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Numerical Solution of Fractional Control Problems via Fractional Differential Transformation

Josef Rebenda, Zdeněk Šmarda

TL;DR

The paper tackles numerical solution of linear fractional control problems with constant state delays under Caputo derivatives of order $0<\nu\le 1$. It introduces a numerical method that combines the method of steps with Fractional Differential Transformation (FDT) to convert the delayed fractional system into recurrence relations and represent the solution as a truncated fractional power series on each interval. The approach is demonstrated on a two-dimensional system with commensurate delays, showing exact matches for certain rational $\nu$ and convergence to the classical integer-order solution as $\nu \to 1$. The method is efficient, robust, and extendable to delays in control and to multi-order or non-constant delays, offering a practical tool for solving fractional control problems.

Abstract

In the paper we deal with linear fractional control problems with constant delays in the state. Single-order systems with fractional derivative in Caputo sense of orders between 0 and 1 are considered. The aim is to introduce a new algorithm convenient for numerical approximation of a solution of the studied problem. The method consists of the fractional differential transformation in combination with general methods of steps. The original system is transformed to a system of recurrence relations. Approximation of the solution is given in the form of truncated fractional power series. The choice of order of the fractional power series is discussed and the order is determined in relation to the order of the system. An application on a two-dimensional fractional system is shown. Exact solution is found for the first two intervals of the method of steps. The result for Caputo derivative of order 1 coincides with the solution of first-order system with classical derivative. We conclude that the algorithm is applicable, efficient and gives reliable results.

Numerical Solution of Fractional Control Problems via Fractional Differential Transformation

TL;DR

The paper tackles numerical solution of linear fractional control problems with constant state delays under Caputo derivatives of order . It introduces a numerical method that combines the method of steps with Fractional Differential Transformation (FDT) to convert the delayed fractional system into recurrence relations and represent the solution as a truncated fractional power series on each interval. The approach is demonstrated on a two-dimensional system with commensurate delays, showing exact matches for certain rational and convergence to the classical integer-order solution as . The method is efficient, robust, and extendable to delays in control and to multi-order or non-constant delays, offering a practical tool for solving fractional control problems.

Abstract

In the paper we deal with linear fractional control problems with constant delays in the state. Single-order systems with fractional derivative in Caputo sense of orders between 0 and 1 are considered. The aim is to introduce a new algorithm convenient for numerical approximation of a solution of the studied problem. The method consists of the fractional differential transformation in combination with general methods of steps. The original system is transformed to a system of recurrence relations. Approximation of the solution is given in the form of truncated fractional power series. The choice of order of the fractional power series is discussed and the order is determined in relation to the order of the system. An application on a two-dimensional fractional system is shown. Exact solution is found for the first two intervals of the method of steps. The result for Caputo derivative of order 1 coincides with the solution of first-order system with classical derivative. We conclude that the algorithm is applicable, efficient and gives reliable results.

Paper Structure

This paper contains 8 sections, 1 theorem, 27 equations.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Considered Problem
  4. Fractional Differential Transformation
  5. Elimination Of The Delays From The State
  6. Determining The Order Of The Fractional Power Series
  7. Applications
  8. Conclusion

Key Result

Theorem 1

Assume that $\{ F_{\alpha} (k) \}_{k=0}^{\infty}$, $\{ G_{\alpha} (k) \}_{k=0}^{\infty}$ and $\{ H_{\alpha} (k) \}_{k=0}^{\infty}$ are differential transformations of order $\alpha$ at $t_0$ of functions $f(t)$, $g(t)$ and $h(t)$, respectively, and $r, \beta >0$.

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1