A Minimum Phase Output in the Exact Tracking Problem for the Nonminimum Phase Underactuated Surface Ship

This note considers the output tracking problem for a nonminimum-phase underactuated surface ship, where the actuation is provided by a stern propeller and rudder. As tracking output, we choose the position of a point placed on the longitudinal symmetry axis, at a positive distance from the position of the ship's geometric center. We show that the exact tracking problem for a smooth reference trajectory always admits a solution for this particular output and that the state space of the zero dynamics features an invariant region that lies in an arbitrarily small neighborhood of zero, provided that the reference trajectory is such that its speed does not vary too much with respect to an assigned reference value and its second derivative is sufficiently small. We present an output tracking controller which guarantees local asymptotic stability of the origin of the tracking error dynamics. Finally, we provide bounds on the zero dynamics' state by means of a positive invariant region of a discontinuous comparison system. We do not take explicitly into account environmental disturbances but present a simulation in presence of a nonrotational current.