## 3 Consensus Control of Master-Slave Systems
### Consensus of Master-Slave Systems with Unknown Parameters  
If the parameters \(a,b,c\) of master system (1) are unknown, then the slave system with estimated parameters \(\hat{a},\hat{b},\hat{c}\) and linear controllers can be given as follows:  
\[\begin{cases}\hat{x}_{1}=z_{1}+(y_{1}-\hat{a})x_{1}+u_{x},\\ \hat{y}_{1}=1-\hat{b}y_{1}-x_{1}^{2}+u_{y},\\ \hat{z}_{1}=-x_{1}-\hat{c}z_{1}\end{cases} \tag{6}\]  
where \(\hat{a},\hat{b},\hat{c}\) are the estimations of parameters \(a,b,c\), respectively. And \(u_{x},u_{y}\) are linear controllers. If the errors are \(e_{x}=x-x_{1}e_{y}=y-y_{1}e_{z}=z-z_{1}\), \(e_{a}=a-\hat{a}\), \(e_{b}=b-\hat{b}\), and \(e_{c}=c-\hat{c}\), then the error system is obtained by  
\[\begin{cases}\hat{e}_{x}=e_{z}-ae_{x}-x_{1}e_{a}+ye_{x}+x_{1}e_{y}-u_{x},\\ \hat{e}_{y}=-be_{y}-y_{1}e_{b}-(x+x_{1})e_{x}-u_{y},\\ \hat{e}_{z}=-e_{x}-ce_{z}-z_{1}e_{c},\end{cases} \tag{7}\]  
where \(u_{x}=k_{x}e_{x},u_{y}=k_{y}e_{y}\).  
**Theorem 2**.: _If the suitable gain coefficients \(k_{x}\) and \(k_{y}\) are chosen such that_  
\[k_{x}>-a+y+1,k_{y}>-b+x^{2}/4, \tag{8}\]  
_and the adaptive laws for estimated parameters \(\hat{a},\hat{b},\hat{c}\) are given by_  
Figure 4: Errors \(e_{x},e_{y},e_{z}\) of master–slave systems (1) and (3).  
\[\hat{a}=-x_{1}(x-x_{1}),\hat{b}=-y_{1}(y-y_{1}),\hat{c}=-z_{1}(z-z_{1}), \tag{9}\]  
_then the master-slave systems (1) and (6) are globally synchronized._  
Proof of Theorem 2.: According to the system (7), if the Lyapunov function is given by  
\[V=(e_{x}^{2}+e_{y}^{2}+e_{z}^{2}+e_{a}^{2}+e_{b}^{2}+e_{c}^{2})/2\]  
then its derivative is  
\[\dot{V} =e_{x}\dot{e}_{x}+e_{y}\dot{e}_{y}+e_{z}\dot{e}_{z}+e_{a}\dot{e}_{ a}+e_{b}\dot{e}_{b}+e_{c}\dot{e}_{c}\] \[=e_{x}(e_{z}-ae_{x}-x_{1}e_{y}+e_{x}+x_{1}e_{y}-u_{x})+e_{y}(-be_{ y}-y_{1}e_{b}-(x+x_{1})e_{x}-u_{y})\] \[\quad+e_{z}(-e_{x}-ce_{z}-z_{1}e_{c})+e_{a}e_{a}+e_{b}\dot{e}_{b}+ e_{c}\dot{e}_{c}\] \[=e_{x}^{2}(1-a+y-k_{x})+e_{y}^{2}(-b+x^{2}/4-k_{y})-ce_{z}^{2}-x_ {1}e_{x}e_{a}+e_{a}\dot{e}_{a}-y_{1}e_{y}e_{b}+e_{b}\dot{e}_{b}\] \[\quad-z_{1}e_{z}e_{c}+e_{c}\dot{e}_{c}.\]  
If the following conditions are satisfied, i.e., \(-x_{1}e_{x}e_{a}+e_{a}\dot{e}_{a}=0\), \(-y_{1}e_{y}e_{b}+e_{b}\dot{e}_{b}=0\), \(-z_{1}e_{z}e_{c}+e_{c}\dot{e}_{c}=0\), then one has \(e_{a}=0,e_{b}=0,e_{c}=0\), or \(\hat{a}=-x_{1}(x-x_{1}),\hat{b}=-y_{1}(y-y_{1})\), \(\dot{\hat{c}}=-z_{1}(z-z_{1})\).  
Therefore, one has \(\dot{V}<0\) by the condition in Equation (8). According to the Lyapunov stability theory, the system (7) is globally exponentially stable about the origin if \(k_{x}>-a+y+1,k_{y}>-b+x^{2}/4\), i.e., the master-slave systems (1) and (6) are globally synchronized.  
The gain coefficients \(k_{x}=4.1,k_{y}=2.2\) according to Theorem 2.1, and the initial values of master-slave systems are given by \((x(0),y(0),z(0))=(0.1,0.2,0.1)\), \((x_{1}(0),y_{1}(0),z_{1}(0))\)\(=(0.2,0.1,0.3)\), and \((\hat{a}(0),\hat{b}(0),\hat{c}(0))=(0.8,0.2,0.7)\), respectively.  
The synchronization errors \(e_{x},e_{y}\), and \(e_{z}\) are shown in Figure 4(a). The master-slave systems have reached consensus as \(t\to+\infty\). In Figure 4(b), the parameters \(\hat{a},\hat{b},\hat{c}\) of slave system (6) can track to the \(a,b,c\) of master system (1) as \(t\to+\infty\).