# Improved Sliding Mode Finite-Time Synchronization of Chaotic Systems with Unknown Parameters
## 2 Fundamental Lemmas  
In this section, the system description is presented, the synchronization problem is explained, and the relevant definitions and several necessary lemmas on finite-time synchronization are introduced.  
The following two \(n\)-dimensional chaotic systems are considered:  
Master system:  
\[\left\{\begin{array}{c}x_{1}(t)=f_{1}(x_{1},x_{2},\ldots,x_{n})\\ \bullet\\ x_{2}(t)=f_{2}(x_{1},x_{2},\ldots,x_{n})\\.\\.\\ \bullet\\ x_{n}(t)=f_{n}(x_{1},x_{2},\ldots,x_{n})\end{array}\right., \tag{1}\]  
Slave system:  
\[\left\{\begin{array}{c}\bullet\\ y_{1}(t)=g_{1}(y_{1},y_{2},\ldots,y_{n})+u_{1}(t)\\ \bullet\\ y_{2}(t)=g_{2}(y_{1},y_{2},\ldots,y_{n})+u_{2}(t)\\.\\.\\.\\.\\ \bullet\\ y_{n}(t)=g_{n}(y_{1},y_{2},\ldots,y_{n})+u_{n}(t)\end{array}\right.. \tag{2}\]  
In the above equation, \(f_{i}(x_{1},x_{2},\ldots,x_{n}),i=1,2,\ldots,n\) and \(g_{i}(y_{1},y_{2},\ldots,y_{n}),i=1,2,\ldots,n\) are nonlinear functions, and \(u_{i}(t),i=1,2,\ldots,n\) is a controller with unknown parameters. Equations (1) and (2) both represent chaotic systems.  
This work aims at designing a suitable controller with particular unknown parameters independent of the system model to synchronize chaotic systems (described by Equations (1) and (2)) in a finite time while the systems remain chaotic. To achieve finite-time synchronization, a system error is defined as follows:  
\[e_{i}(t)=x_{i}(t)-y_{i}(t), \tag{3}\]  
The error dynamics equation is obtained as a derivative of the equation resulting from subtracting Equation (3) from Equation (2). Thus:  
\[\left\{\begin{array}{c}\bullet\\ e_{1}(t)=f_{1}(x_{i})-g_{1}(y_{i})-u_{1}(t)\\ \bullet\\ e_{2}(t)=f_{2}(x_{i})-g_{2}(y_{i})-u_{2}(t)\\.\\.\\.\\ \bullet\\ e_{i}(t)=f_{i}(x_{i})-g_{i}(y_{i})-u_{i}(t)\\.\\.\\.\\.\\.\\ \bullet\\ e_{n}(t)=f_{n}(x_{i})-g_{n}(y_{i})-u_{n}(t)\end{array}\right.. \tag{4}\]  
**Definition 1**.: _Let the master system be defined with Equation \((1)\) and the slave system with Equation \((2)\). Suppose there is a constant \(T>0\) for which \(\underset{t\to T}{\lim}[e_{i}(t)]=0,i=1,2,\ldots,n\) and \(\big{|}e_{i}(t)\big{|}\equiv 0,i=1,2,\ldots,n\) when \(t>T\). Then, the chaotic synchronization of systems \((1)\) and \((2)\) is achieved in finite time \(T\)._  
Achieving the finite-time stabilization of the error system \((4)\) is equivalent to realizing the finite-time synchronization of chaotic systems \((1)\) and \((2)\).  
**Lemma 1**.: _[_26_]__. Consider a system \((5)\)_  
\[\overset{\bullet}{\mathbf{x}}=\mathbf{f}(\mathbf{x}),\mathbf{f}(0)=0, \mathbf{x}\in R^{n}, \tag{5}\]  
_where \(\mathbf{f}:D\to R^{n}\) is continuous on an open neighborhood and \(D\subset R^{n}\). Assuming that there is a continuous differential positive-definite function \(V(\mathbf{x}):D\to R\), and the real numbers \(p>0\) and \(0<\eta<1\) satisfy:_  
\[V(\overset{\bullet}{\mathbf{x}})+pV^{\eta}(\mathbf{x})\leq 0,\forall\mathbf{x }\in D, \tag{6}\]  
_then, the system \((5)\) is a finite-time stable system whose stabilization time depends on the initial state \(\mathbf{x}(0)=\mathbf{x}_{0}\),(here \(\mathbf{x}(0)\) represent vectors, for example \(\mathbf{x}=(x_{1},x_{2}\ldots x_{i})\), so \(\mathbf{x}(0)=\mathbf{x}_{0}=(x_{1}(0),x_{2}(0)\ldots x_{i}(0))\), the initial conditions are simple initial conditions) and satisfies:_  
\[T(\mathbf{x}_{0})\leq\frac{V^{1-\eta}(\mathbf{x}_{0})}{p(1-\eta)}. \tag{7}\]  
**Note 1**.: _Throughout the paper, symbols written in boldface represent vectors._  
**Lemma 2**.: _For every \(a_{1},a_{2},\ldots a_{n}\in R\), the following equation is satisfied:_  
\[|a_{1}|+|a_{2}|+\ldots+|a_{n}|\geq\sqrt{{a_{1}}^{2}+{a_{2}}^{2}+\ldots+{a_{n}} ^{2}}. \tag{8}\]