  This manuscript focuses on the $\mathcal{H}_\infty$ observer design for a
class of nonlinear discrete systems under the presence of measurement noise or
external disturbances. Two new Linear Matrix Inequality (LMI) conditions are
developed in this method through the utilization of the reformulated Lipschitz
property, a new variant of Young inequality and the well-known Linear Parameter
Varying (LPV) approach. One of the key components of the proposed LMIs is the
generalized matrix multipliers. The judicious use of these multipliers enables
us to introduce more numbers of decision variables inside LMIs than the one
illustrated in the literature. It aids in adding some extra degrees of freedom
from a feasibility point of view, thus enhancing the LMI conditions. Thus, the
established LMIs are less conservative than existing ones. Later on, the
effectiveness of the developed LMIs and observer is highlighted through a
numerical example and the application of state of charge (SoC) estimation in
the Li-ion battery model.
