## 1 Introduction  
The mitigation of vibrations of civil structures is vital for ensuring the safety and comfort of their users. Passive tuned mass dampers (TMDs) are commonly employed to reduce vibrations of buildings [1; 2], bridges [3], and other structures. A notable example is the 660-ton TMD installed in the Taipei 101 Tower [4], which also serves as a popular attraction, offering visitors a chance to observe its operation. The classical TMD [5] consists of a moving mass, a spring, and a viscous damper, all tuned to a single vibration mode of the structure. The effectiveness of the TMD depends on its mass, more specifically, the mass ratio which defines the TMD's mass in relation to the modal mass. Using a sufficiently large TMD mass ensures that vibrations are reduced to levels compliant with both technical standards and occupant comfort requirements [6]. The mass of TMDs installed in tall buildings typically amounts to about 1%, or less, of the modal mass. While this percentage may seem small, it translates to a TMD mass that is often in the order of hundreds of tons [1]. Installing large TMD masses is not only expensive but also difficult due to the limited space available for installation. One example is the 432 Park Avenue residential tower in New York, USA, where the required TMD mass of 1200 tons had to be split into two 600-ton TMDs with an innovative design to fit into two narrow spaces available on each side of the tower core [7]. As a result, there is a significant demand for innovative solutions that can provide the required vibration damping efficiency with less mass than a standard TMD. This challenge also serves as the primary motivation for the research presented in this work. Achieving the desired level of vibration reduction using less mass than a traditional TMD is possible through the application of active and semi-active tuned mass dampers (ATMD and STMD, respectively). ATMD systems are essentially TMDs of varying designs that incorporate an active actuator. On the other hand, STMDs employ semi-active dampers. The exploration of these systems in civil engineering was initiated roughly 50 years ago by the structural control concept of [PERSON] [8] and pioneering research activities of [PERSON] [9]. One of the earliest ATMD designs was presented by [PERSON] and [PERSON] in 1980 [10]. Pioneering implementations of ATMDs took place in Japan, with the world's first application of two ATMDs in a ten-story office building in Tokyo in 1989 [11]. Two decades later, in 2009, the list of implementations in buildings in Japan included 40 active and 13 semi-active control systems of various designs [12]. Research progress worldwide has been reviewed in, e.g., [13; 14]. Most recent studies have demonstrated the effectiveness of ATMDs in mitigating the dynamic response of tall buildings subjected to strong winds [15; 16]. Although numerous ATMDs have been implemented, further research into their design, control algorithms, and performance continues. For passive TMDs, increasing their mass naturally results in a change in the force exerted on the structure. To achieve an efficiency for ATMD or STMD comparable to that of a heavier TMD, it is evident that the total force (comprising both the control force generated by actuating elements and the force from the passive elements) in the ATMD or STMD system must act on the structure in a manner similar to a larger-mass TMD. This principle was first applied in [PERSON]'s research group, by [PERSON] [17]. They introduced an optimal acceleration feedback control for ATMD using the classical \(H_{\infty}\) synthesis method (the so-called equal-peak design), consistent with the approach used in the optimal TMD tuning [5; 18; 19; 20]. It is noticeable that their optimal ATMD exhibits a frequency response characteristic like that of a TMD with a larger mass. Subsequent research indicated that a similar ATMD performance can be achieved using a different set of feedback signals.