## 3 Main Results  
### Leader-Following Tracking Control under Fixed Topology  
In this section, consider an MAS where the communication topology is fixed. We will propose a distributed adaptive tracking control protocol to make the above system stable. Before that, following assumption should be satisfied. **Assumption 2**. : _The node 1 which represents the leader agent is a globally reachable node in the undirected graph \(\overline{G}\)._  
For a traditional distributed static consensus protocol to make the MAS achieve consensus, the input of each agent is:  
\[u_{i}=c\mathbf{K}\sum_{i=1}^{N}a_{ij}\big{(}x_{i}-x_{j}\big{)},\ i=1,2,3 \cdots N \tag{4}\]  
where \(c>0\) is the coupling weight between the two neighbor agents, \(\mathbf{K}\in\mathbb{R}^{m\times n}\) is the feedback gain matrix and \(a_{ij}\) is the entry of the adjacency matrix. **Lemma 3**. : _[_19_]_ _Supposing that the graph \(G\) is connected, the N agents could achieve consensus (i.e., \(\lim\limits_{t\rightarrow\infty}\|x_{i}-x_{j}\|=0\), \(\forall\ i,j=1,2,3,\cdots N\)) under the protocol (4) with \(K=-B^{T}P^{-1}\) and the coupling weight \(c>\frac{1}{\lambda_{2}}\), where \(\lambda_{2}\) is the smallest nonzero eigenvalue of the Laplacian matrix \(L\), and \(P>\)0 is the solution of the following linear matrix inequality (LMI):_  
\[A^{T}P+PA-2 BB^{T}<0 \tag{5}\]  
The conditions for the multi-agent system to achieve consensus have been given in Lemma 3. As one of the consensus conditions, the coupling weight c should be greater than or equal to \(\frac{1}{\lambda_{2}}\). Hence, if we want to design the consensus protocol by the method above, the value of \(\lambda_{2}\) is necessary, which means every agent should obtain the entire communication graph to compute the Laplacian matrix and its smallest nonzero eigenvalue. Therefore, the protocol above given in Lemma 3 cannot be implemented by each agent in a fully distributed fashion. On the other hand, in some extreme conditions (e.g., the sensors or the communicators malfunction), the communication between the neighbor agents may be interrupted, and the entire communication topology will also change, as well as the smallest nonzero eigenvalue \(\lambda_{2}\). The multi-agent system would be unstable if we still design the consensus protocol by Equation (4). In order to avoid these limitations, we propose the following distributed consensus protocol with an adaptive time-varying coupling weight \(c(t)\), which could be adjusted by the information from the neighbor agents to drive the agent system to achieve the desired time-varying formation. \[u_{i}(t)=K\bigg{[} \begin{array}{c}\sum\limits_{j\in N_{i}}a_{ij}c_{ij}(t)\big{(} x_{i}(t)-h_{i}(t)-\big{(}x_{j}(t)-h_{j}(t)\big{)}\big{)}\\ +b_{i}c_{i}(t)(x_{i}(t)-h_{i}(t)-x_{0}(t))\end{array} \bigg{]}+v_{i}(t) \tag{6}\]  
\[\dot{c}_{ij}(t)=a_{ij}\big{(}x_{i}(t)-h_{i}(t)-\big{(}x_{j}(t)-h_{j}(t)\big{)} \big{)}^{T}\Gamma\big{(}x_{i}(t)-h_{i}(t)-\big{(}x_{j}(t)-h_{j}(t)\big{)}\big{)} \tag{7}\]  
\[\dot{c}_{i}(t)=b_{i}(x_{i}(t)-h_{i}(t)-x_{0}(t))^{T}\Gamma(x_{i}(t)-h_{i}(t)-x_ {0}(t)) \tag{8}\]  
where \(c_{ij}(t)\) is the time-varying coupling weight between the agent \(i\) and agent \(j\); because the communication topology is an undirected graph, we can obtain \(c_{ij}(t)=c_{ji}(t)\). \(c_{i}(t)\) denotes the corresponding weight between the agent \(i\) and the leader agent. Symbol \(v_{i}(t)\) represents the compensational input, which is used to expand the feasible formation set. We will ascertain the compensational input later. Matrix K \(\in\mathbb{R}^{m\times n}\) is the feedback gain, and \(\Gamma\) is an n \(\times\) n dimension square matrix. Under Assumption 1, the pair (A, B) is stabilizable, and there must exist a symmetric positive definite matrix \(P\), which is the solution of the following Riccati inequation:  
\[A^{T}P+PA-2 PBB^{T}P+I<0 \tag{9}\]  
thus, we let the matrix K and \(\Gamma\) equal to \(-B^{T}P\) and PB\(B^{T}P\), respectively. In protocol (6), we notice that the compensational input \(v_{i}(t)\) is still uncertain.