Wave Excitation Force Estimation for a Multi-DoF WEC via a Cubature Kalman Filter: Improved Design and Results
Authors: Zhu Jiamin , Tona Paolino , Mérigaud Alexis 
Published: September 2023

In this paper, we develop a multiple degree-of-freedom (multi-DoF) wave excitation force estimator based on the square-root cubature Kalman filter (SCKF) [1] combined with an online empirical covariance matching technique for a CETO-like, submerged wave energy converter (WEC) device with three PTOs.
Though wave excitation force estimation is required by many advanced WEC control approaches, the literature is still quite scarce for multi-DoF devices. Indeed, there are several difficulties in designing a multi-DoF excitation force estimator, especially when the full motion of the WEC is considered. The high dimension of the underlying dynamic system leads to difficulties in on-line implementation and in estimator parameter-tuning. Moreover, nonlinearities may significantly deteriorate estimation performance if a linearized model is used by the estimator to reduce computation time. Papers [2-4] present solutions to multi-DoF wave excitation force estimation, ranging from (extended) Kalman filters to feedforward neural networks, but only up to three DoFs, the surge, heave and pitch directions. In [5], results for a CETO-like WEC are presented for all the six DoFs, based on a CKF wave estimator, which uses a linear state-space model. Undesirable large estimation errors on the yaw direction were observed for several sea states, mainly caused by the model errors introduced by linearizing the kinematics linking buoy motion to PTOs motions. Moreover, even though the CKF is known to be relatively easy to calibrate, guessing covariance matrices of the system and measurement model noises remains troublesome.
In this paper, we propose to use the SCKF, together with an empirical online covariance matching to over overcome the difficulty of parameter tuning. In our estimator, a nonlinear dynamic system of dimension 72 is used to predict the system dynamics at the cubature points, and the 6-dimensional excitation force is estimated. In addition, we show that a careful system state scaling can substantially reduce estimation errors on the yaw direction, even when using the linearized dynamic system. Moreover, thanks to the moderate computation complexity of the SCKF, and its parallelizability, online implementation can be achieved by taking slightly larger sampling time step, at the price of a small degradation in performance.
REFERENCES
[1] I. Arasaratnam and S. Haykin. “Cubature kalman filters”. In: IEEE Transactions on automatic control 54.6 (2009).
[2] Ling, B. A., 2015. “Real-time estimation and prediction of wave excitation forces for wave energy control applications”. Master’s thesis, Oregon State University.
[3] Hillis, A., Yardley, J., Plummer, A., and Brask, A., 2020. “Model predictive control of a multi-degree-of-freedom wave energy converter with model mismatch and prediction errors”. Ocean Engineering, 212, p. 107724.
[4] Bonfanti, M., Hillis, A., Sirigu, S. A., Dafnakis, P., Bracco, G., Mattiazzo, G., and Plummer, A., 2020. “Real-time wave excitation forces estimation: An application on the ISWEC device”. Journal of Marine Science and Engineering, 8(10).
[5] Nguyen, H.-N., Tona, P., Mérigaud, A., Cocho, M., Pichard, A. “Wave Excitation Force Estimation for a Multi-DoF WEC via a Cubature Kalman Filter: Design and Preliminary Results”, In Proc. of OMAE 2021, virtual, online.