Stabilization of Sets Parametrized by a Single Variable: Application to Ship Maneuvering

We consider the problem of stabilizing sets parametrized by a single variable. Our approach is based on the solution to the maneuvering problem. In this problem, the state of the system is driven to a path that coincides with the set, and the particular location on the path is determined dynamically from the value of the state. Within this control structure, we induce a separation of time scales between the task of selecting the point on the path closest to the state, and the task of driving the state towards the path. In addition to uniform global convergence to the path, which is achieved without a separation of time scales, the separation of time scales allows us to achieve near forward invariance of the path from a large range of initial conditions. This idea is first illustrated on a simple double integrator with the path corresponding to the unit circle, and then it is discussed for systems in strict feedback form. Finally, the theory is applied to the problem of maneuvering a ship into a harbour.