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Top1. Introduction
Fractional-order based systems and controllers are most widely used in many engineering applications for effective modeling and robust control applications (Caponetto, 2010; Dalir & Bashour, 2010; Kishore, Ibrahim, Karsiti, & Hassan, 2018; Li, Liu, Dehghan, Chen, & Xue, 2017; Pan & Das, 2012; Shah & Agashe, 2016; Dingyü Xue, 2017). The fractional-order PID (FOPID) controller is a variant of conventional PID controller achieved through fractional-ordering of integral and derivative actions (Shah & Agashe, 2016; Tepljakov et al., 2018). The FOPID controller provides more robust control performance over variation in system parameter and provides stable performance for higher-order systems (Shah & Agashe, 2016; Tepljakov, 2017). Furthermore, the controller can easily attain iso-damping property. However, a key issue with the practical realization or equivalent circuit implementation of such controllers in a finite-dimensional integer-order system is the approximation of the fractional-order parameters (Efe, 2011; Krishna, 2011; Li et al., 2017; Valério, Trujillo, Rivero, Machado, & Baleanu, 2013; Vinagre, Podlubny, Hernandez, & Feliu, 2000). In recent years, researchers have developed many numerical approximation techniques. However, it is very difficult to determine the best method. This is because, some of these methods can be more advantageous over the others in certain situations such as the order of approximation, the accuracy of frequency and time responses.
Among the proposed techniques, the Oustaloup approximation is the most widely used. However, for practical applications, this technique cannot fit the wide range of desired frequencies. Thus, a modified or refined Oustaloup approximation was proposed (Merrikh-Bayat, 2012; Dingyu Xue, Zhao, & Chen, 2006). However, the modified version of Oustaloup produces a higher-order transfer function. Therefore, the authors of Senol and Yeroglu (2014) proposed a model order reduction technique using pole-zero cancellation techniques. On the other hand, Matsuda proposed an approximation technique using continued fraction expansion techniques. However, in this method, if the order of the approximation is chosen as an odd number, the approximated transfer function will be improper (Deniz, Alagoz, Tan, & Atherton, 2016; Vinagre et al., 2000). Similarly, the approximation techniques proposed by Charef and Carlson has involved a lot and error for selection the approximation parameters (Oprzędkiewicz, 2014). These approximation techniques are based on continuous time realization of fractional-order parameters.
On the other hand, researchers also proposed several digital time realization methods based on power series expansion, Taylor series expansion, continued fraction expansion etc. It should be noted that that the continued fraction expansion methods are converging more rapidly than power series expansion methods. Thus, the continued fraction expansion-based realization techniques have been studied with the use of generating functions like Euler, Tustin, Simpson, Al-Alaoui, Chen-Vinagre Schneider operators (Saptarshi Das & Pan, 2011). Other approximation techniques based on stability boundary locus, vector curve fitting is also proposed (Deniz et al., 2016; Du, Wei, Liang, & Wang, 2017). A key issue with these methods is that they are quite complex hence difficult to implement. The summary of these approximation techniques including the MATLAB commands for implementing the algorithms is given in Table 1.