## 4 Consensus of Leader-Follower Multi-Agent Systems with Unknown Parameters
### Consensus of Leader-Follower Multi-Agent Systems
Theorem 4: _If the suitable gain coefficients \(k_{ij}^{(x)}\) and \(k_{ij}^{(y)}\)\((i,j=0,1,\cdots)\) are chosen such that_  
\[k_{ij}^{(x)}>1-\dot{a}_{i}+y_{i},k_{ij}^{(y)}>-\hat{b}+x_{i}^{2}/4, \tag{17}\]  
_and the adaptive laws for estimated parameters \(\dot{a}_{j},\dot{b}_{j},\dot{c}_{j}\) are given by_  
\[\dot{a}_{j}=-x_{j}e_{ij}^{(x)},\dot{b}_{j}=-y_{j}e_{ij}^{(y)},\dot{c}_{j}=-z_{ j}e_{ij}^{(z)}, \tag{18}\]  
_then the \(i\)th follower in system (11) and \(j\)th follower in Equation (15) are globally synchronized._  
Proof of Theorem 4: According to the error system (16), the synchronization of \(i\)th follower and \(j\)th follower is proved by the Lyapunov stability theorem, and the Lyapunov function of the \(i\)th follower and the \(j\)th follower is given by  
\[V_{ij}=((e_{ij}^{(x)})^{2}+(e_{ij}^{(y)})^{2}+(e_{ij}^{(z)})^{2}+(e_{ij}^{(a)} )^{2}+(e_{ij}^{(b)})^{2}+(e_{ij}^{(c)})^{2})/2,\]  
and the \(a_{0i}=0,k_{ij}^{(x)},k_{ij}^{(y)}>0\); then, its derivative is  
\[\dot{V}_{ij} =e_{ij}^{(x)}e_{ij}^{(y)}+e_{ij}^{(y)}e_{ij}^{(y)}+e_{ij}^{(z)} \dot{e}_{ij}^{(z)}+e_{ij}^{(a)}\dot{e}_{ij}^{(a)}+e_{ij}^{(b)}\dot{e}_{ij}^{(b )}+e_{ij}^{(c)}\dot{e}_{ij}^{(c)}\] \[=e_{ij}^{(x)}(e_{ij}^{(z)}-\dot{a}_{i}e_{ij}^{(x)}-x_{j}e_{ij}^{(a )}+y_{i}e_{ij}^{(x)}+x_{j}e_{ij}^{(y)}-u_{ij}^{(x)})\] \[\quad+e_{ij}^{(y)}(-\hat{b}_{i}e_{ij}^{(y)}-y_{j}e_{ij}^{(b)}-(x_ {i}+x_{j})e_{ij}^{(x)}-u_{ij}^{(y)})\] \[\quad+e_{ij}^{(z)}(-e_{ij}^{(x)}-\hat{c}_{i}e_{ij}^{(z)}-z_{j}e_{ ij}^{(c)})+e_{ij}^{(a)}\dot{e}_{ij}^{(a)}+e_{ij}^{(b)}\dot{e}_{ij}^{(b)}+e_{ij}^{( c)}\dot{e}_{ij}^{(c)}\] \[=(e_{ij}^{(x)})^{2}(1-\dot{a}_{i}+y_{i}-k_{ij}^{(x)}a_{ij})+(e_{ij }^{(y)})^{2}(-\hat{b}_{i}+x_{i}^{2}/4-a_{ij}k_{ij}^{(y)})-\hat{c}_{i}(e_{ij}^{ (z)})^{2}\] \[\quad-e_{ij}^{(x)}\sum\limits_{l=1,l\
eq j}^{N}a_{il}k_{il}^{(x)} (x_{i}-x_{l})-e_{ij}^{(y)}\sum\limits_{l=1,l\
eq j}^{N}a_{il}k_{il}^{(y)}(y_{i }-y_{l})\] \[\quad-x_{l}e_{ij}^{(x)}e_{ij}^{(a)}+e_{ij}^{(a)}\dot{e}_{ij}^{(a) }-y_{j}e_{ij}^{(y)}e_{ij}^{(b)}+e_{ij}^{(b)}e_{ij}^{(b)}-z_{j}e_{ij}^{(z)}e_{ ij}^{(c)}+e_{ij}^{(c)}\dot{e}_{ij}^{(c)}.\]  
If the following conditions are satisfied, i.e.,  
\[\dot{a}_{j}=-x_{j}e_{ij}^{(x)},\dot{b}_{j}=-y_{j}e_{ij}^{(y)},\dot{c}_{j}=-z_{ j}e_{ij}^{(z)}, \tag{18}\]0  
then one has \(e_{ij}^{(a)}=0,e_{ij}^{(b)}=0,e_{ij}^{(c)}=0\), or \(\dot{a}_{j}=-x_{j}e_{ij}^{(x)},\dot{b}_{j}=-y_{j}e_{ij}^{(y)},\dot{c}_{j}=-z_{ j}e_{ij}^{(z)}\). The \(i\)th follower is synchronized with the leader as \(t\to+\infty\), so the \(i\)th follower will synchronize with the \(l\)th follower as \(t\to+\infty\), i.e.,  
\[\left\{ \begin{aligned} &\lim\limits_{t\to+\infty}(x_{i}-x_{l})=0, \lim\limits_{t\to+\infty}(y_{i}-y_{l})=0,\lim\limits_{t\to+\infty}(z_{i}-z_{l})= 0,\\ &\lim\limits_{t\to+\infty}(\dot{a}_{i}-a_{l})=0,\lim\limits_{t\to+ \infty}(\dot{b}_{i}-b_{l})=0,\lim\limits_{t\to+\infty}(\dot{c}_{i}-c_{l})=0. \end{aligned} \right.\] ( \[l\
eq j\] )  
As the \(a_{il}\), \(k_{il}^{(x)},k_{il}^{(y)},k_{il}^{(z)}\) are bounded, one has  
\[\lim\limits_{t\to+\infty}\sum\limits_{l=1,l\
eq j}^{N}a_{il}k_{il}^{(x)}(x_{i}-x_ {l})=0,\lim\limits_{t\to+\infty}\sum\limits_{l=1,l\
eq j}^{N}a_{il}k_{il}^{(y)}(y_{i }-y_{l})=0.\]Therefore, one has \(V_{ij}<0\) by the condition in Equation (17). According to the Lyapunov stability theory, the error system (16) is globally exponentially stable about the origin if \(k_{ij}^{(x)}>1-\hat{a}_{i}+y_{i},k_{ij}^{(y)}>-\hat{b}_{i}+x_{i}^{2}/4\), i.e., the \(i\)th follower and \(j\)th follower in Equation (11) are globally synchronized.