On the transformation of FOTF system to fractional order LTI model

Abstract

This article addresses modeling systems using fractional order derivatives, highlighting three basic approaches: differential equations, operator methods, and state space representations. Each approach carries different advantages and limitations in the context of time invariant linear systems (LTI) of fractional order. The focus of the article is on Fractional Order Transfer Functions, which represent a special subject of interest because they offer practical utility with available simulation libraries (e.g., CRONE, FOMCON, NiNteger) and approximation techniques (e.g., Oustaloup, CFE, Thiele, Padé).

This paper describes a transition between fractional order transfer functions (FOTF) and pseudo-rational representations of such systems. While the existing literature contains the basics of fractional differential equations and operator theory, it often omits explicit formulas for conversion between different representations. The paper fills a significant gap by proposing a novel algorithm for converting FOTF models to fractional LTI representations. Unlike previous works that implicitly assume such transitions exist, this paper provides formulas for coefficient transformations and demonstrates its effectiveness by minimizing the degree of the system.