DUARTE VAL ´ ERIO AND JOS ´ ES ´ A DA COSTA 1 Symposium on Fractional Signals and Systems Lisbon09 M. Ortigueira et al. (eds.) Lisbon, Portugal, November 4-6, 2009 V ARIABLE - ORDER FRACTIONAL DERIVATIVES AND THEIR NUMERICAL APPROXIMATIONS II – COMPLEX ORDERS Duarte Val´ erio and Jos´ e S´ a da Costa IDMEC/IST, Technical University of Lisbon Lisbon, Portugal e-mail: duarte.valerio@ist.utl.pt, sadacosta@dem.ist.utl.pt Abstract. This paper addresses complex, variable-order fractional derivatives, enlarging the definitions for the real case. Implementations combining discretised Crone approximations using fuzzy logic and interpolation are also addressed. Keywords: fractional calculus, complex derivatives, fractional controller, variable- order derivatives. I. I NTRODUCTION The concepts of differentiation and integration of a real function can be extended to allow for orders that are not positive integers, as part of an area of Mathematics called fractional calculus 1 . A companion paper [1] addressed variable-order derivatives when the order remains real (and presented an overview of existing bibliography). In this paper the definitions therein given are enlarged for the case when the order may be complex, and approximations are built using fuzzy logic, as was done for real time-varying orders. Once more, the Matlab code developed is available at Matlab’s file exchange site or through the first author’s webpage 2 . The paper is organised as follows. Section II defines variable-order derivatives for the complex case. Section III presents the fuzzy-based approximations. Section IV gives some numerical results by way of illustration. Conclusions are drawn in section V. 1 Fractional orders were the target of the first generalisation attempts, hence the name. 2 http://web.ist.utl.pt/duarte.valerio/ninteger.htm