Global stabilization of unstable ship dynamics using PD control

The particular properties of unstable ships force the use of nonlinear mathematical models. A first-order nonlinear model of a ship with a proportional-derivative (PD) controller used as an autopilot is investigated. Thus, a second-order closed-loop system is obtained. Stability properties of nonlinear systems are, in general, difficult to prove. The method adopted here is Lyapunov´s direct method as given by W. Hahn, (1967). This method is particularly suited to second-order systems; however, normally local stability only can be proved by a Lyapunov function. In this case it is possible to find a one-parameter family that proves global asymptotic stability. As a result, the nonlinear system is stable whenever the linearized system is stable. In addition to this an estimation of the exponential decay rate is given