## 3 Main Results
### Leader-Following Tracking Control under Switching Topology  
In this section, we will focus on the formation tracking control of the MAS under switching topology, which means the graph \(\overline{G}\) will change over time. Hence, while the system is running, all the possible connection situations of the swarm system should be considered. Define set \(\left\{\overline{G}:p\in\mathcal{P}\right\}\) as a collection for a system connection graph, where \(p\) and \(\mathcal{P}\) represent the index and the index set to all possible graphs of \(\overline{G}\), respectively. To describe the switching time, we introduce the switching signal \(\sigma(t):[0,+\infty]\rightarrow\mathcal{P}\), at time t, and the connection topology among the follower agents is \(\overline{G}_{\sigma(t)}\). As the topology of the communication changes with time, the neighbors of each agent also change with it. Hence, the tracking protocol we designed is:  
\[u_{i}(t)=K\left[ \begin{array}{c}\sum\limits_{j\in N_{i}(t)}a_{ij}(t)c_{ij}(t )\big{(}x_{i}(t)-h_{i}(t)-\big{(}x_{j}(t)-h_{j}(t)\big{)}\big{)}\\ \hskip 14.226378 pt+b_{i}(t)c_{i}(t)(x_{i}(t)-h_{i}(t)-x_{0}(t))\end{array}  \right]+v_{i}(t) \tag{19}\]  
\[\dot{c}_{ij}(t)=a_{ij}(t)\big{(}x_{i}(t)-h_{i}(t)-\big{(}x_{j}(t)-h_{j}(t) \big{)}\big{)}^{\top}\Gamma\big{(}x_{i}(t)-h_{i}(t)-\big{(}x_{j}(t)-\mathrm{h }_{j}(t)\big{)}\big{)} \tag{20}\]  
\[\dot{c}_{i}(t)=b_{i}(t)(x_{i}(t)-h_{i}(t)-x_{0}(t))^{\top}\Gamma(x_{i}(t)-h_{i }(t)-x_{0}(t)) \tag{21}\]  
where \(a_{ij}(t)\) and \(b_{ij}(t)\) are the elements of the adjacency matrix and leader adjacency matrix. Because the topology is time-varying, the two matrixes above also change over time, and we denote them by \(A_{\sigma(t)}\) and \(F_{\sigma(t)}\), respectively. The tracking error \(e_{i}(t)\) under switching topology could be written as:  
\[ \begin{array}{rl}e_{i}(t)&=x_{i}(t)-h_{i}(t)-x_{0}(t)\\ &=Ae_{i}(t)-BB^{T}P\left[\sum\limits_{j\in N_{i}(t)}a_{ij}(t)c_{ij}(t)\big{(} e_{i}(t)-e_{j}(t)\big{)}+b_{i}(t)c_{i}(t)e_{i}(t)\right]\end{array}  \tag{22}\]  
Consider a series of time intervals \([t_{m},t_{m+1}],\;m\in\mathbb{N}\), which are infinitely bounded and continuous. In each time interval \([t_{m},t_{m+1}]\), we divide it into several subintervals, and there is no overlap among them. To make the problem clear, use the following formula to describe each of them:  
\[\left[t_{m}^{0},t_{m}^{1}\right),\cdots,\left[t_{m}^{i},t_{m}^{j+1}\right), \cdots,\left[t_{m}^{n_{k}-1},t_{m}^{n_{k}}\right),t_{m}^{0}=t_{m},t_{m}^{n_{k} }=t_{m+1}\]  
where \(t_{m}^{j+1}-t_{m}^{j}\geq\tau,\;\tau>0\) is a given constant, \(n\in\mathbb{N}^{+}\). In every subinterval the communication topology of MAS does not change. **Definition 2**. : _The union of a collection of graphs is a graph whose vertex and edge sets are the unions of the vertex and edge sets of the graphs in the collection. We say that such a collection is jointly connected if the union of its members is a connected graph. The graphs are said to be jointly connected across the time interval \([t,t+T],\;T>0\), if the union of graphs \(\left\{\overline{G}_{\sigma(s)}:s\in[t,t+T]\right\}\) is jointly connected._  
**Assumption 3**. : _For each time interval \([t_{m},t_{m+1}]\;m=0,1,2\cdots\), the topology graphs are jointly connected for (1) and (2)._  
Before the stability analysis of the MAS, we should label the eigenvalues of the matrix \(H_{\sigma(t)}=L_{\sigma(t)}+F_{\sigma(t)}\). Specific labeling methods are described in detail in [30]; we will explain the method briefly in the following paragraphs. First, find all the independent components of each subgraph \(G_{\sigma(t)}\) and mark them with the following symbols \(S^{1}_{\sigma(t)},S^{2}_{\sigma(t)},\cdots S^{r}_{\sigma(t)},1\leq\tau\leq n\).