# Earthquake Hazard Mitigation for Uncertain Building Systems Based on Adaptive Synergetic Control
## 2 Dynamic Model  
Figure 1 shows a one-story building structure model supported by an MR damper. The MR damper is fitted to the ground and a brace mounted on the first floor. Previous studies showed that fitting the MR damper on the first floor leads to more powerful control forces than does fitting it on upper floors [34; 35]. Seismically, this structure is susceptible to acceleration \(\ddot{x}_{g}\). The general dynamic motion for the structural system shown in Figure 1 can be described by  
\[M_{s}\ \ddot{x}+B_{s}\ \dot{x}+K_{s}\ x=M_{s}\ \ddot{x}_{g}+f_{MR} \tag{1}\]  
where the variables \(x\), \(\dot{x}\), and \(\ddot{x}\) represent the displacement, velocity, and acceleration of the base floor, respectively. The parameters \(K_{s}\), \(\mathrm{C}_{s}\), and \(M_{s}\) represent the stiffness, damping, and mass of the structural system, respectively. The earthquake data function is represented by \(\ddot{x}_{g}\), while the input force generated by the MR damper is indicated by \(f_{MR}\). The state variable representation of Equation (1) can be described as follows:  
\[\dot{x}_{1}=x_{2}\]  
\[\dot{x}_{2}=-\frac{B_{s}}{M_{s}}\ x_{2}-\frac{K_{s}}{M_{s}}x_{1}+\frac{1}{M_{s }}v+\ddot{x}_{g} \tag{2}\]  
where the control signal is denoted by \(v\) and the earthquake data function is denoted by \(\ddot{x}_{g}\). Figure 2 shows the constituting elements of the MR damper. As shown in the figure, the conventional damper changes its viscosity according to level of current flowing into the actuating coil. The actuating current is established by the control voltage excited by the controller. The nonlinear model of an MR damper was firstly established and developed by [PERSON]. Then, other models of MR dampers based on differential equations were developed by other researchers. These updated versions of MR damper models have been validated and verified in experimental tests [36; 37]. Figure 1: One-story building structure supported by an MR damper and vibrated by an earthquake. As shown in Figure 3, the Bouc-Wen model was used to augment the dynamic properties of the MR device. According to Figure 3, the dynamic equations characterizing the MR device can be written as [38]  
\[f_{MR}=c_{1}\,\dot{y}+k_{1}(x-x_{o}) \tag{3}\]  
\[c_{1}\,\dot{y}=c_{o}\,\left(\dot{x}-\dot{y}\right)+k_{o}(x-y)+\alpha\,z \tag{4}\]  
\[\dot{z}=-\gamma z\,\left|\dot{x}-\dot{y}\right|\,\left|z\right|^{m-1}-\beta \,\left(\dot{x}-\dot{y}\right)\left|z\right|^{n}+A\,\left(\dot{x}-\dot{y}\right) \tag{5}\]  
\[\dot{y}=\frac{\left(\alpha\,z+c_{o}\,\dot{x}+k_{o}(x-y)\right)}{c_{o}+c_{1}} \tag{6}\]  
where \(f_{MR}\) represents the output force resulting from the MR damper. The linear displacement and velocity of the damper are, respectively, represented by the state variables \(x\) and \(\dot{x}\). The variable \(z\) is an evolutionary variable that represents the hysteretic behavior of the output force in terms of the MR damper's displacement and velocity. The coefficients \(k_{o}\) and \(c_{o}\) denote, respectively, the accumulator stiffness and viscous damping at low velocity coefficients, while the coefficients \(c_{1}\) and \(k_{1}\) represent the damping and stiffness at high velocities. The parameters \(\gamma\), \(\beta\), \(A\), and \(n\) characterize the scale and shape of the hysteresis loop. It is worth noting that the above-mentioned parameters concerning the MR damper are related to the applied voltage \(v\)[38]:  
\[\alpha=\alpha_{a}+\alpha_{b}\,u,\,c_{o}=c_{oa}+c_{ob}\,u,\,c_{1}=c_{1a}+c_{1b} \,u \tag{7}\]  
\[\dot{u}=-\eta\,\left(u-v_{a}\right) \tag{8}\]  
where the coefficients \(\alpha_{a}\), \(\alpha_{b}\), \(c_{oa}\), \(c_{ob}\)\(c_{1a}\), and \(c_{1a}\) represent the components of damping coefficients \(\alpha\), \(c_{o}\), and \(c_{1}\). The parameter \(\eta\) is a constant related to the time response, \(v_{a}\) is the control commanded voltage applied to the damper control circuit, and \(u\) is the phenomenological variable representing the system dynamics. \[\dot{x}_{1}=x_{2}\]0 Modified Bouc–Wen model of the MR damper.