## 3 Controller Design for Active Vibration Suppression
### Hybrid Repetitive Control  
Figure 4 presents the hybrid RC framework adopted in this study, where \(C_{1}\) and \(C_{2}\) respectively represent the AFSMC controller and the repetitive controller. The sum of the control variable of the repetitive controller, \(u_{RC}\), and the control variable of the AFSMC controller, \(u_{AFSMC}\), is the total control input variable of the system, \(u_{sum}\). In the hybrid control framework used in this study, we designed  
Figure 3: Membership function of AFSMC. the controllers separately with the aim of having them work synergistically. The controller designs are similar to those in [14]; however, we replace PD control with AFSMC control to handle the complex vibration behaviors found in actual flexible structures more effectively. Periodic disturbances from the external environment or the motions of mechanical structures can damage the system via resonance. Repetitive controllers have proven highly effective in handling periodic signal trajectories and disturbances [19]. Thus, we employed a repetitive controller to eliminate disturbances caused by the periodic motions of the motor. The design principle of a repetitive controller involves adding the internal model of exogenous signals into the feedback control system. According to the internal model principle, asymptotic error tracking can be achieved as long as the closed loop system remains stable [19]. Figure 5 presents a block diagram of the RC adopted in this study. In Figure 5, \(C_{\mathrm{r}}\) denotes the repetitive controller; \(G\) represents the actual system, and \(G_{\mathrm{e}}\) symbolizes the mathematical model of the system. As can be seen, the relationship between disturbance \(w_{\mathrm{d}}\) and output \(y\) in the system of the RC framework is  
\[y=\frac{1-C_{\mathrm{r}}G_{\mathrm{e}}}{1-C_{\mathrm{r}}(G_{\mathrm{e}}-G)}w_ {\mathrm{d}} \tag{17}\]  
When the mathematical model \(G_{\mathrm{e}}\) approximates the actual system \(G\), Equation (17) can be rewritten as  
\[y=(1-C_{\mathrm{r}}G_{\mathrm{e}})w_{\mathrm{d}} \tag{18}\]  
In this way, the original feedback control issue can be converted into a feedforward control issue. To mitigate the impact of disturbance \(w_{\mathrm{d}}\), a cost function can be established:  
Figure 4: Block diagram of the proposed hybrid AFSMC/RC system. Figure 5: Block diagram of the applied repetitive control system. \[J_{\rm rc}=1-C_{\rm r}G_{\rm e} \tag{19}\]  
The design objective of this control problem is to derive \(C_{\rm r}\), the solution to stable plant inversion [20], while satisfying the constraint conditions that accompany periodic signals serving as the exogenous input. For systems with repetitive signals, the internal model \(D_{\rm r}\) can be expressed as \(D_{\rm r}=1-z^{-N}d\) where \(N_{\rm d}\) is the length of the time delay, which is determined by the period of the input signal. If the cost function \(J_{\rm rc}\) is written as  
\[1-C_{\rm r}G_{\rm e}=R_{\rm r}D_{\rm r}\quad or\quad C_{\rm r}G_{\rm e}+R_{ \rm r}D_{\rm r}=1 \tag{20}\]  
then Equation (20) becomes the renowned \"Bezout Identity\" [21], of which (\(R_{\rm r}\), \(C_{\rm r}\)) is a solution, and \(G_{\rm e}\) are \(C_{\rm r}\) are coprime. As the piezoelectric vibration suppression system in this study adopts a single-input-single-output control design, the mathematical model of this system, \(G_{\rm e}\), can be written as  
\[G_{\rm e}=G_{\rm d}G_{\rm i} \tag{21}\]  
where \(G_{\rm d}\) is the minimal phase of \(G_{\rm e}\), and \(G_{\rm i}\) denotes the non-minimal phase of \(G_{\rm e}\). Substituting Equation (21) into Equation (20) gives  
\[ \begin{array}{l}R_{\rm r}D_{\rm r}+CC_{\rm i}=1\\ C=C_{\rm r}G_{\rm d}\end{array}  \tag{22}\]  
Based on the concept of stable plant inversion in [20], a solution (\(R_{\rm r}\), \(C_{\rm r}\)) represented in discrete-time can be obtained to fulfill Equation (22):  
\[