  A deep-sea construction vessel is used to install underwater parts of an
off-shore oil drilling platform at the designated locations on the seafloor. By
using extended Hamilton's principle, a nonlinear PDE system governing the
lateral-longitudinal coupled vibration dynamics of the deep-sea construction
vessel consisting of a time-varying-length cable with an attached item is
derived, and it is linearized at the steady state generating a linear PDE
model, which is extended to a more general system including two coupled wave
PDEs connected with two interacting ODEs at the uncontrolled boundaries.
Through a preliminary transformation, an equivalent reformulated plant is
generated as a 4*4 coupled heterodirectional hyperbolic PDE-ODE system
characterized by spatially-varying coefficients on a time-varying domain. To
stabilize such a system, an observer-based output-feedback control design is
proposed, where the measurements are only placed at the actuated boundary of
the PDE, namely, at the platform at the sea surface. The exponential stability
of the closed-loop system, boundedness and exponential convergence of the
control inputs, are proved via Lyapunov analysis. The obtained theoretical
result is tested on a nonlinear model with ocean disturbances, even though the
design is developed in the absence of such real-world effects.
