Differences between Riemann–Liouville and Caputo calculus definitions in analyzing fractional boost converter with discontinuous‐conduction‐mode operation

Abstract

SummaryFractional‐order calculus is an extension of the integer‐order calculus in that its order is not limited to integers but can be noninteger as well, thus making the application of fractional‐order calculus more flexible. It has been shown that actual reactive elements such as capacitors and inductors have fractional‐order properties and that the theory of fractional‐order calculus can be utilized to more accurately characterize circuits and systems containing reactive elements. However, there is no uniform definition of fractional order calculus. Different definitions will yield different results in solving the same fractional‐order circuits and systems, and it is worth exploring whether these differences have an impact on the accuracy of solving fractional‐order systems. Taking a fractional‐order boost converter running in discontinuous conduction mode (DCM) as an example, this paper focuses on the differences between the Riemann–Liouville (R–L) and the Caputo fractional calculus definitions. Effects of the two definitions on the division of the converter's CCM (continuous conduction mode) and DCM operating regions are analyzed, as well as their impact on the characterization of a fractional‐order boost converter in DCM operation, including the DC (direct current) operating point, current ripple, and voltage gain. Simulations and experiments show that the analytical results based on the R–L definition can more accurately describe the operation of a real fractional‐order converter than those based on the Caputo definition.