## 3 Control Scheme Design
### Design and Analysis for the Reaching Law
Based on the idea of the ERL, a novel FRL is proposed in this paper as follows:\[t_{r}=\int_{s_{0}}^{s_{r}}\frac{1}{-\varepsilon s-ksgn(s)}ds \tag{20}\]0  
where \(\alpha\), \(\beta\), \(\mu\), and \(k\) are the gain matrices of the FRL, \(0<\alpha<1\), \(\beta>1\), \(\mu>0\), and \(k>0\). \(tanh(\cdot)\) is the hyperbolic tangent function. \(s_{0}\) is the original position of the state variable, i.e., the state variable at the time \(t_{0}=0\). The FRL, like the ERL principle, increases the convergence speed of the system state when it is farther away from the sliding surface and decreases the convergence speed of the system when it is closer to the sliding surface. The advantage of (29) is that the convergence speed decreases faster than (19) when the system state is close to the sliding surface, which reduces the effect of system inertia when the system state passes through the sliding surface. Assuming the time at which the state variable of the system first reaches the sliding mode surface is \(t_{c}\), its position at this time is \(s_{c}\), and according to the SMC theory, it can be shown at this time that \(s_{c}=0\). Thus, the reach time \(t_{c}\) can be obtained by integrating (29) as follows:  
\[t_{r}=\int_{s_{0}}^{s_{r}}\frac{1}{-\varepsilon s-ksgn(s)}ds \tag{20}\]1  
As with the ERL, the value of \(s_{0}\) is unknown, and therefore must be considered for the following three cases: \(s_{0}>0\), \(s_{0}=0\), and \(s_{0}<0\). Case 1: If \(s_{0}>0\), according to SMC theory, \(s>0\) in the time interval \([t_{0},t_{c}]\). At this point, we can obtain the following equation from (30):  
\[t_{r}=\int_{s_{0}}^{s_{r}}\frac{1}{-\varepsilon s-ksgn(s)}ds \tag{20}\]2  
Case 2: If \(s_{0}=0\), according to SMC theory, \(s=0\) in the time interval \([t_{0},t_{c}]\). At this point, the state variable of the system is located on the sliding mode surface; therefore,  
\[t_{r}=\int_{s_{0}}^{s_{r}}\frac{1}{-\varepsilon s-ksgn(s)}ds \tag{20}\]3  
Case 3: If \(s_{0}<0\), according to SMC theory, \(s<0\) in the time interval \([t_{0},t_{c}]\). At this point, we can obtain the following equation from (30):  
\[t_{r}=\int_{s_{0}}^{s_{r}}\frac{1}{-\varepsilon s-ksgn(s)}ds \tag{20}\]4  
To summarize, the FRL response time can be expressed as follows:  
\[t_{r}=\int_{s_{0}}^{s_{r}}\frac{1}{-\varepsilon s-ksgn(s)}ds \tag{20}\]5  
Consideration of sliding motion features as the system state variable moves onto the sliding mode surface, (29) can be simplified to (35) as follows:  
\[t_{r}=\int_{s_{0}}^{s_{r}}\frac{1}{-\varepsilon s-ksgn(s)}ds \tag{20}\]6  
When the state of the system is far from the sliding surface, it is converged as shown in (29), and when the system is close to the sliding surface, the value of \(s\) is so small that it can be neglected. Thus, (35) can be regarded as a special stage of (29). Compared to (35), (29) is more concise and can reduce the complexity of the later equation for chattering amplitude. This helps in the subsequent analysis of the effect of each parameter on the chattering amplitude. The same processing used for the ERL is used to obtain the chatter amplitude \(\Delta_{c}\) of the FRL using the following discretization method:\[t_{r}=\int_{s_{0}}^{s_{r}}\frac{1}{-\varepsilon s-ksgn(s)}ds \tag{20}\]7  
where \(T_{s}\) is the sample period. By analyzing (34) and (36), it can be found that the value of \(\mu\cdot ln(1+\mu^{-1})\) increases with \(\mu\) when \(\mu\in(0,\infty)\). By increasing the value of \(\alpha\), the response time \(t_{c}\) will increase, but the chatter amplitude \(\Delta_{c}\) will consequently decrease. By decreasing the value of \(\alpha\), the response time \(t_{c}\) will decrease, but the chatter amplitude \(\Delta_{c}\) will subsequently increase.