A Novel Vieta–Fibonacci Projection Method for Solving a System of Fractional Integrodifferential Equations

This research project focuses on developing a mathematical model that allows us to understand the behavior of the balancing loops in system dynamics in greater detail and precision. Currently, simulations give us an understanding of the behavior of these loops, but under the premise of an ideal scenario. In practice, however, accurate models often operate with varying efficiencies due to various irregularities and particularities. This discrepancy is the primary motivation behind our research proposal, which seeks to provide a more realistic understanding of the behavior of the loops, including their different levels of efficiency. To achieve this goal, we propose the introduction of fractional calculus in system dynamics models, focusing specifically on the balancing loops. This innovative approach offers a new perspective on the state of the art, offering new possibilities for understanding and optimizing complex systems.

“…we can find the solution by x(t) applying the inverse Laplace transform using the formula of Equation ( 10), so we have Equation (26).…”

Section: Solution Of the Fractional Order Model Per Caputomentioning

confidence: 99%

“…Another innovation involves presenting a new approach to numerically solving systems of fractional integro-differential equations [26]. Vieta-Fibonacci polynomials are used as essential functions, and the projection method for the fractional Caputo order is derived for the first time.…”

Section: Introductionmentioning

confidence: 99%

Fractional Calculus to Analyze Efficiency Behavior in a Balancing Loop in a System Dynamics Environment

Barrios-Sánchez,

Baeza-Serrato,

Martínez-Jiménez

2024

“…Undoubtedly, mathematics holds immense value across various disciplines, including physics, economics, and engineering [1][2][3][4][5][6][7][8]. The concept of a difference equation becomes essential when describing the evolution of a phenomenon over time.…”

Section: Introductionmentioning

confidence: 99%

The Generalized Discrete Proportional Derivative and Its Applications

Pandurangan,

Shanmugam,

Rhaima

et al. 2023

The aim of this paper is to define the generalized discrete proportional derivative (GDPD) and illustrate the application of the Leibniz theorem, the binomial expansion, and Montmort’s formulas in the context of the generalized discrete proportional case. Furthermore, we introduce the generalized discrete proportional Laplace transform and determine the GDPLT of various functions using the inverse operator. The results obtained are showcased through relevant examples and validated using MATLAB.

“…Furthermore, in practical applications, it is necessary to further research fractional-order MRCs to find a better universal MRC design [25]. There are also many studies on the application of fractional-order control (FOC) in inverters [26,27] and in some other fields [28,29], and many methods have also emerged for solving and analyzing fractional-order models [30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Fractional-Order Phase Lead Compensation Multirate Repetitive Control for Grid-Tied Inverters

Liang,

Lee,

Zhang

2023

To reduce computational load and memory consumption, multirate repetitive control (MRC) with downsampling rates provides a flexible and efficient design for proportional-integral multi-resonant repetitive control (PIMR-RC) systems for grid-tied inverters. However, in MRC systems, repetitive controllers with low sampling rates produce low delay periods, and integer-order phase lead compensation may cause undercompensation or overcompensation. These imprecise linear phase lead compensations may result in deteriorated control performance. To address these problems, based on an infinite impulse response (IIR) filter, a fractional-order phase lead proportional-integral multi-resonant multirate repetitive control (FPL-PIMR-MRC) is proposed for grid-tied inverters in this paper. The proposed method can provide a suitable fractional phase lead step to achieve a wide stability region, minor tracking errors, and low hardware costs. The IIR fractional-order lead filter design, stability analysis, and the step-by-step parameter tuning of the FPL-PIMR-MRC system are derived in detail. Finally, simulation performed confirms the feasibility and effectiveness of the proposed scheme.