The new repetitive learning for the control of the periodic motion has already been developed. The structure and parameters of the dynamic system don't need to be known exactly by using this controller, in advance. And it will be shown the new repetitive learning controller is more efficiency than the traditional controller. In this study, it will be proved the system using new repetitive learning controller is global asymptotic stability under the ideal condition. It will be also shown the system is robust stability under the nonideal condition. Lastly, the noncircular cutting system that will be illustrated to show the efficiency of stability for the repetitive learning controller is conducted.