# Noninteracting Control Design for 6-DoF Active Vibration Isolation Table with LMI Approach
## 4 Simulation Studies
### Simulations  
The performance of the proposed noninteracting controller was evaluated through two simulation tasks: vibration isolation and platform positioning. The results of these simulations were compared to those of the integral-type servo (ITS) control system. For the former task, the top plate of the AVIS has to be isolated from any vibration source. In practical applications, most of these vibrations are transmitted through the floor and are mainly the result of machines operating in the same workspace. Thus, the control system needs to suppress the ground vibrations in any direction and, while doing so, the actuators should not generate any unwanted motions in the other directions. Meanwhile, in the latter task, the top plate of the AVIS is expected to approach a set-point in any direction without affecting the motion of other directions. In other words, the matrix representing the relationship between the reference and the output should be diagonal. Figure 3 shows the vibration isolation frequency response against the vibration from the bottom plate that is fixed to the ground. Each column of graphs represents the motion of the top plate in respect to the vibration of the bottom plate in that specific direction. As illustrated in this figure, the mass-spring-damper systems, in their nature, are capable of suppressing vibrations, especially high-frequency vibrations. Unfortunately, the resonance phenomenon is unavoidable with the passive systems--the peak in each frequency response. Moreover, a vibration along the \(x_{b}\)-axis of the ground results in a translation in the same direction and an additional rotation about the \(y\)-axis of the top plate, as shown in the first column of Figure 3. Similarly, vibrations along the \(y_{b}\)-axis generate a translation along the \(y\)-axis and a rotation along the \(\varphi_{x}\) direction. Using both control strategies, the performance of the system at low-frequency ranges had improved significantly. Unfortunately, the integral-type servo controller, similar to the uncontrolled case, generated unwanted rotations while suppressing the translations from the bottom plate along the \(x\)- and \(y\)-axes. Furthermore, undesired translations appeared during the suppression of rotational vibrations, as shown in the fourth and fifth columns of Figure 3. On the other hand, the proposed noninteracting controller provided good isolation performance without creating any unwanted motions. Therefore, in Figure 3, the response of the noninteracting control system appears only at the graphs on the main diagonal line. In Figure 4, the time response showcases further the efficiency of the proposed control system. The bottom plate is supposed to perform step-type movements of 0.01 (m) or 0.01 (rad) in different directions at different times. The mutual interference between the DoFs of the AVIS can be clearly seen in the bottom-left sub-figure of Figure 4a, with rotations appearing along with the translational movements. Aside from using the ITS control system, unwanted motions can be seen in the zoomed sub-figures in Figure 4b. On the other hand, unwanted motions of any direction affected by other directions' motions were not or hardly generated when the noninteracting control was used. The differentiations of these results were caused by the resulting system configuration decoupling the mutual interference within the system with the noninteracting control law. In other words, the  
Figure 3: Frequency response of the vibration isolation performance. proposed controller isolated the vibrations and decoupled the interactions of the system effectively and swiftly without generating any undesired motions. Figure 5 shows the frequency response of the system for the positioning task. Each column illustrates the Bode magnitude plots of the system response in every direction in response to the corresponding reference. Accordingly, the \(i^{th}\) entry of the graphs on the main diagonal line depicts the complementary sensitivity from the reference \(r_{i}\) and the output \(Y_{i}\).