Stable Limit Cycles Generating in a Class of Uncertain Nonlinear Systems: Application in Inertia Pendulum

This paper considers the problem of stable limit cycles generating in a class of uncertain nonlinear systems which leads to stable oscillations in the system’s output.This is a wanted behavior in many practical engineering problems. For this purpose, first the equation of the desirable limit cycle is achieved according to shape, amplitude and frequency of the required output oscillations. Then, the nonlinear control law is designed such that the phase portrait of the closed-loop system includes this stable limit cycle. The design of controller is based on the Lyapunov stability theorem which is suitable for stability analysis of the positive limit sets (the stable limit cycle is a positive limit set for the nonlinear dynamicl system). The proposed robust controller consists of two parts: nominal control law and additional term which guarantees the robust performance and vanishing the effect of uncertain terms. Finally, to show the applicability of the proposed method, an inertia pendulum system (with parametric uncertainties in its dynamical equations) is considered and the robust output oscillations are achieved by creating the desirable limit cycle in the close-loop system.