## 1 Introduction  
A tuned mass damper (TMD) [1; 2; 3; 4; 5], an active mass damper/driver (AMD) [6; 7; 8; 9] and an active tuned mass damper (ATMD) [10; 11; 12; 13] are often applied to reduce the structural dynamic responses in civil engineering. The performance of an AMD system is theoretically superior to other forms. However, its applications are relatively few. The main unfavorable factor restricting its development is the time-delay effect. Simultaneously, the AMD system with an auxiliary mass needs an over-capacity driving equipment. Instead, the AMD system with multiple subsystems is more applicable in the civil engineering structures [14; 15].  
The research on time-delay compensation has aroused widespread attention. For instance, the stability of a time-delay control system was analyzed through a linear matrix inequality (LMI) approach [16; 17]. According to a pole-assignment method, a compensation controller was proposed for the systems with certain time delays [18]. However, a time-delay should be regarded as a time-varying variable [19; 20; 21; 22], mainly resulted from the structural response delays, the monitoring time delays of sensors, etc. A reduced-order controller with guaranteed cost control (GCC) algorithm was performed to compensate for the long control force calculation time delays of high-rise buildings [23]. Moreover, the compensation method for the actuator response time delays constituted the important research content [22]. The control-structure interaction (CSI) effect causes the actuator response time delays. This effect has time-varying characteristics and exists between the active control systems and their target structures [24]. In short, the current research focus on the compensation of a single time-delay system. It is essential to design a suitable algorithm to compensate for the time-varying delays.  
In addition to studying the time-delay compensation of a single time-delay system, appropriate compensation control gains are also needed to consider the multi-time-delaycharacteristics. For example, the Lagrange equation of a dissipative system was used to investigate the stability and bifurcation of a system with multi-time delay feedback [25]. A robust control strategy was programmed for discrete-time systems with non-equal time delays, and the input delays were considered in the control law synthesis [26]. Based on Takagi-Sugeno fuzzy controller design techniques, to derive the stability conditions of a multiple time-delay system, a global state-feedback nonlinear controller was constructed [27]. According to linear matrix inequalities (LMIs), the stability and robust H\({}_{\infty}\) controller for time-delayed systems were investigated [28]. The reference [29] was concerned with the H\({}_{\infty}\) synchronization control problem for a class of chaotic systems with multiple delays. However, the time delay of each subsystem is not equal since the performance of each subsystem is different. Thus, the performance of the whole system is reduced due to each subsystem works asynchronously. In conclusion, it is required to design a suitable control gain for the multi-time-delay characteristics of the AMD control system with multiple subsystems. Specially, the method with an H\({}_{\infty}\) control law, which enhances both performance and robustness of these systems, needs to be widely studied.  
For the impact analysis of multi-time-delay on the control parameters, the AMD systems with a single-controller or with multi-controllers are established for buildings. Then, based on an H\({}_{\infty}\) control law, a new compensation gain is devised for a multi-time-delay system. A multi-story frame with an AMD system is conducted to prove the efficiency of the proposed compensation controller.