A Universal Framework of the Generalized Kalman–Yakubovich–Popov Lemma for Singular Fractional-Order Systems

The well-known generalized Kalman–Yakubovich–Popov lemma is widely used in system analysis and synthesis. However, the corresponding theory for singular systems, especially singular fractional-order systems (SFOSs), is lacking. Therefore, many control problems for this type of systems cannot be optimized in limited frequency ranges. In this article, a universal framework of the finite frequency band generalized Kalman–Yakubovich–Popov lemma for SFOSs is established, the bounded real lemma in the sense of the
${L_{\infty }}$
-norm is derived for different frequency ranges, and the corresponding controller is designed to improve the
${L_{\infty } }$
performance index of SFOSs. Three illustrative examples are given to demonstrate the correctness and effectiveness of the theoretical results.