## 1 Introduction
[PERSON] [21] presented an analytical solution for \(H_{\infty}\) tuning of the ATMD utilizing state feedback from the ATMD mass. [PERSON] et al. [22] employed the \(H_{\infty}\) design to an ATMD with absolute displacement feedback and, in another variant, also with relative displacement feedback. [PERSON] et al. [23] used absolute displacement and relative velocity feedbacks in their optimal ATMD design. However, none of the existing studies present the optimal equal-peak design of ATMD using only internal feedback, specifically from relative displacement and relative velocity, without relying on absolute displacement or acceleration. In this work, the control approach by [PERSON] et al. will be adapted to implement an ATMD using exclusively a relative displacement and velocity feedback. This concept will also be applied to the STMD. Regarding STMDs, a significant advantage of these systems is their substantially lower power consumption compared to ATMDs. Additionally, STMDs are inherently stable. The STMDs consist of a mass, a passive spring, and a semi-active damper, for example a magneto-rheological (MR) damper. A salient feature of MR dampers is their short response time [24; 25]. Various designs and operational principles of MR dampers are well-documented [26]. Rotary MR dampers have been successfully implemented in 12 STMDs, each weighing 5200 kg, to mitigate bridge vibrations [27; 28]. MR damper-based STMDs have also been employed in vibration reduction systems for tall buildings [29; 30]. Numerous semi-active control approaches for STMDs have been developed. In the context of harmonic vibrations, effective STMD concepts are based on the semi-active emulation of positive or negative dynamic stiffness, aiming to tune the STMD's operation to the actual frequency of vibration [31; 32]. The semi-active control force of STMDs can be determined using a clipped linear quadratic regulator (LQR) [33]. Other methods include ground hook control schemes [34; 35], on-off phase control algorithms [36; 37], linear quadratic Gaussian control (LQG) [38], nonlinear optimal-based control [39; 40], PID control [41], and more. In previous work [42], the principles of the optimal active control by [PERSON] et al. [17] are utilized to formulate a semi-active control approach for the STMD with absolute acceleration and relative motion (displacement and velocity) feedback. As a result,an optimized STMD was proposed, denoted as STMD-K, in which the parameter \(K\) defines the efficiency of STMD in such a way that for a given \(K\) value, the damping efficiency for STMD-K is the same as for TMD with a mass \(K\) times larger. In [43], a preliminary experimental validation of the STMD-K concept is presented. It was observed that the efficient implementation of the STMD-K concept requires a high-quality acceleration signal. In practice, while the acceleration signal is easily accessible, it may contain high-frequency components that make proper control implementation challenging. The purpose of this paper is to adapt the optimal acceleration feedback control by [PERSON] et al. for its alternative implementation in both ATMD and STMD, using only relative displacement and relative velocity feedback, without relying on acceleration feedback. Simultaneously, following the approach in [42], for a specified design parameter \(K\), we aim to calibrate the control force parameters for both ATMD and STMD such that the efficiency of both systems matches that of a TMD with a \(K\)-times larger mass. The subsequent section provides an essential introduction to [PERSON]'s TMD, viewed as a reference mass damper. In Section 3, a control algorithm for ATMD is presented. The section starts with the definition of its control force, then introduces the control objective and the equations of motion. This is followed by a reformulation of [PERSON]'s acceleration feedback control, aiming for its alternative implementation without acceleration feedback and using only relative displacement and velocity feedback.