## 4 Modeling and Control of MR Dampers for Structural Buildings  
### Modeling of MR Dampers  
The structure of a building is exceptionally vulnerable to seismic loads, especially seismic loads at higher frequencies [172; 173]. Additionally, the control systems used are widely discussed [174; 175]. This section discusses models and controls that have been used in MR dampers in previous studies [176; 177]. Small-scale MR dampers are mostly described using the Bingham model due to its simplicity. This model consists of dashpot and friction elements connected in parallel, while the damper force is formulated as shown in Table 1.  
The control scheme based on the Bouc-Wen model presented by [PERSON] allows for real-time tracking of the desired control style. This control scheme is intended for MR damping devices without feedback from the force sensor. [PERSON] used several Bouc-Wen models to estimate the MR damper force by parallel computation with a constant current. The MR damper's current is determined by a piecemeal linear interpolation scheme. Numerical and experimental validations were carried out. Tests show that the real-time control scheme is numerically stable and a force tracking error of not more than 0.078 indicates acceptable accuracy. One of the model studies conducted by [PERSON] is shown in Figure 8 [178]. [PERSON] et al. conducted a MR damper modeling study to develop an alternative method for MR dampers. It used fuzzy self-tuning based on neural techniques as well as a control system that was validated using a number of simulations. This case study used the Bingham model to describe the damper's behavior. The Bingham model's characteristic parameters were determined depending on the needs of the experiment. Figure 9 shows the predicted damping force generated in the Bingham model in comparison with the experimental responses. When the acceleration had a negative value, the measured force had a positive value. Conversely, when the acceleration had a positive value, the measured force had a negative value. This occured when the velocity was zero. The self-tuning fuzzy model, a general model commonly used to improve MR dampers, was then utilized [177]. \begin{table}
**Model** & **Equation** & **Figure** \\ \hline \multirow{8}{*}{Bouc-Wen Model} & \(f=c_{1}\dot{y}+k_{1}(x-x_{0})\) & \\  & \(c_{1}\dot{y}=\alpha z+k_{0}(x-y)+c_{0}\big{(}x-\dot{y}\big{)}\) & \\  & \(\dot{z}=-\gamma|\dot{x}-y|z|z|^{n-1}-\beta(\dot{x}-\dot{y})|z|^{n}+A\big{(}\dot{x }-\dot{y}\big{)}\) & \\ \cline{1-1}  & \(k_{1}\) represents the damper stiffness of the accumulator & \\ \cline{1-1}  & and \(c_{0}\) is the dashpot coefficient associated with & \\ \cline{1-1}  & viscous damping at higher velocities. A dashpot \(c_{1}\) & \\ \cline{1-1}  & was included in the model to produce the roll-off & \\ \cline{1-1}  & observed in the experimental data at a low velocity. \(k_{0}\) & \\ \cline{1-1}  & is the stiffness control at a higher velocity, and \(x_{0}\) is the & \\ \cline{1-1}  & initial displacement of the spring \(k_{1}\) in relation to the & \\ \cline{1-1}  & nominal damper force due to the battery [180]. & \\ \hline \hline \end{tabular} \
\begin{tabular}{c c} \hline \hline \multirow{2}{*}{BingMax Model} & \(f(t)=k\;\int_{0}^{t}\exp\big{(}-\frac{t-T}{\lambda}\big{)}\dot{x}(\tau)d\tau+f_ {y}sgn\big{(}x(t)\big{)}\) \\  & where \(a=c/k\) is the quotient of the dashpot c and the spring \(k\), and \(f_{y}\) is the friction force on the slider [180]. \\ \hline \hline \end{tabular}
\begin{tabular}{c c} \hline \hline \multirow{2}{*}{LuGre Model} & \(F_{ss}(v)=g(v)sgn(v)+f(v)\) \\  & where \(g(v)\) is the Coulomb friction and the Stribeck \\ \cline{1-1}  & effect, and \(f(v)\) represents viscous friction. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._[PERSON] et al. also used a self-tuning fuzzy model and a Bingham model in their study on MR dampers. The study proposed a new parametric dynamic model for MR fluid absorbers. It adapted the Bingham model and was set up to accurately reproduce the hysterical behavior of the damper.