Bounded Real Lemma for Singular Caputo Fractional-Order Systems

• Published in: IEEE access Vol. 12; pp. 106303 - 106312
• Main Authors: Lin, Ming-Shue, Wu, Jenq-Lang, Arunkumar, Arumugam
• Format: Journal Article
• Language: English
• Published: IEEE 2024
• Subjects: Caputo fractional-order singular systems; Circuit stability; Complexity theory; generalized bounded real lemma; Generalized Lyapunov theorem; Linear matrix inequalities; linear matrix inequality; Lyapunov methods; Mathematical models; Numerical stability; Stability criteria; Transfer functions
• ISSN: 2169-3536; 2169-3536
• DOI: 10.1109/ACCESS.2024.3434729

In this paper, we introduce an innovative generalized Lyapunov theorem and a novel bounded real lemma designed for continuous-time linear singular systems with Caputo fractional derivative of order <inline-formula> <tex-math notation="LaTeX">\alpha </tex-math></inline-formula>, with the constraint <inline-formula> <tex-math notation="LaTeX">1 \leq {\alpha }\lt 2 </tex-math></inline-formula>. We initially present a condition that is both necessary and sufficient for establishing the admissibility of singular fractional-order systems (SFOSs). This condition is articulated through strict linear matrix inequalities (LMIs). Following this, we demonstrate that a SFOS satisfies <inline-formula> <tex-math notation="LaTeX">{H_{\infty }}- </tex-math></inline-formula>norm requirement if and only if two strict LMIs are feasible. The key advantage of the presented LMI conditions is that only one matrix variable needs to be solved. Ultimately, this paper concludes by presenting illustrative examples that highlight the practical effectiveness of our theoretical findings.