  In this paper, we investigate the problem of source seeking with a unicycle
in the presence of local extrema. Our study is motivated by the fact that most
of the existing source seeking methods follow the gradient direction of the
signal function and thus only lead to local convergence into a neighborhood of
the nearest local extremum. So far, only a few studies present ideas on how to
overcome local extrema in order to reach a global extremum. None of them apply
to second-order (force- and torque-actuated) nonholonomic vehicles. We consider
what is possibly the simplest conceivable algorithm for such vehicles, which
employs a constant torque and a translational/surge force in proportion to an
approximately differentiated measured signal. We show that the algorithm steers
the unicycle through local extrema towards a global extremum. In contrast to
the previous extremum-seeking studies, in our analysis we do not approximate
the gradient of the objective function but of the objective function's local
spatial average. Such a spatially averaged objective function is expected to
have fewer critical points than the original objective function. Under suitable
assumptions on the averaged objective function and on sufficiently strong
translational damping, we show that the control law achieves practical uniform
asymptotic stability and robustness to sufficiently weak measurement noise and
disturbances to the force and torque inputs.

Optimization and Control