Recent history of fractional calculus

This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date

Symbolic Representation for Analog Realization of A Family of Fractional Order Controller Structures via Continued Fraction Expansion

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.This paper uses the Continued Fraction Expansion (CFE) method for analog realization of fractional order differ-integrator and few special classes of fractional order (FO) controllers viz. Fractional Order Proportional-Integral-Derivative (FOPID) controller, FO[PD] controller and FO lead-lag compensator. Contemporary researchers have given several formulations for rational approximation of fractional order elements. However, approximation of the controllers studied in this paper, due to having fractional power of a rational transfer function, is not available in analog domain; although its digital realization already exists. This motivates us for applying CFE based analog realization technique for complicated FO controller structures to get equivalent rational transfer functions in terms of the controller tuning parameters. The symbolic expressions for rationalized transfer function in terms of the controller tuning parameters are especially important as ready references, without the need of running CFE algorithm every time and also helps in the synthesis of analog circuits for such FO controllers

A novel ARX-based discretization method for linear non-rational systems

This paper presents a novel, simple, flexible and effective discretization method for linear non-rational systems including arbitrary linear fractional order systems (LFOS). The discretization algorithm relies on the direct integration in the complex domain and application of ARX (AutoRegressive eXogenous) model. Parameters of ARX-model are obtained by numerical inversion of Laplace transform from the set of input/output data from recorded step response to model of non-rational system. Numerical simulations of several representatives of LFOS (e.g. fractional order PID controller, fractional logarithmic filter, fractional oscillator etc.) are used to demonstrate the effectiveness of the proposed discretization method, both in the time and frequency domains. The obtained results indicate that the proposed ARX-based discretization method is adequate technique for obtaining digital approximation of LFOS