## 2 Related Works  
Tuned Mass Dampers (TMDs) have proven highly effective across various applications, particularly in the construction of buildings and bridges. In recent decades, they have also been utilized to mitigate wind turbine vibrations. Typically, TMDs are mounted in the nacelle of bottom-fixed turbines, both offshore and onshore. In contrast, modern floating offshore wind turbines (FOWTs) have experimented with various designs, employing single and multiple TMDs installed in the tower, platform, or nacelle. Most designs use passive TMDs, which, when properly tuned, significantly reduce vibrations' amplitudes. There exist several published examples of passive TMDs applied to FOWTs. For example, the study in [3] compares tuned TMDs for a 5 MW barge-type NREL FOWT, where the spring and damper parameters are optimized using genetic algorithms (GAs) and the natural frequency. Optimizing the TMD parameters with GA improves the vibration suppression rate, using the standard deviation of the tower top displacement (TTD) as the cost function. In [8], passive TMDs installed on the platform and in the nacelle of a semi-submersible FOWT are optimized using GA across three different configurations, aiming to reduce TTD. [PERSON] et al. [12] developed a reduced dynamic model of a Multi-TMD (MTMD) equipped FOWT platform, considering tower bending and barge degrees of freedom. Using GA, they optimized three TMD configurations: one in the nacelle, one on the platform, and the proposed MTMD. The results showed that the MTMD design outperformed the others in reducing vibrations from external loads. [PERSON] et al. [13] analyzed the performance of a 5 MW barge-type FOWT using a TMD optimized with GA. This study included stops to limit the TMD stroke, which were incorporated into the design of a passive structural controller installed in the nacelle. The optimized FOWT showed reduced collective pitch-tower bending mode vibrations. In [14], a TMD was designed and used for a 5 MW NREL monopile FOWT. In this case, GA and the interval approach were used to tune the passive dampers, demonstrating improved vibration attenuation, as indicated by a reduction in the mean square of rotor acceleration and displacement. An alternative approach was presented in [9], where the TMD damping and stiffness parameters are modeled as static controllers, with gains adjusted according to specific FOWT performance requirements. These criteria are formulated using a convex optimization program and solved with an iterative Linear Matrix Inequality (LMI) process. In this case, the TMD was installed in the nacelle. Although semi-active and active controllers are less commonly used, ref. [15] explores their application. Research on semi-active controls is limited, especially in wind energy [4]. However, as shown in [16], a semi-active TMD can perform comparably to an Hoo-based active TMD control, surpassing the passive TMD approach. The study used a model-free adaptive control strategy with a quadratic programming module, adjusting the damping coefficient and spring stiffness of the semi-active TMD in real time. The semi-active TMD was mounted facing upwind, reducing vibrations by minimizing stress on the floating platform's pitch angle. [PERSON] and [PERSON] [17] proposed several semi-active magnetorheological liquid column dampers (MR-TLCDs), designed to maximize energy dissipation and reduce vibrations. The multi-blade coordinate transformation employed in the structural control design converted system matrices into a non-rotating frame, followed by temporal averaging for the application of linear control rules. In [PERSON]'s Ph.D. dissertation [4], a semi-active control for a tower TMD is developed using ground hook (GH) control. This sub-optimal control strategy is straightforward to implement, as it relies on structural feedback. The TMD damper operates similarly to magnetorheological (MR) dampers. In [18], an MR damper modeled using the Bouc-Wen approach is adjusted for an offshore wind turbine.