Nonlocal Nonholonomic Source Seeking Despite Local Extrema

In this article, 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 toward a global extremum. In contrast to previous extremum-seeking studies, we do not approximate the gradient of the objective function but the gradient of a certain local spatial average of the objective function. 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.