# Improved Sliding Mode Finite-Time Synchronization of Chaotic Systems with Unknown Parameters
## 3 Design of Sliding Mode Surface and Controller  
This section introduces a new sliding mode controller that realizes the chaotic synchronization of two different chaotic systems in a finite time. In doing so, the two main procedures include designing a sliding surface and guaranteeing the finite-time stability of the system on a sliding surface. First, an adaptive finite-time controller with unknown parameters independent of the system model is designed. Then, convergence to sliding surface within finite time is guaranteed, and the adaptive law of unknown parameters is designed. The designed novel sliding surface is:  
\[s_{i}(t)=c_{i}e_{i}(t)+\int_{0}^{t}\text{sgn}(e_{i}(\tau))\big{|}e_{i}(\tau) \big{|}^{\alpha}d\tau+e_{i}(t)\text{sgn}(e_{i}(t)),i=1,2,\ldots,n, \tag{9}\]  
where \(s_{i}(t)\in R\), and \(c_{i}\) and \(\alpha\) are constants for which \(c_{i}>0\) and \(0<\alpha<1\). Taking the first-order derivative of the sliding surface, one obtains:  
\[s_{i}^{\bullet}(t)=c_{i}\overset{\bullet}{e_{i}}(t)+\text{sgn}(e_{i}(t))\big{|} e_{i}(t)\big{|}^{\alpha}+\overset{\bullet}{e_{i}}(t)\text{sgn}(e_{i}(t)),i=1,2, \ldots,n. \tag{10}\]  
Based on the sliding mode control principle [28],  
\[e_{i}^{\bullet}(t)=-\frac{\text{sgn}(e_{i}(t))}{c_{i}+\text{sgn}(e_{i}(t))} \big{|}e_{i}(t)\big{|}^{\alpha}. \tag{11}\]Let  
\[c_{i}+\text{sgn}(e_{i}(t))=d_{i}>0,\]  
Then  
\[s_{i}^{\bullet}(t)=d_{i}e_{i}^{\bullet}(t)+\text{sgn}(e_{i}(t))\big{|}e_{i}(t) \big{|}^{\alpha},i=1,2,\ldots,n, \tag{12}\]  
\[e_{i}^{\bullet}(t)=-\frac{\text{sgn}(e_{i}(t))}{d_{i}}\big{|}e_{i}(t)\big{|}^{ \alpha}. \tag{13}\]  
**Theorem 1**. : _Consider the error dynamics Equation (13). Under the introduced sliding surface, the system is finite-time stable, and the time \(T_{1}\) required to reach the equilibrium \(e_{i}(t)=0\) is determined by:_  
\[T_{1}\leq\frac{(1/2\sum\
olimits_{i=1}^{n}e_{i}^{2}(0))^{\frac{1-\alpha}{2}}} {(1-\alpha)\rho 2^{\frac{1-\alpha}{2}}}, \tag{14}\]  
_where \(\rho\) is the minimum value among \(\frac{1}{d_{i}}\), \(\forall i=1,2\ldots,n\)._  
**Proof of Theorem 1. **_Consider the Lyapunov function: \(V(t)=\frac{1}{2}\sum\limits_{i=1}^{n}e_{i}^{2}(t)\)._  
_The function's first-order derivative relative to time equals: \(V(t)=\sum\limits_{i=1}^{n}e_{i}(t)e_{i}^{\bullet}(t)\)._  
_Then, using (_13_), one obtains: \(V(t)=\sum\limits_{i=1}^{n}e_{i}(t)(-\frac{1}{d_{i}}\text{sgn}(e_{i}(t))\big{|} e_{i}(t)\big{|}^{\alpha})\),_  
\[V(t)=-\sum\limits_{i=1}^{n}\frac{1}{d_{i}}\big{|}e_{i}(t)\big{|}^{\alpha+1}\leq -\rho\sum\limits_{i=1}^{n}\big{|}e_{i}(t)\big{|}^{\alpha+1},\]  
_Utilizing Lemma 2, it follows: \(V(t)\leq-\rho\sum\limits_{i=1}^{n}\big{|}e_{i}(t)\big{|}^{\alpha+1}\leq-2^{ \frac{\alpha+1}{2}}\rho(\frac{1}{2}\sum\limits_{i=1}^{n}e_{i}^{2}(t))^{\frac{ \alpha+1}{2}}=-2^{\frac{\alpha+1}{2}}\rho V^{\frac{\alpha+1}{2}}(t)\)._  
_Therefore, \(V(t)+2^{\frac{\alpha+1}{2}}\rho V^{\frac{\alpha+1}{2}}(t)\leq 0\), where \(2^{\frac{\alpha+1}{2}}\rho>0\) and \(0<\frac{\alpha+1}{2}<1\). Since Equation (6) is satisfied, following the Lemma 1 one obtains: \(T_{1}\leq\frac{(1/2\sum\
olimits_{i=1}^{n}e_{i}^{2}(0))^{\frac{1-\alpha}{2}}} {(1-\alpha)\rho 2^{\frac{1-\alpha}{2}}}\)._  
_Hence, the error \(e_{i}(t),i=1,2,\ldots n\) converges to zero in finite time \(T_{1}\). In other words, systems \((\ref{eq:1})\) and \((\ref{eq:2})\) synchronize in a finite time. The proof is finished. \(\Box\)_  
_Once the sliding surface is designed, one needs to design a controller with unknown parameters independent of the system model and ensure the error system converges to the sliding surface within finite time. Therefore, the existence of a sliding surface, as well as the convergence of the error trajectory \(e_{i}(t),i=1,2,\ldots n\) to sliding surface \(s_{i}(t)=0,i=1,2,\ldots,n\) should be guaranteed.