# Containment Control of First-Order Multi-Agent Systems under PI Coordination Protocol +
## 3 Main Results
### Symmetric Follower Topology  
By means of some existing results [40; 41], we present the convergence condition, which is less conservative than (7), of system (3) under symmetric follower topology. Theorem 2: _The leader-follower and follower topologies of multi-agent system (3) satisfy Assumptions 1 and 3, respectively. With \(\sum_{i=m+1}^{n}r_{i}(0)=0\), all the followers converge to an agreement value \(\frac{\sum_{i=m+1}^{n}\sum_{j=1}^{n}a_{ij}x_{j}}{\sum_{i=m+1}^{n}\sum_{j=1}^{ n}a_{ij}}\) that lies in convex hull spanned by the leaders, if \(\gamma_{1}\) and \(\gamma_{2}\) satisfy_  
\[\gamma_{2}\leq\gamma_{1}\leq\frac{2}{\theta_{i}}, \tag{29}\]  
_and_  
\[\rho(L_{F})(\gamma_{2}-2\gamma_{1})-2\gamma_{1}\theta_{i}+4>0,\]  
_for \(i=m+1,\ldots,n\), where \(\theta_{i}=\sum_{j=1}^{m}a_{ij}\)_Proof. : We have already known that the characteristic Equation (9) has a root as \(z=1\), so we only consider the situation that \(z\
eq 1\). From Lemma 3, Equation (9) equals to  
\[\det((z-1)((z-1)I+\gamma_{1}D)+(\gamma_{1}(z-1)+\gamma_{2})L_{F})=0. \tag{30}\]  
When \(z\
eq 1\), (30) is reformulated as  
\[\det(I+G(z)L_{F})=0, \tag{31}\]  
where  
\[G(z)= \begin{bmatrix}\frac{\gamma_{1}(z-1)+\gamma_{2}}{(z-1)(z-1+\gamma_{1} \theta_{m}+1)}\\ &\ddots\\ &&\frac{\gamma_{1}(z-1)+\gamma_{2}}{(z-1)(z-1+\gamma_{1}\theta_{m})}\end{bmatrix} ,\]  
and \(\theta_{i}=\sum_{j=1}^{m}a_{ij},i=m+1,\ldots,n\) is the \((i,i)\) entry of the diagonal matrix \(D\). Let \(F(z)=\det(I+G(z)L_{F})\). According to the generalized Nyquist stability criterion [41], the zeros of \(F(z)\) are in the unit circle when \(G(z)\) does not have poles out of the unit circle, if \(\lambda(G(e^{\mathrm{j}\omega})L_{F})\) does not enclose the point \((-1,\mathrm{j}0)\) for \(\omega\in[-\pi,\pi]\). Because of the symmetry of the Laplacian matrix \(L_{F}\), it follows from Lemma 4 that  
\[\lambda(G(e^{\mathrm{j}\omega})L_{F})\in\rho(L_{F})Co(0\cup g_{i}(e^{\mathrm{ j}\omega})),i=m+1,\ldots,n, \tag{32}\]  
where  
\[g_{i}(e^{\mathrm{j}\omega})=\frac{\gamma_{1}(e^{\mathrm{j}\omega}-1)+\gamma_{ 2}}{(e^{\omega}-1)(e^{\mathrm{j}\omega}-1+\gamma_{1}\theta_{i})}. \tag{33}\]  
For calculating conveniently, we assume that \(\gamma_{2}\leq\gamma_{1}\). In order to ensure that all the poles of \(G(e^{\mathrm{j}\omega})\) are within or on the unit circle, we have  
\[0\leq\gamma_{1}\theta_{i}\leq 2. \tag{34}\]  
Equation (33) is rewritten as  
\[g_{i}(e^{\mathrm{j}\omega})=\frac{\gamma_{1}\cos\omega+\gamma_{2}-\gamma_{1} +\mathrm{j}\gamma_{1}\sin\omega}{a+\mathrm{j}b}, \tag{35}\]  
where \(a=2\cos^{2}\omega+(\gamma_{1}\theta_{i}-2)\cos\omega-\gamma_{1}\theta_{i}\) and \(b=\sin\omega(2\cos\omega+\gamma_{1}\theta_{i}-2)\). To analyze the intersections of \(g_{i}(e^{\mathrm{j}\omega})\) on the real axis, we get  
\[ \begin{split} Im(g_{i}(e^{\mathrm{j}\omega}))=&(2 (\gamma_{1}-\gamma_{2})\cos\omega+(\gamma_{1}-\gamma_{2})(\gamma_{1}\theta_{i }-2)\\ &-\gamma_{1}^{2}\theta_{i})\sin\omega=0.\end{split}  \tag{36}\]  
For \(\omega\in(0,\pi]\), Equation (36) has only one solution \(\omega=\pi\) and  
\[g_{i}(e^{\mathrm{j}\pi})=\frac{\gamma_{2}-2\gamma_{1}}{4-2\gamma_{1}\theta_{i }}. \tag{37}\]  
It is evident that \(\rho(L_{F})Co(0\cup g_{i}(e^{\mathrm{j}\omega}))\) does not enclose the point \((-1,\mathrm{j}0)\), if all the points \((\rho(L_{F})g_{i}(e^{\mathrm{j}\pi}),\mathrm{j}0)\) are on the right side of the point \((-1,\mathrm{j}0)\) for \(i=m+1,\ldots,n\), i.e.,  
\[\rho(L_{F})\frac{\gamma_{2}-2\gamma_{1}}{4-2\gamma_{1}\theta_{i}}>-1. \tag{38}\]  
Hence, \(\lambda(G(e^{\mathrm{j}\omega})L_{F})\) does not enclose the point \((-1,\mathrm{j}0)\) and \(G(z)\) has no poles out of the unit circle, if we choose the gain parameters \(\gamma_{1}\) and \(\gamma_{2}\) satisfying \(\gamma_{2}\leq\gamma_{1}\leq\frac{2}{\theta_{i}}\), and \(\rho(L_{F})(\gamma_{2}-2\gamma_{1})-2\gamma_{1}\theta_{i}+4>0\), for \(i=m+1,\ldots,n\). Thus, \(F(z)\) has all zeros within the unit circle, which means that the roots of characteristic Equation (9) are within or on the unit circle.