  We consider the problem of safe control design for a class of nonlinear,
control-affine systems subject to an unknown, additive, nonlinear disturbance.
Leveraging recent advancements in the application of Koopman operator theory to
the field of system identification and control, we introduce a novel fixed-time
identification scheme for the infinitesimal generator of the
infinite-dimensional, but notably linear, Koopman dynamical system analogous to
the nonlinear system of interest. That is, we derive a parameter adaptation law
that allows us to recover the unknown, residual nonlinear dynamics in the
system within a finite-time independent of an initial estimate. We then use
properties of fixed-time stability to derive an error bound on the residual
vector field estimation error as an explicit function of time, which allows us
to synthesize a provably safe controller using control barrier function based
methods. We conduct a quadrotor-inspired case study in support of our proposed
method, in which we show that safe trajectory tracking is achieved despite
unknown, nonlinear dynamics.
