# Impulsive Multiple-Bipartite Consensus Control for Networked Second-Order Multi-Agent Systems
## 3 Multiple-Bipartite Consensus of Second-Order MASs
[31], under Assumptions 1 and 2, matrix \(\Phi\mathcal{L}\Phi\) has \(k\) zero eigenvalues, and all its other eigenvalues  
\[\lambda_{1},\lambda_{2},\cdots,\lambda_{n-k} \tag{6}\]  
have positive real parts, where  
\[\Phi=\text{diag}\{\Phi_{1},\Phi_{2},\cdots,\Phi_{k}\}. \tag{7}\]  
Moreover, the \(k\) linearly independent left eigenvectors of \(\Phi\mathcal{L}\Phi\) associated with a zero eigenvalue can be taken as  
\[\xi_{1} =\left[\mu_{1}^{\text{T}},0_{n-h_{1}}^{\text{T}}\right]^{\text{T}}, \tag{8}\] \[\xi_{2} =\left[(\sigma_{1}^{(2)})^{\text{T}},\mu_{2}^{\text{T}},0_{n-h_{2 }}^{\text{T}}\right]^{\text{T}},\] \[\vdots\] \[\xi_{k} =\left[(\sigma_{1}^{(k)})^{\text{T}},(\sigma_{2}^{(k)})^{\text{ T}},\cdots,\mu_{k}^{\text{T}}\right]^{\text{T}},\]  
satisfying \(\sigma_{j}^{(i)}\in\mathbb{R}^{n_{i}}\), \((\sigma_{j}^{(i)})^{\text{T}}\mathbf{1}_{n_{i}}=0\), \(\mu_{i}=[\mu_{i,1},\mu_{i,2},\cdots,\mu_{i,n_{i}}]^{\text{T}}\in\mathbb{R}^{ n_{i}}\), \(\mu_{i,l}\geq 0\) (\(l=1,2,\cdots,n_{i}\)) and \((\mu_{j}^{(i)})^{\text{T}}\mathbf{1}_{n_{i}}=1\), for \(i=1,2,\cdots,k\), \(j=1,2,\cdots,k-1\). With the above preparation, we can now give our main result. **Theorem 1**. : _Suppose that all \(\mathcal{G}_{i}\) are the structurally balanced directed graphs with acyclic partition, and Assumptions 1 and 2 hold. If \(h<\min_{1\leq i\leq n-k}\frac{4\beta^{2}\mathbf{Re}(\lambda_{i})|\lambda_{i}|^{ 2}-2\beta^{2}|\lambda_{i}|^{4}}{44\mathbf{Im}^{2}(\lambda_{i})+\alpha\beta^{2 }|\lambda_{i}|^{4}}\), then by using control protocol (3), system (1) can reach multiple-bipartite conserves, \(\iota_{c}\),_  
\[\lim_{t\to\infty}r_{i}(t)=\chi_{i}(t),\quad\lim_{t\to\infty}v_{i}(t)=\kappa_{ i}(t),\quad i\in\mathcal{P}_{i}^{(1)};\]  
\[\lim_{t\to\infty}r_{i}(t)=-\chi_{i}(t),\quad\lim_{t\to\infty}v_{i}(t)=-\kappa _{i}(t),\quad\forall i\in\mathcal{P}_{i}^{(2)},\]  
_where \(i=1,2,\cdots,k\) and \(\chi(t)\), \(\kappa_{i}(t)\) is explicitly expressed by_  
\[\chi_{l}(t)=\left[(\xi_{l}^{\text{T}}\Phi)\otimes I_{N}\right]r(0)+\Big{[}( \xi_{l}^{\text{T}}\Phi)\otimes I_{N}\Big{]}v(0)t,\]  
\[\kappa_{l}(t)=\Big{[}(\xi_{l}^{\text{T}}\Phi)\otimes I_{N}\Big{]}v(0),l=1, \cdots,k,\]  
_in which \(\lambda_{i}\), \(i=1,2,\cdots,n-k\) and \(\xi_{l}\), \(l=1,2,\cdots,k\), are defined in (6) and (8), respectively._  
Proof. : Let  
\[\Phi=\text{diag}\{\Phi_{1},\Phi_{2},\cdots,\Phi_{k}\}. \tag{7}\]0  
and  
\[\Phi=\text{diag}\{\Phi_{1},\Phi_{2},\cdots,\Phi_{k}\}. \tag{7}\]1  
The multiple-bipartite consensus error equation for system (1) is then  
\[\left( \begin{array}{c}e_{r}(t)\\ e_{0}(t)\end{array} \right)=\left( \begin{array}{c}r(t)\\ v(t)\end{array} \right)-\left( \begin{array}{c}\chi(t)\\ \
u(t)\end{array} \right), \tag{9}\]  
where \(r(t)=\big{[}r_{1}^{\text{T}}(t),r_{2}^{\text{T}}(t),\cdots,r_{n}^{\text{T}}(t )\big{]}\) and \(v(t)=\big{[}v_{1}^{\text{T}}(t),v_{2}^{\text{T}}(t)\cdots,v_{n}^{\text{T}}(t )\big{]}\). Note that Laplacian matrix \(\mathcal{L}\) is defined by considering the effects between agents from both the same group  
\(p_{1}=(\mathbf{1}_{n_{1}}^{\mathrm{T}},\mathbf{0}_{n_{-1}}^{\mathrm{T}})^{\mathrm{ T}}\), \(\cdots\), \(p_{k}=(\mathbf{0}_{n_{-n_{k}}}^{\mathrm{T}},\mathbf{1}_{n_{k}}^{\mathrm{T}})^{ \mathrm{T}}\). Subblock \(J\) corresponds to the nonzero eigenvalues of matrix \(\Phi\mathcal{L}\Phi\); therefore, \(-J\) is Hurwitz stable. Considering that \(\mathcal{L}\Lambda=\Lambda\mathcal{L}\) and \(\Lambda^{2}=\Lambda\), then \(\Psi^{-1}\Lambda\Psi=\mathrm{diag}\{I_{k},\mathbf{0}_{(n_{-k})\times(n-k)}\}\). Therefore, we have  
\[ \begin{split}\Omega_{j}=&\mathrm{diag}\{\Psi^{-1}, \Psi^{-1}\}\mathrm{Ydiag}\{\Psi,\Psi\}\\ =&\left(\begin{array}{cc}I_{n}-\Psi^{-1}\Lambda \Psi&h(I_{n}-\Psi^{-1}\Lambda\Psi)\\ -\alpha\Gamma&I_{n}-\alpha\hbar\Gamma-\beta\hbar\Gamma-\Psi^{-1}\Lambda\Psi\\ \end{array}\right).\end{split}