Article Transactions of the Institute of Measurement and Control 1–11 Ó The Author(s) 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0142331216687020 journals.sagepub.com/home/tim Reduced model of linear systems via Laguerre filters Ameni El Anes 1 , Kais Bouzrara 1 and Jose ´ Ragot 2 Abstract In this paper, we propose a technique to reduce the complexity of an existing (initial) model via the Laguerre filters. We present an analytical method for the parameter identification of the Fourier coefficients of the Laguerre model. This technique is based on the bilinear discrete transformation and in which the Fourier coefficients are expressed in recurrent form in terms of the Laguerre pole. This latter is estimated by an iterative technique, based on the Newton algorithm. This identification technique is after extended to the case of the ARX-Laguerre model and the MISO-ARX-Laguerre model and its performances are illustrated by numerical simulations. Keywords Order reduction, ARX-Laguerre model, bilinear transformation, pole optimization, Laguerre filters Introduction The modeling of a system consists of three complementary stages. The first is related to the choice of a structure that can represent effectively the behavior of the system to be modeled, this structure depending on a certain number of parameters. The second phase is related to the parameter identification. The third phase, known as validation phase, tests the quality of the obtained model in term of representativeness of charac- teristics of the real system. Modeling by using orthonormal bases such as FIR (Finite impulse response) base, the Laguerre base, the Kautz base and the generalized orthogonal base (GOB) present sev- eral advantages with respect to classical linear models such as, Auto Regressive with eXternal inputs (ARX), Auto Regressive Moving Average with eXogenous Inputs (ARMAX) and so forth. Among these advantages, we notice:  Their independence of the sample time and no need of any explicit knowledge about system time constant and time delay for model development.  Good compromise between Finite Impulse Response and Infinite Impulse Response.  Good approximation can be obtained with a small num- ber of model coefficients for asymptotically stable systems owing to orthogonality property of the used functions. We note that the Laguerre model has received interest in the literature of many fields, such as Tanguy et al. (2000), Aoun et al. (2007),Wahlberg and Mkil (1996), Tuma and Jura (2015) and Wang and Jiang (2011) for the system identi- fication; Asad and Hasan (2012), Mahmoodi et al. (2009), Wang (2004), Yakub and Mori (2014) and Abdullah and Idres (2014) for control system; Horng and Chou (1988), Sachinn et al. (2005), Ding et al. (1990), Anfinsen and Aamao (2015) and King and Paraskevopoulos (1977) for system diag- nosis; Chou and Horng (1986) for state estimation; Masnadi- Shirazi and Aleshams (2003) for filter design. In what follows, we assume that model of the system is known and we want to reduce complexity by expressing it as Laguerre filters. The Laguerre functions have the property of being com- pletely characterized by one free parameter entitled Laguerre pole and its optimal identification results a significant reduc- tion of the model parameter number. The optimization of the Laguerre pole has been widely discussed in the specific litera- ture; see, for example, Fu and Dumont (1993), den Brinker and Sarroukh (2004). When expanding the ARX coefficients on Laguerre bases, an easy representation and a good approx- imation capability of complex linear system is given. The expansion of the ARX model on Laguerre bases was first sug- gested in Bouzrara et al. (2012). The resulting model is enti- tled ARX-Laguerre model. We propose in this article to represent Single Input Single Output (SISO) and Multiple Input Single Output (MISO) systems by using ARX-Laguerre model where the Laguerre coefficients are obtained by applying the bilinear transforma- tion and the Laguerre poles are optimized by an iterative algorithm based on the Newton Raphson’s approximation 1 Research Laboratory of Automatic, Signal and Image Processing, National School of Engineers Monastir, Tunisia 2 Research Centre for Automatic Control of Nancy, France Corresponding author: Ameni El Anes, Research Laboratory of Automatic, Signal and Image Processing, National School of Engineers, Rue Ibn el jazzar, Monastir, 5019, Tunisia. Email: elanes.ameni@yahoo.fr