Research article New methods of Laguerre pole optimization for the ARX model expansion on Laguerre bases Tawfik Najeh, Abdelkader Mbarek n , Kais Bouzrara, Lotfi Nabli, Hassani Messaoud Research Laboratory of Automatic Signal and Image Processing, National School of Engineers of Monastir, University of Monastir, 5019, Tunisia article info Article history: Received 15 January 2014 Received in revised form 14 April 2017 Accepted 21 May 2017 Keywords: ARX-Laguerre model Pole optimisation Newton-Raphson method Genetic algorithms abstract The ARX-Laguerre model is a very important reduced complexity representation of linear system. However a significant reduction of this model is subject to an optimal choice of both Laguerre poles. Therefore we propose in this paper two new methods to estimate, from input/output measurements, the optimal values of Laguerre poles of the ARX-Laguerre model. The first method is based on the Newton- Raphson's iterative technique where we prove that the gradient and the Hessian can be expressed analytically. The second method is based on Genetic Algorithms. Both proposed algorithms are tested on a numerical example and on a heating benchmark. & 2017 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Recently an upsurge in research relating to Laguerre filters for use in system modelling and reduced order models have been provided [1–7]. The Laguerre functions have the property of being completely characterized by one parameter entitled Laguerre pole and its optimal identification results in a significant reduction of the model parameter number. Many works have dealt with La- guerre modeling, among them we remind the works of Fu and Dumont (1993) [8] who established an optimality condition of the Laguerre pole using the system impulse response and those of Tanguy et al. (2000) [9] where the optimal pole is expressed in terms of Laguerre model coefficients. Some researchers suggested an off-line method to estimate the position of the stationary points of the squared error curve of a truncated Laguerre series or La- guerre filter [10]. Other orthonormal function bases (OFB) have been proposed in the literature such as generalized Laguerre functions and Meixner functions [11–15], Kautz orthonormal basis [16] and generalized orthonormal basis (GOB) [17]. We note that these bases are characterized by a set of poles; the choice of which strongly influences the parsimony of the expansion. The problem of optimum choice of the free parameter of some orthogonal functions (Laguerre, Kautz, Meixner) for the conventional system modeling has already been reported by many researchers [5–7,11– 15,18,19]. Some authors were interested by the optimization of the GOB poles, Den Brinker et al. [20] developed a method to recur- sively determine GOB poles. Malti et al. [21] and Oliveira e Silva [22] used the gradient algorithm to minimize output quadratic error. When expanding the ARX coefficients on two Laguerre bases, an easy representation and a good approximation capability of complex linear system is given. Expansion of the ARX model on Laguerre bases was first suggested by Bouzrara et al. [1]. The re- sulting model is entitled ARX-Laguerre model. The parsimony of the expansion is strongly linked to the choice of the poles defining both Laguerre bases. Bouzrara et al. [3] have proposed an iterative algorithm to optimise Laguerre poles based on an analytical so- lution which depends on the coefficients defining the ARX-La- guerre model. In this paper, we focus on Laguerre pole optimization of the ARX-Laguerre model by proposing two different methods. The first one uses an iterative algorithm based on the Newton-Raphson's approximation technique that deals with black box context where only the response of the system to a persistently exciting input signal is known. Moreover this methods needs initial values of the Laguerre poles in the iteration procedure which results is a local minimum. We suggest a procedure in the black box context, that yields the optimal values of the Laguerre poles and the least squares estimates of the ARX-Laguerre model coefficients. The second proposed method uses the Genetic Algorithms (GA) that are widely adopted in recent years and which are very powerful in stochastic system modelling and have no special requirement on the form of the objective function. The processing of these algo- rithms can be achieved in many different ways by using some Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2017.05.015 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved. n Corresponding author. E-mail addresses: najehtawfik@gmail.com (T. Najeh), abdelkader.mbarek@enim.rnu.tn (A. Mbarek), kais.bouzrara@enim.rnu.tn (K. Bouzrara), lotfinabli@yahoo.fr (L. Nabli), hassani.messaoud@enim.rnu.tn (H. Messaoud). Please cite this article as: Najeh T, et al. New methods of Laguerre pole optimization for the ARX model expansion on Laguerre bases. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.015i ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎