String Periodic Steady-State Stabilization via Boundary Sliding Mode Limit Cycle Generation

Periodic motion generation in flexible cables via boundary actuation aims to address a practical challenge in resolving issues related to cable entanglement or its trapping. Thus motivated, the present research explores the use of self-wave propagation based on the Van der Pol reference model approach to generate a limit cycle at the actuated boundary. To address the problem in question a hyperbolic PDE (partial differential equation) string model with an appropriate dynamic boundary actuation is adopted as a case study. For generating a desired in-domain periodic steady-state motion a boundary vander-Pol-based reference model, possessing a limit cycle, is brought into play. A SOSM (second-order sliding mode) algorithm is then applied to track the proposed reference model at the actuated boundary. The resulting closed-loop system is demonstrated to asymptotically converge to a periodic steady-state motion. The performance and robustness features of the developed design are supported by numerical simulations.