This work aims at designing control algorithms for an under-actuated compliant adaptive prosthetic hand to perform a complex task of grasping and manipulating of an unknown object. As multiple objectives are considered, if one controller is designed for one control objective, a hybrid controller is needed to coordinate different controllers to perform complex tasks. This work thus tries to apply recently developed framework of System of Funnels to the underactuated compliant adaptive prosthetic hand to design this hybrid controller. In order to apply System of Funnels, each controller has to have the flexibility to obtain any given domain of attraction by tuning its parameters. This paper proposed nonlinear observer based PID controllers for Grasp Approach and Grasp Stabilization. The nonlinear observer estimates the needed velocity of the under-actuated compliant adaptive prosthetic hand. By tuning parameters of PID controller and nonlinear observers appropriately, any desired domain of attraction for each control objective can be achieved. Our next step is to design some manipulation controller with adjustable domain of attraction in order to apply the framework of System of Funnels.