Reduced-Order Model Approximation of Fractional-Order Systems Using Differential Evolution Algorithm
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Reduced-Order Model Approximation of Fractional-Order Systems Using Differential Evolution Algorithm Bachir Bourouba1 · Samir Ladaci2
· Abdelhafid Chaabi3
Received: 3 September 2016 / Revised: 21 November 2017 / Accepted: 22 November 2017 © Brazilian Society for Automatics–SBA 2017
Abstract In this paper, we authors propose to use an optimization technique known as Differential Evolution (DE) optimizer for the approximation of fractional-order systems with rational functions of low order. Usual integer-order models with eleven unknown parameters are optimized to represent non-integer-order systems using the DE algorithm. Four numerical examples have illustrated the efficiency of the proposed reduced-order approximation algorithm. The results obtained from the DE approach were compared with those of Oustaloup and Charef approximation techniques for fractional-order transfer functions. They showed clearly that the proposed approach provides a very competitive level of performance with a reduced model order and less parameters. Keywords Differential Evolution (DE) · Parameters optimization · Fractional-order systems · Reduced-order systems · Approximation method
1 Introduction Fractional-order systems are gathering a huge interest by the scientific research community of various engineering domains (Diethelm and Ford 2001; Koeller 1984; Ladaci and Charef 2006; Reyes-Melo et al. 2004; Neçaibia et al. 2015; Rabah et al. 2016). Good reviews devoted specifically to the subject are available in the literature (Miller and Ross 1993; Oldham and Spanier 1974; Oustaloup 1995). Fractional-order mathematical models have proved to be more accurate for description of many physical phenomena such as electrochemical processes (Ichise et al. 1971),
B
Samir Ladaci [email protected] Bachir Bourouba [email protected] Abdelhafid Chaabi [email protected]
1
Department of Electrical Engineering, Sétif 1 University, 19000 Sétif, Algeria
2
Depart. of E.E.A. Of.: 447, National Polytechnic School of Constantine, Nouvelle ville Ali Mendjli, BP 75 A, 25100 Constantine, Algeria
3
Department of Electronics, University Mentouri, 25000 Constantine, Algeria
long distributed lines (Heaviside 1922), dielectric polarization (Sun et al. 1984), viscoelastic materials (Bagley and Calico 1991), colored noise (Mandelbrot 1967) and chaos (Khettab et al. 2017). Indeed, using an integer model instead of the fractional one to characterize these processes requires a high-order models or the neglecting of some physical phenomena like the diffusion phenomena (Yakoub et al. 2015). Unfortunately, identifying a fractional-order system is not as easy as for the integer-order case because it requires estimation of both model coefficients and fractional orders. Different approaches have been proposed for their modelization both in time and frequency domains: Based on the ability to define systems using continuous-order distributions, Hartley and Lorenzo (2003) showed that frequency domain fractionalorder system identification can be performed. Least-square
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