# Ordered-Bipartite Consensus of Multi-Agent Systems under Finite Time Control
## 2 Preliminaries
### Problem Formulation  
Consider a _MAS_ consisting of the following \(N\) identical agents described in dynamics (1):  
\[\hat{x}_{i}(t)=u_{i}(t),i=1,2\dots,N, \tag{1}\]  
where \(x\in\mathbb{R}^{n}\) and \(u\in\mathbb{R}^{n}\) denote the state vector and the input vector of agent \(v_{i}\), \(i=1,2,\dots,N\), respectively. **Assumption 1**. : _Graph \(\mathcal{G}\) has a acyclic partition \(\{\mathcal{V}_{1},\mathcal{V}_{1},\dots,\mathcal{V}_{p}\}\)._  
Under this assumption, \(\mathcal{G}_{i}\) is a subgraph of \(\mathcal{G}\) made up of all the points in \(\mathcal{V}_{i}\) and the edges between those points. Finally, the network topology of acyclic-partition is given an intuitive algebraic expression in the form of the following adjacency matrix \(A\). \[A= \begin{pmatrix}A_{11}&0&\cdots&0\\ A_{21}&A_{22}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ A_{p1}&A_{p2}&\cdots&A_{pp}\end{pmatrix} _{N\times N}, \tag{2}\]  
\(A_{ii}\) represents the adjacency matrix of \(\mathcal{G}_{i}\), and \(A_{ij}\) represents the information transmission from \(\mathcal{G}_{i}\) to \(\mathcal{G}_{j}\). **Remark 1**. : _It is worth noting that \(A\) is the upper triangular block matrix, and \(A_{ij}=0,i<j\) means that the information transmission from topology \(\mathcal{G}_{i}\) to topology \(\mathcal{G}_{j}\) can only be carried out when \(i<j\). In fact, this algebraic condition comes from the property of the acyclic partition structure itself. Acyclic partition can reduce information redundancy and improve the information transfer efficiency of the multi-agent system because of the properties of directed transmission with respect to inter-group information._  
**Assumption 2**. : _Each \(\mathcal{G}_{i}\) has a spanning tree._  
If \(\mathcal{G}_{i}\) is only a digraph with non-negative (or non-negative) weights, the eigenvalue 0 of \(\mathcal{G}_{i}\) has its right eigenvector \(1_{n_{i}}\) and its left eigenvector \(\omega_{i}=[\omega_{n_{i-1}+1,i},\omega_{n_{i-1}+2,i},\dots,\omega_{n_{i-1}+n,i}]^{T}\) under Assumption 2. It is well known that the result of this has been widely used in the _MAS_. Article [22] discussed the problem of finite-time consensus of first-order systems with (unilateral and ungrouped) directed topology and confirmed that the existence of spanning trees is sufficient to achieve finite-time consensus. If \(\mathcal{G}_{i}\) has a bipartite structure, gauge transformation is needed before solving the problem. **Assumption 3**. : _The row sum of each block \(\Phi_{i}L_{ij}\Phi_{j}\) is zero._  
Assumption 3 shows that the effect of all agents in another different group on a certain agent is trivial to zero in the sense of \"group\", and the network topology constructed based on this assumption reflects the internal requirements of \(\mathcal{GC}\). **Remark 2**. : _For a general matrix \(A\), one can define \(L(A_{ii}),i=1,\dots,p\), \(L(A)\) by the way defined in the last subsection. Note that \(L(A_{ii})\) is not necessarily equal to \(L(A)_{ii}\). However, under Assumption 3, there is \(L(A_{ii})=L(A)_{ii}\). In fact, Assumption 3 is, on the one hand, an algebraic description of the network topology for acyclic partition, and on the other hand allows the definition of acyclic partition structure to be well-defined. It is specifically reflected in the proof process in the next section--that is, Assumption 3 resolves the solution of a certain GC control problem into acomplete consensus problem. From this aspect, it can be said that the addition of the acyclic partition topology to the finite-time GC problems makes it possible to have a solution for practical problems such as the management of intelligent agents._  
**Lemma 2** ([13]).