# On the Delayed Scaled Consensus Problems
## 2 Preliminaries  
The communication topology of a multi-agent system can often be characterized by a weighted directed graph [1]\(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})\) with a node set \(\mathcal{V}=\{1,2,\cdots,n\}\) representing \(n\) agents, an edge set \(\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}\) describing the information exchange among them and a nonnegative adjacency matrix \(\mathcal{A}=(a_{ij})\in\mathbb{R}^{n\times n}\). We assume \(a_{ij}>0\) if and only if \((j,i)\in\mathcal{E}\), representing the information flow from agent \(j\) to agent \(i\). Because there are no self-loops in \(\mathcal{G}\), we have \(a_{ii}=0\) for all \(i\in\mathcal{V}\). The degree of \(i\) is defined as \(d_{i}=\sum_{j=1}^{n}a_{ij}\). Given a sequence \(i_{1},i_{2}\cdots,i_{k}\) of distinct nodes, a directed path in \(\mathcal{G}\) from \(i_{1}\) to \(i_{k}\) consists of a sequence of edges \((i_{j},i_{j+1})\in\mathcal{E}\) for \(j=1,\cdots,k-1\). If there is a node (called root) in \(\mathcal{G}\) from which all other nodes can be reached along directed paths, then we say that it has a spanning tree. Furthermore, \(\mathcal{G}\) is said to be strongly connected if between any pair of distinct nodes \(i,j\), there exists a directed path from \(i\) to \(j\). The graph Laplacian matrix of \(\mathcal{A}\), \(\mathcal{L}=(l_{ij})\in\mathbb{R}^{n\times n}\), is defined by \(l_{ii}=-\sum_{j\in\mathcal{V}\setminus\{i\}}l_{ij}=d_{i}\) and \(l_{ij}=-a_{ij}\) for \(i\
eq j\). Clearly, \(\mathcal{L}1_{n}=0\), where \(1_{n}\in\mathbb{R}^{n}\) represents the \(n\)-dimensional vector with elements being all ones. It is well known that \(\mathcal{L}\) plays a significant role in the convergence analysis for consensus seeking [1]. By convention, the determinant, trace and adjugate of a square matrix are denoted by \(\det(\cdot)\), \(\mathrm{tr}(\cdot)\) and \(\mathrm{adj}(\cdot)\), respectively.  
Here, we consider a group of \(n\) agents, labeled from one to \(n\), composing a fixed directed network \(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})\). The dynamical behavior of each agent is described by a continuous-time system:  
\[\hat{x}_{i}(t)=u_{i}(t),\quad i\in\mathcal{V}, \tag{1}\]  
where \(x_{i}(t)\in\mathbb{R}\) is the information state and \(u_{i}(t)\in\mathbb{R}\) is the control input of the agent \(i\) at time \(t\). In the vector form, we denote \(x(t)=(x_{1}(t),\cdots,x_{n}(t))^{\mathsf{T}}\in\mathbb{R}^{n}\) and let \(x(0)=(x_{1}(0),\cdots,x_{n}(0))^{\mathsf{T}}\in\mathbb{R}^{n}\) be the initial value. Given any scalar scale \(\alpha_{i}\
eq 0\) for the agent \(i\), the system (1) is said to achieve scaled consensus to \((\alpha_{1},\cdots,\alpha_{n})\) if \(\lim_{t\to\infty}(\alpha_{i}x_{i}(t)-\alpha_{i}x_{j}(t))=0\) for all \(i,j\in\mathcal{V}\) and all initial conditions \(x(0)\)[6].  
**Remark 1**: _The notion of scaled consensus is a generalization of the standard consensus [2], cluster consensus [7] and bipartite consensus [8]. Note that the scales \(\alpha_{i}\) associated with each individual \(i\) can be positive or negative, adding to the flexibility of the consensus-seeking process. Scaled consensus can also be defined in terms of the global asymptotic stability of the manifold defined by the above condition \(\lim_{t\to\infty}(\alpha_{i}x_{i}(t)-\alpha_{i}x_{j}(t))=0\), both of which are known to be equivalent for linear processes [6]._