  Many existing region-of-attraction (ROA) analysis tools find difficulty in
addressing feedback systems with large-scale neural network (NN) policies
and/or high-dimensional sensing modalities such as cameras. In this paper, we
tailor the projected gradient descent (PGD) attack method developed in the
adversarial learning community as a general-purpose ROA analysis tool for
large-scale nonlinear systems and end-to-end perception-based control. We show
that the ROA analysis can be approximated as a constrained maximization problem
whose goal is to find the worst-case initial condition which shifts the
terminal state the most. Then we present two PGD-based iterative methods which
can be used to solve the resultant constrained maximization problem. Our
analysis is not based on Lyapunov theory, and hence requires minimum
information of the problem structures. In the model-based setting, we show that
the PGD updates can be efficiently performed using back-propagation. In the
model-free setting (which is more relevant to ROA analysis of perception-based
control), we propose a finite-difference PGD estimate which is general and only
requires a black-box simulator for generating the trajectories of the
closed-loop system given any initial state. We demonstrate the scalability and
generality of our analysis tool on several numerical examples with large-scale
NN policies and high-dimensional image observations. We believe that our
proposed analysis serves as a meaningful initial step toward further
understanding of closed-loop stability of large-scale nonlinear systems and
perception-based control.
