Solution Analysis and Novel Admissibility Conditions of SFOSs: The 1 &lt; <i>α</i> &lt; 2 Case

Abstract

This article considers the regularity, nonimpulsiveness, stability, as well as admissibility of singular fractional-order systems (SFOSs) with the fractional-order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (1,2)$ </tex-math></inline-formula> . First, the existence and uniqueness of time-domain solutions of the systems are analyzed by using the Kronecker equivalent standard form, and then the necessary and sufficient condition for the regularity is proposed. Second, the explicit time-domain solutions of the systems are presented, which are the basis of the necessary and sufficient conditions for the nonimpulsiveness and stability. Third, two novel necessary and sufficient conditions for the admissibility of the systems are proposed in terms of the nonstrict linear matrix inequalities (LMIs) and strict LMIs, respectively. Finally, two numerical examples about the SFOSs are provided to illustrate the validity of the obtained conclusions.