Available online at www.sciencedirect.com Systems & Control Letters 52 (2004) 387–393 www.elsevier.com/locate/sysconle Inverting fractional order transfer functions through Laguerre approximation Guido Maione ∗ Dipartimento di Elettrotecnica ed Elettronica, II Facolt a di Ingegneria –Politecnico di Bari, Viale del Turismo, 8, 74100 Taranto, Italy Received 29 October 2002; received in revised form 11 December 2003; accepted 20 February 2004 Abstract This paper concerns the approximate inversion of fractional order transfer functions. The approximating solution is of the form of a weighted sum of Laguerre functions. The weights of the sum are found from a simple generating expression. As case study, the inversion of the Laplace transform of the Abel’s equation is considered. c  2004 Elsevier B.V. All rights reserved. Keywords: Fractional order transfer functions; Laguerre modeling; Approximate inversion of Laplace transform; Abel’s-type equation 1. Introduction In the last few decades, many authors have intro- duced fractional calculus as modeling tool for describ- ing properties of materials and processes [5,12,17]. In many cases, indeed, models in form of fractional order dierential equations (FODE) give better insight into physical processes than integer order dierential equa- tions. In particular, Caputo and Mainardi [7,11] pro- vided fundamental considerations in favor of FODE modeling in some kind of diusion phenomena. In ad- dition, FODE have apparent advantages in describing neuronal behavior [2], Chua’s circuits [3], biological systems [19], etc. In the area of electronics and control, the idea of using fractional systems for modeling ideal loop transfer functions dates back to Bode [4,6]. He showed that, to reduce the eects of disturbances and * Tel.: +39-080-596-4271; fax: +39-080-596-4229. E-mail address: gmaione@poliba.it (G. Maione). uncertainties on the closed loop system performance, the loop gain must have a frequency behavior de- scribed as a fractional order transfer function (FOTF). After the Bode’s seminal work, fractional systems did not receive considerable attention in control the- ory and applications. Only in the 1990s, there was a renewed interest for applying fractional systems to control problems. In particular, Oustaloup and Mathieu [18] showed the advantages of a fractional order controller known as commande robuste d’ordre non entier (CRONE) respect to classical devices. Today, even if the applications of fractional calculus to control problems represent a rapid emerging eld attracting the attention of researchers and practition- ers, many obstacles to the development of this area still remain. For example, as pointed out in [19], the study of fractional order models in the time domain has been usually avoided. On the other hand, the lack of simple Laplace inversion formulas in tables and/or the complicate expressions necessary to provide ex- plicit analytical solutions for the common (unit step or ramp) responses are an obstacle to the development of 0167-6911/$ - see front matter c  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2004.02.014