Article
On the fractional closed-loop linear parameter varying system identification under noise corrupted scheduling and output signal measurements
Transactions of the Institute of Measurement and Control 1–13 Ó The Author(s) 2018 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/0142331218821409 journals.sagepub.com/home/tim
Zaineb Yakoub, Messaoud Amairi, Mohamed Aoun and Manel Chetoui
Abstract It is well known that, in some industrial process identification situations, measurements can be collected from closed-loop experiments for several reasons such as stability, safety, and performance constraints. In this paper, we investigate the problem of identifying continuous-time fractional closed-loop linear parameter varying systems. The simplified refined instrumental variable method is developed to estimate both coefficients and differentiation orders. This method is established to provide consistent estimates when the output and the scheduling variable are contaminated by additive measurements noise. The proposed scheme is evaluated in comparison with other approaches in terms of a simulation example.
Keywords Fractional calculus, instrumental variable, linear-parameters-varying, least squares, nonlinear optimization
Introduction Studies of fractional order models have been evolved rapidly over the last two decades and they have been used in many fields of engineering and applied sciences (see Battaglia et al., 2000; Dalir and Bashour, 2010; Oustaloup, 1995: and references therein). Hence, a reasonable amount of literature on stability analysis (Aoun et al., 2007; Chen et al., 2017; Matignon, 1998), controller design (Lin et al., 2000; Luo et al., 2010; Podlubny, 1999; Zamani et al., 2017) and system identification (Cois et al., 2001; Cui et al., 2017; Malti et al., 2008; Sabatier et al., 2006) has been made available. This deep interest is aroused by the fact that several complex physical systems such as thermal systems and biological systems can be suitably depicted by fractional order models as they provide a reliable modelling tool allowing the illustration of their properties. In such practical situations, fractional systems must be identified using data collected from closed-loop experiments. This need cannot be avoided when the process is unstable and requires control or when the opening of the loop is difficult or even impossible owing to the reasons of production or safety. Furthermore, one of the main factors that make fractional identification from closed-loop experiments more important than the open-loop system identification is the correlation between the input and the output noise of the plant which may be amplified due to the long memory aspect of the fractional systems (Yakoub et al., 2015a). Therefore, during the last few years, the fractional closed-loop linear time invariant (LTI)
system identification has been a well-studied research topic and significant efforts have been spent on developing efficient approaches. Two mainstream approaches exist for the fractional closed-loop identification of LTI systems, namely the direct approach and the indirect approach. In a brief historical review in 2014, Tavakoli-Kakhki and Tavazoei (2014) proposed a graphical method using a closed-loop step response data to identify the order and the coefficients of an unstable fractional order system with input time delay. Subsequently, in 2015, an iterative simplified refined instrumental variables (SRIV) method has been extended to the fractional closedloop case by Yakoub et al. (2015a). Then, the fractional order optimization closed-loop bias eliminated least squares method has been addressed to surmount the bias problem caused by the correlation between the input signal and the output noise signal (Yakoub et al., 2015b). In 2017, the motivation for the fractional closed-loop system identification to obtain a modelbased fractional order controller to achieve the desired performances was established by Yakoub et al. (2017). Despite their great efficiency, the extension of these approaches to the fractional linear parameter varying (LPV) University of Gabes, National Engineering School of Gabes, Tunisia Corresponding author: Zaineb Yakoub, Universite´ de Gabe´s, E´cole Nationale d’Inge´nieurs de Gabe`s, Rue Amor Ibn elkattab, zrig Eddakhlania, Gabes 6029, Tunisia. Email:
[email protected] 
