  In many learning based control methodologies, learning the unknown dynamic
model precedes the control phase, while the aim is to control the system such
that it remains in some safe region of the state space. In this work our aim is
to guarantee safety while learning and control proceed simultaneously.
Specifically, we consider the problem of safe learning in nonlinear
control-affine systems subject to unknown additive uncertainty. We model
uncertainty as a Gaussian signal and use state measurements to learn its mean
and covariance. We provide rigorous time-varying bounds on the mean and
covariance of the uncertainty and employ them to modify the control input via
an optimisation program with safety constraints encoded as a barrier function
on the state space. We show that with an arbitrarily large probability we can
guarantee that the state will remain in the safe set, while learning and
control are carried out simultaneously, provided that a feasible solution
exists for the optimisation problem. We provide a secondary formulation of this
optimisation that is computationally more efficient. This is based on
tightening the safety constraints to counter the uncertainty about the learned
mean and covariance. The magnitude of the tightening can be decreased as our
confidence in the learned mean and covariance increases (i.e., as we gather
more measurements about the environment). Extensions of the method are provided
for Gaussian uncertainties with piecewise constant mean and covariance to
accommodate more general environments.
