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.<>