## 2 Magnetorheological Smart Isolation System
### Control Algorithm and Strategy  
The HSA method is a structural dynamic time history analysis method, which has the characteristics of high accuracy and simple calculation consumption. In real-time control scenarios, the issue of time delay is often encountered. When the system uses only one step in time delay, the control force produced by the HSA method will have minimum time delay. When the delay time is equal to the multiple (e.g., \(N\)) integral steps, the method can estimate the state vector of the (\(N\)\(-\)1)th step, then generate the optimal control force of the \(N\)th time step for the structure by state vector calculation. The formula for the structure dynamic response can be expressed by the following HSA method:  
\[\left. \begin{array}{l}x_{n+1}=\mathbf{G_{11}}x_{n}+\mathbf{G_{12}}\dot{x}_{n }-\mathbf{Q_{12}}\ddot{x}_{n}+\mathbf{R_{12}}(\mathbf{1}\ddot{x}_{g,n+1}- \mathbf{M^{-1}}\mathbf{B}\mathbf{U}_{n+1})\\ \dot{x}_{n+1}=\mathbf{G_{12}}x_{n}+\mathbf{G_{22}}\dot{x}_{n}-\mathbf{Q_{22}} \ddot{x}_{n}+\mathbf{R_{22}}(\mathbf{1}\ddot{x}_{g,n+1}-\mathbf{M^{-1}} \mathbf{B}\mathbf{U}_{n+1})\\ \ddot{x}_{n+1}=-\mathbf{1}\ddot{x}_{g,n+1}+\mathbf{M^{-1}}(\mathbf{B}\mathbf{ U}_{n+1}-\mathbf{C}\dot{x}_{n+1}-\mathbf{K}x_{n+1})\end{array} \right\} \tag{1}\]  
The performance index function is defined as follows:  
\[\mathbf{J}=\frac{1}{2}\mathbf{x}^{T}W_{1}\mathbf{K}\mathbf{x}+\frac{1}{2} \mathbf{x}^{T}W_{2}\mathbf{M}\dot{x}+\frac{1}{2}(\mathbf{B}\mathbf{U})^{T} \mathbf{K^{-1}}\mathbf{B}\mathbf{U} \tag{2}\]  
where \(W_{1}\) and \(W_{2}\) are the weight parameters of adjustment control effect for the displacement and velocity response of the isolation layer, where I is the identity matrix. The specific parameter values can be determined by simulation calculation. Detailed calculation procedure is given in the literature [26]. By setting \(\delta\mathbf{J}_{n+1}=0\), yields  
\[\mathbf{B}\mathbf{U}_{n+1}=\mathbf{D_{1}}x_{n}+\mathbf{D_{2}}\dot{x}_{n}- \mathbf{D_{3}}\ddot{x}_{n}+\mathbf{D_{4}}\ddot{x}_{g,n+1} \tag{3}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Physical Quantity** & **Symbol** & **Dimension** & **Similar Coefficient** \\ \hline length & \(S_{l}\) & L & 1/5 \\ elastic modulus & \(S_{E}\) & ML\({}^{-1}\)T\({}^{-2}\) & 1 \\ stiffness & \(S_{k}\) & MT\({}^{-2}\) & 1/5 \\ acceleration & \(S_{a}\) & L/T\({}^{2}\) & 1 \\ time & \(S_{T}\) & T & \(1/\sqrt{5}\) \\ velocity & \(S_{v}\) & L/T & \(1/\sqrt{5}\) \\ displacement & \(S_{x}\) & L & 1/5 \\ mass & \(S_{m}\) & M & 1/25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The similar coefficient of the model. In order to realize the closed-loop control, by giving up \(\mathbf{D}\ddot{\mathbf{i}}\tilde{x}_{g,n+1}\), so each \(\delta\mathbf{J}_{n+1}\) is no longer zero. This is only an approximate optimization:  
\[\mathbf{BU}_{n+1}=\mathbf{D}_{1}x_{n}+\mathbf{D}_{2}\dot{\tilde{x}}_{n}- \mathbf{D}_{3}\ddot{\tilde{x}}_{n} \tag{4}\]  
where \(\mathbf{D}_{1}\), \(\mathbf{D}_{2}\) and \(\mathbf{D}_{3}\) are all constant matrices, and \(\mathbf{U}\) is the optimal control force vector for each single time step. The constant matrix \(\mathbf{B}\) indicates the location or configuration of the dampers within the structure [16]. \[u(i)= \begin{cases}u_{\max}&\text{ if }\quad f_{opt}\times\dot{x}_{b}<0\\ u_{\min}&\text{ if }\quad f_{opt}\times\dot{x}_{b}\geq 0\end{cases}  \tag{5}\]  
For an MR control system with variable damping, the control strategy is usually divided into two types. One is similar to the switch control mode. The voltage regulation for the MR damper is either maximum or minimum in temporal series. The other is the continuity control mode, which considers the characteristics with continuously adjustable damping force of the MR damper. The control strategy is used to regulate the optimal control force in real-time [27; 28; 29]. \[u(i)=\left\{