H∞ controller design for Nabla discrete fractional-order systems

This article aims to provide several criteria with sufficient conditions for Nabla discrete fractional-order systems to satisfy a certain $H_{\infty }$ performance index. Firstly, the instability region is approximated by a series of union sets of fan-shaped domains, and the mathematical expression of each fan-shaped domain is given. Secondly, by using the above approximation method and the generalized Kalman-Yakubovič-Popov lemma, some criteria with linear matrix inequality (LMI) formulations for the boundedness of the $H_{\infty }$ norm of the transfer function matrix for Nabla discrete FOSs are derived. Thirdly, an LMI-based $H_{\infty }$ state feedback controller design method is provided by using the projection lemma, which is aimed to stabilize the system and optimize its $H_{\infty }$ performance index simultaneously. Finally, the effectiveness of the proposed approach is verified by the accurate numerical solutions and system state responses obtained from two simulation examples.