Title: A Nonlinear Observer Design for the Discrete-time Systems: Exploiting Matrix-Multiplier-based LMI Approach

Abstract: This letter focuses on the $\mathcal{H}_\infty$ observer design for a class of nonlinear discrete systems under the presence of measurement noise or external disturbances. A novel Linear Matrix Inequality (LMI) condition is developed in this method through the utilisation 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 LMI is the generalised matrix multipliers. The deliberate 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 proposed LMIs are less conservative than existing ones. Later on, the effectiveness of the developed LMIs and observer is highlighted through the numerical example and an application of state of charge (SoC) estimation in the Li-ion battery model.