# Containment Control of First-Order Multi-Agent Systems under PI Coordination Protocol +
## 3 Main Results
### General Directed Follower Topology  
**Theorem 1**. : _Consider the multi-agent system (3) with leader-follower and follower topologies satisfying Assumptions \(1\) and \(2\), respectively. With condition \(\sum_{l=m+1}^{n}q_{i}r_{i}(0)=0\), all the followers reach containment-rendezvous asymptotically that the followers converge to an agreement value in the convex hull spanned by the leaders, if \(\gamma_{1}\) and \(\gamma_{2}\) satisfy_  
\[\gamma_{1}-\gamma_{2}>0,\]  
\[(\gamma_{1}-\gamma_{2})^{2}(\gamma_{2}-2\gamma_{1}+\frac{4 Re(\kappa_{i})}{| \kappa_{i}|^{2}})-\frac{4\gamma_{2}Im(\kappa_{i})^{2}}{|\kappa_{i}|^{4}}>0,\]  
_for \(i=m+1,\ldots,n\), and_  
\[|\lambda(\gamma_{2}D\Theta^{-1}(e^{\mathrm{j}\omega}))|<1 \tag{7}\]  
_hold with \(\omega\in(0,\pi]\), where \(\Theta(e^{\mathrm{j}\omega})=(e^{\mathrm{j}\omega}-1)^{2}I+\gamma_{1}(e^{ \mathrm{j}\omega}-1)(L_{F}+D)+\gamma_{2}(L_{F}+D)\) and \(\kappa_{i},i=m+1,\ldots,n\) represent the eigenvalues of the matrix \(L_{F}+D\)._  
Proof. : According to the properties of the Kronecker product, we set the agents' state dimension as \(p=1\) in the following proof, and system (6) is written as  
\[\hat{Y}(k+1)= \begin{bmatrix}I_{n-m}-\gamma_{1}(L_{F}+D)&\gamma_{2}I_{n-m}\\ -L_{F}&I_{n-m}\end{bmatrix} \hat{Y}(k). \tag{8}\]  
Meanwhile, we divide the proof into two steps including convergence analysis and the analysis of final rendezvous state. **Step 1**: To analyze the convergence performance of system (8), we investigate the characteristic equation of (8) as follows,  
\[\det(zI- \begin{bmatrix}I-\gamma_{1}(L_{F}+D)&\gamma_{2}I\\ -L_{F}&I\end{bmatrix} )=0, \tag{9}\]  
and it can be reformulated from Lemma 3 as  
\[\det((z-1)^{2}I+\gamma_{1}(z-1)(L_{F}+D)+\gamma_{2}L_{F})=0. \tag{10}\]  
Evidently, it is obtained from Lemma 1 that the Equation (10) has a root at \(z=1\). Before analyzing Equation (10), we first pay attention to the following equation  
\[\det((z-1)^{2}I+\gamma_{1}(z-1)(L_{F}+D)+\gamma_{2}(L_{F}+D))=0,. \tag{11}\]  
Obviously, (11) is equivalent to  
\[z^{2}+(\gamma_{1}\kappa_{i}-2)z+(\gamma_{2}-\gamma_{1})\kappa_{i}+1=0,i=m+1, \ldots,n. \tag{12}\]  
Instead of studying Equation (12) directly, we apply the bilinear transformation \(s=\frac{z+1}{z-1}\) to it and get  
\[\gamma_{2}\kappa_{i}s^{2}+2(\gamma_{1}-\gamma_{2})\kappa_{i}s+(\gamma_{2}-2 \gamma_{1})\kappa_{i}+4=0. \tag{13}\]  
Thus, Equation (12) has all roots within the unit circle, if and only if Equation (13) has all roots in the open left half complex plane. Then, we reformulate Equation (13) as  
\[s^{2}+(p_{1}+\mathrm{j}q_{1})s+p_{2i}+\mathrm{j}q_{2i}=0,i=m+1,\ldots,n, \tag{14}\]  
where \(p_{1}=\frac{2(\gamma_{1}-\gamma_{2})}{\gamma_{2}}\), \(q_{1}=0\), \(p_{2i}=\frac{\gamma_{2}-2\gamma_{1}}{\gamma_{2}}+\frac{4 Re(\kappa_{i})}{\gamma _{2}|\kappa_{i}|^{2}}\), \(q_{2i}=-\frac{4 Im(\kappa_{i})}{\gamma_{2}|\kappa_{i}|^{2}}\). Based on Lemma 2, the polynomial has all zeros in the open left half complex plane if and only if the gains \(\gamma_{1}\) and \(\gamma_{2}\) satisfy  
\[ \begin{split}& p_{1}>0,\\ & p_{1}^{2}p_{2i}+p_{1}q_{1}q_{2i}-q_{2i}^{2}>0,i=m+1,\ldots,n.\end{split}  \tag{15}\]  
which is equivalent to  
\[ \begin{split}&\gamma_{1}-\gamma_{2}>0,\\ &(\gamma_{1}-\gamma_{2})^{2}(\gamma_{2}-2\gamma_{1}+\frac{4 Re( \kappa_{i})}{|\kappa_{i}|^{2}})-\frac{4\gamma_{2}Im^{2}(\kappa_{i})}{|\kappa_ {i}|^{4}}>0,\\ & i=m+1\ldots,n.\end{split}  \tag{16}\]  
Thus, the roots of Equation (11) all lie inside the unit circle if and only if there exist gain parameters \(\gamma_{1}\) and \(\gamma_{2}\) satisfying condition (16). Back to Equation (10), we reformulate it as  
\[\det(I-\gamma_{2}D\Theta^{-1}(z))=0, \tag{17}\]  
where \(\Theta(z)=(z-1)^{2}I+\gamma_{1}(z-1)(L_{F}+D)+\gamma_{2}(L_{F}+D)\). Under condition (16), it is obvious that if condition \(|\lambda(\gamma_{2}D\Theta^{-1}(\hat{e}^{\omega}))|<1\) holds with \(\omega\in(0,\pi]\), the roots of Equation (17) lie inside the unit circle except for one root at \(z=1\).