Bounded Real Lemma for Singular Caputo Fractional-Order Systems

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 $\alpha $ , with the constraint $1 \leq {\alpha }\lt 2$ . 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 ${H_{\infty }}-$ 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.