## 3 Control Strategy Description
### Control Strategy for the Engine  
To avoid the disturbance torque generated from the large speed difference of the clutch, the engine was operated in the mode of speed regulation in this study. The driving plate of the clutch should be maintained to follow the speed of the driven plate.  
As far as we know, conventional PI controllers with fixed parameters are not suitable for regulating the engine speed because of their complicated dynamic characteristics [36; 37; 38]. The parameters of the controller must be adjusted according to the actual operating state of the engine. Here, a fuzzy algorithm was introduced and applied in the PI controller to form an adaptive method, rather than the method with fixed parameters. The engine speed control principle diagram is depicted in Figure 6.  
To improve the dynamic response of the engine speed regulation, eliminate integral saturation phenomenon, and avoid overshoot and oscillation of the engine speed, the fuzzy algorithm was adopted to correct parameters of the PI controller. The PI controller can be expressed as:  
\[u(t)=K_{\mathrm{P}}\cdot e(t)+K_{\mathrm{i}}\int_{t}^{0}e(\tau)d\tau \tag{20}\]  
\[e(t)=\omega_{\mathrm{e,cmd}}-\omega_{\mathrm{e}}(t) \tag{21}\]  
where \(K_{\mathrm{P}}\) denotes the proportional coefficient, \(K_{\mathrm{i}}\) denotes the integral coefficient, \(\omega_{\mathrm{e,cmd}}\) denotes the target speed for the engine, and \(e(t)\) denotes the speed error at the time \(t\).  
The parameters \(K_{\mathrm{P}}\) and \(K_{\mathrm{i}}\) of the controller can be normalized by Equations (22) and (23); then, the normalized parameters \(K^{\prime}_{\mathrm{P}}\) and \(K^{\prime}_{\mathrm{i}}\) in the interval \([0,1]\) are generated, which can be represented by the fuzzy sets {NB, NS, ZO, PS, PB}. Figure 7 shows their corresponding membership functions.  
Figure 5: Computational procedure of the sliding mode control based on group-preserving scheme.  
Figure 6: Architecture of the adaptive PI controller.  
Figure 7: Membership functions of parameters: (**a**) \(K_{\text{p}}^{\prime}\) and (**b**) \(K_{\text{i}}^{\prime}\).  
\[K^{\prime}_{\text{P}}=\frac{K_{\text{P}}-K_{\text{p,min}}}{K_{\text{P}\_\text{max} }-K_{\text{P}\_\text{min}}} \tag{22}\]  
\[K^{\prime}_{\text{i}}=\frac{K_{\text{i}}-K_{\text{i,min}}}{K_{\text{i,max}}-K_{ \text{i,min}}} \tag{23}\]  
where \(K_{\text{P}\_\text{max}}\) is the maximum of the parameter \(K_{\text{P}}\), \(K_{\text{P}\_\text{min}}\) is the minimum of the parameter \(K_{\text{P}}\), \(K_{\text{i,max}}\) is the maximum of the parameter \(K_{\text{i}}\), and \(K_{\text{i,min}}\) is the minimum of the parameter \(K_{\text{i}}\).  
The speed error \(e\) and its derivative \(\dot{e}\) are regarded as the inputs, which can be expressed by the fuzzy sets {NB, NM, NS, ZO, PS, PM, PB}. Their membership functions are shown in Figure 8. The fuzzy inference rules for parameters \(K^{\prime}_{\text{P}}\) and \(K^{\prime}_{\text{i}}\) are listed in Tables 2 and 3, respectively. Finally, accurate parameters for the PI controller were achieved by the defuzzification. In this study, the height method was applied to calculate PI controller parameters, which can be expressed as [39]:  
\[y(\text{x})=\frac{\sum\limits_{n=1}^{M}y^{n}\prod\limits_{i=1}^{2}F_{i}^{n}(X_ {i})}{\sum\limits_{n=1}^{M}\prod\limits_{i=1}^{2}F_{i}^{n}(X_{i})} \tag{24}\]  
where \(F_{i}^{n}\) is fuzzy sets, \(y^{n}\) is fuzzy singletons, \(x_{i}\) is the input variable, and \(M\) stands for the number of inference rules.