This paper aims to design the controller for the system with dynamic uncertainty and unknown dynamics, where the system states are not measurable. The neural networks are employed to solve the unknown nonlinear problem of systems. The dynamic surface control (DSC) technique is employed to avoid the repeated differentiation in virtual control, simplifying stability analysis and the controller design. The Luenberger observer is introduced to estimate the unknown state variables, and its effectiveness is verified. By introducing the state error transformation in the Lyapunov stability analysis, the dynamics uncertainties are well solved, and the uniformly ultimate boundedness of all states within the closed-loop system can be guaranteed. Finally, both theoretical analysis and simulation results prove that all signals are uniformly ultimately bounded and converge to zero rapidly, proving the effectiveness of the proposed approach.