H∞ dynamic observer design for a class of Lipschitz nonlinear discrete-time systems with time varying delays

This study explores the development of  dynamic observer (HDO) for discrete-time nonlinear systems (DTNLS) with time-varying delay (TVD) and disturbances. The approach is to construct an augmented Lyapunov–Krasovskii function (LKF) with double summation terms, using the generalized reciprocally convex matrix inequality (GRCMI), as well as the Jensen-based inequality (JBI) and the Wirtinger-based inequality (WBI). These lead to less conservative time-dependent conditions, represented as a set of linear matrix inequalities (LMIs) that can be efficiently solved using the LMI or YALMIP toolboxes. In addition, the proposed observer includes the widely used proportional observer (PO) and proportional integral observer (PIO) as specific cases. Two examples are presented to demonstrate the validity and effectiveness of the results.