7 þ 0:0029wÀ8
þ0:001wÀ9 þ 0:0003wÀ10 þ Á Á Á
By using (60), we compute the Fourier coefficients of the Laguerre
model and we obtain the following expansion of G1(z) on Laguerre
basis:
G1 ðzÞ ¼ 7:5481L0 ðzÞ þ 2:078L1 ðzÞ þ 0:5964L2 ðzÞ
þ 0:1781L3 ðzÞ þ0:055L4 ðzÞ þ 0:0175L5 ðzÞ
þ 0:0057L6 ðzÞ þ 0:0019L7 ðzÞ
þ 0:0006L8 ðzÞ þ 0:0002L9 ðzÞ þ Á Á Á
Since na¼nb ¼2 we keep two values of the truncating orders Na,
Nb¼1, 2. The equation systems obtained for Ni, i¼a,b, from (46)
or (48) are summarized as
For Ni ¼ 1

For Ni ¼ 2

8

g 0,i ¼ hi ð1Þ

Ti

g x ¼ hi ð2Þ
0,i i
:
Ti

8

g 0,i Àg 0,i xi ¼ hi ð1Þ

Ti

hi ð2Þ
2
g 0,i xi þ g 1,i ½1À2xi Š ¼ T

i

: g x2 þg ½2xi À3x3 Š ¼ 0
0,i i
1,i
i

The results of the resolution of the optimization problem (53) are
summarized in Table 1.
We summarize in Tables 2 and 3 the performances of the
Laguerre model and the ARX-Laguerre model respectively.
We note that for the representation of a damped system
the performances of the Laguerre and ARX-Laguerre models
are similar in terms of the NMSE and the PRR. The values of the
criteria Ja and Jb characterizing the estimation degrees of the
decomposition of the polynomials AðzÀ1 Þ and BðzÀ1 Þ on the
independent Laguerre bases tend to zeros. Then, an optimal
approximation of the ARX model by the ARX-Laguerre model is
guaranteed which explains the lower values of the NMSE.
Furthermore, according to Table 1 we can verify the relation
(52) about the parameter number reduction of the ARX-Laguerre
model with respect to the ARX model. To validate the obtained
results in Tables 2 and 3 by using the input signal of Fig. 2, we
draw in Fig. 3 the outputs of the ARX model and the ARX-Laguerre
Table 1
Example 1: parametric identification of the ARXLaguerre model.

0
−0.5

Truncating orders
Na¼ 1
Nb¼1

−1
−1.5
−2

Na ¼2
Nb¼2

Laguerre poles

0

100

200

300

400 500 600 700
Number of iterations

800

900 1000

Fig. 2. Sequence of gaussian input u(k) on a window of measures H ¼1000.

xa ¼ À0:3873
xb ¼ À0:78

xa ¼ 0
xb ¼ 0

Fourier coefficients
[1.6812, 7.9905]

[1.55, À 0.6, 5, À 3.9]


40

Table 2
Example 1: performances of the Laguerre model.
PRR (%)

1
2
3
4

75
50
25
0.0

30

NMSE (%)
11.631 Â 10 À 1
3.07 Â 10 À 3
5.9 Â 10 À 5
0.0

20
Amplitude

N

Nb

1
1
2
2

1
2
1
2

PRR (%)
50
25
25
0.0

NMSE (%)
1.3752 Â 10
0.3068 Â 10 À 5
0.0
0.0

0
−10
ARX model
ARX − Laguerre model

−30
Ja

À4

10

−20

Table 3
Example 1: performances of the ARX-Laguerre model.
Na

855

Jb

0.0
0.0
0.0
0.0

0.0
0.0
0.0
0.0

−40

0

100

200

300

400 500 600 700

800

900 1000

Fig. 3. Example 1: validation of the ARX-Laguerre model, Na ¼1 and Nb¼ 1.

obtained with truncating orders Na ¼1 and Nb¼1. We plot in
Fig. 4 the outputs of ARX model and the Laguerre model with a
truncating order N¼ 2.

40

Example 2 (Oscillating System). We consider the ARX model
with six parameters represented by the following transfer
function:

20

¼

ðzÀ2:67ÞðzÀ0:82Þ
ðz þ 0:5ÞðzÀ0:7ÞðzÀ0:9Þ

Amplitude

À0:6z2 þ 2:094zÀ1:314
G2 ðzÞ ¼ 3
z À1:1z2 À0:17z þ 0:315

30

10
0
−10
−20

i.e. na¼nb ¼3 with ha ð1Þ ¼ 1:1, ha ð2Þ ¼ 0:17, ha ð3Þ ¼ À0:315 and
hb ð1Þ ¼ À0:6, hb ð2Þ ¼ 2:094, hb ð3Þ ¼ À1:314.
The Laguerre pole computed by the Fu and Dumont algorithm
is x ¼ 0:15. Then, the bilinear transformation of G2(z) is given by:
ðwÀ1 Þ ¼

À3:778wÀ3 þ 5:277wÀ2 À0:299wÀ1 À0:176
wÀ3 À1:126wÀ2 À2:721wÀ1 þ 3:103

ARX model
Laguerre model

−30
−40

0

100

200

300

400 500 600 700

800

900 1000

Fig. 4. Example 1: validation of the Laguerre model, N ¼ 2.

and by using (60) we obtain:
G2 ðzÞ ¼ À0:3748L0 ðzÞ þ 1:5330L1 ðzÞ þ 0:0048L2 ðzÞ
þ0:6814L3 ðzÞ þ 0:1054L4 ðzÞ þ0:3383L5 ðzÞ
þ0:1154L6 ðzÞ þ 0:19L7 ðzÞ þ 0:0995L8 ðzÞ
þ0:0421L15 ðzÞ þ 0:0344L16 ðzÞ þ Á Á Á

Since na¼nb ¼3 we keep three values of the truncating orders Na,
Nb¼1, 2, 3. The equation systems obtained for Ni, i¼ a,b, from (46)
or (48) are summarized as:
For Ni ¼ 1

For Ni ¼ 2

8

g ¼ hi ð1Þ
0,i

Ti

h ð2Þ
g 0,i xi ¼ i
Ti

hi ð3Þ

2
g 0,i x ¼

i
:
Ti

8

g Àg x ¼ hi ð1Þ
0,i 1,i i

Ti

h ð2Þ
2
g 0,i xi Àg 1,i ð2xi À1Þ ¼ i
Ti

2
3
g 0,i x Àg 1,i ð3x À2xi Þ ¼ hi ð3Þ

i
i
:
Ti

For Ni ¼ 3
8

g Àg x þ g x2 ¼ hi ð1Þ
0,i 1,i i
2,i i

Ti

g x Àg ð2x2 À1Þ þ g ð3x3 À2x Þ ¼ hi ð2Þ
0,i i 1,i
i
2,i
i
i
Ti

2
3
4
2
g 0,i x Àg 1,i ð3x À2xi Þ þg 2,i ð6x À6x þ 1Þ ¼ hi ð3Þ

i
i
i
i

Ti

g x3 Àg ð4x4 À3x2 Þ þ g ð10x5 À12x3 þ3x Þ ¼ 0
: 0,i i
i
1,i
2,i
i
i
i
i
The resolution of the problem (53) yields to the results are
Tables 5 and 6 reveal an important reduction of the parametric
complexity of the ARX - Laguerre model with respect to the
Laguerre model without altering the quality of the approximation
deﬁned by the NMSE. Indeed, for the Laguerre model the same
performance in terms of NMSE with respect to the ARX-Laguerre
model is guaranteed by increasing the truncating order N. Therefore, it is clear that the ARX-Laguerre model is adequate to
the representation of oscillating systems with a remarkable

856


Table 4
Example 2: parametric identification of the ARX-Laguerre model.
Truncating orders

Na ¼ 1
Nb¼ 1

Na ¼2
Nb¼ 2

Na ¼ 3
Nb¼ 3

Laguerre poles

xa ¼ 0:097
xb ¼ 0:7256

xa ¼ À0:4017
xb ¼ À0:3291

xa ¼ 0
xb ¼ 0


[1.1088, 0.1837]

[0.8811, 0.7971, À 1.3769, 2.2521]

[1.1, 0.17, À 0.315, À 0.6, 2.094, À 1.3136]

8

Table 5

6

PRR (%)

NMSE (%)

N

PRR (%)

NMSE (%)

1
2
3
4
5
6
7
8

83.33
66.66
50
33.33
16.66
0
–
–

96.1206
43.304
43.1219
21.011
18.3215
10.9226
8.8475
5.9037

9
10
11
12
13
14
15
16

–
–
–
–
–
–
–
–

4.6032
3.2590
2.4934
1.8192
1.3791
1.0214
0.7709
0.5748

4
2
Amplitude

N

0
−2
−4
−6
ARX model

−8
−10

Table 6
Na

Nb

PRR (%)

NMSE (%)

Ja

Jb

1
1
2
2
1
2
3
3
3

1
2
1
2
3
3
1
2
3

66.66
50
50
33.33
33.33
16.66
33.33
16.66
0.0

9.2479
7.076
0.672
8.0474 Â 10 À 5
7.0398
1.1177 Â 10 À 4
0.6728
0.0
0.0

0.0366
0.0366
0.0
0.0
0.0366
0.0
0.0
0.0
0.0

2.1467
0.0
2.1467
0.0
0.0
0.0
2.1467
0.0
0.0

0

100

200

300

400 500 600 700

800

900 1000


The Laguerre pole computed by the Fu and Dumont
algorithm is x ¼ À0:32 and the bilinear transformation of G3(z)
is given by:
ðwÀ1 Þ ¼
17:0986wÀ5 À88:7453wÀ4 þ 258:3915wÀ3 À397:5374wÀ2 þ 264:71wÀ1 À51:5931
wÀ5 À15:2194wÀ4 þ 26:1620wÀ3 þ 19:9369wÀ2 À69:7422wÀ1 þ 38:2186

parametric reduction compared to the Laguerre and ARX models.
These performances of the ARX-Laguerre model is explained by
the fact that the approximation of the polynomials AðzÀ1 Þ and
BðzÀ1 Þ of the ARX model on the Laguerre bases is guaranteed. In
fact, this result is confirmed by the lower values of the criteria Ja
and Jb. According to Tables 5 and 6, we illustrate in Figs. 5 and 6,
the evolution of the outputs of ARX-Laguerre and Laguerre
models respectively for the same number of parameters equals
to 3. We note the correspondence between outputs in Fig. 5 and
the discordance in Fig. 6.
Example 3 (Highly oscillating system). We consider the ARX
model with ten parameters represented by the following transfer
function:
3:8z4 À3:71z3 þ 0:3995z2 À0:2439z þ 0:254
G3 ðzÞ ¼
1À0:85z5 À0:555z4 þ 0:616z3 À0:0733z2 À0:0612
¼

ðzÀ0:82ÞðzÀ0:5Þðz þ 0:17Àj0:35Þðz þ 0:17 þj0:35Þ
ðzÀ0:9ÞðzÀ0:5Àj0:3ÞðzÀ0:5 þj0:3Þðz þ0:8Þðz þ 0:25Þ

ha ð2Þ ¼ 0:555, ha ð3Þ ¼
i.e.
na¼nb ¼5
with
ha ð1Þ ¼ 0:85,
À0:616, ha ð4Þ ¼ 0:0733, ha ð5Þ ¼ 0:0612 and hb ð1Þ ¼ 3:8, hb ð2Þ ¼
À3:71, hb ð3Þ ¼ 0:3995, hb ð4Þ ¼ À0:2439, hb ð5Þ ¼ 0:254.

By using (60), we obtain:
G3 ðzÞ ¼ 3:9968L0 ðzÞÀ0:723L1 ðzÞ þ 2:3405L2 ðzÞÀ0:1521L3 ðzÞ
þ 1:012L4 ðzÞÀ0:0687L5 ðzÞ þ 0:4018L6 ðzÞÀ0:0454L7 ðzÞ
þ 0:1615L8 ðzÞÀ0:0105L9 ðzÞ þ 0:0896L10 ðzÞ
þ 0:03L11 ðzÞ þ 0:0813L12 ðzÞ þ 0:0649L13 ðzÞ þ Á Á Á

Since na¼nb ¼5 we keep five values of the truncating orders Na,
Nb¼1, 2, 3, 4, 5. The equation systems obtained for Ni, i¼a, b,
from (46) or (48) are summarized as:
For Ni ¼ 1

8

g 0,i ¼ hi ð1Þ

Ti

g x ¼ hi ð2Þ
0,i i

Ti

h ð3Þ
2
g 0,i xi ¼ i
Ti

g 0,i x3 ¼ hi ð4Þ

i

Ti

g x4 ¼ hi ð5Þ
0,i i
:
Ti

For Ni ¼ 2

8


Ti

g x Àg ð2x2 À1Þ ¼ hi ð2Þ
0,i i 1,i

i

Ti

h ð3Þ
2
3
g 0,i xi Àg 1,i ð3xi À2xi Þ ¼ i
Ti

g 0,i x3 Àg 1,i ð4x4 À3x2 Þ ¼ hi ð4Þ

i
i
i

Ti

g x4 Àg ð5x5 À4x3 Þ ¼ hi ð5Þ
0,i i
:
1,i
i
i
Ti


857

For Ni ¼ 3

For Ni ¼ 5

8

g 0,i Àg 1,i xi þg 2,i x2 ¼ hi ð1Þ

i

Ti

g x Àg ð2x2 À1Þ þ g ð3x3 À2x Þ ¼ hi ð2Þ

i
2,i
0,i i 1,i
i
i

Ti

h ð3Þ
2
3
4
2
g 0,i xi Àg 1,i ð3xi À2xi Þ þ g 2,i ð6xi À6xi þ 1Þ ¼ i
Ti

g 0,i x3 Àg 1,i ð4x4 À3x2 Þ þg 2,i ð10x5 À12x3 þ 3xi Þ ¼ hi ð4Þ

i
i
i
i
i

Ti

g x4 Àg ð5x5 À4x3 Þ þg ð15x6 À20x4 þ 6x2 Þ ¼ hi ð5Þ
0,i i
:
1,i
2,i
i
i
i
i
i
T

8
g Àg x þ g x2 Àg x3 þg x4 ¼ hi ð1Þ
0,i 1,i i

2,i i
3,i i
4,i i

Ti

g x Àg ð2x2 À1Þ þ g ð3x3 À2x Þ
0,i i 1,i

i
2,i
i
i

h ð2Þ

4
2
5
3

Àg 3,i ð4xi À3xi Þ þ g 4,i ð5xi À4xi Þ ¼ i

Ti

g 0,i x2 Àg 1,i ð3x3 À2xi Þ þg 2,i ð6x4 À6x2 þ 1Þ

i
i
i
i

Àg ð10x5 À12x3 þ 3x Þ þ g ð15x6 À20x4 þ6x2 Þ ¼ hi ð3Þ

i
3,i
4,i
i
i
i
i
i

Ti

g x3 Àg ð4x4 À3x2 Þ þ g ð10x5 À12x3 þ3x Þ

0,i i
i
1,i
2,i
i
i
i
i

6
4
2
Àg 3,i ð20xi À30xi þ12xi À1Þ

þ g 4,i ð35x7 À60x5 þ30x3 À4xi Þ ¼ hi ð4Þ

i
i
i

Ti

0,i i

1,i
2,i
i
i
i
i
i

Àg 3,i ð35x7 À60x5 þ 30x3 À4xi Þ

i
i
i

þ g ð70x8 À140x6 þ 90x4 À20x2 þ1Þ ¼ hi ð5Þ

4,i
i
i
i
i

Ti

0,i i

1,i
2,i
i
i
i
i
i

Àg ð56x8 À105x6 þ 60x4 À10x2 Þ

3,i
i
i
i
i

: þ g ð126x9 À280x7 þ 210x5 À60x3 þ 5x Þ ¼ 0

i

For Ni ¼ 4
8
g Àg x þg x2 Àg x3 ¼ hi ð1Þ
0,i 1,i i

2,i i
3,i i

Ti

2
3
g x Àg ð2x À1Þ þ g ð3x À2x ÞÀg ð4x4 À3x2 Þ ¼ hi ð2Þ
0,i i 1,i
i
2,i
3,i

i
i
i
i

Ti

g x2 Àg ð3x3 À2xi Þ þ g ð6x4 À6x2 þ 1Þ
0,i i
1,i
2,i
i
i
i

Àg ð10x5 À12x3 þ3x Þ ¼ hi ð3Þ

i
3,i
i
i
Ti

g x3 Àg ð4x4 À3x2 Þ þg ð10x5 À12x3 þ 3x Þ
0,i i

i
1,i
2,i
i
i
i
i

hi ð4Þ

6
4
2

Àg 3,i ð20xi À30xi þ 12xi À1Þ ¼

Ti

g 0,i x4 Àg 1,i ð5x5 À4x3 Þ þg 2,i ð15x6 À20x4 þ 6x2 Þ

i
i
i
i
i
i

Àg ð35x7 À60x5 þ30x3 À4x Þ ¼ hi ð5Þ

:
i
3,i
i
i
i
Ti

8
6
4

Amplitude

2
0
−2
−4
−6

ARX model
Laguerre model

−8
−10

0

100

200

300

400 500 600 700

800

900 1000

Fig. 6. Example 2: validation of the Laguerre model, N¼ 3.

4,i

i

i

i

i

i

The results of the resolution of the optimization problem (53) are
For such a system, the challenge of the ARX-Laguerre model is
clearly emphasized. Indeed, from Tables 8 and 9 the ARX-Laguerre
model guarantees a significant parametric reduction compared to
the Laguerre model by assuring a better representation of the
system. These results are expressed in terms of NMSE and PRR,
which illustrates an important reduction contrary to the Laguerre
model. This latter must increase the parameter number in order
to achieve the performances of the ARX-Laguerre model. Then,
the Laguerre model is handicapped by the overparametrization
problem. These results confirm the assumption stated above that
is the Laguerre model remains relevant only for damped systems
the poles of which are neighboring. The performances of the ARXLaguerre model are obtained according to the decomposition of
the polynomials AðzÀ1 Þ and BðzÀ1 Þ on the independent Laguerre
orthonormal bases. This is confirmed by the small values of
criteria Ja and Jb. These interpretations show the efficiency of
the proposed identification approach of the parameters (Fourier
coefficients and Laguerre poles) of the ARX-Laguerre model.
Moreover, the simulations prove that if NiZ ni, we have xi ¼ 0,
i¼a,b, and the NMSE is almost zero. These results coincide with
the theoretical demonstration of the relation (52).
According to Tables 8 and 9, we illustrate in Figs. 7 and 8, the
evolution of the outputs of ARX-Laguerre and Laguerre models
respectively for the same number of parameters is equal to 4. We

Table 7
Example 2: parametric identification of the ARX-Laguerre model.
Truncating
orders

Na ¼2

Na ¼ 3

Na ¼4

Na ¼5

Nb¼ 1
Laguerre
poles

Na ¼ 1

Nb¼2

Nb¼ 3

Nb¼4

Nb¼5

xa ¼ 0:2548

xa ¼ À0:292

xa ¼ À0:0357

xa ¼ 0:1128

xa ¼ 0

xb ¼ À0:5409 xb ¼ À0:0617
Fourier
[0.9212, 2.5]
coefficients

[0.6063, 0.9897,
4.0228, À 3.4927]

xb ¼ À0:0616

xb ¼ À0:2234

xb ¼ 0

[0.8289, 0.6277, À 0.5776,
4.0226, À 3.4929, À 0.0002]

[0.8989, 0.3, À 0.6489, 0.3064, 4.5345,
À 2.6348, À 0.8221, À 0.5849]

[0.85, 0.555, À 0.616, 0.0733, 0.0612, 3.8,
À 3.71, 0.3995, À 0.2439, 0.254]

858


10

Table 8
PRR (%)

NMSE (%)

N

PRR (%)

NMSE (%)

1
2
3
4
5
6
7
8
9

90
80
70
60
50
40
30
20
10

93.0263
57.4786
29.5694
14.8041
8.0585
7.6589
6.0988
5.8714
5.3552

10
11
12
13
14
15
16
17
18

0
–
–
–
–
–
–
–
–

5.3052
4.9331
4.4287
4.7951
4.1456
3.7365
3.2735
2.5289
1.8918

5

Amplitude

N

0
−5
−10
−15

Table 9
Nb

PRR (%)

NMSE (%)

Ja

1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
5

1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
5

90
70
60
50
70
60
50
40
60
50
40
30
50
40
30
20
40
30
20
10
0.0

18.5417
7.8916
7.891
7.8681
12.4052
5.6443
5.6402
5.6409
0.7905
0.5945
0.5946
0.5731
0.6856
0.0058
0.0058
0.0012
0.674
0.0064
0.0064
0.0
0.0

0.114
0.114
0.114
0.114
0.0147
0.0147
0.0147
0.0147
0.3105
0.3105
0.3105
0.3105
9.8915 Â 10 À 5
9.8915 Â 10 À 5
9.8915 Â 10 À 5
9.8915 Â 10 À 5
0
0.0
0.0
0.0
0.0

1.9118
0.0212
0.0212
0.0
1.9118
0.0212
0.0212
0.0
1.9118
0.0212
0.0212
0.0
1.9118
0.0212
0.0212
0.0
1.9118
0.0212
0.0212
0.0
0.0

0

100

200

300

Jb

note the correspondence between outputs in Fig. 7 and the
discordance in Fig. 8.

400 500 600 700

800

900 1000


10
5

Amplitude

Na

ARX model

0
−5
−10
−15

ARX model
Laguerre model
0

100

200

300

400 500 600 700

800

900 1000

Fig. 8. Example 3: validation of the Laguerre model, N ¼ 4.

6.2. Application to the benchmark: Feedback’s Process Trainer PT326
The Feedback’s Process Trainer PT326 is a benchmark system for
identiﬁcation as described by Ljung [4]. The device’s function is like
a hair dryer: air is fanned through a tube and heated at the inlet. The
air temperature is measured by a thermocouple at the outlet. The
input u is the voltage over a mesh of resistor wires to heat the
incoming air; the output T is the outlet air temperature (Fig. 9).
Three hundred input/output observations were collected from
the process at a sampling time of 0.08 s. The process input u was
chosen to be a binary random signal shifting between 3.41 V and
6.41 V. Figs. 10 and 11 show the output temperature T and the
input voltage u respectively.
The benchmark can be described by the ARX model (1) with na¼5
and nb¼5 which ensures that NMSE¼0.0255%. This model contains
ten parameters which can be estimated using least-square methods
and we obtained the following polynomials AðzÀ1 Þ and BðzÀ1 Þ:
AðzÀ1 Þ ¼ 1:014zÀ1 À0:07561zÀ2 À0:0295zÀ3 À0:1129zÀ4 þ0:05486zÀ5
BðzÀ1 Þ ¼ À0:0002191zÀ1 þ0:002143zÀ2 þ0:06647zÀ3 þ 0:05827zÀ4 þ 0:0154zÀ5

From the polynomial BðzÀ1 Þ we notice that the system delay is d¼ 2.
Elsewhere, the writing of polynomial AðzÀ1 Þ raises an oscillating
system as it contains two complex poles and one negative real pole.
Therefore, its description by a classical Laguerre model leads to a

u (V)
(Voltage)

Feedback's
Process Trainer
PT326

T (°C)
(Temperature)

Fig. 9. Block system of the Benchmark.

vanishing model. However, its description by an ARX-Laguerre model
with two combinations of truncating orders (Na¼Nb¼1) and
(Na¼Nb¼2) supplies a perfect description. In fact, for Ni, i¼a,b, the
equation systems are summarized from (46) as:
For Ni ¼ 1
8

g 0,i ¼ hi ð1Þ

Ti

g x ¼ hi ð2Þ
0,i i

Ti

hi ð3Þ
2
g 0,i xi ¼
Ti

g 0,i x3 ¼ hi ð4Þ

i

Ti

hi ð5Þ

4

g 0,i xi ¼
:
Ti

For Ni ¼ 2
8


Ti

g x Àg ð2x2 À1Þ ¼ hi ð2Þ
0,i i 1,i

i

Ti

h ð3Þ
2
3
g 0,i xi Àg 1,i ð3xi À2xi Þ ¼ i
Ti

g 0,i x3 Àg 1,i ð4x4 À3x2 Þ ¼ hi ð4Þ

i
i
i

Ti

hi ð5Þ

4
5
3

g 0,i xi Àg 1,i ð5xi À4xi Þ ¼
:
Ti


6.5

Table 10
Parametric identification of the ARX-Laguerre model.

6

Truncating orders

Na ¼ 2
Nb¼ 2

xa ¼ À0:0715
xb ¼ 0:0054

xa ¼ À0:083
xb ¼ 0:21


5

Na ¼1
Nb¼ 1

Laguerre poles

5.5
T (°C)

859

[1, À 2.0569 Â 10 À 4]

[1.0166, 0.012, 0.0036, 0.0351]

4.5
4

Table 11
Performances of the ARX-Laguerre model.

3.5
3

Na

5

10

15

20

24

Nb

PRR (%)

NMSE (%)

Ja

Jb

1
2

1
2

80
60

0.7824
0.0271

0.0814
0.0034

6.7128 Â 10 À 1
5.6771 Â 10 À 4

Time (s)
6.5

Fig. 10. Output signal T (1C).

6
7

5.5

6.5

5

T (°C)

6

4.5

u (V)

5.5

4

5

3.5

4.5

Benchmark system

3

4

5

10

15

20

24

Time (s)

3.5

Fig. 12. Validation of the ARX-Laguerre model, Na ¼2 and Nb¼ 2.

3

5

10

15

20

24

7. Extension to multivariable (MIMO) case

Time (s)
Fig. 11. Input signal u(V).

Let us consider an MIMO linear system with p inputs and m
outputs which can be represented by an input–output model as
MðzÀ1 ÞYðzÞ ¼ NðzÀ1 ÞUðzÞ

The Laguerre poles and the Fourier coefficients are summarized in
Table 10 and the performances of the ARX-Laguerre model in
Table 11.
From Table 11 we conclude that the proposed ARX-Laguerre
model obtained for truncating orders Na¼ Nb¼2 describes the
benchmark with the same NMSE obtained when the latter is
represented by an ARX model. However, the proposed model
parameter number is smaller than that defining the ARX model
and the complexity reduction evaluated by the PRR reaches 60%.
In Fig. 12 we plot the output of the ARX-Laguerre model with four
parameters and the output of the benchmark. We notice the
concordance between both outputs.
To confirm the deficiency of the classical ARX model, we
estimate this latter having only four parameters as:

with
UðzÞ ¼ ½u1 ðzÞ . . . up ðzÞŠT

ð66Þ

YðzÞ ¼ ½y1 ðzÞ . . . ym ðzÞŠT

ð67Þ

À1

2
Nðz

BðzÀ1 Þ ¼ 0:001377zÀ1 þ0:04261zÀ2
We plot in Fig. 13 the output of the estimated model with of the
benchmark. We note the difference between both outputs and the
increase in the NMSE to 0.5124%.

À1

and Mðz Þ and Nðz Þ are polynomial matrices with respective
dimension (m Â n) and (m Â p) defined as
2
3
M11 ðzÀ1 Þ . . . M 1m ðzÀ1 Þ
6
7
^

^
ð68Þ
MðzÀ1 Þ ¼ 4
5
À1
À1
M m1 ðz Þ . . . Mmm ðz Þ

À1

AðzÀ1 Þ ¼ 1:681zÀ1 À0:7476zÀ2

ð65Þ

6
Þ¼4

N 11 ðzÀ1 Þ

...

N 1p ðzÀ1 Þ

^

^

À1

Nm1 ðz

Þ

...

À1

Nmp ðz

3
7
5

ð69Þ

Þ

where M ii ðzÀ1 Þ, M ir ðzÀ1 Þ and Nit ðzÀ1 Þ, i¼ 1, y, m, r¼ 1, y, m, t¼1,
y,p, are polynomials given by
Mii ðzÀ1 Þ ¼ 1Àaii ð1ÞzÀ1 À Á Á Á Àaii ðni ÞzÀni
Mir ðzÀ1 Þ ¼ Àair ð1ÞzÀ1 À Á Á Á Àair ðnir ÞzÀnir ,

ð70Þ
ia r

ð71Þ

860


6.5
6

T (°C)

5.5
5
4.5
4
3.5

Benchmark system
ARX model

3

5

10

15

20

24

require the use of a set of input/output collected from the system.
We prove that the parameter number of the ARX-Laguerre model
never goes more than that of the ARX model.
In order to validate the ARX-Laguerre model and the proposed
parametric identification approach, a comparative study with the
classical Laguerre model is presented in simulations. We note that
the ARX-Laguerre model and the Laguerre model have similar
performances in terms of the Normalized Mean Squared Error and
the Parametric Reduction Rate when the damped systems are
considered. However, these performances become different for
the representation of the oscillating systems. In fact, to achieve
the same performances of the ARX-Laguerre model, the Laguerre
model needs a very high parameter number. These results are
obtained according to the parametric identification of the ARXLaguerre model by the proposed optimization problem approach
which influences the parsimony of the expansion.

Time (s)
References

Fig. 13. Validation of the ARX model, na¼ 2 and nb ¼ 2.

Nit ðzÀ1 Þ ¼ bit ð1ÞzÀ1 þ Á Á Á þ bit ðni ÞzÀni

ð72Þ

with the integers ni and nir are structure parameters.
The ith subsystem can be written as an ARX model:
yi ðkÞ ¼

nir
m
XX

air ðjÞyr ðkÀjÞ þ

r ¼1j¼1

p
ni
XX

bit ðjÞut ðkÀjÞ

ð73Þ

t ¼1j¼1

and the reduction of its parameter number can be achieved by
expanding each of the coefficients air ðjÞ, j ¼ 1, . . . ,nir and
bit ðjÞ, j ¼ 1, . . . ,ni on two independent Laguerre bases.
8. Conclusion
In this work, a new black-box linear ARX-Laguerre model has
been introduced to overcome the handicap of the overparametrization problem in the ARX model. This is obtained by proposing
the expansion of the ARX model parameters associated with the
input and the output on two independent Laguerre orthonormal
bases. This new linear representation is characterized by filtering
the input and the output as well of the system represented by the
ARX model. To identify the parameters (Fourier coefficients and
the two Laguerre poles) of the ARX-Laguerre model, we propose
an approach based on the approximation of the ARX model
polynomials by solving an equation system which is formulated
as an optimization problem. This identification approach does not

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