## 3 Control Scheme Design
### Design and Analysis for the Reaching Law  
In SMC, the switching control law \(\tau_{sw}\) is determined by the reaching law. In this section, an innovative fast-reaching law (FRL) is presented to analyze the chatter characteristics. This law is able to deal with the conflict between the convergence time and the chatter amplitude in a conventional SMC scheme. To clearly analyze the superiority of the FRL, it is required to introduce the CSMC ERL [18]. In SMC, the ERL can be described as follows:  
\[\dot{s}=-\varepsilon s-ksgn(s) \tag{19}\]  
where \(\varepsilon\) and \(k\) are the gain matrices of the ERL, \(\varepsilon>0\), \(k>0\), and \(sgn(\cdot)\) is the function of the sign. The sign function is a discontinuous function that causes discontinuity in the output of the switching control term, which, in turn, leads to chattering in the system. The time for the state variable \(s\) to reach the sliding mode surface is defined as the response time \(t_{r}\) of the ERL. \[t_{r}=\int_{s_{0}}^{s_{r}}\frac{1}{-\varepsilon s-ksgn(s)}ds \tag{20}\]where \(s_{0}\) is the sliding mode surface value at the original instant (\(t_{0}=0\)), \(s_{r}\) is the state variable at the first time the system reaches the sliding mode surface, and \(s_{r}=0\). Because the value of \(s_{0}\) is uncertain, (20) needs to be examined for the following three cases: \(s_{0}>0\), \(s_{0}=0\), \(s_{0}<0\). Case 1: If \(s_{0}>0\), according to SMC theory, \(s>0\) in the time interval \([t_{0},t_{r}]\). At this time, the following equation can be obtained from (20):  
\[t_{r+}=\frac{1}{\varepsilon}ln(\frac{\varepsilon}{k}s_{0}+1) \tag{21}\]  
Case 2: If \(s_{0}=0\), according to SMC theory, \(s=0\) in the time interval \([t_{0},t_{r}]\). At this point, the state variable of the system is on the sliding mode surface; therefore,  
\[t_{r}=0 \tag{22}\]  
Case 3: If \(s_{0}<0\), according to SMC theory, \(s=0\) in the time interval \([t_{0},t_{r}]\). At this time, the following equation can be obtained from (20):  
\[t_{r-}=\frac{1}{\varepsilon}ln(-\frac{\varepsilon}{k}s_{0}+1) \tag{23}\]  
In summary, the response time of the ERL can be expressed as follows:  
\[t_{r}=\frac{1}{\varepsilon}ln(\frac{\varepsilon}{k}|s_{0}|+1) \tag{24}\]  
To analyze the chattering by discretizing this expression, in digital control, (19) can be rewritten as follows:  
\[\frac{s(N+1)-s(N)}{T_{s}}=-\varepsilon s(N)-ksgn(s(N)) \tag{25}\]  
where \(T_{s}\) is the sample period. When the state variable of the system reaches the sliding mode surface in limited time. Following the SMC theory, the system state variable will oscillate around the sliding mode surface, namely, \(s(N)\to 0^{+}\) or \(s(N)\to 0^{-}\). Moreover, \(-\varepsilon s(N)\) in (25) can be neglected when the state variable of the system reaches the neighborhood of the sliding surface. Thus, when \(s(N)\to 0^{+}\), the following is obtained:  
\[s(N+1)=-kT_{s} \tag{26}\]  
When \(s(N)\to 0^{-}\), the following is obtained:  
\[s(N+1)=kT_{s} \tag{27}\]  
Therefore, the chatter amplitude of the ERL is as follows:  
\[\Delta_{r}=2 kT_{s} \tag{28}\]  
By analyzing (24) and (28), it can be observed that increasing \(\varepsilon\) will decrease the response time \(t_{r}\) and vice versa when the system state variable of the system is distant away from the sliding mode surface \(s\). However, changing the values of \(\varepsilon\) when the state variable of the system is approaching the sliding mode surface \(s\) has a negligible effect on the response time \(t_{r}\); moreover, increases in the gain \(k\) will decrease the response time \(t_{r}\) but will increase the chatter amplitude \(\Delta_{r}\), and decreases in the gain \(k\) will decrease the chatter amplitude \(\Delta_{r}\) but will increase the response time \(t_{r}\). Therefore, in conventional sliding mode control using the ERL, an irreconcilable contradiction exists between the response time of the system and the magnitude of the chatter.