# Comparative Study Between Active AMD and ABS Devices by Using \(\mu\)-Synthesis Robust Control
## 4 Results
### Displacements and Accelerations Analysis  
* **AMD Control System** In this case, recall that \(\mu_{\text{C-MD}}\) refers to the controller designed with the AMD system, with which Figure 7a-c was obtained. They represent the three floors' responses from the bottom to the top of the structure. The three floors recorded the displacements, demonstrating the devastating impact of an earthquake without protection, and the significant reduction in the structure through \(\mu_{\text{C-MD}}\) control. When compared to the first, second, and third floors, the displacement peaks without control were 0.1036 m, 0.1847 m, and 0.2345 m for each floor, respectively. However, after applying \(\mu_{\text{C-MD}}\) control, these peaks decreased to 0.0540 m, 0.0881 cm, and 0.0975 cm.  
The percentage of these considerable reductions reached about 71% for the first floor, 73% for the second floor, and 75% for the last floor. This constituted an encouraging result, which led us to continue the evaluation of the \(\mu_{\text{C-MD}}\) controller by computing the structure accelerations.  
The accelerations plotted in Figure 8a-c showed the three floors' responses from the bottom to the top of the structure, before and after control. The figures clearly demonstrated an improvement in the structure's acceleration responses, with the peak accelerations of the floors before control reaching 3.331 (m/s\({}^{2}\)), 3.4229 (m/s\({}^{2}\)), and 4.59 (m/s\({}^{2}\)) from the bottom to the top, respectively, and reducing to 2.25 (m/s\({}^{2}\)), 3.12 (m/s\({}^{2}\)), and 2.1 785 (m/s\({}^{2}\)) after control.  
The acceleration reduction percentages reached nearly 48%, 56%, and 67% for the first, second, and third floors, respectively, confirming the controller's efficiency and providing satisfying reduction results.  
Figure 7: Displacements of the structure with AMD System (\(\mu_{\text{C-MD}}\)): (**a**) first floor displacement, (**b**) second floor displacement, (**c**) third floor displacement.  
Figure 8: Accelerations of the structure with AMD System (\(\mu_{\text{C-MD}}\)): (**a**) first floor acceleration, (**b**) second floor acceleration, (**c**) third floor acceleration.  
* **ABS Control System**  
In Figures 9 and 10, one can see the three floor displacements and the accelerations of the structure responses before and after control with the controller \(\mu_{\text{C-BS}}\), which was made for the ABS control system. In these figures, one can see a significant attenuation of the three floors' displacements and accelerations.  
Figure 10: Accelerations of the structure with ABS (\(\mu_{\text{C-BS}}\)): (**a**) first floor acceleration, (**b**) second floor acceleration, (**c**) third floor acceleration.  
Indeed, the application of the \(\mu_{\text{C-BS}}\) controller reduced the peak displacements in Figure 9a-c to 0.0617 m, 0.0787 m, and 0.1063 m. In Figure 10a-c, one can see that after control, the acceleration peaks were also reduced to 3.3860 (m/s\({}^{2}\)), 5.0413 (m/s\({}^{2}\)), and 5.7201 (m/s\({}^{2}\)) for the first, second, and third floors, respectively.  
The percentages of the displacement attenuation achieved were about 67%, 71%, and 72%, while the percentages of the acceleration attenuation were about 32%, 54%, and 59% reached by the first, second, and third floors, respectively.  
The time domain measurements of the three floor movements and accelerations, following the implementation of the AMD and ABS control systems with the \(\mu_{\text{C-BS}}\) and \(\mu_{\text{C-MD}}\) controllers, indicated that both the systems effectively achieved the performance objectives, with the \(\mu_{\text{C-MD}}\) controller demonstrating marginally superior attenuation results.