ℒ1 adaptive and extremum seeking control applied to roll parametric resonance in ships

This paper proposes a robust control approach to stabilize roll parametric resonance, a dangerous nonlinear resonance phenomenon for ships. To that matter, we extend a recent extremum seeking speed and heading control strategy to stabilize roll parametric resonance in ships, aiming at robustness. Based on previous results where extremum seeking was shown to be capable of effectively reducing parametrically excited roll motions by a combined variation of the ship´s forward speed and heading angle, two major modifications are suggested. Firstly, the speed and heading controllers are formulated in the framework of ℒ₁ adaptive control which guarantees robustness while still having fast adaptation. By doing so, robustness is increased with respect to model uncertainty, lack of knowledge, and bounded external disturbances. Secondly, the extremum seeking loop is modified towards limit cycle minimization, which replaces the objective function of the previously presented control approach, thus relaxing the assumption that the frequency range for roll parametric resonance is a priori known. In simulations, the proposed overall robustification approach is shown to effectively stabilize the roll oscillations due to roll parametric resonance.