• This is a Time-varying Continuous Approach.

The dynamics of an underactuated ship is

The coordinate transformations

are first used to transform the ship system to

where and . Then the following nontrivial coordinate transformations

are applied to the transformed ship system to obtain the new system:

Let and be strictly positive constants such that . The control and are designed in the paper as

where

It is proven that the closed loop system is GAS at the origin in the paper.

Remark: The stabilizer design mentioned above is nontrivial, it is difficult to extend the control design scheme to solve a trajectory-tracking problem, and the convergence of the stabilizing errors to zero is slow. In addition, the physical meaning of feedback is not clear.

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@inproceedings{mazenc2002global,
title={Global uniform asymptotic stabilization of an underactuated surface vessel},
author={Mazenc, Fr{\'e}d{\'e}ric and Pettersen, Kristin Ytterstad and Nijmeijer, Henk},
booktitle={Proceedings of the 41st IEEE Conference on Decision and Control, 2002.},
volume={1},
pages={510--515},
year={2002},
organization={IEEE}
}