  In this work we propose and analyze a new framework to learn feedback control
policies that exhibit provable guarantees on the closed-loop performance and
robustness to bounded (adversarial) perturbations. These policies are learned
from expert demonstrations without any prior knowledge of the task, its cost
function, and system dynamics. In contrast to the existing algorithms in
imitation learning and inverse reinforcement learning, we use a
Lipschitz-constrained loss minimization scheme to learn control policies with
certified robustness. We establish robust stability of the closed-loop system
under the learned control policy and derive an upper bound on its regret, which
bounds the sub-optimality of the closed-loop performance with respect to the
expert policy. We also derive a robustness bound for the deterioration of the
closed-loop performance under bounded (adversarial) perturbations on the state
measurements. Ultimately, our results suggest the existence of an underlying
tradeoff between nominal closed-loop performance and adversarial robustness,
and that improvements in nominal closed-loop performance can only be made at
the expense of robustness to adversarial perturbations. Numerical results
validate our analysis and demonstrate the effectiveness of our robust feedback
policy learning framework.
