## 3 Controller Design for Active Vibration Suppression
### Adaptive Fuzzy Sliding Mode Control  
The nonlinearity of the mathematical model in the previous section makes it difficult to obtain the relevant state variables. Generally, the format of the fuzzy rule bank in a fuzzy controller is fixed, and the parameters are not updated when the system changes. Consequently, it is difficult to predict control performance in the event of changes to the system parameters. This led researchers to develop AFSMC [15], which combines the advantages of fuzzy control, sliding mode control, and adaptive control. In a control system with unknown parameters, analyzing the input and output data can further understanding of the control system in order to establish language rules. Furthermore, adaptive control means that the rule bank can be adjusted online, and the robustness can be enhanced via the sliding mode. Figure 2 displays the AFSMC framework, in which \(gs\) and \(gu\) denote the input and output scaling factors, respectively. During the design process of the controller, the sliding surface must be defined. In this study, we used a second-order sliding surface, which can be mathematically presented as  
\[S(e)=ce+\lambda e \tag{9}\]  
where \(\lambda\) is the weight of the sliding mode. Using the sliding surface, we can convert the two-dimensional input variables (\(e\), \(ce\)) of the original fuzzy controller into a one-dimensional variable (\(s\)). With appropriate scaling based on \(gs\), the input variable can then be input into the fuzzy controller as follows:  
\[s(e)=S(e)\times gs \tag{10}\]  
As shown in Figure 2, the concept of the controller is to describe physical quantities using linguistic variables and identify the corresponding membership grade. With the adaptive law, the rule table can be adjusted and defuzzified online, which enables the rule bank to begin learning from zero rules. This in turn revises the defuzzified control rules. The control variable can then be obtained using the center of gravity defuzzification method, and then the control input variable is adjusted based on output scaling factor \(gu\). With regard to fuzzy control, we employed a triangular membership function with seven one-dimensional fuzzy variables for the input variable \(s\) and defuzzified variable \(u\), as shown in Figure 3, where NB is the negative big, NM is the negative medium, NS is the negative small, ZO is the zero, PS is the positive small, PM is the positive medium, and PB is the positive big. In sliding model control, the sliding surface reaching condition is  
\[s\dot{s}<0 \tag{11}\]  
When \(s>0\), the control variable must be increased to reduce \(s\dot{s}\); if \(s<0\), then the control variable must be decreased to reduce \(s\dot{s}\). Input variable \(s\) and control variable \(u\) can thus be designed to satisfy the reaching condition, where rule i can be written as  
\[R_{\mathrm{i}} \tag{12}\]  
Figure 2: Block diagram of adaptive fuzzy sliding mode control system. where \(S_{i}\) is the rule i of the component in question prior to fuzzification; \(U_{i}\) denotes the rule i of the component following defuzzification; and \(C_{i}\) indicates the central position of rule table i of the component following defuzzification. Using the defuzzified \(U_{i}\) corresponding to the ith membership grade obtained from component \(S\) before fuzzification, the control variable can be written as  
\[u=\frac{\sum_{i=1}^{m}\mu_{i}U_{i}}{\sum_{i=1}^{m}\mu_{i}}=\frac{\sum_{i=1}^{m }\mu_{i}C_{i}}{\sum_{i=1}^{m}\mu_{i}} \tag{13}\]  
where m is the number of rules, with i equaling 1, 2,..., and m; the initial values of \(C_{i}\) are all 0, and they are therefore referred to as zero rules. During the control process, we can use the adaptive law to update \(C_{i}\), the mathematical formula of which is  
\[\dot{C}_{i}(t)=-\gamma\frac{\partial s(t)\dot{s}(t)}{\partial C_{i}(t)} \tag{14}\]  
where \(\gamma\) denotes the adaptive rate. Using the chain rule, we can rewrite Equation (14) into  
\[