This article introduces a novel algorithm based on the Nash differential game approach for designing observer gains and developing an observer-based linear quadratic regulator (LQR) controller for systems with unknown model uncertainties. By formulating the observer error dynamics as a two-player game and defining two quadratic performance indices with infinite horizons, the Nash equilibrium is determined to optimize observer gains. The proposed approach effectively addresses model uncertainties by constructing an observer-based LQR controller for systems with uncertain dynamics. Sufficient conditions ensuring the asymptotic stability of the closed-loop system are analytically derived. Mathematical expectation is employed to address the challenges posed by unknown initial conditions in practical applications, ensuring robust performance. The applicability and efficiency of the proposed method are validated through comprehensive simulations on a two-degree-of-freedom (2-DOF) robotic system, confirming its robustness and optimal performance under uncertainty.