  Linear-quadratic regulators (LQR) are a well known and widely used tool in
control theory for both linear and nonlinear dynamics. For nonlinear problems,
an LQR-based controller is usually only locally viable, thus, raising the
problem of estimating the region of attraction (ROA). The need for good ROA
estimations becomes especially pressing for underactuated systems, as a failure
of controls might lead to unsafe and unrecoverable system states. Known
approaches based on optimization or sampling, while working well, might be too
slow in time critical applications and are hard to verify formally. In this
work, we propose a novel approach to estimate the ROA based on the analytic
solutions to linear ODEs for the torque limited simple pendulum. In simulation
and physical experiments, we compared our approach to a Lyapunov-sampling
baseline approach and found that our approach was faster to compute, while
yielding ROA estimations of similar phase space area.
