## 3 Consensus Control of Master-Slave Systems  
### Consensus Control of Master-Slave Financial Systems  
The master system is given by financial system (1), and the slave system with linear controllers \(u_{x},u_{y}\) can be obtained as follows:  
\[\begin{cases}\dot{x}_{1}=z_{1}+(y_{1}-a)x_{1}+u_{x},\\ \dot{y}_{1}=1-by_{1}-x_{1}^{2}+u_{y},\\ \dot{z}_{1}=-x_{1}-cz_{1},\end{cases} \tag{3}\]  
where \(a=0.9,b=0.1,c=1\), \(u_{x}\), and \(u_{y}\) are linear controllers. If one defines the errors \(e_{x}=x-x_{1},e_{y}=y-y_{1},e_{z}=z-z_{1}\), then the error system is obtained by  
\[\begin{cases}\dot{e}_{x}=e_{z}-ae_{x}+ye_{x}+x_{1}e_{y}-u_{x},\\ \dot{e}_{y}=-be_{y}-(x+x_{1})e_{x}-u_{y},\\ \dot{e}_{z}=-e_{x}-ce_{z},\end{cases} \tag{4}\]  
where \(u_{x}=k_{x}e_{x},u_{y}=k_{y}e_{y}\).  
**Definition 1**.: _If the errors \(e_{x},e_{y},\) and \(e_{z}\) of Equation (4) tend to zero as time \(t\to+\infty\), then the master-slave systems (1) and (3) have achieved synchronization or consensus._  
**Theorem 1**.: _If the suitable gain coefficients \(k_{x}\) and \(k_{y}\) are chosen such that_  
\[k_{x}>-a+y+1,k_{y}>-b+x^{2}/4, \tag{5}\]  
_then the master-slave systems (1) and (3) are globally synchronized._  
Proof of Theorem 1.: According to the error system (4), if the Lyapunov function is  
\[V=(e_{x}^{2}+e_{y}^{2}+e_{z}^{2})/2,\]  
then its derivative is  
\[\dot{V} =e_{x}\dot{e}_{x}+e_{y}\dot{e}_{y}+e_{z}\dot{e}_{z}\] \[=e_{x}(e_{z}-ae_{x}+ye_{x}+x_{1}e_{y}-u_{x})+e_{y}(-be_{y}-(x+x_{1 })e_{x}-u_{y})+e_{z}(-e_{x}-ce_{z})\] \[=(-a+y-k_{x})e_{x}^{2}+(-b-ky)e_{y}^{2}-ce_{z}^{2}-xe_{x}e_{y}\] \[=(-a+y+1-k_{x})e_{x}^{2}+(-b+x^{2}/4-k_{y})e_{y}^{2}-ce_{z}^{2}-( e_{x}+xe_{y}/2)^{2}.\]  
According the condition in Equation (5), one has \(\dot{V}<0\). According to the Lyapunov stability theory [38; 39], the system (4) is globally exponentially stable about the origin if  
Figure 3: Attractor of financial system (1) with \(c=2\). (**a**) Phase diagram of \(x\) vs. \(y\). (**b**) Phase diagram of \(y\) vs. \(z\).  
\(k_{x}>-a+y+1\), \(k_{y}>-b+x^{2}/4\), i.e., the master-slave systems (1) and (3) are globally synchronized.  
**Example 1**.: _According to the attractor of system (1), the variables \(x,y\) satisfy \(|x|<3,y<4\), so the gain coefficients of the controllers \(u_{x}\) and \(u_{y}\) can be given by \(k_{x}=4.1,k_{y}=2.2\) based on Theorem 1. And the initial values of master-slave systems are \((x(0),y(0),z(0))=(0.1,0.2,0.1)\), and \((x_{1}(0),y_{1}(0),z_{1}(0))=(0.2,0.1,0.3)\), respectively._  
The synchronization errors \(e_{x},e_{y}\), and \(e_{z}\) are shown in Figure 4 by Matlab R2020b. Obviously, the master-slave systems have reached consensus as \(t\to+\infty\).  
**Remark 2**.: _Consensus control usually refers to the consensus problem among multiple systems. In this paper, only the synchronization control method for master-slave systems is proposed, because the topological network structure between multiple systems is not provided. However, this control method can be extended to multiple slave systems to synchronize with the master system (1), and then the consensus control among multiple systems is easily obtained. Furthermore, the new slave systems can also be designed to synchronize with the slave system (3), so the consensus control for multiple systems is also achieved in the end._