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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]

This article proposes a new strictly bounded real lemma (BRL) for continuous‐time singular Markovian jump systems (SMJSs) in terms of strict linear matrix inequalities (LMIs). While the previous studies have been done for SMJSs that do not have a path from disturbances to desired output, the proposed BRL covers SMJSs with disturbance‐contaminated output by introducing additional free matrix variables. Based on the proposed BRL, the sufficient condition for state‐feedback stabilization is successfully obtained in terms of strict LMIs. The result leads to find the minimal ℋ∞ performance by solving the optimization problem subject to the suggested LMIs. Two examples show the less conservatism of proposed method compared with the previous work. [ABSTRACT FROM AUTHOR]

Summary: This paper considers a dynamic output‐feedback H ∞ control for continuous‐time singular Markovian jump systems, whereas the existing research studies in literature focused on state‐feedback or static output‐feedback H ∞ control. While they have only provided the sufficient conditions, this paper successfully obtains the necessary and sufficient condition for the existence of the dynamic output‐feedback H ∞ control. Furthermore, this condition is expressed with linear matrix inequalities by the so‐called replacement technique. Two numerical examples show the validity of the resulting control. [ABSTRACT FROM AUTHOR]