  This paper proposes a novel unified interval-valued observer synthesis
approach for locally Lipschitz nonlinear continuous-time (CT) and discrete-time
(DT) systems with nonlinear observations. A key feature of our proposed
observer, which is derived using mixed-monotone decompositions, is that it is
correct by construction (i.e., the true state trajectory of the system is
framed by the states of the observer) without the need for imposing additional
constraints and assumptions such as global Lipschitz continuity or contraction,
as is done in existing approaches in the literature. Furthermore, we derive
sufficient conditions for designing stabilizing observer gains in the form of
Linear Matrix Inequalities (LMIs). Finally, we compare the performance of our
observer design with some benchmark CT and DT observers in the literature.
