The design of dynamic observer (DO) for non-linear Lipschitz systems with multiple time-varying delays and disturbances is studied. Sufficient conditions for the existence of these observers are presented in the form of rank equality. Compared to previously published work, the system under consideration includes non-linearity, non-commensurable delay, and external disturbance. Through the use of the Wirtinger inequality and the extended reciprocally convex matrix inequality, new and less conservative delay-dependent conditions in terms of linear matrix inequalities (LMIs) are derived based on the Lyapunov–Krasovskii functional method. Solving these LMIs makes it possible to obtain DO that satisfies an performance index. Through two numerical examples in which the comparison with the proportional observer (PO) and the proportional–integral observer (PIO) shows the efficiency of the proposed DO synthesis condition. Furthermore, the results indicate that the DO developed in this paper is more resilient to parameter perturbations.
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