Discrete-Time ${\cal H}_\infty$ Control Problem for Nonlinear Descriptor Systems

This note presents an explicit solution to the problem of disturbance attenuation with internal stability for discrete-time nonlinear descriptor systems. Both the static-state feedback and dynamic output feedback cases are considered. In particular, we characterize a family of H
_infin
controllers solving the problem locally around a neighborhood of the origin. To do this, we first derive two stability criteria for discrete-time nonlinear descriptor systems, and then, a version of a bounded real lemma is also developed based on the concepts of dissipation inequality and differential game. After that, the results are used to derive the H
_infin
control theory for nonlinear discrete-time descriptor systems. The approach taken is mainly algebraic, and hence is simple and clear.