Abstract
This research provides a solution to the challenging problem of enforcing a periodic or cyclic motion in underactuated systems, even when the upper position is an unstable equilibrium point of the open-loop system. Here, we present the solution to the tracking control problem of an underactuated cart-pendulum system, where the pendulum is forced to oscillate around its upward position. A two-relay controller was used to generate a desired trajectory, which is a locally asymptotically stable limit cycle, for the system in question. The tuning of the two-relay controller to produce the desired frequency and amplitude is also discussed. Furthermore, a twisting controller is designed to ensure the robustness properties of the closed-loop system, which is verified from Lyapunov stability analysis. The positions and velocities of the cart and the pendulum were assumed to be available for feedback. Numerical and experimental results validated the theoretical analysis.
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