## 5 Simulation and Experimentation
### Active Control Method with the Piecewise Strategy  
Although the active control method with the equivalent stiffness and damping coefficient can effectively control vibration and provide desired dynamic performance indexes, there are overshoot and oscillations in simulation and experimental results. For realizing the variable control parameters of the electromagnetic isolation system and eliminate overshoot and oscillations to keep the stability of the system, an active control method based on the equivalent stiffness and damping coefficient with the piecewise strategy is proposed. From Figure 4, it can be seen that the equivalent damping coefficient shortens the response time of the electromagnetic isolation system, reduces the overshoot and the number of oscillations. However, a smaller equivalent damping coefficient may lead to the overshoot and oscillations of the system, a larger equivalent damping coefficient may increase the settling time, which is not conducive to the real-time requirements in engineering practice. Therefore, the following piecewise strategy is proposed based on the model of electromagnetic isolation system:  
\[k_{2}=540\ \mathrm{N/m},\,d_{2}=30\ \mathrm{N}\cdot\mathrm{s/m},\,\text{ when}\ x<4.00\times 10^{-5}\ \mathrm{m}; \tag{2}\] \[k_{2}=540\ \mathrm{N/m},\,d_{2}=70\ \mathrm{N}\cdot\mathrm{s/m},\, \text{ when}\ x\geq 4.00\times 10^{-5}\ \mathrm{m}. \tag{1}\]  
The plate will produce the vibration displacement \(x\) after the plate of the electromagnetic vibration isolation system suffer an external force, the controlled current is then aroused by solving algebraic Equation (23) with the equivalent stiffness and damping coefficient based on the piecewise strategy to offset the external force and keep the stability of the system. The control diagram with the active control method based on the piecewise equivalent stiffness and damping coefficient is shown in Figure 7. According to the vibration displacements are measured by the displacement sensor, various equivalent stiffness and damping coefficients could be conducted. With Equations (1) and (2), the controlled currents are then achieved by solving algebraic equation with the vibration displacement in Equation (23). In order to verify the advantages of the active control method based on the piecewise equivalent stiffness and damping coefficient, the the method based on the fixed equivalent stiffness \(k_{2}=540\ \mathrm{N/m}\) and fixed equivalent damping coefficient \(d_{2}=70\ \mathrm{N}\cdot\mathrm{s/m}\) is utilized for comparison purpose. Figure 8 shows the comparison results, when the electromagnetic isolation system is controlled by active method with the piecewise equivalent stiffness and damping coefficient, the vibration displacement \(x\) is stable at \(5.09\times 10^{-5}\ \mathrm{m}\) which is close to \(0\ \mathrm{m}\), the rise time \(t_{r}\), the peak time \(t_{p}\) and the settling time \(t_{s}\) are both equal to \(0.32\ \mathrm{s}\), the overshoot \(\sigma\%\) and the number of oscillations \(N\) are both equal to \(0\). When the electromagnetic isolation system is controlled by active method with the fixed equivalent stiffness \(k_{2}=540\ \mathrm{N/m}\) and fixed equivalent damping coefficient \(d_{2}=70\ \mathrm{N}\cdot\mathrm{s/m}\), the overshoot \(\sigma\%\) and the number of oscillations \(N\) are both equal to \(0\), and the vibration displacement \(x\) is stable at \(5.09\times 10^{-5}\ \mathrm{m}\); however, the rise time \(t_{r}\), the peak time \(t_{p}\), and the settling time \(t_{s}\) are both equal to \(0.58\ \mathrm{s}\). The simulation results show that the equivalent damping coefficient can reduce the overshoot and the number of oscillations, and ensure the stability of the system; however, an excessive equivalent damping coefficient may increase the settling  
Figure 7: The control diagram with the piecewise strategy. time.