# Ordered-Bipartite Consensus of Multi-Agent Systems under Finite Time Control
## 2 Preliminaries
### Problem Formulation
: _If \(\mathcal{G}\) is a signed graph satisfying Assumption 1, Assumption 3, and Assumption 4, Assumption 2 holds if and only if \(0\) is the \(p\)-multiple eigenvalue of \(L\), while all other eigenvalues of \(L\) have positive real parts. Furthermore, the \(p\) linearly independent right eigenvectors corresponding to eigenvalue \(0\) are_  
\[\left((\Phi_{1}\mathbf{1}_{k_{1}})^{T},\underbrace{0,\ldots,0}_{n-k_{1}}\ \right)^{T},\,\ldots,\,\left(\underbrace{0,\ldots,0}_{i-1},\,(\Phi_{i}\mathbf{ 1}_{k_{i}})^{T},\,\underbrace{0,\ldots,0}_{n-\sum\limits_{j=1}^{j}k_{j}}\right)^ {T},\,\ldots,\,\left(\underbrace{0,\ldots,0}_{i-1},\,(\Phi_{i}\mathbf{1}_{k_{ i}})^{T}\right)^{T}.\]  
_The \(p\) linearly uncorrelated left eigenvectors belonging to eigenvalue \(0\) are_  
\[ \begin{split}\varpi_{1}&=(v_{1}^{[1]},\ldots,\,v_{k_ {i}}^{[1]},\,\underbrace{0,\ldots,0}_{n-k_{1}})\,,\\ &\vdots\\ \varpi_{i}&=(\varrho_{1,1}^{[i]},\,\ldots,\,\varrho_{1,k_ {1}}^{[i]},\,\ldots,\,\varrho_{i-1,1}^{[i]},\,\ldots,\,\varrho_{i-1,k_{i-1}}^ {[i]},\,v_{1}^{[i]},\ldots,\,v_{k_{i}}^{[i]},\underbrace{0,\ldots,0}_{n-\sum \limits_{j=1}^{i}k_{j}})\,,\\ &\vdots\\ \varpi_{p}&=(\varrho_{1,1}^{[p]},\,\ldots,\,\varrho_{1,k_ {1}}^{[p]},\,\ldots,\,\varrho_{p-1,1}^{[p]},\,\ldots,\,\varrho_{p-1,k_{i-1}}^ {[p]},\,v_{1}^{[p]},\ldots,\,v_{k_{j}}^{[p]})\,,\end{split} \]  
_where \(\Phi_{i}\,v^{[i]}\) are non-negative vector satisfying \((\Phi_{i}\,v^{[i]})^{T}\mathbf{1}_{k_{i}}=1\) and \((\Phi_{j}\varrho_{j}^{[i]})^{T}\mathbf{1}_{k_{j}}=0\,(j=1,\,2,\ldots,i-1)\) for \(i=1,\,2,\ldots,p\)._  
Similar to the previous Lemma 2, the graph associated with this lemma is given by Figure 2. The partition represented by this figure is the same as the previous one, with the difference that the groups contain positive and negative weights and are marked in red. **Definition 5**. : _The system (1) is said to reach GC in finite time in \(p\) groups if there exists \(T\in\mathbb{R}_{+}\) such that_  
\[\left|\phi_{t}x_{i}(t)-\phi_{t}x_{j}(t)\right|=0\text{ when }\overline{j}= \overline{i}, \tag{6}\]  
_where \(\{\ \overline{1},\ldots,\ \bar{p}\ \}\) is an acyclic partition of \(\mathcal{V}\) and any \(t\geq T\)._  
_Furthermore, if there exists one non-trivial \(\Phi_{k}(1\leq k\leq p)\), i.e., there exists a \(\Phi_{k}(1\leq k\leq p)\) with positive and negative entries, the system (1) will be said to reach GBC in finite time in \(p\) groups._  
**Remark 4**. : _Compared with complete consensus (CC), which refers to a unique convergence state, GC refers to multi-agent systems with multiple convergence states. There is no (directed) spanning tree in the communication topology of MAS, and the systems cannot achieve complete consensus. With this consideration, the logic of GC is to achieve multi-consensus by dividing the communication topology into a partition. The result is that the agents partitioned into the same group have the same convergence state (i.e., CC). Thus, GC is an extension of CC. Further, if the communication networks of each group under such partitioning is a bipartite topology, the GC implemented by the systems is called GBC. In other words, GBC means that the system is characterized by multi-pair convergence states, and each pair of convergence states is symmetric._  
**Remark 5**. : _The abovementioned expression (6) will be describe as the following equation if \(\Phi\) is nontrivial (i.e., \(\Phi\
eq I_{N}\)),_  
\[ \begin{cases}\left|x_{i}(t)-x_{j}(t)\right|=0,\text{ the weight of }\mathcal{P}_{ij}\text{ is positive,}\\ \left|x_{i}(t)+x_{j}(t)\right|=0,\text{ the weight of }\mathcal{P}_{ij}\text{ is negative}\end{cases} \ \ \ \ \ \overline{j}=\overline{i}\text{ for any }t\geq T. \tag{7}\]  
_The bipartite consensus problems require that the agents on the same side reach consensus and the agents from different sides converge to the same value but in the opposite direction [14]._  
Consider the control protocol of each \(i\)-th agent given by (8). This paper will explore the consensus problem of the _MAS_ based on these control protocols.