The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils

Iqbal M Batiha; Reyad El-Khazali; Ahmed AlSaedi; Shaher Momani

doi:10.3390/e20060400

The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils

Entropy (Basel). 2018 May 23;20(6):400. doi: 10.3390/e20060400.

Authors

Iqbal M Batiha¹, Reyad El-Khazali², Ahmed AlSaedi³, Shaher Momani¹

Affiliations

¹ Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan.
² ECCE Department, Khalifa University, Abu-Dhabi 127788, United Arab Emirates.
³ Department of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University (KAU), Jeddah 21589, Saudi Arabia.

Abstract

This paper introduces a general solution of singular fractional-order linear-time invariant (FoLTI) continuous systems using the Adomian Decomposition Method (ADM) based on the Caputo's definition of the fractional-order derivative. The complexity of their entropy lies in defining the complete solution of such systems, which depends on introducing a method of decomposing their dynamic states from their static states. The solution is formulated by converting the singular system of regular pencils into a recursive form using the sequence of transformations, which separates the dynamic variables from the algebraic variables. The main idea of this work is demonstrated via numerical examples.

Keywords: Adomian decomposition; Mittag–Leffler function; Schur factorization; descriptor fractional linear systems; fractional calculus; regular pencils.