Stabilizing control for an inverted pendulum with restricted travel

For the problem of stabilizing an inverted pendulum with restricted travel, a saturating control law is developed that satisfies the amplitude constraint on the cart and has a large region of attraction. The explicit expression of the region of attraction is also obtained. The analysis and design are performed based on the linearized model of the system, as in the study of Lin et al. The control law has two design parameters: T and k, which are both positive. As T approaches 0, the region of attraction approaches the maximal one. These parameters are chosen to optimize the transient response of the closed-loop system considering the input constraint, observation noise, etc. The effectiveness of the control law is demonstrated by simulations and experiments.