# Impulsive Multiple-Bipartite Consensus Control for Networked Second-Order Multi-Agent Systems
## 3 Multiple-Bipartite Consensus of Second-Order MASs  
To realize the multiple-bipartite consensus of second-order MASs, the distributed control protocol with local instantaneous interaction is given in the following. \[ \begin{split} u_{i}(t)&=\sum_{k=1}^{\infty}\Big{\{} \alpha\big{[}\sum_{j\in i}a_{ij}\big{(}r_{j}(t)-\text{sgn}(a_{ij})r_{i}(t)\big{)} \\ &\quad+\sum_{j\in\bar{i}}a_{ij}\big{(}r_{j}(t)-\phi_{j}a_{ij}r_{i} (t)\big{)}\big{]}\\ &\quad+\beta\big{[}\sum_{j\in\bar{i}}a_{ij}\big{(}v_{j}(t)-\text{ sgn}(a_{ij})v_{i}(t)\big{)}\\ &\quad+\sum_{j\in\bar{i}}a_{ij}\big{(}v_{j}(t)-\phi_{j}a_{ij}v_{i }(t)\big{)}\big{]}\Big{\}}\delta(t-t_{k}),\end{split}  \tag{3}\]  
where \(\delta(t)\) is the Dirac delta function that satisfies \(\delta(t)=0,\ t\
eq 0\), and \(\int_{-\infty}^{+\infty}f(t)\delta(t)dt=f(0)\). The time sequence \(\{t_{k}\}_{k=0}^{\infty}\) satisfies \(0=t_{0}<t_{1}<\cdots<t_{k}<\cdots\), with \(\lim_{t\to\infty}t_{k}=+\infty\) and \(h=t_{k+1}-t_{k}\), where \(k=1,2,3,\cdots\). Denoting \(r_{i}(t_{k}^{+})=\lim_{t\to t_{k}^{+}}r_{i}(t)\), \(r_{i}(t_{k}^{-})=\lim_{t\to t_{k}^{-}}r_{i}(t)\), \(v_{i}(t_{k}^{+})=\lim_{t\to t_{k}^{+}}v_{i}(t)\) and \(v_{i}(t_{k}^{-})=\lim_{t\to t_{k}^{-}}v_{i}(t)\); here, we assume that \(v_{i}(t)\), \(i=1,2,\cdots,n\) is left-continuous, that is, \(v_{i}(t_{k}^{-})=v_{i}(t_{k})\), \(i=1,2,\cdots,k\). \(\alpha\) and \(\beta>0\) are the control gains to be designed. By using the property of the Dirac delta function, second-order MASs (1) with the control protocol (3) have the impulsive differential equation form of:  
\[ \begin{split}\begin{cases}\dot{r}_{i}(t)=v_{i}(t),\ t\
eq t_{k},\\ \dot{v}_{i}(t)=0,\ t\
eq t_{k},\\ \Delta r_{i}(t_{k})=r_{i}(t_{k}^{+})-r_{i}(t_{k})=0,\\ \Delta v_{i}(t_{k})=v_{i}(t_{k}^{+})-v_{i}(t_{k})=\alpha\big{[}\sum_{j\in\bar{ i}}a_{ij}\big{(}r_{j}(t_{k})\\ \quad-\text{sgn}(a_{ij})r_{i}(t_{k})\big{)}+\sum_{j\in\bar{i}}a_{ij}\big{(}r_{j} (t_{k})-\phi_{j}a_{ij}r_{i}(t_{k})\big{)}\big{]}\\ \quad+\beta\big{[}\sum_{j\in\bar{i}}a_{ij}\big{(}v_{j}(t_{k})-\text{ sgn}(a_{ij})v_{i}(t_{k})\big{)}\\ \quad+\sum_{j\in\bar{i}}a_{ij}\big{(}v_{j}(t_{k})-\phi_{j}a_{ij}v_{i }(t_{k})\big{)}\big{]}.\end{split} \end{split} \tag{4}\]  
To facilitate the analysis and design, the vector form of system (4) should be constructed. Accordingly, let \(r(t)=[r_{1}(t)^{\text{T}},r_{2}(t)^{\text{T}},\cdots,r_{n}(t)^{\text{T}}]^{ \text{T}}\). Furthermore, \(v(t)=[v_{1}(t)^{\text{T}},\)\(v_{2}(t)^{\text{T}},\cdots,v_{n}(t)^{\text{T}}]^{\text{T}}\). Then, system (4) can be reformulated as  
\[ \begin{split}\begin{cases}\left(\begin{array}{c}\dot{r}(t)\\ \dot{v}(t)\end{array}\right)=\left[\left(\begin{array}{cc}\mathbf{0}_{n\times n }&I_{n}\\ \mathbf{0}_{n\times n}&\mathbf{0}_{n\times n}\end{array}\right)\otimes I_{N} \right]\left(\begin{array}{c}r(t)\\ v(t)\end{array}\right),\ t\
eq t_{k},\\ \left(\begin{array}{c}\Delta r(t_{k})\\ \Delta v(t_{k})\end{array}\right)=\left[\left(\begin{array}{cc}\mathbf{0}_{n \times n}&\mathbf{0}_{n\times n}\\ -\alpha\mathcal{L}&-\beta\mathcal{L}\end{array}\right)\otimes I_{N}\right]\left( \begin{array}{c}r(t_{k})\\ v(t_{k})\end{array}\right),\ t=t_{k},\end{cases}\end{split} \end{split} \tag{5}\]  
where \(\Delta r(t_{k})=r(t_{k})^{+}-r(t_{k})\), \(\Delta v(t_{k})=v(t_{k})^{+}-v(t_{k})\), and \(\otimes\) is the Kronecker product. Furthermore, if all \(\mathcal{G}_{i}\) are the structurally balanced directed graphs with acyclic partition, then \(\Phi_{i}\mathcal{L}_{i}\Phi_{i}\) is a standard Laplacian matrix form. Therefore, according to Ref.