## 1 Introduction
In this study, a state feedback linearization control scheme is proposed for an unconventional hourglass-shaped heaving point absorber. An hourglass-shaped, point absorber buoy was considered in three recent articles by [PERSON] et al. [17; 20; 21] and is the motivation for this paper. In [20], the authors developed the hourglass-shaped buoy's nonlinear hydrostatic force term and compared its energy extraction performance to a cylindrical buoy. Using the results from [35], where a cubic term was inserted into the cylindrical buoy's control strategy, the authors recreated the effect naturally through the hourglass shape of the buoy. They showed that the hourglass buoy provided the same reactive power management as the nonlinear control of [35] using a velocity feedback control law. In [17], the hourglass buoy control was extended further using the Hamiltonian surface shaping and power flow control (HSSPFC) technique. The model for the hourglass buoy was the same as [20] where again, the dynamic Froude-Krylov force was not included. The energy extraction performance of the hourglass buoy was compared to a cylindrical buoy with an artificially applied cubic nonlinearity. A key result was that the hourglass buoy, containing the natural cubic nonlinearity, outperformed the cylindrical buoy with the artificial cubic term. In [21], a rate feedback control strategy was used for an hourglass-shaped buoy point absorber capable of adapting its cone angle based on incident wave conditions. Using an irregular wave example, it was shown that the energy extraction increased by 50% when the cone angle was varied. The model used for the simulation was the same as described previously. The goal of this paper is to explore the effect of the nonlinear dynamic Froude-Krylov force, derived using the approach of [PERSON] and [PERSON] [27], on an hourglass-shapedbuoy point absorber. We first examine the buoy's simulated response to regular waves with and without the dynamic FK force in the model. It is shown that neglecting this term can lead to underprediction of the buoy motion in some cases. Thus, it should be included in the analysis and model-based control system design. Adjusting the draft could potentially increase the energy extracted, as shown on a spherical buoy in [36]. However, for the hourglass-shaped buoy, varying the draft leads to additional terms in the static and dynamic Froude-Krylov force expressions. Therefore, in this study, the considered buoy's draft is at its vertical midpoint in all cases. A feedback linearizing controller is used to investigate further the effect of including the dynamic FK force term in the hourglass-shaped buoy's PTO control law. It should be noted that this control strategy is provided merely as an example of how the nonlinear model can be used in a model-based control strategy. A sinusoidal reference trajectory is used but is not presumed to be optimal from an energy extraction perspective. Finally, the closed-loop energy extraction performance of hourglass and spherical buoys are compared using the previously mentioned feedback linearizing controller, where the nonlinear models are used for each of their control laws. This is performed to illustrate the unique behavior of the hourglass shape to help explain why it extracts greater energy than a similarly sized spherical buoy. This article has three contributions. First, the closed-form dynamic Froude-Krylov force expression for an hourglass-shaped buoy was developed. Second, we show that the dynamic Froude-Krylov force of an hourglass-shaped buoy is not negligible and should be considered during analysis and control system design. Third, the closed-loop energy extraction of an hourglass point absorber is greater than a similarly sized spherical device. The remainder of this paper is organized as follows. Section 2 describes the model used by the simulation and the control law, including developing the hourglass buoy's FK forces. Section 3 shows the contributions of the static and dynamic FK forces to the buoy's free response.