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

TL;DR

This paper introduces a method using Continued Fraction Expansion for the analog realization of various fractional order controllers, providing symbolic transfer functions that facilitate circuit synthesis and eliminate the need for repeated approximation algorithms.

Contribution

It presents a novel CFE-based approach for the analog realization of complex fractional order controllers, offering explicit symbolic transfer functions in terms of tuning parameters.

Findings

Provides symbolic transfer functions for FO controllers

Enables easier analog circuit synthesis

Eliminates repeated CFE computations

Abstract

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.