## 2 Single-Degree-of-Freedom Control Design  
Consider the single-degree-of-freedom (SDOF) subjected to the actuator installed in series with a damper and a spring, as shown in Figure 1d. The equation of motion can be written as:  
\[m\ddot{x}+c\dot{x}+kx=c\dot{u}+ku+d \tag{1}\]  
where \(\cdot=d/dt\) represents the time-derivative. Dividing Equation (1) by \(m\) obtain:  
\[\ddot{x}+2\zeta\omega_{n}\dot{x}+\omega_{n}^{2}x=2\zeta\omega_{n}\dot{u}+\omega_ {n}^{2}u+\omega_{n}^{2}\overline{d} \tag{2}\]  
where:  
\[\omega_{n}=\sqrt{\frac{k}{m}},\ \zeta=\frac{c}{2m\omega_{n}},\ \overline{d}= \frac{d}{k} \tag{3}\]  
In contrast to general equations of motion subjected to force and disturbance, the right-hand side of Equation (2) consists of the displacement and velocity of the displacement actuator. Hence, the active vibration control algorithms introduced earlier that produce the desired force cannot be directly applied to Equation (2). In this study, we propose the following control algorithm for the system given by Equation (2). \[\ddot{u}+2\zeta_{f}\omega_{n}\dot{u}+\omega_{n}^{2}u=g\Big{(}2\zeta_{f}\omega _{n}\dot{x}+\omega_{n}^{2}x\Big{)} \tag{4}\]  
The control algorithm given by Equation (4) is the reciprocal of Equation (2) except for the gain and damping factor. The transfer function of the compensator given by Equation (4) can be also expressed as:  
\[\frac{U(s)}{X(s)}=H(s)=\frac{g\Big{(}2\zeta_{f}\omega_{n}s+\omega_{n}^{2}\Big{)} }{s^{2}+2\zeta_{f}\omega_{n}s+\omega_{n}^{2}} \tag{5}\]  
where \(X(s)=\mathcal{L}\{x(t)\}\), \(U(s)=\mathcal{L}\{u(t)\}\), and \(\mathcal{L}\) is the Laplace operator. The compensator given by Equation (5) will be referred to in this paper as the PIPO controller. Unlike the PPF controller, the output of the PIPO is the desired displacement of the actuator installed in series with the damper and the spring. The novelty of the proposed single-input and single-output (SISO) PIPO controller given by Equation (5) is that its form is very simple and can be easily implemented using analog circuits or digital controllers. Of course, a sensor capable of precisely measuring the displacement of the table and an actuator capable of accurately realizing the desired displacement are required. The smaller the damping factor of the compensator, the greater the damping effect, but the narrower the controllable frequency band. Figure 2 shows the Bode diagram of the PIPO controller for various \(\zeta_{f}\), where \(\omega_{n}=1\ \mathrm{rad}/\mathrm{s},\ g=0.1\) are used. The magnitude plot shown in Figure 2a shows the typical second-order low-pass filter, whereas the phase plot shown in Figure 2b indicates that the phase at the filter frequency is 90 degree, which implies active damping. Hence, it can be said that the control algorithm given by Equation (5) can affect the system by increasing the damping at the target frequency. It is desirable to give the proposed control algorithm a larger frequency band. So, it seems preferable to use \(\zeta_{f}=0.3\) as in the case of the PPF controller. As stated above, the compensator transfer function has 90-degree shift at the filter frequency, so that it can create active damping for the addressed problem. Combining Equation (2) with Equation (4) results in the following closed-loop matrix equation of motion. \[\left\{ \begin{matrix}\tilde{x}\\ \tilde{u}\end{matrix} \right\}+\left[ \begin{matrix}2\zeta\omega_{n}&-2\zeta \omega_{n}\\ -2g\zeta_{f}\omega_{n}&2\zeta_{f}\omega_{n}\end{matrix} \right]\left\{ \begin{matrix}\dot{x}\\ \dot{u}\end{matrix} \right\}+\left[ \begin{matrix}\omega_{n}^{2}&-\omega_{n}^{2 }\\ -g\omega_{n}^{2}&\omega_{n}^{2}\end{matrix} \right]\left\{ \begin{matrix}x\\ u\end{matrix} \right\}=\left[ \begin{matrix}\omega_{n}^{2}\\ 0\end{matrix} \right]\overline{d} \tag{6}\]  
Applying the Laplace transform to Equation (6) obtains:  
\[\left[