# Containment Control of First-Order Multi-Agent Systems under PI Coordination Protocol +
## 3 Main Results
### General Directed Follower Topology
Thus, the characteristic Equation (9) has all roots within or on the unit circle and we have finally proved the asymptotic convergence of the system under Assumption 1, i.e. \[\lim_{k\to\infty}\hat{Y}(k)=\lim_{k\to\infty}[X_{F}^{T}(k),\hat{R}^{T}(k)]^{T} =[X_{d}^{T},\hat{R}_{d}^{T}]^{T}, \tag{18}\]  
where \(X_{d}\in R^{n-m}\) and \(\hat{R}_{d}\in R^{n-m}\) are constant vectors. **Step 2**: We will prove that all followers reach an agreement value in convex hull spanned by leaders. According to Equation (4), we obtain with \(p=1\)  
\[\lim_{k\to\infty}L_{F}X_{F}(k)=0. \tag{19}\]  
Under Assumption 1,we get from Lemma 1 and Equation (19)  
\[\lim_{k\to\infty}x_{i}(k)=x_{d},i=m+1,\ldots,n, \tag{20}\]  
where \(x_{d}\in R\) is a constant. From Equation (20), it is clear that followers converge to the same value. Taking the \(z\) transformation of Equation (4) with \(p=1\), we get  
\[ \begin{split} zX_{F}(z)-zX_{F}(0)=&(I-\gamma_{1}L_ {2})X_{F}(z)+\gamma_{2}R(z)\\ &-\frac{z}{z-1}\gamma_{1}L_{1}X_{L}.\\ zR(z)-zR(0)=& R(z)-L_{F}X_{F}(z)\end{split}  \tag{21}\]  
Re-express Equation (21) as  
\[ \begin{split} X_{F}(z)=&[(z-1)I+\gamma_{1}L_{2}+ \gamma_{2}\frac{L_{F}}{z-1}]^{-1}\\ &[zX_{F}(0)+\frac{z}{z-1}\gamma_{2}R(0)-\frac{z}{z-1}\gamma_{1}L _{1}X_{L}]\\ =&[(z-1)I+\gamma_{1}D+(\gamma_{1}+\frac{\gamma_{2}}{ z-1})L_{F}]^{-1}\\ &[zX_{F}(0)+\frac{z}{z-1}\gamma_{2}R(0)-\frac{z}{z-1}\gamma_{1}L _{1}X_{L}].\end{split}  \tag{22}\]Since the convergence of the system has been proved, using the final value theorem yields  
\[ \begin{split}\lim_{z\to 1}(z-1)X_{F}(z)=&[(z-1)I+ \gamma_{1}D+(\gamma_{1}+\frac{\gamma_{2}}{z-1})L_{F}]^{-1}\\ &[z(z-1)X_{F}(0)+z\gamma_{2}R(0)-z\gamma_{1}L_{1}X_{L}]\\ =& x_{d}[1,1,\ldots,1]^{T}.\end{split}  \tag{23}\]  
Letting \(W(z)=(z-1)X_{F}(z)\), we have \(\lim_{z\to 1}W(z)=x_{d}[1,1,\ldots,1]^{T}\) and  
\[ \begin{split}&[(z-1)I+\gamma_{1}D+(\gamma_{1}+\frac{\gamma_{2}}{z- 1})L_{F}]W(z)\\ &=[z(z-1)X_{F}(0)+\gamma_{2}zR(0)-z\gamma_{1}L_{1}X_{L}].\end{split}  \tag{24}\]  
Multiplying \(Q=[q_{m+1},q_{m+2},\ldots,q_{n}]\) on both sides of (24), we get  
\[ \begin{split}& Q[(z-1)I+\gamma_{1}D]W(z)\\ &=Q[z(z-1)X_{F}(0)+\gamma_{2}zR(0)-z\gamma_{1}L_{1}X_{L}],\end{split}  \tag{25}\]  
where \(Q\) is the left eigenvector of \(L_{F}\) corresponding to eigenvalue 0. Then, we take the limit of Equation (25) as \(z\to 1\) and have  
\[Q\gamma_{1}Dx_{d}[1,1,\ldots,1]^{T}=Q[\gamma_{2}R(0)-\gamma_{1}L_{1}X_{L}]. \tag{26}\]  
Equation (26) is rewritten as  
\[ \begin{split}\gamma_{1}x_{d}\sum_{i=m+1}^{n}\sum_{j=1}^{m}q_{i} a_{ij}=&\gamma_{1}\sum_{i=m+1}^{n}\sum_{j=1}^{m}q_{i}a_{ij}x_{j}\\ &+\gamma_{2}\sum_{i=m+1}^{n}q_{i}r_{i}(0),\end{split}  \tag{27}\]  
and we finally get  
\[x_{d}=\frac{\sum_{i=m+1}^{n}\sum_{j=1}^{m}q_{i}a_{ij}x_{j}}{\sum_{i=m+1}^{n} \sum_{j=1}^{m}q_{i}a_{ij}}, \tag{28}\]  
with \(\sum_{i=m+1}^{n}q_{i}r_{i}(0)=0\). Apparently, \(x_{d}\) is in the convex hull formed by leaders. When \(p\
eq 1\), the proof is the same while the final state \(x_{d}\) is a constant vector instead of a constant. Hence, we can draw the conclusion that all followers will eventually reach an agreement value in the convex hull spanned by leaders under our proposed protocol. Theorem 1 is proved.