Normalized finite fractional differences: Computational and accuracy breakthroughs

Abstract

This paper presents a series of new results in finite and infinite-memory modeling of discrete-time fractional differences. The introduced normalized finite fractional difference is shown to properly approximate its fractional difference original, in particular in terms of the steady-state properties. A stability analysis is also presented and a recursive computation algorithm is offered for finite fractional differences. A thorough analysis of computational and accuracy aspects is culminated with the introduction of a perfect finite fractional difference and, in particular, a powerful adaptive finite fractional difference, whose excellent performance is illustrated in simulation examples.

A Comparative Analysis of Laguerre-Based Approximators to the Grünwald-Letnikov Fractional-Order Difference

“…Similar results are reported in other FIR-based approximations to FD [5,6]. New time-domain modeling concepts for FD have been introduced in [7,8].…”

Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: New stability criterion for FD-based systems

“…The steady-state accuracy analysis has shown [13] that steady-state error-free output modeling (with respect to FD) can be obtained for the NFFD-based system only.…”

Modeling and Identification of Fractional-Order Discrete-Time Laguerre-Based Feedback-Nonlinear Systems

“…However, to obtain high modeling accuracies we have to use high implementation lengths of the FFD approximation. This inconvenience can be essentially reduced by making use of our computationally more efficient approximations to FD, that is AFFD, PFFD [32], FLD and, in particular, FFLD [19], the task being a subject of our current and future research works. …”