## 5 Sliding Mode Variable-Structure Controller Based on Piecewise Power Reaching Law
### Design of Piecewise Power Reaching Law  
This paper proposes a new piecewise power reaching law to accelerate the convergence rate further. This law sets 1 as the boundary point and divides the approach into two phases: \(|S|\geq 1\) and \(|S|<1\). Utilizing the properties of power functions, the system has a high convergence rate in both phases, and the convergence rate at the equilibrium point (\(|S|=0\)) is zero. The structure of the piecewise power reaching law is as follows:  
\[\dot{S}= \begin{cases}-k_{1}S-k_{2}S^{\frac{p}{q}}&|S|\geq 1\\ -k_{3}S-k_{4}S^{\frac{m}{q}}&|S|<1\end{cases}  \tag{22}\]  
where \(k_{1},k_{2},k_{3},k_{4}>0\) and \(k_{1}+k_{2}=k_{3}+k_{4}\), \(\frac{p}{q}>1\), \(0<\frac{m}{n}<1\), and p, q, m, n are all odd numbers. Like general power laws, the designed reaching law is continuous at any stage and does not exhibit chattering theoretically. To allow for a flexible design of \(k_{2}\) and \(k_{4}\), two linear terms are added (otherwise, to ensure the continuity of the reaching law \(k_{2}=k_{4}\), which restricts parameter selection). From the expression of the approaching law, it can be seen that the motion process of the system is divided into two stages, far away from the sliding surface and near the sliding surface. When moving away from the sliding surface, i.e., \(|S|\geq 1\), the approach law is dominated by the first term, and when approaching the sliding surface, i.e., \(0<|S|<1\), the approach law is dominated by the second term. From the properties of power functions, it can be inferred that the designed convergence law has a significant convergence rate at any stage. The design of a piecewise function has several advantages:  
1. The system has a faster convergence rate at each stage. 2. The law governing \(|S|\geq 1\) and \(0<|S|<1\) can be designed separately, so their parameters do not affect each other. 3. Two parameters, \(k_{1}\) and \(k_{3}\), are added, making parameter adjustment more flexible. Theoretically, it can be proven that the proposed piecewise power reaching law has the following two properties in terms of convergence time. **Property 1. **_Let the initial value of \(S\) be \(S_{0}\). A system using the piecewise power reaching law will reach the sliding surface in finite time, given by:_  
\[t= \begin{cases}t_{1}+t_{2}&|S_{0}|\geq 1\\ t_{3}&|S_{0}|<1\end{cases}  \tag{23}\]  
_where_  
\[t_{1}=\frac{q}{k_{1}(p-q)}\left[\ln(k_{1}+k_{2})-\ln(k_{1}S_{0}^{\frac{q-p}{ q}}+k_{2})\right] \tag{24}\]  
\[t_{2}=\frac{n}{k_{3}(m-n)}\left[\ln(k_{3}+k_{4})-\ln k_{4}\right] \tag{25}\]  
\[t_{3}=\frac{n}{k_{3}(m-n)}\left[\ln(k_{3}S_{0}^{\frac{n-m}{n}}+k_{4})-\ln k_{4 }\right] \tag{26}\]  
**Proof. ** If \(|S_{0}|\geq 1\), the reaching process is divided into two phases: \(|S|\) moves from \(|S_{0}|\) to \(1\) and then from \(1\) to \(0\). When \(1\leq|S|<|S_{0}|\):  
\[\dot{S}+k_{1}S+k_{2}S^{\frac{p}{q}}=0 \tag{27}\]  
Equation (26) is a Bernoulli differential equation, with a solution given by:  
\[S^{\frac{q-p}{q}}=Ce^{k_{1}(\frac{p-q}{q})t}-\frac{k_{2}}{k_{1}} \tag{28}\]  
where \(C=S_{0}^{\frac{q-p}{q}}+\frac{k_{2}}{k_{1}}\). Since p and q are both odd, p - q is even, and \(S_{0}^{\frac{q-p}{q}}=(-S_{0})^{\frac{q-p}{q}}\). Let \(S=\pm 1\), then the following expression is obtained:  
\[t_{1}=\frac{q}{k_{1}(p-q)}\left[\ln(k_{1}+k_{2})-\ln(k_{1}S_{0}^{\frac{q-p}{ q}}+k_{2})\right] \tag{29}\]  
When \(0<|S|<1\):  
\[\dot{S}+k_{3}S+k_{4}S^{\frac{m}{n}}=0 \tag{30}\]  
Following a similar solution process, where \(S_{0}=\pm 1\), and let \(S=0\); then, the following expression is obtained:  
\[t_{2}=\frac{n}{k_{3}(m-n)}[\ln(k_{4})-\ln(k_{3}+k_{4})] \tag{31}\]  
Therefore, the time required to reach the sliding surface is \(t_{1}+t_{2}\). If \(0<|S_{0}|<1\), then the approaching process consists of only one phase, where \(S\) moves from \(S_{0}\) to \(0\).