Lyapunov inequality and bounded real lemma for discrete-time descriptor systems

A generalised Lyapunov inequality is described which can be reformulated as nonstrict linear matrix inequalities (LMIs), for checking the regularity, impulse immunity, and stability of discrete-time descriptor systems simultaneously. Based on this inequality, a bounded real lemma in nonstrict LMIs is obtained which characterises properties of such descriptor systems, including regularity, impulse-free property, stability, and H_∞ norm bound condition. The proofs are purely algebraic, therefore they are simple and definite. These results could play a key role in the LMI-based H_∞ controller design for discrete-time descriptor systems