An innovative bounded real lemma for singular fractional-order systems

This article modifies the bounded real lemma for singular fractional-order systems (SFOS). First, an admissible set is constructed by using a single real variable, leading to a generalized S-procedure. Next, H_∞ norm inequalities converted into frequency domain inequalities (FDIs) and are aligned with the generalized S-procedure's left hand-side. This allows replacing complex variables with real ones in the bounded real lemma for normal fractional-order systems (FOSs). Subsequently, by using full-rank decomposition of the descriptor matrix, a single real-variable bounded real lemma for SFOSs is derived. This reduces conservatism in the design of the H_∞ feedback controller. Finally, three simulation examples are proposed to validate the effectiveness of the proposed method. All the simulation results illustrate that the required variables in the bounded real lemma can be changed from complex to real ones without loss of validity of the bounded real lemma.