## 3 Performance Indexes of the Electromagnetic Isolation System
### Relationship between Current and Stiffness and Damping Coefficient  
Due to the fact that the two electromagnets with the electromagnetic repulsion force can be simplified as a linear spring or a spring and a damper, the electromagnetic isolation system can be regarded as a  
Figure 2: The model of the electromagnetic vibration isolation system. parallel structure consisting of two linear springs or two linear springs and two dampers. In other words, the electromagnetic force could be equal to the following two cases:  
1. \(f_{e1}=k_{2}x\), in which \(k_{2}\) is the equivalent stiffness of the two electromagnets. It implies that the two electromagnets with the electromagnetic repulsion force can be simplified as a linear spring. Then, Equation (2) can be rewritten as follows: \[M\ddot{x}+d_{1}\dot{x}+(k_{1}+k_{2})x=f_{d}\] (5) The transfer function is achieved by Fourier transforming Equation (5) as follows: \[G_{1}(s)=\frac{1}{Ms^{2}+d_{1}s+(k_{1}+k_{2})}\] (6)  
Based on the above transfer function, the damping ratio \(\zeta\) which is hoped to satisfy \(0<\zeta<1\) and the undamped natural frequency \(w_{n}\) are respectively obtained as follows:  
\[\zeta=\frac{d_{1}}{2*M*w_{n}} \tag{7}\]  
\[w_{n}=\sqrt{\frac{(k_{1}+k_{2})}{M}} \tag{8}\]  
Due to the fact that an object acts on the system is equal to a step input to the electromagnetic vibration isolation system which is a second-order oscillating system. The dynamic performance indexes of the system can be calculated to design the controller as follows:  
\[t_{r}=\frac{\pi-\arccos\zeta}{w_{n}*\sqrt{1-\zeta^{2}}} \tag{9}\]  
\[t_{p}=\frac{\pi}{w_{n}*\sqrt{1-\zeta^{2}}} \tag{10}\]  
\[\sigma\%_{\rm{}}=e^{-\pi\epsilon\zeta/\sqrt{1-\zeta^{2}}}*100\% \tag{11}\]  
\[t_{s}=\frac{4}{\zeta*w_{n}} \tag{12}\]  
\[N=\frac{2*\sqrt{1-\zeta^{2}}}{\pi*\zeta} \tag{13}\]  
where \(t_{r}\) is the rise time, \(t_{p}\) is the peak time, \(\sigma\%\) is the overshoot, \(t_{s}\) is the settling time, and \(N\) is the number of oscillations. Given the desired scopes of performance indexes of the second order system as follows:  
\[\left\{ \begin{aligned} 0<\zeta<1\\ 0<t_{r}<0.1\\ 0.05<t_{p}<0.2\\ 0<\sigma\%<10\%\\ 0.1<t_{s}<2\\ 0<N<5\end{aligned} \right. \tag{14}\]Based on the above inequalities, the equivalent stiffness \(k_{2}\) of the two electromagnets can be calculated within a certain range as follows:  
\[0<k_{2}<1151 \tag{15}\]  
After the equivalent stiffness \(k_{2}\) is selected in the above range, the relationship between the equivalent stiffness \(k_{2}\) and coil current \(i\) and the vibration displacement \(x\) can be built as follows:  
\[ \begin{split} f_{e1}(x,i)&=8259.7863*x^{2}+2.4055*i ^{2}-317.9956*x\\ &-4.0467*i+311.2811*x*i+2.4351=k_{2}x\end{split}  \tag{16}\]  
Equation (16) indicates that the values of the equivalent stiffness \(k_{2}\) can be satisfied by controlling the coil currents when the gaps are measured by two displacement sensors. Then, the relationship between current and stiffness is achieved based on the above equation:  
(2) \(f_{e2}=k_{2}x+d_{2}\dot{x}\), in which \(k_{2}\) is the equivalent stiffness of the two electromagnets, \(d_{2}\) is the equivalent damping coefficient of the two electromagnets and \(\dot{x}\) is the derivative of \(x\). The \(\dot{x}\) can be calculated by \([x(k)-x(k-1)]/T\), in which \(x(k)\) is the vibration displacement at time \(k\) and \(T\) is the control cycle of controller which is equal to 0.01. It implies that the two electromagnets with the electromagnetic repulsion force can be simplified as a linear spring and a damper.