Damping a pendulum's swing by string length adjustment - design and comparison of various control methods

Abstract

A novel nonlinear control theory based feedback controller is proposed to damp the oscillations of the suspended load (pendulum) using the active modification of the length of the suspension string. This setting is a highly nonlinear one since the approximate linearization around the equilibrium working point is neither controllable, nor asymptotically stabilizable. The nonlinear design of the control law is therefore based on the conveniently selected control Lyapunov function. The resulting control law is then compared to the previously developed timedelay feedback control law, both in simulations and using the laboratory experimental realization of the suspended load system. Despite the fact that in the simulations the time-delay feedback control law suppresses the oscillations better than the nonlinear control law, in the experiments the performance of the time-delay feedback and of the nonlinear control law are rather similar. Moreover, the former keeps the pendulum string length oscillating, the latter stabilizes the nominal string length as well. Finally, the numerical optimization shows that the ideal damping would be provided by the impulsive-like control producing piece-wise constant string length dynamics.

Algorithms for cable-suspended payload sway damping by vertical motion of the pivot base

“…. Note that analogous result has not been achieved so far for the alternative option when the base is fixed and the damping is achieved by varying the cable length [27].…”

“…The robustness in sway damping of the proposed solution was thoroughly studied, including experimental validation. An alternative control design by Lyapunov methods was presented in [26], see also a recent study [27] and references therein. Note that by the Lyapunov approach, the amplitude decay does not achieve exponential character, as it is the case for the former method.…”

Controlling the variable length pendulum: Analysis and Lyapunov based design methods

“…To start with, [24] and the references therein use the following representation ẋ1 = x 2 , ẋ2 = −x −1 3 (2x 2 u + g sin x 1 ) , ẋ3 = u, (4) x 1 := φ , x 2 := φ , x 3 := l, u := l.…”

“…which is neither controllable, nor stabilizable. Yet, (4) was asymptotically stabilized by a smooth feedback in [24,25]. Next, consider the following two-dimensional state space model…”

“…Yet, the variation of the gravity center position was attenuated there in some indirect way only. A Lyapunov based nonlinear control design was also used in [24], which considered a three-dimensional state space model and included a cable length penalty term into the Lyapunov function candidate. For the same three-dimensional model, a control strategy was developed in [25], relying on the concept of passivity, which allows to generate a broad set of Lyapunov function candidates providing various mutually complementary damping performances.…”