We consider safety in simultaneous learning and control of discrete-time
linear time-invariant systems. We provide rigorous confidence bounds on the
learned model of the system based on the number of utilized state measurements.
These bounds are used to modify control inputs to the system via an
optimization problem with potentially time-varying safety constraints. We prove
that the state can only exit the safe set with small probability, provided a
feasible solution to the safety-constrained optimization exists. This
optimization problem is then reformulated in a more computationally-friendly
format by tightening the safety constraints to account for model uncertainty
during learning. The tightening decreases as the confidence in the learned
model improves. We finally prove that, under persistence of excitation, the
tightening becomes negligible as more measurements are gathered.Comment: Accepted in NeurIPS 2021 Workshop on Safe and Robust Control of
Uncertain System