On bounded realness for ℋ∞ control design in discrete‐time descriptor systems.

This article suggests a new necessary and sufficient condition of bounded real lemma (BRL) for discrete‐time descriptor systems (DTDSs). First, a Lyapunov function for DTDSs is designed by using a positive definite matrix whose the dimension equals to the rank of a singular matrix in DTDSs, whereas the existing work in the literature has considered the dimension which equals to the length of the state. By considering the zero constraint which comes from the singular matrix in DTDSs, two slack variables are introduced into the difference of Lyapunov function. Then, a set of linear matrix inequalities (LMIs) is provided to ensure the necessary and sufficient condition of the BRL for DTDSs. Next, an inversion formula is given to apply the proposed BRL to a stabilization problem of DTDSs. Based on the inversion formula, the BRL for the closed‐loop system with state‐feedback ℋ∞$$ {\mathscr{H}}_{\infty } $$ control is obtained in terms of non‐convex conditions. Therefore, a sufficient condition of the non‐convex conditions is provided in terms of LMIs. Three numerical examples show the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]