The main theme of this thesis is the design of a full order nonlinear deterministic dynamic observer for estimation of state space from its nonlinear input-output dynamic control system. The quadratic Lyapunov function stabilization approach has been adapted and developed. The sufficient conditions for existence of the dynamic observer for some class of nonlinear input-output dynamic system have been presented and discussed. Useful linear transformations have also been adapted to design a stable controller based on the suggested dynamic (deterministic) observer system. Computational algorithms based on the presented theorems for design a deterministic stable controller have also been discussed and developed. Illustrations are presented to demonstrate the validity of the presented procedure.