Damping Angular Oscillations of a Pendulum under State Constraints

This paper considers the problem of damping the oscillation of a plane pendulum by moving the pivot in the vertical plane and the weight along the rod of the pendulum, with limited amplitudes. Conditions for stability of the motion of the pendulum are derived using energybased methods, and based on them energy-based controls satisfying the amplitude constraints are developed, which move the pivot and weight sinusoidally and achieve an excellent control performance by controlling the variables with constant amplitudes. When the oscillation of the pendulum becomes sufficiently small, each of the energy-based controls is taken over by a linear or saturating control, which also satisfies the amplitude constraint, to regulate the entire state of the system to the nominal point. The saturating control is used to move the pivot horizontally, which also effectively damps the residual oscillation of the pendulum. Numerical and experimental results are given to demonstrate the effectiveness of the proposed control laws.