Orbital Stabilization of Inverted-Pendulum Systems via Coupled Sliding-Mode Control

In this paper, we propose a coupled sliding-mode control (SMC) method for the periodic orbit generation and the robust exponential orbital stabilization of inverted-pendulum systems. We first design an SMC law to force a coupled sliding surface to be reached in finite time, such that the zero dynamics are generated in the form of a second-order undamped and forced nonlinear differential equation. Through the stability analysis, it is shown that there exist exponentially stable periodic solutions of the resulting zero dynamics (i.e., limit cycles around either the upright or downward equilibrium), even in the presence of the matched disturbance. Second, we design a target orbit stabilization control law by further introducing an auxiliary control law to the designed SMC law. This auxiliary control law utilizes the general integral of the autonomous zero dynamics, which preserves its zero value along the given target orbit, and thus, it can contribute to the exponential stabilization of the general integral. To demonstrate the validity of the proposed method, both the periodic orbit generation and target orbit stabilization control of the cart-pendulum, as an example among inverted-pendulum systems, are performed in numerical simulations.