Generation of Stable and Robust Limit Cycle in the Uncertain Nonlinear Systems Using Sliding Mode Controller

Generation of stable oscillations in the dynamical systems is one of the most important practical issues that have been addressed in this paper with a new approach. For this purpose, a novel approach is proposed in designing the sliding surface, such that moving on the sliding surface leads to achieving stable limit cycle and consequently stable oscillations in the output of the uncertain closed-loop system. This is done without periodic reference signal (and its derivatieves) which is a common approach in solving the tracking problem in dynamical systems and is done using the concept of positive limit sets. For this purpose, considering a limit cycle as an invariant and positive limit set, the Lyapunov function that is suitable for stability analysis of the invariant sets (instead of stability analysis of equilibrium point) is selected in a different way depend on the geometric shape of the target limit cycle. Next, based on this Lyapunov function, the suitable virtual control input is designed in order to create the considered stable limit cycle in the reduced order equations. After that, the suitable sliding surface is selected according to this virtual control input and appropriate control input is designed to reach the sliding surface in the finite time and create the admissible oscillations in the output response. Simulation results on single link flexible joint robot demonstrate the proficiency of the proposed controller in generating stable limit cycle in the closed-loop robot system and creating stable oscillations in its output.