Influence of models approximating the fractional-order differential equations on the calculation accuracy

Abstract

In recent years, the usage of fractional-order (FO) differential equations to describe objects and physical phenomena has gained immense popularity. Due to the high computational complexity in the exact calculation of these equations, approximation models make the calculations executable. Regardless of their complexity, these models always introduce some inaccuracy which depends on the model type and its order. This article shows how the selected model affects the obtained solution. This study revisits seven fractional approximation models, known as the Continued Fraction Expansion method, the Matsuda method, the Carlson method, a modified version of the Stability Boundary Locus fitting method, and Basic, Refined, and Xue Oustaloup filters. First, this work calculates the steady-state values for each of the models symbolically. In the next step, it calculates the unit step responses. Then, it shows how the model selection affects Nyquist plots and compares the results with the plot determined directly. In the last stage, it estimates the trajectories for an example of an FO control system by each model. We conclude that when employing approximation models for fractional integro-differentiation, choosing the appropriate type of model, its parameters, and order is very important. Selecting the wrong model type or wrong order may lead to incorrect conclusions when describing a real phenomenon or object.