Abstract
This paper proposes new stabilizing control laws for a variable length pendulum. By introducing a Lyapunov-type nonnegative function representing the magnitude of the angular oscillation of the pendulum, a stabilizing condition for the pendulum is derived that is more general than the one proposed by Yoshida et al. Two stabilizing control laws are developed on the basis of this condition. One is a control law that changes the velocity of the weight along the pendulum sinusoidally, which can reduce the load on the actuator and has good robustness properties to the nonlinearities and noise of the system. The other is a control law that performs a bang-bang control of the acceleration of the weight to approximate the optimal control law that maximizes the damping ratio of the angular oscillation of the pendulum. Furthermore, a formula is derived to evaluate the ratio between adjacent amplitudes of the oscillation for a given velocity profile of the weight, and then the ratios for the proposed control laws are evaluated. Numerical and experimental results are given to show the validity of the proposed methods.
