# Improved Sliding Mode Finite-Time Synchronization of Chaotic Systems with Unknown Parameters
## 3 Design of Sliding Mode Surface and Controller
In the model \(\stackrel{{\bullet\bullet}}{{\phi}}+\mu_{1}\stackrel{{ \bullet}}{{\phi}}+\mu_{3}\stackrel{{\bullet\bullet}}{{\phi}}+ \omega_{0}{}^{2}(\phi+\alpha_{3}\phi^{3}+\alpha_{5}\phi^{5}+h_{0}\phi\cos( \omega_{1}t))=K_{e}\sin(\omega_{1}t+\delta_{0})\), \(h_{0}\) is parameter excitation amplitude and \(\omega_{1}\) is parameter excitation frequency, which can affect the chaotic state of the system. This method can be seen in references [24] and [25]. In the method discussed in this paper, a ship's parametric excitation rolling system is regarded as a master system. The controller's chaotic system with unknown parameters independent of the system model is regarded as a slave system. The master system and slave system are based on the ship parametric excitation rolling system with the same fundamental structure. In other words, the chaotic control is achieved over the same structure systems. Master system:  
\[\left\{ \begin{array}{l}\stackrel{{\bullet}}{{x}}_{1}=x_{2}\\ \stackrel{{\bullet}}{{x}}_{2}=-0.069x_{2}-0.5367{x_{2}}^{3}-1.25 x_{1}+0.9926{x_{1}}^{3}-0.0998{x_{1}}^{5}\\ -2.875x_{1}\cos 2.236t+0.16\sin 2.236t\end{array} \right., \tag{19}\]  
Slave system:  
\[\left\{ \begin{array}{l}\stackrel{{\bullet}}{{y}}_{1}=y_{2}+u_{1 }(t)\\ \stackrel{{\bullet}}{{y}}_{2}=-0.069y_{2}-0.5367{y_{2}}^{3}-1.25 y_{1}+0.9926{y_{1}}^{3}-0.0998{y_{1}}^{5}\\ -2.875y_{1}\cos 2.236t+0.16\sin 2.236t+u_{2}(t)\end{array} \right., \tag{20}\]  
\[\text{Error system}:\,\left\{ \begin{array}{l}e_{1}=x_{1}-y_{1}\\ e_{2}=x_{2}-y_{2}\end{array} \right.,\]  
Error dynamics equation:  
\[\left\{ \begin{array}{l}\stackrel{{\bullet}}{{e}}_{1}=x_{2}-y_{ 2}-u_{1}(t)\\ \stackrel{{\bullet}}{{e}}_{2}=-0.069x_{2}-0.5367{x_{2}}^{3}-1.25 x_{1}+0.9926{x_{1}}^{3}-0.0998{x_{1}}^{5}\\ -2.875x_{1}\cos 2.236t+0.16\sin 2.236t+0.069y_{2}+\\ 0.5367{y_{2}}^{3}+1.25y_{1}-0.9926{y_{1}}^{3}+0.0998{y_{1}}^{5}\\ +2.875y_{1}\cos 2.236t-0.16\sin 2.236t-u_{2}(t)\end{array} \right., \tag{21}\]  
Using Equation (9), the sliding mode surface is designed as follows:  
\[\left\{ \begin{array}{l}s_{1}=c_{1}e_{1}(t)+\int_{0}^{t}\text{sgn}(e_{1}(t)) \big{|}e_{1}(t)\big{|}^{\alpha}d\tau+e_{1}(t)\text{sgn}(e_{1}(t))\\ s_{2}=c_{2}e_{2}(t)+\int_{0}^{t}\text{sgn}(e_{2}(\tau))\big{|}e_{2}(\tau)\big{|}^ {\alpha}d\tau+e_{2}(t)\text{sgn}(e_{2}(t))\end{array} \right., \tag{22}\]  
Figure 3: The phase diagram of \(x_{1}\) and \(x_{2}\) about ship parametric excitation rolling system. According to Equation (15), the controller with unknown parameters independent of the system is designed as:  
\[\left\{ \begin{array}{l}u_{1}(t)=x_{2}-y_{2}+\frac{1}{c_{1}+\text{sgn}(e_{1}(t))} \big{[}\text{sgn}(e_{1}(\tau))\big{|}e_{1}(\tau)\big{|}^{\alpha}+\hat{k}_{1} \frac{|\underline{s}_{1}|}{s_{1}}\big{]}\\ u_{2}(t)=-0.069x_{2}-0.5367x_{2}{}^{3}-1.25x_{1}+0.9926x_{1}{}^{3}-0.0998x_{1} {}^{5}\\ -2.875x_{1}\cos 2.236t+0.16\sin 2.236t+0.069y_{2}+0.5367y_{2}{}^{3}\\ +1.25y_{1}-0.9926y_{1}{}^{3}+0.0998y_{1}{}^{5}+2.875y_{1}\cos 2.236t\\ -0.16\sin 2.236t+\frac{1}{c_{2}+\text{sgn}(e_{2}(t))}\big{[}\text{sgn}(e_{2}(\tau) )\big{|}e_{2}(\tau)\big{|}^{\alpha}+\hat{k}_{2}\frac{|\underline{s}_{2}|}{s_{2 }}\big{]}\end{array} \right., \tag{23}\]  
From Equation (16), the adaptive law of unknown parameters is:  
\[\left\{ \begin{array}{l}\overset{\bullet}{\hat{k}_{1}}=|s_{1}|-k_{1}\frac{| \underline{k}_{1}-k_{1}|}{(\underline{k}_{1}-k_{1})}\\ \overset{\bullet}{\hat{k}_{2}}=|s_{2}|-k_{2}\frac{|\underline{k}_{2}-k_{2}|}{ (\underline{k}_{2}-k_{2})}\end{array} \right.. \tag{24}\]  
In the simulation, the step size was set to 0.01 s, and the initial values were \(x_{1}(0)=3,x_{2}(0)=3,y_{1}(0)=1,y_{2}(0)=1\), and \(c_{1}=8,c_{2}=9,\). In addition, the initial values of unknown parameters were set to \(k_{1}=0.1\) and \(k_{2}=0.2\).