Robustness of Vibrational Control in the Presence of Additive Disturbances

Vibrational control seeks to stabilize an unstable system by judiciously injecting a state-free high-frequency dither. This paper presents some robustness properties of vibrational control with respect to additive disturbances in a nonlinear system. We assume that, without disturbances, the appropriately averaged system is regionally asymptotically stable. Using perturbation techniques, our first result shows that the stabilization realised through vibrational control is robust with respect to additive disturbances of sufficiently small amplitude. Indeed, the perturbed system of vibrational control has a robustness feature similar to input-to-state stability in a local region. In the case of periodic disturbances, our second result indicates that vibrational control naturally dampens disturbances of sufficiently high frequency, which allows for relatively high amplitude disturbances. A tight bound for the effect of such periodic disturbances on the ultimate deviation of states from the desired equilibrium is presented. Simulation results from a planar manipulator support the theoretic analysis.