This paper focuses on the analysis of the nonlinear automatic gain control (NAGC) in order to make a mass oscillate at its natural frequency with a specified amplitude. Despite their widespread use in Micro Electro Mechanical Sensor gyroscopes, ensuring the stability of this oscillation remains challenging. In this paper, we investigate the stability of two versions of the NAGC: an ideal and an implementable one, using appropriate change of variables and transverse contraction criterion from orbital stability theory. The results are validated through a numerical example obtained from a real gyroscope system.