Swing-up and Balance Control of Rotary Double Inverted Pendulum

In this paper, we study the swing-up and balance control for the rotary double inverted pendulum (RDIP) system. In the swing-up phase, we design an energy-based swing-up control for the RDIP system and discuss the convergence of energy. By using the LaSalle’s invariance principle and some properties of the RDIP system, we prove that its total mechanical energy will converge to that at the upright equilibrium point (UEP) and the rotary arm will approach a desired position with zero angular velocity, or the system will remain at one of the up–down, down–up, and down–down equilibrium points. Moreover, we prove that these three equilibrium points are unstable. In balance control, with the aid of the properties of the mechanical parameters of the double pendulum, we prove that the system is linearly controllable at the UEP without any assumption on the mechanical parameters, which allows us to design a linear state feedback controller to balance the system around the UEP. The simulation results demonstrate the effectiveness of the proposed controller.