Growth rate of a pendulum with a time-varying length

Stability of non-autonomous system is in general difficult to analyze and the frozen in time assumption is known to break down in high dimensional system. This work proposes another criterion to analyze the stability of non-autonomous system by obtaining its upper bound on the energy growth. The computation of upper bound is then formulated as a convex optimization problem based on linear matrix inequality. We also perform pendulum experiments with a linear varying or oscillating length to analyze its stability. The experimental results demonstrate that the time-varying length of pendulum will lead to a growing oscillation amplitude of pendulum and the associated growth rate also agree well with the upper bound prediction based on the linear matrix inequality. This work provides a new analysis framework to analyze stability of non-autonomous system.