## 2 Establishment of the Multi-Time-Delay Control System  
### Time-Delay Sources  
As shown in Figure 1 about the workflow of a control system, the time-delay resulted from different types of time-delays. A new reduced-order controller for buildings is conducted to compensate for the time-delay \(d_{4}\)[23]. Moreover, the compensation method for the time-delay \(d_{6}\) configurated the important research content [22]. The others (\(d_{1}\), \(d_{2}\), \(d_{3}\) and \(d_{5}\)) are described in this section. For a building, owing to the limited in-plane stiffness of a floor, the stress wave generated by a horizontal external excitation propagates in each floor. The propagation time is the structural response delay \(d_{1}\). The state-space equation is described as,  
\[\left\{ \begin{array}{c}\dot{Z}_{str}(t)=A_{str}Z_{str}(t)+B_{str}w(t)\\ Y_{str}(t)=C_{str}Z_{str}(t-d_{1})+D_{str}w(t-d_{1})\end{array} \right. \tag{1}\]  
where \(A_{str}\) is the state matrix, \(B_{str}\) is the excitation matrix, \(C_{str}\) is the state output matrix and \(D_{str}\) is the direct transmission matrix of a building. \(Z_{str}\) is the state vector and \(Y_{str}\) is the output vector. \(w\) is an external excitation. The structural response delay of each floor of a high-rise building is not equal, so the input equation of its control system cannot be simply described as the second equation of Equation (1) which in this article only describes this related parameter. In addition, the first equation of Equation (1) representing the dynamic equilibrium of the structure cannot consider the addition of the time delay \(d_{1}\), since it is contained directly in the equilibrium equation. In result, the structural response delay is not considered in this paper. Figure 1: The model of a control system. Taking a force balanced accelerometer as an example, the process of monitoring structural responses takes time which is called the monitoring time-delay of sensors \(d_{2}\). The state-space equation of a monitoring system is,  
\[\left\{ \begin{array}{c}\dot{Z}_{sen}(t)=A_{sen}Z_{sen}(t)+B_{sen}a_{str}(t)\\ U_{sen}(t)=C_{sen}Z_{sen}(t-d_{2})+D_{sen}a_{str}(t-d_{2})\end{array} \right. \tag{2}\]  
where \(A_{sen}\) is the state matrix, \(B_{sen}\) is the excitation matrix, \(C_{sen}\) is the state output matrix and \(D_{sen}\) is the direct transmission matrix of a monitoring system. \(Z_{sen}\) is the state vector and \(U_{sen}\) indicates the output vector of sensors. \(a_{str}\) is structural acceleration responses. The distribution of sensors in a building is scattered. There is a certain distance between the sensors and its control room. The required time from the output signals of sensors to their acquisition device is called the time-delay \(d_{3}\). The transmitted time from the control signals to the actuator is called the time-delay \(d_{5}\). According to the reference [30], since the speed of electromagnetic waves in a line is fast enough, which is about 2.45 m/s, the time delays (\(d_{3}\) and \(d_{5}\)) can be ignored. The measurement of acceleration signals is easier to be conducted than the displacement and velocity signals [31]. The acquisition device converts the received feedback voltage signals into the acceleration signals, and the measurement of acceleration signals is,  
\[a_{sen}(t)=U_{sen}(t)/K_{a} \tag{3}\]  
where \(K_{a}\) is the sensor sensitivity coefficient. In view of the acceleration feedback signals, a new state observer in the reference [31] is adopted to estimate the whole state vectors of an observer-based control system. The control force calculation time-delay \(d_{4}\) consists of two parts (\(d_{41}+d_{42}\)). The time required for the estimation process is called the time-delay \(d_{41}\). The state-space equation of the observer-based control system is,  
\[\left\{ \begin{array}{c}\dot{Z}_{obs}(t)=A_{obs}Z_{obs}(t)+B_{obs}a_{sen}(t) \\ Y_{obs}(t)=C_{obs}Z_{obs}(t-d_{41})+D_{obs}a_{sen}(t-d_{41})\end{array} \right.