On the use of dynamic mode decomposition for time-series forecasting of ships operating in waves

In order to guarantee the safety of payload, crew, and structures, ships must exhibit good seakeeping, maneuverability, and structural-response performance, also when they operate in adverse weather conditions. In this context, the availability of forecasting methods to be included within model-predictive control approaches may represent a decisive factor. Here, a data-driven and equation-free modeling approach for forecasting of trajectories, motions, and forces of ships in waves is presented, based on dynamic mode decomposition (DMD). DMD is a data-driven modeling method, which provides a linear finite-dimensional representation of a possibly nonlinear system dynamics by means of a set of modes with associated frequencies. Its use for ship operating in waves has been little discussed and a systematic analysis of its forecasting capabilities is still needed in this context. Here, a statistical analysis of DMD forecasting capabilities is presented for ships in waves, including standard and augmented DMD. The statistical assessment uses multiple time series, studying the effects of the number of input/output waves, time steps, time derivatives, along with the use of time-shifted copies of time series by the Hankel matrix. The assessment of the forecasting capabilities is based on four metrics: normalized root mean square error, Pearson correlation coefficient, average angle measure, and normalized average minimum/maximum absolute error. Two test cases are used for the assessment: the course keeping of a self-propelled 5415M in irregular stern-quartering waves and the turning-circle of a free-running self-propelled KRISO Container Ship in regular waves. Results are overall promising and show how state augmentation (using from four to eight input waves, up to two time derivatives, and four time-shifted copies) improves the DMD forecasting capabilities up to two wave encounter periods in the future. Furthermore, DMD provides a method to identify the most important modes, shedding some light onto the physics of the underlying system dynamics.