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

A generalised Lyapunov inequality is described which can be reformulated as nonstrict LMIs, for the check of 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(infinity) 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(infinity) controller design for discrete-time descriptor systems.