ISA Transactions 51 (2012) 848–860

Contents lists available at SciVerse ScienceDirect

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

BIBO (Bounded-Input Bounded-Output) stability criterion of the
sy...
850

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

b

Ia ¼ f‘n gNaÀ1 and Ib ¼ f‘n gNbÀ1 , where
n¼0
n¼0
qffiffiffiffi...
K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860

851

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

and the ...
852

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

provides a global minimum. Therefore, the standard parameter...
K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860

polynomial of the left hand side of (50) does not contain a
const...
854

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

the Laguerre basis
Ja ¼

na
X

NaÀ1
X

!2

1
a
ha ðjÞÀ
g n,a...
K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860

40

Table 2
Example 1: performances of the Laguerre model.
PRR (%...
856

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

Table 4
Example 2: parametric identification of the ARX-Lague...
K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860

857

For Ni ¼ 3

For Ni ¼ 5

8

 g 0,i Àg 1,i xi þg 2,i x2 ¼ hi ð...
858

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

10

Table 8
Example 3: performances of the Laguerre model.
P...
K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860

6.5

Table 10
Parametric identification of the ARX-Laguerre model....
860

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

6.5
6

T (°C)

5.5
5
4.5
4
3.5

Benchmark system
ARX model

...
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Decomposition of an ARX model on Laguerre orthonormal bases

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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.

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Decomposition of an ARX model on Laguerre orthonormal bases

  1. 1. ISA Transactions 51 (2012) 848–860 Contents lists available at SciVerse ScienceDirect 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
  2. 2. 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
  3. 3. 850 K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 a b Ia ¼ f‘n gNaÀ1 and Ib ¼ f‘n gNbÀ1 , where n¼0 n¼0 qffiffiffiffiffiffiffiffiffiffiffiffi X 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 qffiffiffiffiffiffiffiffiffiffiffiffi 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: qffiffiffiffiffiffiffiffiffiffiffiffi 8 2 1Àxa X 0,y ðzÞ ¼ YðzÞ zÀxa ð28Þ 1Àxa z X n,y ðzÞ ¼ X nÀ1,y ðzÞ : zÀxa qffiffiffiffiffiffiffiffiffiffiffiffi 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: #! qffiffiffiffiffiffiffiffiffiffiffiffi X 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 qffiffiffiffiffiffiffiffiffiffiffiffi 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: qffiffiffiffiffiffiffiffiffiffiffiffi 8 x ðk þ1Þ ¼ x x ðkÞ þ 1Àx2 yðkÞ 0,y a 0,y a qffiffiffiffiffiffiffiffiffiffiffiffi 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 qffiffiffiffiffiffiffiffiffiffiffiffi 2 2 : þ ð1Àxa ÞxnÀ1,y ðkÞ þ ðÀxa Þn 1Àxa yðkÞ ð23Þ nb X j¼1 BðzÀ1 Þ ¼ #! qffiffiffiffiffiffiffiffiffiffiffiffi X 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 qffiffiffiffiffiffiffiffiffiffiffiffi 8 x ðk þ 1Þ ¼ x x ðkÞ þ 1Àx2 uðkÞ 0,u b 0,u b qffiffiffiffiffiffiffiffiffiffiffiffi 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 qffiffiffiffiffiffiffiffiffiffiffiffi 2 2 : þ ð1Àx ÞxnÀ1,u ðkÞ þ ðÀxb Þn 1Àx uðkÞ b b ð31Þ
  4. 4. K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 851 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 ðkފT 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  Na) and (Na  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 qffiffiffiffiffiffiffiffiffiffiffiffi 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
  5. 5. 852 K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 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 g x5 Àg ð6x6 À5x4 Þ þ g ð21x7 À30x5 þ10x3 Þ 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
  6. 6. K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 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 g n,b ‘n ðj, xb Þ 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: qffiffiffiffiffiffiffiffiffiffiffiffi 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
  7. 7. 854 K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 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 g n,b ‘n ðj, xb Þ 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 ðkÞÀyðkފ NMSE ¼ kPH m 2 k ¼ 1 ½ym ðkފ 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]
  8. 8. K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 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 Number of iterations 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 Number of iterations 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 þ0:1191L9 ðzÞ þ 0:0793L10 ðzÞ þ 0:0807L11 ðzÞ þ0:0612L12 ðzÞ þ 0:0574L13 ðzÞ þ 0:0466L14 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 summarized in Table 4. We summarize in Tables 5 and 6 the performances of the Laguerre model and the ARX-Laguerre model respectively. 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
  9. 9. 856 K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 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 Fourier coefficients [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 Example 2: performances of the Laguerre model. 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 ARX − Laguerre model −8 −10 Table 6 Example 2: performances of the ARX-Laguerre model. 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 Number of iterations 800 900 1000 Fig. 5. Example 2: validation of the ARX-Laguerre model, Na ¼2 and Nb¼ 1. 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 g 0,i Àg 1,i xi ¼ hi ð1Þ 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
  10. 10. K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 857 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 g x4 Àg ð5x5 À4x3 Þ þ g ð15x6 À20x4 þ6x2 Þ 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 g x5 Àg ð6x6 À5x4 Þ þ g ð21x7 À30x5 þ10x3 Þ 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 Number of iterations 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 summarized in Table 7. We summarize in Tables 8 and 9 the performances of the Laguerre model and the ARX-Laguerre model respectively. 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]
  11. 11. 858 K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 10 Table 8 Example 3: performances of the Laguerre model. 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 Example 3: performances of the ARX-Laguerre model. 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 Number of iterations 800 900 1000 Fig. 7. Example 3: validation of the ARX-Laguerre model, Na ¼3 and Nb¼ 1. 10 5 Amplitude Na ARX model ARX − Laguerre model 0 −5 −10 −15 ARX model Laguerre model 0 100 200 300 400 500 600 700 Number of iterations 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 g 0,i Àg 1,i xi ¼ hi ð1Þ 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
  12. 12. K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 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 Fourier coefficients 5 Na ¼1 Nb¼ 1 Laguerre poles 5.5 T (°C) 859 [1, À 2.0569  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  10 À 1 5.6771  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 ARX − Laguerre model 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 ðzފT ð66Þ YðzÞ ¼ ½y1 ðzÞ . . . ym ðzފT 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  n) and (m  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
  13. 13. 860 K. Bouzrara et al. / ISA Transactions 51 (2012) 848–860 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 [1] Arnold CR. Laguerre functions and the Laguerre network. Their properties and digital simulation. Technical Note. No. 28, Massachusetts Institute of Technology, Lincoln Laboratory; 1966. [2] Dolan ED, Lewis RM, Torczon VJ. On the local convergence of pattern search. SIAM Journal on Optimization 2003;14(2):567–83. [3] Fu Y, Dumont GA. An optimum time-scale for discrete time Laguerre network. IEEE Transaction on Automatic control 1993;38(6):934–8. [4] Ljung L. System Identification: Theory for the User. Prentice-Hall; 1987. ´ [5] Malti R. Representation de systemes discrets sur la base des filtres ´ orthogonaux—Application a la modelisation de systemes dynamiques multi variables. These de doctorat a l’INPL: Institut National Polytechnique de Lorraine, France; 1999. [6] Malti R, Ekongolo SB, Ragot J. Dynamic SISO and MISO system approximations based on optimal laguerre models. IEEE Transactions on Automatic Control 1998;43(9):1318–23. [7] Ninness B, Gustafsson F. Unifying construction of orthonormal bases for system identification. IEEE Transactions on Automatic Control 1997;42(4): 515–21. [8] Oliveira GHC, Amaral WC, Favier G, Dumont GA. Constrained robust predictive controller for uncertain processes modeled by orthonormal series functions. Automatica 2000;36(4):563–71. ´ [9] Tanguy N, Morvan R, Vilbe P, Calvez LC. Online optimization of the time scale in adaptive Laguerre-based filters. IEEE Transactions on Signal Processing 2000;48(4):1184–7. [10] Wahlberg B. System identification using Laguerre models. IEEE Transactions on Automatic Control 1991;36(5):551–62. [11] Wahlberg B. System identification using Kautz models. IEEE Transactions on Automatic Control 1994;39(6):1276–81. [12] Wang L. Discrete model predictive controller design using Laguerre, functions. Journal of Process Control 2004;14(2):131–42. [13] Zhang H, Chen Z, Wang Y, Li M, Qin T. Adaptive predictive control algorithm based on Laguerre functional model. International Journal of Adaptive Control and Signal Processing 2006;20(2):53–76.

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