## 3 Main Results
### Leader-Following Tracking Control under Fixed Topology
The value of \(v_{i}(t)\) plays an important role to achieve the desired time-varying formation. It is a judging condition which is used to estimate whether the expected formation could be realized. In this article, we define \(v_{i}(t)\) as the following formulation [29]:  
\[Ah_{i}(t)-\dot{h}_{i}(t)-Bu_{0}(t)+Bv_{i}(t)=0,i=1,2,3,\cdots N \tag{10}\]  
**Theorem 1**. : _Consider an MAS with a leader represented by (1) and (2). Suppose the Assumptions 1-2 hold and the offset vector \(h_{i}(t)\) satisfies the formation tracking feasibility condition (10), then the multi-agent system could achieve the desired time-varying formation by using protocol (6) under fixed communication topology._  
Proof of Theorem 1.: Let \(e_{i}(t)=x_{i}(t)-h_{i}(t)-x_{0}(t)\), \(i=1,2,3,\cdots N\). The \(e_{i}(t)\) denotes the formation tracking error of the \(i\)th agent, and \(\mathbf{e}(t)=[e_{1}(t),e_{2}(t),e_{3}(t),\cdots e_{N}(t)]^{T}\). \[ \begin{array}{rl}\dot{x}_{i}(t)&=Ax_{i}(t)+Bu_{i}(t)\\ &=Ax_{i}(t)+Bv_{i}(t)\\ &-BB^{T}P\left[\begin{array}{c}\sum\limits_{j\in N_{i}}a_{ij}c_{ij}(t)\big{(} x_{i}(t)-h_{i}(t)-\big{(}x_{j}(t)-h_{j}(t)\big{)}\big{)}\\ +b_{i}c_{i}(t)(x_{i}(t)-h_{i}(t)-x_{0}(t))\end{array}\right]\end{array}  \tag{11}\]  
according to (10) and (11), we could obtain:  
\[ \begin{array}{rl}\dot{e}_{i}(t)&=\dot{x}_{i}(t)-\dot{h}_{i}(t)-\dot{x}_{0} (t)\\ &=Ae_{i}(t)-BB^{T}P\left[\begin{array}{c}\sum\limits_{j\in N_{i}}a_{ij}c_{ij} (t)\big{(}e_{i}(t)-e_{j}(t)\big{)}+b_{i}c_{i}(t)e_{i}(t)\end{array}\right] \end{array}  \tag{12}\]  
Consider the following Lyapunov function candidate:  
\[V(t)=\sum\limits_{i=1}^{N}e_{i}^{T}Pe_{i}+\sum\limits_{i=1}^{N}\sum\limits_{j \in N_{i}}\frac{\big{(}c_{ij}(t)-\beta\big{)}^{2}}{2}+\sum\limits_{i=1}^{N} \sum\limits_{j\in N_{i}}(c_{i}(t)-\beta)^{2} \tag{13}\]where \(\beta\) is a positive constant--we will determine its value later. Then, if we evaluate the derivative along the trajectories of Equation (12), we can obtain:  
\[ \begin{array}{ll}\dot{V}(t)&=2\sum\limits_{i=1}^{N}e_{i}^{T}P\dot{e}_{i}+\sum \limits_{i=1}^{N}\sum\limits_{j\in N_{i}}\bigl{(}c_{ij}(t)-\beta\bigr{)}\cdot \dot{c}_{ij}(t)+2\sum\limits_{i=1}^{N}\sum\limits_{j\in N_{i}}\bigl{(}c_{i}(t) -\beta\bigr{)}\cdot\dot{c}_{i}(t)\\ &=2\sum\limits_{i=1}^{N}e_{i}^{T}P\dot{e}_{i}+2\sum\limits_{i=1}^{N}\sum \limits_{j\in N_{i}}b_{i}(c_{i}(t)-\beta)\cdot e_{i}^{T}\Gamma e_{i}+\sum \limits_{i=1}^{N}\sum\limits_{j\in N_{i}}a_{ij}\bigl{(}c_{ij}(t)-\beta\bigr{)} \cdot\bigl{(}e_{i}-e_{j}\bigr{)}^{T}\Gamma\bigl{(}e_{i}-e_{j}\bigr{)}\\ &=e_{i}^{T}\bigl{[}I_{N}\otimes\bigl{(}A^{T}P+PA\bigr{)}-2\beta(L+F)\otimes \Gamma\bigr{]}e_{i}\end{array}  \tag{14}\]  
According to Lemma 2, we can rewrite (14) as following form:  
\[\dot{V}(t)=e_{i}^{T}\Bigl{[}I_{N}\otimes\Bigl{(}A^{T}P+PA\Bigr{)}-2\beta H \otimes\Gamma\Bigr{]}e_{i} \tag{15}\]  
From Lemma 2, we know that H is a symmetric positive definite matrix, so there must exist an orthogonal matrix \(\Upsilon\in\mathbb{R}^{N\times N}\) such that:  
\[THT^{T}=diag(\lambda_{1},\lambda_{2},\lambda_{3},\cdots\lambda_{N}) \tag{16}\]  
where \(\lambda_{1},\lambda_{2},\lambda_{3},\cdots,\lambda_{N}\) are the eigenvalues of H and for any \(i=1,2,3,\cdots N\), we have \(\lambda_{i}>0\). Let \(\widetilde{e}=(T\otimes I_{n})e\):  
\[\dot{V}(t)=\widetilde{e}^{T}\Bigl{(}I_{N}\otimes\Bigl{(}A^{T}P+PA\Bigr{)}-2 \beta\Lambda\otimes\Gamma\Bigr{)}\widetilde{e}=\sum\limits_{i=1}^{N} \widetilde{e}_{i}^{T}\Bigl{(}A^{T}P+PA-2\beta\lambda_{i}\Gamma\Bigr{)} \widetilde{e}_{i} \tag{17}\]  
Because the eigenvalues \(\lambda_{1},\lambda_{2},\lambda_{3},\cdots,\lambda_{N}\) are non-negative, there must exist a positive number \(\beta\) to make \(\beta\lambda_{i}>1,\ i=1,2,3,\cdots,N\).