The present work is part of the "Hand of Hope" project, which seeks to develop low-cost robotic hand prostheses, with the aim of contributing to the social and labor inclusion of people with upper limb motor disabilities living in developing countries. The objective of this work is to design and deploy the grasp control system to enable a robotic prosthesis to perform object catch functions. For purposes of evaluating the system, five grasp elemental functions are defined: Relaxed hand, Cylindrical grip, Pinch grip, Thumb adduction, and Index finger extended. In the design phase it was estimated and compared the mathematical model of the unstable system based on two experimental methods: model based on a reference transfer function, and model based on the inverse Laplace transform of the output signal. In the implementation phase, a PI control function based on Ziegler-Nichols closed-loop method was tuned. Finally, the results were evaluated in a prototype of robotic hand prosthesis at UTPL-Ecuador, the results obtained are within the defined control requirements: the settling time is equal to 1.075 seconds, the overshoot, and the steady state error are zero.