First Order Plus Fractional Diffusive Delay Modeling: interconnected discrete systems

TL;DR

This paper introduces a novel First Order Plus Fractional Diffusive Delay (FOPFDD) model with a fractional delay term, enhancing the modeling of delay-dominant interconnected discrete systems in both frequency and time domains.

Contribution

The work generalizes the FDD term to any real fractional order, develops numerical methods for inverse Laplace transform, and demonstrates improved accuracy over existing models for complex diffusive systems.

Findings

FOPFDD accurately models delay-dominant systems.

The fractional delay term improves frequency response tracking.

The model outperforms existing models in simulating diffusive-like responses.

Abstract

This paper presents a novel First Order Plus Fractional Diffusive Delay (FOPFDD) model, capable of modeling delay dominant systems with high accuracy. The novelty of the FOPFDD is the Fractional Diffusive Delay (FDD) term, an exponential delay of non-integer order $\alpha$, i.e. $e^{-(Ls)^{\alpha}}$ in Laplace domain. The special cases of $\alpha = 0.5$ and $\alpha = 1$ have already been investigated thoroughly. In this work $\alpha$ is generalized to any real number in the interval $]0,1[$. For $\alpha=0.5$, this term appears in the solution of distributed diffusion systems, which will serve as a source of inspiration for this work. Both frequency and time domain are investigated. However, regarding the latter, no closed-form expression of the inverse Laplace transform of the FDD can be found for all $\alpha$, so numerical tools are used to obtain an impulse response of the FDD. To establish the algorithm, several properties of the FDD term have been proven: firstly, existence of the term, secondly, invariance of the time integral of the impulse response, and thirdly, dependency of the impulse response's energy on $\alpha$. To conclude, the FOPFDD model is fitted to several delay-dominant, diffusive-like resistors-capacitors (RC) circuits to show the increased modeling accuracy compared to other state-of-the-art models found in literature. The FOPFDD model outperforms the other approximation models in accurately tracking frequency response functions as well as in mimicing the peculiar delay/diffusive-like time responses, coming from the interconnection of a large number of discrete subsystems. The fractional character of the FOPFDD makes it an ideal candidate for an approximate model to these large and complex systems with only a few parameters.