Justifying the Stabilization of a Marginally Stable Ship

As a precursor to capsize, marginal stability, resulting from incorrect loading conditions and crew negligence, poses a serious danger to ships. Therefore, as a benchmark problem for preventing capsize, the use of an actively controlled pendulum for the stabilization of a marginally stable ship was analyzed. Lyapunov stability criteria and closed loop eigenvalues were used to evaluate the extent to which a proposed pendulum controller could cope with different ship stability conditions. Equations of motion were solved to observe the controller’s performance under different damping conditions. The behavior of the controller yielded the following results: a marginally stable ship can be stabilized, as long as there is no right hand plane zero; energy dissipation is key to the stabilization of a marginally stable ship; the controller must have knowledge of the ship’s stability to prevent controller-induced excitation; and a stabilized tilted ship is more robust to external disturbances than a stabilized upright ship. Nomenclature m Ship Mass including Pendulum Mass Ixx Ship Rotational Inertia mcw Pendulum mass g Gravitational Acceleration φ Roll Angle φ̇ Roll Anglular Velocity of Ship θcw Pendulum Angle θ̇cw Pendulum Angular Velocity θcw,re f Reference Pendulum Angle broll Ship Roll Damping bpend Pendulum Damping Tpend Torque on the Pendulum Tpend,eq Pendulum Torque that Satisfies Equilibirum zg z-coordinate of the Center of Mass (COM) ρw Density of Displaced Fluid A Ship Displaced Volume BoG Distance btw. Centers of Mass and Buoyancy at φ = 0◦ BoMx Distance btw. Buoyancy Center and Metacenter GMx Metacentric Height GZ Righting Arm Lcw Length of Pendulum Lo f f Pendulum Offset k1 Nonlinear Feedforward Reference Gain k2 Linear Feedback Gain kp Proportional Feedback Gain zship Potential Right Half Plane Zero of Ship pship Right Half Plane Pole of Ship Emech Total Mechanical Energy KE Total Kinetic Energy PE Total Potential Energy