# Earthquake Hazard Mitigation for Uncertain Building Systems Based on Adaptive Synergetic Control
## 4 Computer Simulation  
In this section, the effectiveness of the classical and adaptive synergetic controllers is verified; a comparison study of their performance was conducted to show which controller could perform better in terms of vibration suppression during an earthquake. The numerical simulations were conducted within the MATLAB/SIMULINK environment. The \"ODE45\" numerical solver with variable-step time was chosen. The variable hit ranged between min-step-size \(10^{-4}\) and max-step-size \(10^{-3}\). The Simulink modeling of the modified Bouc-Wen model is shown in Figure 4. As shown in the figure, this Simulink model was utilized to solve the equation of the z-variable. All constituting elements of the model solver of the z-variable were supplied by the SIMULINK library. Table 1 lists the numerical values of the MR damper's parameters. In the case of conventional SC, the design parameters of the controller were chosen to be \(T=1\), \(c=10\). In the case of adaptive SC, the selected design parameters of the controller were \(c=10\), \(T=1\). Moreover, the parameters of the adaptive rates were set as follows: \(\gamma_{1}=\gamma_{1}=1\). On the other hand, the parameters \(M_{s}=93.3\) Kg, \(C_{s}=175\) N\(\cdot\)s/m, and \(K_{s}=12\times 10^{5}\) N/m were used for the single-degree-of-freedom building structural system. In this study, the earthquake data applied for numerical simulations were based on an earthquake disaster that occurred at Imperial Valley (Brawley) in May 1940. The magnitude of the earthquake was estimated at approximately 7.1 on the Richter scale. Figure 5 displays the behavior of the recorded earthquake. Figure 6 depicts the dynamic behaviors of building displacements due to earthquake vibration under the classical synergetic controller and adaptive synergetic controller. According to Figure 6, both the conventional and adaptive control approaches could successfully suppress the effect of earthquake shaking as compared to an uncontrolled system with the support of an MR damper. On the  
Figure 4: The Simulink model of modified Bouc–Wen model. The Root Mean Square (RMS) was chosen as the index of evaluation, and a comparison between SMC and ABSMC was made in terms of displacement deviation. In the case of an uncontrolled building structure, the RMS value in displacement deviation over the time of the earthquake was evaluated to be \(13.21\times 10^{-5}\), while the conventional SC recorded an RMS value of \(4.23\times 10^{-4}\). The lowest value of displacement RMS was calculated when involving adaptive SC over the entire time of the earthquake, at \(1.017\times 10^{-3}\). As such, adaptive SC showed better rejection capability to suppress the level of vibration on the building structure due to the earthquake. Figure 7 shows the acceleration generated by the adaptive synergetic controller and the conventional synergetic controller. According to Figure 7, one can notice that the adaptive synergetic controller resulted in a lower acceleration value in the quaked building, as compared to the larger acceleration due to the classical SC. Thus, the adaptive synergetic controller is better able to suppress building vibration due to earthquake as compared to the conventional synergetic controller. The control signals generated by the classical and adaptive synergetic controllers are indicated in Figure 8. The figure indicates that the control effort in the case of classical SC was less than the control effort produced by ASC. Of course, this the price paid by ASC for its better capability in terms of vibration suppression in the building structure. Figure 8: Control signals generated by the SC and ASC. Figure 7: Acceleration responses of the earthquakes building based on SC and ASC. Figures 9 and 10 show the behaviors of uncertainty estimates, generated by the adaptive laws, for both coefficients of building damping viscosity and stiffness, respectively.