## 4 Consensus of Leader-Follower Multi-Agent Systems with Unknown Parameters
### Consensus of Leader-Follower Multi-Agent Systems  
As the financial system (1) behaves with a complex motion trajectory when it is at a state of chaos, so it is chosen as the leader system, i.e.,  
\[ \begin{cases}\dot{x}_{0}=z_{0}+(y_{0}-a)x_{0},\\ \dot{y}_{0}=1-by_{0}-x_{0}^{2},\\ \dot{z}_{0}=-x_{0}-cz_{0}.\end{cases}  \tag{10}\]  
The \(i\)th (\(i=1,2,\cdots\)) follower system with estimated parameters \(\hat{a}_{i},\hat{b}_{i},\hat{c}_{i}\) and two linear controllers \(u_{0i}^{(x)},u_{0i}^{(y)}\) is given as follows:  
\[ \begin{cases}\dot{x}_{i}=z_{i}+(y_{i}-\hat{a}_{i})x_{i}+u_{0i}^{(x)},\\ \dot{y}_{i}=1-\hat{b}_{i}y_{i}-x_{i}^{2}+u_{0i}^{(y)},\\ \dot{z}_{i}=-x_{i}-\hat{c}_{i}z_{i},\end{cases}  \tag{11}\]  
where \(\hat{a}_{i},\hat{b}_{i},\hat{c}_{i}\) are the estimations of parameters \(a,b,c\), respectively. And \(u_{0i}^{(x)},u_{0i}^{(y)}\) are the linear feedback controllers of \(i\)th follower. If the errors of the leader and \(i\)th follower are given by  
\[e_{0i}^{(x)}=x_{0}-x_{i},e_{0i}^{(y)}=y_{0}-y_{i},e_{0i}^{(z)}=z_{0}-z_{i},e_{ 0i}^{(a)}=a-\hat{a}_{i},e_{0i}^{(b)}=b-\hat{b}_{i},e_{0i}^{(c)}=c-\hat{c}_{i},\]  
then the error system between the leader and \(i\)th follower is obtained by  
\[ \begin{cases}e_{0i}^{(x)}=e_{0i}^{(x)}-ae_{0i}^{(x)}-x_{i}e_{0i}^{(a)}+y_{0}e_ {0i}^{(x)}+x_{i}e_{0i}^{(y)}-u_{0i}^{(x)},\\ e_{0i}^{(y)}=-be_{0i}^{(y)}-y_{i}e_{0i}^{(x)}-(x_{0}+x_{i})e_{0i}^{(x)}-u_{0i}^ {(x)},\\ e_{0i}^{(z)}=-e_{0i}^{(x)}-ce_{0i}^{(z)}-z_{i}e_{0i}^{(c)},\end{cases}  \tag{12}\]  
where \(u_{0i}^{(x)}=a_{0i}k_{0i}^{(x)}e_{0i}^{(y)}+\sum\limits_{j=1}^{N}a_{ij}k_{ij} ^{(x)}(x_{i}-x_{j})\), \(u_{0i}^{(y)}=a_{0i}k_{0i}^{(y)}e_{0i}^{(y)}+\sum\limits_{j=1}^{N}a_{ij}k_{ij} ^{(y)}(y_{i}-y_{j})\), \((i=1,2,\cdots)\). The \(a_{ij}\) denotes the connection between \(i\)th and \(j\)th agents. If there is an edge between followers \(i\) and \(j\), then \(a_{ij}=1\). And \(a_{ij}=0\) if there is not any edge between the \(i\)th and \(j\)th followers. \(k_{ij}^{(x)}\) and \(k_{ij}^{(y)}\) are the feedback gains, and the \(k_{ij}^{(x)}=k_{ij}^{(y)}=0\) in the case of \(a_{0i}=1\). Obviously, it is easy to obtain the following Theorem 3 based on Theorem 2. **Theorem 3**. : _If the suitable gain coefficients \(k_{0i}^{(x)}\) and \(k_{0i}^{(y)}\)\((i=0,1,\cdots)\) are chosen such that_  
\[k_{0i}^{(x)}>-a+y_{0}+1,k_{0i}^{(y)}>-b+x_{0}^{2}/4, \tag{13}\]  
_and the adaptive laws for estimated parameters \(\hat{a}_{i},\hat{b}_{i},\hat{c}_{i}\) are given by_  
\[\hat{a}_{i}=-a_{0i}x_{i}(x_{0}-x_{i}),\hat{b}_{i}=-a_{0i}y_{i}(y_{0}-y_{i}), \hat{c}_{i}=-a_{0i}z_{i}(z_{0}-z_{i}), \tag{14}\]  
_then the leader-follower systems (10) and (11) are globally synchronized._  
Similarly, the \(j\)th (\(j=2,\cdots\)) follower system with estimated parameters \(\hat{a}_{j},\hat{b}_{j},\hat{c}_{j}\) and linear feedback controllers \(u_{ij}^{(x)},u_{ij}^{(y)}\) is given as follows:  
\[ \begin{cases}\dot{x}_{j}=z_{j}+(y_{j}-\hat{a}_{j})x_{j}+u_{ij}^{(x)}\\ \dot{y}_{j}=1-\hat{b}_{j}y_{j}-x_{j}^{2}+u_{ij}^{(y)}\\ \dot{z}_{j}=-x_{j}-\hat{c}_{j}z_{j}\end{cases}  \tag{15}\]  
If the \(j\)th follower is connected with the \(i\)th follower, and the errors between the \(i\)th and \(j\)th followers are defined as  
\[e_{ij}^{(x)}=x_{i}-x_{j},e_{ij}^{(y)}=y_{i}-y_{j},e_{ij}^{(z)}=z_{i}-z_{j},e_{ ij}^{(a)}=a_{i}-\hat{a}_{j},e_{ij}^{(b)}=b_{i}-\hat{b}_{j},e_{ij}^{(c)}=c_{i}- \hat{c}_{j},\]then the error system between the \(i\)th and \(j\)th followers is obtained as follows  
\[\left\{ \begin{aligned} &\dot{e}_{ij}^{(x)}=e_{ij}^{(z)}-\dot{a}_{i}e_{ij}^{(x )}-x_{j}e_{ij}^{(a)}+y_{i}e_{ij}^{(x)}+x_{j}e_{ij}^{(y)}+u_{0i}^{(x)}-u_{ij}^{(x )},\\ &\dot{e}_{ij}^{(y)}=-\hat{b}_{i}e_{ij}^{(y)}-y_{j}e_{ij}^{(b)}-(x _{i}+x_{j})e_{ij}^{(x)}+u_{0i}^{(y)}-u_{ij}^{(y)},\\ &\dot{e}_{ij}^{(z)}=-e_{ij}^{(x)}-\hat{c}_{i}e_{ij}^{(z)}-z_{j}e_ {ij}^{(c)},\end{aligned} \right. \tag{16}\]  
where \(u_{ij}^{(x)}=\sum\limits_{j=1}^{N}a_{ij}k_{ij}^{(x)}(x_{i}-x_{j}),u_{ij}^{(y) }=\sum\limits_{j=1}^{N}a_{ij}k_{ij}^{(y)}(y_{i}-y_{j})\) (\(i=1,2,\cdots\)).