Effect of Delay on Control of Direct Resonance of Ships in Beam Waves Using a Proportional–Derivative Controller With Delay

A harmonically excited, single-degree-of-freedom time-delay system with cubic and quintic nonlinearities is studied. This system describes the direct resonance of a ship with an actively controlled antiroll tank (ART) that is subjected to beam waves. We consider low-, medium-, and high-freeboard ship models. A proportional–derivative (PD) controller with a constant time delay is assumed to operate the pump in the active ART system. The delay originates from the time required to pump fluid from one container to another, the inertia of large impeller blades and linkages, and the measurement and processing time of the roll-sensing unit. The stability boundary of the system, in the parametric space of the control gain and the delay, is derived analytically from the characteristic equation of the linearized system. We show that the area of the zero equilibrium region is inversely related to the derivative time constant of the PD controller; thus, we focus on a strictly proportional-gain controller. The spectral Tau method is used to identify the eigenvalues associated with the zero equilibrium since the rightmost eigenvalues determine the system's robustness to perturbations in the initial conditions. We use the method of multiple scales and harmonic balance to obtain the global bifurcation diagram in the space of the applied frequency and the amplitude of the response. Numerical simulations verify our analytical expressions. Study of the dynamics, stability, and control of the roll motion of ships is critical to avoid dynamic instabilities and capsizing.