# Improved Sliding Mode Finite-Time Synchronization of Chaotic Systems with Unknown Parameters
## 3 Design of Sliding Mode Surface and Controller
Then, Systems \((\ref{eq:1})\) and \((\ref{eq:2})\) reach chaotic synchronization._  
_The designed controller is as follows:_  
\[u_{i}(t)=f_{i}(x_{i})-g_{i}(y_{i})+\frac{1}{d_{i}}[\text{sgn}(e_{i}(t))\big{|} e_{i}(t)\big{|}^{\alpha}+\hat{k}_{i}\frac{\big{|}s_{i}(t)\big{|}}{s_{i}(t)}], \tag{15}\]  
_where \(i=1,2,\ldots,n\), \(d_{i}=c_{i}+\text{sgn}(e_{i}(t))>0\), and \(k_{i}>0\). Here \(k_{i}\) denotes the unknown parameters that are greater than zero and independent of the synchronized system model. Symbol \(\hat{k}_{i}\) denotes the estimated value of unknown parameter \(k_{i}\). Note that the unknown parameter \(k_{i}\) is not within the system model \((\ref{eq:1})\) and only depends on the sliding surface. For the conventional approach, the unknown parameters are selected in the  
Thus,_  
\[V(t)=-\sum_{i=1}^{n}\left[k_{i}\big{|}s_{i}(t)\big{|}\right]-\sum_{i=1}^{n}k_{i} \big{|}\big{|}\hat{k}_{i}-k_{i}\big{|}\leq-\gamma(\sum_{i=1}^{n}\big{|}\big{|}s_ {i}(t)\big{|}\big{|}+\sum_{i=1}^{n}\big{|}\big{|}\hat{k}_{i}-k_{i}\big{|}\big{|}).\]  
_where \(\gamma\) denotes the minimum value among \(k_{i}\). Using Equation (8) in Lemma 2, it follows:_  
\[ \begin{array}{l}-\gamma(\sum\limits_{i=1}^{n}\big{|}\big{|}s_{i}(t)\big{|} \big{|}+\sum\limits_{i=1}^{n}\big{|}\big{|}\hat{k}_{i}-k_{i}\big{|}\big{|}) \leq-\gamma(\sum\limits_{i=1}^{n}\big{|}s_{i}(t)\big{|}^{2}+\sum\limits_{i=1}^ {n}\overline{k_{i}^{2}})^{\frac{1}{2}}=-2^{\frac{1}{2}}\gamma(\frac{1}{2}\sum \limits_{i=1}^{n}\big{|}s_{i}(t)\big{|}^{2}+\frac{1}{2}\sum\limits_{i=1}^{n} \overline{k_{i}^{2}})^{\frac{1}{2}}\\ =-2^{\frac{1}{2}}\gamma V^{\frac{1}{2}}(t)\leq 0\end{array} \]  
_where \(2^{\frac{1}{2}}\gamma>0,0<\frac{1}{2}<1\). Since Equation (6) is satisfied, Lemma 1 gives:_  
\[T_{2}\leq\frac{(\frac{1}{2}\sum\limits_{i=1}^{n}\big{|}s_{i}(0)\big{|}^{2}+ \frac{1}{2}\sum\limits_{i=1}^{n}\overline{k_{i}^{2}}(0))^{\frac{1}{2}}}{2^{- \frac{1}{2}}\gamma}.\]  
Therefore, the error trajectory \(e_{i}(t)\) converges to sliding mode surface \(s_{i}(t)=0\) in finite time \(T_{2}\). The proof is finished. In summary, the block diagram of synchronization and control scheme is as follows:  
**Note 2. **_Following Theorems 1 and 2, the sliding mode controller \((\ref{eq:  
After the above parameters are determined, the nonlinear mathematical model for the roll system of a ship with parametric and forced excitation is changed to the following:  
\[\overset{\bullet\bullet}{\phi}+0.069\overset{\bullet}{\phi}+0.5367\overset{ \bullet}{\phi}^{3}+1.25\phi-0.9926\phi^{3}+0.0998\phi^{5}+2.875\phi\cos 2.236t=0.16 \sin 2.236t\]  
Let \(x_{1}(t)=\phi(t),x_{2}(t)=\dot{\phi}(t)\), and, by conversion to state equations, it is obtained as follows:  
\[\left\{ \begin{array}{l}\overset{\bullet}{x_{1}}=x_{2}\\ \overset{\bullet}{x_{2}}=-0.069x_{2}-0.5367x_{2}{}^{3}-1.25x_{1}+0.9926x_{1}{ }^{3}-0.0998x_{1}{}^{5}\\ -2.875x_{1}\cos 2.236t+0.16\sin 2.236t\end{array} \right.. \tag{18}\]  
The parametric excitation roll system of a ship is a chaos system that can be proved by MATLAB simulation results. The history charts of \(x_{1}\) and \(x_{2}\) of the parametric excitation roll system of the ship are shown in Figures 1 and 2, respectively, in which it is clear that the system is in chaos. The phase diagram of the parametric excitation roll system of the ship is shown in Figure 3, which further proves that the system is a chaos system and similar to a Duffing chaos system. Figure 1: The time response of \(x_{1}\) about ship parametric excitation rolling system Equation (18). Figure 2: The time response of \(x_{2}\) about ship parametric excitation rolling system Equation (18). In the traditional methods, \(h_{0}\) and \(\omega_{1}\) are usually chosen as unknown parameters.